ALPHA PARTICLE mom 01-“ 18o Thesis £09 Hm {Dog-rec 0? D“. D. RECHEGAN STATE UNWERSITY Bing W. Peon I974 This is to certify that the- ‘ thesis entitled 16 Alpha Particle Model of 0 presented by Bing W. Poon has been accepted towards fulfillment of the requirements for Ph. D. Physics degree in .72 “4 of {If , t f Major professor N . 107k Date 0v 5’ " 0-7639 ABSTRACT a Particle Model of 160 BY Bing W. Poon 16 The low lying even parity states of O are investi- gated by the coupled-channel method in the weak-coupling model of the a particle and 12C nucleus. Different phenomenological a-a potentials are used in determining the effective a- 12C potential. In calculating the effective a- 12C potential, no intrinsic excited states of 12C are considered, but their effects on different physical properties are calculated. The problem of wave function symmetry is considered. Correct binding energies of the bound states can only be obtained when we use a deep o-a potential. From our results we show that the a-a scattering data do not sufficiently determine the interaction to make predictions on the structure of more complicated nuclei. The interactions with repulsive cores give a very smooth structure to the states and tend to underbind the ground state. Also the transition rates to the 0; state are too large. The deep attractive interactions suffer from Opposite defects. The states are now overbound, and differ Bing W. Poon from each other to such an extent that the transition rates are an order of magnitude too small. Of the two types of interaction, we prefer the deep potential, because the structure of the states matches the more fundamental cluster model. a PARTICLE MODEL OF 160 By . - \ x -‘-‘-.l I Cbmt‘lBing—WicPoon A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 197” ACKNOWLEDGEMENT I would like to thank Professor George Bertsch for his guidance and support during the time that this work was performed. Thanks is also due to Dr. K. Kolltveit for the encouragement and support during the early period of my Ph. D. program, to Professor B. H. Wildenthal for some helpful suggestions. Lastly, I would like to thank my wife, Viota, without her patience and understanding this work would not have been possible. ii ACKNOWLEDGEMENTS . . . . . . . LIST OF TABLES . . . . . . . . LIST OF FIGURES. . . . . . . . INTRODUCTION. . . . . . . . . Chapter I . II III IV V TABLE OF CONTENTS Historical Review of the a Model . Formalism. . . . . . . . . Weak Coupling Model Wave Function. Problem of Wave Function Symmetry. Matrix Elements of Electromagnetic Transition Operators . . . . Quadrupole Rate with Full Exchange Mean Square Radius with Full Exchange Width of the Resonance . . . . Numerical Details . . . . . . Energy Spectra, Wave Functions and Properties. . . . . . . . Summary and Conclusion . . . . REFERENCES . . . . . . . . . . . APPENDIX APPENDIX A O O O O O C O O O O O B O O O O O O O O O O 0 iii Physical Page ii iv 14 1” 17 20 22 27 30 31 3M H3 US H9 52 Table 5a 5b 10a 10b 11 LIST OF TABLES Mixing ratios from TTM's calculation and present calculation. . . . Volume integrals of various potentials (Vd being defined in (#8)). ‘ 12 Ground stage properties of C and from Brink . . . . . . . Computation times for the CCE program . Mixing ratios of various 1 in G.S. and 0; state for Ali and Bodmer potential, also the mixing ratios calculated by Noble and Coelh031.. . . . . . . Mixing ratios of various 1 in G.S. and 05 state for Vary and Dover's folded potential Exchange matrix elements . Ground state binding energies for different potentials . . . . . . . . Excitation energies for different potentials . . . . . . Root mean square radii . . B(E2) transition rates Comparison of Our B(E2) and Brown's B(E2)” a decay width . . . . . iv 16 Page 5” 55 56 56 57 57 58 58 58 59 60 51 LIST OF FIGURES FIGURE Page 1 Spectra from experiment, TTM's calculation and present calculation using TTM's potential . . . . . . . . . . . . 62 2 Spectra from experiment, calculation using Ali and Bodmer potential and calculation using adjusted Ali and Bodmer potential . . 63 3 Spectra from experiment, calculation using folded potential and calculation using Neudatchin potential. . . . . . . . . 64 H Spectra from experiment and calculation using adjusted folded potential . . . . . . . 65 5 Nodes of a+a system is ground state (both particles are located along 2 axis). . . . 65 6 Nodes of the a+ 120 in ground state for Neudatchin potential (the 12C are on the x—y plane) . . . . . . . . . . . . . 67 7 Nodes of the a+ 12C in 0; state for Neudatchin 68 potential . . . . . . . . . . . . 8 Nodes of the a+ 12C in ground state for the adjusted folded potential . . . . . . . 59 9 Nodes of the 0+ 120 in 0: state for the adjusted folded potential . . . . . . . . . . 70 10 The plots of P (6) for ground state versus cos (9) for different potentials. . . . . 71 11 The plots of P (e) for 0: state versus cos(e) for different potentials . . . . . 72 12 Ground state wave function for Ali and Bodmer potential . . . . . . . . . . 73 13 0+ state wave function for Ali and Bodmer pgtential . . . . . . . . . . . . 74 Page FIGURE 1% Ground state wave function for Folded p0tentia1 o o o o o o o o o o o 75 15 0+ state wave function for Folded patential . . . . . . . . . . . ' 76 vi INTRODUCTION The low lying even parity states of 160 have been difficult to understand in the language of shell model since their energies are much lower than expected on the basis of single particle energies.l Also the electromagnetic pro- perties of these states have further emphasized the complex nature of this nucleus. For example, the existence of a rotational band among these states requires strong deforma- tionz’3 and the strong E2 transition from the 2+ state at 6.91 MeV requires states mixing in the ground state. A 16O at 6.06 shell model representation of the 0+ state of MeV is discussed by Brown and Green“ using mainly Hp-hh state. This state has a particular mixture of shell config- urations giving a triaxially deformed density distribution. The number of configurations needed to describe this state is very large, and an even greater number would be needed to calculate the energy of the state?9 To obtain a wave function usable for discussing the electromagnetic properties, mixing is required between this state, a similar 2p-2h state, and the closed shell state. Bertsch and Bertozzis, using a particle model and making suitable approximations on the wave function of four interacting particles, found the radius of the ground state and the energies of the ground and first 0+ 1 state agree very well with experimental values. Since the a particle model provides a much more transparent description of the structure of 16O, a more detailed investigation of a particle model on 16O is desired. We start by assuming the 16O is made up of a 120 core plus an a particle. Since we know the realistic a-d potential 120,6,7,8 and the structure of the we.can calculate the effective a- 12C potential. We put the wavefunction of the 9 system in a weak coupling model form, use the effective d- 120 potential, and find a set of coupled channel equations from the Schroedinger equation. After we solve these equa~ tions, we obtain the spectrum and the wave functions. We use the wave functions to calculate the electromagnetic transition rates and other physical properties of these states. In Chapter I, we will outline a brief historical review of the a model. In Chapter II, we will show the formalism we needed in our calculation. In Chapter III, we will present the numerical method we used in our calculation. The spectra, wave functions and some physical properties will be shown in Chapter IV. In Chapter V, we will present the summary and conclusion of our calculation. CHAPTER I Historical Review of the a Model The a particle model of the nuclei was first introduced by Gamow in 193010 as a extension of his successful investi- gation of a decay. At that time, nuclei were thought to be composed almost entirely of c particles or protons and electrons. The idea went out of favor when the neutron was discovered and Heisenberg and Majorama had developed their simple and satisfying theories of nuclear structure. However, many physicists felt that the central field approximation needed for the shell model was probably not valid for light 11 nuclei. In 1936 this feeling was augmented by Bohr who showed that the binding energies for most nuclei could be 12 reproduced quite well with a liquid drop model. Subsequently the a model was revived in three different forms: 1) The first approach was introduced by Margenau13 in 19ul, 1D and initial investigations weremade by Brink and by Biel.15 In this approach, the wave function of the H N nucleons is ;9— ll 'j> (b > . ll Z where A __ l —- 3 3 [J t lt/z The N vectors Rl to RN are parameters representing the center of the N a particles. The wave function ¢i describes the motion of a single nucleon in a l s harmonic oscillator orbit centered at the point Ri' A H—N nucleon state may be constructed from these N orbital wave functions by requiring that each orbit state should be occupied by 2 protons and 2 neutrons and then forming the corresponding u-N particle normalized Slater—determinant wave function T(Rl,....RN). Using the variational principle 3 <~llHli> I 0. Gal?) we can obtain the R and b. 1 Using this approach, Brink, et al.6 found the maximum binding for 160 when R are at the corners of a regular 1 tetrahedron. The value of the binding energy they obtained (Table l) is 9u.u MeV as compared to the experimental value of 127.6 MeV. They also found that the excitation energies of different a configurations is very sensitive to the nature of the 2-body force. Because the forces they used are not considered realistic enough, they could not get a definite assignment of d configurations to observed levels. This type of approach requires the center of each a cluster to be fixed, and as a result the Slater determinant states are not eigenstates of angular momentum and often not eigenstates of parity either. Also a recent investiga— tion by Irvine and Abulaffio16 found an c-cluster configura— tion is more bound than the optimum regular tetrahedron and there is nosharp minimum in the energy of the ground state of 16O in this model. This suggests that the ground state must be a mixture of c—clusters configurations, or that it is important to treat the cluster coordinates as dynamical variables. 2) A second c-particle model which considers the dynamical 17 motion to some extent was introduced by Wefelmeiser and developed by Wheeler,18 Dennison,19 Kameny20 and Perry and Skyrme.21 It is assumed that the N o nuclei may be treated as a system of N alpha particles which obey Bose-Einstein statistics (the a particle considered here has no internal structure). This approach actually is the extension of Wheeler's c-cluster model,22’23’2u’25 the added assumption (c with no internal structure) is to simplify the anti- symmetrization of the wave function in Wheeler's c-cluster model. In this model, the total Hamiltonian is written as H=HM+HM where H = vibrational Hamiltonian vib Hrot = rotational Hamiltonian. By applying the results of molecular theory to determine the allowed rotational and vibrational quantum states of the nucleus, one can get for 16O:18 E = E; (%)(3(“°)+ mm, + Mug M. u, + e- where E0 is the ground state binding energy of the nucleus I is the moment of inertia of the nucleus nlhw2 is the single vibrational energy nzhw2 is the doubly degenerate vibrational energy njhw3 is the triply degenerate vibrational energy a is the vibrational energy that one a going through or going around the other 3a to form a symmetric state. By choosing suitable parameters, one can get a spectrum that agrees very well with the experimental result. Unfortunately 16 + l (6.92 MeV) states are 15 to 20 times shorter than the experi- 20 they found for o the mean life of the 03 (6.06 MeV) and 2 mental result. One may say that because this model has undetermined parameters which are fitted to the data, the validity of the a model is not really put to a test. But how bad it is to represent the particle as a structureless particle? In a recent research made by Abul-Magd,26 the lower bound to the ground state energy of a system of Na particles interacting via a potential determined by c—a scattering experiments is compared with the energies of nuclei with 2 N protons and 2 N neutrons. It is concluded that the Na system can in principle be represented as systems of rigid and structureless(xparticles. So now the remaining question is how things will happen if we put the realistic c-d potential into the a particle model and treat the a coordinates as dynamical variables. 3) The third a-particle model which not only considers the dynamical motion but also uses a real interaction is a 160. To solve this, there are three A-particles problem for different techniques. The first of these is by Bertsch and Bertozzi.5 They approximated the N—particles wave function of 16O to reduce the problem to a one dimensional Schroedinger equation which they can solve. The second technique is by Mendez and Seligman.27 They used the harmonic oscillator wave functions as the basis of the system, and used the method developed for A particles harmonic oscillator wave function28 to solve the problem. The third technique is by Terasawa, TanifuJi and Mikoshiba.9 They separated the A particles into a core (120) plus an a particle. In other words, they simplified the u-particles problem to a 2—particles problem. Using the weak coupling model wave functions as the basis of the system, they can find the spectrum and the wave functions. Now let us discuss these three different techniques a little bit further. a) Bertsch and Bertozzi5 approximate the ground state wave function of 16O by (a - I? 49H“) and the first excited state wave function by t!) I Z I ‘- ‘ .. f6- .1' (““133 (Ian) The Hamiltonian of the system is ‘3. H=-— V+Zv(f.) law where V(riJ) is the interaction between a particles. In terms of the internal coordinates, the intrinsic Hamiltonian is 5 t? l a . a . = — _ "'1 —— f —_ ._ Ha Z; m, h a “s + Mp) - £1- t?- £1. 14% a 9—— gal I They considered the a—configuration of 16O as a tetrahedron composed of 2 triangles lying on a common base with apices Joined by the coordinate r13 and they used the angle 6 between the two triangles to describe the tetrahedron. Transforming the r13 in the Hamiltonian to 6 and simulating the integrations over riJ by replacing r1,j by d-an average separation distance, they found the Hamiltonian for the 6 variable becomes IlmJ‘—— 110 + ‘1 where ‘ e __ I d \+ ’w“‘03) .Q_ + EL Ho — ”gig )4: A: be VLJZAMI H, = -51. Mgfmg Ad 3 DM‘ mu ‘6 0‘ 3(3‘ 39 -E l m n a: lj By neglecting entirely the coupling of the potential energy with the coordinates and treating the off diagonal matrix elements of the coupling in the second order perturbation theory, they found the effective Hamiltonian for the 9 coordinates becomes H = H + Z (43(rnn) H, ¢,({nn)>z ”ll 0 on E ‘w EA’ 10 They treated ¢(rmn) as oscillator functions and got _.£ 3 9 a . Hm— m‘ be 3A‘(\+U)‘-g-) 30 +V(EAM%> Using Ali and Bodmer's a-d potential,29 they found that both the interaction energy of the c particles and the root mean radii of 160 agree pretty well with the experimental results except the .5 They also found the peak of the wave function for the ground state is at the regular tetrahedron shape and the excited 0+ state (6.06 MeV) has a peak at the shape of a plane diamond. b) Mendez and Seligman27 also did a similar calculation on 120 and 160. They used harmonic oscillator wave function as the basis of the system and the Hamiltonian is given by H=H,+ H, where n 1 l ‘l. = Z 1:. + W“ ‘2', 1f (3. AM‘ a isi‘l ‘1 V‘ M‘ 55‘ 9‘ 2 H. =- Z \l.. _ Z- {1" £h>ltt (l) lu 15 where YLn is the spherical harmonic function representing the angular part of the a particle in the Ln state d is the internal wave function of the core in the In state and satisfies with on being the excitation energy of the 12 th C 1 th t t . - n en sae Htéi."wn§1,\ Thus the center of mass energy of the 0 particles which th leaves the core in its n state is given by Iin:= ii " Rim Putting our wave function into the Schroedinger equation, multi l in both sides b and inte ratin over p y s y (he ck)” s s all coordinates except the r, we obtain 5: .93: _ L..(L.+\) 3 _ ' ' a. m w >+ El no field V.-u.l..> ‘J K K.I (v) (2) I fi where (L, L),1 =‘= (1.0 32.“), M u is the reduced mass of our system. The effective potential V can be defined to be35 a-12C 3 (A) V“; An. <1 “ d- c 16 where ?(r) is the a density function of the 12C ' = Va-a Va—d + Vcoul Va-a is the phenomenological a-c potential Vcoul is the Coulomb potential between the two a. Now let us expand the potential in spherical harmonics v,_.. 2 vim 7,3915) (3) c Bun H m I Z V, n) Dmgelclg.) 7330.49) Junk where A m I I ’ V = 6’,¢’ refer to the body fixed system e,¢ refer to the a coordinates in the space fixed system 61,¢1 refer to the Euler angles between the body fixed and the space fixed syStems. Equation (2) can be rewritten as (Appendix A) 17 ‘k‘ A‘ Law“) I \3—(17! _ Y‘ > + a] Run“) = Z Z V“(v) A(L.1.L;1.:(V\KIKI)S. R (r) \umk 1.11.: (5) I I I ) 6‘3””Au"”bk‘laleXMIHMBLCH) A(t.l. 1.1.: n K «a = ) w; WNW”) .( u. LL00}! o)(1.:l K’mll. K) U0. 1.1.111332) (5a) where U(LI,L'I';J£) is the Racah U—coefficient (J1J2M1M2IJM) is the Clebsch Gordon coefficient Solving these coupled channel equations, we can find the energy En and the wave function. Problem of Wave Function Symmetry The alphaparticles must obey Bose-Einstein statistics. In an exact a theory, the Hamiltonian would be symmetric with respect to d coordinates, and the wave functions would have definite symmetries, so all that would be necessary would be to select out the totally symmetric states. This is obvious in the 2d problem, where all that is necessary is to throw out the odd parity solutions. In our model with c+ 12C, symmetries are no longer evident and there will be 18 extra states that are not obviously spurious. However, we can construct a model for the 12C as c+ 8Be, and calculate the matrix element of the a exchange operator. Ideally, this matrix element would be +1 for the true states, and all other states could be rejected. The details of the construction of this operator are as follows. Let us define the exchange matrix elements as I EQ ._. (Jags agave,” qupeanan a.» Here the notation\¥J(dl++02+c3+au) means the wave function of the number 1 a particle moving relative to the core which made up of the three a particles a2 a and on. P is the 3 12 exchange operator exchanging a1 and a2. Now let us decouple the core into one a moving around a small core ( 8Be), so the wave function becomes Renew» = w, a. .. use.» = Z MlYta.)®<}13w+ 854)]: ll. I“ (t) Z 9n. LL 7(a) e [Z 41(jta.)af‘t'3.))] . I R kw) u flit) “21-: Y. I 1 7|... (9%) 6 [703) Qfltc] Sq) 1.11 n 3 19 where r is the distance between the C.M. of 120 and l the a2 r2 is the distance between the C.M. of 8Be and 12 the a2 in C system. Here RLnI (r1) represents the radial part of the wave function n when a is moving around the core 12C, and U (r ) represents 1 1112 2 the radial part of the wave function when d2 is moving around 8 the core Be. We can then express the wave function with particle l and particle 2 interchanged as Pu \Ckl (4‘9 d‘+‘8*)> = \qu (“’HU‘V" ‘B¢)> R.;;(fi) K:'(fi) = z. t: ‘2. ilk“®.l71:wz‘x;"fi«> fl Using the Racah Coefficient,36 we can recouple this wave function into Pu \ ‘13, (“'Bdlflgtv K' 'Ul) (It) =23, 4—,— L 2 util.’ 31.8110 “‘8 1: I; x 171;“) a (7L: (4.) 697(1;(‘B.))i)3 20 Now we can evaluate the overlap of this wave function with a wave function in our standard representation by neglecting the difference between the CM of the 88c and 120 EC: = <~PI<¢.~«.+‘3.>\ mwzxw ow 83.)) = Z' Z) R ’ ml. 1.1.. k1 1:1: L; 1. ,u.) “1.15““; R150 “111; (m M. x ULLLICJI.’ 31.11) To obtain this result, we require (yd‘fl) 0 Q3 (‘;*33()‘ 71:. (“JG éafidz+83()> 5 g ’ SL1 Ln L. where ¢§ng(a2+8Be) is the angular part of the core's wave n function with angular momentum equal to In. By inspecting the exchange matrix element, we can find the symmetric states of our system. We will come back to this problem in Chapter IV. Matrix Elements of Electromagnetic Transition Qperators To find the electric transition rate, we start with the equation37 B(E1.L)=[Kfij_+—\) (433 “ Eje— Egg Y;— 7LH(91 4”) “‘ID] 21 where n is the total number of a particlesin the nuclei in our case n = N eJ is the charge of the a particle (i.e. =2e). J Now let us separate the right hand side into two terms 3m, L) = “M“ 9% r where I o. = W Hz. u mimic“ 9t) and | 4 L 0'1: Jill-4r t) (‘13: “ 11-7; T3 7.5% #3)“ ‘1’.) Physically we can say the lec comes from the three a particles in the core, and Qa comes from the extra a particle. Using our form of the wave function (Eqn. 1), we can get (Appendix B) co *’ I I L 3; In+LK+;I Q“ = Z l [J Riff) ( Rl 1”(r) ark—I) f uni“ o “ X (upomw) w (413 ') LLo) um: If; ; L 1.) (:1an 22 Q = Z. POL” U133,- QU‘XW‘B “c. L11; 1" (AU |) (M1? \) U (1‘1“ If J, 3 LL.) i 3* 33 1 x1 RLLLU) RLLILM M1 Quc(1.."*1n) where [Ql2 (In'+In)]2 is the electric transition rate C between the states In' and In of the core 120. In our calculation, we use the experimental values and the values predicted by the rotor model as our values of [Q12 (In'+In)]2. C Quadrupole Rate with Full Exchange As we mentioned before, ideally the exchange matrix elements would be +1 for the true states, but in our calcula- tion we found all of them are very much smaller than +1 except the one for the ground state; the reason for this is presumably because of the inadequacy of basis: no intrinsic excited states of 120 are being considered. Because the exchange matrix elements are smaller than one, we no longer can rely on the formula mentioned in the previous section in calculating the quadrupole rate (Eqn. (6) and Eqn. (6.a)). A better approximation for the wave function is 23 ' PHIL) u . 6:...) = NVHP. +P. «PJZL “‘ f ” “)\7.“"WU.E°‘I>°M.) I... . l n 3 Y; E N: C” fiz*fifi‘fiw) \C‘JJ> t J' v_ where NJ can be written as N =(14+12xECJ ) % From Appendix B we can find UP H w.) 716+) R. \‘P >= 2 Z. Z 1 (239“me RIM] 1...}..[1 LM :1, 1‘ 1’ *1 S mu.) Rm: MAR] (131.3» "‘2. \3“) (11W; 1" WWW) I I ’ ’ ’ M M: X U0“ 1"] 1‘ 31‘1“) (yhk(d')\ 7?“) \ 71,1 (01.)) and ( [5577" M 7, n P.

\|

+g<491m ma» M) “ (qu “2 7:69!) (“I * “X + ‘7») H (F) + (*J“ “12*“? P's Y: 7J0") “ CF!) ». (cm i P.” v; m} \\P") + M {RH v3 7.6m} M") * (4’3“ i Pu, {£71010} M75 where {A,B}E A+B + B+A and use has been made of (4?“ P? r.‘ 7. n.) P..- “4?) 2 («3“ PI r3 nu,» H P’) a: (4’ |\ PI r: 7. u.) \l GP") where i i J and i # 2, J # 2. 25 From the previous section, we know @ f; (if: “ ti17’zé’t‘3) “ (Y!) = Q m m, n .3 7. w n P.» = and scam) = [KL—1 002 \\-‘- 2e r ,ymuP, )] we can get B(EQ, A) = “33'th ‘r 0.35%: (WW? 7’. (“3 WWI) + A Q“C <4): ‘ Rah}; ”1963—40 (4):“ 6Q“; 712(2) “CV3 +9 <¢J‘Yu‘¢3> Q“c]k (5) where use has been made Of E. W): E. W) =2 P. PP’> 26 (i.e., assuming the 12C core is totally symmetric) and _3~ q. 3 z ‘3, I, 3’ gm ,2},(

Here is Just ECJ defined in the previous section, 12I so we can write 15039.3) = {NINE [‘l’ o. + 40%». (15%qu We.) ml 6P”) + g Qucx 5cI + [$3 (43“ $1:an 7, cot.) My) Z + 62sz $1..C 1} where NJ is defined in the previous section. Note that the matrix element reduces to the unsymmetrized I matrix element in the limits where Pl2|¢J > vanishes, i.e., B(ELD) " 10. +’ Que-ll (6a) and if we approximate (43“ e P. HP") =2- (we nab") Ea" (cm P..6|H>‘) 2 503 (+‘uew‘3/if 28'...“ , ' 1 we get B(Ec’a)1‘: {NTN‘J i‘r + IQ E6}x[9,+ Q“c]} and (6b) 27 Mean Square Radius with Full Exchange Using the same notations we used in the previous section, .. .... f... _ <41, \.3 m = .3.“ [$wa + E} . o‘mm. n.) W; + 1 by} W,» <43: { P.,r:3\+">] = N3." [of wl P") + :2 «P3 \n‘ PPS + 3 <6? w M 495 . 3 UP“ \ $13145 + 3(4’Pr.‘\+"><¢"m.\¢"> + 3 (4’). fi.‘ if) < (7) where use has been made of «PW ¢‘>= <¢’\r.‘\4>’> = <«Flr.‘\¢’> and <43 \ rim \+"> :2 «PW r: PF} <49. m \ 43> 28 Since PPS-=2, “y” WP > 2 RIM “1 1‘:% {1) L v, ‘7..(°0(®(7'1(°>W}(1(8«))> we can get (41: PM) Z [SPLMN .m 44] 111.111; ‘ I / I * l S Muff») 2131’ (K) At} U(L’1.:J I: 311 > 8L1,’ Note that Eqn. (7) reduces to the unsymmetrized matrix element in the limits where P12|¢J'> vanishes, i.e., I Q, 2 I emit)?- fluhn‘W» 35W. \¢‘>j ma) and if we approximate <4? w P... \+’> = «PW 3453 Ea:f (*3, Pu {.2 ‘¢J'> = ECI'<¢J\{PI\+T’> we can get 29 1 ' J 3' q. 3 2 3' <°rx\n‘\ = N [<4 M4 >+ 3:“ 1n \4’ > +3(<¢’N\¢’>+ <4‘\n’\¢">) a" +3 84314445» (4’; r3 14>">) 53] (7,, Here the ri should represent the distance between the center of mass of the 16O and the iEll a particle. However, in our calculation, we calculate the distance between the center 12 of mass of the C and the iELI a particle. In order to transform the from the center of mass system to our system, we need the following transformation, .. 3 ¢H "' "ii: 4f.) cn ZJC-ée; + ((2; ; 4‘ a; ‘ To include the finite size of the a particle, we use the following equations, 2 ._ z 2 «42.4» - am.- \»> + «mo 8..., where Rd is the radius of the a particle (1.7 fm). In our calculation, we put the value of the radius of the 120 core as our value of , (i f l) 30 Width of the Resonance If the interior wave function is calculated and Joined smoothly into the exterior solution at r=a, the phase shifts can be expressed in terms of the logarithmic derivatives at the boundary r=a: .. (4.. AR: where Hz is the interior wave function with angular momentum 1. If the exterior solution is the Coulomb wave, then we can introduce the real parameter Si by where F2 is the regular Coulomb wave Gm is the irregular Coulomb wave k is the wave number. If the rapid change of the phase shift 82 in a small energy range can be represented by a linear approximation: flue) = u b E. then the width of the resonance can be written as38 S ”4-6- CHAPTER III Numerical Details The calculation of the potential matrix elements is straight forward. We first project the effective potential V 120 (Eqn. 3) onto the basis Y? and find each component Va—(Eqn. h). In integrating Equation (N), we used Simpson' 8 Rule with 20 points in.‘bdirection (from O to w) and with #0 points in 6 direction (from O to g). The size of basis that we require to give a good representation will be discussed in the next chapter. After we found the V?, we can use Equation (5a) to calculate the right hand side of Equation (5). Let us rewrite it to be R 3'” =2 140:1; (0R:1 L“1“U where RLnIn (r) is the second derivative of R: I (r) with Luann “ " respect to r, and we have combined the -———§———-and En terms r LnI into QL.I,(r) 31 32 Remember the matrix elements are functionsof r. So if we have five channels for the wave function and we want to calculate the wave function between 0 and 7 fm in the step of 0.05 fm, we will have 5x5x6%6§-= 3,500 matrix elements to calculate. For our Sigma 7 computer, it takes about H minutes to calculate all these if V _120 is independent of ¢, otherwise it takes about 25 minuges to calculate. Once all matrix elements are calculated, we put them in file for later use in the coupled channels equations (CCE). To solve the CCE, we first have the boundary conditions for each channel wave function at r=0 and r=rmax (where rmax is the outside limit of the wave function, in our calculation rmaxg 7 fm). Let us symbolize these wave function as Einner(r) where the superscription "inner" denotes that they solve the Schroedinger equation in the inner region with 0>' IIC match 1 (r )) . If m - n+1, we can define an (m—l) match dimensional vector g from the following equation: I t inner ( ) _ outer ( ( I”: rmatch)) x E = 0 rmatch Once g is found, the first (m-l) channel wave functions nner nne f = I: x u1 r outer : B outer and V x "2 will Join smoothly at the match point. The last channel wave function usually will not Join smoothly unless En in Eqn. (5) is indeed the eigenvalue. So in order to deter- mine the eigenvalue En and solve the CCE, we have to vary the En until this channel wave function Joins smoothly. There are a few tricks we needed in using the program in order that the En converge quickly to the eigenvalue. First, we match the wavewhich contributes most to the total wave function. Secondly, we choose the match point close to the peak of the wave. The program usually searches H times until it converges to within 10-“ MeV of the eigenstate. In Table (R), we show the time needed for each search versus different number of channels. CHAPTER IV Energy Spectra, Wave Functions and Physical Properties In Chapter II, we showed that if we know the structure 0 and the a-d potential, we can calculate the a-lZC effective potential. The simplest d- model of 12C is a 6,7,8 of 12 static triangle formed by the 3d particles. Let us choose the Z axis to be the symmetric axis of the triangle,140 then the 0(0) in Eqn. (3) becomes ((11) = C E: 3(9)») 4(9’ltg‘)§(<\?- ”(g—'3) T where C satifies S ‘07-) A-Q = 3 12 and r0 is the radius of the C in a—particle model (we choose r0 = 1.85 fm). Because of the structure of the 12 C, the sum over 1 and m in Eqn. (u) is limited to those which satisfy the symmetry 0 ‘ m properties of a triangle (i.e., YLseven or Y1, where l+m = even; m=3xn and n is interger number). In determining how many terms we needed in the summation to give a good representation when we used the A11 & Bodmer potential, we 3” 35 found that with Y (L = O, 2, u, 6, 8, 10, 12), Y L,0 3,i3’ Y5 +3, and Y6 :6 terms, Eqn. (fl) gives a good representation ’- 3 of the effective potential (in a sense that the projected potential differs from the actual potential by less than 10% at the valley of the potential and less than 1% at other point of the potential). Later when we used the stronger 33 3M potential--Neudatchin potential and Vary & Dover potential, we only used U (L = 0, 2, h, 6, 8, 10, 12) terms in L,0 Eqn. (h). For the a—d potential, we first used the Ali and Bodmer potential.29 This potential has a short—range repulsion and L-dependence which come from the exclusion effect. Because , this potential is L-dependent, there will be some difficulties in carrying out the calculation. First, in the expansion of the potential (Eqn. (fl)), we expand it in the basis of Y lm’ where l is the angular momentum of the relative motion between the a particle and the center of mass of the 12C, not the angular momentum of the relative motion between the individual a—a pair. Secondly, we don't know the magnitude of each component of the potential. For the first problem, we assume 12 the C core is rigid enough such that we can approximate the Y1 m in Eqn. (h) to be the relative motion of the a-a pair. 3 For the second problem, we first calculate the wave function of the system by assuming the relative motion is in the S state only. After we find the wave function, we can find the mixing of the potential. Using this information we can 36 find a new potential and recalculate the wave function. In principle we can do this repeatedly until we get a self- consistent potential. In Figure 2, we show the spectra for Ali-Bodmer potential (only the states that have positive exchange matrix elements (see Chapter II) are shown) and the experimental result. It can be seen that although the states are in the right order, the binding energy of the ground state is only about 71% of the experimental value. The excitation 0+ state is unbound by 1.7 MeV while in experiment it is bound by 1.N8 MeV. In 27 they also encountered Mendez and Seligram's calculations, the underbound problem and they concluded that the a-a potential determined by the phase shift methods is not strong enough to give the right binding of the nucleus. To investigate the potential, we first increased the depth of the potential without changing other parameters and we show the result in Figure 3. It can be seen the ground state now becomes overbound if we want the first excited O+ state to be bound at the right amount of energy. We also tried Benn “1 and Chien's potential.”2 But these & Scharf‘s potential two potentials are similar to the A11 and Bodmer's, they give results similar to Ali and Bodmer's. Since we believe the underbinding of the nucleus is due to the fact that the potential is not strong enough, so in the next step, we use the deep potentials suggested by Neudatchin33 and by Vary and Dover.3'4 This type of potential 37 is somewhat simpler for it is L—independent. But because it does not include the exclusion effect explicitly, it usually induces a lot of spurious states and we have to be very careful in choosing the true states. In the resonating— “3 group calculation on two alpha particles system, the wave function for the lowest state that describes the relative motion of the two alpha particles has two nodes. To show that this argument is consistent with the shell model,u3’lu first let us describe the spatial behavior of the two a clusters by 4" Mr l-fiaé‘: {(42.32} H S .7 r-M I ~\— 5: :2 I 3" “N 1...! Ch. where R1 and R2 are the position vectors of the center of mass of the two a clusters respectively and a is the width of the oscillator well. Secondly, let us describe the relative motion of the two alpha clusters by the function P '1. .. 9 > 1 2 "“V 3(0- CYP- EM + 1%“ “3‘6 where C is the normalization factor. 38 Using these notations, we found the usual shell model wave function describing the lowest configuration (ls)u(lp)u can be written as t = A 14:42 1‘? gar/n] (*0 where A is the antisymmetrization operator and'§(o,r) denotes the appropriate charge-spin function. It is clear the function g(r) has two nodes and we have shown the argument is consistent with the shell model. Since we know in c-particle model the a configuration of 16O in ground state is a regular tetrahedron,5 c+120 system if we put the 120 on the x—y plane, the extra so in the a particle should be peaked along the Z axis. If we consider 12 the C as a static triangle formed by the 3d particles, the 12 wave function describing the relative motion of the c+ C should have M nodes along the Z axis. This is how we identify the ground state of 160. To identify the 0; state, we follow the argument given 2 by Terasawa, et al.9 In the 0Ne(d,6Li) 160 reaction, both the 0: and 0: states are strongly excited. This can be inter— preted as the pick up of four particles from the 2s-1d shell for the 0; state, and the pick up of four particles from the lp shell for the 0: state. Since our model describes 16O as an a particle plus a 120, the a-particle in the 0; consist of four nucleons in the 2s-ld shell. This four- state will particle excitation gives four for the number of nodes of the 39 a-particle wave function in this state. Also we know that the a—configuration of 16O in the 0: state is a plane diamond.5 If the 12 C is lying on the x-y plane, the extra a particle should be peaked on the x-y plane and the number of nodes of the wave function describing the relative motion of the a+ 120 should be four along the x-y plane. This is how we identify the 0: state of 160. To project our wave function into the coordinate space, we approximate the internal wave function of the 120 as the rotor model wave function. Because we put the 120 core on the x—y plane, the wave function in coordinate space can be written as R (r) 43:0“ 0) 7:; .JiSY— YLO (006) where r,6 refer to the a coordinates in the space fixed system. and the probability of the a particle located along 9 degree is 12(6) = \l>(r,e)\l Y‘dv where Hts) == \ QMUMV No attempt has been made to identify the 2+ and N+ states, because we no longer have enough basis to describe the internal U0 wave function of the 120. But by looking at the B(E2) transition rates and the a decay widths, we can identify the physical 2+ and h+ states. In Figure 3, we show the spectrum we got for the Neudatchin potential33 and the folded potential by Vary 3H and Dover. We can see the bound states in both spectra are overbound. For the folded potential, the depth of the potential was originally adjusted to fit the a+ l6O elastic scattering data, therefore it is very likely that the potential is too deep for the a-a potential. We adjust the potential's depth to fit the binding energy of the ground state and get the spectrum shown in Figure h. The over all fit of the spectrum is good. We plot the probability of the a particle versus the angle between the a particle and the 12 C core in Figure 10 and Figure 11 (in these figures, solid lines represent the folded potential by Vary and Dover, dashed lines represent the A11 and Bodmer potential and dotted-dashed lines represent the Heudatchin potential). Figure 10 shows that the a configuration of the 16O in ground state is a tetrahedron. Figure 11 shows that the a configuration of 16O in the first excited 0+ state tends to be a plane the diamond for the deep potential, but tends to be a mixture between the plane diamond and the tetrahedron for the All and Bodmer potential. This is because the Ali and Bodmer potential is too weak to produce a bound 0: state. This is another evidence that the A11 and Bodmer potential is not suitable in the structure calculation. H1 The ground state wave function and the 0: state wave function for Ali and Bodmer potential are shown in Fig. 12 and Fig. 13. The ground state wave function and the 0: state wave function for the Folded potential are shown in Fig. lu and Fig. 15. In these figures, only the two most important components of the wave function are shown. In Table 5 we show the probabilities of the wave function for different potentials and the probabilities calculated by Noble and Coelho.31 In Table 6 we show the exchange matrix elements for the states of different potentials. Here we only show those with positive sign, those with negative sign that violate the symmetric property of the nucleus are not shown in the table. The mean square radii (Eqn. 7) (Eqn. 7a) (Eqn. 7b) and the B(E2) transition rates (Eqn. 6) (Eqn. 6a) (Eqn. 6b) are shown in Table 9 and Table lOa. All mean square radii agree very well with experimental results. The mean square radii for the repulsive core potential are larger than those for the deep potentials. That is because the different shapes of the two potentials. The B(E2) transitions for the repulsive core potentials are larger than those for the deep potentials. The repulsive core potential predicts an excessively large transition strength to the 0: state, while the deep potentials predict too small a transition rate as expected. We compare the B(E2) predicted by the deep potential and the B(E2) from Brown's shell model calculation in Table U2 10b. The shell model predicts an excessively large B(E2), while the a model with a deep d-a potential predicts too small a B(E2) We use the formalism mentioned in Chapter II to calculate the width of resonance. To check the result, we also calculate the width by using the R-matrix formalism by Arima and Yoshida.”6 It turns out that both results agree within 10%. Since the width is very sensitive to the energy of the state, in order to compare the width with the experi- mental value,u7 we have to adjust the a-c potential by the method we mentioned before to get the energy of the state close to the experimental value. We list our results together with the experimental results in Table 11. The width we got for the N+ state is only about 50% of the experimental value,“7 while the width for the 2+ state is almost three times bigger than the experimental result. But considering the crude calculation we had for the decay width, it is quite satisfactory that the results are in the right order of magnitude. CHAPTER V Summary and Conclusion We start by assuming the l60 is made up of a 12C core plus an a particle. We use different realistic a-a potentials to determine the effective a- 120 potential. In determining the effective a- 120 potential, we only consider the pair wise a-c potential and neglect all the higher order interactions (e.g., three-body interaction). For the structure of the 120, we assume the 3d particles are in a triangular configuration. The size of the triangle is deter— mined by electron scattering. No intrinsic excited states of 120 are being considered. Knowing the structure of the 12C and the d—a potential, we can find the effective a- 12C potential. We put the wave function of the system in a weak coupling model form, use the effective a-lzc potential, and find a set of coupled-channel equations from the Schroedinger equation. By solving the coupled-channel equation, we find that the bound states of the nuclei are underbound for the weak 0—0 potentials and become overbound when we use the deep a-c potentials. By adjusting the depth of the folded potential of Vary and Dover, we can find the correct binding of the nuclei and still get a spectrum that fits the experi- mental reSult reasonably well. Therefore, the deep potential M3 uu is qualitatively more realistic than the weak potential for structure calculations in 0- particle model. Unfortunately, one problem of the deep potential is that it induces a lot of spurious states. We find the physical states by choosing the states that have the right number of nodes in their wave functions and satisfy the Bose- Einstein statistics. In considering the problem of wave function symmetry, we find the exchange matrix elements (see Chapter II) are smaller than +1; the reason for this is presumably because of the inadequacy of basis: no intrinsic excited states of 12 C are being considered. To remedy this defect, we derive some formulas in Chapter II to calculate different physical properties (except the a-Width). From our results we show that the a-a scattering data do not sufficiently determine the interaction to make predictions on the structure of more complicated nuclei. The interactions with repulsive cores give a very smooth structure to the states and tend to underbind the ground state. Also the transition rates to the 0: state are too large. The deep attractive interactions suffer from opposite defects. The states are now overbound, and differ from each other to such an extent that the transition rates are an order of magnitude too small. 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Reichstein, W. McClure, and Y. C. Tank, Phys. Rev. l§§(l969)l351. J. V. Noble, Phys. Lett. §l§(1970)253. J. Borysowicz, Phys. Lett. 3§B(l97l)ll3. A Arima ands. Yoshida, Nucl. Phys. ggggj197u)u75. C. M. Jones, §t_al,, Nucl. Phys. 31(1962)l. G. W. Greenlees, G. J. Pyle, and Y. C. Tang, Phys. Rev. 171(1958)1115. A. P. Zuker, B. Buck, J. B. McGrory, Phys. Rev. Lett. 2_1_(1968)39. 48 50. F. Ajzenberg-Selove, Nucl. Phys. Al66(197l)l. APPEN DI CES 49 APPENDIX A (01):“ 2;, 7mm») 17:" (9.3%) \(L’1’)SH> L+I+L’*1' M, L 1 3’ LI 1’ 3 = .. -| ( \) (83h) 9213;: ( ) ( H)( , > M! M1 " “(M1 -H “‘ t “HQ 9 (1 \’n,’» Represent the \IH) wave function by a rotational matrix § Dix ' the <1M‘\D:M\1’w\;> can be solved, i.e., I W: X i , <1m,\D:M\1’M;)_-.-. *2; 8n (.9 * (131071541) "KW xm—x M‘mm; o o a *(L 4 (XL! L') (Aw)(a.(+~)(aL’+) 2 and «mm 12. 7 (M) bf (9 W) ’),.,> :.- (-‘Suflnbng Z Z 2 8t j(:‘*‘)(°*+')fetf+u) xx. InK :32‘ Mfi-Ifix .(-0 L13 (1,330 1 L, LR L' M.M"M fin); -H "n, M h.’ 0 0 o x I, {1 1} I) 3,. ML Km, x "I 'X xwyx 50 51 1 Normalize the wave > (i e D1 ‘9 D“ ) ‘1“ . . I “X ‘WI and make use of the following relation (31H) \ \ , I I \ . ~ 1+1” +834 +(+M‘tM+M+M‘+M/+M 1‘ ‘1 on = Z (.‘)\ i l t 3 l 3 1 3 I. ‘1 ‘3 “9“)“, M‘th; x (1! )‘jl)(lx 1‘14: ll 3‘13)((\ {‘ 73) fi‘ "'2 ”I 4‘; M; 'M; “A; 'm‘ “’3 mt! “M1,, ”M5 we can get ((L1)3H\:ZK 7“”) 9:46.30] (L' 1' A“) = Z K-\)l'+t+w\l-\..+L+l /(JL*')(OY+|)(&1tl) ' L1 HT (&I+|)Dl+l) (““4“) l M X' x (fix'wlu) U(L1,L'L') 3!) gym, x APPENDIX B N >= Z; Rum uh; Th) Hi" MW” 91“)"; 723(91):, a.\~17=- ,2, WI” “if“ Wm Mm 71,0917]. 731 1:1 _ 9‘15“") an“) . I , , -27, -—-LY—‘——Ll;_-'——-U(L'1‘Ilg}il) * I 7’1; (3') ‘9 (7)16“) 49 7&1,’(98’)>i>3’ [Ii—:3- (QI “ ‘3 74 “ thxb 2W) E; 621:” Z [3 Rafi") 21:1; (0 {.9 0”,} x 1 gumtu‘) 2111' (mow, ] U (1." 1 3 1,711) x ( 7:01.70 ( 717.57 6911,)1 "I! “ 71.4““) 9(7u‘“‘wfli>i> 52 53 We know (7’35 0891 ll 73“) n 71; (9") 6‘ a?!) U1 flu -.-.- (A) J GIMME i; 13" i k x (Wynn 7! (“J w 71" (4')) 51,1: ELL, / (YL (’1‘) \\ Y( (‘11) “ Y1: (6.3) ‘ W I ' Y; Y). = (A) 1‘ I (1‘ I t>1(31'+')(ufl)} (ALHB 0 0 L?“ and U011: ,J’l:; i '1) L'H'+3'+1' ' a- . y )1“ .‘ L 13,1} (0 fl )(w) {1; 3 1, we can get 9 1(T“)(q m 7‘“ Pd: )= Z2: filgkufi') 21.35““: M] 1. X& .:W I X [S U“ (K) R“, U2) M1] (‘ (n+1 HA.1*:J(’1*‘)01':H)93HY&1H) ‘ \ WK WWW“) Y1 1 1'} 1;; Q Q ”351.1917. 1:31 54 TABLE l.--M1x1ng ratios from TTM's calculation and present calculation. Terasawa's Results Present Results n + + + + + + + + J 02 21 22 23 02 21 22 23 sao+ 85.3 8n du0+ 88.6 0.03 N.A. 8a 0.02 o.u 81:2+ 1.7 29 N.A. 3 31 28 duz+ 1a 3.3 61.6u N.A. 16 5 61 5 4. gn2 6.3 9 N.A. 8 8 6.7 55 TABLE 2.-—Volume integrals of various potentials (Vd being defined in (#8)). Type of Potential Vd (MeV-fm3) A.B.* (L=0) —2o.25 A.B.* (L=2) 162.38 A.B.‘ (L34) u87.06 A.B.' (L=sc)+ 79.3 Chien (L=SC)+ 117.13 Folded 318.2 Neudatchin “37-38 Folded (adjusted) 260.9 § AB 8 Ali and Bodmer +sc - Self-consistent 56 TABLE 3.--Ground state properties of 120 and 160 from Brink.6 Nucleus B Bhf Bexp R fihf Rexp (MeV) (NeV) (MeV) (fm) (fm) (fm) 12 C (triangle) 62 92.2 2.62 2.37 16o (tetrahedron) 9u.u 92.9 127.6 2.62 2.71 2.6M TABLE u.--Computation times for the CCE program. Number of channels 2 h Time for Each Search .0“ sec .17 sec .27 sec 57 + for Ali and Bodmer potential, also the ratios calculated by Noble and Coelho. TABLE 5a.--Probabilities of various 1 in G.S. and $5 state xing :1 11 _____ __ N.C. A.B. Potential J1T G.S. G.S. 0; sno+ N.A.+ 0.u33 0.831 da2+ 0.375 0.511 0.125 guu+ 0.15 0.036 0.021 fu3‘ 0.075 0.011 0.022 has“ N.A.+ 0.008 0.0008 . 31 N.C. = Noble and Coelho +N.A. - not available TABLE 5b.--Probabilities of various 1 in G.S. and 0; state for Vary and Dover's folded potential. --.——-.. -. -_... _— Folded Potential n + J G.S. 02 .9110+ 0.0856 0.698 du2+ 0.357 0.23 gah+ 0.50u 0.063 11:6+ 0.052 0.008 ka8+ 0.0016 0.000u 58 TABLE 6.—-Exchange matrix elements. Potential State G.S. 0: 2: Ali and Bodmer 0.828 0.25 0.11 Neudatchen 0.126 0.12 0.283 Folded (Adjusted) 0.27 0-245 0-0“" TABLE 7.--Ground state binding energies for different potentials. E(MeV) Experiment —7.16 A11 and Bodmer -5.l Neudatchen -22. Vary (adjusted) -7.2 TABLE 8.--Excitation energies for different potentials. + + 02 21 Experiment -l.1l MeV -0-1N2 MeV -0.8 Nev 1.2 MeV . Vary (adjusted) 59 .uxoa mom coomooamcoo weaon ohm oaoappmm a on» no mean opficam onea mm.m ms.~ ms.m mo.m mm.~ mm.m mH.m mo.m mo.m Asov AmO_mm_mowfi mo.~ ~.o mfl.o m~.s m.a oe.a mo.o Hm.m eo.o e.m ANEMV Amo_mm_.m.ov om.m Hm.~ Hm.m mo.m om.~ om.m ms.m mo.~ mm.m sm.m Ashe A.m.o_~m_.m.ov~\ Asv Ansv Aosv Asv Ansv Aosv Asv Ansv Aosv Ammxm .cdm .cum .cum .cvm .cum .cvm .cvm .cvm .sdm cozopwosoz AUoBmSnomv covaom nosoom cam HH¢ m.afiomn opmSUm coozln.m mqm¢e 60 . m H m.m m.e ~o.m ma.ma .mo o.ee oa.mm .mma mo.~o ma+oo +o++m . . . H see. me. ooo. moo. em.o mm.o he.H H.oa m~.e H+m m o++m loo Aoov Aoov gov Aoov Aoov Aoy loos Aoov .oeo A eomovAmmvm .cvm .cvm .cum .sum .mwm .cum .cvm .cum .fimm cm : donooooaoz Aooooanoov oooaom doaoom ode Had .oopoa gossamcoae Ammvmts.ooa mamas 61 TABLE 10b.--Comparison of our B(E2) amd Brown's B(E2)4. * + 2 4 50 Folded A+B 4 B(E2)(e fm ) exp (Eqn.(6b) (Eqn.(6b) Brown 2146.8. 811 0.94 0.53 5.3 2;»0; 40115 68. 39.6 103. * Folded (adjusted) +Ali and Bodmer (adjusted) (see figure.2) TABLE 11.-- cc decay width. State Exp50 Vary(adjusted) 2 1 Rev 2.6 kev 4 33 kev 16.3 kev 62 23 _ .. 5 - \ (2‘) 4 - a: J!” 3 - “~ 2 b E I P 2,‘ x’ -| - 02 -2- \\ #// EXP TERESAWA PRESENT FIG.l.--Spectra from experiment, TTM's calculation and present calculation using TTM's potential. 63 ZW' I u {fr 8\ h ' ll ‘\‘ I o L + 9’”, " ----------- “\‘t E———— , L -2 - " 81: I -4 r- F' r I -(5- I I EXP ALI-BODMER ADJUSTED FIG.2.--Spectra from experiment, calculation using Ali and Bodmer potential and calculation using adjusted Ali and Bodmer potential. 64 3 ' EXP NEUDATCHIN 4t g: 4 H * Ozzdx _4_ Bil \‘1 1.: 3,93 ‘1 1. 5 ’8' '1 § '1 42- g \ '. 46- '1 "‘1. -20- E. FIG.3.--Spectra from experiment and calculation using Neudatchin potential. 65 ’I 401' I5 2'; I, O I 0‘: ”,2 Bil EXP FOLDED FIG. 4.--Spectra from experiment and calculation using adjusted folded potential. 66 FIG.5. --Nodes of a+d system in ground state, dashed line represents the region' where the two K particles overlap. 67 +The place where the wave function most concentrates on. FIG. 6.——Nodes of the 0+ 120 in ground state for Neudatchin potential, the dashed lines represent the locations of the 0( particles in the"C core \ FIG.7.--Nodes of the a+ 12C in 05 state for Neudatchin potential. 69 12 adjusted folded potential. FIG.8.--Nodes of the a+ C in ground state for the 70 12 folded potential. FIG.9.—-Nodes of the a+ C in 0: state for the adjusted 71 _ 1 Vary 0-8 ’ — — - - A. B. —- — - - Neudatchin . POW) I 0.75 050 025 0 con 6 FIG.10.—-The plots of Po(0) for ground state versus cos (0) for different potentials. 72 0.2 P- .1 Vary — - — - A. B. /‘ — -— -— Neudatchin /' POM) 1 I f 0.75 0.50 0.25 0 cone FIG-ll---The plots of Po(0) for 0; state versus cos (0) for different potentials. 1‘ (fm) 1 \V FIG.lZ. Ground state wave function for All and Bodmer potential. d<92 FIG.l3.--O; state r (fm) wave function for Ali and Bodmer potential. up.2 v.1 75 I-I'“ \F' W! r (fm) 010 mur 1.1, FIG.l4.--Ground state wave function for adjusted Folded potential. \V Ho FIG. lS.--0; state wave function for adjusted Folded potential.