MODEL DEPENDENCE OF THE SENDING ENERGY OF NUCLEAR MATTER Thesis for the .Degree of Ph. D. MICHTGATT STATE UNIVERSITY MECHAEL DAVID MILLER 1 9 6 9 This is to certify that the thesis entitled MODEL DEPENDENCE OF THE BINDING ENERGY OF NUCLEAR MATTER presented by Michael David Miller has been accepted towards fulfillment of the requirements for Ph. D. degree mm Major profes ~ . r Due September 16, 1969 0-169 1’ amnma av ‘5 P am a sun: ,- BODK BINDERY INC. 5 - Inna-v .Iunrnc ABSTRACT MODEL DEPENDENCE OF THE BINDING ENERGY OF NUCLEAR MATTER BY Michael David Miller It is possible to produce hard core, soft core, finite core, and momentum-dependent potentials which fit the two- nucleon elastic scattering data equally well and which are, therefore, indistinguishable from that standpoint. This thesis examines the feasibility of using nuclear matter calculations to distinguish between potentials which cannot be distinguished through two-nucleon elastic scattering experiments. Using the method described by Brueckner and Masterson for the static potentials and, with modifications, for the momentumedependent potentials, the mean binding energy per nucleon in nuclear matter is calculated for each of the S, P, and D states using several phenomenological two-nucleon potentials which have identical on-energy-shell matrix elements. The binding energy is found to be very sensitive to the form of the short range repulsion in the potential. Replacing a hard core potential by a short range momentum- dependent one having identical on-energy-shell matrix elements is found to increase the contribution of the S and P states to the binding energy of nuclear matter. There was little change in the contribution of the D states. With the proper Michael David Miller form for the momentum dependence, increases in the binding energy of over 12 MeV per particle were obtainable. The hard core Hamada-Johnston potential predicts a binding energy of 8.5 MeV per particle for a Fermi momentum of 1.4 F-l. This is fairly typical of hard core potentials which fit the two-nucleon elastic scattering data fairly well, but is in poor agreement with the empirical value of 16 MeV per nucleon. Replacing the S state potential alone by an equivalent momentum—dependent one results in a cal— culated value for the binding energy which is in excellent agreement with the empirical value. K. A. Brueckner and K. S. Masterson, Jr., Phys. Rev. 128, 2267, (1962). MODEL DEPENDENCE OF THE BINDING ENERGY OF NUCLEAR MATTER BY Michael David Miller A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1969 60/77? 1/~27~70 ACKNOWLEDGMENTS I would like to thank Professor Peter S. Signell for his help in selecting and solving this problem. For her help in the preparation of the manuscript, I would like to thank my wife who, along with my children, Matthew and Marna, also provided greatly appreciated companionship. ii TABLE OF CONTENTS LIST OF TABLES .0...0.0.0.0.0..........OOOOOOOOOCOOOOOO LIST OF FI GURES 0.00.0.0...0..........OOOOOOOOOOOOOO... I. INTRODUCTION 00......O.........OOOOOOOOOOOOOOOOO. II. THEO A. B. C. RY 00.0.0000.........CCOOOOOOOOOO00.0.0000... PrOperties of Nuclear Matter ................ Summary of Brueckner-Goldstone Theory ....... Baker TranSfom ............OOOOOOOOOOOOOOOOO III. POTENTIALS ..........OOOOOOOOOOOOOOOOO00.0.0.0... IV. CALC A. B. C. ULATIONS I.........OOOOOOOOOOOOOOO0.0.0.0.... Static Potential Calculations ............... Momentum-Dependent Potential Calculations ... Phase Shift Calculations .................... V. RESULTS 0.000000000000000.........OOOOOO......OOO APPENDIX . REFERENCES iii iv \o \o H < 11 22 30 39 39 44 50 51 68 74 Table II. III. IV. LIST OF TABLES Page 1 )5 for various phenomenological two- nucleon potentials............................ 4 Ramada-Johnston potential parameters.......... 38 Mean bipding energy per nucleon predicted by the So potentialSOOOOOOOOOOOOO0.0.0.000... 53 Percent increase of the 1S K matrix elements over the values ogtained with the 0.4 F hard core potential................. 55 Mean binding energy per nucleon predicted by momentum-dependent potentials generated from the Ramada-Johnston potential............ 56 iv Figure 1. LIST OF FIGURES Page 1S phase shifts predicted by the static pogentials and the values deduced from two-nucleon scattering experiments............ 7 Static lS0 potentials......................... 31 Momentum-dependent 180 potentials. ......... ... 34 Momentum-dependent lD2 potentials. ............ 59 Momentum-dependent 3Dl potentials. ........... . 60 Momentum-dependent 3D2 potentials ....... . ..... 61 1Pl phase shift vs. energy.................... 63 3P0 phase shift vs. energy.................... 64 3Pl phase shift vs. energy........ ....... ..... 65 3P2 phase shift vs. energy...... ..... .. ....... 66 SECTION I INTRODUCTION Ever since the discovery that the atomic nucleus was composed of neutrons and protons, there have been attempts to explain the properties of nuclei in terms of the inter- action of these particles. Although it is now clear that an accurate theory of the nucleus will involve more than the nonrelativistic Schroedinger equation for neutrons and protons interacting through a two-nucleon potential, this is at least a good starting point. Even if the two body forces prove to be unable to account for all of the nuclear data, their failure should at least provide some clue as to the nature of the many body forces. The two-nucleon potential need only be valid up to about 350 MeV in the laboratory frame. Above this energy pion production becomes significant and the use of the non- relativistic Schroedinger equation is probably not justified. However, since the tap of the Fermi sea in nuclear matter corresponds to an energy of only about 160 MeV, the 350 MeV upper limit on the theory should not be too restrictive., It is generally agreed that if a two-nucleon potential is to fit the two-nucleon elastic scattering data, it must assume the form of the one-pion-exchange potential at large 1 2 distances. At small distances it must have some form of short range repulsion, at least in the S and 3Po states. Within this broad framework there have been many attempts to produce a purely phenomenological two-nucleon potential which not only fit the two-nucleon elastic scattering data, but also produced reasonably good results in nuclear cal- culations. The phenomenological potentials are best classified according to the form used for the short range repulsion. The most common types have been the infinitely repulsive hard core, the finite core, the soft core, and the momentum- dependent repulsion. The ability of these potentials to describe the two-nucleon interaction has differed widely. The best way of comparing the quality of one phenomenolog- ical potential with another is to compare the goodness-of- fit parameter, :k‘, obtained when each of the potentials is used to predict the same set of experimental data. For N data, 1‘. fi<fir£1f Ila] 6" where 6" is the experimental standard deviation associated with the experimental datum E; , and I"; is the value pre- dicted by the potential.1 The closer the predicted values are to the experimental values, the smaller 1‘ will be. The more accurate data with their lower uncertainties will be more senSitive to variations in the potential parameters than the less accurate data. 3 Table I shows some of the more pOpular potentials and their associated If values for the current set of statisti- cally valid two-nucleon elastic scattering data below 350 MeV. The Bressel-Kerman finite core potential2 seems to be the one preferred by the elastic scattering data. However, the hard core Ramada-Johnston potential3 is five years older and is still almost as good a fit to the data. Probably only some small changes in the values of the potential parameters in order to account for the additional, more accurate data would result in very close agreement between these two potentials. In fact, previous work with the very accurate data available at 210 MeV4 has shown that at that energy the majority of the difference in the two potentials could be eliminated by changes in the 3P2 state alone. Presumably, similar results would be found at other energies. Thus, at the present time, the two-nucleon elastic scattering data does not appear to be able to distinguish between properly parameterized hard core and finite core potentials. This was emphasized for the 180 state when a family of hard, soft, and finite core potentials was gener- 1 ated, each of which gave a precise fit to the So phase shifts deduced from the elastic scattering data below 330 MeV.5 The same thing could probably be done for the other states. Furthermore, Baker has shown that given any hard core potential, a family of momentum-dependent potentials can be generated which will produce elastic scattering phase shifts which are identical at all energies to those 2 Table I. x.for various phenomenological two-nucleon potentials. ,_ a Potential Type 2,’ for X for 648 pp 952 np data data Bressel-Kermana finite core 1382 2031 Hamada—Johnstonb hard core 1929 2149 Bethe-Reidc hard core 1763 4249 Yaled hard core 2471 2511 a Reference 2. Reference 3. c d R. V. Reid, quoted in Bhargava and Sprung, Reference 2. K. E. Lassila, M. H. Hull, Jr., H. M. Ruppel, F. A. MacDonald, and G. Breit, Phys. Rev. 126, 881 (1962). 5 produced by the original hard core potential.6 It has been suggested that the many-body problem might be helpful in distinguishing between potentials that are identical from the standpoint of two-nucleon elastic scat- tering.7 The advantage of the many-body problem is that it allows for scattering off the energy shell. This is scat- tering in which the two-body interaction occurs between two particles in excited states at an energy different from that of the initial and final states. There have been previous attempts to use nuclear matter calculations to distinguish between hard core and momentum-dependent potentials.8 Although the results indi- cated a desirable increase in binding for the momentum- dependent potentials, they were not conclusive qualitatively. For one thing, the potentials used did not have identical on-energy-shell matrix elements, although it is clear from Baker's work that it is possible to develop potentials of these two forms that do. Thus one was left to wonder whether the nuclear matter calculations merely emphasized the on-energy-shell differences in the potentials, which could be distinguished through sufficiently accurate elastic scattering experiments, or if the differences in the nuclear matter calculations were really due to off-energy-shell processes, which are not involved in two-body scattering. To further complicate matters, the momentumrdependent potential calculations were carried out using perturbation theory while the hard core potentials were treated using Brueckner's method. The different approximations made in the two methods alone could easily result in discrepancies of over 2 MeV per nucleon in the average binding energy, which is more than the difference between some models when a consistent procedure is used. Both of these sources of uncertainty have been avoided here. In the treatment of the 1 So state alone, the momen- tum-dependent potentials were derived from the 0.4 F hard core 180 potential of Reference 5 using Baker's transform- ation. As shown in Figure l, the 180 phase shifts below 330 MeV obtained from the soft core and finite core poten- tials.agree with those obtained from the 0.4 F hard core potential to within a small fraction of the standard devia- tion obtained from the experimental data. Above 330 MeV they begin to diverge slowly. Of course, the 180 phase shifts obtained from the momentum-dependent potentials are identical to those produced by the 0.4 F hard core potential at all energies. So for use in nuclear matter calculations these potentials can be considered to have identical on- energy-shell matrix elements. In working with the other states, the momentum-dependent potentials were obtained from and compared with the Hamada-Johnston hard core poten- tial alone. In calculating the binding energy for nuclear matter, the method described by Brueckner and Masterson9 was used for all of the potentials. The modifications made in treating the momentumrdependent potentials involved no 60'- ’30 Phase 3min (Degrees) 1 A 1 h 1 O '00 200 300 EMB (MeV) Figure l. 1S phase shifts predicted by the static potentials and the values deduced from two-nucleon scattering experi- ments. See Reference 5. 8 additional approximations above those involved in the calculations using the other potential forms. Thus the results obtained here should be an accurate quantitative description of the ability of nuclear matter to distinguish between potentials that cannot be distinguished through two- nucleon elastic scattering experiments. SECTION II THEORY A. Properties of Nuclear Matter Nuclear matter is a hypothetical system of % neutrons and % protons where A is allowed to become infinitely large. If the coulomb repulsion of the protons is ignored, it is thought that in its lowest energy state a stable configura- tion would result, characterized by a uniform density, /4 , and a mean binding energy per particle, «k . Because nuclear matter is assumed to be homogeneous and isotropic, the single particle wave functions of nuclear matter are just plane waves. Thought of as a first approximation to a heavy nucle- us, nuclear matter provides a medium in which the ability of a two-nucleon potential to predict some of the properties of a nucleus can be tested without all of the difficulties inherent in the calculation of the prOperties of a finite nucleus. A potential which fails to satisfactorily predict the binding energy and density of nuclear matter need not be dragged through a finite nucleus calculation. The average binding energy per particle that nuclear matter would be expected to have if it really existed is deduced from the semi-empirical mass formula.10 According to this formula, the binding energy of a nucleus is 10 -£ r s _ ‘ _, E’quqfatflr*aJ(‘§‘Z)‘/4‘+q,Z/4:+asfl (II’l) The first term is the volume energy term, the second a correction to account for surface tension, the third is the symmetry energy. The fourth term represents the contribution of the coulomb repulsion between protons, and the last term is a small correction which accounts for pairing effects. With properly chosen constants, this equation predicts the binding energies of nuclei with a standard deviation of 2.61 MeV per nucleon.11 However, the deviations from the mean values have systematic trends which are evidence of shell structure and no attempt was made to account for this prOperty of nuclei in equation (II-l). For nuclear matter,the number of nucleons, A, is infin- itely large, so the surface and pairing effects are negli- gible. As stated before, coulomb effects are ignored, and equal numbers of neutrons and protons are assumed. The result of these assumptions is that the expected binding energy of nuclear matter is just the volume energy term of the semi-empirical mass formula. This quantity has been 11, 12 determined to be about -16 MeV per nucleon. The corre- sponding energy for finite nuclei is about -8 MeV per par- 13 The difference is due solely to the inclusion of tic1e. coulomb, surface, symmetry, and pairing effects in finite nuclei. In addition to the binding energy of nuclear matter, the saturation density must also be calculated. The latter 11 is determined from potentials by plotting the binding energy as a function of particle density. The minimum of the curve determines the saturation density and the asso- ciated binding energy. These can then be compared with the empirically determined quantities. Starting from the density of the interior of a heavy nucleus and taking into account the effects of coulomb repulsion and surface tension, the saturation density of nuclear matter has been determined to be about 0.170 F-3.14 Because nuclear matter is treated as the ground state of a Fermi gas which is perturbed by two- body interactions between the nucleons, a more convenient quantity to use in describing the density of nuclear matter is the Fermi momentum, 4, , which is related to the satura- tion density, /0, , by 4 3 4 3 NEG-’57“) =23; F./ B. Summary of Brueckner-Goldstone Theory15 The theory of nuclear matter begins with A nucleons composing a Fermi gas in its ground state, which is assumed to be non-degenerate. The density of the Fermi gas is,/“‘fig' where.n.is the volume of a very large box in which all the nucleons are contained. If the particles do not interact, then the energy of the distribution is just the sum of the kinetic energies of the particles 12 To obtain the ground state of nuclear matter, a two- nucleon potential, PQ', acting between states i and‘/ is introduced. The total Hamiltonian then becomes .4 ._ .4 Isl I", "’ which can be written H= xx, r-M (“‘2’ where The quantity Va.) is the single particle potential acting on particle i and is dependent only on the momentum of the particle on which it acts. It is introduced only in order to simplify the calculation by improving the convergence of the series expansion and will have no effect on the energy of the distribution. The actual form of the single particle potential will be discussed later. If f is the perturbed ground state wave function, then the energy per particle, ET, is found by solving #1}: E? (II-3) For the unperturbed ground state this reduces to 444.2225. where ‘2? is a Slater determinant of the single particle states, y; . The normalizations of 1p and fl are chosen 13 such that <£/¢>:/ <é/f>=/ Then from equations (II-2) and (II-3) OWE/11>“; sér/mxf> CHM/14> giving 5- E. = «UM/15> Equation (II-3) can be rewritten (£-M)/f>=H,/1V.> (II-4) Adding the homogeneous equation {(5. ”If. M.>=o to the right hand side of equation (II-4), one obtains M»- : ,Z‘M. > 4W) = /¢2>-—b— E ~5°/2£> —_:’,;~///1/7> Then using equation (II-4), /1P>= M. 54,, ,_.—_’ -—-—;-//.//> = /fi,>+¢._l:o H/f) (II-5) where Pa /- /[.><é/ is an operator which prevents the unperturbed ground state from occuring as an intermediate state. Using equation (II—5) in equation (II-4) and l4 multiplying on the left by (é/ , the energy shift is found to be 4—15. =<95./d/é.>+ @Mg—fi- MM} which can be expanded as a perturbation series in /£ giving 55.: <é/M/4:> +<¢2///, 5/; hf/é.>+-~ (11-6) to second order. At this point it becomes more convenient to let the subscripts refer to states rather than particles. Subscripts a through j refer to states above the Fermi sea, k through n to states in the Fermi sea. The rest refer to any state. Sums over these subscripts imply sums over the entire range of states covered by the subscripts. Since each momentum state below the Fermi momentum will contain two neutrons and two protons (one of each spin), there will be four distinct states of each momentum up to the tOp of the Fermi sea for a total of A filled states. A particle will have a momentum greater than A; only if it has been excited, leaving a hole in the state it originally occupied below the Fermi sea. Equation (II-6) can be rewritten as an expansion in terms of the single particle potential, V; and the two-body potential, 2%,. Then in the notation of second quantization (see Appendix) equation (II-6) becomes to first order I 5-2:. = 1L2 Ova/m» - 4-2: - 244/ w.» ...- ," n’" " (II-7) 15 l The factor 5» is needed because the summation is carried out over all states with the result that each distinct matrix element is counted twice. A problem immediately arises because of the strong short range repulsion of many two-nucleon potentials. Matrix elements of these potentials will be extremely large and equation (II-7) will not converge even though the energy, shift, E'- Ea , may be small. Brueckner's solution to this problem was to replace the expansion in terms of the two- nucleon potential by an expansion in terms of the reaction matrix defined by K(W)= 7+, - 24., {-23} KM) (II-8) where Q is the Pauli Operator with the pr0perties CAFf> if]? and 7 are both excited states ($égff1: above the Fermi sea K. 5 otherwise and 807'): ((54+5f’ W)/ff> where h/ is the starting energy. The energy denominator is determined by calculating the sum of the particle energies minus the sum of the hole energies. The starting energy is obtained by subtracting é}+£}. from the result. Introducing the reaction matrix is equivalent to regrouping the matrix elements of the two-nucleon potential betWeen particles in the original eXpansion in such a way that each term in the new series makes a small enough con- tribution to the total energy that the series quickly l6 . converges. Interactions between holes are not included in this partial summation because their momenta are restricted to IfsA; whereas the momentum of a particle can be many times that. Hence the phase space available to the holes is much more limited than that available to the particles with the result that interactions between particles are expected to be the major contributors to the energy shift. The grouping of terms is probably more easily explained in terms of diagrams. Any series of interactions between particles of the form ————— + + ..-..- +0.0 -—-- -“ is replaced by one diagram containing the interaction repre- sented by The dashed line represents the two nucleon potential and the wavy line the reaction matrix. The diagrams to first order in the reaction matrix and single particle potential are The contribution of the first order diagram in the single particle potential cancels the corresponding term arising from lhfi regardless of the form of the single particle potential. The second of the first order diagrams 17 represents the sum of diagrams involving the twornucleon potential m: Q---Q £19:ng- The third is just the exchange diagram All second order diagrams of the reaction matrix are either redundant or do not conserve momentum, so they do not contribute to the energy of nuclear matter. For example, the diagram ---- “=0 0 + 0--Of . 0*... is clearly included in the first order diagram An example of a second order diagram that does not conserve momentum is 18 Since the state ca is above the Fermi sea and state re is in it, momentum conservation is clearly violated between the two interactions. It is possible to choose the single particle potential in such a way that most of the important three-body diagrams will cancel. If this is done the contribution of these and higher order diagrams to the energy of nuclear matter is 16 Even if expected to be only about 1 MeV per particle. some other form is chosen for the single particle potential, the contribution to the energy which is not included in the first order diagrams is not expected to exceed 4 MeV.16' 17 This is only about 10% of the total potential energy, and if it is not considerably more model dependent than the contri- bution of the first order diagrams it will not alter the conclusions drawn on the basis of the first order diagrams alone. Because only first order diagrams were considered here, the single particle potential was chosen in such a manner that the contribution of the two reaction matrix diagrams were cancelled by the diagram involving the single particle potential. That is <”/ V/”> 5 % <""'// ~ ;. ("M/k /nm> Thus the energy shift will be just the sum of the matrix elements of the unperturbed Hamiltonian, #4 , between all single particle states in the Fermi sea. This is just the sum of the kinetic energies and single particle potential energies of each particle in its lowest state. 19 In order to calculate the reaction matrix elements, it is convenient to define a correlated two-body wave function, g2 . The unperturbed two-body wave function is just the product of two unperturbed single particle wave functions. if g €53 E/ff> As stated before, the unperturbed single particle wave func- tions are just plane waves. If the nuclear matter under consideration is assumed to be enclosed in a large box of volume.fl-, then the single particle wave functions are " ..- 54,4 é(€)=-—7&;e so the two-body unperturbed wave functions are d -_. ... .J I £11,": ily'rs $03". =Jz- e e This can be rewritten $5 (‘ " - —-—-—’ . ' i f}. 4,4. ‘J'L e e (II-9) where =2’-<'P.'+7i) F: 7,"- K flaw/E 121(41‘27) These new quantities are the center of mass coordinate of m (II-10) ...-D the two particles, I? , and the total momentum K, . The other two quantities are the relative position and momentum. Equation (II-9) can be rewritten 20 _. 1?’ ¢( (4, .) = 35-6 ” 2%, (M?) (11-11) This defines g&}0§29 as that part of the unperturbed two- body wave function which is a function only of the relative position and momentum of the two particles. The two-body correlated wave function is then defined 4. a .444, 3 0‘ g”) £7 Multiplying both sides by the two-nucleon potential and using the definition of the reaction matrix, equation (II-8), results in -.w- 52 .. ”0“” ”60% ‘ KP}; So equation (II-12) can be rewritten Lazé --E? )«e V ,7, (II-13) Because the operators 3’, e , and C? all conserve total ...) momentum, the dependence of if” and ’7' on ..w. and I? will be the same. The correlated wave function can then be written J 75 '1? 2?. =fiL€ f 3/4, (II-14) where 7;}, is a function of the relative position vector and momentum only. 21 Using equations (II-ll) and (II-l4) in equation (II-l3), one finds, as expected, that the reaction matrix elements .3 are independent of J2. and K . (”UK/ff) .../4(7) 7/0: k) 3%, (F) I? (11-15) where gfi‘k 9,4, (ii/6,, (if?) Mr; I.) V” (.7ij (11-16) and 17-0-1?) 6’ (’3’ ’9- ##2e We I (11-17) I r ' J P 2n GAO) As stated previously, the total momentum, *9; , is conserved in the two-nucleon interaction; however, the relative .3 momentum, A , can change to some new value, I”. The Pauli Operator, C? , eliminates from the integrand all transitions _J .4 to occupied states in the Fermi sea. If k. and A3. are the momenta of particles 1 and 2 in some intermediate state, then 4:0 if k,k, and 43k.- where, by equation (II-10), ”a "/r a. A- The energy denominator, e , is dependent not only on the ut3>t Fh bk 2? kinetic energy of the states involved, but also on the single particle potentials associated with them. However, 22 the single particle potentials are functions of momentum which are determined by the reaction matrix elements. Thus one must assume some initial value for the single particle potentials and calculate the reaction matrix. New single particle potentials can be calculated from the reaction matrix, and these are then used in recalculating the energy denominator. This procedure is repeated until the single particle potentials generated by the reaction matrix agree with those used to calculate that reaction matrix. If it were not for the complicated dependence of the operatorch and <3 on the total and relative momenta, nuclear matter calculations would be greatly simplified. In fact, if one sets Qzl and defines the energy denominator to be just the kinetic energy of the particles involved, as is the case for the free two-nucleon interaction, one finds that be(€39 can be calculated analytically and equation (II-16) becomes I :kj? r/ ‘7%fi)3 ¢QVU"§;V/’e/; fi'l DMVJ'7%{) a/r' ..3 where A; is the relative momentum of the two nucleons before scattering. This is the integral equation for two- particle scattering. C. Baker Transform The Schroedinger equation for two nucleons interacting through a momentumrdependent potential can be shown to have solutions which are identical at large distances to those produced by the Schroedinger equation for two nucleons interacting through a static potential outside a hard core.6 23 The Schroedinger equation for the momentum dependent poten- tial can be written - t {f;[/(df‘*1/-/uwfi+fl/‘W]+ K 0)} m -- E 5’09 (II-18) where/uzl for a repulsive force and ”00:0. M is the nucleon mass, and K809 is a static potential. The form of the term involving the momentum operator is required for hermiticity and time reversal invariance.18 If/«USI for all values of r, then equation (II-18) just reduces to the Schroedinger equation for a static potential. Using /f=L$VC equation (II-18) can be written /(r) l7 1’044-7 7/4.“) vfif)+ 4350) 7/2“») 1" %[c‘~ VJJWJV‘W = 0 x Since/a. and K are functions of I“ only, equation (II-19) (II-19) may be separated into radial and angular components. Writing the equation for a particular partial wave and noting that the angular component is identical to the one obtained in the static problem, the radial equation becomes J‘y fi?’) 1. 51/0) ; (r + ,. 27 t J (107).? (IP20) 4‘“) "’ [W K ‘WJ {LE-flif‘fiw yaw) 2r ”0) #0) Mt,‘ where /"’)?;‘é-/u") Now make the transformation 24 l‘s rg/Q £0); {9049) So .31.... , .4 "" T ”’V) «94 where rho); i];- ’9') Then equation (II-20) becomes Misc! law/2g; ) [WK/)1 4/” W21 WW age/m) Zu_,u__ ) /a___’____ I 200 - I , I I I I.“ I I I X 0 \ -200 .J _J l J l 0 0.25 0.5 0.75 I.0 I.25 r (F) —> Figure 2. Static 180 potentials. See Reference 5. 32 less sensitive to variations in the mesh size around the core edge in numerical calculations. Potentials were available which had hard cores of radii 0.1 F to 0.4 F and which fit the 1 80 phase shifts equally well. The potential having a hard core radius of 0.1 F was classified as a soft core potential because of its Yukawa repulsion at small distances which joined smoothly onto the hard core at 0.1 F. Presumably the Yukawa repulsion could have been continued into the origin, but it is so large at 0.1 F that there probably would have been little difference in the resulting potential outside 0.1 F. The 400 MeV finite core potential was one of a group of potentials having core heights ranging from 400 MeV to 2000 MeV. These were generated in the same manner as the hard core potentials. The parameterizations of the three static potentials used here are given below:5 Hard core 1SO potential V - -1X ’y‘ - i‘ 00- Va,” -(..5'52:_%. - /o.r¢.7§‘.. +£20.59. for r>a.w= K = as for r50.~//‘-' "Soft“ core 150 potential '1" «7; V“): Mpg-IV/A3—f—r- *SZXI.7%— fox-P) an: 3 .0 forr‘salF 400 MeV finite core 180 potential - I __,_4‘.. I VII): (Mffi‘3,02&‘§::/0723e- ”9p- '(fl)+ $1006 \k) X T 33 where X = £35- 1. - 197. 322. MeV-F -‘ "n3 [35.0 M.V e Von-p"0'08"t‘7' x¢= 0.79597 The effective momentum-dependent potentials shown in Figure 3 were all obtained from the 0.4 F hard core poten- tial using equation (II-30) and should not be thought of as being completely equivalent to the three static potentials. They may be used in the Schroedinger equation with an effective wave function which agrees with the true wave function only in the region outside the range of the momen- tum dependence. 1 The form of f0) used here was «0)? {0) where ~55 {093/1- (ct/)8 A n30 I / This form is convenient because it allows the integrals involved in calculating f6”) and a to be solved analyti— cally, giving /0) = r- OI P S‘IHO‘OJ) ‘ RW' “‘ Taking the Baker transform of the 0.4 F hard core potential with an.) as described above, one finds I V0) =l Km. {(0‘) a ”6’. 5525' 6 {Cr} " u, .. 41g - :45] kc - 0 ‘7' - ~ u .w--_.- /.S"r’.7e [09 4472.2”): Mic/{(0fb) + n‘t‘({2f)*l)(/+4qzk‘/n) ‘7‘4‘ (‘0‘) M (III—1) 34 I60 MeV ”=3 ¢——0.4F HARDCORE 600 . 0 MeV +-n=5 —> 400 - — WI) (MeV) I60 MeV 20° n=l.4 0M0 p— _200 l ' I I l L 0 0.25 0.5 0.75 l.0 I25 r(F)—' Figure 3. Momentum-dependent lSo potentials. Generated from 0.4 F hard core potential. 35 where "ha as #c /?(r)=r+%/(n[fcr)] a. : 0.7/33 and M is the nucleon .mass. Care must be taken in treating the coulomb interaction in the effective potential. The coulomb potential must be included in the potential for iso- spin triplet states before the transformation is made. This results in an additional factor [V377 ,ucfl -z -z a e fay = H37? {0) - which must be added to the term in the square brackets of equation (III-1). The resulting effective potential is the correct one to use for the pp interaction. -For nucleon- nuoleon interactions which do not involve the coulomb poten- tial one must then subtract 6:, - 41:91 can!) " r/«(d from the effective potential for the isospin triplet states. The /axc) in the denominator is required because of the form of the effective potentials and wave functions. The longest range momentum dependence used in the 1So state was one in which the range parameter, ’7, in the equation for {0) was set equal to 1.4. This produced a momentum dependence with the same range as that used in the Bryan-Scott potential for this state. The effective momentumrdependent potential for this value of the range parameter and a shorter-ranged one with the value of the range parameter set at 3 are included in Figure 3. All of 36 the potentials are in good agreement beyond about 1.4 F, except for the intermediate range momentum-dependent one. It is significantly less attractive than the others beyond about 0.8 F. For the 381, P, and D states there was no set of static potentials comparable to the one available for the 180 state. Consequently, for these states it was only possible to compare hard core and momentumrdependent potentials. The hard core potential used was the Hamada-Johnston potential which consisted of a sum of fOur terms:3 VG’)’ K094- W(r)S"+ K; (r) (Z5) + V“(’9 L”. The subscripts refer to the central, tensor, linear spin- orbit, and quadratic spin orbit contributions, respectively. The individual contributions are given by v.0): ace (who? iXJ-r- 6:) M /+ at. Ya» A. no] Vr(r)= o.oe(2’-”x)(f-Pl} Za)[/+ar Va). 6.- flu] v..w= m. 6.. ya) [n 6.. 70)} V“ (r): ”I! 61.4. X”: Eu)[/+a“ You +' A“ X210] where 7%x)5 5%: aha/$4.) 7..) fish x: #‘1 37 The pion mass, ’Mk , is 139.4 MeV, and the hard core radius is 0.4855 F for all states. The parameters a , b , and 6, which are state dependent, are given in Table II. The operators 52‘ , L-f, and l,“ are defined by (5;: (fi? _, '9 ~ <07- ) J“ Equivalent momentumrdependent potentials were generated from the Hamada-Johnston potential numerically. 38 Table II. Ramada-Johnston potential parameters. Parameter Singlet Triplet Triplet Singlet even odd even odd ac 8.7 -9.07 6.0 -8.0 bc 10.6 3.48 -l.0 12.0 aT -1029 ‘005 bT 0.55 0.2 GLS 0.1961 0.0743 bLS -7012 -001 GLL -0.000891 -0.000891 0.00267 -0.00267 aLL 0.2 -7.26 1.8 2.0 b -0.2 6.92 -0.4 6.0 SECTION IV CALCULATIONS The binding energy per nucleon to first order in the reaction matrix was calculated using the method described by Brueckner and Masterson9 (BM) for static potentials and modified as outlined by Ingber19 for momentum-dependent potentials. The approximations made in BM simplify the calculations considerably, but at the expense of 1 or 2 MeV in the accuracy with which the mean binding energy per 16 This does not seriously detract nucleon is determined. from the usefulness of the method, however, because the main object here is to compare potentials, not to obtain the best possible value from each one. It is more important, in this case, that the same approximations be made for each poten- tial. A. Static Potential Calculations The first step was to calculate the Green's functions for each partial wave. These were obtained by expanding equation (II-17) in partial waves. In order to simplify the calculation of the Green's functions, it was assumed in BM9 that the energy denominator was independent of the total momentum and was rewritten 39 40 5., us, - 4::— 5;" = Z[€C/rm) - 5*(k.,)] (IV-1) where Fan. and E; are the self-consistant energies of par- ticles moving in the Fermi sea and E: and 5‘“ are the self- consistent energies of excited states above the Fermi sea. The quantity on the right hand side of equation (IV-l) is dependent only on km and If“ , the relative momenta of states in and n and states a and A respectively. This approximation is accurate if z; is essentially a quadratic function of k, or if the relative momenta are large compared with the total momenta. In line with this approximation, the total momentum was replaced by its average value for a given relative momentum, /< , where .05.... A" I?" 2744;1(l- %)(erg) for k‘ks (“31%) (IV-2) :: O for k%/frt The Green's functions were then given by , / P771001“) 03,) @cc’)=xn‘/2[5(B_€xa§j——/ (”‘3’ for on-energy shell propagation. For off-energy-shell prOpagation the denominator of the integrand was replaced by .2[£‘(k)-"(£')J— [I where A is the mean excitation energy, taken to be 564;.) 1710) . The Pauli step function,/(/3A’j , excluded values of k” which corresponded to filled states in the Fermi sea. It was given by20 /O3/«9= o , “our .7. : / I A"- II‘I‘" > A“; (IV-4) = k, ’7’)- ié’ otherwise, k’P where f9 is the average momentum of equation (IV-2). In the actual calculation the integral of equation (IV-3) was split into two parts: N /‘° ._ I/‘r ... I/.. 2x‘ . 271'“ a Zn‘ 1: OUI’ where Ah” was chosen so that the denominator of the inte- t. 0" ~ grand could be approximated by 1;:- for k aka" >/<, . For k"> k,- ,//(/9’ A7: / , so the second part of the integral could be written M o q ° 4' I ” ..zz‘ti Jr“ ”)J‘ U‘ r) .44 INT But this could be rewritten without further approximation as Q fl " zfi£l1(“)./l (£99.12 ‘= - fiégfifi; a $91.4 " M ‘3" +217“. ‘Jléi';le‘(kvr')J*. The first integral on the right side was evaluated analyti- cally, giving for on-energyéshell propagation 42 . M r. 1 61-03,): 71:» $711+!) (’3) (IV-5) ’ ' ”r0 . ~ . fl ku‘ (6“2__ ' +1?//1'(« I')J),(/< ’) ‘3 + [ o9] EJA 214:1 k)'c-‘Rt where I; is the lesser and I“, the greater of t‘ and r" . The wave functions were calculated by iteration of the equation J’ /" z ‘ ‘~ I . r. X" (‘3’) ' f‘(kr);‘v +7/c‘é-4wdfi' f‘CrQZ‘,¢w) fray“) (IV-6) IL . , . . where ““09 1.3 the static two body potential and 5".(10‘) = J;(A’)-[J‘(£r‘) 61(4 4)/6:‘(4-I")] 6:0,"): 592(4”)'[6¢C4") (91(1’31'9/520": 4-)] The forms of 5.0 and 5 resulted from the requirement that the wave function vanish at r; , the radius of the hard core.20 For potentials having no hard core, The wave function in the integrand of equation (IV-6) was initially set. equal to flag 5“. . Thereafter the values of the wave functions calculated on the previous iteration were used in the integrand. This was repeated until for any value of I' the change in the wave function on two succes- 3 sive iterations was smaller by at least a factor of 10- than the wave function itself. 43 The reaction matrix elements were then calculated using if (k«) J1 k a < // gig}... I my ”a (Iv-7) + V/r/“é &(£'?¢:,V 11' (’y 511’ (IQ/r) where C _ £2Ju)(27+/) 313‘. 3 J is the total angular momentum and T is the isospin. For potentials which do not have a hard core, the first term in the square brackets fs zero. Finally, the single particle potential was calculated from (kp “)/3 my £2 #77070» va&)é ‘ -—-A"./A = 71.2 2 c ./ 47‘0")! 1(r’r)/’,&Ck,r) (IV-10) 49‘ 14:4? Both of these equations now involve derivatives of the wave function. These are difficult to calculate because the wave function is calculated numerically by an iterative process. However, by integrating equations (IV-9) and (IV-10) by parts, it is possible to eliminate the wave function 45 derivatives in favor of derivatives of Bessel functions and Green's functions, both of which can be expressed analyti- cally.19 In order to carry out this modification it was neces- sary to know something about the form of the potential, ”4' (7' V). Defining a function 60(fi)3‘f<;u(¢)-4) the Schroedinger equation for a momentum-dependent potential, equation (II-18), can be rewritten 2. tr 1 r .. _. . gfi+ UC’)% / ”()+ K(,.)§¢(p):[: yca (IV—11) where “(')EVC")+_ “1’9 U___C__"d is the static part of the potential. The quantity 4409 will in general be a function of 07,.1 , and s . However, for convenience, these subscripts as well as those on 16(0 and [‘09 will be suppressed. In the actual calculations the same Ada) was used for all states, although there is certainly no reason to expect this to be true. This will be discussed in greater detail later. The term [480) in the static part of the potential in equation (IV-11) can be evaluated by comparing equations (II-20) and (II-29) with the result .5 .’ __“_,,amd ‘/u'ov‘:Uud I magmas—2419!”; ,——«.—-]+ . ,0) Km-..) 46 Using /ar 1’- , if rér‘ Olif r’Q/ The equation for the reaction matrix elements was also integrated by parts to eliminate the derivatives of the wave function. In this case recursion relations were used to eliminate the second derivative of the Bessel function leaving 962-1“ ’1" /.;/ if my!» J. . (“K/k) 2 Wig-1.2.7., ch. , ’ 1%,?7u'U‘I’) “,{rCA’cWch} ‘ J: y + Kl (19,)2 {:4 MO) “(baby-11? r9wér) ”2410 «I2 r) J. . - wwbgflc) *Zk “"0 £4 “Id/w (“)1 (”'22) With these changes it was possible to extend the method described in BM to momentum-dependent potentials. Because of the necessity of calculating the first derivative of the Green's function and the need to extend the region of inte— gration all the way in to the origin instead of just to the hard core radius (0.4 F to 0.5 F usually), the computer time required for momentum-dependent potentials was about two to four times as great as that required for hard core poten- tials. llllllllilll'lllll ill-lulllllll lllllllll II 50 C. Phase Shift Calculation As a check on the validity of the equations and the programing, one further modification was made in the equa- tions for the momentum-dependent potentials. By changing the denominator of the Green's functions and their deriva- tives (equations(IV-5) and (IV-21)) to be just the differ- ence of the relative kinetic energies , L , at .z[E(;/<.)-E*Ck'7]3 1174*: - i/Tqé' and setting flip, 2475 1 for all P and A” , the Green's functions and their derivatives for nuclear matter became those associated with the free two-nucleon elastic scattering problem. The singularity within the range of integration results in complex Green's functions and therefore complex wave functions. From the equations thus modified, the phase shifts, 11$?! , for the uncoupled states could be calculated using ‘ a 1_2Mk. & 31.,(2 z“ 6): 7" “/7 (W) V070W: (‘1’) (Iv-23) and compared with the known values of those phase shifts. The integrand of equation (IV-23) is of the same form as that of equation (IV-10) which was evaluated as shown in equation (IV-22). SECTION V RESULTS Before using the program to calculate the binding energy of nuclear matter, the Green's functions were modified as described previously so that the elastic scattering phase shifts could be calculated. When these were compared with their known values they were found to differ in the third significant figure. This difference was due almost entirely to the method of evaluating the Green's function integral around the singularity. When the free two-nucleon Green's function integral was evaluated analytically, the phase shifts calculated by the two methods were in even better agreement. Returning to nuclear matter calculations, the Hamada- Johnston hard core potential was used for all states in order to find the self-consistent single particle potential and the binding energy per nucleon predicted by this model for a Fermi momentum of 1.4 FSl. The resulting binding energy per nucleon, 8.5 MeV, is fairly typical of the values pre- dicted by hard core potentials which fit the elastic scat- tering data9 and is clearly in poor agreement with the empir- ical value of 16 MeV. The study of the model dependence of the 180 state was then carried out using this single particle 51 52 potential as a starting point. The Hamada-Johnston 180 potential was replaced by the 0.4 F and 0.1 F hard core potentials, the 400 MeV finite core potential, and finally the momentum-dependent lSO potentials. The results are shown in Table III. From the calculations involving the three static potentials, it was clear that the contributions of the 3 51' P, and D states were not significantly affected by the changes introduced in the single particle potential as a result of using different 180 potentials. Consequently it was not necessary to recalculate these states when the momentum-dependent 180 potentials were used. The results shown in Table III confirm the expectation that potentials having soft cores, finite cores, or a momentum-dependent repulsion all would produce a desirable increase in the binding energy above that predicted by a longer range hard core potential, at least for the 180 state. The maximum amount of binding occurred with the momentum- dependent potential having the range parameter about equal to 3. This can perhaps be understood by comparing the effective potentials for n=l.4, 3, and 5, as shown in Figure 3. As the value of the range parameter, n, increases, the size of the short range repulsion increases. This is accompanied by a corresponding increase in the intermediate range attraction. As n increases past 3, the increase in attraction in the intermediate range is not sufficient to counteract the increased short range repulsion seen by the wave function. Similarly as n decreases from 3 down to 1.4, Table III. Mean binding energy per nucleon lSo potentials. 53 predicted by the 180 potential Potential energy for 1 Total energy* (MeV) (MeV) 0.4 F hard core -15.8 ~9.3 0.1 F hard core -16.8 -lO.3 400 MeV finite core -l7.2 ~10.7 n=l.4 momentum dep. -l7.4 -10.9 n=3 momentum dep. ~18.2 —1l.7 n=4 momentum dep. —17.8 -ll.3 * Using Hamada-Johnston potential for all other states and including a kinetic energy of 24.39 MeV per nucleon. 54 very little short range repulsion remains, but there is also virtually no attraction in the intermediate region. This is borne out by Table IV which compares the values of the 1S0 reaction matrix elements for each potential to the values obtained when the 0.4 F hard core potential was used. In all cases the gain in the size of the reaction matrix elements increases with momentum, indicating that the size of the short range repulsion is the dominant factor. The momentum- dependent potential with n=l.4 is especially interesting. Although at high momenta the K matrix elements are consider- ably larger than those of the 400 MeV finite core potential, at small momenta they are somewhat smaller. This seems to verify the previous remark regarding what appears to be an undesirably weak attraction in the intermediate region of this potential. The 2.4 MeV increase in the average binding energy per nucleon of the 1So state for the n=3 momentum-dependent potential over the value obtained with the 0.4 F hard core potential is a change in the right direction. But combined with the hard core Hamada-Johnston potential for the other states it still leaves the total of the contributions of all the states short of the 16 MeV empirical value. This clearly indicates the necessity of examining the effects of the various potential forms on the other states. Table V shows the contribution of each state to the binding energy of nuclear matter for several momentum- dependent potentials and for the Ramada-Johnston potential 55 Table IV. Percent increase of 150 K matrix elements over values obtained with 0.4 F hard core potential. Potential k/kF= 0.1 0.3 0.5 0.7 0.9 0.1 F hard core 4.1 5.0 6.3 9.3 17.7 400 MeV finite core 5.7 6.8 8.8 13.1 27.2 n=1.4 momentum dep. 4.7 6.1 9.7 18.8 47.1 n=3 momentum dep. 9.0 11.1 15.2 24.2 50.8 56 Table V. Mean binding energy per nucleon predicted by momentum-dependent potentials generated from the Hamada- Johnston potential. State Hard Momentum—dependent H-J core H-J n=l.4 n=3 n25 (MeV) (MeV) (MeV) (MeV) l a 30 -15.09 -16.30 —19.66 -20.08 351 -16.05 -12.94 -19.55 -21.21a 1P1 3.68 3.21 3.53 3.61 3PO -3.47 -3.71 -3.69 -3.65 3P1 11.13 10.07 10.40 10.41 3p2 -7.13 -4.73 -7.45 -8.91 102 -3.09 -2.86 -3.07 -3.11 301 1.57 1.56 1.57 1.57 302 -4.47 -4.40 -4.48 -4.48 Total -8.54 —6.64 -16.61 -18.69 aThese values are for n=3.7. bTotal for each column is the sum of the S states from that column, the P and D states from the hard core H-J column, and a kinetic energy of 24.39 MeV per nucleon. 57 from which they were generated. The increase in the binding energy of the 180 state when the hard core in that state was replaced by a momentum-dependent repulsion was greater than that obtained when the 0.4 F hard core was replaced by a momentum-dependent repulsion. The 381 state shows even more sensitivity to the form of the short range repulsion than the 180 produced about 3 MeV less binding in the 381 state than the state. The intermediate range momentum dependence hard core Hamada-Johnston potential. The short range momentum dependence characterized by n=3.7 produced about a 5 MeV increase in binding. Combined with the increase of 5 MeV in the 1 So state and using the hard core Hamada-Johnston potential for the P and D states, the n=3.7 momentum depend— ence in the BSl state produced too much binding. This could be corrected by adjusting the value of the range parameter for either or both of the S states to reduce the binding energy to the desired value. Setting n=3 for both states results in good agreement with the empirical value. In any event, a deviation of 1 or 2 MeV from the empirical value is not a cause for concern since the method used in the calcula- tion could have introduced errors of this magnitude. The D states showed some loss of binding with the inter- mediate range momentum dependence. With the shorter range momentum dependence, n=3, the contribution of these states to the binding energy was brought into close agreement with the values obtained from the hard core potential. This was 58 perhaps to be expected since the D waves see virtually nothing of the interior region of the potential. What is seen, however, is the decrease in attraction in the inter— mediate region of the n=l.4 momentum-dependent potential and the fairly close agreement between the hard core and the shorter range momentum-dependent potentials in the inter— mediate region as shown in Figures 4, 5, and 6. Thus it makes little difference whether a hard core or short range momentum-dependent potential is used in the D states. 1 3 3 The P1, P0, and P states together are about 1.8 MeV 1 more attractive when the n=l.4 momentum dependence is used ' to replace the hard core. This increase in binding is reduced to half this value when the range parameter is set equal to 5. The 3P2 state more than cancels the increase in binding produced by the other P states for n=l.4. For that value of the range parameter, the 3P2 state produces 2.4 MeV less attraction than it does with the hard core potential. This changes to about 1.8 MeV of added attraction when n is increased to 5. As n increases above 5, the increase in attraction levels off and will then gradually return to the value obtained with the hard core potential. Setting the range parameter equal to 3 for all of the P states results in 1.4 MeV more binding energy than is obtained when the hard core Hamada-Johnston potential is used. The 3P0 phase shift is very poorly defined by experiment at energies between about 60 MeV and 130 MeV, a region which 59 Wr) (BeV) r(F) Figure 4. Momentum-dependent lD2 potentials. Generated from the Ramada-Johnston potential. 6o l.5 I60 Me V OMeV ////////// I.O #- ///,/ ’x ’/ Wr) (BeV) 0-5 ’- I60 MeV V‘V ‘7 ......... ~ . ............... V‘N ....vvv—v—v CCCCCCCCCCCCCC ..‘s I... ooooooooooooo ... ....... ’0'... ho. 00 0.5 I.O r(F) Figure 5. Momentumrdependent 3D1 potentials the Hamada-Johnston potential. :03 ’.4:-‘.’.‘ ° L 0 MOVL’ 79.11 ...“ - ----- . Generated from 61 3 02 ' l.0 - ‘--n= 5 and HJ I60 MeV ll'l'mlu' <— n= 3 -\ || " 13 ll'ou v.\ m ||l a I \. I' I _\ "It: “1! \\ ~’ ..... fil" \ g, 0. 5 ”H“ ”h... ..... x160 MOV \‘h 0 M V \\.::- n = I. 4 ——§..\ ‘3‘3. ‘25. O \°"~-‘.":.°pr- — n=5 and HJ l l l 0 0.5 LO l.5 ' r (F) Figure 6. Momentum-dependent 3D2 potentials. Generated from the Ramada-Johnston potential. 62 is very important in nuclear matter. The uncertainty in this phase shift results in an uncertainty of about 0.4 MeV per nucleon in the contribution of this state to the binding energy of nuclear matter.21 The 3P1 and 3P2 phase shifts are pinned down somewhat better than the 3P0, but at energies above about 100 MeV the Ramada-Johnston potential does a rather poor job of fitting these phase shifts.* Using the phase shift approximation for the reaction matrix, the Hamada-Johnston potential was found to produce about 0.4 MeV per nucleon less binding than the experimentally determined phase shifts in the 3P0 state and 0.6 to 0.8 MeV per nucleon more binding in each of the other states. Consequently the numerical results obtained for the P states should be inter— preted with caution. In conclusion, the binding energy of nuclear matter seems to be quite sensitive to the form of the short range repulsion used in phenomenological two-nucleon potentials. Even two-nucleon potentials which have identical on-energy- shell matrix elements may predict mean binding energies per nucleon for the S and P states which differ by several MeV. The binding energies of both the S and the P states showed a desirable increase when a hard core potential was replaced by a short range momentum-dependent one. Using a momentum- dependent potential for the 8 states and a hard core poten- tial for the P and D states resulted in a mean binding * See Figures 7 through 10. 63 o \ '\ ‘ \ \ \ \ \ \ ..\ 3.40 .. 8 Dr \ ‘ 3 \ s—BRESSEL- \. \ KERMAN .. \ 3: \ 53-: 5 — P \ . :6 \ LL \ -20 — \ \ HA MA DA - JOHNSTON —-——>\\ \ \ ’25T‘ \\ \ \ \ \ . \ _, I ‘ l 1 l 30o 50 IOO . ISO 200 I ELAB (MBV) Figure 7. 1'P phase shift vs. energy. Reference 22, quares from Reference 4. Circles are from 64 l5 IO - .\ U) Q) Q) B. m 5 - Q \— 9E 4: 2 x ‘8 ° " \ 4: CL \ é \ \ V ..5 _. "'C) l I l I 0 50 I00 I50 200 ELAB (MGV) Figure EL 3P phase shift vs. energy. Circles are from Reference 22, gquares from Reference 4, and triangles from Reference 21. - 65 O _ _. 3 5 P, .\ BRESSEL - KERMAN 840 — ””d g HAMA DA -JOHNS TON C» Q) Q \— :75 c-Is L— U) Q) u, 3 Q. -20 - ..255.. "3() l l l J O 50 IOO ISO 200 ELAB (MBV) Figure EL, 3P phase shift vs. energy. Circles are from Reference 22, squares from Reference 4, and triangles from Reference 21. 66 l5- ‘ 'HAMAOA- / JOHNSTON / .\ U) Q) ‘8 8’ 9 IO ~ § E c—BRESSEL - KERMA N U) Q) U) 2 o. 5 _. (3 1 i1 L l ' o 50 IOO ISO 200 ELAB (MCV) Figure 10. 3P phase shift vs. energy. Circles are from Reference 22, quares from Reference 4, and triangles from Reference 21. ‘ 67 energy per particle which was in good agreement with the empirical value of 16 MeV. It would be desirable to repeat 3 the work on the 51’ P, and D states with a set of potentials similar to the ones used for the 1So state rather than the Hamada-Johnston potential which is not always in agreement with the phase shifts obtained from analyses of the two- nucleon elastic scattering data. APPENDIX APPENDIX DIAGRAMS The unperturbed ground state of nuclear matter is represented by a Slater determinant of the unperturbed single particle wave functions corresponding to the A single particle states of lowest energy. That is ..1 A ‘ .3 p44!) a 7T¢.(r.-) ’ in ‘ where C? is the antisymmetrizing Operator. The unperturbed wave function, £5 , is normalized = Let $0?) and 4(3) be two single particle states interacting through the two-nucleon potential. Then the product of these states is written /,a;> —.— gar) an?) {/fljh‘) 2125:) Véxjjéa") J2; Jag 68 69 The matrix element of the single particle potential is similarly defined ; flay V ,1; 0;) a/x Define the Fermion creation and annihilation operators with the properties that car creates a particle (annihilates a hole) in state f , and a, annihilates a particle (creates a hole) in state f’. These operators satisfy the anticomr mutation relations “2 “P “r ”/ =0 ) “/74”; “fr“ ' “/‘rr*“;“/ =5” Diagrams provide a convenient way of illustrating the effect of a particular term in the Brueckner-Goldstone expansion. In order that they be used in a consistent way, it is necessary to specify a few rules for drawing or inter- preting a diagram. The direction of increasing time will be toward the tOp of the page. Any state not specifically included in a diagram is assumed to be as it was in the unperturbed state. Thus above and below the diagram all states below the Fermi level are filled and all states above are empty. The first interaction results in two particles in the Fermi sea being excited to states above the Fermi sea, leaving holes in the originally occupied states. Continuing in the direction of increasing time, particles are repeatedly scattered into unoccupied states until the final interaction results in their being scattered back into their original states, leaving the wave function as it was before the first interaction took place. An upward directed line represents 70 a particle above the Fermi sea and a downward directed line represents a hole in the Fermi sea. Since particle number is conserved, there must always be one hole in the Fermi sea for every particle above it. The matrix element (ff/V/”J> is represented by a horizontal dashed line connecting the intersection of lines representing the states /o and r‘ at one end with the intersection of lines representing states 7. and ,s at the other end. The matrix element .. The particles in this state are destroyed by the operator Gr‘h . The Operator 4; a; creates particles in the final state The matrix element corresponding to this interaction is (“b/V /0m> and the energy denominator for the intermediate state resulting from this interaction is-(clre‘.-é’,.. ’6‘)" . The minus sign preceding this term will later be combined with other factors of -l as described in the final rule in the previous paragraph. Thus the effect of the first inter- action is written affirm a" (5*Fs‘€~*€q) The interaction with the single particle potential results in the scattering of a particle in state A? in the Fermi sea into the unoccupied state hr also in the Fermi sea, leaving a hole in state ,[7. It can also be thought of as the scattering of the hole in state 0; into state I . This interaction contributes the additional factor - W”? (cafl'b' 5,11) to the term resulting from the first interaction. Although this interaction involves scattering from state In to,17, the energies of states a , é , and I: also affect the 72 contribution of this interaction. This is an example of a process taking place off the energy shell. The final inter- action scatters the particles in excited states a and L back into the holes left in states ’7 and .1 as the result of the previous interactions. After this interaction the particles are all back in their unperturbed ground states. The total contribution of this diagram is (”1 /V/a ‘>ga L /V/""') (64w: -€,, —c—1)(€. *éz-fw‘fn) X<£ /4:4;a‘~ a6 04. 61:61:41“. a... /fa> The quantity / f I" f 4, f .. is just *1, depending on the order of the creation and destruction operators. Using the commutation relations for the operators and remembering that a/f‘V/fl>: 67;.) if [05/4 (in the Fermi sea) 0 if lp>A (above the Fermi sea) the expectation value of this particular set of operators is found to be +1. This agrees with the result obtained by using the last rule for the diagrams. There are three hole lines, two closed loops, and one interaction involving the 3+2+1=+1 . single particle potential, giving a sign of {-1) The two minus signs from the energy denominators cancel, leaving as the contribution of this diagram to the energy of nuclear matter 73 §nUV/a L> (fiflurfl-ég-E;)(ffl.rdi"5k‘£3) To find the total contribution of all the distinct diagrams of this type, one must calculate the sum of these terms where ayand L. are allowed to take on all possible values greater than A, and m, n , and I take on all possible values between 1 and A. The sum is then multiplied by %lto account for the fact that since <,«;/u/~> = «HM-s» each distinct combination of states has been calculated twice. REFERENCES 1r 10. 11. 12. 13. 14. 15. REFERENCES P. Cziffra and M. J. Moravcsik, University of E California Radiation Laboratory Report UCRL-8523 Rev. “ C. B. Bressel and A. K. Kerman quoted in P. C. Bhargava and D. W. L. Sprung, Annals of Physics (N. Y.), 12, 222 (1967). T. Hamada and I. D. Johnston, Nucl. Phys. 21, 382 (1962» l M. D. Miller, P. S. Signell, and N. R. Yoder, Phys. Rev. 176, 1724 (1968). M. S. Sher, Ph. D. thesis, Michigan State University, 1969. G. A. Baker, Jr., Phys. Rev. 128, 1485 (1962). R. E. Peierls, Proceedings of the International Conference on Nuclear Structure, Kingston, University of Toronto Press, Toronto, 1960. A. M. Green, Nucl. Phys. 33, 218 (1962) and A. M. Green, Phys. Lett. 1, 136-T1962). K. A. Brueckner and K. S. Masterson, Jr., Phys. Rev. 128, 2267 (1962) hereafter referred to as BM. J. Orear, A. H. Rosenfeld, and R. A. Schluter, Nuclear Physics, University of Chicago Press, Chicago, 1967, p. 7. P. A. Seeger, Nucl. Phys. 2;, l (1961). A. E. S. Green, Phys. Rev. 9;, 1006 (1964). D. Halliday, Introductory Nuclear Physics, John Wiley & Sons, Inc., New York, 1962, p. 261. B. H. Brandow, Ph. D. thesis, Cornell University, 1964 quoted in Reference 16. Much of the material for this section was obtained from Reference 16. 74 16. 17. 18. 19. 20. 21. 22. 75 B. Day, Rev. Mod. Phys. 39, 719 (1967). M. W. Kirson, Nucl. Phys. A99, 353 (1967). S. Okubo and R. E. Marshak, Annals of Physics 4, 166 (1958) quoted in P. Signell, The Nuclear Potential, in Advances in Nuclear Physics, M. Baranger and E. Vogt, ed., Vol. 2, Plenum Press, New York, 1969. L. Ingber, Ph. D. thesis, University of California, San Diego, 1967. K. A. Brueckner and J. L. Gammel, Phys. Rev. 109, 1023 (1958). P. S. Signell and M. D. Miller, Phys. Rev. 178, 2377 M. H. MacGregor, R. A. Arndt, and R. M. Wright, University of California preprint UCRL - 70075 (Part x).