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Ilr 4. la. . " 3‘3. 'u»: “13116: in. . ,. . u I ‘G 4.4 4... 3,- -" "91$, . ,. , 3 .n 5144.11 .: x, - n4 .- 3.41:: “‘ —‘. ‘«..‘~_. mm‘ ' 4 .. ‘ ,_ I". 141:" - ‘ " ' ‘- F‘ 7 . ‘ ' “:"“r::.:“ L... 9555.9 1;” ‘ ’ .‘L N ”a 4112‘ ' .nmv..:1.’::"1 - 5.7:. ; 431'”, .. .: ”5 4.2.24 "1 '4 r3; ' -' 4;. , 5,-4.4“ -'-.\|4"4 .g- V o"- . .J... ”3,7. 10 mrlugw 4 574:: (SAN SYATE UNIVERSITY HBR'ARIF‘; NM“ NM}? \‘Slil ll. M H 3 1293 00599 l LIBRARY Michigan State University This is to certify that the dissertation entitled STUDIES OF THE BETA DECAY OF LIGHT PROJECTILE FRAGMENTS FAR FROM STABILITY presented by David Mikolas has been accepted towards fulfillment of the requirements for DOCTOR OF PHILOSOPHY degree“, PHYSICS ;7 VEW‘ Major-prof Date Sept. 27L 1989 MS U i: an Affirmative Action/Equal Opportunity Institution 0 12771 025.; 03320 PLACE IN RETURN BOX to remove thle checkout from your record. TO AVOID FINES return on or before dete due. DATE DUE DATE DUE DATE DUE fiffl MSU Is An Affirmetive AetiorVEquel Opportunity tnetttuion STUDIES OF THE BETA DECAY OF LIGHT PROJECTILE FRAGMENTS FAR FROM STABILITY By David Nikolas A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1989 ABSTRACT STUDIES OF THE BETA DECAY OF LIGHT PROJECTILE FRAGMENTS FAR FROM STABILITY By David Mikolas Nuclei far from stability can be produced by the fragmentation of beams at the National Superconducting Cyclotron Laboratory at Michigan State University, and separated by the Reaction Product Mass Separator. Since these fragments maintain a significant fraction of the momentum of the beam broadened by the reaction dynamics, the beta decay of these nuclei can be observed only after they come to rest in a thick layer of material such as an active silicon detector. Here decay products such as the emitted 8 particle and any B-delayed charged particles can be detected and often their energy determined. New techniques to study beta decay and beta delayed particle emis- sion from projectile fragmentation have been developed. Problems such as the determination of absolute partial and total decay rates, con- tamination by other isotopes, background and daughter activity, fitting procedures where only a small number of events are available or where data are contaminated with other activity or background and energy calibration are discussed. Techniques for their management are presented. The Nuclear Shell Model, with an empirical Hamiltonian based upon tabulated nuclear structure information from stable or close to stability nuclei, will benefit particularly from information of the structure of’nuclei far from stability for two reasons. At first, ob- servables such as beta decay rates can be measured and compared to predictions. Any deviations might give insight into the limitations of the present Hamiltonians. Later as more information is gathered, a significant body of excited states will be compiled facilitating a new fit of the matrix elements of the Hamiltonian including far-from- stability configurations. It is however difficult to extract the standard quantities such as beta decay matrix elements from the data in a form comparable to the predicted values. Many levels will be unbound to nuclear decay, often to the emission of two or more particles. The distribution in phase space of all of these particles must somehow be interpreted with a reasonably simple model in order to extract beta decay matrix elements. In this thesis, a simple model is presented in lorder to interpret such data. This model incorporates the generalized density of states concept into the standard model of beta decay to bound levels. Copyright by David Mikolas 1989 To My Favorite Scientist and my Love, Hiroko Takayasu ACKNOWLEDGMENTS I was watching an episode of NOVA a while ago. Two senior scien- tists were reminiscing an evening which took place in the early days of radar. On that night they were graduate student and advisor, peering into space at a wavelength of a cm or so, seeing fer the first time the universe calling out at ll°K. Their two faces became almost those of children as they recreated in their minds the excitement of that night. The experience of gazing into the controls of some contraption in the middle of the night with my advisor at my side, learning something new about our universe is one which I shall not ever forget. Thank you, Jerry Nolen. I want also to point out the miracle that my parents Eugene and Ailene Mikolas worked in nurturing ten billion base-pairs of DNA (a few picograms) into what I am today. How. They are responsible for put- ting me in college and making sure I stayed there. Writing this thesis has been a long process. I hope that it was worth it. I therefore thank its audience Rick Harkewicz in advance, for his (hopefully) reading it. This whole thing started when my grand-advisor Ruby Sherr was rum- maging around in the p shell looking for trouble. He found it. The beta decay branching information for ’C in AJzenberg-Selove left a lot to the reader's imagination. He called up Jerry Nolen who told him where to put the ’C, and there began the practice of implanting delayed vi charged particle radioactivity in active Silicon detectors here at NSCL. Thanks Ruby. I give special thanks to the members of my PhD. conmittee. They have been both helpful and patient. In particular, Alex Brown's tui- tion in Nuclear Physics gave to me a sense of purpose for this project. I hope that the techniques discussed in this thesis will ultimately repay him with lots and lots of B(GT)'s! Halter Benenson proved to be a source of guidance and support in both Physics and in the English language itself. I should also thank Jack Hetherington for simply being himself. I want to emphasize that EVERYONE at NSCL is to be thanked for creating a warm, friendly environment in which to do research. I am always proud of this point when I describe this laboratory to others. If you have a question or need help, one only need wander down the hall and ask. Dick Blue, Ed Kashy, Dave Morrissey and Brad Sherrill have all provided essential help at some point. The NSCL Computer group deserves a standing ovation for maintaining an incredibly reliable net- work of computers and a damn good data aquisition environment which as far as I can tell surpasses that at any other Nuclear Physics Laboratory. Michael Maier has been a consistent source of information and perspective. His door has always been cpen for questions - including those dumb 'emergency' questions. He has been an important resource for me throughout my stay here at NSCL. Finally, I would like to thank Hiroko Takayasu for all kinds of things. If I did so, it would require another chapter. Instead, I'lJ. Just say 'Thanks Hirot' vii TABLE OF CONTENTS I. Introduction: .................................................. 11. Beta Decay to Bound States: ................................... 11.1 Transition Rates in Nuclear Beta Decay: ................. 11.2 lepton Phase Space: ..................................... 11.3 Branching Ratios and Half Lives: ........................ II.” Empirical Coupling Constants: ........................... 11.5 Gamow-Teller Quenching: ................................. 111. Beta Decay to Unbound States: ................................ 111.1 Structure of Unbound and Loosely Bound levels: ......... 111.2 Excitation Energy and Unbound Levels: .................. 111.3 Beta Decay Rate: ....................................... 111.“ Beta Delayed Three Body Final States: .................. 111.5 Acceptable Level of Approximation: ..................... IV. Beta Decay of ’C: ............................................. 1V.1 The MSU Experiment: ..................................... 1V.2 Three Body Simulation for ’C: ........................... V. Other Cases of Beta Delayed Charged Particle Emission: ......... VOI 1.": 00000000 .0... 00000000000000000000 O 0000000 000000 0000000 V1. The Michigan State Reaction Product Mass Separator (RPMS): ... V1.1 E x B Recoil Isotope Separation (RIS): .......... . ....... V1.2 Design and Construction: ................................ V1.2.1 Phase-Space Acceptance: ........................... V1.2.2 Scattering Angle: ................................. V1.3 Beginning a Typical Experiment: ......................... v1.3.1 Start-up Procedures: .............................. V1.3.2 Tuning Up: .... .................................... VI.3.2.1 Scaling in m/q: v1.3.2.2 Scaling in velocity: V1.3.2.3 Scaling in Dispersion: VI.” Monochromation: ......................................... VII. General Experimental Considerations: ........................ VII.1. Projectile Fragmentation: .............................. VII.2 Target Parameters: . ............ .. ............. .. ..... .. V11.2.1 Choice of Target Thickness: ....................... V11.2.2 Scattering Angle: .. ........ . .............. . ....... V11.2.3 Target Material: .................................. 111.3 Half Life Determinations: .............................. VII.3.1 Choice of Detector Thicknesses: ................... VII.3.2 Background Problems: ...... . ................. . ..... V11.3.2.1 Daughter Activity: VII.3.2.2 Cosmic Rays: viii 18 V11.3.2.3 Thermal Neutrons: VII. 3. 2. A Electrical Noise: VII.3.3 Decay Curves: ................ .................... 130 VII. 3.3.1 One Atom at a Time Operation: V11.3.3.2 Fixed Cycle Operation: VII.3. 3 3 Daughter Activity: VII.3.A Fit tting Procedures: .... ....... ...... ........... .. 1AA VII.3.A.1 Least Squares: VII.3. A. 2 Maximum Likelihood: VII.3. A. 3 Use of All Available Information: VII.3. A. A Finite Channel Size: V11.3.A.5 Uncertainties in the Fitted Parameters: VII.A Beta-Delayed Charged Particle Measurements: ............ 15A VII.A.1 Normalization: ................ . ................... 155 V11. A. 2 Energy Calibration: ............................... 157 V11. A. 3 Pulse Height Defect: ..... ....... ....... 160 VII. A. A Line Shape Distortion from Beta Energy Loss: ...... 162 VII. A. 5 Line Shape Distortions from Escaped Charged Particles: ............................................. 16A V11. A. 6 Multi-Element Detector Systems, Coincidences: ..... 165 APPENDIX A Reprint of MIKOLASBBa, "Branching Ratios of ’C to Low Lying States in ’B": ........ . .......... ... .................... 168 APPENDIX B Beta Delayed Particle Decay Thresholds for Light Nuclei: ...... . .......... . ..... ........... ..... . ......... 18A REFERENCES: . ................. . ........... . ........................ 19A ix LIST OF FIGURES Figure 1.1 Year of publication for the data used in the compilation Of. BROHNBSe eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee eeeeeeee Figure 1.2 Trends in beta decay energies and nucleon decay thresholds of daughters. In both cases data are plotted for N-Z of the parent. Graphs are arranged with beta decay energies and nucleon separation energies on alternate pages so that one can look through two consecu tive pages into a source of light and visualize the relationship to distance from stability. ................ Figure 1.3 Beta decay energies and neutron decay thresholds in Mass 19. ... ............ ............. ..... .................... Figure 1.A Beta delayed particle decay thresholds for 22Al. ....... Figure 2.1. The behavior of various expressions related to nuclear beta decay rates as a function of beta decay energy for three values of Z for the daughter. See text for complete discussion. ................. . ........ .... ...... Figure 3.1 Energy level diagram for the beta delayed alpha emission of “Be. Branching ratios are from ALBURGER81 and MILLENER82. ............. ........ ........................ Figure 3.2a Density of states p(E) for a hypothetical state in 8Be which decays by s-wave (2:0) emissiog of two alpha particles. A spectroscopic factor 6 _ of 1.0 was used, which is close to the actual value f F 'Be decay into this channel. Values for E, of 0.0, 0.2, 6?3, 0.5 and 1.0 MeV are used. Note the ghost structure for E0 = 0.0 MeV. ...... ........ ........................... . ......... Figure 3.2b, same as Figure 3.2a but with a logarithmic scale. .... Figure 3.3 Area of p(E) (examples of which are in Figures 3.2a,b) integrated from 0 to 3 MeV as a function of E.. Note that this area is either significantly greater or less than unity under most circumstances. For E0 = 0.0 MeV, the area of the sharp spike shown in Figures 3.2a,b is 1.0, and the remaining 0.37 is contained in the high energy ghost. ................. ........ .......... ........ Figure 3.A The beta decay rate per unit energy A(E) dB for 'B decay to a hypothetical state in ‘Be with E0 = 3.0, 9.0 and 12.0 MeV. The dotted line represents the shape of the beta neutrino phase space factor fA(N.,A) evaluated at each excitation energy Ex‘ .... ...... ................ 11 1A 26 142 A3 A3 AA A7 Figure 3.5 Schematic representation of a beta delayed three body decay. .................. . .................. . ....... ..... A9 Figure 3.6 Illustration of various energy variables. The excitation energies E , E and E of the daughter, intermediate and finaI nu leus rgspectively are defined in the text. Arrows point from A to B where EzB-A. Thus A is negative. The beta decay energy from parent to dgughter E , and particle decay energies from the daughter t8 intermediate nucleus E , and from the intermediate to final nucleus E aAe used to calculate the penetrabilities P1(E ) and 62(E ). These then give the local decay rates P 2E ) and P TE ) in the daughter and intermediate nucleug. Energy dif erences are calculated by subtracting the value at the tip of the arrow from that at the origin. While two decay channels are illustrated for the decay of the intermediate nucleus, only one particle decay channel of the daughter is shown fer the sake of simplicity. . ......... .... ...... 50 Figure A.1 Enlargement of a nuclear emulsion track from a fragment produced in an energetic collision of a relativistic proton with a nucleus of the emulsion. This product was identified as ’C by its unusual decay mode; B+20+p. This is the first atom of ’C ever to be observed in decay. The energies of the charged particle imply that the ’C decay populated a level in ’B at 12.2 MeV, which then decayed by the emission of a proton and two high-energy alpha particles which could not have come from the ground state of 'Be. This decay mode is the subject of this chapter. .................. ........... ................... 56 Figure A.2 Energy level diagram for the decay of ’C. From Figure 1. of Appendix A (MIKOLASBBa). ......... ................. 58 Figure A.3 Monte Carlo simulation of the decay of ’C to a level in ’B at 12.1 MeV. The ’C is uniformly distributed in a A00um thick silicon detector. In this situation 81% of the events produce counts below 10 MeV because of the escape of the proton from the silicon before losing its full energy. The disagreement in the 12.1 MeV peak position is within the quoted uncertainty in energy in MIKOLASBBa. The energy assignment comes from ESTERL72. 61 Figure A.A Charged particle sum energy spectrum for the beta delayed 20(+n) decay of ’Li separated with the RPMS. The uncharged neutron deposits no energy in the silicon; thus the energy of the two alpha particles can be determined separately from the total breakup energy. ............... 63 xi Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure A. 5 Line shape for the ground state (0 ) and first excited state (2, 2. 9 MeV) of 'Be used in the calculation of the breakup of ’B through these channels. ........ ........... A.6 Line shape for the ground state (3/2') of 5Li. ......... A.7 (Solid) Density of states function p(E) for the 12.1 MeV state in ’B populated by the decay of ’C. The spectroscopic factors have been adjusted to reproduce the observed FNHM of A00 keV. (Dotted) The decay rate l(E) to ’8 obtained by multiplying p(E) by the phase space factor for the beta neutrino pair. .. ..... ............... A. 8 Components of the calculated beta delayed proton spectrum. The three solid curves represent alpha decay to the 3/2 ground state of ’Li which then emits a proton in flight, proton decay to the 2+ excited state of 'Be near 3 MeV and proton decay to the 0 ground state of 'Be, listed in increasing order in average energy. Diamonds represent the sum of the three contributions, and the histogram is the data of ESTERL72. .............. A. 9 The contribution from the long high-energy tail of the 1/2 level at 2. 9 is included. The sum of all four components is shown by the large diamonds. The histogram is the same as in Figure A.8.~ ............... . ....... ... 5.1 Partial Decay Scheme for "N beta delayed alpha emission. Information take from AJZENBERG-SELOVE87, NAPSTRA86, OLNE8382 and ZHA089. . ........ . ...... .... ..... 5.2 Histogram of energy recorded in a 50um thick silicon detector during the 'Beam Off' periods following “N implantation. The assignment of each peak is indicated. ZHA089 suggest that the structure labeled A is actually a group of known but unresolved 1 levels. The structures labeled B and c are likely to be 1’ or 2' levels in H0 which have not yet been identified. ..................... 5.3 Decay scheme for ‘30 delayed charged particle emission. Compare to ’C in Figure A.2. ............. ............... 5.A Preliminary decay energy spectrum for "0 ions implanted in a A00um thick silicon detector. The lowest peak is most likely due to beta decay to bound states where only the beta particle deposits ionization in the silicon. ... 5.5 Beam off time spectra for the three peak structures in Figure 5.A and fer the highest energy part. Once a half life is established these curves can be used to determine xii 66 66 67 7O 70 78 79 81 82 the fraction of the area under each peak which is due to real ‘30 decay. ...................... ..... ...... ........ 83 Figure 6.1 Illustration of the Reaction Product Mass Separator; a) side, and b) top. 6) illustrates that changes in scattering angle are facilitated by only small rotations of the RPMS about the center of the Cornell magnet. ..... 90 Figure 6.2. Schematic representation of a histogram of vertical position as a function of the square root of the Particle Identification function (PID.) The square root of PID is used for the horizontal axis in order to keep the groups more evenly spaced. The boxes represent typical resolutions in m/q and in PID. ....... . ........... ....... 96 Figure 7.1 Illustration of various spectra generated by the Monte Carlo simulation discussed in the text. Rays represent the energy of fragments in the target and detector as a function of depth of penetration. Bold line represents the primary beam. dN/dX represents the rate of production per unit thickness in the target, dN/dE is the rate of emission per unit energy from the back of the target. R is implantation range in the silicon detector, and dN/dR is the rate of implantation per unit thickness in the silicon. ........... . ............................. 110 Figure 7.2 Simple illustration of various rates considered in the text. a) Gaussian gbstribution (bold) and two weighting functions E and E ‘ b) Same Gaussian distribution and weighted distributions. ................. . ........ ... 111 Figure 7.3 Series of simulations of distributions of fragments as a for increasingly thick targets. Left and right groups correspond to the two reactions shown in Figure 7.A (next page). In the center column, the wide rectangle of variable height represents the target thickness, while the two narrow rectangles behind the target represent the (fixed) range of the beam with respect to the target thickness. When the target rectangle just covers the narrow rectangle, the target is just thick enough to stop the beam. The diamonds represent the remaining energy of the beam after passing through target. Target are multiples of 0.0025 inch thick Ta. ......... . ............ 115 Figure 7.A Peak heights and peak locations for those distributions in Figure 7.3 (previous page) as a function of target thickness. As stated in the text, E is kinetic energy per nucleon. ............................................ 116 Figure 7.5 Illustration of the spatial separation of fragments at xiii the focal plane of the RPMS. The vertical axis corresponds to vertical position at the location of the detectors, which is inversely proportional to the mass to charge ratio m/q. The horizontal axis corresponds to implantation depth in the silicon detector(s) ........... 12A Figure 7.6 Decay rates of the various components from an initially pure source of "B, including ‘78, ‘7C, "N, “C, “N and ‘5C activity. See text for details. .................... 1A0 Figure 7.7 Same as Figure 7.6, but shown on a compressed time scale. .................................................. 1A1 Figure 7.8 Primary ‘7B decay rate and total decay rate for two sources of "B. The lower curves represent an initially pure source of ‘78. The upper curves represents an initially equilibrated source with a "B production rate of 100 per second. The total decay rate of the initially equilibrated source deviates strongly from a the decay rate of 1"B itself, and illustrates the difficulty in extraction of a correct half-life when the production rate is too large! ...................................... 1A2 xiv Table 1.1 Table A.1. Table 6.1. Table 7.1 LIST OF TABLES Some maximum excitation energies and inferred temperatures following beta decay. .......... . ........... Spectroscopic Factors fit to the beta delayed proton spectrum of Esterl et al. The last column represents the calculated efficiency for collection of the full energy peak for ’C activity in the A00um thick detector used in the present experiment. ................................. Scaling Exponents for RPMS Settings. 02 through 023 represent the four quadrupole magnets, B the Cornell Dipole, E and B the electric and magneEic fields in the Nein filtgr, andwet the angle of the tail with respect to the axis of the Riin filter. .... ............ ......... Some useful narrow beta delayed charged particle lines of known summed charged particle energies which might be incorporated in an energy calibration. Not listed is the isotope ’Li. The beta delayed 20 + neutron decay channel of the 5/2' level in ’Be at 2.3A MeV produces a broad charged particle energy spectrum which collapses to a narrow peak if the neutron energy is detected and added. This state also decays through n + aBe, giving a narrow neutron energy peak. "C decay also provides narrow neutron lines for calibration of neutron detectors. The nucleus "N is a candidate for narrow neutron lines - see ZHA089. .... ......................... XV 12 69 99 SECTION I 1. Introduction: The beta decay properties of nuclei provide a view on the internal structure of the nucleus predominantly by the association of decay rates with the overlaps of calculated wave-functions and known or em- pirical operators. The study of the decay properties of nuclei far from stability allows us to put theories of nuclear structure and decay to the test. This is because most of the information used in the con- struction cfl'these theories represents properties of nuclei very close to the line of stability‘. Often, the beta decay will populate unbound levels. In this case further study of the nuclear structure can be extracted from the nuclear decay properties following beta decay. A recent compilation of beta decay data was produced by BROWN85. While the data spans decades of research, a clear peakziJl the number of 'new' nuclei measured and B(GT) (see Equation 2.A) occured. The number of newly accessible nuclei has diminished as the current production and detection technologies have matured. This trend is il- lustrated in Figure 1.1. The characteristics of beta decay of nuclei far from stability differ from those closer to stability in many ways. The major dif- ference lies in the greater amount of energy released in each decay. The available phase space for the electron and neutrino behaves roughly 1. The stability line is drawn on the Z,N axis. 2 and N represent proton and neutron number respectively. The nuclei which define the stability line are those with the ratio of protons to neutrons which provide the lowest energy configuration for a given number of total nucleons (2+N). 2 as the fifth power of decay energy, and thus half lives become dramati- cally shorter for those nuclei farther from stability. Many more excited states become energetically accessible witnl the increase in available energy, and the binding energy of the daughter nucleus simultaneously decreases. This is illustrated in the series of graphs in Figure 1.2. Here the beta decay energy and nucleon separa- tion energy of the daughter are graphed for each element (Z) as a function of 'neutron richness' (N-Z). With the availability of more isotopes for decay studies, higher production rates and improved and innovative measurement techniques, new decay phenomena and un-expected variances with theoretical models will be discovered as the study of nuclei far from stability continues. One area of particular growth will be the observation and understanding of delayed multi-particle emission. The probability of observing such processes rapidly increases farther from stability. The case of the decay of "B is illustrated in Figure 1.3, where the beta decay energies and neutron decay thresholds in Mass 19 are shown. Here, up to Seyen_neutrons can be emitted following the beta decay of ‘98. In other fields of nuclear physics, the breakup of light systems is studied as they are emitted from hot nuclear sources. Table 1.1 gives some examples of high energy beta decays which can provide be- tween 1 and 2 MeV of excitation energy per nucleon in the daughter. The associated nuclear temperature T which one might assign to this ex- citation energy per nucleon from a simple Fermi gas model is given by, T . «fig/T. [1.11 F +- 1 3 15¢ : i g : e 1 '3 10." " 1 :3 » — 1 z t r“ ‘ c5 5: l 1 “ ‘ Al. 1 0 l LIA ill—1 in ALL - - -714 A; 80' "I' ' U f'fi—UV ' l' '—'I m . g , 1 . F1 60" -4 d 1 _—1 . t> : = 1 A . En 40' . 3 : _ 1 m ‘ f ‘ . 207 1 C) r . ‘3 : fl : O -iil- 11 i11 -L--4 1965 1970 1975 1980 1985 Figure 1.1 Year of publication for the data used in the compilation of BROWN85 . 25-! I I I I I 1 I I 1 I I I I I r A I 2 % 20r A 2 - _ v _ .. >. r .9 ~ DD - l 3" 1.5— 'fl 2 ' DH . 12:] I .31 >. ‘ XL! ‘ 10— -- g - OBe : Q r- 'I'B : 3 5‘ "j 0 P _ m I - r- ‘1 o l I l l l L1 1 1 I l l l l l 1— —5 o 5 10 N—Z Figure 1.2 Trends in beta decay energies and nucleon decay thresholds of daughters. In both cases data are plotted for N-Z of the parent. Graphs are arranged with beta decay energies and nucleon separation energies on alternate pages so that one can look through two consecu- tive pages into a source of light and visualize the relationship to distance from stability. g 25- I I I I I I 'l ‘1’ ' I 2 ' . v - “d 20:- ‘j 13 _ d g ’— .1 U) .. - 2: 15- ".1 5'5 - DHe : 2 o: XLi J 2% 1 - OBe : 2; : If E! : 0. 5___ _: 'U P 4 a) .. >5 __ a '2 _ .. a 0 I] .T. L l 1 1| — -5 O 10 Figure 1.2 (continued) q--—d-1q—-uHE—Hfi-u_‘-qufi I Olfllll bbP—anP—r————--—-Fb-F % m m m 5 o 926 mmuonm Amoco 30m 10 fiqfiq—qqfiq—111-—111-—1fi14qfi 1 e .. OFN I _u X. nu F Lin-FPP-np—l-prP—P—nn—bpbp— % m m m 5 o 905 mmeocm havoc 30m 10 Figure 1.2 (continued) qfiqdqqqqd‘Tfiqq—uuqq—q+qfi— 1 L DC XN 00 pprbl—P-up—nppP—-bn———-P_ J0 a m ...” m 5 o 935 Renata. manta 8.839 «Qu-_u‘-q—-qdq—.-q4—qufidlfi r- I I I U L 00 XF ONe an—P-pP—h-pb—P—pr—nnbbF 5 0 5 on on 1i mw :u nu :63 2285:. Santa 338 10 Figure 1.2 (continued) qd__ qua—qu-fi—q-ufi_fiud+— I .u l .l P. 0 II. .I M q. . . I. T .J .“ .. i .l fih’ l /.. v . .l T . . / I HP... 1 a g 0 - N M M q... . I .U X. nu .w an. 1 L pp-P—nP-n—hP-n —P-b—rer— 5 O 5 O 5 O 2 2 1 1 936 mmsoam Sacco comm 10 Jfi-u—Jfifi———fi11—udqfidfiuqfi L rl n 0 ll 4 so . i n 0’ I U 1’9 l I I... . .l n..\, .. P S m M I. L D X 0 m. no I 1| ..1 P P P P — P — rbll—I ~ P P n P n h b — E n p— m. m m 5 o 905 Ammonm income 30m 10 Figure 1.2 (continued) r 5 2 925 seamen; manta BASS --_b—r-—P—P-—b-Pr—P-r-— nu ma nu cu nu 2 1 1 dun—du—q—-qqufiqqqd—q-1— .III 1 I L I I1 -PSQM - enxoe L I L I IL ppp——-——-_pn—P—FP-n—nnb-F 5 0 5 0 5 0.. 2 2 1 1 $35 Scenes: seated season 10 Figure 1.2 (continued) 10 __ -60 —6n _ 50 _5n —4n 440 =3n 2n —4r1 _ n —3r1 19 C 1 30 —2n — n —2n ”N — n _ n - 10 19 — O - 0 MeV Figure 1.3 Beta decay energies and neutron decay thresholds in Mass 19. 11 Table 1.1 Some maximum excitation energies and inferred temperatures following beta decay. Parent Daughter Maximum Ex Maximum Inferred Isotope Isotope (MeV) Temperature (MeV) ‘He 'Li 10.65 3.3 'L1 ‘Be 16.00 A.0 '3 'Be 16.96 A.1 ’Li ’Be 13.61 3.5 ’C ’B 15.A8 3.7 “L1 “Be 20.73 3.9 “Be “B 11.51 2.9 ‘iBe ‘28 11.71 2.8 ‘18 "C 13.37 3.0 "N ‘20 16.32 3.3 ‘30 "N 16.7A 3.2 “Be ‘“B 17.36 3.1 ‘“B 1"C 20.6A 3.A ‘98 "C 19.10 3.2 "B "C 22.98 3.3 "B "C 26. 3.3 22A1 22Mg 17.A1 2.5 12 The level density parameter a is set to 8 MeV in this case (from NAYAK88). A great deal of research is presently underway at NSCL and other laboratories to infer the conditions which occur in a heavy ion reac- tion-from the break-up of light nuclei emitted from the reaction environment. The observation of the breakup of these nuclear systems in the 'clean environment' of beta decay (no nuclear spectators or con- tamination from other processes when beam strikes target) may improve our understanding of such processes, and may indeed yield important spectroscopic information not easily obtainable in any other way. Some beta delayed particle decay channelswill be quite complex, or compete with two or more other particle decay channels. In Figure 1.A the beta delayed particle decay channels for 22A1 are shown. In Appendix B maximum beta decay energies and particle decay thresholds for all energetically accessable channels are tabulated for nuclei with protons and neutrons in the 0p and 031d shells. While it will become more difficult to extract detailed sthucture information from delayed particle distributions as the decay process becomes more complex (more particle decay channels and greater multiplicities), the large number of isotopes which will be available for study in the future will make these types of investigations very attractive and productive. Projectile fragmentation will be a predominant source of unstable nuclei for nuclear structure studies. However the study of the decay properties of projectile fragments beyond half-life is still in it's infancy. This reaction provides it's 13 0.00 ZZMg Figure 1.A Beta delayed particle decay thresholds for 22Al. 1A own advantages and disadvantages to the experimenter with a particular observable in mind. New techniques are presently being developed, and more will no doubt be necessary in the filture to push the quantity and quality of nuclear structure information obtainable from nuclei far from stability to their limit. This thesis is the result of a fusion between two goals. The first goal is to provide a discussion of a set of studies of the decay properties of a particular nucleus ’0, which involved the development of a formalism with which to study the beta delayed multi particle final states and extract information which can be compared to shell model calculations. The second is an attempt to create a 'how to manual' for investigators interested in the study of the beta decay properties of nuclei produced by projectile fragmentation. This work is then naturally divided in two halves. The first half dealing with specific experiments, results, and comparisons to theoretical calcula- tions. These are given in Sections 11 through IV. The second half is represented by the remaining sections, which contain detailed dis- cussions of the various elements of an experiment and subsequent analysis which have been worked out over the years 1 have been at NSCL. In section II, the standard formalism for connecting 'observables' such as beta decay half-life and branching ratio with 'calculables' such as nuclear wave ftmctions, beta decay operators, and phase space factors for the beta-neutrino pair are presented. In section 111, this connection is rederived for the situation where the final state is un- bound. While various approximations are made, a system is developed to make preliminary interpretation of beta-delayed breakup spectra, even 15 in the extreme case where the nucleus 'breaks-up' into three or more parts. In section IV, these models are applied to data from a recent ex- periment on the decay of ’C. The experiment its self is described in MIKOLASBBa, which has been included here as Appendix 1. A summary of the information gained from the study of the beta-delayed three body final state of ’C is given in section IV.3 The remaining sections of this thesis are devoted to the various experimental and analytical techniques which have been developed in or- der to record the 'observables' with enough precision in order to begin to address the phenomena discussed in sections 11 through IV. The Conclusion section sumarizes what has been learned about the decay of nuclei far from stability, and how this differs from ex- perience to date on those nuclei closer to stability. The final Recommendations section suggests what new work, both theoretically and experimentally must be carried out in order to address these issues in more precise and calculable ways in the future. 16 SECTION I I 11. Beta Decay to Bound States: The rate of beta decay of a nucleus can be calculated in terms of a factor representing available phase space, and an overlap of the final and initial nuclear wave functions including the appropriate operators and coupling constants. The necessary overlaps and operators are briefly discussed in Section 11.1, and the determination of lepton phase space is outlined in Section 11.2. In Section 11.3, the relationship between calculated quantities (such as B(GT) and lepton phase space) and measured quantities (such as half life and branching ratios) is derived for discrete final states. When states are unbound and broad, this derivation does not apply. That situation is discussed further in Section III. Empirical values for the coupling constants derived from measured nuclear beta decay rates are discussed in Section ILA. Finally, an observed reduction in some types of nuclear beta decays rates with respect to that of the free neutron (Gamow-Teller Quenching) is dis- cussed in Section 11.5. Good discussions of all facets of beta decay can be found in many texts. KONOP1NSKI66, SCHOPPER69, and MORITA73 are especially useful, with the latter text the most advanced of the three. 11.1 Transition Rates in Nuclear Beta Decay: The original proposals of Fermi (FERMI33) and of Gamow and Teller (GAMOW36) have remained as the two known forms of allowed (see below) nuclear beta decay, and are referred to as Fermi and Gamow-Teller beta decay. Fermi first proposed that the electron and neutrino were emitted such that their spins (S=1/2) were anti-parallel and thus 18 coupled to 3:0. (The neutrino was postulated at that time by PAUL133, but had not been observed until much later by REINES60.) Thus since the two leptons couple to spin 0, no change in the spin of the nucleon or nucleus results. In a simple model for the structure of the nucleon, Fermi beta decay is just the change in identity of one of the quarks within the nucleon, such that [uud] <-> [udd] (i.e. proton <-> neutron), while the spin of the quark remains unchanged (VENTOBA) . It then fellows that the spin of the whole nucleon is unchanged, as is the isospin and spatial wave function including parity. The change in the quark results in a change in T2, and thus only the charge of the nucleus is altered in Fermi beta decay. Pure Fermi decay will then only connect two states of practically identical wave functions in all respects except for T2, the charge. There is only one such state in the daughter nucleus, called the analog state. All of the Fermi strength is directed toward this state where isospin is a good quantum number fer the states involved. Fermi decay could explain only a fraction of the known beta decays. The Gamow-Teller theory of beta decay could explain many more, by requiring the electron and neutrino couple to 8:1. The spin of the quark and thus the nucleon must then flip, which allows the nucleon to mix with other spatial and isospin (but not other parity) configura- tions. The operator permits a change of up to one unit of spin and of isospin of the nucleus. Thus Gamow-Teller decay populates many levels in a nucleus, including but not limited to the analog state. The selection rules for Fermi and Gamow-Teller beta decay are as follows. Fermi: AT = 0; A.1“ = 0“°, Gamow-Teller: AT = 1,0,—1; A.1“ = 1,0,4? 19 (except 0+ » 0+). At low excitation energy in the daughter where the phase space for beta decay is large, most of the levels have the same isospin as the ground state. Since this is of course one unit dif- ferent from that of the parent nucleus, the half life of a nucleus is predominantly determined by Gamow-Teller decay. Both Fermi and Gamow-Teller are called allowed beta decay because the orbital angular momentum of both emitted leptons is 0. Forbidden beta decay is more complex of a process than allowed decay to ac- curately calculate, since the leptons must carry angular momentum. The small mass of the leptons requires them to be emitted at a great dis- tance from the nucleus in order to carry one or more quanta of angular momentum, and this results in a strong inhibition of decays where A2 2 1. Thus the name forbidden decay. The degree of forbiddeness is defined as the number of orbital angular momentum quanta which must be carried by the leptons. Thus A2 = 1 is first forbidden, A2 = 2 is second ferbidden, etc. Forbidden decays provide a mechanism to connect states of different Parity not connected by allowed beta decay processes (viz. Au=(-1)A2. Thus 3/2' -> 1/2+ can proceed through first forbidden decay and ‘5/2+ -> 1/2+ through second since A2 2 1 and AI = no. 5/2' -> 1/2+ may proceed by first forbidden decay, since A2 = 1 and AI = yes (the electron neutrino pair are coupled to J = 1.) Allowed beta decay rates can be calculated to a very good ap- proximation by the simple application of Fermi's Golden Rule, 1 = 32,-000 I<1b|0|¢1>12. [2.1] J J The two allowed beta decay operators, Fermi and Gamow-Teller, will have different intrinsic strengths independent of spatial overlap. 20 These have been parameterized as gA and gv, the Axial-Vector and Vector coupling constants, respectively, which represent the strength of the Fermi and Gamow-Teller (I and 01) operators. With these coupling con- stants and all other coefficients included, the beta decay rate can be expressed as 2 2 2 2 11 = c rA(w,,2) [avgvl<¢J|I|Ii>| + QGTgAIl ], [2.2] where, 7 c - 1“°)2 5 , [2.3] c (m.c ) Here fA is the dimensionless phase space factor evaluated specifically for Gamow-Teller decay, and 6V (Equation 2.17) is fV/fA, a correction factor of order 1:0 (0 ~ 1/137) which represents the slight difference between the phase space factors for Fermi and Gamow-Teller decay. W, represents the total decay energy, and is given in Equation (2.22). The evaluation of the phase space factor is discussed in more detail in Section 11.2. QGT is an empirical quenching factor for the Gamow- Teller decay, and is discussed further in Section 11.5. me2 is the energy equivalent of the rest mass of the electron. The squares of the transition matrix elements for Fermi and Gamow- Teller (abbreviated as F and GT respectively) are traditionally expressed as the dimensionless quantities, B(F) = I12. J B(GT) = (SA/8V)2 I|2. [2.u1 J The formalisms used in the evaluation of the matrix elements are described by BROWN85, and in references therein such as MAYERSSS. The nuclear beta decay rate becomes, 21 2 11 - c gv rA(w,,2) [6vB(F) + QGTB(GT)], [2.5] With C explicitly evaluated, and a value for g: obtained as described in Section II.A, the expression for the decay rate becomes, in 2) -1 1 = ‘311653’ rA(w,,2) [5VB(F) + QGTB(GT)] s , [2.6] J The factor of ln(2) comes from the traditional discussion cm? the half‘ life of a nucleus rather than the decay rate. This is discussed more in Section 11.3. 11.2 lepton Phase4§pacez The present theory of the Electroweak interaction states tnult the process of nuclear beta decay is actually a sequential pair of two body decays. Thus for B' decay of a nucleus (A,Z), (A,Z) -> (A,Z+1) + w', W- -) e- + V. [2.7] The mass of the W boson has recently been measured to be roughly 90 GeV (ARNISON83 and BANNER83; see also WATKIN886 for casual reading.) 'Ihis is four or five orders of magnitude more massive than typical nuclear beta decay energies. Thus the W is created in a virtual state, and lasts such a short time that one does not need to consider it as an in- dependent particle in the kinematics. The available phase space can therefore be calculated by the simple approximation of direct three body decay (daughter nucleus and two leptons). Since the mass of the nucleus is much greater than that of the leptons, the approximation can be made that all of the kinetic energy is carried by the leptons. The available phase space is the proportional to 2 2 p = pedpedfle pvdpvdflv, [2.8] 22 with the appropriate delta functions inserted. The solution of Equation 2.8 can be found in any introductory text on beta decay, such as those mentioned in the beginning of SectMNIII or MARMIER60. The result is a dimensionless phase Space factor for the lepton pair in the absence of any Coulomb effects upon the electron, We r(w.,Z=0) pw(w,-w)2 dW 1 £512w3-9WE-81p. + fiw.ln . [2.9] Here W, is the total decay energy of the two leptons (including their rest mass) divided by m°c2, W is the total energy of just the electron, and, p = (w2_1)1/2 . [2.10] The calculation of the total decay energy W, is discussed in Section 11.3, and given explicitly in Equation 2.22. The effect of the Coulomb field on the accessible states in phase space is significant at all energies, and can have orders of magnitude effects for heavy, highly charged nuclei. The effect can be ap- proximated by simply multiplying the phase space factor for 2:0 by the ratio of the squares of the amplitudes of electron wave functions at infinity, to that at the nucleus. Thus, I>|2 l|2 where u is the electron wave functnwlfvpm the solution of the r(w,,2) = f(W,,Z:0) [2.11] SchrOdinger or Dirac equations in the nuclear Coulomb potentiad”. 'The radii“: R must be chosen carefully for an accurate determination of the phase space calculation. The finite size and internal structure of the :nucleus would require, for a complete solution, the phase space factor 23 to be weighed by the expectation value of the beta decay transitdxnl as a function of R. This however is most likely inordinately rigorous for most applications, and usually one assumes the weight to be uniformly distributed over the volume of the nucleus. This subject has been ap- proached in light nuclei by TOWNER73b and WILKINSON7A, The Coulomb correction has been extensively calculate by Wilkinson and Macefield (WILKINSON7A), where the ratio of the squares of the amplitudes in Equation 2.11 for the solution of the Dirac equation is evaluated for a uniformly charged nucleus as a uniform source of lep- tons. The Coulomb field is corrected for the screening of the atomic electrons, such that the field does not have a strict r.2 dependence, knit instead drops to zero at a finite distance. A 'radiative‘ correc- tion is evaluated, taking into account a type of bremsstrahlung radiation as the electron suddenly leaves the nucleus, and the charge distribution within the nucleus is re-arranged. This process can in- volve both the production of real photons, and emission and re- absorbtion of virtual photons. It has been evaluated to order a in W1LKINSON7A. The various types of radiative corrections are discussed in detail in SCHOPPER69. Corrections for the finite mass of both the nucleon and the nucleus are also included. The overall correction fac- tor is expressed as an analytic function of beta energy of four coefficients. This correction, termed 8 after the authors, (in keep- WM ing with the notation of BROWN85) is given as, SWM = exp[ X an[ln(E,)n] ] , [2.12] n=0,3 where the coefficients c1n are those given in tabular form in WILKINSON7A. E, is the maximum kinetic energy of the beta, such that 2A a, = (w,.1)m,e2 . [2.13] Wilkinson and Macefield state that the correction SWM’ applied to Equation 2.6 will provide fA (see below) to an accuracy better than 0.1% for electron endpoint energies E, from 10 keV to 20 MeV, and for 2:1 to 101. Brown also includes two more correction factors. The first is an additional radiative correction to orders 202 and Z203, as described in WILKINSON73, 7A, and is given as, A 62 5 = 1 + 3.67x10‘ 2 + 3.60x10‘ 2 . [2.1A] R The second correction in BROWN85 allows for the 'diffuseness' of the nuclear charge distribution (<5WM is calculated for a hard sphere), and is discussed in W1LKINSON78. It is given as, 4521.36 6 5 = 1 + 1.8x10 - 1.2x10‘ zw,. [2.15] D The differences in the mechanisms of Fermi and Gamow-Teller beta decay require slightly different choices of R, and involve slightly different forms of some of the other corrections. This is discussed in detail in WILKINSON7A. The phase space factor for Gamow-Teller decay f is first calculated, A fA(W,,Z) = 508R5WMf(w°’z:O) . [2.16] “The factor {V is then obtained for e1 decay by a small correction fac- tor 5v, _ 3.11.5 31.11.12 6v ‘ 1 i [1‘5)(197]G‘Z ‘ [1051(197] ’ [2'17] such that, 25 286 Z38 “010 b V 77" fl? 'V‘V'l V V Y‘VU'II Y ‘ 10.0 P fYYYV' 1 V YTY—Tv'l Y r‘T VVVVV I V : : : ...: 3 .— -I . 1.006 : I g: ,. 4 I- --------------- J .2 L- ‘ 1- ‘ d 1.000 ““““ O _ fl- 4 o : A: 2 ~..‘ g : ‘1 3 “ __________ 0.996 '- - 8 : _ _ _ '6. 1 u 1.0 : ’ — ___________ t— : _ _ _ < .n : ’ a ’ 1 53‘ 0.000 :- 60 J g 0.0 - 13’ . P 6; 4 —‘ p Z I :1 ’ 0. - " .. o :......l . .....I 2--....1 3 l ni......1 ..I 0.9” n nnnnn 10" 10° 101 0.1 10‘1 10° 10x Edeoey (Kev) Edeeey (M8v) Z I 20 Z a 20 10010 :7 :Vj - L :Y-YV—Y-TVL .1 .7 I-Y V-V'l -'-: 10.0 :v v rvevl f V Y YIYV'I r v I v 7111‘ 1 : 1 3 3 1.000 3' 7 I: 5'° C \\ p’ I : 1-2 1- ‘ \ 1 * rrx-r ---.- ------- ‘ ‘5 ‘2 o 1.000_fi+m , ” I 1“‘_ , " " A- T ” ~ - .2 t : 3 ‘6 0.9“ r _ _ _ ‘6 ‘fi U 1.0 ,, 4 0 I- . 4 : '4 h C - - -. I .0 1- , «— ------ I 3 0.000 _— 5D 1 8 0.5 t I ,’ 1 ° : '6: : 3 . , 0000 ' 6* ‘ 8 1’ 8+ ' : ' 1 0 ’ I ‘ I , ‘ I bLA I‘ll L L AJUAI LA Al k‘ “‘1 “*1 “1 cm . Ann L n nnn AA LLLL 10"1 10° 101 0 1 10"1 10° 101 E“! (MeV) Edeeey (MeV) . V 2'” - L v.1;IT-VVL .1 .1 1.71.“: -1- ‘ EjYVVUV‘ V V V 'VVIII Y I Y YVY'V‘ V _ 4 : d I.“ r . -.‘ q I: ‘ s ‘ p- < d o l ‘ \ ~ I , g to r - ‘ ‘ ‘ 'i q Lm ;.::::j:...i' 2“ 2 ’ “ 2 p ‘ . > 1 3 o ’ g _ I—--w. I 0 1° = Q 4 - ...... F - - - A D ’ ’ I t a.” _ an —I a p I I O - - ‘ ’ I ‘ U . ......... 6' ‘ a ‘4 I + h ......... 6' ‘ 8 E I ‘ I 1 0.” — “ ' I " ‘ I J - n nnnnnl L n Lnnnnnl n n n Annnnl ;‘ 10-2 A (AIAAl L A LA AIAAl L A LA lenl 10" 10° 101 10"1 10° 101 a“, (HOV) Em, (KEV) Figure 2.1. The behavior of various expressions related to nuclear beta decay rates as a function of beta decay energy for three values of Z for the daughter. See text for complete discussion. 26 rv(w,,2) = svrAIw,,2) . [2.18] In Figure 2.1 the behavior and magnitudes for various expresSions above are illustrated as a function of beta decay energy. They are evaluated for a daughter 2 of 6, 20 and 50. Further discussion of lepton phase space can be found in any of the texts listed in the beginning of this section. The subject is com— prehensively, exhaustively and rigorously examined by BEHREN382. In the future, as beta decay rates are determined for isotopes further from stability, a re-evaluation of the phase space correction factors will become necessary. The corrections listed above are all given as only a function of Z. Implicit in the numerical coefficients are evaluations for the most stable A. The gentle rise in the mass surface on the neutron rich side provides a large range in N whiifll are particle stable, for a given 2. n For example, it may be possible to compare such isotopes as 3"Ca and 70Ca. Here the nuclear mass changes by a factor of 2. and the radius by 26%, while the nuclear charge remains constant. Further, the diffuseness will be different for neutron rich, near stability and neutron deficient nuclei. Further corrections are necessary, but become increasingly com- plex” In a review article on the subject, HOLSTEIN7A reviews a set of corrections collectively referred to as 'recoil effects.‘ Briefly, the :finite mass of the nucleon and thus nucleus with respect to the decay energy implies that the recoil momentum of the nucleon/nucleus nunSt be explicitly considered. In an interesting discussion on beta decay, SAMSONENK089 discuss the importance of treating the mesons within time nucleus in the beta decay process. 27 11.3 Branching Ratios and Half Lives: The principal quantities which one can directly calculate in al- lowed nuclear beta decay are the decay matrix elements B(GT) and B(F) , and their corresponding phase space factors fA(W,,Z) and fV(W,,Z). However, the observables in beta decay are usually the overall half- life (t) of a nucleus and branching ratios (BR) to well defined discrete final states, such that, {BR = 1. [2.19] J J The decay rate is related to the half-life by, l = 4%(3-1. [2.20] For each excited state j, a partial half life tJ can be defined in terms of the total half life t and branching ratio as, t = l_n.1.2_) = ..t_. [2.21] j *j BRJ The phase space factor is a function of the decay energy of the th 1 state to the jth state. 0 is defined as the difference between B the atomic mass excess of the ground state of the parent nucleus, and that of the daughter. The atomic mass excess of a nuclear state repre- sents the difference between the mass of the atom (the nucleus in that state plus 2 electrons bound in their atomic ground state) and M12 times the mass of a neutral atom of ‘ZC. Many subtleties exist in the details of the energy differences when the beta decay of ions or fully stripped atoms are dealt with in beta decay in astrophysical environment, or in storage rings such as the SIS/ESR project underway at 681 in Darmstadt. The change in bind- ing energies of each of the electrons as the nuclear charge is changed must be taken into account, as well as decay rate to competing 28 processes. One such process is the beta decay to bound electronic states such that the electron is captured huxlan un-occupied atomic orbital. For most beta decay experimental situations, the parent nucleus is at rest, imbedded in matter so that it is a neutral atom. In this case, the decay energies for e- and e+ beta decay are just, W,_ QB + (Exi - Ex ) , J W, (QB - 2m,02) + (Exi - Ex + In light nuclei, Ex ) . [2.22] J i is usually 0 for states whose beta decay can be observed. The low spins and large level separations of light nuclei discourage isomeric levels with half-lives long enough to compete with photon emission. However, occasional isomeric states do occur, and the beta decay rate from levels such as “N (Jn=0-, tzSus?) have been measured (GAGLIAR0183, MINAMISONO83, HAMEL85 and HEATH85). Isomeric states with much longer half lives are found in isotopes of Na auui Al. The symbol ft has been agreed upon as a simple way to express the strength of a given measured decay rate. It is defined such that, ft = rA(w,,Z) t [2.23] j . The combination of Equations 2.6, 2.21, and 2.23 yields the relation- ship, ft [8VB(F) + QGTB(GT)] = 61701A s . [2.2u] Often, ft values are reported as their logarithm in base 10. Thus the symbol log(ft) is most often reported. Values of log(ft) ffiwnn 3.3 to 6 are typical for CT beta decay. 29 11.A Empirical Coupling Constants: The vector coupling constant gV can been empirically determined from the analysis of the decay rates of many nuclear 0+-->0+ transi- tions, where only Fermi decay contributes. In these cases, both the matrix elements B(F) and the phase space factors fV can be calculated very precisely, and thus the observed decay rates are presumed to be accurate determinations of the coupling constant gv. Such an analysis is discussed in WILKINSON78a, and it is the results of this analysis which is incorporated into the numerical coefficient of Equation 2.6. A current summary of experimental and theoretical work associated with + -) 0+ nuclear transitions and their relationship to more fundamental 0 concepts has been compiled by HARDY89. While thelbeta decay of the free nucleon (neutron) seems the most fundamental source of beta decay infcrmation, it is useful only after a value for gv has been obtained since the neutron can undergo both Fermi and Gamow-Teller beta decay. The ratio (SA/3V)2 can be estimated as follows. The accepted half life of the neutron is presently 10.6 minutes. See FREEDMAN85, FREEDMAN86. With a QB of 0.7823 MeV (WAPSTRA86), the phase space factor for both Fermi and Gamow-Teller transitions is 1.637. The B(F) for a free nucleon is 1, and l<01>| is 3 for the free nucleon. The ratio (gA/gv) can be determined from Equation 2.2Aq saith the quenching factor QGT explicitly set to unity for neutron decay (see Section 11.5). Thus, (1.637)(636.)[1+3(gA/gv)2] = 6170, [2.25] which gives, 30 (gA/gv) : 1.28 . [2.26] This is close to the value of 1.251(9) ofWILKINSON73, which has been adopted by BROWN85, and the newer value of 1.2606(75) of W1LKINSON78. See also FREEDMAN85 and FREEDMAN86. 11.5 Gamow-Teller Quenching: The strength of the Vector coupling constant gv has been empiri- cally determined from a set of carefully chosen and measured 0+~>0+ nuclear beta decays, and the Axial-Vector coupling constant gA has been evaluated for the case of the free neutron where the strength is all to one final state. Nuclei with more than a single nucleon can not be used in determinations of the coupling constants. This is because un- like Fermi decay, Gamow-Teller connects an initial nuclear state with many final states, and all of the Gamow-Teller strength to these final states must be suumed up in order to calculate 3A' This is usually im- possible, since many of the final states sharing the Gamow-Teller strength lie above QB’ and thus can not be populated by beta decay. Thus gA is ultimately obtained from the neutron decay half-life alone, and the value thus obtained is called the free nucleon value. A careful study of hundreds of experimentally determined ft values and M1 gama ray decay rates for nuclei in the ad shell was carried out by BROWN85. A clear trend in the predicted versus measured ft values was observed. The experimental ft values were consistently larger than the predicted values. The theory could be brought into agreement with the experimental values when a quenching factor QGT of 0.6 was intro- duced. This Gamow-Teller quenching has been the subject of many investigations. It at first appears that the nucleon somehow changes 31 in the nuclear environment, such that the 8A is reduced by a factor 0.61/2~.77 with respect to the free nucleon value! However, it is now believed that a much simpler explanation ex- ists, without requiring a renormalization of the Axial-Vector coupling constant. Most likely the quenching is actually an artifact of some of the simplifying assumptions in the shell model theory used to predict the B(GT) values. The Gamow-Teller strength has not disappeared, but instead is distributed over more levels than was previously thought. The present shell model configurations do not include enough degrees of freedom to predict all of these states, and thus assign extra strength to the states which it does predict. Calculations using larger model spaces will be useful (if difficult and time consuming) in order to confirm the presence of the missing strength in higher lying levels. A second source of quenching involves mixing of the predicted states with the A resonance, representing mesonic degrees of freedom not included in the shell model. BROWN87 estimates that roughly 2/3 of the quenching is a result of configuration mixing, and the other 1/3 is a result of mixing with the A resonance. As discussed in the Introduction Section, Conclusions Section, and Recommendation Section, decay studies of nuclei far from stability might increase the infbrmation on Gamow-Teller quenching. This will be made possible by exploring QGT in nuclei with greatly different N/Z ratios and greater W, than those which have been studied to date, where the behavior of the quenching might shed some light upon the underlying mechanism. 32 SECTION III 111. Beta Decayito Unbound States: When the final state of a nuclear beta decay is unbound to still further decay modes such as nuclear particle emission, the connection between the beta decay matrix elements (e.g. B(GT), B(F)) and the ob- served decay rate becomes more complex than the relationship derived in Section 11. This is primarily because the final state is now no longer discrete but is instead spread out in energy. In some cases states populated by beta decay are sufficiently excited to break up into three or more nuclear particles. In this situation the connection between the observed distribution in n-body phase space to the initial beta decay matrix elements is even more complex. While the connection between observed and directly calculable quantities is complex and will continue to involve some approximations, it will become an increasingly popular problem as more and more infor- mation is gathered on the decay of exotic nuclei. Decay properties are the predominant source of information on the structure of nuclei when only thousands to millions of atoms of a particular isotope are availablez. The determination of most of the B(GT) in the decay of nuclei very far from stability will involve the observation of delayed particle emission. The initial states are often close to being unbound and most 2. The only other piece of information which can be obtained from a sample of a thousand to a million nuclei is the ground-state mass. In the future, techniques involving high intensity lasers and storage techniques such as ionic traps or storage rings may be able to probe other ground state properties such as magnetic moment and RMS matter and charge radius of the nucleus when only a small number of nuclei are available. 3A of the final states are indeed unbound. There are two important com- plications which arise when the final state of a beta decay is unbound to particle emission; 1) the radial wave functions must somehow be modified since there is a finite expectation value for finding some part of the nucleus well beyond a radius inferred from a standard Harmonic Oscillator (HO) wavefunction, and 2) the problem becomes time dependant and thus the excitation energy of the state is no longer well defined. 111.1 Structure of Unbound and Loosely Bound levels: The first point actually applies to levels which are loosely bound as well as those which are unbound. The most dramatic example of this might be found in “Li. This nucleus is believed to be bound to two neutron emission by only 197 i: 110 keV (WAPSTRA85). TANIHATA85a, b, TANIHATA88 and KOBAYASH188 report a nuclear radius extracted from total reaction cross section measurements which is roughly 20% larger than that of its beta decay daughter “Be! If this is indeed a valid inter- pretation of the reaction crossection, it is the most dramatic example of variation of a nuclear radius with binding energy. In most situations the radial wave functions will not show such a dramatic difference between parent and daughter. However, slight changes in the radial wave functions induced by changes in binding energy have been studied for many years. THOMASS1 and EHRMAN51 studied the coulomb energy shift of single particle levels which could be rep- resented by a single nucleon and an N=Z core. The difference in mass between the mirror ground states represents the coulomb interaction of 35 the proton with the core, under the assumption that all other interac- tions are the same for .the nucleon whether it is a neutron or proton. Because of the mutual Coulomb repulsion, the proton rich nucleus usually becomes unbound to particle decay first. Thus the radial wave function of the proton expands more rapidly than that of the neutron as a function of excitation energy, which in turn decreases the overlap of the proton's wave function with the core. This results in a smaller coulomb energy shift than that for the ground state, shifting the ex- citation energy of the proton-rich level leSs than that of the neutron- rich level. This shift is usually referred to as a Thomas-Ehrman shift3. It is experimentally observed as a lower excitation energy for a state (relative to the ground state) in the nucleus with a proton excess than the mirror state in the nucleus with a neutron excess. Most wave filnctions needed to obtain B(GT) will not be of single particle nature, and thus binding energy effects can not be easily cal- culated. ORMAN085, 86, and 89 are developing an Isospin Non-Conserving (INC) Shell Model Hamiltonian which partially addresses this issue. Here binding energy effects will be indirectly represented as an Isospin dependence in the interaction which has the potential for some success because of the correlation between binding energy and the T2 of the nucleus. 3. A systematic study of shifts in mirror levels was carried out by NOLEN68. They found that a small fraction of the shift could not be accounted fer by any known phenomena. No simple solutions have yet been found to explain the residual shift. This residual shift is often referred to as the Nolen- Schiffer anomaly. 36 A different approach was taken by TOWNER73 to calculate the effect of binding energy upon the wave functions of nuclei. Here, each state was expanded into a series of nucleon + A-1 configurations where ex- cited states in the A-1 nucleus were included. The binding energy effects were calculated for each configuration. B(GT) were then calcu- lated for mirror decays both to the same final state (such as ”B, "N decay to ”C) and to mirror final states (such as "0 —>"N, and "B- >‘ ’C.) The observed differences in the mirror B(GT) could usually be reproduced, serving the dual purpose of explaining the observed asym- metry in B(GT) and providing a way to incorporate binding energy effects into the wave functions of levels which could not be repre- sented as single particle configurations. 111.2 Excitation Energy and Unbound Levels: As pointed out above, unbound levels will have a finite width. This implies that the beta decay may populate the level at more than one energy. The beta decay rate must now take into account not only the density of states available to the beta neutrino pair in phase space, but the internal density of states of the final nucleus. The internal density of states for the unbound system can be ob- tained from a Breit-Wigner parameterization of the line-shape. It can be derived in various forms from first order perturbation theory. A complete treatment of R-matrix theory can be found in LANEAS. This representation has a wide range of application in resonant phenomena where an unbound state can remain static long enough so that its decay can be treated in first order perturbation theory.- One form often used to represent scattering states is 37 F(E)/2u 2 2 (P(E)/2) + (E-Eo) Here the local width I‘(E) represents the decay rate of the level at p(E) dE = dE. [3.1] energy E. E, is often quoted as the excitation energy of the state, but actually is best described as the energy at which the phase shift would be 90° if the resonance were produced in a scattering situation with no other scattering processes interfering. When describing an ob- served state without a complete phase-shift analysis, it is simply an adjustable parameter which gives a best fit. If the state is relatively narrow, such that I‘(E) is nearly con— stant over the region (E,-I‘/2) S E S (E,+I‘/2), then the full width at half-maximum (FWHM) of the distribution is just 1‘. However this is true only when P is constant and thus the distribution symmetric. P(E) actually represents the size of the transition matrix element for the decay of the state being described at each energy, and its ability to describe the FWHM of the line shape is true only in certain limits. In the same fashion that the beta decay rate was obtained in Section II, the nuclear decay rate P(E) can be expressed as the product of an overlap between initial and final states, and a phase space fac— tor. However, one must keep in mind that the emission of a nuclear particle from a nucleus is a complex process, and only approximate tools exist for the treatment of such problems. The dynamics of how a complex particle and residual nucleus form out of a single initial nucleus, and of how they penetrate the nuclear potential barrier is a difficult and poorly understood topic, and will not be dealt with in detail here. The concept of a spectroscopic factor from shell model overlaps, and a penetrability from standard R-Matrix theory will be 38 used. The calculation of spectroscopic factors is discussed in Appendix A. (MIKOLASBBa). The spectroscopic factor is usually expressed as a fraction of a Wigner single particle unit, 2 Y2 : 431591-2- [3.2] mRo where u is the reduced mass of the two products of the decay. The decay can often proceed through more than one angular momentum 2, so that a spectroscopic factor is usually expressed as 0 for each partial 2 wave. If one or both particles have a non-zero spin, then a second quantum number, channel spin, must also be sunmed over. This will be ignored in the present derivation for simplicity. The final state is actually the two nuclear particles separated at large distances. Thus the complete overlap of final and initial states includes the expectation value of finding the two particles at a separation r as row. However, this factor is traditionally included in the penetrability factor discussed below. The phase space factor is in fact obtained in a manner very similar to the phase space factor for beta decay. Here the Schrédinger equation is solved for relative angular momentum 2 in the region r>R, for the relative kinetic energy of the decay channel of interest, giving the ratio of intensities of the wave function at rzR, and r=0. Using the standard Wigner parameters p and n, p = krv z z e2 n : %, [3.3] where, 39 k . A“? [3.u] velocity, v J2E/u, [3.5] and E, the decay energy in the center of mass with u as the reduced mass, the penetrability P for a given decay into two spin-zero par- 2 ticles is, P = p , [3.6] Q F§(p.n) + Gi(o.n) where F and G are the regular and irregular coulomb wave functions, and can be obtained either from iterative computer programs (e.g. BARNE'I'I'BZu) or from tables (e.g. MAR10N68 and SHARPSS.) For neutron decay, n=0 and F and G reduce to Bessel functions. The local width of the state is finally given as 2 0i 12 p F,(p.n) + G,(0.n) or in the more compact form f(E) = 2 a: Y2 92(5). [3.8] The state is often unbound to more than one value of 2, and some- times to even more than one type of particle decay channel. In order A. At low decay energies (below roughly 100 keV for light nuclei) the program discussed in this reference sometimes fails to converge in certain energy windows, while converging both below and above this window. An error message is returned tozaotify of such failure. Adjusting the constant ACCUR (~10 ) will move the window around enough to calculate any desired value. If extensive low-energy calculation is important, then this problem should be investigated more thoroughly. A0 to avoid the proliferation of indices, a 'master' equation will not be written down. Instead, the expression, NR) = £213,559. [3.9] i 2 illustrates that the total decay rate is simply the sum of all of the partial decay rates into the partial waves 2 and particle decay channel 1. In Figure 3.1, a level diagram of the beta delayed alpha decay from ‘ ‘Be is shown. The level in “B decays by the emission of an al- pha particle to both the ground state and first excited state of 'Li. This situation illustrates the use of Equation 3.9 in calculating the total decay rate as a sum of the partial decay rates. One fact which becomes important in the comparison of beta decay rates to the beta decay matrix elements is the area of the Breit-Wigner distribution. For a constant I‘ the area of Equation 3.1 is unity when integrated over the unlikely limits of -w to a. However, if the penetrability is varying over the line shape under consideration, then the are is no longer normalized to unity. Figure 3.2 shows various line shapes for a hypothetical state in 'Begs for different values of E,. In Figure 3.3, the areas of these distribution are plotted as a function of E,. Two important points can be extracted from this exer- cise. First, the areas can vary significantly from unity which has a direct effect upon the beta decay rates and thus half-life of a nucleus which populates this state. Secondly, the line shape may become so asymmetrical that a second maximum, or 'ghost' appears at higher ex- citation energy. Ghosts like this often appear near threshold, where the penetrability is varying exponentially and can A1 II 32"). 1/2 IOZSO 3/2’ 9875 3Q: 7973 ’5 was/2r 7286 "2’6793 l I 66 30 4.| P— 3’2 5021 5/2 1 4445 ZIZS Ha 4': 312' FIG. 4. Proposed decay scheme of “Bo. New information from the present work includes the B—ray branching inten- my to the 9875-kov state and a branching rationto the ground and 478-koV states of 'Li. Figure 3.1 Energy level diagram for the beta delayed alpha emission of “Be. Branching ratios are from ALBURGER81 and MILLENERBZ. 112 6. 5&- 5" : '> 4:— g : E3 : ‘ci 2:" 1&- LI. 0 7| 101 E— . p 5 | .. > 0 0 10 =— 5 E A r a - Q 10'1 5‘" 10—2 A l l l l L L L I I L l J 1 1 Pl 0 0.5 1 1.5 2 E, OBe (MeV) Figure 3.2a Density of states p(E) for a hypothetical state in °Be which decays by s-waye (i=0) emission of two alpha particles. A spectroscopic factor 6 _ of 1.0 was used, which is close to the actual value for 'Be decay’Qnto this channel. Values for E, of 0.0, 0.2, 0.3, 0.5 and 1.35M“ are used. Note the ghost structure for E. = 0.0 MeV. « Figure 3.2b, same as Figure 3.2a but with a logarithmic scale. 43 1,50leITIIU1[YlIlUIIIITIIIII fP(Ex)dEx D Q C” IIIIIITIII11ll111IIIIIIIIIII lllllllllllllllllllllLJlllll P 0.00 llLLlllllllllLLlllllllll 0 0.25 0.5 0.75 1 E0 (MeV) Figure 3.3 Area of p(E) (examples of which are in Figures 3.2a,b) in- tegrated from 0 to 3 MeV as a function of E,. Note that this area is either significantly greater or less than unity under most cir- cumstances. For E. = 0.0 MeV, the area of the sharp spike shown in Figures 3.2a,b is 1.0, and the remaining 0.37 is contained in the high energy ghost. 141% change by many orders of magnitude. This 'ghost' of the ground state in 'Be has been been observed by BECHETTIBi. It is interesting to note that the reaction 12C(a,Y)“0, which is the predominant source of “0 production in the Helium burning in starts, proceeds almost exclusively through the high energy tail of the 7.115 MeV 1- state in “0, which is actually bound with respect to this channel by M1 keV. See Section 5.5 of CLAYTON68 for a further dis- cussion. III.3 Beta Decay Rate: Equations 3.8 and 3.1 give p, the internal density of final states. The external density of states for the beta decay has been discussed previously, and is given in Equation 2.11. The beta decay rate will then be a product of these two densities of state. Thus the beta decay rate i dB at an energy E is given by 7 2 (he) 8V l(E) dE = 2 5 rA(w.,2) [8VB(F) + QGTB(GT)] p(E) dE, [3.10] c (m.c ) O!“ r (w.,2) [5 B(F) + Q B(GT)] P(E)/2n l(E) dE = éfigéia A V GT dE, [3.11] (run/2)2 + (3-2.)2 where the results of Section II are combined with equation 3.10. In the limit of small and constant P, Equation 3.11 reduces back to Equation 2.6. However because of the strong energy dependence of the electron neutrino phase-space factor, a Breit-Higner lineshape will be populated asyumetrically, with the lower energy tail producing more beta decay rate than the higher energy counterpart. Figure 3.4 illustrates the effect of the beta neutrino phase—space factor upon the shape of the distribution. The closer a state is to 45 the endpoint energy, the more dramatic the asymmetry, up to the point where a second maximum ('ghost') at lower energy is produced. Likewise, the larger the width, the stronger the 'ghost'. This type of ghost phenomena was probably first described by HATT6H, while observing the decay of 'B to the lower 16.67 MeV level of the Isospin mixed doublet in l'Be. In that example the decay fed not only the peak, but even more strongly to a tail at lower energy which rapidly increased at lower excitation energy. It is important to point out that something like an ft value should no longer be defined, since f is no longer evaluated only at a single point. Neither should a branching ratio be defined. A more complete analysis of multi-level final states (see WARBURTON86) shows that interference between two final states of the same spin and parity can actually reduce the beta decay rate. However in spite of the warnings above, in this simple model a branching ratio might be defined as 13121 = 171% dB 111(8) [3.12] if interference effects are ignored (see ZHA089 for example.) Here t is the beta decay halflife of the parent. III.” Beta Delayed Three Body Final States: Two particularly dramatic examples of beta delayed multiple par- ticle emission were described in the Introduction (Figures 1.2 and 1.3), those of "B and of 2"iii. In these examples many of the states populated by beta decay will result in final states which involve three or more nuclear particles“ In the model described above, the beta decay rate per unit energy into one of those levels at a specific #6 :l I I' ,l.l .I l I l I lfrT I I 1 T I l I I I 1 r1 I l I: 100 '5" “a T 3 : ”a? 1 ' 1 % 1° E’ 1 a C 9.0 3 " '1 E3 10_2 E ‘2' 72’ E 12.0 5 I 1 10":3 7::- 10-4 1 I I l I I I I J_l I I I 14 LLI LLL I I I l 0 2.5 5 7.5 10 12.5 15 Ex ”Be (MeV) Figure 3.” The beta decay rate per unit energy l(E) dB for ‘B decay to a hypothetical state in 'Be with Eo = 3.0, 9.0 and 12.0 MeV. The dotted line represents the shape of the beta neutrino phase space fac- tor fA(H.,A) evaluated at each excitation energy Ex’ 117 energy can be obtained as the product of three quantities; 1) the phase space factor for beta decay to that energy, 2) the beta decay matrix element and their associated coupling constants, and 3) the internal density of states at that energy p(E) as given in Equation 3.1. The internal density of states at energy E was calculated from the local width of the state at that energy and E,. Thus independent of the type or complexity of the particle decay, once the total particle decay rate of the state at each specified energy is evaluated by calculation or experiment, the beta decay rate can then be evaluated. In Figures 3.5 and 3.6, a schematic representation of a beta delayed three body decay is shown. ‘The particle decay rate of the level in the beta decay daughter can not be calculated in a straight- forward manner, since the particle decay energy and thus penetrability depends upon what energy in the intermediate state the primary particle decay feeds. In the spirit of Equation 3.9 where the total decay rate is Just the sum of all possible partial decay rates, one can imagine that the decay has a differential rate to each excitation energy in the intermediate nucleus proportional to the penetrability at that energy times the internal density of states in that intermediate nucleus. Thus with the intermediate state described by, F1(Ei)/2n 91(31) dis:i = 2 2 dEi, [3.13] and r(g)=292Y2p(g) [3111] i 1 2 2 2 ° as prescribed in Equations 3.2 through 3.8, the local decay rate of the state in the daughter populated at E becomes d '48 5Li+cx 2a+p Figure 3.5 Schematic representation of a beta delayed three body decay. H9 1?— 7r E1 Q, .... _ Ea "1F ‘11 E, E3 J ‘— ‘17 — “—73 "11' E: 1__ ____ I 153—— Figure 3. 6 Illustration of various energy variables. The excitation energiesE and E of the daughter, intermediate and final nucleus respectivglye re de ined in the text. Arrows point from A to B where E: B- A. Thus A is negative. The beta decay energy from parent to daughter E and particle decay energies from the daughter to inter- mediate nucgeus E , and from the intermediate to final nucleus E are used to calculate the penetrabilities P (E ) and P (E ). These then give the local decay rates P( ) and F “(E ) in the dgu ggter and inter- mediate nucleus. Energy diff Eigences are alculated by subtracting the value at the tip of the arrow from that at the origin. While two decay channels are illustrated for the decay of the intermediate nucleus, only one particle decay channel of the daughter is shown for the sake of simplicity. 50 2 2 I‘d(Ed) dEd = 2 92 Y p(Ei) dEIi P2(EZ)' [3.15] The integral in Equation 3.15 represents an average penetrability, weighted by the internal density of states. The internal density of states in the daughter is then, ( ) Pd(Ed)/2n E [3 16] pE dE = d. . d d d (Pd(Ed)/2)2 + (Ed-E.d)2 d This expression finds its roots in the generalized density of states concept described in HENLEY60. In the present notation, the widths are all functions of excita- tion energy in some nucleus, while the penetrabilities are functions of particle decay energies. As illustrated in Figure 3.5, the relation- ship between these variables are, EB = moczwo = Q8 - Ed , E1 : Ed + A1 - E1 and £2 : Ei + A2 - Bf . [3.17] QB is the atomic mass excess of the parent minus that of the daughter ground state, A1 is the atomic mass excess of the ground state of the daughter minus that of the intermediate nucleus, and A2 is the atomic mass excess of the ground state of the intermediate nucleus minus that of the final nucleus. Ef is the residual excitation energy in bound or narrow levels in the final nucleus. 111.5 Acceptable Level of Approximation: The extrapolation of predicted nuclear properties from nuclei close to stability become increasingly unreliable as they are applied farther from stability, and therefore so does the acceptable degree of approximation in the interpretation of the data. Thus even though the 51 beta-delayed particle emission channels become more complex, the re- quirements on the thoroughness of the interpretation decreases. As discussed above, the shell model itself involves some ap- proximations which limit the degree of agreement one would expect with experiment. As discussed in Section II, the apparent quenching of the Gamow-Teller operator is most likely due in part to GT strength to con- figurations not included in the calculation. It represents on the average a reduction in the observed rate to 0.6 of the calculated value. This value would actually be somewhat different for each in- dividual B(GT), with the value of 0.6 representing an average. It is therefore likely that a large body of experimental B(GT) extracted from delayed particle emission data from nuclei far from stability would be - very useful even though they may contain random uncertainties up to perhaps 20% and systematic uncertainties of perhaps 5%. Unless binding energy effects upon the radial wave functions can be understood better in the future, it is possible that this might even preclude the study of CT quenching far from stability even if the ex- perimental uncertainties mentioned above are achieved. In a beta delayed three body decay, there are at the minimum five parameters which can be used to fit a given branch; one beta decay branch with a matrix element B(GT), which feeds a level defined by E‘d’ which decays with a spectroscopic factor 9? to a level in the inter- mediate nucleus defined by E°i’ which then decays with a spectroscopic factor a: to the ground state of the final nucleus. In a realistic situation there may be five or ten times this number which contribute significantly to the total beta decay rate of a nucleus. with good 52 spectroscopy these can be decoupled into groups so that all of them would not be simultaneously adjustable. However, it is probably cer- tain that some residual uncertainty in the B(GT) will remain. The level density in the daughter and in the intermediate and final nucleus will determine the difficulty with which reliable B(GT) will be ex- tracted. When states are broad enough and close enough so that they overlap, they will interfere. In this situation a more complex version of Equation 3.1 should be recovered from the appropriate limit in R- Matrix theory in order to include all phase shifts. An example of such analysis fOr two body decay can be found in WARBURTONBG. The first place to apply this method is a very light system where the level density is low as are the particle decay multiplicity and complexity. The following Section describes a study of beta decay in Mass 9. 53 SECTION IV IV. Beta Decay of ’C: Figure 11.1 shows the first record of the decay of an atom of ’C from SHAHISG. It was found in a routine analysis of tracks in nuclear emulsion. The decay of 'Li and 'B to 'Be has often been observed as 'hanmer tracks,‘ with the long ionization path of the parent nucleus ending in a short back-to-back 'hammer head' of two alpha particles from the breakup of 'Be(2+). This first event of ’C was conspicuous in that there were three charged particles emitted following beta decay. The incoming track of the ’C ion and outgoing tracks of the beta par- ticle, two alpha particles as well as that of a proton are all resolved. There energies were calculated from the density and length of the tracks. The analysis of the decay suggests that by the emission of an electron of 11 MeV (suggesting a 08 above 16 MeV), the ’C decayed to a state in ’B at 12.2 MeV, which decayed by the emission of a proton of 7 MeV, the rest of the energy shared by the two alphas. Remarkably this is exactly the decay sequence which while dominating the high energy decay particle spectrum of ’C has been most difficult to study. It has never again been directly observed! In this section, I will briefly sumarize an experiment in which the beta delayed charged particle emission of ’C was studied here at Michigan State University (MSU) by implanting the ’C activity directly into silicon. This experiment has been thoroughly described elsewhere (MIKOLAS88a and Appendix A.) I will then focus on the beta decay to, and particle decay of a state in ’B at 12.1 MeV which is most likely the decay process illustrated in Figure 11.1. This requires using the model described in Section III to simulate the beta delayed three body 55 FIG. 1. A mgmtognph of an event interpreted as the beta decay ofC' 111M tucknmproducedinséar(A)and disinteya into a proton, two particles, an a positron (Uncksi, 2, 3, and!» respectively.) Tun: I. Character-ism: of C' deny. Range Ener ln Truck mien»:i Identification My Angles F 6.2 C’ 9. 4 65° 1 341.0 P 7. 6 115° 2 9.9 He‘ 2.7 3 7.8 He‘ 2.1 136.5" 4 . . . 3.1 11. Figure “.1 Enlargement of a nuclear emulsion track from a fragment produced in an energetic collision of a relativistic proton with a nucleus of the emulsion. This product was identified as ’C by its un- usual decay mode; B+2a+p. This is the first atom of ’C ever to be observed in decay. The energies of the charged particle imply that the ’C decay populated a level in ’B at 12.2 MeV, which then decayed by the emission of a proton and two high-energy alpha particles which could not have come frdm the ground state of 'Be. This decay mode is the subject of this chapter. 56 sequential decay of ’B following the beta decay of ’C. I will compare the MSU data with that taken at the 88 inch cyclotron in Berkeley (HARDY65, ESTERL72.) In each situation only an inclusive energy spectrum exists, with no correlations between pairs of particles. In the MSU data, the total energy of all three emitted particles is col- lected and summed for each event. In the Berkeley data, only the energy of the proton is recorded. By means of the Generalized Density of States model described in Section III, and comparison with shell model predictions, these two sets of data can be used to produce a single model which explains all data and is consistent with all predictions. 1V.1 The MSU Experiment: As pointed out above, the experiment is described in detail in MIKOLAS88a which is reproduced in Appendix A. Only the relevant points will be outlined here. The Reaction Product Mass Separator (RPMS) at the National Superconducting Cyclotron Laboratory (NSCL) at MSU was used to separate fragments of a "C beam produced by collisions with a Nickel target. ’C fragments were separated from all other activity and implanted within active silicon detectors. When the ’C atoms decayed, the beta particle left the silicon depositing on the average only a few hundred keV of ionization. However, the state in ’B would always decay by the breakup into three particles, two alphas and a proton. A level diagram for ’C decay is given in Figure 11.2. Since the proton and alphas have no energetically accessible excited states in this reaction, the sum of the kinetic energies of the three particles is equal to the excitation 57 3/2' 16.50 9C 3/2' 14.66 T-3/2 12.1 —— -—P= 0'4 I": 5 — 9.1 Eli.l/a— '1' a 1/2‘ 2.9 Pa: 1,3 I‘= 1.5 \ — 2-9 899.24- ‘1' p 5/2‘ 2.36 / 1"“I 1-5 __ 18 5L1 - ° 3/2- 'I' a 3/2 0.00 — —§— -—— -\— -o.19 “sea. + p B -0.28 201 + p Figure ".2 Energy level diagram for the decay of ’C. From Figure 1. of Appendix A (MIKOLASBBa). 58 energy in the ’B nucleus plus the energy associated with the breakup of the ground state of ’B. If all three particles come to rest within the silicon detectors, their ionizations are collected together and the a signal is produced proportional to the total breakup energy of the sys- tem. In figure 3 of Appendix A, the energy spectrum recorded in a silicon surface barrier detector of “00m thickness is shown. The two prominent peaks at low energy represent the population of the ground state and 5/2' excited state of ’B at 2.36 MeV. A small peak near 12 is also observed. The number of ’C ions implanted was compared to the area of these peaks, and branching ratios were extracted under the as- sumptions that these levels were narrow and their shapes calculable directly from a simple model of the energy loss of the beta particle as it is emitted in different direction. The branching ratios so ex- tracted are summarized in Table 1 of Appendix A. They are also converted to B(GT) under the assumption that the levels are narrow and formulae of Section II can be used. The experimental B(GT) are com- pared to those predicted by the shell model, and those extracted from various published observations of ’Li decay. Overall agreement between all three sets of B(GT) is good. However, for the branch to the 5/2' level the B(GT) extracted from ’C decay is over a factor of two lower than that observed for ’Li. As pointed out in Section VII of Appendix A, if these values are verified it represents the strongest mirror asymnetry observed to date. A peak is also observed near 12 MeV resulting most likely from the decay of a level in ’B at 12.1 MeV. If it were to decay by proton 59 emission to the ground state of 'Be, the resulting protons would have a kinetic energy of 11 MeV, and a corresponding range of almost 1 mm in silicon - twice the thickness of the detector used. Only the protons emitted close to the the plane of the silicon would come to rest within the silicon and result in complete collection of the ionization. Many more would pass out the sides such that a significant fraction of their energy would be missing from the spectrum. These protons would produce a tail at lower energy in the spectrum. The results of a Monte Carlo simulation of such a situation are shown in Figure 14.3 where they are compared to the experimental data. The simulation represents the beta decay of ’C to a state in ’B at 12.1 MeV which decays 1001 by proton emission to the ground state of ‘Be. The peak lies at a slightly higher energy because of the energy loss of the beta particle in the data and in the simulation. The simulation has been normalized to the data in such a way that the areas within 800 keV of the peak are equal. Under these assumptions only 201 of the events occur within the normalized area, with 801 of the events gener- ating counts at lower energy. This efficiency implies a B(GT) for the 12.1 MeV near 3.0, which is very large, containing 201 of the Ikeda sum rule strength of (gA/gv)2 3 lN-Zl. However, as can be seen below, this is an extreme upper limit. IV.2 Three Body Simulation for ’C: Three factors taken together strongly suggest that the level at 12.1 MeV in ’B decays most often through some channel which distributes the energy much more evenly between the three particles than does the 'Be(gs) channel. The first is the beta delayed alpha and two alpha 60 103 I I I I I I I I l I I I I I I I rI’l I I I I I I I I Li > 1 0) 34102 1 O 3 o .1 "" I \ - 2101 .3 Z : D I I 8 1 | ' 10° ‘ 1 1 “2: I LI I l I I I LJ ILL LJ I I I I H hi 1 q o 2.5 5 7.5 10 Energy (MeV) Figure “.3 Monte Carlo simulation of the decay of ’C to a level in ’B at 12.1 MeV. The ’C is uniformly distributed in a “00m thick silicon detector. In this situation 81% of the events produce counts below 10 MeV because of the escape of the proton from the silicon before losing its full energy. The disagreement in the 12.1 MeV peak position is within the quoted uncertainty in energy in MIKOLASBBa. The energy as- signment comes from ESTERL72. 61 spectra of ’Li from NYMAN81 and LANGEVIN81. In these experiments ’Li was deposited on a thin carbon foil, and the alpha activity counted in separate silicon detectors. A strong component in the beta decay resulted in single alpha particles up to 6 MeV, and two alpha sumed energies of 9 MeV. Clearly a process not involving 'Be was contribut- ing these high energies. ’Li activity was also implanted in silicon at MSU. In Figure A.A, the beta delayed two alpha spectrum for ’Li decay is shown. Since the decay of ’Be is to two alpha particles plus a neutron instead of'a proton in the case of ’B decay, the fraction of the energy shared by the alphas could be determined. Again a significant fraction of the events result in high energy alpha particles. The second factor suggesting channels other than proton + 'Be(gs) is the shape of the delayed proton spectrum from ’C decay of ESTERL72. Here in addition to two small peaks at 9 and 12 MeV, a large continuum in the proton spectrum rises to lower energies, containing more than 951 of the strength above 3 MeV in the proton spectrum. No such in- tense continuum is seen in the MSU ’C data (Figure 3 of Appendix A.) Thirdly, the behavior of the predicted spectroscopic factors for states near 12 MeV in ’B is quite erratic, often favoring ’Li channels over those involving ‘Be. A complete table of spectroscopic factors for six decay channels is given in Table 2 of Appendix A, and they are discussed in Section VI of that reference. 62 r- I I I I l I I I I I I I I f] I I I I I I I I I l I I I I 4 103 : “E > I 3 0) t - M E ‘ O 2 a 10 g- j . . \ I I U) *- . E" 1 Z 10 5- .5 r: s a O _ . D r _ 100 r l .1 : I I I I I I I I I I I I I I I , I I I I I L_L I I : O 2.5 5 7.5 10 12.5 15 Energy (MeV) Figure 11.11 Charged particle sum energy spectrum for the beta delayed 2n(+n) decay of ’Li separated with the RPMS. The uncharged neutron deposits no energy in the silicon; thus the energy of the two alpha particles can be determined separately from the total breakup energy. 63 Finally, the single event of SNAM156, while not statistically overwhelming, does suggest that channels for the decay of the 12.1 MeV state involving ’Li (suggested by lower energy protons) are M up- $.11. In order to investigate the role of 2’Li +a channels in the decay of the 12.1 MeV level, a computer program was written to calculate the decay probabilities for states unbound to three body decay. The program is generalized and easy to use. The proposed excitation energies, B(GT) and spectroscopic factors for each 2 are entered in a small table. The complete 3 body decay is simulated, and inclusive and correlated two particle energy spectra are generated for the beta decay of interest. The data can be compared directly with experiment and the agreement evaluated. The calculation is complex, and takes up to many minutes to complete a calculation of a delayed particle spectrum using a VA)! 8530 computer. Thus it is not well suited for a fitting proce- dure which varies each of the parameters. In the case of ’C decay the number of fitted parameters far outweighs the number of features in the data. None-the-less, little is known about the decay of this level, and it's basic nature can be explored with such an analysis. For the sake of simplicity only three channels were chosen for the initial calculation; 'Be(0+) + p; (2:1), 'Be(2+) + p; (2:1), ’Li(3/2') + a; (2:2). The 2:2 channel for ’Li+a was chosen as the one which tended to show the strongest spectroscopic factor for those levels near 12 MeV calculated in Appendix A. The first step in the procedure is to reproduce the shape of the intermediate states in 'Be and ’Li. In Figure 11.5, the ground state and first excited states of 611 'Be near 3 MeV are shown as calculated from a simple Breit-Nigner. The spectroscopic factors and E.s are varied to reproduce published widths. In Figure “.6 the shape of the ground state (3/2-) of 5L1 is shown. Here only the spectroscopic factor is varied, as E, is defined as zero for the ground state. Next, the width Pd(Ed) of the daughter ’8 is calculated as a func- tion of energy. At each energy Ed the Width is calculated from Equations 3.13 through 3.15 by evaluating the probability of particle decay into each of the channels. Finally the shape of the 12.1 MeV level in ’B is calculated. It is shown as the solid curve in figure 14.7. The dotted curve in that figure is the same shape, but weighted by the phase space factor for the beta neutrino pair. A long low energy tail emerges which represents the competition between the decrease in internal density of states in the ’B nucleus off-resonance, and the rapid increase of density of states in phase space for the beta neutrino pair. The curve falls to zero at the particle decay threshold for ’B. At each energy E in the ’8 level the fraction of particle decays d which feed each energy in each of the intermediate states is now calcu- lated. These fractions are normalized, inverted and then used by a Monte-Carlo routine to produce the final spectra. An important element in the Monte-Carlo simulation is the decay in flight of the inter- mediate nucleus. For example if the ’B nucleus decays to ’Li + a, the ’Li takes I4/9 of this primary decay energy as kinetic energy. The recoiling sLi then emits the proton with some angular distribution, and the laboratory kinetic energy of the proton depends upon this angle. 65 1.0K) q : Ffr I I I I I I l' I I I I I I I I I I I I I3 0.50 i i .5" I I l E i ‘ ‘-’ (L11) :- _: a E E v 0.05 P " c. - . - 4 0.01 J I I I II I I I I I I I I I I I I I I I I o 2.5 5 7.5 10 E, aBe (MeV) Figure “.5 Line shape for the ground state (0+) and first excited state (2 , 2.9 MeV) of 'Be used in the calculation of the breakup of ’B through these channels. 1.00 IIIITIIIIIIIIIIITII—IIIITIIIII CLEK) I IIIIII I I I IJJI I J 021(} p(E) (MeV") CLCHS I I IIIIII I IIIIIII I 1 0.01 IIIIIIIIJIIIIILILIIL LIIIII -4315 (J 1315 ES '715 it) E, 51.1 (MeV) Figure A.6 Line shape for the ground state (3/2') of 5Li. 66 ...a D ..a -1 IIIIlIIIIIIIIIIIIIIIIIjIIIITI10 I IIIIIII I 11111 100 5" 110-2? E : 1 E 3’10"1 .— —= 10-3 a A E 1 A no : : ca ‘Ei " — - 72’ .- /"'.-_ .1 / 10"2 —- / 10-4 I / Z / - I IILJIIIILIIIII 1 0 2.5 5 7.5 10 12.5 15 E, “B (MeV) 10-3 IILlLIILIIIIII 0—5 Figure 11.? (Solid) Density of states function p(E) for the 12.1 MeV state in ’B populated by the decay of ’C. The spectroscopic factors have been adjusted to reproduce the observed FHHM of “00 keV. (Dotted) The decay rate ME) to ’B obtained by multiplying p(E) by the phase space factor for the beta neutrino pair. 67 While the program will accommodate an angular distribution with respect to the primary decay axis, none is used in this simulation again for reasons of simplicity. The simulation of the total charged particle energy spectra for the decay is shown in Figures ”.8 and “.9. The agreement of the proton inclusive spectrum with that of ESTERL72 is excellent. In Figure 11.9 the same proton spectrum is superimposed on the Berkeley data. A branch to the broad 1/2' level in ’B at 2.9 MeV is also included to represent the lowest energy portion of the data. The agreement is ex- cellent. Even with the simple assumptions above the complete spectrum can be explained by the strong population of the ’Li channel and weak and nearly equal population of the channels involving 'Be. The results of the fit are summarized in Table “.1. The process by which the final set of 63 were obtained was an in- teractive process between the researcher and the computer. For example, when structures are isolated, first approximations to some parameters can be obtained in a nearly uncoupled mode. Adjusting cer- tain parameters to fit certain peaks is the quickest way to pin down some parameters. Others require many iterations of trial and error. A second version of the program is being written which should run in a much shorter period of time, perhaps a few CPU seconds per iteration. When this is complete, automated iteration can be implemented. However, the number of fitted parameters will be large, and a large amount of interactive work will still be necessary in most cases to ‘keep the program honest.‘ 68 Table 4.1. Spectroscopic Factors fit to the beta delayed proton spectrum of Esterl et al. The last column represents the calculated efficiency for collection of the full energy peak for ’C activity in the HOOum thick detector used in the present experiment. Channel Spectroscopic Factor Partial Width at Full Energy Peak Wigner SP units Resonance (MeV) Efficiency (AOOu) 5Li + 0 0.07089 0.531 0.88 'Be(gs) + p 0.0010u 0.027 0.27 'Be(2+) + p 0.00363 0.047 0.u6 TOTAL: 0.606 TOGETHER: 0.79 69 103 101 COUNTS 10° I I I I I I 5 10 15 E1) (MeV) O FFH'II—I Ilnrrll i Illl :3 Figure 14.8 Components of the calculated beta delayed p_roton spectrum. The three solid curves represent alpha decay to the 3/2 ground state of ’Li which then emits a proton in flight, proton decay to the 2- ex- cited state of 'Be near 3 MeV and proton decay to the 0 ground state of 'Be, listed in increasing order in average energy. Diamonds repre— sent the sum of the three contributions, and the histogram is the data of ESTERL72 . l 03 .0... ‘. a, 3 O ...“. ’3... O F .0 ‘ ’. 102 o 0.. '. ‘ (I) ‘ ’0 {-0 °. . I g ’ .... I U o“ O 0.00 O 100 ' o I I I I I I I I II o I L I I ”I (3 5 10 15> IE:p (MeV) Figure 4.9 The contribution from the long high-energy tail of the 1/2' level at 2.9 is included. The sum of all four components is shown by the large diamonds. The histogram is the same as in Figure 4.8.“ 70 In order to use such model to extract accurate B(GT) from delayed multi-body breakup, better coincidence data will be needed. In the same spirit that three-moving-source fits are performed to inclusive date where double-differential cross-sections are available, two par- ticle angular correlations will significantly improve the reliability with which spectroscopic factors and B(GT) can be determined from beta delayed multi-particle data. A further discussion of such pos- sibilities can be found in the final Recommendations section. IV.3 Conclusion on ’C study: A technique for the study of the beta decay branching ratios of beta delayed particle decay precursors has been developed. The tech- nique takes takes advantage of the recoil energy of projectile fragments to implant them within the volume of an active silicon detec- tor where the sum energy of the three particles is recorded. This technique, coupled with the intermediate energy heavy ion beams avail- able at NSCL and the mass separation of projectile fragments with the RPMS, provided a situation in which the decay of the exotic nucleus ’C to the ground state and first two negative parity states of ’8 could for the first time be observed. Branching ratios and B(GT) have been extracted for these states. Higher lying states are open to complex decay channels which some- times produce high energy protons. In particular, a state at 12.1 MeV in ’B is strongly populated by ’C decay (log(ft)=3.6). While the par- ticle breakup of this level produces a distinct peak at 12 MeV in the data, the branching ratio is not immediately available from a straightforward analysis of the spectrum. A strong proton decay branch of this state to the ground state of 'Be would result in an efficiency 71 of only ~25; for containing the total energy of the 12 MeV (cm) proton since its range is much greater than the thickness of the detector. On the other hand, the delayed proton data of Esterl show that the decay predominantly proceeds through channels which result mostly in the emission of low energy protons. A model has been developed to understand the particle decay of broad levels populated in beta decay. The model follows sequential particle decay where the final state involves three separate nuclear particles - in this case two alpha particles and a proton. A simple Breit-Wigner representation of the internal density of states in the primary (’8) and intermediate ('Be and ’Li) resonances coupled to an external density of states function generated by the Coulomb wave- functions. A generalized program has been written to carry out such calculations, providing various density of states functions and the resulting inclusive and coincident laboratory energy spectra of each of the resulting particles. Thus experimental data of both inclusive and kinematically more complete nature can be fit with a few simple parameters; beta decay and particle decay matrix elements and resonance energies. The beta delayed proton data of Esterl were fit with this model. The data show two distinct peaks which correspond to two particle decay channels; ‘Be(gs)+p and 'Be(2+)+p. The area of these structures are almost exclusively the result of decays through these channels, and thus represent a nearly independent measure of these branches. The third channel to be considered is ’Li+a. This channel appears to dominate the particle decay spectrum of the 12 MeV state. The spectro- scopic factors for the three particle decay channels were adjusted to 72 match this spectrum. As each channel dominated a given region of the proton. energy spectrum, a straightforward fitting could be carried out in this particular case without a least-squares fitting procedure. Since the overall width of the state is experimentally known to lie be- tween 1100 and 600 keV, only two parameters were varied; the ratios of the spectroscopic factors of the two 'Be+p channels to that of the ’Li channel. The three spectroscopic factors were always renormalized to maintain a constant width in the peak. Table 11.1 gives the final spectroscopic factors obtained in the fit to the data of Esterl. The certainty of the two 'Be values is a factor of two in either direction, while that of the ’Li is closer to 30$ - the certainty of the width of the 12.1 MeV peak in the present NSCL data. Uncertainty due to inadequacies of the model itself are difficult to assess. More detailed examination of empirical phase shift data in 0-0 and p-a scattering may provide a slightly better rep- resentation of these resonant states. The calculated laboratory proton spectrum is then used to produce a predicted total energy spectrum as would be recorded in a AOOum thick silicon detector as used in the present experiment. This calculation shows that 25% of the events which are in the 12.1 MeV peak in ’B produce a sum energy well below this value because the protons physi- cally escape the silicon detector before depositing all of their energy. Efficiencies for the proton from each of the three channels considered are also given in Table 4.1. The calculated efficiency of 75%, obtained from the model using' the parameters fit the data of Esterl can then be used to extract a B(GT) for the 12.1 MeV state. A value of 1.7 is obtained for this 73 matrix element (log(ft)z3.6). This represents 13% of the total.(rf sum rule strength for ’C decay. See Figure 14.10. The value compares very favorably to the predicted value of 1.9 to the third 3/2" level in 98 using the 6-16TBME interaction. The three other interactions give predicted values between 1.4 and 1.8. That reference gives other evidence which supports the assignment of the observed 12.1 MeV level to the third 3/2- state in ’B. Thus the agreement of the experimental B(GT) with the theoretical values is reassuring, especially when the theoretical values are relatively interaction-independent. Future studies of the decay of "O to particle unbound levels in "N will involve a mrther technical extension (four particles in the final state) as well as a mrther extension of the model itself. This experiment is outlined in the following chapter. More kinematically complete data (multiple detectors measuring different particles) may help distinguish between competing particle decay channels. Other techniques, such as two a plus neutron detection for the mirror decay of ’Li will also be helpful in separating the various particle decay channels and thus improving the reliability of the fitting. Computational improvements on the program which calculates the spectra will bring the iteration time down to roughly one per second. At this point automated search and fitting routines will be able to produce fits where the channels are not so clearly separate in the in- clusive data as they are for ’C decay. 711 | ‘3/2‘ 16.65 E 9C I Lt: .q . 1 x 5 Q .— _é_ __ Figure 9.10 Experimental and Theoretical beta decay B(GT) distributions and calculated particle decay threshold energy for ’C. predicted 3/2' is assigned to the strongly populated state observed at Good agreement is found between the predicted and new experimental B(GT) to the three lowest negative parity states, as well as for the third 3/2'. 12.1 MeV as discussed in the text. 913 75 / 2.9 8362+ + P —0.19 2a + p The third SECTION V V. Other Cases of Beta Delayed Charged Particle Emission: V.I "N: While the method of implanting beta delayed charged particle emit- ting precursors inside silicon was developed for the study of ’C decay, it was even better suited to the study of the delayed alpha emission from the beta decay of "N. While ’0 produced high energy protons which could have a range up to 1 m in Si, ”N activity produces beta delayed 0 particles only up to a few MeV which have a ten times shorter range. Thus for this study the detectors could be much thinner (50pm vs. liOOum) and thus the effect of the beta energy loss on the delayed charged particle spectrum could be minimized. Figure 5.1 shows a partial decay scheme for ”N. A more complete summary of decay to bound and narrow (I' as I‘Y) states can be found in OLNESSB2. Two levels are of special interest in this experiment. The 13 level (Ex = 7.619) and 1; level (Ex = 8.039) are 1.392 and 1.812 MeV respectively above a decay threshold in "O at 6.227 MeV. They are predicted by OLNESSBZ to be strongly populated by Gamow-Teller decay of "N. In a future experiment ZHAO88 plan to determine the parity non- conserving a decay of a 0" level (Ex : 6.880) at 0.653 MeV above alpha decay threshold. Since the alpha spectrum is dominated by the allowed alpha decays, it was critical to understand the allowed beta delayed alpha spectrum befbre finalizing the design of the PNC experiment. In the present experiment "N activity was implanted in a series of 501m thick silicon surface barrier detectors. The range distribu- tion covered about 300nm, and thus to contain most of the 77 1- 13.90 12.19 “0+2n 8.04 1"O-i-n 6.23 “C-i-a 0* 0.00 Figure 5.1 Partial Decay Scheme for "N beta delayed alpha emission. Information take from AJZENBERG-SELOVE87, WAPSTRA86, OLNESS82 and ZHA089. 78 1 inunlL J Figure 5.2 Histogram of energy recorded in a 501m thick silicon detec- tor during the 'Beam Off' periods following "N implantation. The assignment of each peak is indicated. ZHAO89 suggest that the struc- ture labeled A is actually a group of known but unresolved 1" levels. The structures labeled B and C are likely to be 1' or 2 levels in "O which have not yet been identified. 79 activity the silicon detectors were inclined at an angle of 115° with respect to the axis of incidence. The entire detector assembly includ- ing the position sensitive proportional counter were kept in a helium atmosphere to minimize the number of "N which could stop in the air between detectors and introduce uncertainties in the efficiency. The beam was run in Fixed Cycle mode (Section VII.3.3.2). The beam was turned on for roughly one mean life of ”N, and then off for the same period (during which time the decays were recorded). Figure 5.2 shows a decay energy spectrum recorded in one of the silicon detectors. The large low-energy tail labeled ALTERNATE beta is produced by background and by beta decay to bound states. The beta from these decays occasionally passes through a large amount of silicon and therefore produces a large signal in the detector. Two peaks labeled 1; and 1; are clearly visible, confirming the predicted strong branches to these two levels. The results of this experiment and in- terpretation are summarized in ZHA089. V.2 1’0 Decay: The case of ”O decay is potentially even more complicated than that of ’C. As shown in Figure 5.3, "O is a candidate for beta delayed four body break-up. The half life and ground state branching ratios are also important for studying mirror asynmetry in beta decay, and for the potential of observing second class currents (WILKINSON71b, TOWNER72, WILKINSON7uc, ASAHI89. A very preliminary decay energy spectrum for "0 recorded in a ‘100m thick silicon detector is shown in Figure 5.9. Here again the lowest energy peak is due to background and beta decay to bound states. 80 3/2‘ 17.77 130 U”— ...-a (.0 then 2 expt. U 7 / \\ //// 18.3 14.3 12.39 11.86 11.28 11. ? 9.60 9.50 9.22 6.38 1.94 8380+ '1' 6Lil/2_ 8862+ '1' 5183/2... 9B1/2_ + a 8830+ '1' bids/2.... 1/2+ + a 1200+ + P 301 + p 1fiCz+ + P 2 1 C0++P Figure 5.3 Decay scheme for "0 delayed charged particle emission. Compare to ’C in Figure 11.10 81 3 10 ITIIIIIIIITIIIIIIIIIIIIII 102 E a .3 1 3 G I - 5 - 8 1" 101 t— —5 E " ‘ r 100 5— :J I I pl I I I L I I L LIJ I I I I I I I I I 0 50 100 150 200 250 Energy Figure 5.9 Preliminary decay energy spectrum for "0 ions implanted in a .400m thick silicon detector. The lowest peak is most likely due to beta decay to bound states where only the beta particle deposits ionization in the silicon. 82 ...-A CD (.0 m ~: 3 E M I 2 Z. . .3 10 g :1 : " 8 101 I 1 1‘ o e as HAIL”-.- 11.‘ 0 20 40 60 Time (ms) Figure 5.5 Beam off time spectra for the three peak structures in Figure 5.9 and fer the highest energy part. Once a half life is estab- lished these curves can be used to determine the fraction of the area under each peak which is due to real "0 decay. 83 The data was taken in the Fixed Cycle mode with the beam on time roughly 1 1/2 mean lives and the beam off roughly 2 mean lives. In Figure 5.5, time spectra for each of the three peak structures and fkn~ the high energy continuum are shown. Once a half-life is established from a thorough examination of all data from all runs, the exponential part of each curve can be extracted and thus the contribution from real "O decay in each of the energy bins can be measured. A summary of the shell model predictions using the16-16TBME interaction, and measure- ments of ESTERL7O is shown in Figure 5.3. 814 SECTION VI VI. The Michigan State Reaction Product Mass Separator (RPMS): V1.1 E x B Recoil Isotppe Separation (RIS): A Recoil Mass Separator uses electric and/or magnetic fields to separate isotopes which are moving at a velocity imparted by the nuclear reaction. Since recoils from nuclear reactions are almost al- ways stripped of at least some of their electrons, they carry a charge and thus may be manipulated by the fields. Reactions which produce nuclei far from stability often result in a spread in velocities; the separating device must therefore disperse in (for example) m/q while remaining achromatic in velocity. This can be done with the use of two forces whose effects have different dependence on the velocity of the ion. In electromagnetic ion beam manipulation, the strength of the electric and magnetic fields are loosely analogous to the refractive index of the glass elements of an optical lens. There are differences though. The deflection of a beam by an electromagnetic field can depend both on the energy of a particle, and its mass to charge ratio (m/q). Below is a simple derivation of a device which can disperse particles in m/q, while cancelling the velocity dependent parts of the electric and magnetic forces. Here it will be assumed that a parallel beam of particles is deflected by thin electric and magnetic devices superimposed in space. The fields are perpendicular so that the forces are parallel. Each particle is described by a mass m, charge q and forward momentum pp. The forces will be in the transverse direction resulting in transverse momentum pt. All calculations are carried out in the non-relativistic 86 limit and for small deflections, such that p p: The forces are, Magnetic Terms: Electric Terms: dp dp tB tE FB = dt = qu, FE = dt = qE. [6.1] L If the duration of the impulse At : V’ where L is the length of the device, then the resulting impulse Apt is given by, =dp dp t3A 1 _ th This impulse deflects the particle by an angle 0, (small angle approximation) 6 :——=-—g—BL o [6.3] Thus magnetic devices bend particles by an angle proportional to q/p, and electric by q/T, where T is the kinetic energy. Now when the two fields are superimposed, the angle of deflection is just 6 = ‘63' BLB + 5%; ELE . [6.4] It is clear that with static fields there can be no electromagnetic separation of particles of identical m/q, so we define x : m/q. Now consider a group of particles with a small spread in velocities and mass to charge ratios. Define 6v : (v-v.)/v0 and 8x : (x-x.)/x,. [6.5] If Equation 6.4 is expanded to first order in 5v and 6x, then 1 6: xovo’ [ v.BLB (1-5x-5V) + ELE (1-5x-25V) ]. [6.6] 87 5 Case 1: If EL : -v.BL then 005x)» 0 and only 0(6V) remains . This E B device distinguishes particles of different velocities, independent to first order of their m/q. It was first described by Wien and is thus usually called a Wien filter. All particles with v : v, pass through undisturbed, and 0 ~ 8V. Case 2: If the magnetic field is doubled so that 2EL E B’ 0(5v) - 0 and O(-6x) remains. This provides separation in x, independ- ent of velocity to first order. 0 ~ (1-5x)/2 so that all particles of the same x are deflected by the same amount. For an isotope separator to separate particles in such a way that once they come to rest they are physically separate, it needs to dis- perse them in position, not just angle. This can be accomplished with the addition of’lenses which establish a point-to-parallel condition at the beginning of the device, and the reverse - parallel-to-point - at the end. With theta zero following the first lens, the final deflected position y at the end is just the sum of the products of the various an- gular dispersions and their distances from the final focus; y : X Ayi ~ X 61 di . [6-7] 1 i V1.2 Design and Construction: In the simple model of Section V1.1, a device which separates par- ticles according to m/q - independent of velocity - can be built if the dispersion Di : 3V : d1 '3‘,— [6.8] 5. 0(8) indicates terms of order 6. 88 due to the magnetic field is twice that of the electric field. When the RPMS was constructed, a velocity filter (E : vo B) was available on loan from the Lawrence Berkeley Labs. It was used as a meson separator at the BEVALAC. The device consists of a set of electrostatic plates oriented one above the other, providing a vertical electric field. Coils are included to generate a horizontal magnetic field in the same volume. Since the plates are 5m long and are separated by only 10cm, the device is best utilized when the electric and magnetic forces al- most cancel, and the particles pass straight between the plates without deflection. This condition is satisfied in Case 1 of Section V1.1, for a velocity filter. The angle of deflection is proportional to 8v, and is zero for v : v. where E : voB. To make a mass separator at NSCL, a magnetic dipole was added, following the Wien filter, which provides a term proportional to (1 -5v-8x). This cancels the velocity dispersion, and provides dispersion in mass-to-charge ratio. A magnetic quadrupole doublet near the front of the RPMS following the target provides a parallel beam for the Wien filter and dipole, and a second doublet refocuses the parallel envelopes of each m/q to different vertical posi- tions after the dipole. The second quadrupole doublet and detector system are all mounted on a tail platform which pivots up from horizon- tal about the center of the dipole magnet. This allows the experimenter to move the detectors and intercept different m/q isotope groups, while the dipole field and Wien filter fields remain fixed. The RPMS is il- lustrated in Figure 6.1. More can be learned about the RPMS in NOLEN84 and the upcoming PhD thesis of HARKEWICZ. 89 “SUI-02‘ 2|! Figure 6.1 Illustration of the Reaction Product Mass Separator; a) side, and b) top. b) illustrates that changes in scattering angle are facilitated by only small rotations of the RPMS about the center of the Cornell magnet. 90 VI.2.1 PhasegSpace Acceptance: The RPMS is equipped with four selectable apertures to constrain the particles to certain envelopes. The three momentum dimensions are represented by velocity and vertical and horizontal scattering angle (phi, theta). The three spatial dimensions are represented by the ver- tical and horizontal position at the target plane, and position of the target location along the axis of the RPMS. This last dimension is es- sentially fixed for thin solid targets.6 The first device to limit the envelope of particles to be separated is called the 'half-aperture.‘ It is a thick block of metal whose top surface cuts across the center of the optical axis of the front of the RPMS, about 80 cm downstream from the target. This axis is the exten- sion of the beam-line axis, passing through the center of the beam-spot on the target and through the axis of the first quadrupole doublet of the RPMS. The only particles which can pass by this half-aperture are in the upper half plane - only those with a vertical scattering angle greater than 0°. The second device is the solid-angle defining aperture. There are five of these in a vertical ladder, roughly 120 cm from the target. The apertures define the shape of the envelope of the particles entering the 6. This was not the case when a gas target was used (GIL88). A chamber filled with Helium gas was used as a target, and was 10cm in length. In this case, the sixth element in phase space occupied by the reaction products was also extended. This could have a significant effect on the resolution of the RPMS, because the first quadrupole doublet will produce a parallel envelope within the RPMS only if the particles emerge from the focus; a point on the axis which is appropriate to their velocity. 91 first quadrupole doublet. One can choose either a small solid angle, which has limited phase-space and should provide better overall m/q resolution, or a large solid angle which accepts more phase space and thus gives a higher count-rate. Particles near 0° can be eliminated with an annular aperture, which blocks the center while allowing those particles scattered to larger angles to pass. Four apertures have been used to date: 4°x2°, 2°x1°, 2°x1/2°, and 1°x1/2°, (horizontal x vertical). It is important to point out that there is a vertical offset in the axis of the RPMS between the first quadrupole doublet and the Wien fil- ter. The center of the Wien filter is 2 inches above the axis of the doublet, so that it is coaxial with the center of the upper half-plane of the doublet. This offset, coupled with the half-aperture, makes use of the large 8 inch diameter of the first quadrupole doublet in order to reduce the effect of the chromatic aberration in the size of the final image at the end of the RPMS. For a fixed m/q dispersion, the size of the image determines the resolution of the device and thus its ability to separate isotopes with nearby m/q. The third element is a pair of jaws between the Wien filter and Cornell magnet which move in the vertical direction. Since the envelope is parallel here, the vertical position of the particle is correlated not only with the velocity, but with the vertical scattering angle. Thus these jaws can not actually select a particular velocity, but can only impose a triangular envelope in velocity space. Finally, the fourth device consists of a 'pair of horizontal and vertical slits, usually found directly in front of the detectors after 92 the last quadrupole doublet and the window separating the vacuum from atmosphere. The vertical slits are used to define the m/q distribution which passes into the detectors, and is the functional device in the physical separation of the isotopes. They are placed so that the center of each pair of slits corresponds the center of the detector, and also to the image of the isotope of the desired m/q. m/q is then adjusted by the motion of the entire tail of the RPMS. The vertical slits can be widened to allow more values of m/q to enter the detector while tuning up, then narrowed to allow only one isotope or a few isotopes to enter the detector when a decay measurement is underway. V1.2.2 Scatteringggngle: Some experiments demand operation at a non-zero scattering angle - typically 2° to 5°. The scattering angle of the RPMS is defined as the angle between the optical axis of the RPMS and the direction that the beam would travel if it continued through the target. A scattering angle of 0° and a thin target would allow the beam to travel directly down the axis of the RPMS and possibly strike one of the electrostatic plates within the Wien filter. The most trivial way to change the scattering angle is to pivot the entire RPMS about the target. However this would require the 12m length of the RPMS would subtend too much floor space. Instead, the RPMS is held fixed at one point (the center of the Cornell Dipole) about which it is rotated by a few degrees. An inflector magnet is introduced in the beam-line just before the target. The incoming angle of the beam, position of the magnet, and small back and forth motion of the RPMS are 93 chosen so that at a deflection of the beam by the inflector magnet of - 15° allows the beam to strike the target at a scattering angle of 0°, and a deflection of +15° gives a scattering angle of slightly over 30° . A diagram of the RPMS as viewed from above is shown in Figure 6.1b. V1.3 Beginning a Typical Experiment: VI.3.1 Start-up Procedures: The start-up procedure for an experiment with the RPMS is of a tedious and technical nature, and involves details specific to the RPMS in its present configuration. I The RPMS is presently being reconfigured for Phase II operation at NSCL, and thus this information is not given here. The interested reader is directed to the future dissertation of HARKEWICZ, and to the RPMS users handbook to be completed in late 1990. The procedure outlined is designed to obtain some count rate of near- stability isotopes from a 'cold start,’ without ever presenting the silicon detectors or position sensitive proportional counter with a high count-rate of particles that might damage them. Once a variety of particles are reaching the telescope, one should try to identify the particles with the on-line analysis program - tradi- tionally SARA. First, a two-dimensional histogram of AE (vertical) vs. E will give some indication of what particles are reaching the detec- tors. Identification can not be certain from just this histogram alone. Next, a one-dimensional histogram of the function PID (AE, E) (Section VIII.4.1) should be displayed and compared to the plot of AE - E. Small changesin the dipole current will bring different isotopes in and out of view this way, and one could in principal learn something about the 94 particle identification by trial and error. However, a simpler method exists. A position-sensitive gas proportional counter should be used to measure the vertical position of the particles reaching the focal plane. The detector that has been used to date is described in CURTIN85 and ORMAND84. A new detector which can be Operated in vacuum is under con- struction. See MIKOLAS88b. It is useful to monitor a two dimensional histogram of vertical position (Y) vs. PID (X). A map of all particle stable light isotopes which can be made by the fragmentation of a “Ne beam is shown in Figure 6.2 for reference. Tuning of the RPMS continues with a histogram of vertical position vs. energy, gated on only one isotope. If the dipole is set too low, and is not completely cancelling the velocity dispersion of the Wien filter, the particles of higher velocity (and therefore energy) are below those of lower velocity. If the dipole is too high, the opposite case is true. When set correctly, the correlation between velocity an position is completely cancelled. When the RPMS is tuned properly, isotopes should appear as small groups in a histogram of position vs. PID. However, it may not be im- mediately obvious M isotopes are being observed. One must 'move around' in m/q space and in velocity space to reach the desired m/q and optimize the yield. When moving in m/q space, it is helpful to remember the 'holes,‘ those isotopes which are not themselves stable, but the isotopes with one more and one less neutron are stable. Some of these are ’He, 1He, '°Li, °Be, "Be, ’8, “B, and ‘°B. 95 L5 F° c: m/q (vertical) 9° UI 9° c: 135 Figure 6.2. IHo 0C ”B "k3: 1‘3 17 786 , ”N 1 ‘C~1fiq “0‘ 17 :3 “1111-16911 munch: “0:17:21; 1y}JL .14r49F-- ° ‘:B‘§B” 1 1%) “fi' 6 7L1 9 ’00 52 ,9 afi? ‘fi; 1%, O ”a. uc ”0 Ifia 1%; "LL“B 193 is” ' e IqB‘AR: ”Hindu l'BeL’B—“C "13 /“Be ifij/ I-l. I I I I In I I I I I T I I I I—IT "' a D -1 ' n a a I: :1 I .. D‘o~ :3 ~ 1- D‘D—a : r—u-n D-—D-—4}——D-—43-—C3—-- 1- " 43—D—D "' ’ 0”5_—43 0 :1 C’ C: ‘ . r” I: D c1 c1 . .- u D .1 -- 0 D D — . a 1:1 . .. I D 0 a !- \D\D d +- l—ll - u —u-—0 — D a I 1 1 A_‘7l I 1 1 I 1 I 1 I I I_ 10 20 30 in order to keep the groups more evenly spaced. typical resolutions in m/q and in PID. 96 Schematic representation of a histogram of vertical posi- tion as a function of the square root of the Particle Identification function (PID.) The square root of PID is used for the horizontal axis The boxes represent VI.3.2 Tuning Up: While an experiment begins with some initial settings, either ffiwnn a previous run or from a calculation based on calibrations of the various settings of the components of the RPMS, simple scaling laws can be used to move from one set of settings to another. There are four quadrupole field strengths, two magnetic dipole fields and an electric dipole field, and the angle of the tail. Scattering angle, slit posi- tion and target height and angle are adjusted less frequently. Once point-to-parallel-to-point conditions have been established, (somewhat different optical conditions may be used for special cases), the quadrupole magnets should be scaled as a unit, with their fields proportional to the magnetic rigidity, p/q. Thus they are linear in both v and m/q in the non-relativistic limit. The ratio of magnetic to electric fields in the velocity filter is varied only when changing velocities. The relation, E = -v.B [6.9] must be held, so that E/B is linear in velocity, and independent of m/q. In order to maintain zero dispersion in velocity, the field in the dipole must always follow the magnetic field in the velocity filter. It has been simplest experimentally to keep the power supplies for the high voltage field in the Wien filter fixed. They have usually been kept fixed at the highest voltage obtainable without sparking in order to maximize the dispersion. When this is the case, the two magnetic dipole fields must always be set in the same ratio, which is determined by the velocity. Changes in m/q are simply changes in the quadrupoles, and in the tail angle. 97 Non-relativistic scaling laws for three parameters - m/q, velocity, and dispersion - are discussed below. They are also summarized in Table 6.1 for quick reference. Here, Be represents the field in the dipole (obtained from Cornell University), Bw and Ew are the magnetic and VI.3.2.1 Scaling in m/q: When the RPMS is functioning to separate isotopes, they are dis- persed in position at the final focal plane with the smaller m/q isotopes deflected mrther up than the heavier ones. Simply moving the tail down will increase the value of m/q which can pass through the slits and enter the detectors. However, when the m/q is increased the isotopes are also more rigid and therefore difficult to focus. Thus the quadrupoles must then be in- creased to maintain the same envelope within the Wien filter and dipole, and bring them back to a focus. Therefore the quadrupoles scale linearly with m/q and the tail angle inversely. VI.3.2.2 Scaling in velocity: From Equation 6.9 it is clear that there are two independent ways to change the center of the velocity band-pass of the RPMS - to vary ER, proportional to velocity, and to vary Bw and Be together inversely to velocity. In order to maintain the same m/q at the same vertical posi- tion, only the second choice will work. Since B0 91’. ~ W , [6.10] Be must decrease if m/q and 9 are to remain constant while velocity decreases. 98 Table 6.1. Scaling Exponents 7 for RPMS Settings. resent the f0ur quadrupole magnets, B the Cornell Digole, E and28 electric and magnetic fields in the fiein filter, ande the tail with respect to the axis of the Wein filter. tail Device(s) Parameter(s) m/g v D Quadrupoles 020, 021, 022, 023 +1 +1 0 Tail Angle etail -1 0 +1 Magnetic Dipoles B“, BC 0 -1 +1 Electric Dipole Ew 0 0 +1 7. For example, to scale the magnetic field in the Wien filter, from its previous value which produced (m/q, v, D)o sgttings to new settings which give (m/q, v, D)ne w’ use Ifle relation: :wnew = [qunew]0 x ("newl'1 Dnew)+ wold m qo x I Dold 99 through Q23 Ihe angIe of As with m/q scaling described in the previous Section, the quad- rupoles must also be varied with the magnetic rigidity and therefore velocity. VI.3.2.3 Scaling in Dispersion: Here the velocity and m/q are to remain fixed, and only the separation between various groups of m/q (and thus tail angle) is to be varied. This means that the quadrupoles are not varied. With the relationship in Equation 6.9 maintained, there is still one degree of freedom. E B , Bc’ are varied together, proportional w ’ w to dispersion D. When Bc is raised, and m/q and velocity are fixed, then the tail angle must also increase. V1.4 Monochromation: In both half-life determinations and beta-delayed charged particle spectroscopy the broadness of the distribution in ranges of the par- ticles in the silicon detectors is often a limiting factor. The suppression of sensitivity to unwanted activity can be accomplished through the use of differential ranges described in Section VII.2.1. In delayed charged particle spectroscopy the detector usually must be much thinner than the spread in ranges of the isotope of interest in order to limit the contamination from the beta particle (Section VII.3.4). Both of these experimental situations can benefit greatly from a reduction in the spread of range distributions. In order to compress the range distribution, the RPMS is first tuned to a momentum focus as described in Section VI.3.1. This gives a spot size for a given isotope of approximately 4 mm at the detector. The dipole current and tail angle are adjusted slowly together so that 100 the mv/q dispersion of the dipole is varied while the average m/q reach- ing the detector remains fixed. This 'detuning' introduces a mis-match between the velocity dispersion of the dipole and that of the Wien fil- ter, and the particles with higher velocity end up above those with lower velocity. The image is now enlarged vertically, and the dipole should be de-tuned until the maximum aperture of the telescope is filled - typically 20 m for a standard 300 um2 silicon surface barrier detec- tor. The vertical vs. energy histogram will now show a clear correlation between position and energy, with a resolution of 4/20 or 201. A graded absorber is then introduced having a thickness which in- creases with vertical position. The gradient of the thickness is chosen to compensate for the larger velocity (and range) of the ions entering the top of the absorber so that the velocity (and range) of those exit- ing are compressed before entering the silicon. This distribution is narrower, and can place those ions which would normally fill 250 um of silicon into one detector only 50um thick. The reduction of the spread in energies of the particles passing through the degrader is referred to as monochromation. This was a particularly useful technique in the case of 'He decay. The range of the 'He fragments collected by the fragmentation of a 22Ne beam by a target just thick enough to stop the beam is roughly 5 mm. The experiment required the thickness of the detectors to be under 50 um, only a few percent of the spread in ranges of the °He's collected! While many detectors of 50 um thickness could be placed in series, the efficiency could approach 501 only by careful monochromation of‘ the ‘He's. 101 Since the compression is accomplished by a deliberate degradation of the m/q resolution, this technique is limited to very light isotopes, such as those of helium through Boron - where the distance in m/q space is large from one isotope to the next of a particular element. Ideally a self-contained system could be added to the RPMS following the m/q selection by slits at the focal plane. This then would prevent the momentum dispersion from degrading the m/q separation. 102 SECTION VII VII. General Experimental Considerations: There are at least three general questions which one must address when choosing experimental conditions at the target.location. The first question is the choice of which target material will be used and what thickness it will be. The second question is which beam particle to use and at what kinetic energy. The third question is at what scat- tering angle will the fragments be collected. However, before these can be answered a brief discussion of the projectile fragmentation reaction is necessary. VII.1. Projectile Fragmentation: The dynamics of nuclear reactions are complex and beyond the scope of this dissertation. However with the goal of the optimization of a production rate of a given isotope in mind, it is helpful to consider a few qualitative features of projectile fragmentation. It is important to note that projectile fragmentation is ascribed to processes which take place at relativistic energies (many hundreds of MeV per IMMJIGOD) and that it is not obvious that they should occur at energies presently available here at NSCL (30 to 50 MeV per nucleon.) Review of ex- perimental data representing projectile fragmentation at relativistic energies are given in GREINER75 and VIYOGI78, and at lower energy by CASKEY88. Theoretical discussions are given in GOLDHABER74, OLIVEIRA79, and MORRISSEY79, and reviews of fragmentation processes by GOLDHABER78, HUFNER85, FRIELANDER87 and LYNCH87. Reviews of reactions which lead to the production of nuclei far from stability can be found in WESTFALL79, WOODSBS and ROECKL88. The process of projectile fragmentation shows several 104 characteristics which are distinct from other processes. The fragments of the beam nucleus continue to move forward with nearly the beam velocity. There is usually a spread in velocity and direction compared to the incident beam. This can be almost completely described by a Gaussian spread in momentum in all directions with the centroid at the beam momentum-per-nucleon. Often a slight shift to lower momentum in the lab is observed, and can be explained as a 'frictional' type of energy loss as bonds in both nuclei are broken. The width of the distribution in momentum increases with the num- ber of nucleons which are removed. In the model of GOLDHABER74, this spreading can be thought of as the sampling of the Fermi momentum of the individual nucleons. In the frame of the projectile with nucleon number Ap, the momentum of all of the nucleons sum to zero. If sud- denly n nucleons are removed so that a fragment with nucleon number Af= Ap-n is produced, that fragment must have a recoil momentum -p, where, n p = I‘Dfemim. 17.1] 1: in order to maintain a total momentum of zero in this frame. In this approximation the momentum of each nucleon in a given direction is a Gaussian distribution and uncorrelated to the motion of all other nucleons, except that the total momentum of all nucleons in the nucleus sums to zero. Thus the distribution P of momentum in a given direction (longitudinal used here) is P(p ) = e . [7.2] 105 They show that for a simple situation where nucleons are suddenly removed at random, 0 can be obtained from, A (A - A) 02 : 03 pr- 1f , [7.3] P where 2 (P > P 08 = '39— : £2122 [7.4] A typical value for 0",,ch4 which reproduces the fragmentation data at relativistic energies is 80 MeV. At lower energy the value which provides the best fit to the data is larger. This is most likely due to a breakdown of the model. In slower collisions other processes can occur which can absorb a significant amount of kinetic energy. CASKEY88 indicate that while values of 0.20“ of 50 to 100 MeV reproduce the high-energy part of the spectrum, a long low energy tail seems to be generated by other processes. Rough examination of the published spectra indicates that a value for 0.2cu of 100 MeV will reproduce the FWHM of the energy distribution of fragments where three nucleons are removed. This value will be used in all further calculations as a simple way of simulating a representative momentum distribution only. The cross-section for a particular isotope is difficult to calcu- late, but as pointed out by WESTFALL79 and LYNCH87 it seems to be fairly independent of the particle which induces the fragmentation (the target nucleus in the case of projectile fragmentation.) Immediately after the fragmentation event the fragment cools and relaxes to 106 (nearly) some ground state configuration predominantly by the evapora- tion of nucleons and light nuclei (e.g. alpha particles). Far from stability, the separation energy of the type of nucleon in excess is usually much lower than that of the other type, and thus the evapora- tion tends to bring the fragment closer to the stability line. The measured yields of a given isotope are therefore quite removed from the primary yields of the fragmentation process. It is important to stress that fragmentation is not necessarily the ideal name for the dominant processes which produces nuclei far from beta stability which continue close to the frame of the beam. Deeply in-elastic and particle transfer processes may significantly contribute to yields from reactions at 30 to 50 MeV per nucleon and no data exists at present which could clearly distinguish contributions from these processes. (See section VII.2.1) VII.2 Target Parameters: VII.2.1 Choice of Target Thickness: There are two opposing factors which must be balanced when choos- ing a target thickness. First, the production rate of a given isotope in the target monotonically increases with target thickness until the beam has lost so much energy that it is below the threshold for the production of that isotope. Second, the energy loss of the fragments tends to spread them out in energy, since the lower energy fragments loose energy faster than the higher energy fragments. In order to bet- ter understand the yield of fragments from a thick target, a simulation routine was written which models the production and energy loss of 107 fragments in a given target. Simple assumptions are made wherever pos- sible to reproduce the important behaviors of the various phenomena such as the spread in momentum from the reaction, energy loss in the reaction (friction), beam energy dependence of the cross-section, and energy loss of both the projectile and the fragment in a thick target. In thick targets, the projectiles have a different velocity at each location in the target. When a fragment is produced, it continues to loose energy at a new rate until either it comes to rest in the tar- get or leaves the target. The calculation provides for a loss of total kinetic energy of the projectile at the beginning of the fragmentation before nucleons are removed or momentum is transferred as a simple way of including losses to friction. In practice this is set to zero, as this seems to give general agreement with the data of CASKEY88. The energy dependence of the cross-section is unknown. At relativistic energies it seems to be constant and fairly independent of entrance channel, however incidental information at lower energy indi- cates that the cross-section for nuclei far from stability significantly increases between 25 and 40 MeV per nucleon. In order to simulate some sort of dependence of the production rate on beam energy a simple power—law parameterization of the produc- tion rate as a function of beam energy above some threshold, [E-E E > E 9 thresh] thresh E S E:thresh [7.5] o. 2: to c> rn—+ 108 where q is some exponent chosen to best represent whatever data might be available. q = 0 implies a rate independent of energy above threshold, while q = 1 implies a rate proportional to energy above threshold. C is a normalization constant given by, beam 0 = ggég) dE [7.6] E:thresh In this chapter E always represents the kinetic energy per nucleon. If the projectile remains in the target at all times when its energy is between E and E beam thresh’ the probability is 100% that the simulation will deliver a fragment. The production rate N can then be scaled to a known cross-section at one or more energies when these become avail- able. For each event, a location in the target is chosen at random in order to reproduce the behavior as given in Equation 7.5. The beam momentum per nucleon is calculated at that point in the target through the simple range-energy relations, A R(E) = CR —§' Ep, [7.7] 2p [R(Ebeam)-X] z2 1/p E(X) = . A C p , [7.8] p R 109 filé I %|% 1 i a .X(nnn) fig 33(flflm) Target RPMS Silicon Detector Figure 7.1 Illustration of various spectra generated by the Monte Carlo simulation discussed in the text. Rays represent the energy of frag- ments in the target and detector as a function of depth of penetration. Bold line represents the primary beam. dN/dx represents the rate of production per unit thickness in the target, dN/dE is the rate of emis- sion per unit energy from the back of the target. R is implantation range in the silicon detector, and dN/dR is the rate of implantation per unit thickness in the silicon. 110 I 1 W V r 1 V T V I I V Y ' l I Y r r Figure 7.2 Simple illustration of various rates considered in the+text. a) Gfussian distribution (bold) and two weighting functions E and E ' . b) Same Gaussian distribution and weighted distributions. 111 where CR and p are chosen to reproduce tabulated or measured ranges over the region of interest. X is the depth in the target as measured from the front and E is always the kinetic energy per nucleon. Once a location has been chosen, the fragment (Af,zf) is intro- duced with a fOrward momentum as described by Equation 7.2, centered at the beam momentum per nucleon. The energy upon exiting the target, or location in the target at which the fragment stops are then determined from Equations 7.6 and 7.7. A schematic representation of this process distributions of the beam and resulting fragments is shown in Figure 7.1 for a typical projectile fragmentation-like reaction near 35 MeV per nucleon. The lighter fragments with a smaller nuclear charge (2) than that of the beam have a smaller energy loss per unit thickness (dE/dX), and there- fore loose energy at a slower rate than does the beam. For the thinnest target the fragments leave the target with a velocity only slightly lower than the beam. However, as the thickness increases the beam loses energy more rapidly than the lower 2 fragments, and soon is decelerated far below the velocity of the fragments. The RPMS has typically been run with a target Just thick enough to stop the beam. As illustrated in Figure 7.1, the RPMS has an acceptance window in energy of roughly 201. The count rate at the detector following the RPMS is sensitive only to that part of the yield which is within this window. The size of the window is fixed in relative energy, not ab- solute. Therefore the yield per percent energy is more important than yield per MeV. The difference is illustrated in Figure 7.2. Here a simple Gaussian is used to represent the energy spectrum of fragments. 112 When the Gaussian is multiplied by the straight line shown in 7.2a, the resulting function E dN/dE [ = dN/d(ln(E)) ] reaches a maximum at a slightly higher energy that does dN/dE itself. In order to maximize the count rate at the focal plane a device with a fixed relative energy band-pass such as the RPMS would be tuned above the peak yield in the energy (dN/dE) spectrum. As discussed in later sections, beta-delayed particle spectroscopy often requires the implantation of the fragments in a thin silicon detector. Here thin means much narrower than the range distribution of the fragments at the focal plane after transmission by the RPMS. In this situation the count rate per unit range should be optimized. Using, d" - 1’! 25 dB ‘ (as) / [as] ' [7.9] and Equation 7.7, fl - 1-p 2! dR - E dE' [7.10] A value of p near 1.7 is typical, and thus E.()'7 and E'O'7dN/dE are also included in Figures 7.2a and 7.2b respectively. Once again the distribution differs considerably from the original Gaussian, only this time the RPMS should be tuned LOWER than the location of the highest count rate in order to maximize the rate of implantation in a thin detector. The singularity at very low energy disappears if instead of plotting a differential quantity one integrates over a detector of“ finite thickness AR. In Figure 7.3 a series of target thicknesses have been used to calculate the various count rates for two reactions. The outgoing energy of beam particles which do not undergo a reaction is illustrated 113 for comparison. In the left group neutron-rich fragments leave the target with half of the energy per nucleon of the original beam when the target is sufficiently thick to stop the beam. In the right group neutron deficient fragments slow down quicker than the primary beam and fail to emerge from a target which is thick enough to stop the beam. In Figure 7J4 the results of these simulations are summarized. For each reaction, location of the peak of each of the three distributions are given in the lower graphs while the heights of the peaks are shown in the upper graphs. It is clear that the count rate per MeV, and per percent energy are optimized with targets which are slightly less than half as thick as the range of the beam, while the count rate per unit final range implanted in silicon is fairly flat and shows a gradual maximum at a target thickness near the range of the beam. It is also clear that targets Just thick enough to stop the beam produce a rate which are within a factor of two of the optimal rate. While a certain count rate may be optimal with a thinner target, it is often helpful to stop the beam in the target for other reasons. If it passes beyond this point, it can activate other components of the RPMS (make them radioactive) and it can scatter from the plates of the Wien filter and other surfaces within the RPMS, so that a small frac- tion of scattered beam and induced secondary particles reach the detectors without proper trajectories. However for some reactions such as “O . "O, ”C ¢ ’C, the fragments have a shorter range than the primary beam and therefore both must be allowed to pass through the target. The beam which passes through the target may still be 114 EEEEDJDJDDE] out. .0 Figure 7.3 Series of simulations of distributions of fragments as a for increasingly thick targets. Left and right groups correspond to the two reactions shown in Figure 7.“ (next page). In the center column, the wide rectangle of variable height represents the target thickness, while the two narrow rectangles behind the target represent the (fixed) range of the beam with respect to the target thickness. When the target rectangle Just covers the narrow rectangle, the target is Just thick enough to stop the beam. The diamonds represent the remaining energy of the beam after passing through target. Target are multiples of 0.0025 inch thick Ta. 115 3 i- ‘E 2' ‘3 L J L > a a a i r 1 5' O I I . 0- r 1 r zzNe eloN 160 #130 (E/A=35neV) (E/A=40MeV) A gmi 'I""l""l"": : a 30; ‘3 ‘1 01 'r 4 - J j . mm 3 - E :E _: j 166mg/cm' . 3 oPu-Jsuetyaetcua ‘ “-rh—H- 9- 0 0.2 0.4 0.0 0.8 Target Thickness (mm) Figure 7.“ Peak heights and peak locations for those distributions in Figure 7.3 (previous page) as a function of target thickness. As stated in the text, E is kinetic energy per nucleon. 116 prevented from entering the RPMS by moving away from a scattering angle of 0°. VII.2.2 Scattering Angle: While operation at a scattering angle (Section 11.1.2) of 0° is usually desirable, (the fragmentation yield is most likely a maximum here), it is sometimes necessary to use a thinner target. One can then take advantage of the angular spread of fragments from the reaction to help separate the primary beam from the fragments of interest. While this is very dependent on initial and final energies and the number of protons and neutrons removed, it is safe to say that the distribution at 35 MeV per nucleon is fairly flat over many degrees, and for more violent collisions and heavier target nuclei may extend past 5° before a significant drop in cross-section is observed. If the scattering angle of the RPMS is moved away from O° the beam can be prevented from entering the RPMS while the rate of fragments entering may remain nearly constant. The angular distribution of beam particles leaving the target is a function of two things, the divergence of the beam striking the target and the scattering of the beam within the target. The divergence of the beam is physically limited by the ratio of the 10 cm diameter aper- ture of the last quadrapole of the beam-line before the target to the roughly 2.5m distance between it and the target. Thus the half-angle of the incident cone can be no more than 1°. The scattering of the beam within the target is a statistical process. Multiple small-angle (large impact parameter) coulomb scat- tering with the nuclei in the target can scatter a small fraction of 117 the nuclei many degrees. Since multiple coulomb scattering is statis- tical in nature, the RMS spread should behave like the square-root of the target thickness. On the other hand, the shape of the distribution due to occasional large-angle scattering from single small-impact- parameter collisions is independent of the target thickness, while it's intensity varies linearly with thickness. While this process is dif- ficult to model explicitly because of the statistical nature of the process and uncertainties in the exact screening effects of the atomic electrons upon the nuclear coulomb potential, it can be estimated by, 111.1 MeV -———- 1 arms - B pea zinc .IL/LR [1 + 9 log1O(L/LR)], [7.11] where zinc is the charge of the incident particle, p is its forward momentum and L/LR is the thickness of the foil in radiation lengths. Equation 7.11 was taken from the Partice Properties Data Booklet, AGUILAR-BENITEZBS. For Be, Si and Ta the radiation length is 35.3, 9.36 and 0.111 cm respectively, others can be found in various compila- tions. The equations, pa2 = A J 22 + 1863 E , [7.12] and c2 8 : EL [7.13] tot are helpful. Here Etot is the total energy of the particle Etch : A (931.5 + E), [7.1“] and as always E is the kinetic energy per nucleon. For a typical target of thickness 0".015 used to stop a “Ne beam at 35 MeV per nucleon, the multiple scattering in the target 118 redistributes "N fragments produced in the front of the target by an EMS angle of 0.035 rad, or about 2°. For reference, the grazing angle is about 7°. In a recent experiment where “O beam nuclei with a kinetic energy of '40 MeV per nucleon were fragmented into "0, thin targets were needed to allow the “0 fragments to pass through the target. Operation at a scattering angle of 2° was possible with a Beryllium target (2:11, B =O.6° lab), but the rate of scattered “O was too grazing high for scattering angles below 5.5° when a tantalum target (2:73, egrazing:6°u° lab) was tested. The yield of exotic isotopes from the proJectile fragmentation reaction fills a cone with a half angle of at least 3° for E/A = 35 MeV “Me on Ta even before scattering through a thick target. Since most of the primary beam leaves the target within a much smaller cone the goal of increasing the scattering angle of the RPMS is to avoid the high rate of beam particles with traJectories near 0°. The apertures of the RPMS are typically larger in horizontal angle than vertical. Since the scattered beam, and to a lesser extent the fragments drop in yield at larger angles, more solid angle can be obtained if the scat- tering angle is chosen in the vertical direction, where the width of the aperture is narrower. A small permanent magnet dipole has been built, to be placed Just before the target of the RPMS. The effect is a vertical downward deflection of the beam, and since the mgnet is placed very close to the target, the deflection in angle involves little physical displace- ment of the beam-spot on the target. Since the acceptance of the RPMS is typically a factor of four smaller in the vertical direction than 119 the horizontal and extends from O° only upward, a 1° downward deflec- tion is sufficient to keep the primary beam out of the Wien filter, while maintaining most of the aperture of the RPMS illuminated by the roughly 3° cone of the reaction. A photograph of the completed as- sembly can be found in MIKOLAS88b. The magnet provides a nearly flat field of 6 k6 over a volume 2.0 inch tall, by 0.6 inch wide, by 2.1 inch long. This would result in a 1° deflection of a fUIly stripped beam of “Ne at E/A = 35 MeV. This can be calculated from the convenient formula qu = 58.7 ./ 5:2 + 2Em. c2 , [7.15] when q is the charge state of the ion, B is the magnetic field in Tesla, p is the radius of curvature of the orbit in meters, E is the kinetic energy per nucleon in MeV, and me2 is the equivalent energy of the rest-mass in MeV. The constant 299.9 is actually the speed of light c in m/s times 10'6. V11.2.3 Target Material: The choice of the target material itself is important for the production of nuclei far from stability. One trivial parameter is the melting point. Tin and Copper targets were eliminated as choices be- cause of local melting from the 10-30 w of power that the beam deposited as it stopped in the target. This power can be quickly ob- tained as the product of the beam current (as measured by a charge integrator on a faraday cup or target) and the effective beam voltage (the DC voltage that would be necessary to accelerate the particles to their present energy). The equivalent voltage of a beam is Just E/A * m/q, 35 * 22/7 ~ 110 MV for the beam mentioned above. 120 From a reaction standpoint the mass of the nucleus and 'neutron- richness' also seem to play a role. A heavy nucleus such as Ni or Ta seems to be more effective in removing many nucleons from the beam than a light nucleus such as Be. This effect seems to be strong enough to offset the larger number of Be nuclei that can be put into a target of a given thickness with respect to the energy loss of the beam. (i.e. Be has more nuclei for a given number of electrons and therefore for a given energy loss). A large N/Z ratio of the target seems also to be important for the production of the most neutron-rich fragments. There is no clear rule. Using a beam of 22Ne at E/A : 60 MeV, DUFOUB87 report that "’B was produced by a tantalum (2:73, N:108) target at a much higher rate than with a Beryllium (2:4, N:5) target. However, ‘°C and "C were produced at a higher rate with the Beryllium target. It is important to note that while both "B and "C can be produced by simply removing nucleons from the “Ne beam, "C can be produced only by accepting a neutron from the target. Since proJectile fragmentation targets are extremely thick compared to traditional targets, it is pos- sible that a fragment may undergo a second nuclear reaction in the target before leaving. Thus the 22We may loose three nucleons in the first reaction and become "N, then pass a second nucleus and exchange a pion somewhere else in the target, becoming "C. It is not clear what conclusions could be drawn from the results Just mentioned, except that the choice of target is difficult, and should be made during the experiment! 111.3 Half Life Determinations: Half-lives of many light neutron-rich isotopes have been measured with the RPMS at NSCL (SAMUELBB, CURTIN86). We have used the same 121 technique as WESTFALLBO to measure the half-life of proJectile frag- ments. Once an isotope of interest is identified in a silicon detector telescope, the beam is shut off and the gain of the silicon detector preamplifiers is raised. That proJectile fragments have such a long range - much longer that the recoils from target fragmentation or fu- sion - is an important element of this technique. The detectors can be quite thick, sometimes many mm of silicon. This is important since the beta particle is approximately minimum-ionizing, and loses only 0.5 MeV per mm in silicon. Now that an atom has been identified and is known to lie within a given silicon detector, and the time of the arrival is recorded, it can be directly linked to any 'decay' event corresponding to a beta decay within that detector. Four points are now discussed that must be considered both when designing a particular half-life experiment, and when analyzing the data afterward. First is the choice of detector thicknesses. Next are sources of background events. Third is the functional form of decay curves fitted to the data under different conditions, and fourth is techniques for fitting these curves to the data. VII.3.1 Choice of Detector Thicknesses: There are a few considerations involved in the choice of detector thickness. The sensitivity of a detector to minimum-ionizing beta ac- tivity is proportional to the distance that a beta travels in the silicon. Therefore thicker detectors give bigger signals. However, thicker detectors will tend to stop other isotopes which might con- taminate the decay curve. Thin detectors, on the other hand, have the 122 advantage that they allow separation of activity beyond m/q, on the basis of range (see below). There is no combination of electric and magnetic fields that will. separate two isotopes of the same m/q. However, interaction with mat- ter will allow such separation. For example, if the RPMS is tuned to m/q = 3, the isotopes ’H, ‘He, ’Li, ‘zBe, ‘58 and "C will all pass through with approximately the same distribution in velocity. Using Equation 7.7, it can be shown that the range of fragments vary roughly as (A/22)-(E/A)1'7, or (A/22)-v3'u. Thus for fixed m/q and a velocity window of 101, two species with a difference in 2 greater than about 301 will not overlap in range. In other words, one isotope will com- pletely stop in one volume of silicon, while the other will stop completely in a distinctly separate volume. This method has separated "B and "N which are only 11 different in m/q and otherwise difficult to completely separate with the RPMS. It would also help reduce the rate of one of the pair of ”Be and ”B with respect to the other. These have nearly identical half-lives and identical m/q. The measurement of the half-life of "N serves as a good example. The large flux of 'Li, “Be and ”B (m/q : 2.67, 2.75, 2.8 respectively) which "leaked through" the RPMS when measuring the half- life of "N (m/q : 2.71) mostly passed through the detector which was used for decay studies because of the difference between the ranges of these particles and that of "N (SAMUEL88a). A simulation of the ver- tical position and range (longitudinal position) distribution of these isotopes is shown in Figure 7.5. This separation was very helpful. However, the decrease in the thickness meant a smaller signal produced by the beta in the silicon. A trigger efficiency of 501 was 123 on _ 173 1‘Be I I I l I I y I y I U I 11 ' 1.5 -1 1 A. _ 1“ 3.0 ° : 7:. :0 o a a E = 15XA MeV « 8 2.0 ran 0 D—_ D -é — .F. On '4 4-) a n a “Don [I D C! q a b 0 O u D D D : 2.5 — u n D ._ U. . U U .4 \ - n U E] 1:1 . E Z n D ‘ 3.0 1‘— D—Cl— D .5— 3.5 t I L l 1 1 l 1 1 ABA I L m4 .1 0.5 1 1.5 Range in Si (mm) Figure 7.5 Illustration of the spatial separation of fragments at the focal plane of the RPMS. The vertical axis corresponds to vertical position at the location of the detectors, which is inversely propor- tional to the mass to charge ratio m/q. The horizontal axis corresponds to implantation depth in the silicon detector(s) 1211 obtained in this experiment for detection of the beta decay. VII.3.2 Background Problems: There have been at least four sources of signals that resemble beta-decays of the isotope of interest, but are in fact of other origin and must be identified and in most cases eliminated before good half- lives can be extracted. Three of these; daughter and other residual activity, cosmic rays, and thermal neutron-induces activity result in actual ionizing events in the silicon. The fourth, electronic noise, originates in the amplification and transmission of signals from the experimental vault to the data-U. To illustrate the sensitivity to background, consider a hypotheti- cal situation with a fixed number of atoms and a fixed rate of back- ground (e.g. 1000 atoms initially and 1 count per second of background. It is always easier to measure the half-life accurately when it is short. If the half-life is 1000 seconds, there will be over 3000 back- ground counts in the time that it takes for 900 of the atoms to decay. However, if the half-life is short, say 1 second, then there will only be 3 or 4 counts due to background before 900 decays have occurred. If the atoms are not produced at once, but spread out in time, then the situation gets worse. If the rate of production is much less than one atom per half-life, then the sensitivity to background is roughly one thousand times worse! So, if the half-life is one second, then 3 or 11 counts of background will contaminate the decay of each atom, and if the half-life is 1000 seconds, then the hopeless situation 125 of over 3000 background counts per real decay eliminates the pos- sibility of measuring the half-life! VII.3.2.1 Daughter Activity: When a nucleus is said to be far from stability, this almost al- ways means that two or more sequential beta decays are necessary for it to reach stability. Since the decay energy and therefore the decay rate tends to increase with distance from stability, the half-lives of the daughters are almost always much longer than the initial isotope produced. For example, while the half-lives of ”B and "N are 5 and 250 mS respectively, the first daughter half-lives ("C and "O) are 30 and 100 times longer. For the very short-lived activities, a single atom will usually decay before the next is produced, and therefore the beam is shut off after each ion is detected and does not resume until many (usually five or six) half-lives of the atom have passed and it has had sufficient chance to decay. However, the period that the beam is shut off is often short compared to the half-life of the daughter activity produced by the first decay, and therefore a build-up of daughter activity in the detector may occur. One solution is to reduce the beam intensity and therefore production rate to keep the decay spectrum clean, however this would lead to a dramatic drop in the num- ber of decays observed and consequently a drop in the statistical integrity of the data. This problem is discussed in more detail in Section VII.2.3.3, and experimental solutions are described in Section VIII.5 and VIII.6. VII.3.2.2 Cosmic Rays: Cosmic ray muons are a second source of ionizing background ac- tivity in the silicon detectors. These have a flux of about one or two 126 counts per second per cm3. See AGUILAR-BENITEZBB for a more complete breakdown of cosmic ray rates, energies and angular distributions. The muons are the decay products of high-energy pions, which are themselves produced by the collision of very high energy cosmic ray protons with the nuclei of atmospheric atoms many kilometers above the surface of the earth. Since the muons stop in many meters of earth, they are not isotropic, but instead are from only the upper hemisphere, predominantly within 115° of the zenith. Since the unions are moving relativistically, and are approximately minimum ionizing, the distribu- tion of energy losses in a detector is proportional to the distribution of path-lengths. The muons will generate signals not very different from a source of minimum ionizing beta particles from implanted ac— tivity, and therefore have a very strong probability of generating background at a rate of the order of one count per second. Because of their high average energy, a muon will rarely come to rest within any given detector system of reasonable size. It would instead pass right through, losing about 1.5 MeV for every g/cm2 areal density. If a 'box' made of some material capable of detecting muons is placed around the silicon, the muon signal in the silicon can be rejected based on a coincidence with this 'anti' detector. VII.3.2.3 Thermal Neutrons: While cosmic rays have no structure in time, a third source of ionizing background in silicon detectors at the focal plane of the RPMS has posed a significant problem to the determination of only the shortest of half-lives. Figure 2 of SAMUEL88 shows an accumulated time spectrum of beta-like events in the first 50 ms following the arrival of the (relatively) long lived isotope ‘He (t :800 ms). After the 1/2 127 ion arrives the cyclotron beam is shut off within 110 us. In addition to the flat background plus slow decay of ‘He there is a very rapidly decaying component within the first 10 ms after the beam is shut off. This component can be fit with an exponential function with a 'half- .life' of‘fl tz1 ms. CURTIN86 has attributed the time structure of this component to the diffusion and absorbtion of thermal neutrons within the experimental vault. Since the target (the source of the neutrons) is about 15m from the detectors and the velocity of a neutron with a kinetic energy of 0.025 eV (roughly kT at STP) is about 2 m/ms, this explanation is plausible8. A similar problem has been encountered at GANIL, where the drift spaces are larger and the 'half-life' of the background is 6 ms. They have adopted the same explanation. The mechanism by which the neutrons induce ionizing signals in the silicon is not clear. Thermal neutron capture is possible. The ‘Li in the lithium-drifted silicon detectors has a cross-section for capture of thermal neutrons resulting in the emission of an alpha particle of about 1000 barns. However these should show up as a peak in the energy spectrum of the silicon detector of about 11.75 MeV, which has not been observed so far. The experiment of SAMUEL88a made use of only surface barrier detectors without lithium doping, and still the u ms 'activity' was present. More likely, the neutron is captured by a silicon nucleus (cross section of the order of 0.1 barn) or other material nearby, and a high-energy (5 to 8 MeV) gamma ray is emitted. These photons can 8. The only other reasonable explanations are fast-neutron induced Ams beta or isomeric gamma decay (E > 0.5 MeV) in the vault. It is not likely that either of these processes could be so prolific. 128 produce a small shower of lower energy e+e- pairs and e'Y Compton pairs, and can then easily give a broad low-energy distribution. The short time-structure and high multiplicity (one count for every eight beam-off periodsg) interferes with half-life determina- tions less than about 20 ms, but does not significantly affect the longer-lived activities if the first 20 ms of the data is ignored. See SAHUELBB. VII.3.2.“ Electrical Noise: A fourth source of background counts is not related to ionization in the silicon, but instead is electronic in origin. The elimination of this problem is not simple, and has required extensive tests and modification of the original techniques of Curtin and Westfall. Most of these modifications are in the electronic set-up of the experiment, and will be described in Section VIII.3. The only points that can be made involving the physical environment of the detectors is their shielding to electromagnetic radiation, both 'RF' (60 Hz through 1 GHz) and visible light, and grounding. The silicon detectors are photo-sensitive. Visible photons from fluorescent room lights produce electron-hole pairs in the detector - a reverse biased diode) and induce a complex 120Hz modulathnn This results in huge signals from the pre-amplifier. The solution here is 9. This represents the rate of "u ms" events in a silicon surface barrier detector of 0.6 mm thickness and an area of 300 mm”, with a constant fraction discriminator threshold of roughly 0.5 MeV (as determined by a '°’Bi electron conversion source). The detector was surrounded by roughly 300g of Aluminum as a detector holder. The average beam intensity was about 200uA of E/A : 35 MeV ‘zNe , impinging on a tantalum target thick enough to stop the beam - roughly 0.015 inches. 129 obvious - light shielding is necessary. It does not however, have to be as complete as one would need for an active device with multiplica- tion such as a photo-multiplier tube. The silicon detectors are also effectively large-area bare capacitors hooked to an extremely sensitive charge amplifier. Any stray electric or magnetic fields will induce the motion of charges and therefore big signals from the preamp. Good results have been obtained with a Faraday cage in the fOrm of an aluminum detector holder and thin aluminized mylar windows. Attention should be paid to skin depth. While the preamplifier system is not so sensitive to the low frequency components of induced currents, they are typically intense at NSCL, and not necessarily shielded by the thin metalized window. The use of a thick metal collimator and the placement of detectors far from the hole inside the ’cage' will help shield at all frequencies to which the sys- tem is sensitive (except light), since the hole will be much smaller than a quarter wave-length. The gas proportional counters for position detection are also quite sensitive to induced fields. A sheet of aluminum foil or other thin conductive material between the gas detector and beam-line ground such as the m/q slits dramatically reduces induced 60Hz noise. One should take care that particles of interest pass through unimpeded. VII.3.3 Decay Curves: VII.3.3.1 One Atom at a Time Operation: If one atom, or an instantaneously produced group of atoms were to decay to a stable daughter with a decay rate A, then the rate of change of the number of atoms, and the total number atoms at time t respec- tively are 130 3% = -)\N, N = Noe-kt; 1'. 2 0. [7.16] The instantaneous rate of decay is then I‘(t) = - $3 = AN = Woe-At. [7.17] If this decay is recorded as a function of time one would measure B(t) : er(t), [7.18] where e is the efficiency for detecting decays. In this notation, R represents the experimentally observed count-rate, while r represents the actual decay rate. Now, if m atoms are produced at known but different times t m -A(t-ti) r(t) = i X e 0(t-ti), [7.191 i=1 i, where 9 is the step function to insure that only those atoms which have already arrived before time t are counted. In the limit of a very large number of atoms, the rate of arrival can be written as a con- tinuous function A(t) and the decay rate is just the convolution t . p(c) = i dt' e’m"t ) A(t'). [7.20] For very exotic species, the half-lives tend to be very short (large A) and the times between arrivals tend to be long (many times 1/A). The cyclotron is then shut off after each arrival, and the decay of the single atom is observed. The arrival time of all atoms are re- corded, so that a complete history can be reconstructed. For such a sparse time-structure, only the most recent previous ion (nearest neighbor) correction is usually necessary. The decay rate after the arrival of atom N is then -A(t-t ) -l(t-(t -t -t )) r(t,t2tN) = ie N + ie N "°" °ff , [7.21] 131 th where t represents how much beam-on time was needed produce the N Non atom after the end of the N-1fl cycle, and toff is the length of the time that the beam is off after the arrival of any atom. Simplified, this gives -A(t-tN) r(t,t2tN) = ie [1 + e -A(t +t ) Non off 1- [7.22] Now the data usually consists of hundreds or thousands of such iterations, and an average decay curve can be built up by summing the decay curves following the arrival of each atom, making all the pre- vious atoms the history of the detector at that point. If, during the beam-on periods, atoms are produced at an instantaneous rate AP, then the distribution of tom's is just given by the interval distribution 'Apton P(ton) = Ape , [7.23] when the production rate is stable in time10. If Equation 7.23 applies, then the measured decay rate with its nearest-neighbor correction can be averaged over all possible values of t . on -At a -A(ton+toff) -A ton r(t) = ie (1 + dt e i e p ], [7.2A] 0 on p which becomes, A -it p -lt _ off p(t) - Ae (1 + A+Ap e ]. [7.25] So, for example if the off period is three half-lives, and the atoms are produced on the average one every half-life (when the beam is 10. The production rate could vary due to fluctuations in beam current, drift in beam-line and RPMS elements, and changes in target thickness due to sputtering, evaporation and softening. It is therefore better to determine P(ton) directly from the data if possible. 132 on), then when an atom arrives, there is a 51 chance that the atom from the previous cycle is still there. If only the decay rate of a single isotope were to be determined, (this is equivalent to a half-live measurement of a pure source with no daughter activity) then this carry-over would not have any significant effect. However, there is usually more than one isotope passing through the WC; defining slits and reaching the detectors. It is then important to understand how much of once species is contaminating the decay spectrum associated with the another. For instance, any deter- mination of the half-lives of ‘zBe, "B and "C (all m/q : 3 wiini t1/2 ~ 22ms, 10ms, and 70ms respectively, from CURTIN86 and MUELEBBB will require the understanding of how each species contaminates the decay curves of the others. The experimental decay spectra are usually generated for each isotope. This is done by sunning together the distributions of all of the decay events directly associated with (following the arrival of) each type of atom. If the atom of type i has just arrived in the detector, then m A -A t - _21 J off x B(t) - eiri(t) + J§1 [AptIEJPJ(t)e Apt a -A t' -A t' , J on pt on 0 dton e e m e A —A t : eiri(t) + J§1 [A:;:§j)€jrj(t)e J Off. [7.26] where m is the total number of types of atoms, 6 is the efficiency, A J J is the decay rate, A is the partial rate of production of the j-t-r-1 P3 133 type of atom, Apt is the total rate of production of all atoms, such that m Apt = ié A . [7.27] and rJ(t) = A e 3 . [7.28] The efficiencies a could be different for different species for two reasons, their range distributions in the silicon can differ (Section VII.3.1), as can the energy loss and multiple scattering of the beta. See Section VII.A.A. If the beam-off time toff is made to be different for each isotope, (see Section VIII.5), then the subscript j can be added to boff' VII.3.3.2 Fixed Cycle Operation: When the species is produced at a rate much higher than the decay rate, it is not efficient any more to turn off the beam each time an atom arrives. In this cases, it is better to have switch the beam on and off continuously at a fixed rate. This was the case for a recent experiment to study the beta- delayed alpha emission from "N (see Section V), where the half life is near 1 second and the production rate near 20 atoms per second. The experimental cycle was divided into four periods. The first was the beam-on period, next a short wait, then a beam-off period during which the decay of "N was studied, then a second wait period. These will be referred to as ton’ w1, toff’ and w2 , and were set to 1.0, 0.05, 1.0 and 0.10 seconds respectively. The total cycle period to is just the sum of these four periods. 1311 In this experiment, the absolute time of arrival of each ”N atom was recorded, as well as the time of each decay (see Section VIII.1) . For the branching ratio to be determined, it was first necessary to count the number of alpha decays detected, and the number of ‘ °N atoms collected during a given run. However, the ratio of these two numbers is not yet the branching ratio, since some decays occur when the beam is on, and therefore go undetected. This ratio must be divided by the efficiency for detecting an event to obtain a branching ratio. The ef— ficiency can contain many terms related to experimental conditions, but the term representing the fraction of the decays that occur during a beam-off period dominated, and is derived below. If “N nuclei were produced in only a single beam-on period, then one can easily show that Equation 7.20 gives -At -Aw t -A(t -t') rs(t) = Ae e 1 °" dt' A(t')e °“ , [7.29] 0 where rs is the decay rate of the group of atoms produces in a single beam-on pulse. If instead H'N is produced in an infinite number of previous beam-on periods, then an extension of Equation 3.29 gives a ~aAtc r8(t) X e 9 a:0 ”J t) rs(t) = T . [7.30] 1-e c If A(t) is nearly constant in time within the beam-on period, then the integral in Equation 7.29 can be evaluated. Then, if r“ is itself integrated over the beam-off period, the efficiency for detecting the alpha particles becomes 135 -Aw -At -At t 1 off on = Offdt' r (t') = 6 (1-8 JU‘6 ) . [7.31] cyc 0 a (1 -Atc) -e VII.3.3.3 Daughter Activity: In Equation 7.26 a correction was shown to be necessary if an ex- periment intended to determine the half-life of an isotope can not absolutely differentiate decay signals from the isotope of interest from those of other isotopes implanted. As pointed out in section VII.3.2.1, the decay of a nucleus which is far from stability is usually followed by the decay of one or more daughter nuclei. So it is now also important to consider the effects of daughter activity, since this will create an extension of the distribution of counts beyond what one would expect for a simple decay, and will usually make the measured half-life too long if not properly taken into account. See below. If a source of parent nuclei is created at time t:0, then the rate of decay of the parent is -A t r1(t) = A1e ‘ . [7.32] The rate of decay of the daughter nucleus (the "grow in" curve) is just the convolution t -A1t' -A2(t-t') P2(t) 2 dt' A e e 1 II >’ ‘_ A1 A2 -A t -A t *1 ' *2 If the daughter has in turn a daughter of its own, then a second [7.33] convolution of r2 with the tertiary decay rate gives r3, A A t -A t' -A t' -A (t-t') 1 2 2 1 3 3 3 A1 ' *2 o = (1%}? Al—Ate-ABt’e-A2t] - 111 le-A3t-d‘tl 1. [7.311 1 2 2 3 1 3 136 The decay rate of the parent r1 is a maximum at t:0, and has a slope equal to -A1. Both r2 and r3 are 0 at t:0, but while r2 has a positive slope at t:0, the slope of r3 is zero at t:0. Equation [7.311] can be easily generalized to the situation where there is more than one daughter activity following the decay of a parent, and each of these daughter nuclei will themselves be followed by more than one tertiary decay. The decay rates, per atom, are -A1t r1(t) = A.I e , A A -A t -A t 121 21 1 r .(t) - [ e - e 1 , 21 1'12: AA A - t -A 1; Ian”) ‘ —‘LIA 2:111 [——_A -A [e 311* 21] 1 21 21 311 -A . t -A t -——1 (e 31‘) - 1 ) ] . [7.35] A1-A3iJ The rate of events recorded in the experiment following the production of a single parent nucleus B(t), is then m1 m21 B(t) ‘ €1r1(t) +i§1w21 (521”21(t) *Jé1w31153ijr31J(t)l’ [7.38] where 021 is the fraction of beta decays of the parent which result.iJ1 nucleus 1 as a daughter, w31J is the fraction of beta decays of nucleus 1 which result in nucleus J as a tertiary product, 61, £21, and e 31J are the appropriate efficiencies for detecting a decay, and m and 1 m21 are the number of daughters and of tertiary products considered for each of those daughters, respectively. The decay of the isotopes "Be and ”B serve as good examples to explore experimental problems associated with daughter activity. In the case of ”Be with a half-life of roughly 0.022 seconds, the 137 daughter ”B has a nearly identical half-life of 0.020 seconds. Since both have a decay energy of about 12 MeV, their decays should look similar in the silicon detector and therefore be difficult to dis- criminate. A simple fit to a decay curve containing events from both ‘zBe and its daughter activity will always give a half-life longer than the correct value for ”Be. CURTIN86 approached this problem by re- quiring two pulses to occur within a preset time interval, and then assigning the first recorded pulse to the decay of ”Be and the second to that of ”B. While this method could slightly bias the extracted value for the half-life, it was successful in separating the two ac- tivities. A more sophisticated method with which to treat the order of the pulses is discussed in section VII.3.”.3. The decay of "B is even more complex. The beta decay of "B will not always be followed by the decay of ”C. Since the decay energy of ”B is so great, it can populate levels in ”C which emit a neutron, leaving “C. There is then “C activity associated with "B decay. "B can also feed very high-lying levels in ”C which are unbound to two or more neutron emission. The decay of these levels can then produce “C, ”C, "'C and ”C by beta-delayed one, two, three or four neutron emission. Most of these daughter nuclei are radioactive, and some are themselves beta-delayed neutron emitters. Thus the decays of ”C, ”N, “C, “N, ”C, and even ”C can sometimes follow "’B decay. Values for half-lives and delayed neutron probabilities relevant to the decay of ”B have been taken from various experimental results and shell model calculations. Since they were compiled for illustrative purposes only, they are not given here. 138 The half-life of ”B was first measured by SAMUEL88. In this ex- periment, the rate of production of "’B was low enough (less than 100 atoms per hour) so that there was little interference from the daughter decays in the decay curve of ”B. At higher production rates however, the buildup of these activities would have caused a significant problem for the analysis. In Figure 7.6, the total beta-decay rate probability from a single atom of "B produced at time t:0 is shown. The individual components of "B, secondary and tertiary products are illustrated, as well as the total rate of all isotopes together. Figure 7.7 shows the same infor- mation, but has been extended to a much longer time scale to better show the decay of the longer lived products. In Figure 7.8, the simulated decay spectrum of "B and its products has been generated for a production rate of 100 atoms per second for a period sufficiently long (more than one minute) so that the decay of the products is fully equilibrated with the production rate of "B (one approximately every two half-lives.) The primary ("B only) and total decay rates for this case is compared to those of a single atom of ”B as calculated for Figures 7.6 and 7.7. While the total decay rate rapidly departs from that of "B alone in the high production rate case, the total decay rate associated with the single atom closely fOllows that of "B alone for many half-lives. From Figure 7.8, one can conclude that if this build-up is allowed to occur, it would be nearly impossible to obtain an accurate half-life without detailed information on the delayed-neutron emission probabil- ities and half-lives of the many isotopes involved, and their detection efficiencies. Thus the full production rate is useless in 139 VIIIIIrUIIIIII1IjIIIIIIU 103 A total 8: primary 7 ----- secondary ft) 101 'U ............ tertiary C1 0 C) Q) m " ———————————————————————— V It... ‘- -------- 4/ 3 l/ a: s a: . ..... z'r’ .................. i14LL;1.1"1|iiilliiiilinii 0 0.02 0.04 0.06 0.08 Time (seconds) Figure 7.6 Decay rates of the various components from an initially pure source of 1"B, including "B, "C, ‘7N, ‘°C, “N and ‘°C activity. See text fOr details. 1H0 total 8: primary llll ----- secondary oooooooooooo l l ILJIIJ Rate (seconds—1) 'c. ........ 0.. Time (seconds) Figure 7.7 Same as Figure 7.6, but shown on a compressed time scale. 1111 Rate (seconds-1) Figure 7.8 Primary "B decay rate and total decay rate for two sources The lower curves represent an initially pure source of ”B. 0 of "B. The upper curves represents an initially equilibrated source with a "B production rate of'100 per second. initially equilibrated source deviates strongly from a the decay rate of ”B itself, and illustrates the difficulty in extraction of a cor- I I T l ‘- "5102 F '5101 5‘ -._: 10° - 1111 0.01 0.02 0.03 0.04 0.05 Time (seconds) rect half-life when the production rate is too large! 1H2 The total decay rate of the this situation, and the beam intensity and thus the total number of decays observed would have to be greatly reduced to insure the accuracy of the measurement. While it is conceivable that one might try to ex- tract delayed-neutron probabilities by building up a 'source' of daughter activity for a long period (long with respect to the half-life of "B itself, but not to that of the daughter activity) and careful measurement of the exact shape of the slow component of the decay spectrum, it would be difficult in practice for many reasons, not the least of which is the large number of free parameters involved. When one requires the activity to stop within the actual device used to detect the beta particle, then little can be done to avoid the daughter build-up except for limiting the count rate. For certain cases, if the maximum energy of the beta particle from the daughter ac- tivity is low enough then the detection system may be made immune to these decays by simply raising the electronic threshold above this energy. CURTIN86 successfully used this technique to remove ‘He ac- tivity from the decay spectrum when measuring half-lives near m/q = 3.0. The decay energy of ‘He is only 3.5 MeV, and most beta particles are less than 2 MeV. All other activities near m/q:3 have decay energies above 10 MeV. The thickness of the detector used (5mm) in- sured that most high-energy betas could loose more than 2 MeV in the silicon, and thus would produce signals above the threshold. However, if one allows the activity to come to rest in something other that the detector itself, and this 'catcher' can be periodically removed and a new catcher put in place of it, then one can avoid build- up of unwanted activity. A device has been constructed to accomplish this. A wheel of thin plastic with a circumference of nearly a meter 1113 is mounted on a fast DC torque motor, and can be moved under computer control. The wheel is placed such that one part of its perimeter is sandwitched between two silicon detectors. Once an atoms passes through the first detector and comes to rest within the wheel, it is rotated to a beta detector. As this occurs, a clean section of the wheel is moved into place to catch the next atom. This device is dis- cussed in detail in section VIII.6. VII.3.” Fitting Procedures: In order to understand the decay data, one fits an appropriate theoretical time distribution to the data. The theoretical distribu- tion will most likely have at least one free parameter that can be adjusted until the fit is considered optimum. For example, if one has a source of atoms for which a half life is required, a distribution something like one of the expressions for B(t) above is used, or per— haps something as simple as N. exp(-At) + b, with N., A, and b (a constant background term), as adjustable parameters. It is not always such a simple task to compare two sets of parameters, and determine which is the 'better' fit in order to con- verge on the 'best' fit. We will use the definition that the better fit is the one which has more probability of having generated the data. Thus the best fit is the set of parameters which has the most probabil- ity of having generated the data, of all available choices in parameter space which are physically meaningful. If the experiment were to be carried out many many times, the data would become smooth (an exact ex- ponential plus constant in this example) and the best fit determined to arbitrarily good accuracy if the fitting function represents the actual physical phenomena responsible for the data. However, statistical 1H“ variation will cause each data-set to differ from the smooth function. The best fit is the one which requires the least amount of statistical variation to account for the difference between the fit and the data. This can now be defined mathematically. In order to calculate which of two functions is more likely to have generated the data, the set pi is defined as the distribution of probability of the measured values yi about a particular trial "best" value y(xi), pi = f(yi. y(xi)). [7.37] where x1 is the 191 value of the independent variable of a measurement (the time associated with the i133- channel of a decay histogram for instance), yi is the measured value at that channel (number of counts for instance), and y(xi) is the fitted function evaluated at that point. VII.3.“.1 Least Squares: Typically, for large number of counts where the normal distribu- tion can be used, (y,-y(xi))2 2 f(y1, y(xi)) = 753%}; e 2°1 , [7.38] where 01 is the RMS deviation of y1 about the "right" value y(xi), and is usually taken for large y(x1) as 01 = /y(x1) . [7.39] It is convenient to assume that the fit is usually fairly good when a fitting routine is near convergence, so most people make the ap- proximation that Oi : / yi 9 [7.110] 1145 and thus draw their error bars on the data. However, it is important to remember that the error bars actually belong on the theoretical function, not the data! The data is simply data points - measurements. One must express how the data can be explained by statistical devia- tions about the fitted value.11 The best fit is determined by maximizing the overall probability P that a function y(x) could have generated this data when the proper form of the statistical deviations is included, and N P : H pi . [7.”1] 1:1 The function y will contain some variable parameters aJ, (e.g. No, A, b) that will be adjusted to maximize P. Thus the set of J equa- tions, ‘g—' P 3 o ’ [70112] “J must be solved simultaneously. Since the logarithm function is monotonic and P is presumably greater than zero, we may also maximize ln(P). If we use the normal distribution Equation 7.38 with the ap- proximation of Equation 7.30, we can simply minimize (note sign change) 2 . [7.113] 2012 If we let all the 01's have the same value, they can be omitted, N -ln(P) = X 1:] and the requirement for the least-squares fit becomes Just, 11. If you don't believe this, consider the result that for a normal probability distribution, the probabilities for measuring 1100 and 900 when the most likely value is 1000 are equal, while the probability of measuring 1000 when the most likely value is 1100 is greater than when the most likely value is 900. 1116 3 2 '3-0—J i 1 (yi-y(x1)) = 0. [7.1-111] Equation 7.113 and 7.1411 represent the standard Least Squares technique. 11 M2 For many situations, the functional form of y(x) can be included ex- plicitly, the derivatives taken, and the resulting simultaneous equations solved analytically. Alternatively, numerical techniques can be used. VII.3.“.2 Maximum Likelihood: Now, if the data is sparse, so that the normal distribution may not be used (i.e. few or zero counts per channel) Equation 7.1111] is no longer valid and should not be used. If the fluctuations about the functional form of the fit are truly statistical, then one must use the Poisson distribution: Y1 -y(x1) f( (x )) - y(xi) e [7 as] Y1: Y i - (Y1)! - - If one takes the log of P, then differentiates with respect to the parameters a of y (Equations 7.37, 7.41 and 7.112), then dropping the J terms with vanishing derivatives the more generally useful condition of Maximum Likelihood is obtained: N %C§ £1 (yiln(y(xi)) -y(xi)) = 0. [7.116] VII.3.“.3 Use of All Available Information: When one tries to extract information from data at the limits of statistical accuracy, it is important to use all of the information at hand. For instance if the decay of many isotopes are measured simul- taneously one should not include independent backgrounds to the fitting functions for each of the isotopes, but instead couple the fits together such that the same background us used throughout. Negative 197 backgrounds and efficiencies greater than 1.0 should be excluded from the fitted parameter space since they are unphysical. However, while one might be tempted to impose the additional requirement that the area of the fitted function over the interval which overlaps with the data equal the total number of counts in the decay curve, this is usually rigorously imposed by the fitting technique itself. In most cases, Least-Squares and Maximum Likelihood automatically require this agree- ment and it should not be separately imposed. There is one source of information that has been avoided here up to this point. It is the event-by-event ordering of the recorded decays following each ion. In our examples we have discarded a wealth of information by summing the event structure associated with each ion into a single histogram. The possibility for recovering additional in- formation by maintaining the ordering of the decay events can be illustrated through the example of ‘zBe decay. Consider a simple model of an ideal experiment in which the half- life and delayed—neutron emission probability (PD) of ”Be is to be determined. The efficiency of the detection of the decay of both ”Be and 123 is in this example 1001, and there are no sources of back- ground. The ”Be ions are the only ions to reach the detector, and they are widely spaced in time so that the beam can be shut off for each ”Be ion separately with no nearest-neighbor contamination. A histogram of the times of all decay events following the arrival of a ‘zBe ion represents the sum of the decay rates of ‘aBe and “B, and is obtained from Equations 7.36, B(t) = r1(t) + 82 r2(t), [7.u71 1118 where r1 and r2 are the decay rates of ”Be and ”B respectively as described in Equations 7.32 and 7.33, and «12 is the delayed neutron probability Pn of ‘zBe. Under the conditions discussed above, a histogram of the times of the first decay event following the arrival of a ”Be ion represents a pure decay curve of ‘zBe only - the first term in Equation 7.373 However, a histogram of the times of the second decay event following the arrival of a ”Be ion does not represent a pure exponential decay curve of ‘ZB, but instead represents a grow-in curve typical of a daughter (the second term in Equation 7.117). If instead the times he- .tgggg the first and second decay events are histogrammed, one is avoiding the convolution used to derive Equation 7.36, and the dis- tribution should be that of a simple exponential decay curve of the daughter ‘zB, whose normalization gives directly the value of P". This technique reduces the fitting requirement from one data set and three unknowns (AnBe, Ana, 1»), to two data sets; the first with only one unknown ”111313) and the second with the two remaining unknown values U”B’ m). Since the reliability of a fit rapidly decreases with in- creasing number of fitted parameters, the reduction of the problem to two independent fits with fewer parameters should be a much more reli- able way to obtain the half-life and delayed neutron probability of ‘zBe. This method is more difficult to apply in practice, since the ideal constraints in the above example are not usually applicable. There is normally a non-unity efficiency (50-90%), a non-zero and even time-dependant background, and sometimes even other longer-lived ac- tivity present in the detectors from previous cycles. The last problem 1‘19 can sometimes be solved by using a partial data set. Regions (in time) of the data imediately following the arrival of a longer-lived species can simply be ignored for many half-lives of the ion. The structure of the background-plus-longer-lived activity component can be derived from the decay curves following the arrival of very long-lived or stable isotopes, or following a randomly generated trigger not associated with any heavy ion. See section VIII.2.5. However, the worst problem here is the non-unity efficiency. This results in the miss-assignment of counts, and the addition of two more parameters - the efficiency for each of the two isotopes. Whereas in the simple example above the first count could always be associated with the decay of ”Be and the second to that of ”B, in the realistic situation both the first and second counts can be from any of the three sources; ”Be decay, ”B decay, or background. If the low count-rate restriction still applies, one may impose the conditions that if the first event is indeed the decay of ”Be then the second count can be only ”B decay or background, and if the first event is the decay of 123 then the second count can only be background. In this situation, the proper form for the theoretical distribu- tion for the histogram of the first counts, second counts, third counts and so forth is difficult to calculate. In the limit of high ef- ficiency (>901) one might derive an expression which contains only first-order corrections to the that which is used for the ideal case discussed above. Thus if high efficiency can be obtained, the mother- daughter structure of the decay sequence may be preserved, increasing the reliability of the fitting procedure by reducing the number of fitted parameters, or at least the number of strongly-weighted 150 parameters in each fit. The high-efficiency requirement should be im- posed on all future detector designs lacking other gating devices (such as gamma-ray or delayed proton or neutron tagging). A further discussion of the use of all available information can be found in LYON889a, b. VII.3.“.A Finite Channel Size: It is important to remember that the decays are discrete in time, and not a continuous, smooth distribution. One does not actually measure the function R(t), but instead an array of’integers repre- senting the number of counts per channel; yi. Counts-per-channel really represents an integral of the rate over the channel size, not the instantaneous rate at a fixed time, and there is only an ap- proximate correspondence 1 iAt R(iAt) z Ri : '3? R(t') dt' . [7.48] (i-1)At In the special case of R(t) : Noe-kt a pure exponential, the relation- ship between the two quantities in Equation 7.118 is simply eMt, and can be absorbed into No if the absolute normalization is not important. However, if any other function is used, even by the addition of only a constant term to the exponential, then the relationship becomes more complex. It is critical fOr the experimenter to determine for the par- ticular nature of the data under analysis whether this effect is indeed significant or can safely be ignored. However this correction will most likely become important only when there are only a few channels per half-life or when the function is very non-exponential. More precisely, the derivative R'(t) should always be much smaller than R(t)/At. 151 VII.3.“.5 Uncertainties in the Fitted Parameters: In addition to the extraction of the most likely values for the fitted parameters, any fitting procedure must also evaluate the cer- tainty with which these values may be taken and used to interpret the underlying physics addressed by the experiment. This certainty is usually evaluated by calculating how far each of the fitted parameters may be altered before the agreement between the function and the data is degraded by a specified amount. Within the framework of the Least- Squares procedure, the agreement between the fitting function and the data points can be evaluated quantitatively by means of the normal probability distribution associated with each data point. This method is usually called the chi-squared method. With it, one can state ex- actly how good a fit is by stating how likely it would be for a second identical experiment (with data presumbly differing from the data of the first experiment only in the manner described by the distribution f of Equation 7.37) to produce data less likely to have been generated by the fitted parameters. In the Maximum Likelihood formalism however, there is no simple interpretation of the actual value of the log-Likelihood function (section VII.3.“.2), other than the fact that the maximum value is ob- tained when the fitted parameters are those most likely to have generated the data. While any other set of parameters give smaller values, the significance of the size of the decrease in terms of how much less probable it is for these values to have generated the data is not imediately evident. In either case it is important not only to vary each of the parameters independently, but also in groups. The surface in parameter 152 space defined either by the sum of the squares (Equation 7.113) or of the Likelihood function (the summation term of Equation 3.116) may be curved in such a way that a long trough of near-extremum values may ex- tend in a direction which is not parallel to any axis. Thus while the second derivative with respect to any one axis such as az/aA2 and 32/362 may be large, suggesting a well defined minimum, another such as 31/313, where A:a§, e:b§, may be very smll. In other words, coupled changes between two or more parameters may provide fits nearly as good as the best fit. This would imply uncertainties in the fitted parameters are much larger than if the parameters were varied independ- ently. Any fitting- routine is usually based upon variations in the parameters (the physics) while keeping the data fixed. In reality, it is the physics which is constant, and the data which varies from ex- periment to experiment. The difference is in most cases subtle, but can be significant. A technique to understand the actual uncertainties in a fit, inde- pendent of the statistical distribution used to generate the fitting conditions has been developed by SAMUEL88a. With the assumption that the data and its statistical deviations are indeed described by the fitting function and the distribution f respectively, the method deter- mines the actual probability distribution of fitted parameters for a given set of "actual" parameters. The method uses a Monte-Carlo simulation to generate a large ensemble of simulated experimental data. The ensemble represents a independent simulations of many atoms, and of all background processes included in the fitting function. For ex- ample, if one obtains from an initial fit a half-life, efficiency and 153 background rate, these three values are used to generate a large en- semble of simulated decay spectra. The fitting routine is used to generate a set of parameters for each member of the ensemble. For each of the parameters the distribution of fitted values represents what one might call the response function of the routine. The distribution should peak at the "correct" value used to generate the ensemble, and the width of the distribution represents the uncertainty. Thus true confidence limits can be set on each of the parameters, and one need not explicitly address the coupling between parameters. VII." Beta-Delayed Charged Particle Measurements: The emission of nuclear particles from a nucleus following beta decay was discussed in Sections III through V. The sum of the kinetic energies of the delayed particle and of the recoil represent the total decay energy, and thus the excitation energy of the state populated by beta decay, minus the threshold energy for this break-up. A careful measurement of the energy deposited in the silicon after beta-decay will give a spectrum with peaks which can be identified with the break- up of states populated by the beta decay. For example, the decay diagram for "N is shown in Figure 5.1, and the delayed alpha spectrum of 2hao89 is shown in Figure 5.2. Two peaks in the spectrum can be directly associated with the breakup of the third and fourth 1" levels in "O. Implantation of delayed charged particle emitters within silicon offers some advantages over other techniques where the activity remains in the target or is attached to a passive surface such as a rotating wheel or tape. One advantage is the ability to count how many ions were implanted, and another is the nearly run 1111 Sr solid angle for 1511 detection of the emitted charged particle. However, both of these ad- vantages are slightly less than perfectly implemented, and there are many experimental considerations involved in producing a spectrum with good branching ratio infbrmation, energy resolution and energy calibra- tion. These are discussed below. VII.A.1 Normalization: The connection between the number of counts in a peak such as those in Figure 5.2, and an actual experimental branching ratio depends on a number of factors. The efficiency for fixed cycle operation as discussed in section VII.3.3.2 will be addressed here. We will use the simple definition of extracted branching ratio: Counts in Decal Peak / EFF @ecay Counting) BR : Counts of ions / EFF (ion Counting) ’ [7‘u9] where EFF is the overall efficiency for the conditions in the sub- script. This efficiency can be broken down into components which must be individually determined, then multiplied together to obtain the to- tal efficiency. The first efficiency is associated with fraction of the time that the system is in a "beam-off" mode so that decays can be observed. This cycle efficiency EFFcyc is usually less than 50% and is given by Equation 7.31. The time-structure of the dead-time of the data aquisition system must also be determined. Since the atoms produced at the beginning of a beam-on period are less likely to decay in the beam-off period than those produced at the end of the beam-on cycle, the dead-time at the end of the beam-on period must carry a higher weight than that near the 155 beginning. Also the beam-off dead-time at the beginning of the period is more important than that at the end. It is actually difficult to measure accurately the dead time at each instant in time. Instead, the average dead-time can be determined both for beam-on and for beam-off. A very close approximation to the time dependence of the dead-time would then be to distribute the dead-time within a period as the in- stantaneous count-rate within the period. Thus since most decays are observed at the beginning of the beam-off period, this is where most of the dead-time should occur. While this is obviously an approximation, if the dead-time correction is small (less than 5% for instance) then this approximation should give correct the count-rate for dead time to better than 1%. While most ions come to rest within the active volumes of the silicon detectors, a small fraction will end up stopping in the front and back dead-layers of the detectors. In this case some of the ionization does not result in collected charge. Some will also stop in any air which might be between the detectors, unless of course the detectors are operated in vacuum. The distribution of particles throughout the various materials depends on many factors, but is typi- cally uniform per unit areal density when the detectors are thin (50pm). One must estimate the efficiency with which the ions have been counted in each detector. It is always difficult to determine whether an ion comes to rest in the very back of one detector, or the dead— layer of the front of the following detector. One must measure or estimate the thicknesses of the dead-layers and extrapolate from the range distributions within the detectors. In the case of delayed alpha activity from ”N, the range of alpha particles is short. However for 156 ‘°N which come to rest near either face of the silixxui, there is a chance that the alpha will be emitted toward the face and escape without depositing all of its energy within the silixxn1. This ef- ficiency will vary dramatically with distance to the face. ZHA089 have used a Monte-Carlo simulation to understand the loss of counts form the peak in the alpha spectrum from this mechanism. Distortions of the peak-shape by the escape of particles are discussed in section VII.A.S. If the stopping of ions within the air between detectors poses a problem to the experiment such as background and uncertainties in the efficiency, one can replace the atmosphere near the detectors with an equal pressure of helium. This can be accomplished by simply placing a large plastic bag around the detectors leaving two openings for the helium. A small flow of helium gas into one opening, and an equal flow of excess helium out of the other passage will maintain a density one roughly eight that of normal air, solving 87% of the problem of absorb- tion in air with 1% of the trouble of a vacuum system. VII.U.2 Energy Calibration: One must also determine the relationship between the energy loss in the silicon and a given pulse height transmitted to the ADC in the acquisition system. In air, or even Helium, alpha sources placed near the silicon will loose too much energy in the gas and not produce pulses of accurately known energy in the silicon. Most alpha sources available will emit alpha particles only in the region of 5 to 8 MeV, while delayed charged particles can be emitted over atmuxnbroader range. The ground-state breakup of ’8 following ’C beta decay releases only (143 MeV, while the decay of the T:O, T:1 doublet in 8Be following the decay of “B can releases over 16 MeV. Table 7.1 contains a list of 157 potential calibration decays of light isotope which if periodically im- planted in the detectors throughout an experiment can provide an internal calibration. However, before the calibration is assumed to represent the limits of the precision in energy of the detector and electronics, one must consider recoil effects from the beta and neutrino emission, and any pulse-height defect. The recoil problen1i45 addressed in WARBURTON87, and the pulse-height defect is discussed in section VII.A.3. As an alternative or supplement to internal calibration of the ex- periment one may use calibrated capacitors. The quantity of charge collected from a given amount of energy deposited in silicon is well know and is a natural property of silicon. By charging a capacitor of‘ known capacitance to a measured voltage one can inject a known amount of charge into the silicon detector pre-amplifier. Since the average energy loss resulting in the production of an electron-hole pair is nearly a constant of the material at the 1% level (See section 111.3.3), capacitors used for such a purpose are usually calibrated in MeV per Volt. The response is dependant on many parameters in the electronics, such as shaping time of the amplifier and capacitance of the detector itself. Long, high count-rate experiments may change the capacitance of the silicon detector from radiation damage and alter the energy calibration. It is therefore good practice if possible tn) con- tinuously check the calibration for experiments which require a high degree of accuracy in the energy measurements. 158 'Table 7.1 Some useful narrow beta delayed charged particle lines of known summed charged particle energies which might be ixuuarporated ir1 an energy calibration. Not listed is the isotope 9Li. The beta delayed 20 + neutron decay channel of the 5/2' level in ’Be at 2.3” MeV produces a.broad charged particle energy spectrum which collapses to a narrow peak if the neutron energy is detected and added. This state also decays through n + °Be, giving a narrow neutron energy peak. ”C decay also provides narrow neutron lines for calibration of neutron detectors. The nucleus H’N is a candidate for narrow neutron lines - see ZHA089. Parent Decay Sum Energy Half life Branch Reference(s) Isotope Channel (MeV) (sec) [2) “Be 7Li + a 1.21 11.8 2.5 ALBURGER81 and , MILLENER82 7Li + (1 0.73 0.11 “B 20 16.76 0.77 small WARBURTON86 ”8 3a 0.38 0.020 1.5 BARNES73 9C 20 + p 0.28 0.13 60 MIKOLASBBa 2.6” 17 12.1 ? ‘ZN 3a 0.38 0.011 2.7 SCHWALM66 "N '~c + a 1.39 0.90 6.8 2HA089, OLNESB2 1.81 1.8 '30 ‘2C + p 1.57 0.008 11 ESTERL70 and ASAHI89 159 VII.4.3 Pulse Height Defect: There are several effects which can decrease the peak height as- sociated with a given amount of energy deposited in the silicon. These are grouped together under the term pulse height defect. The most im- portant effect related to delayed charged particle spectroscopy in light nuclei implanted within silicon is that of energy loss of the delayed particle and recoiling nucleus by non-ionizing processes. When 1"N decays to alpha-unbound levels in ”0 for instance, the decay of the excited state result in an alpha particle and a recoiling ‘“C nucleus. They share the break up energy, in the ratio of approximately 14/18 to 4/18. Thus when the third 1- level in the daughter ‘°0 breaks up giving 1.3 MeV to the particles, the “'C nucleus carries less than 0.3 MeV. It is moving so slow that a significant fraction of its energy loss will be to non-ionizing process such aszelastic atomic scattering from silicon atoms which can knock the atoms out of the lat- tice. This process dissipates a large amount of energy without the production of an electron-hole pair. Energy loss to non-ionizing processes can result in a signal of perhaps 1.2 MeV instead of 1.3, and could result in the mis- identification of a level in some cases. For protons and alphas, the defect will be of the order of a few keV but for heavier particles tine defect rapidly increases. For example, when heavy (A2200) alpha emit- ters are implanted within silicon, the alpha particle is the only to cause ionization; the energy of the recoil is ignored, as it loses es- sentially none of its energy to ionization. Even for the energy loss which does produce ionization there are effects which can prevent this charge from being properly collected and 160 fhwxn properly contributing to the height of the resulting pulse to the ADC. For slow, highly ionizing particles the density of electrons and [holes in the silicon is much greater than for fast, less-ionizing par- ticles. Once the electrons and holes begin to separate they prmxnlce a space-charge which cancels the electric field imposed by the bias of the detector. This prevents further separation of the holes and electrons until the space-charge is reduced by collection of the few charges which have been moved to the outside edges of the cloud. 'Ihis has two effects on the pulse height. The first effect results from a change in the rise and decay times of the charge pulse from the detec- tor. The amplifier is usually adjusted to have a response in frequency space that is peaked at one frequency, and drops off to zero at lower and higher frequency. Thus any change in the peak shape will also change the power and/or phase spectrum at the input of the amplifier and will usually change the output amplitude. This is analogous to the changes in gain observed when the capacitance of the detector is al- tered by radiation damage. The second effect of the high charge density resulting from highly ionizing particles is that of recombination. The longer that the electrons and holes occupy the same volume of silicon, the more chance there is for the destructive recombination of an electron-hole pair into phonons, and a resulting decrease in the pulse height at the ADC. Thus in performing delayed charged—particle spectroscopy where energies must be accurately measured, many effects must be considered when comparing calibration standards or calibrated capacitors to decays where heavier recoils deposit a fraction of the energy with low velocity and high ionization density. 161 VII.4.4 Line Shape Distortion from Beta Energy Loss: In section VII.3.1, the energy loss of the beta particle in the silicon was the source of information for the experiment, and the dis- cussitni centered around how to maximize this signal. Here, the energy loss of the beta is a contaminant and must be minimized. While the contribution can never be completely removed when the activity lies within the active volume of the silicon, there are many ways to reduce the energy loss or subtract it afterwards during analysis of the data. The spectrum of energy loss from a source of beta activity within a silikxni detector of finite thickness is difficult to calculate. The decay produced betas of all energies between zero and the ilLLl avail- able energy. Each energy beta will have different energy-loss and multiple scattering behavior. However for most decays the energy loss of a beta in silicon can be fairly well approximated as 0.5 MeV per mm (1.5 MeV/g/cmz). If one ignores the possibility of polarization or alignment fkwnn the heavy ion reaction being maintained in the silicon resulting in an angular distribution of the betas with respect to the orientation of the silicon wafer (a fairly safe assumptnxibut not guaranteed), then the betas can be assumed to be emitted isotropically. With the angle 9 defined as the angle between the normal to the surface of the detector and the direction of the beta, isotropic emission can be expressed as d2? _ -1 dcosO do ' 4n ' [7°50] With (1? 1 d0 ' 2u' [7.51] Equation 7.50 can be integrated over 0, giving dP -1 dcosO ‘ 2 ' [7.52] 162 When the beta emitted from a point a distance L, from the surface of the detector passes through that surface, it has traveled a distance L and deposited and energy E in, where L:;%;-e-,andE:g-E-L. [7.53] Using the chain rule, the expression for the probability of energy loss is dP dP dcosG dL . dE ' dcose dL dE ' [7'5“] Through substitution, one can obtain dP L dB -1 8E. = .Efg [HY] , [7.45] When expressed as a function of energy only, and the physical limits are included, the final result is E 55% SEE, — = . [7.1%] 0 E < E, The integral of Equation [7.46] gives a total probability of 0.5 for leaving each of the surfaces. In an actual experiment, the beta ennth- ters can be distributed at all values of L, from zero to the full thickness of the detector. This distribution must be taken into ac- count before the actual distortion of the line-shape is simulated. There are two important differences in the degradation of the limua-shape between the beta energy-loss mechanism, and simple Gaussian noise. The first is that the beta energy loss shifts the peak to higher energy by a constant offset depending on the range distribution of the ions and the thickness of the detector, while the simple noise will not alter the peak position in a linear system. The centroid of a line-shape generated from Equation 7.45 can not be defined mathemati- cally, since the energy-weighted integral diverges. However, in 163 practice a centroid will be finite. The maximum amount of energy [unit :3 beta can actually deposit in the silicon is determined by a combina- tion of the maximum energy of the beta itself, the diameter of the silicon (we have assumed an infinite slab) and the irreversible scat- tering of the beta out of the detector before it loses all of its energy. The second important difference is the interference with higher energy weak peaks by a lower energy strong peak. Thinner detectors will decrease the high-energy tailing from the beta energy loss, but the increased capacitance will broaden the symmetric Gaussian line- shape from the electronic noise of the detector system. Monochromation of the ions (see section V1.3) will greatly increase the number of ions which stop in a detector which has been chosen for its thinness. The shape>cd‘the energy loss spectrum of the electrons is depend- ent on the details of the range distribution of the ions. [?igure 7.9 shows a simulation for a source at two positions in the detector, one source is in the center and the other is implanted at a depth of only 101 of the thickness of the detector. For a given beta energy threshold, these two sources would have different efficiencies for triggering an event in a half-life experiment. Techniques to measure and subtract the beta energy loss, or to limit it to a well defined set of energies are discussed in section VII.4.6. VII.4.5 Line Shape Distortions from Escaped Charged Particles: In the case of ”N the range of the alphas in silicon was small compared to the thickness of the detectors used, and the escape of the alphas before losing their total energy was a small effect on the 164 spectrum and required only a small correction to the area of the peaks in the data analysis. In other cases much longer range particles are emitted and require thicker detectors to contain and record the full energy. In the case of 9C beta-delayed protons of almost 11 MeV were ex- pected. These have a range of roughly one millimeter in silicon. This is close to the total range of the incoming 9C ions from the RPMS! If 100% containment were to be achieved a detector well over 2mm thick would be required. However with this thickness the degradation of the spectrum from the beta energy loss would be absolutely unacceptable. A compromise was reached and ’C ions were implanted in botthCL4 mm thick detector and a 1mm detector. As can be seen in Figure~4.3, the shape of the energy loss spectrum for high energy protons-Bragg curve results in the bimodal distribution of energies deposited in the silicon. This is a par- ticularly troublesome distribution, as it causes a second peak to appear which has nothing to do with the decay of a second excited state in the daughter. In Figure 4.3 the actual recorded spectrum of delayed particles from ’C is also shown. A peak at 12 MeV is evident. It con- tains counts from at least three different decay mechanisms which produce two alpha particles and a proton with combined energies of 12 MeV, but shared in very different ways. See section IV fkn~aa dis- cussion of the decay of ’C and distribution of energy between the the three particles. VII.4.6 Multi-Element Detector Systems, Coincidences: In sections VII.4.4 and VII.4.5 above, the difficulties associated with choosing the correct thickness of a detector are outlined. With 165 the addition of two more silicon detectors directly in front and behind the detector containing the decay events both problems can be ad- dressed. Mikolas88a describe a technique to subtract the energy loss of the beta event by event by measuring the energy loss in a parallel detec- tor. The detector arrangement is shown in that reference. During analysis of the data, the presence of a signal in E1, E2 and E3 are all required after a ’C ion comes to rest in EH. Since a signal is ob- served in E3, the beta must pass completely through E2. The distance fuwmn the ’0 position to the back surface of E1 can be determined from the range-energy relationship for the incoming ’C event. If one as- sumes that the dE/dX and direction of the beta is constant then through the use of similar triangles, one can show T - R AE1 : '—E'.:.——X AE2, [7.57] E2 where AE1 and AE2 are the energy loss of the beta particle in E1 and E2, TE1 and TE2 are the thicknesses of the two detectors, and R is the range of the ’C ion measured from the front of the detector. If for each decay event the value of AE1 is subtracted from the energy re— corded in E1, then the long tail of the beta energy distribution can be suppressed. As shown in MIKOLASBBa the spectrum recorded in E1 near 3 MeV is shown without any modification. Also shown is the subtracted spectxunn. While it contains only a small fraction of the total number of counts, the narrow state at 2.36 MeV is better separated from the underlying broad state, which is nearly invisible in the raw data. Part of the suppression comes simply from restricting the direction that the beta particle travels. In order to reach E3 it must leave E1 166 close to the normal of the detector. This would truncate tflua tail to some maxiunmienergy in a simple model. While the determination of the energy loss in E2, and the scaled subtraction from the energy in E1 further improves the peak shape, any method to restrict Just the direc- tion of the beta will significantly improve the resoluticni<3f a beta- delayed charged particle experiment. Multi-element detector arrays can also be used to gain more infor- mation on high energy particles which can leave the silicon in which the decay originates. The efficiency of a proton leaving one detector and entering the next is difficult to accurately determine and is usually of the order of only 20%. This may however aid in the iden- tificaticn1<3f the particle decay modes of a state. In the case of 9C decay, very few events were recorded where more tfiuuw 2 MeV was deposited in each detector and the sum was near 12 MeV. This implies that the state at 12 MeV emits a high-energy and thus long range proton capable of entering another detector at best a small fraction of the time. This is consistent with other observations of this state, as described in Section IV. 167 APPENDICES APPENDIX A BETA DECAY BRANCHING RATIOS OF ’C TO LOW LYING STATES IN ’8 Rowland «on: Physical Rule. 0. Vol. 81. No. 8. February 1000 mddad’cmhwlyhumh’l D.MIAMW.MLKMLW,LLMJI.IM J.WJ.S.MdZ.O.Xh I.” 168 PHYSICAL REVIEW C VOLUME 37. NUMBER 2 FEBRUARY 1988 Branchingratlosof’Ctolowlyingstatesia’B D. Mikolas. B. A. Brown, W. Benenson. L. H. I'Iarwood. E. Kathy, J. A. Nolen. lr.. B. Sherrill. J. Stevenson, I. S. Winfield, and 2. Q. Xie Department of Physics and Astronomy and National Superconducting Cyclotron laboratory. Michigan State University. East Lansing. Michigan 4882‘ R. Sherr Department of Physic. Princeton University. Princeton. New Jersey 08340 [Rweived 1 May 1987) Wereportthefirstobservationofthebetadecayof’Ctothegroundstateandtotwolow-lying excitedstatesof’B. Branchingratiostothesestatesweremeasured.andapreviouslyreported branchtoalevelnear 12 Mveasconfirmed. Thebranchingratiotothel'a-i-‘groundststsis 601:!0%.andthebranchingratiostothenarrow{-'levelst2.36MeVandtothebrosd T'level near2.9 MeVare [7:696 and lltSfi. respectively. Becameofthe thres-hodynatureofths'l decay.the’Cionswereimplantedintheacuvevolumeofsiflcondetectomandthetotaldecayen- ergissofthestatesin’Bweredirectlymeasured. Wecomparethessresultstoshellmodelcelculao timmthemlogoeesyor'uw’neuetome'upm’smeuoe. Ihecomparisoate'ljde- cayindieatmuuymmmyinthsbetsdeaymsuixehmenumthef‘kvdhrguthnuymch asymmetry previously observed. I. INTRODUCTION The beta-delayed proton precursors with T,--§, A-4n+l (ll-2.3“...) have been extensively stud- ied, and delayed proton activity from every member of this group from ’c :6 “0e has been measured.” Most delayed proton spectra from these nuclei show a compli- cated structure with many peaks, reflecting transitions to discrete narrow levels. In the case of ’c. (see Fig. 11 the delayed proton spectrum is dominated by a continuum or proton energies.” In addition. the data below about 3MeVofRefs.3and4werecontaminatedbyelectron pileup.’ This prevented the measurement of most of the experimental beta decay strength to ’3. which is expect- edtoliebelowthisexperimentalcutof. Furthermore. the ’C beta-delayed proton spectrum is diflcult to inter- pretitself.becauseunknownfractiouofthedmyener- gy are carried away by the two alpha particles. The de- csyofthegroundstateof’hgeneratesaprotonwitha kinetic energy of only 0.16 MeV.‘ and the first negative- parity excited state (f‘ at 2.360 MeV). while it is rela- tivelynarrow.decayslessthan 1%ofthetimebypr0ton emissiontothegrotmdstateof'he.’ Itinsteaddecays mainlythroughbroadintermediatestatesleadingtoa continuum of proton energies. Thsproblemofthespreadinprotonensrgiesbecomm worse for the higher excitation energies. More decay channels (which can involve excited states in the inter- mediate fragment) become energetically allowed, and enoughphassspscebecomesavailableformanyofthess channels to be populated with significant probability. In this case the proton energy spectrum does not directly reflect the excitation spectrum in ’3, since all paths in- volve decays to three particles (2n + 9). Even for a fixed excitation in ’B, diluent particle decay channels which 3;: 169 involve states in 'Be and ’IJ will generate difierent pro- tonspectrathatcanbequitebroad. Thebreakupofthe ’3 nucleus promptly into three particles is also possible. Therefore. to measure the branching of ’C decay re- quires a method that detects the population and excita- tion energies of the ’3 levels directly. independent of the decay modes of these states. In the present experiment, we used the technique of implantation of the delayed particle precursor, ’C. withinssilicondetector. Thismethodwasfirstusedby Chen er cl.‘ to investigate the decay of ’Li, the mirror nucleus of ’C. In that experiment, the excited states of ’Dedecsyedto 2a+n, but only thesum ofthe twoa particlsenergieswssmeasured. Thedelayedneutronen- ergy spectrum was recorded in a separate experiment.’ The excitation energy spectrum in the ’Be nucleus could be inferred from the data only with the use of specific as- sumptions about the particle decay for each state. The implantation technique was also used by Barnes and Ni- chols’ to make a precise measurement of the B-delayed breakup of the second excited state of l2C into three a particles. In this case. as in the case of ’B decay, all the decay particles are charged. and as long as they are com- pletelycontainedinthesilicomthetotaldecayenergyof the nucleus is measured. independently of the decay mods. ‘l'heenergydepositedinthedetectorinwhichboth the ’C nucleus decays and the ’B decay fragments stop is the sum of the excitation energy of the ’B nucleus. of the ground state breakup energy of ’B (0.28 MeV). and of a smallcontribution from the energy loss of the beta parti- cle. Because of the low energies of the alpha particles and the proton, the fraction of the energy loss to nonion- icing processes in the silicon must be considered. since these processes will not contribute to the measured ener- 766 fl BRANCHING RATIOS OF 'C TO LOW LYING STATES IN ’3 761 3 2_ (a) (b) 9 , - C 3I'Llié" 3/2‘ use r-sgz T137 VLLSL!’ ' § 11 s r- 0.4 r- s 5“. no r- as — "‘9' ”3"“ rue sl-le'vz. + a 65 -- LL- 12' 29 a is I" 1.5 . r- 1.1 as 12' / \._._ m‘zl’ 'Be,.+p ’Hem.+azs‘&££ =/ / 5/2‘ 236 / _. .. ,5 su —- \ 243 5/2‘ sn- * a ‘Beo. + n 167 7 3/2' coo —- 9— -_ < —o is ’Beoe + p 2a + h is? T— — ooo 3/2' B -020 2a or p 88 ' FIG. 1. Relative mass and excitation energies of known and presumed if, if, and {3 levels, relevant to beta decay in mass 9, and particle decay thresholds of those levels. All energies in (a) are relative to the ground state mass of ’3 (atomic mass excess 12.415810.ml MeV) and those in (b) are relative to the ground state of ’Be (ll.3477t0.(XX)4). All ground state masses are taken from Wapstra ct al. (Ref. 58). Energies and widths of excited states are all from Ajzcnberg-Selove (Ref. 6), except those of the 2.9 and 12.1 MeV levels in ’B, which are discussed in the present work. gy in the silicon. We estimate that this pulse height de- fect should lie between 2 and 8 keV per particle'o'" (de pending on the lab energy of each), and results in only a slight broadening of the peaks that will not significantly elect the following analysis. Since the direction of the particles with respect to the silicon crystal axes cannot be controlled, large pulse height anomalies are possible for particles which channel in the silicon. lattice. We as- sume that the probability of emission of a particle in a direction suflciently close to any channeling axis is small. We have expanded upon this technique by using a multidetector telescOpe. This provides a way to subtract the contribution of the beta particle from the spectrum and thus allows the use of thicker detectors to contain the charged particle energy. It also provides a simple way to check on the containment of charged particles and a potential method to distinguish difierent decay modes. The delayed proton data of Hardy et al.’ and Esterl er cl.‘ contain peaks at roughly 9 and 12 MeV center- of-mass energy, based on the proton decay to the ground state of 'Be. These could originate from the decay of two different states in 9B, or a single state at 12 MeV which proton decays to both the ground state and first excited state of 'Be (as suggested by the authors). Since our technique measures the excitation in the 9B nucleus directly, there can be no ambiguity of this type. The im- plantation technique also allows the number of nuclei imbedded in the silicon to be counted, which permits ab- solute branching ratio measurements. In the following section, we will describe the experi- mental procedure of obtaining the beta-delayed charged-particle spectra for °C decay. In Sec. III we outline the details of the analysis of the data, and methods and assumptions used in obtaining the branch- ing ratios. In Secs. IV and V we discuss previously pub- lished data on the beta decay of ’u and the °Be(p,n)’B reaction, respectively. In Sec. VI, shell model calcula- tionsofbeta-decayntesof’Cand’LLthe’dem)’B relative cross sections, and the beta-delayed particle de- 170 cay modes in the mass 9 system are presented. We dis- cuss the connections between all of these data and calcu- lations in Sec. VII, and conclude with suggestions for further experiments in Sec. VIII. I]. EXPERIMENT A 91 mg/cm2 Ni target was bombarded by an 'E/Aazs MeV "c“ beam from the xssoo cyclotron at the National Superconducting Cyclotron Laboratory. The Reaction Product Mess Separator (RPMS) (Refs. 12 and 13) was used to separate ’C fragments from the oth- er reaction products emerging at a scattering angle of 3'. A thin kapton window separated the vacuum of the RPMS from the focal plane, which is in air. The focal plane of the RPMS was equipped with slits to select par- ticles of a given mass/charge ratio (or /q), a position- scnsitive proportional counter, and a four element silicon detector telescope. The telescope is schematically illus- trated in Fig. 2. It consisted of two 400 pm thick sur- face barrier detectors and two lithium drifted silicon detectors with thicknesses of l and 5 mm. These will be referred to as A5, 51, £2, and E3, respectively. The en- ergy response of the electronics was measured to 5% ac- curacy by means of a calibrated pulser. Approximately 6N0 9C ions stopped in El, and 4000 in 52 in 12 h of data taking with an average beam current of 60 nA. Ions were identified by their energy loss in each detector. The identification was verified by an m /q determination from focal plane position. Other ions which stopped in the detectors were mostly ‘B nu- clei or scattered beam particles (~2ooo and ~sooo events, respectively). The implanation depth profile of the 9C ions in the silicon was adjusted by means of aluminum absorbers in front of the defining slits so that any variation in the containment of high energy protons could be explored. The range distribution of ’C ions cal- culated from the total energy measurement is also illus- trated in Fig. 2. Thecyclotronbeamwasturnedofiwithin‘Opsofthe arrivalofanion(bymeansofafastphaseshifierinthe 768 D. MIKOLAS er al. 31 .1 _1 at: El 82 :3 ad 'c \ P MN!) m A \ 0.4 0.4 1.0 5.0 FIG. 2. A schematic diagram of the silicon detector tele~ scopeusedattheRPMSfoealplane. A’Cionisidentifiedin thetelescope.andcomestorestin£l orE2. When’Cdecays, the B particles, with energies distributed between 0 and 15.5 MeV, usually leave the silicon detector in which the decay occurs, and sometimes enter other elements. The daughter nu- cleus. ’B, always breaks up into two a particles and a proton. The implanted ’C range distribution is indicated in the bottom ot‘El andEZ. Ihethicknessmofthedetectorsinmmareindi- cated under each element. rf transmitter of one of the three “dees” of the cyclo- tron), and the gain of the silicon detector preamplifiers increased by a factor of 10 for a 480 ms period, approxi- mately four half-lives of 9C. The detection of an ion in the focal plane telescope will be referred to as a “beam- on" event, and all events during the following 480 ms period as “beam-oil" events. After the beam was turned of and the preamplifiers allowed to stabilize (16 ms), a beam-off event was triggered by signals above a discrimi- natorlevel,setnear200keV,fromanyoneofthelast three silicon detectors. A sealer module that counted 103 % COUNTS / 100 keV O 3 o 0 3.5 5 7.5 10 Energy (MeV) FIG. 3. Thebeam-ofienergyspectraessociated with 'Cand "CionsrscordedinEl. Thespectrumassocistedwith uChas beennormalizedtothesamenumberofbeam-oneventsasthe ’C spectrum. 12.5 16 171 o toosoosooeoosoo Time(ms) FIGJ. Timespectraofbeem-oleventsafterthearrivalof ’CionsinElandinRinwhichthesignalinthatdetector was above or below 2.0 MeV. For each of the four combina- tions, a histogram ofevents following the arrival ofa "C is also included, to illustrate the background rate. The smooth curvesarenotdirectfitstothesedata. Theyaredescribedin 8ec.III. - Counts / 7.8 ms ° 3 33“... pulsesfromaZfiZkquuartzoscillatorwasclearedbya beam-on event, and was read but not cleared for the sub- sequent beam-ofi' events. Thus the time between the beam-on event and each of the following beam-ofi' events could be determined. The data were recorded event by event on magnetic tape. Each beam-on event could be directly matched with the set of subsequent beam-ofi' events during oil-line analysis. III. ANALYSIS The energy spectrum of beam-ofi' events for which a ’C ion stopped in 51 is shown in Fig. 3. The beam-o! spectrum associated with uC ions, normalized to the same number of beam-on events, is also included in the figure. The RC spectrum was used to define the back- ground spectrum. Sources of background include resid- ual beta activity within and near the silicon detectors and electrical noise. Time spectra of events in each detector associated with 9C or ”C are shown in Figs. “ll-Md). The events are further divided into groups of those for which the beam-ofi' signal in the detector of interest was above and those for which it was below 2 MeV. The area of the corresponding regions in the decay spectra in El and 32, associated with ’C and '2C, above and below 2 MeV were used to generate constantebackground and exponential-plus-constant-background curves, and these results are superimposed on the data in Fig. 4. The area of the appropriate region from the normalized uC asso- ciated energy spectrum, shown in Fig. 3, is used for the constant background measurement, while the difi'erence between the areas from the 9C and the normalized l2C spectrumisusedtodeterminethecoeflcientoftheex- ponential decay. The previously-measured half-life""‘ fl BRANCI'IING RATIOS OF ’C TO LOW LYING STATES IN ’3 769 of 126.5:1.0 ms is used throughout this analysis. As can be seen in Fig. 4, the agreement is good. This indi- cates that the background is indeed independent of time, and that the separate contributions to the energy spectra from 9C and background have been determined correct- ly. The efficiency of the system for the detection of the decay of 9C nuclei was 87% in El and 94% in E2, after correction for finite counting time and dead time. The missing decays are ascribed to the decay of 9B in its ground state, where the total energy released was small, and near the thresholds of the constant fraction discrimo inators used in the experiment. The E2 detector is much thicker than £1, and most of the 9C ions that entered it came to rest close to its front surface. The beta particles leaving £2 through the front traversed only a few hundred microns of silicon, while those leaving the back, a minimum of 800 pm. This makes the peaks in 52 (not shown) broader than their counterparts in 51 (Fig. 3) by shifting half of the events higher in energy by about 300 keV on average. If our description of the threshold problem is correct. about half of the counts in £2 associated with the decay to the ground state should then lie well above the threshold level of the constant fraction discriminator, in a region where the eficiency problem should not occur. This would make the number of undetected decays in 52 roughly half of that in 51, which conforms with the measured efficiencies, and supports the conclusion that the missing counts are associated with the ground state branch. Shell model calculations (discussed in Sec. VI) indicate that the decay of 9C should most often populate the three lowest negative parity states in ’B (including the ground state) with roughly equal Gamow-Teller strength. (See Table I.) However, of these three levels, only the ground state and one excited state branch are evident in Fig. 3. While‘the ground state" and first {7 level7 in ’B are narrow, it is reasonable to believe that the first {7 level in 9B is broad. As discussed in Sec. V, there is evidence for this broad state near 2.7 MeV in the °Be(p,n)’B spectrum. In the mirror nucleus ’Be, the lev- e1 near 2.9 MeV has a width of roughly 1.0 MeV. (See Table II for both nuclei.) The decay of 9c to a level in “B with a width of 1.5 MeV would be obscured in out spectrum by the beta energy loss associated with the nearby if state. Two techniques described below were deve10ped to reduce this efi'ect in order to search for this broad level. A simulation of the beta energy-loss spectrum was generated by a Monte Carlo routine based on the rela- tion ' 5°55 E E P(E)6£=- £1 ’ 2 ° (1) 0, E 100 MeV) is proportional to Gamow-Teller strength. Reviews of this relationship can be found by Goodman et til.Jo and Ted- deucci ct al." The Gamow-Teller component of the ’Belp,n)°B reaction can populate the same levels in 9B as the beta decay of ’c. Since the initial state is now ’3e instead of 9C, and there is no B‘W phase-space weight- ing, the observed intensities will be very difl'erent than for the case of 9C decay. Fazely er al.” and Pugh” report cross sections for the ’Be(p,n)’B reaction, and Pugh describes a detailed analysis of the data. Cross sections for those levels which peak at 0' are listed in Table III, and the reported parameters which describe the broad 4;“ level are in- cluded in Table 11. While this work represents the clearest indication of the broad i' level in ’3 besides the present ’C decay data, there still remains a lar e un- certainty in the nature of this level. Fazely ct al. 3 show a fit to the neutron spectrum at a small angle using a level at 2.6 MeV with a width of 1.65 MeV. However, the more complete analysis of the data by Pughn indi- cates a larger width of about 3 MeV for this 2.6 MeV state. The increase may be related to the introduction of a narrow 31* level at 2.71 MeV in the fit. It is not clear if more levels are contributing to the spectrum. Since beta decay involves only Gamow-Teller transitions, it populates fewer states. Therefore, widths extracted from these experiments are likely to be more reliable than from the 9Be(p,n)"B reaction. We have estimated B(GT) values from the °Be(p,n)°B reaction cross sections, and include the results in Table III. These estimates are based on three simple assump- tions: (1) the cross section at 0' is directly proportional to B(GT) (except between analog states where there is an additional Fermi component), (2) the B(GT) value for the ground-state to ground-state transition is 1.0) (a value close to the four theoretical values—see the following section). and (3) 15% of the cross section of the ground- state to ground-state analog transition is due to the Fer- mi component [based on the distorted wave impulse ap- proximation (DWIA) calculations presented in Ref. 33]. We have chosen the ground-state transition B(GT) as the best reference for normalization. These assumptions give a proportionality constant such that for all nonana- log transitions, B(GT)=0121a(0') . (5) where o is in units of mb/sr. This value is in agreement with the more thorough treatment of this relationship by Goodman still.” and Taddeucci et al." VL SHELL MODEL CALCULATIONS We have calculated the level schemes of the A =9 sys- tem within the full p-shell model space using the shell- TABLE III. Excitation energies, widths. and cm. sections of those levels reported by Pugh (Ref. 33) whose cross section peaks at 0' in the 'Be(p,n)’B reaction. The B(GT) values are from Eq. (5). E. (MeV) FWHM (MeV) a at 00‘ (mb/sr) a at 3.4' (mb/sr) Deduced B(GT) 0.00 a 9.52 :004 7.56 :006 (1.0101' 2.36 a 2.09 :004 1.68 :006 0.253 2.71:0.1 0.71:01 2.83 :036 2.10 :034 (0.342)‘ 2.75:0.3 3.13:0.2 9.73 :029 8.24 :026 1.177 4.3 :02 1.6 :02 2.41 :006 2.15 :006 0.292 1223:01 0.5 :01 0230:0014 0188:0017 0.028 l3.96:t:0 l d 0066:0012 0.049 :0015 0.038 14.60:1:0.1 0.6 :01 0213:0025 0173:0040 0.026 'I'he experimental widths of these peaks are used to define the experimental resolution ( ~04 MeV FWHM). ”This value was fixed in order to find the proportionality constant in Eq. (5). ‘The identity of this state is uncertain, but if it represents strength to a level separate from the broad if at 2.75, it is not a .I '= l ‘, if, or {7 level. according to shell model predictions. This being the case, it would not be populated through the Gamow-Teller channelasdiscussedinthetext,andthusavalueforB(GT)wouldnotbe ' . ‘Theintrlnsic widthofthestatecould mtbedetermindbeyofitheconclunonthuithsmaucompardmtheeapenmenulm lution. 176 .... ... ...8... .z. :8... ...: 88.. .... ...8... ..8... a... 8... .... .2... 3.8... .8. 88... 5.1m... . ...~: 88... .8... 8... a... .... .... 88... ...: 88... .8. 38... .2. .88... .8... ...... ...... ...... .... 88... .... :8... ...... 88... .8: 2.8.. 8..... ...... 9.... « .... ...: 8... .8: .8... ...... :8... .8: ...... .... .5... 8.... ...... 8.... 8. ...... 88... a... 2.8.. .8. 88... ...: ..8... .3. 28... N8... 8.. ...... .... ...: ...... .8: ..8... .... ..8..... .....N. e...... .... ......8... 2.... 8.. an... .... ...... 88... .8: ......o ...: =8... .8. :8... ...: ...8... 8..... ...... 8... m .0. 9.0.0 8: ..N... .3 88... .85 ...... .8... ~88 ...... 8... ...... ...: 88... .89.. ...... .8... 2.... ...... 8... ...... ...: 88... .8... 8... .8... .8... 8... 8.... 8.. ...: 88... .8... 9... ...... 8.... 3s m 88 2.... .8... ...... ...8. 8.... .88. 4.... . ...3. .86 3o... .8... 3.. 88 ...... .8... ...... ....o. E... .88. n... .....n. 8..... 8o... 3.... 3.. .u 88 a... .8... ..N... .8: 8..... .89. n... .83 3.... ..8... 8..... .....v .... 88 ...... .8: .... .8: 8..... .88. a... ...... 8..... ...... 8... 8.... W W 3. ....0.0 m 88 ...... x... .o... .88. ...... ..8.. 8.... ...... m 88 8... n... 8... .83. 2.... 8.... 2.... a... D. 8:. 8... a... ..o... .83. «.... 8.... ..8... 2... 83 8... ...... 3.. .83. «.... o8. 8...... 8... m 2... ...... o... .8... e8... 8. ...... 8... N... «8.. 8..... 8. 2... 8.. a... 8.... ..8... 8.. z... 8... n... .8... ..8... ...... m ... v8... 8... a... ...... : ... ...... 8.... 8..... 8... ...~ 8..... 8... a... 8... ....~. on... 2...... 3.... 8... ..~ 8..... ...... a... 2.. .. a. ...... 8..... 3.... 8.. ... 8..... on... a... 2... .. .N. ...... 8.... 8.. 8... m ... ....0.0 $3.. N". ..u. N u. ..u. . n. .u. Fa... F0... .>o:. .5 5.3.3 .305. 0+ -....3. 0+ La... 0+ (3.... 0+ -33. .— ... Add. a + 1.09. 9.0.0.0? mg em .2 ...oz. ...Om .o. in. 523. ...... 5:00 ...... ....n ...... :83. ....n ...o»: 35...: .2 ...o... m2m~ .... to. ...-...... 0... 5.30 ”.803 9.50:0. 2.. ... .3...» 8. 20:09.8... .30.. 3.. 30.8.00...— ...8. u... ... .... .cm 0.2.. 0353.3 ...... 2......) 2:. .38.... 0.008289% mafiecaaoeso u... ..0 ...»: o... 0. 8855...... ... 09>.» so... ......) 3:38.. u... can .03....— 03 ...—.0... :23... 8.... u... ..0.. 800:. .30.. 0.0.8... 330.? o... ..0.. 80.3.. 0.003033... 2:. .... -... Xcdn—al o .30.» ..o... no. ..0 8030.. 0. 3...: 88.08.. 0.3:... ... 8...: =< 6:. .0 me ... >92 .m .e o: 0. 00.0.09... 23. L. o... 0. ....o .0 0e .0 >83 5.... 3.. 9:. 8...: 2.05 880.82. 2.... .882? on... o... ..0.. 2.0.8.0»... .900... ......m .>. mdmfir 774 177 (keV) Total width 1:2 3 - “.‘n— +a 0.029 (74) 0 l 5 -0 u1/2_ +0 2 l ’ s “In “ +0 0.0019 (79 (Continued). ’11,”. +0 [=0 TABLE IV. [=1 '3‘;+ + P 0.023 (4”) 0++P l=1 ’Be(p,n)’B B(GT) 8101‘) my 5s (MeV) J. BRANCI'IING RATIOS OF ’C TO LOW LYING STATES IN ’8 775 model code OXBASH.“ Four interactions were con- sidered, two of the original interactions of Cohen and Kurath,” i.e., the (6-16) 28MB and the (8-16) POT in. teractions, as well as the more recent interactions of Mil- lener" and Kumar.37 All of these are based upon a least-squares fit of the interaction parameters to some subset of the p-shell binding energies and energy levels. Predicted excitation energies of if, if, and {7 levels in ’B and ’Be, and B(GT) values for the pulation of these levels from the beta decay of 9C and Li, and from the 9Be(p,n)9B reaction are summarized in Tables IV and III, respectively. These theoretical B(GT) values are defined in Ref. 38 and include the (g 4 In)2 factor. They have also been multiplied by a factor of 0.6 to take into account the empirical quenching observed for this operator.” Since binding energy and isospin mixing corrections have not been considered, the predicted exci- lation energies and beta-decay B(GT) values apply to both members of the mirror system. Table IV contains calculated spectroscopic factors for all available two-body decay channels using each of the four p-shell interactions. Channels for which there can be no strength from a pure p-shell model have blank en- tries. The a-particle spectroscopic amplitudes were cal- culated using an 800) cluster for the a particle as de- scribed in Ref. 39, and were cross-checked against simi- lar calculations provided by Millener.”_ We have calcu- lated the partial widths based on the decay of these lev- els in ’0. The resulting widths in key are listed in parentheses to the right of the corresponding spectro- scopic factors in Table IV. The widths are given by"‘“ r, -my‘r,(£, —Q) . (6) whereO}isthespectroscopicfactor,Pisthepenetrabil- ity,E,istheexcitationenergyofthe nucleus,andy2 istheWignersingleparticlereducedwidthgivenby ,_g(sc)1 ’ zuki‘ Here it is the reduced mass. The penetrabilities were calculated from the Coulomb wave functions using the method of Steed as described by Barnett." The channel radius R0 was set to 4.0 fm for the 'Be+ p channels, and 4.5 fm for the 3Li +a channels. The penetrabilities for the first three levels are calculated using the excitation energies of 00), 2.36, and 2.9 MeV, for the if, 9,”, and «1", levels, respectively. For the next two levels, the theoretical excitation energies for each prediction are used. All of the penetrabilities for the last four levels predicted to lie above 8 MeV are cal- culated, assuming the state is at an excitation energy of 12.1 MeV. Where calculated, the penetrabilities for the '30” + p, I'Be;, + p, sLin-Hr, and ’Li:n-+a chan- nels are calculated, assuming Q =—0.19, 2.7, 1.69, and 8.69 MeV, respectively. The last three values represent arbitrary effective centroids for the broad intermediate states in ‘Be and ’Li. For states above the Coulomb plus centrifugal barrier, which lies a few MeV above threshold for p and a emis- sion from light nuclei, the penetrability represents 640 180 670 160 70 310 53 850 0020 (51) 0.0189 (23) 0(1)“ (12) 0.0037 (20) 0012 (64) 5.0[541 (0.3) 0038 (200) ). 3." 2 AAAAAQsA sous-51 (o 4) 0.011 ( l 10 0.024 (240 0.019 (330) 0.0158 (100) 0.0004 (7) 0.0130 (52) 0.019 (330) 0.0137 (65) 0.016 ( l 10) (7) 3.3034] (0.1) 0.0039 (in) 'BelowthisstatetheMillenerlRef. 361interactioaa1sopredictsstatesat 11.98 MeV(§')and 12.40MeV(,’-‘1. 0.0115 (11) 0.11114 (30) 0.024 0.0!) 0.046 0.0!) 0.0!) 0.012 0.019 0 017 2.32 2.48 1 89 2.66 1.63 I 00 1 19 l 42 11 61 9.80 10. I 3 l 3.03' 13.34’ 12 04 10.87 12 18 'BelowthisstatetheKumrlRefJ7linteractiona1sopredictsstateaat 11.36HeV(-§'), 12.04 MeV(%'),and 12.24 MeV( 3.. I I 5 178 776 D. MIKOIAS et al. 37 rou hly the available phase space (e.g., k’dk ask dE as Edi, where k and E are the center-of-mass momen- tum and energy for the decay). Thus for a given spec- troscopic factor, the widths of the states are only slowly varying functions of the actual binding energy of the state. Therefore, once above the Coulomb barrier, the widths for the mirror decays to 'Be + p or it should be similar to each other, as should those for a + sLi or ’I-Ie. Given the uncertainty in the calculated spectroscopic factors, it is not important at the moment to include the small differences between these unbound levels in mirror nuclei. Thus, the widths given for states above 4 MeV in ’B can be treated as estimates of the widths of the same levels in ’Be. The major uncertainties in these width calculations are related to the approximations used in Eq. (6), and to the fact that the small spectroscopic factors will be sensi- tive to admixtures from configurations outside the p- shell model space. We believe that the former is respon- sible for the “factor of 4" discrepancy discussed in the following section. Equation (6) could be replaced with more complete models“ however, this treatment would be beyond the scape of the present work. VII. DISCUSSION The states in Table IV can be divided into three groups: (a) the lowest three states below 4 MeV which have small beta-decay B(GT) values but which dominate in the beta decay of 9C and 9Li because of the large Q value for the decay; (b) the next two states lying between 4 and 8 MeV which also have small beta-decay B(GT) values; and (c) the next few states above 8 MeV which have large beta-decay B(GT) values. We will use this division to organize our discussion. A. Grown): Statssbslew4MeV The states in group (a) have small B(GT) values for beta decay; that is, small compared to the sum-rule value of 0.6m /gy)13(z —N)=-8.5. For these states, the - inal (6-16) 23MB interaction of Cohen and Kurath ’ appears to give the best overall agreement with experi- ment for both energy levels and beta decay B(GT) values. Indeed, these B(GT) values are in remarkably good agreement given the smallness of the values (see Table I). The states in group (a) have large spectroscop- ic factors for the allowed nucleon decays to the ground state and 2* states of 'Be, and allowed a decays to the ground state of ’Li (see Table IV). The calculated B(GT) values for the 9Belp,n)9B reaction of Table IV also agree well with the experimental values of Table III. While the two most recent measurements of the branching ratio for the °Li—-°Be ground state beta de- cay yield B(GT) which agree favorably with the value for ’C decay (see Table I), agreement between measurements for decays to the other two low-lying states is not as good. In the case of the f“ level, the values for the ’Li decay span a range of a factor of S, with our value from 9C decay close to the middle. In the case of the branch 179 to the if level, the three B(GT) values for ’Li decay are all very close, while our value for the decay of 9C is lower than these values by more than a factor of 2. If isospin is conserved, the wave functions for mirror lev- els, and thus the B(GT) values for mirror beta decays, should be identical. Deviations from mirror symmetry have been extensively investigated, and examples of asymmetry in beta decay have been identified.‘“' We define an asymmetry parameter, A=VB(GT)_—VB(GT)+ . (8) as a way to express the absolute difierence in the beta decay matrix elements. Previously the two largest abso- lute asymmetries observed in the p shell were found in the decays of ”B and ”N to the ground state of [2C (A=0.043),‘° and the decays of ‘B and ”o in the ground states of l3C and ”N, respectively (A20.076).’°‘” As given, the data in Table I suggest a value Aa0.09:l:0.03 for the decay to the %' level. Towner” has demonstrated that most of the measured asymmetries can be fairly well accounted for by small differences in the radial wave functions of the mirror states. This difference is due to the Coulomb interac- tion, which introduces an asymmetry in the binding en- ergies of the mirror nuclei. However, for the decay to the {7 level in mass 9, Towner predicts a value for A of only 0.008, an order of magnitude smaller than observed. Inconsistencies between the various measurements of B(GT) values for ’Li beta decay shown in Table l and particle decay branching ratios in Table II cast doubt on the branching ratio measurements for the decay of ’Li t0 the «if level in 9Be. Bach relies on some particular par- ticle decay characteristic of the {T level that is not yet determined with suitable reliability. As an example, let us develop a possible explanation for the observed asym- metry between the B(GT) for the population of the 7‘ level from 9Li and ’C decay, based on a misinterpreta- tion of the beta-delayed particle spectra for ’Li. The shell model calculations shown in Table IV indi- cate that the :17 level in 9Be and ’B has a large spectro- scopic factor for the decay into both the ground state and first excited state of 'ne. If the high-energy tail of this broad level neutron decays through the low energy tail of 'Be;,., these events could contaminate the inter- mediate energy peak in the delayed neutron spectrum discussed in Sec. IV, and as a result cause the branching ratio to the the ,7 level to be overestimated. In order to be more quantitative. let us assume that the beta decay branching ratios of’Li to the 3‘ and 7" levels or’ae are both 15%. We will use 7% at the frac-‘ tion of the decays of the narrow 4}‘ level to the ground state of 'Be (consistent with previous experiments; see Table II). If 5% of the decays of the if state populate the first 2+ state of 'Be instead of the ground state, and these lower energy neutrons are mistaken as neutrons from the decay of the if level to the ground state of 'Be, the inferred branching ratio to the if level will be reduced slightly to about 14%. However, since the strength of the intermediate energy neutron group must l7, BRANCI'IING RATIOS OF ’C TO LOW LYING STATES IN ’3 177 be multiplied by l/0.07=l4.3 in order to obtain the branching ratio to the if level, the inferred branching ratio to this state will be 14.3X[(15%><0.07)+(15%X0.05)]=26% (9) instead of 15%. Here, the first term represents the real branch to the if level, and the second term represents the branch to the 7‘ level that would be mistaken as ad- ditional population of the {3' level. While only an instructive example. .this scenario is consistent with the known data. Adlofi et al.“ have given only an upper limit of 28% to the fraction of the decays of the 3,“ level that do not decay to the ground state of 'Be. (Langevin et al.” suggest a value for this fraction of 70%. However, this value is based in part on the branching ratio to the }‘ level of Chen et al.,' itself a factor of 5 lower than that of Macefield et al.") The results listed in Table IV indicate a large spectroscopic factor for the decay of the if level to the first excited state of 8Be, and so one must expect some fraction of the decays of the high-energy part of the broad 1'3 level to decay thrOugh this channel, despite the Coulomb plus I =1 centrifugal barrier. A possible candidate for a fourth level belonging to ' group (a) has been reported by Esterl er al.‘ They ob- served a delayed proton group from the decay of 9C that would correspond to a state in 9B at 3.25:0.25 MeV with a width of 0.21:0.1 MeV, decaying by proton emis- sion to the ground state of ‘Be. The narrow width of this peak precludes the assignment to the first if level discussed above. Based on the nonobscrvation of the population of this narrow state in our data, we put an upper limit on the value of the 9C beta decay B(GT) value of 0.004 [or a log(ft)_2_6.2]. It is dificult to tell whether or not the peak of Esterl et al. is consistent with this strength. The shell model calculations (see Table IV) do not leave room for any narrow i', if, or 1}" levels between the first if level at 2.36 and the lev- els of group (c) predicted to lie above 9 MeV. This peak cannot be attributed to the nucleon decay of a level near 6 MeV to the first 2+ state of 'Be, since this state in '8: has a width of about 1.5 MeV, and the experimental peak has a width of only a fraction of an MeV. Thus, if this peak represents a state in ’B, it most likely represents a first-forbidden transition to a level near 3.25 MeV with a width of 0.21:0.1 MeV. B. Group“): Statssbstwssn4-18M0V We cannot identify any of the states in group (b) in the present experiment. This again is consistent with the shell-model calculations, which predict a small B(GT) value for the population of these states from beta decay. Both states in this group have strong spectroscopic fac- tors for decay through the 'Be;, + nucleon channel. While the if level is predicted to decay nearly ex- clusively through this channel, the if level is predicted 180 to have some strength to the 'Be” + nucleon and mass 5 + or channels. Pugh” has identified a state in 9B at 4.31:0.2 MeV with a width of l.6:tO.2 MeV, populated in the 9Belp,n)"B reaction. The cross section for this level peaks at 0', and implies a value for B(GT) of about 0.29 (see Table III and Sec. V). This represents an excellent match for the if state of group (b) predicted to lie near 5 MeV. The predicted widths of the if and 3'7 levels of group (a) are both about a factor of 4 greater than ob. served in (compare values for these widths given in Tables II and IV). If the predicted width of the {3 level of group (b) is similarly reduced by a factor of 4, the agreement between the calculated position and width of this level and the level described by Pugh” is nearly per- fect. Esterl er al.‘ have identified a peak in the delayed proton spectrum that could represent the proton decay of a level in “a at 40:03 MeV with a width of 10:02 MeV to the ground state of 'Be. From the calculated partial widths listed in Table IV, we estimate that the if level of group (b) will decay through this channel roughly 10% of the time. It is then reasonable to suspect that beta decay to this if level followed by pro- ton decay to the ground state of 'Be may be the source of the proton peak described by Esterl et al.‘ The population of this if level from the decay of °C would be difficult to observe in our present data, due to its large width. However, if present, it would manifest itself as a broad peak to the higher energy side of the peak due to the if level in Fig. 5. The lack of observa- tion of such a structure in the present data sets an upper limit for the B(GT) of 0.(X)7 [log(ft)=5.9] for the beta decay strength to the state reported by Pugh.” This is consistent with the small predicted B(GT) values for the if state of group (b) listed in Table IV. The if level near 7 MeV is predicted to decay almost exclusively through the 'Be;+ + nucleon. If we assume that the width is overpredicted by the same factor of 4 as discussed above. the width of this state is expected to be near 1 MeV. This level has small B(GT) values for population through both beta decay and the ’de,n) re- action. There is no evidence for such a level in the analysis of Pugh.JJ The same delayed proton peak of Esterl et al.‘ that we have associated with the {-- level of group (b) (see above) cannot be attributed to the decay of the 11" level to the first 2* state of 'Be. This level is expected to have a width of at least 1 MeV, and the 2+ state in 'Be has a width of about 1.5 MeV. These two factors suggest that a delayed proton peak from this lev- el should have a width in the lab of 2 MeV or more, in contrast to the observed width of 1.0 MeV observed by Esterl et al.‘ Langevin et al." observe a beta-decay branch of ’Li to a broad level near 8 MeV. They have associated this strength with a reported level at 7.94 MeV in “Be (see Sec. IV). While this assignment is somewhat uncertain, their measured branching ratio of 1.5:0.5 % implies a B(GT) of 0.06:0.02 [log(ft)~5.0:t0.l] at this energy. This value is consistent with the predicted B(GT) for the %" level of group (b). 773 o. mucous et al. a C. Group (e): States above 8 MeV All of the states in group (c) have large B(GT) values for population by beta decay, and small B(GT) for the 9Belp,n) reaction. They all have very small spectroscop- ic factors for all of the decay channels considered in Table IV. Because of their small predicted widths, they are all candidates for the sharp state we observe near 12 MeV. For all of the interactions the predicted excitation energies are too low by 2-3 MeV. However, this is not an uncommon problem with shell model calculations. We note that all of the calculations also underpredict the excitation energy of the lowest %" =% state, which lies experimentally at 14.7 MeV, by amounts of 1.39, 2.26, 2.41, and 1.05 MeV, for the (6-16) 28MB,” Mil- lener," Km,” and (8-16) POT (Ref. 35) interactions, respectively (this is the same ordering as in Tables I and IV). ‘ Pugh” has reported three states that are candidates for these four predicted levels. All of the predicted and experimental B(GT) values for the °Be(p,n)°B reaction are so small that their uncertainties allow for no unambi- guous assignments of any one of the predicted levels to any one measured level. Hardy et al.3 and Esterl er al.‘ report peaks in the delayed proton spectrum near 9 MeV and 12 MeV in the center of mass. There are at least three different ex- planations for these peaks; that the decay of 9C popu- lates (1) two levels in ’a, the first level near 9 MeV with a width of 1.5 MeV, and the second is near 12 MeV with a width of 0.5 MeV—both levels at least occasionally pro- ton decay to the ground state of 3Be; (2) two levels in 9B near 12 MeV populated in the ra- tio of 1.5:1, the first level decaying to the 2* excited state of 8Be, and the second level to the ground state; (3) a single level in ”a near 12 MeV, which proton de- cays to the broad 2+ level in 'Be at 3 MeV and to the ground state in the ratio of 1.5:1. ' In all cases, the large continuum rising to the lower en- ergy side of the two peaks can still be explained as the dominant decay of one (or both) level(s) through the ’Li" +a channel, generating protons with energies lower than those from the 'Be channels. This is the only decay mode observed by Nyman et al. '7 of one or more levels near 12 MeV in °Be populated from the beta decay of Li. In the present experiment, we observe only a narrow peak near 12 MeV and do not observe the broad 9 MeV peak. We therefore can rule out (1). In order for (2) to be consistent with the single narrow peak in our data, the two levels should have nearly identical excitation en- ergies or very different beta-decay branching ratios. Ex- planation (3) seems the simplest way to explain the two sets of data, and we will adopt this. In Sec. III we determined the single standard devia- tion lower limit of the branching ratio to the narrow lev- el at 12.1:0.6 MeV as 1.8%. This value represents a value for the B(GT) of 0.6. The largest contribution to the uncertainty (and thus contributing to the smallness 181 of the lower limit) results from the uncertainty in the measured excitation energy of the state in our experi- ment. If the state can be assigned the same excitation as that of Esterl et al.} at 12.111010 MeV, then the one standard deviation lower limit for the B(GT) becomes 1.1. Corrections for efiiciency will further increase these lower limits for the B(GT). ' Of the predicted levels in group (c), only the i‘. if, and second J,“ levels have predicted beta decay B(GT) values large enough to be consistent with the measured B(GT). If we assume that there is only one level near 12 MeV that is strongly populated by beta decay, and that it is responsible for both peaks in the delayed proton data, then we must rule out the predicted if level, since it should have no significant branch to the ground state of 'Be. This leaves us with only the 1' and second 7‘ levels of group (c) as candidates for the level populated by beta decay. If the underprediction of the excitation energies of these levels (as discussed in the beginning of this section) is taken into account, the agreement be- tween the if and the observed level at 12 MeV is com- plete, and the second if level is too high by 2-3 MeV. While we cannot associate with certainty the 12.2i0.1 MeV level observed by Pugh” with the level observed in beta decay, the observed strength in the ’Be(p.n)°B reac- tion is consistent with the predicted value for the «13 level. ' The level populated strongly near 12 MeV is clearly a part of the giant Gamow-Teller resonance. If indeed it turns out to be a if level, this level might be described as the “antianalog” state. This term describes the level (or group of nearby levels) which have structure similar to that of the analog state itself, but have the isospin of the ground state. This strong transition can be com- pared to the strong beta decay of '2N and 12B to the 12.7 MeV J'=l+ T=0 level in 12c, with a B(GT) of about 2.0 [log(ft)z3.5].”'“ This level lies 2.4 MeV below the J'=l+, T:1 analog state in 12C. Assuming that our narrow state has a J'={-‘, T=3'- assignment, and that it is the level seen by Esterl et al.‘ at 12.1 MeV, the sep- aration between this level and the analog 1': i“, T =% level at 14.7 MeV in 9B is 2.6 MeV. This is nearly the same separation as that between the T=0 and T=1 1': 1+ levels in "C! VIII. SUMMARY We have observed the population of the ground state and two low lying excited states in 9B from the beta de- cay of 9C, by implanting the radioactive ions in solid state detectors. A technique was deve10ped to remove the beta energy loss contamination of the spectrum, al- lowing for the use of much thicker detectors than has been previously possible. The branching ratios to these levels have been determined, and the results compared to several measurements of the decay of the mirror nucleus 9Li. We have also compared the beta decay data for both nuclei and the 9Betp.n)°ll cross sections to shell model predictions. The 9population of a state near 12 MeV from the decay of C is confirmed, while a previ- 3_7 BRANCHING RATIOS OF ’C TO LOW LYING STATES IN ’B 779 ously reported peak at 9 MeV in the delayed proton data is not seen. This supports the previous suggestion that the 9 MeV peak was generated by a state at 12 MeV de- caying through the first excited state of 'Be.‘ Overall, the agreement between the present data, the mirror decay of °Li, the °Be(p,n)9B reaction, and shell model calculations is quite good. The most striking disagreement is between the mirror B(GT) values for 9C and 9Li decay to the lowest «37 level in 913 and 9Be, re- spectively. The three measurements for the °Li decay are all very close to each other, yet more than a factor of 2 greater than the value we observe for 9C decay. We have suggested a possible systematic misinterpretation of the ’Li decay data that might explain this observed asymmetry. The absolute difi'erence between the ob- served matrix elements is ten times greater than predict- ed by Towner,” who is able to account for the asym- metries observed in other systems, and may be the larg- est observed to date. There are many ambiguities that must be resolved be- fore the level assignment and B(GT) strengths can be un- derstood in mass 9. We feel that a new measurement of the beta decay of ”U is called for. The beta-delayed neu- tron spectrum of ’Li should be remeasured with the de- cay of the 9Li nuclei occurring within a silicon detector or between a pair of silicon detectors. This would pro- vide a highly efficient coincident detection of the delayed 2a energy. The complementary pair of excitation energy spectra in °Be and 9B from the decay of 9Li and 9C will allow for a more complete characterization of the beta decay strengths in this system. The reactions °Li(°Li,’ii) and ‘Li(‘Li,’Ile) may also be used to study the region near 12 MeV of excitation in 9B and 9Be. These reactions have small calculated spectro- scopic factors for the levels at low excitation, and large spectroscopic factors for states between 8 and 15 MeV of excitation in these nuclei. ACKNOWLEDGMENTS We would like to thank J. C. Hardy, D. J. Millener, and G. Bertsch for helpful comments on various subjects discussed in this paper. This work was supported in part by National Science Foundation Grant PHY 86- 11210. ‘ ‘Joseph Cerny and J. C. Hardy, Annu. Rev. Nucl. Sci. 17, 333 (1977). 2M. A. C. Hotchkis. J. E. Rein. D. J. Vieira. F. Blonnigen. r. F. Lang. D. M. Moltz. X. Xu, and Joseph Cerny, Phys. Rev. C 35. 315 (1987). 3J. C. I-Iardy, R. I. Verrall. R. Barton, and R. E. 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APPENDIX B APPENDIX B BETA DELAYED PARTICLE DECAY THRESHOLDS FOR LIGHT NUCLEI Neutron Rich Nuclei: 6HE :) 6LJ MAXIMUM EXCITATION ENERGY FROM BETA DECAY 3.51 a + d = 6L1 + 1.A8 MeV 8HE :> 8LI MAXIMUM EXCITATION ENERGY FROM BETA DECAY 10.65 7L1 + n = 8L1 + 2.03 MeV a + t + n = 8L1 + 9.50 5HE + t : 81.1 + 5.39 a + “H = 8L1 + 7.32 6L1 + 2n = 8L1 + 9.28 6HE + d : 8L1 + 9.78 8LI :> 8BE MAXIMUM EXCITATION ENERGY FROM BETA DECAY 16.00 23 = 8BE + -0.09 MeV 9L1 :) 9BE MAXIMUM EXCITATION ENERGY FROM BETA DECAY 13.61 23 + n = 983 + 1.57 MeV 88E + n = 98E + 1.67 SHE + a = 9BE + 2.47 11LI :> 113E MAXIMUM EXCITATION ENERGY FROM BETA DECAY 20.73 lOBE + n : 11BE + 0.50 MeV 98E + 2n = 118E + 7.32 6HE + a + n : 11BE + 7.91 7HE + a = 118E + 8.36 6HE + SHE : 11BE + 8.81 23 + 3n : 11BE + 8.89 8BE + 3n = llBE + 8.98 SHE + a + 2n = 11BE + 9.78 SHE + SHE + n = 11BE + 10.68 8L1 + t : 11BE + 15.72 7L1 + n + t : 11BE + 17.75 9LI + d : 11BE + 17.92 9LI + p + n : 11BE + 20.19 a + 2t + n = 11BE + 20.22 7L1 + “H : 113E + 20.57 11BE z) 118 MAXIMUM EXCITATION ENERGY FROM BETA DECAY 11.51 7L1 + a = 113 + 8.66 MeV 23 + t = 118 + 11.13 8BE + t = 113 + 11.22 108E + p : 118 + 11.23 103 + n = 113 + 11.45 12BE :> 128 MAXIMUM EXCITATION ENERGY FROM BETA DECAY 11.71 11B + n = 123 + 3.37 MeV (BA 8L1 + a = 128 + 10.00 198E => ‘HH3 MAXIMUM EXCITATION ENERGY FROM BETA DECAY 17.36 138 + n = 198 + 0.97 MeV 128 + 2n = 198 + 5.85 118 + 3n = 198 + 9.22 118E + t : 198 + 11.96 9L1 + a + n = 198 + 11.79 108E + t + n = 198 + 11.96 10LI + a = 198 + 12.59 9L1 + SHE : 198 + 12.68 128E + d : 198 + 19.55 108E + 9H : 198 + 19.78 8L1 + 6HE : 198 + 19.87 8L1 + a + 2n = 198 + 15.85 8L1 + SHE + n : 198 + 16.79 12BE + p + n : 198 + 16.77 7L1 + 6HE + n = 198 + 16.91 7L1 + 7HE : 198 + 17.35 128 => 12C MAXIMUM EXCITATION ENERGY FROM BETA DECAY 13.37 33 = 12C + 7.27 MeV 88E + a = 12C + 7.37 138 :> 13C: MAXIMUM EXCITATION ENERGY FROM BETA DECAY 13.99 120 + n = 13C + 9.95 MeV 98E + a = 13C + 10.65 33 + n = 13C + 12.22 88E + a + n = 13C + 12.31 SHE + 23 = 13C + 13.11 88E + SHE : 13C + 13.21 198 2) 19C MAXIMUM EXCITATION ENERGY FROM BETA DECAY 20.69 13C + n = 19C + 8.18 MeV 108E + a = 19C + 12.01 12C + 2n = 190 + 13.12 98E + a + n : 19C + 18.82 6HE + 2a = 19C + 19.92 88E + 6HE : 19C + 19.51 98E + SHE = 190 + 19.72 3a + 2n = 19C + 20.90 88E + a + 2n = 19C + 20.99 118 + t : 19C + 20.60 158 :> 150 MAXIMUM EXCITATION ENERGY FROM BETA DECAY 19.10 190 + n = 150 + 1.22 MeV 13C + 2n : 15C + 9,39 1188 + a = 15C + 12.73 108E + a + n = 15C + 13.23 108E + SHE = 150 + 19.12 12C + 3n = 15C + 19.39 128 + t = 15C + 18.95 98E 4 6HE : 15C + 19.07 185 17B 16C 17C 18C 19C 20C => 17C 16C 128E + 1088 + 118E 108E 13B 1188 1088 + + + + + z) 16N 15N :) 17N 16N => 18N 17N :) 19N 18N => 20N 19N 16C + MAXIMUM EXCITATION ENERGY FROM BETA DECAY + 15C 19C 13C a 1288 11BE 6HE 19B 108E a a t SHE SHE 12C 158 138 98E n + + + + + + + + + + + + + + + + + + 2n 3n 9n n SHE 6HE n t 7HE 2n 3n n n 2n 5n d 9H 8H5 17C + 0.79 MeV = 17C + 9.99 = 17C + 6.20 : 17C + 19.38 : 170 + 19.59 = 17C + 15.99 : 170 + 10.79 = 17C + 17.29 = 17C + 17.58 : 17C + 17.69 = 17C + 17.71 : 17C + 18.22 = 17C + 18.55 : 17C + 18.61 : 17C + 19.11 = 17C + 19.32 = 17C + 21.08 : 170 + 21.37 = 17C + 21.92 MAXIMUM EXCITATION ENERGY FROM BETA DECAY + n 16N + 2.99 MeV MAXIMUM EXCITATION ENERGY FROM BETA DECAY + lSN 19C 13B 16C I) + + + + I) '0 w U‘KDM 17N + 5.88 MeV : 17N + 8.37 : 17N + 10.10 : 17N + 11.12 : 17N + 13.11 MAXIMUM EXCITATION ENERGY FROM BETA DECAY + 16N 15N 15C 17N 16C 16N 158 18C 15N 18N 17N 17C 16M 1') MAXIMUM 11 MAXIMUM n + + + + + + + + + + + 2n 6 3n a D An 2n 3n t n 9n 18N + 2.83 MeV 18N + 8.71 18N + 11.20 18N + 11.71 EXCITATION ENERGY 19N + 5.32 MeV = 19N + 8.19 : 19" + 12.77 : 19" + 19.02 : 19N + 15.52 : 19N + 16.31 : 19N + 16.51 EXCITATION ENERGY 20N + 1.89 MGV 2ON + 7.16 20N + 9.98 20N + 13.88 20N + 19.62 20N + 15.87 186 FROM BETA DECAY FROM BETA DECAY 22.98 8.01 13.16 11.77 16.89 15.93 18C + d 2 20M + 15.93 16N => 16C) MAXIMUM EXCITATION ENERGY FROM BETA DECAY 10.92 12C + a = 160 + 7.16 MeV 17N =) 17C) MAXIMUM EXCITATION ENERGY FROM BETA DECAY 8.68 160 + n = 170 + 9.19 MeV 13C + a = 170 + 6.36 18N :> 180 MAXIMUM EXCITATION ENERGY FROM BETA DECAY 13.90 19C + a = 180 + 6.23 MeV 170 + n = 180 + 8.09 160 + 2n = 180 + 12.19 19N :> 190 MAXIMUM EXCITATION ENERGY FROM BETA DECAY 12.59 180 + n = 190 + 3.96 MeV 15C + a : 190 + 8.97 19C + a + n = 190 + 10.18 19C + SHE = 190 + 11.08 170 + 2n = 190 + 12.00 2ON z) 200 MAXIMUM EXCITATION ENERGY FROM BETA DECAY 18.30 190 + n = 200 + 7.61 MeV 180 + 2n = 200 + 11.56 16C + a = 200 + 12.32 15C + a + n = 200 + 16.57 19C + 6HE : 200 + 16.82 15C + SHE : 200 + 17.97 19C + a + 2n = 200 + 17.79 21N z) 210 MAXIMUM EXCITATION ENERGY FROM BETA DECAY 17.92 200 + n = 210 + 3.79 MeV 190 + 2n = 210 + 11.39 180 + 3n : 210 + 15.30 17C + a = 210 + 15.32 16C + a + n = 210 + 16.06 16C + SHE : 210 + 16.95 190 => 19F MAXIMUM EXCITATION ENERGY FROM BETA DECAY 9.82 15N + a = 19F + 9.01 MeV 210 :) 21F' MAXIMUM EXCITATION ENERGY FROM BETA DECAY 8.18 20F + n = 21F + 8.10 MeV 220 => 22F MAXIMUM EXCITATION ENERGY FROM BETA DECAY 6.61 21F + n = 22F + 5.19 MeV 230 => 23F MAXIMUM EXCITATION ENERGY FROM BETA DECAY 19.11 22F + n = 23F + 7.55 MeV 21F + 2n = 23F + 12.79 220 + p : 23F + 13.38 187 20F 22F 23F 29F 25F :) => => z) z) 20NE 160 + a = 20NE + 9.73 MeV 22NE MAXIMUM EXCITATION ENERGY FROM BETA DECAY 180 + a = 22NE + 9.67 MeV 21NE + n = 22NE + 10.36 23NE MAXIMUM EXCITATION ENERGY FROM BETA DECAY 22NE + n = 23NE + 5.20 MeV 29NE MAXIMUM EXCITATION ENERGY FROM BETA DECAY 23NE + n = 29NE + 8.87 MeV 200 + a = 29NE + 12.17 22NE + 2n = 29NE + 19.07 25NE MAXIMUM EXCITATION ENERGY FROM BETA DECAY 29NE + n = 2SNE + 9.28 MeV 210 + a = 2SNE + 12.71 23NE + 2n : 25NE + 13.15 MAXIMUM EXCITATION ENERGY FROM BETA DECAY 188 7.03 10.86 8.51 19.70 19.70 Neutron Deficient Nuclei: 88 :) 88E MAXIMUM EXCITATION ENERGY FROM BETA DECAY 16.96 a + a = BBB + -0.09 MeV 9C :) 9B MAXIMUM EXCITATION ENERGY FROM BETA DECAY 15.98 a + p + a = 98 + -0.28 MeV 88E + p = 98 + -0.19 SLI + a = 98 + 1.69 12N z) 12C MAXIMUM EXCITATION ENERGY FROM BETA DECAY 16.32 a + a + a = 12C + 7.27 MeV 88E + a = 12C + 7.37 118 + p : 12C + 15.96 130 :> 13N MAXIMUM EXCITATION ENERGY FROM BETA DECAY 16.79 12C + p = 13N + 1.99 MeV a + a + a + p = 13N + 9.22 88E + p + a = 13N + 9.31 98 + a : 13N + 9.50 SLI + a + a = 13N + 11.18 SLI + 88E = 13N + 11.28 17NE z) 17F MAXIMUM EXCITATION ENERGY FROM BETA DECAY 13.51 160 + p = 17F + 0.60 MeV 13N + a = 17F + 5.82 12C + p + a = 17F + 7.76 12C + SLI : 17F + 9.73 15N + p + p = 17F + 12.73 20NA :) 20NE MAXIMUM EXCITATION ENERGY FROM BETA DECAY 12.87 160 + a = 20NE + 9.73 MeV 12C + a + a = 20NE + 11.90 12C + 888 = 20NE + 11.99 19F + p : 2005 + 12.85 20MG z) 20NA MAXIMUM EXCITATION ENERGY FROM BETA DECAY 9.71 19NE + p = 20NA + 2.20 MeV 150 + p + a = 20NA + 5.73 16F + a = 20NA + 6.26 150 + SLI : 20NA + 7.69 18F + p + p = 20NA + 8.61 21MG :> 21NA MAXIMUM EXCITATION ENERGY FROM BETA DECAY 12.08 20NE + p : 21NA + 2.93 MeV 17F + a : 21NA + 6.57 160 + p + a = 21NA + 7.17 160 + SLI : 21NA + 9.13 22A]. :2) 22MG MAXIMUM EXCITATION ENERGY FROM BETA DECAY 17.91 21NA + p = 22MG + 5.50 MeV 20NE + p + p : 22MG + 7.93 189 160 23AL 29 AL 2981 2581 27F 28F 28S 298 31€L 32CL 18NE + a = 22MG + 8.19 17F + p + a = 22MG + 12.06 + p + a + p : 22MG + 12.66 190 + a + a = 22MG + 13.25 190 + 88E : 22MG + 13.39 17F + SLI : 22MG + 19.03 160 + 68E = 22MG + 19.03 160 + p . 5L1 : 22MG + 19.63 12C + 10C : 22MG + 16.10 19NE + 3HE : 22MG + 17.08 :> 23MG MAXIMUM EXCITATION ENERGY FROM BETA DECAY 22NA + p : 23MG + 7.58 MeV 19NE + a = 23MG + 9.65 :> 29MG MAXIMUM EXCITATION ENERGY FROM BETA DECAY 20NE + a = 29MG + 9.31 MeV 23NA + p = 29MG + 11.69 =) 29AL MAXIMUM EXCITATION ENERGY FROM BETA DECAY 23MG + p : 29AL + 1.87 MeV 20NA + a = 29AL + 9.32 22NA + p + p e 29AL + 9.95 :> 25AL MAXIMUM EXCITATION ENERGY FROM BETA DECAY 29MG + p : 25AL + 2.27 MeV 21NA + a = 25AL + 9.15 20NE + p + a e 25AL + 11.58 :) 2731 MAXIMUM EXCITATION ENERGY FROM BETA DECAY 26AL + p = 2731 + 7.96 MeV 23MG + a = 2731 + 9.39 :> 2851 MAXIMUM EXCITATION ENERGY FROM BETA DECAY 29MG + a = 2881 + 9.98 MeV 27AL + p = 2881 + 11.58 z) 281? MAXIMUM EXCITATION ENERGY FROM BETA DECAY 2731 + p : 28F + 2.06 MeV 26AL + p + p = 28? + 9.53 29AL + a = 28? + 9.53 :) 29P MAXIMUM EXCITATION ENERGY FROM BETA DECAY 2831 + p : 29? + 2.75 MeV 25AL + a = 29P + 10.96 29MG + p + a = 29? + 12.73 :> 315 MAXIMUM EXCITATION ENERGY FROM BETA DECAY 30F + p = 313 + 6.13 MeV 2781 + a = 313 + 9.08 :) 32S MAXIMUM EXCITATION ENERGY FROM BETA DECAY 2831 + a = 323 + 6.95 MeV 190 11 12 11 10. 13 10. 12. 10 11 .22 .86 .79 .72 .31 27 77 .95 .66 32AR 3388 35K 36K 36CA 37CA 38CA 903C 90TI 91TI FROM BETA DECAY FROM BETA DECAY FROM BETA DECAY EXCITATION ENERGY FROM BETA DECAY FROM BETA DECAY FROM BETA DECAY 31P + p : 323 + 8.86 :) 32CL MAXIMUM EXCITATION ENERGY 313 + p : 32CL + 1.57 MeV 30? + p + p = 32CL + 7.70 28F + a : 32CL + 8.59 z) 33CL MAXIMUM EXCITATION ENERGY 328 + p : 33CL + 2.28 MeV 29P + a : 33CL + 6.98 2851 + p + a = 33CL + 9.23 :) 35AR MAXIMUM EXCITATION ENERGY 39CL + p = 35AR + 5.90 MeV 313 + a : 35AR + 6.93 :> 36AR MAXIMUM EXCITATION ENERGY FROM BETA DECAY 323 + a : 36AR + 6.69 MeV 3SCL + p : 36AR + 8.51 :) 36K MAXIMUM 3503 + p = 36K + 1.67 MeV 32CL + a = 36K + 6.52 39CL . p + p = 36K + 7.56 318 + p + a = 36K + 8.09 :) 37K MAXIMUM EXCITATION ENERGY FROM BETA DECAY 36AR + p : 37K + 1.86 MeV 33CL + a = 37K + 6.22 328 + p + a = 37K + 8.50 35CL + p + p : 37K + 10.36 323 + 5L1 : 37K + 10.96 :> 38K MAXIMUM EXCITATION ENERGY FROM BETA DECAY 37AR + p = 38K + 5.19 MeV :) 90CA MAXIMUM EXCITATION ENERGY FROM BETA DECAY 36AR + a = 90CA + 7.09 MeV 39K + p : 90CA + 8.33 :) 903C MAXIMUM EXCITATION ENERGY 39CA . p = 1030 + 0.51 MeV 36K + a = 903C + 5.53 38K + p + p = 903C + 6.30 35AR + p + a = 908C + 7.19 35AR + 5L1 : 903C + 9.16 :> 915C MAXIMUM EXCITATION ENERGY 90CA . p = u1sc + 1.09 MeV ‘ 37K + a = 913C + 6.27 36AR + p + a : 913C + 8.13 39K + p + p = 915C + 9.91 36AR + 5L1 : 913C + 10.09 191 10. 10. 10. 11 10. 13. 11 13 6O 86 .78 .96 62 .72 30 .99 .92 FROM FROM FROM FROM FROM FROM FROM BETA BETA BETA BETA BETA BETA BETA DECAY DECAY DECAY DECAY DECAY DECAY DECAY 92TI => 923C MAXIMUM EXCITATION ENERGY FROM BETA DECAY 91CA + p : 925C + 9.27 MeV 38K + a = 925C + 5.75 93TI :> 9330 MAXIMUM EXCITATION ENERGY FROM BETA DECAY 39K + a = 935C + 9.81 MeV 9201 + p = 938C + 4.93 93V => 93TI MAXIMUM EXCITATION ENERGY 39CA + a : 93TI + 9.97 MeV 923C + p : 93TI + 9.99 91CA + p + p : 93TI + 8.76 90CA + 3HE : 93TI + 9.91 38K + p + e 93TI + 10.23 99V z) 99TI MAXIMUM EXCITATION ENERGY 90CA + a = 99TI + 5.13 MeV 933C + p : 99T1 + 8.65 36AR + a + a = 99T1 + 12.17 99CR :> 99V MAXIMUM EXCITATION ENERGY 93TI + p = 99V + 1.77 MeV 903C + a = 99V + 5.70 39CA + p + a = 99V + 6.29 923C + p + p = 99V + 6.25 39CA + 5L1 = 99V + 8.20 950R :) 95V MAXIMUM EXCITATION ENERGY 99TI . p : usv + 1.61 MeV 913C + a = 95V + 5.66 90CA + p + a = 95V + 6.79 90CA + SLI : 95V + 8.71 u3sc + p + p : usv + 10.26 96CR => 96V MAXIMUM EXCITATION ENERGY 95TI + p = 96V + 5.36 MeV 9708 =) 97V MAXIMUM EXCITATION ENERGY 96TI + p = 97V + 5.17 MeV 96MN :) 96CR MAXIMUM EXCITATION ENERGY 95V + p : 96CR + 9.89 MeV 99TI + p + p : 96CR + 6.50 92TI + a = 96CR + 6.77 918C + p + a = u6cn + 10.51 90CA . p + a + p e 96CR + 11.63 38CA + a + a = 96CR + 12.26 913C + SLI : 96CR + 12.51 90CA + p + SLI = 96CR‘+ 13.59 93TI + 3HE = 96CR + 15.08 938C + p + p + p = 96CR + 15.15 192 10. 12. 11 15. .98 .85 38 73 .56 .39 .58 .93 98 97MN 98MN 99FE 50FE 51FE 51CO 52CO 53NI 59NI 55NI 56CU 57CU => => => :) => => => :> => => => => 1170 95TI 98C R 96V 4. R 99TI 99MN 99TI 97V 98CR + + 50MN 51 51 99CR 99CR MN 5OCR FE SOMN ... 52FE 50CR 51MN + 53CO 51MN 98CR 52FE + ... 59CO 53FE 55CO 59FE 56NI 59FE 55CO + 57NI 56CO MAXIMUM EXCITATION ENERGY FROM BETA DECAY BETA BETA BETA BETA BETA BETA BETA BETA BETA BETA DECAY DECAY DECAY DECAY DECAY DECAY DECAY DECAY DECAY DECAY + p = 97CR + 9.77 MeV 93TI + a = 97CR + 7.66 p + p = 97CR + 10.12 MAXIMUM EXCITATION ENERGY FROM BETA DECAY + a = 98CR + 7.69 MeV 97V + p : 98CR + 8.10 MAXIMUM EXCITATION ENERGY FROM + p = 99MN + 2.08 MeV 95V + a = 99MN + 8.16 p + a = 99MN + 9.78 p + p : 99MN + 10.18 99TI + SLI : 99MN + 11.79 MAXIMUM EXCITATION ENERGY FROM + p = SOMN + 9.59 MeV MAXIMUM EXCITATION ENERGY FROM + p = 51MN + 5.27 MeV MAXIMUM EXCITATION ENERGY FROM + p = 51FE + 9.88 MeV 97CR + a = 51FE + 8.09 p + p = 51FE + 9.97 MAXIMUM EXCITATION ENERGY FROM + p = 52FE + 7.38 MeV 98CR + a = 52FE + 7.99 p + p = 52FE + 12.65 MAXIMUM EXCITATION ENERGY FROM + p = 5300-. 1.60 MeV 99MN + a = 53CO + 7.95 p + p : 53CO + 8.98 p + a = 53CO + 9.59 98CR + SLI : 53CO + 11.5O MAXIMUM EXCITATION ENERGY FROM + p = 59CO + 9.35 MeV MAXIMUM EXCITATION ENERGY FROM + p = 55CO + 5.06 MeV MAXIMUM EXCITATION ENERGY FROM + p = 56NI + 7.17 MeV 52FE + a : 56NI + 8.00 p + p = 56NI + 12.23 MAXIMUM EXCITATION ENERGY FROM + P = 57NI + 7.33 MeV 53FE + a = 57NI + 7.56 193 10.88 12.58 12.12 7.00 11.78 13.01 12.21 7.78 7.67 19.38 7.68 LIST OF REFERENCES AGUILAR-BENITEZ88 AJZENBERG-SELOVE86 LIST OF REFERENCES Aguilar-Benitez, M. et al. 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