' . . . . , .. _ . , . _. . . pm. ‘ m. .-. .... . .M-in ‘ .. .. q... "Atarsx‘v ._._.’........’.‘... -u‘- .‘.“-.vu ..m... . ,. v-..,.._,..._...,. ..y.:.... Tm“..- ._...».-.A- ....,,...._...;.‘. ... .3 m... . w” .. lg?“ «W7? $95-36.“ ‘ . .. . _,,,r 4 . r. ‘ .‘1'. .3.93.1. ”a: w: '.'."".‘ L'J".-".'f"'i.‘. '5 " '.""‘.‘.'.z " -.‘ -> 9‘ -“”- - ‘ . - .. " .‘ZM; ..x 1', .. . “x ..‘.. gum ”ivy-1,,“ _ ._ -.. . _..J< . ._. _ V. ,v. __ , .v . q .. - - .. - ~ . . ... .. . W... A _ . .. VAuon OF THE MANYBODY APPROXIMATIONS _. ’ " m SPHERICAL mucus: f Thesis for the Degree of Ph. D. MICHlGA‘N STATE UNIVERSITY SHAKIR M. MUSTAFA 19}? [Htibln This is to certify that the thesis entitled VALIDITY OF THE MANY-BODY APPROXIMATIONS IN SPHERICAL NUCLE I presented by Shakir M. Mustafa has been accepted towards fulfillment of the requirements for _Bh..D_._degree in _Eh.y.sa.cs_' Major professor Date Feb.22, 1972 0-7639 ABSTRACT VALIDITY OF THE MANY-BODY APPROXIMATIONS IN SPHERICAL NUCLEI BY Shakir M. Mustafa Reduced transition probabilities B(E2), and Spectro- scopic factors Sj for stripping (d,p) reactions are calcu- lated in the quasi-boson approximation (QBA), Random Phase approximation (RPA), Tamm-Dancoff approximation (TDA), and improved Random Phase approximation (IRPA). The effect of the Pauli principle on the J=O pairs is shown to be negli- gible, while it is not for J#O pairs. The effects of these approximations on the B(E2) and Sj systematics are studied by comparing the averages of the absolute deviations from experimental values for each approximation. These studies have been carried out for a large number of spherical nuclei. The problem is formulated by using double-time Green's function techniques. VALIDITY OF THE MANY-BODY APPROXIMATIONS IN SPHERICAL NUCLEI BY Shakir M. Mustafa A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1972 TO MY WIFE KHALIDA, DAUGHTER AND SON DINA AND MAZIN ii ACKNOWLEDGEMENTS I like to express my sincere gratitudes to Professor Truman O. Woodruff for his many helps and discussions during this work. Further gratitudes are expressed to the Department of Physics at Michigan State University, and to the Government of Iraq, who made it possible for me to continue this work. Thanks go to Mrs Jean Strachan for typing the first three pages and to Mrs Debbie Bruchmann for typing the last two pages of the references. I am deeply indebted to my wife for the years of patience and encouragement and thank her for the typing of this thesis. iii TABLE OF CONTENTS Page LIST OF TABLES .......................... l vi LIST OF FIGURES ................. ..... ... viii Chapter I INTRODUCTION ................... 1 II a. Double-Time Green's functions 7 b. Quasi-Particle Scheme (BCS- approximations) .............. 11 c. The pairing interaction ...... 14 d. The Hamiltonian .............. 18 e. Collective Operators ......... 25 f. The Reduced Transition Probability B(E2)............. 27 III The Approximation Methods ...... 28 a. The Quasi-Boson Approximation (QBA) .. ....... .............. 28 b. The First O+ Seniority 0 State 29 c. The Collective Motion ........ 35 d. Random Phase Approximation (RPA) 42 iv e. Tamm-Dancoff Approximation(TDA) IV Deuteron Stripping Reaction ....... V Numerical Results ................ Appendix A .......................... ..... Appendix B ............................... Appendix C ............ . .................. Appendix D .............................. . Appendix E ............................... Appendix F ............................... Appendix G ............................... References ............................... 49 59 67 72 73 76 77 77 82 84 93 LIST OF TABLES Table Page 1. The values of n' for Sn124 and Hg202 isotopes . . . . . . . . . . . . 45 110 2. The values of Nj and Nj/Oj for Sn isotope . 48 3. B(E2) values calculated for the parameters given in tables 11-14 . . . . . . . . 53 4. B(E2) values calculated for the parameters given in tables 15-20 0 o o o o o o o 55 5. The absolute diviations from experimental values of B(E2) from table 4. . . . . . 58 6. The absolute diviations from experimental values of B(E2) from table 3. . . . . . 58 7. The values of S for (d,p) reactions calculated fo% the parameters given in tables 11—14 0 o o o o o o o o o o 62 vi Table Page 8. B(E2) values from KS. ......... 63 9. S£ values from ref. 19. ....... 66 10. The radial integrals RaB' ..... 76 11-15. The values of the parameters An’ AP' An’ and Ap calculated for the set of single particle energies given by KS. 85 16-21. The values of the parameters An, Ap, An and AP calculated for the set of single particle energies given in ref. 10. ... ..... .................... 89 vii LIST OF FIGURES Figure Page 1. Schematic illustration of the Shell-Model 5 basic act of truncation 2. O 6 The first excited (J"=0+ and seniority zero) states in Sn isotopes plotted against the active number of neutrons N. 3. . 34 The values of na and 2va are plotted against20 (ea-l) for Sn in (b, d) and for H92 in (a,c) 4. . . 47 The values of An and A are plotted against the mass number A fog Sn isotopes. . . 71 viii I. Introduction In the fermi gas model of a nucleus, the forces between pairs of nucleons are neglected, and the nucleus assumed to be contained in a sphere of definite volume V with radius R=rOAl/3. While this degenerate gas model give a qualitatively correct picture of the nucleus, actual numerical results for the energy levels are far from accurate. Progress in understanding the structure of the nucleus began after the conception of the shell model of the nucleus by Mayer, Haxel, Jensen and Suessls. The great successes of the shell model, in which the nucleons are assumed to move independently in a certain average potential, showed that the main part of interaction between the nucleons can be reduced to a spherically symetrical, self-consistent field acting on all of them. The next important step was made by Bohr and Mottelsonlz’l3 who proposed the unified nuclear model, in which an additional self-consistent part of the interaction is extracted which is non-spherical and time dependent. With this model it is possible to explain many of the regularities in the low-lying nuclear levels in the language of collective motions. The pairing-plus-multipole-multipole forces model represents a further development of these models. The accumulation of data on the transition rates in even-even nuclei, from the first 2+ state to the 0+ ground state shows enhancement by about a factor of 40 on the average 2 over the single particle estimate, which can be explained by assuming a collective motion of the nucleons. Before the unified model was suggested by Bohr and Mottelson, it was believed that the independent particle and the collective models represented two opposite limits for nuclear physics. The tendency of Fermi particles to pair was rec0gnized 45 (1942) to introduce many years ago. This idea motivated Racah the seniority coupling scheme. Since the J=0 pair is more bounded than the J#O pair, and also J=0 pairs behave like bosons, which makes it possible for these pairs to be close to each other, therefore the ground state of the system can have a minimum energy if all particles are paired. These ideas led Mayer (1950)42 to explain J=0 in the ground state of even—even nuclei. In 1956 Cooper43 showed that two fermions coupled to J=O form a bound state, and behave like bosons. This explained why a fermion system (electrons) can exhibit superconductivity and a boson system (He4) superfluidity. In 1957 Bardeen, Cooper, and Schrieffer (BCS)44 used this idea in establishing their remarkably successful theory of superconductivity. In 1958 Bohr, Mottelson, and Pines introduced for the first time this idea into nuclear physics to explain the energy gap in the spectrum of even-even nuclei. The super- conducting solution of the nuclear problem is characterized by a depressed ground state energy and an energy gap in the spectrum of quasi-particles. The BCS treatment of the pairing part of the Hamiltonian provided a good approximation to the real situation; the other part of the Hamiltonian which contains the long range of the interaction can be expanded in multipole-multipole interactions. This pairing plus quadrupole approximation leads to a fairly good understanding of the prOperties of the nucleus and its quadrupole vibrations. The descriptions of various properties of the nucleus depends upon the model for the nuclear force and the approximation scheme. Several studies have been done for various approximations used in nuclear physics. In most of these studies solvable models were used. In these models N particles are assumed to occupy single level j, and the single particle energy is taken to be zero. These particles are assumed to interact via a pairing interaction. For this kind of interaction the solvable model turns out to give a fairly good description of this part of interaction. This similarity between the degenerate system and the actual systems will be pointed out, whenever it arises, as we study the actual systems. In spite of this similarity, different correlations have been neglected in such models, and, therefore, a study of various many—body approximations in actual systems might be worthwhile. The pairing-plus- quadrupole model has been used successfully by KS for the collective motion in spherical nuclei. In this study a phenome— nological approach is used, in which the single particle energies are taken from experiment. In this method the nucleons in closed shells form an enert core, and only those nucleons outside this core are assumed to be active nucleons occupying partially filled active orbitals as is shown in figure (1). The effect of faraway orbits is assumed to produce an effective charges and effective interaction. This phenomenological treatment has the advantage of redusing the dimensionality of the problem, and it makes it easy to study the different approximations at once and systimatically for a large number of nuclei. In this study I consider the same Spherical nuclei studied by KS and tabulate the average deviations from experiment for each type of approximation.‘ The effect of the Pauli principle is eSpecially noticeable in the random phase approximation. The quantities studied here are the reduced transition probabilities, the spectroscOpic factors for the stripping (d,p) reactions on odd mass isotopes, and the energies of the first exited J=O states, which are calculated in the QBA only, since the numerical calculations showed that the Pauli principle effect is negligible for this part of the interaction. It might be important to notice here that if one does not take the effective charges and instead uses the real charges (ep=1, en=0), one needs to include the faraway orbitals. This will give the same results for the reduced transition probability but requires more labOI'Bu'o Figure 1. Schematic illustration of the shell- model's basic act of truncation. Figurel The active orbits (partially filled) v§¢0 u§¢0 tho Filled orbits (Inert Core) II. as Double-Time Green's Functions Various many-body techniques have been applied in efforts to treat the two-body residual interaction between the nucleons in the outer shells of the nucleous, with these nucleons assumed to move in a self-consistent field produced by the core and to have single-particle energies Ea determined to fit the eXperimental data. The Green's function method furnishes justifications and physical insights into these methods; its physical meaning emerges from the fact that it describes the propagation of one or a few particles in a system of many particles. The hierarchy of equations of motion for the Green's functions provides an exact solution to the Hamiltonian for the problem under consideration; the major approximation enters the decoupling of these equations. Following Zubarev6 we define double-time Green's functions as follows: Ga(t,t') = i 8(t,t')<[A(t),B(t')]> , (II.l) Gr(t.t') =-i B(t.t')<[A(t),B(t')]> r (11.2) where A and B are two operators of the system at different times, and > 6(T) = {3 L3 r and a stand for retarted and advanced Green's functions respectively. The commutator [A.B] AB-nBA , where = {-1 Fermion n +1 Boson , its fourier transform with respect to time G(w) is given by: +00 —l— f dTe 2N in G(w) G(T) << A(t)| B(t') >>w . (11.3) A higher order Green's function P(w) defined by F(w) = << C(t) A(t)| B(t')>>w , can be reduced to the lower order G(m) by means of the RPA decoupling procedure: T(w) = w <>w= wG(w).(II.4) This approximation can be improved by evaluating + m in the first 2 state (IRPA) instead of the ground state (RPA) . The equation for G(w) is given by: mG(w) = <[A(t;%B(t)]>w + <<[A(t),H]| B(t')>>w I where H is the Hamiltonian. Similar equations can be obtained for all higher order Green's functions from which a complete set of coupled equations is obtained. The time correlation functions are +w _. _ , FAB(t,t') = f J(w)e8we lw(t t )dw = , (11.5) +” -iw(t-t') FBA(t,t') = f J(w)e dw = . where J(w) is the spectral function and B = E%_. , K is the Boltzmann constant, and T is the absolute temperature. G(w + iy) - G(w - iy) eBw_ I Y=O' J(w) = i T'l Exploiting the analytical property of G(w) in the complex plane, we can write +oo cm) = —%—1; f (eBE-n) J(E)g—%. (11.6) In the nucleus the residual interactions are prevented from destroying the independent particle picture by the presence of the energy gap in the spectrum of quasi-particles. Therefore the Spectral function J(w) can be writen as; J(w) = I(wi) (Hm-mi) + I(-wi) (Hawaii) . Using the above expression for J(w) the expression for G(w) takes the following form; 10 8w -Bw _ l e - e - _ - le’ - Er [ 5:5;3— 1(wi) + ‘aizgé’ I‘ mi) 1 The Residues (Res.) of G(w) at the poles (w=iwi) are given by 8w. Res.G(m) = Limit(m-wi)c(w)=§%(e l-n)1(wi), w=wi w+wi -Bw (11.8) _ . . _1 i _ Res.G(w) — L1m1t(w+wi)G(w)—§?(e -n)I( mi). w=-wi w+-wi In many cases we need to evaluate the expectation values of the product of two operators at the same time. Setting t=t', equations 7 take the forms: +00 ‘ +00 = f J(w)dw , = f J(w)edew . (11.9) Eqns. 9. 10, and 11 then yield: l Res.G(wi) Res.G(-wi) 2F = Bwi + 'Bwi , (e -n) (e -n) (11.10) , Res.G(w.) Res.G(-w.) §FKAB> = -B:- + 8w.1 (l-ne 1) (l-ne 1) 11 Taking the limit T + 0 (Since the thermal energy is very small compaired to the nuclear energies) we obtain = 1&1 Res.G(w) n w=-w i (II.10) = +2n Res.G(w) w=+wi If Green's function is written in the form G(w)= N/D(w) , then the eigen values are given by the poles in the Green's function and the eigenvalue equation is given by D(wi) = 0. Expanding D(w) around the pole mi , the residue of G(w) at the pole is given by N I Res.G(gl = IEBTETT (II.11) w—-wi 8w w=iwi where = constant. b: Quasi-Particle Scheme (BCS approximation) The elementary excitations in a system of non-interacting (or independent) particles are simply desicribed by €(p)=p2/2m; as the interaction is turned on, the propagation of a particle in the system is affected by the presence of the other particles, and as the particle moves it will push and pull other particles. Thus the particle plus its surrounding environment will behave like a new particle, characterized by a new normalized 12 parameter m* (effective mass); the effect of the interaction with other particles is all included in the new effective mass, and the elementary excitations are now described by the free-particle-like expression; s(p)=p2/2m* . In this manner the original interacting particles are transformed into independent quasi-particles. It is important to notice here that the quasi-particle picture is based on a mathematical device; its main object is to keep the independent particle picture from being destroyed by the interaction. The independent particle idea is very important for the Hartree—Fock treatment of the nucleus in which the nucleons are described by independent single particle wave functions. The quasi-particles we face in nuclear physics are determined by the short-range interactions (such as the pairing interaction), and in this case the quasi-particle is just a mixture of particle and hole near the Fermi surface. The transformation to a quasi-particle description is effected by means of the Bogoliubov- Valatin (B-V) transformation: 1. = - s v E a m - E -m aa uaca a ac~a , a ( , a) , a (a, a). (II.12) t a— a+ = u c - s v c , s =(-) ma a a a a -d a where a:, aa are the quasi—particle creation and annihilation Operators, cl, Ca ; are those for particles, and a stands for (a,ma) , and -a for (a'-ma)’ with a=Ja , 13 the angular momentum. In order for this transformation to be canonical (i.e. preserves the commutation relation), 11a and va must satisfy the following: 2 2 a + Va = 1; - , (11.13) with this transformation the energies of the elementary excitations or quasi-particles are given by: 2 1/2 Ea =1 (ea-x)2+A 1 . where ea is the independent-or single-particle energy, A is the chemical potential (Fermi energy) and A is the energy gap. The point is that the Spectrum of elementary excitations of the quasi-particles has an energy gap. This energy gap is very important in determining the validity of the independent- particle picture. The B-V transformation by itself is an exact unitary transformation. The major errors come from the neglect of certain terms in the transformed Hamiltonian, which are assumed to be small, in the approximation usually referred to as the BCS approximation after Bardeen Cooper and Schrefer, who introduced it in the theory of superconductivity. In this approximation, the wave function is not an eigenfunction of the number of particles Operator, i.e.. [H,Nl 7‘0 . 'with a resulting small effect on the energy eigenvalues of the low-lying states. This can be seen from the exact solution 14 of the pairing Hamiltonian in the in degenerate system, i.e. Ev - E0 which is independent of N. Here v is the seniority no.. = % v(2a-v+2), (11.15) 9 is the pair degeneracy. g is the strength of the pairing interaction, E is the ground state energy;'4Kl . The quantity 0 v: represents the average occupation number of the original particles, which interact with a short-range interaction, which mixes the particle and the hole, so that this distribution has a diffuse Fermi surface in contrast to the situation for independent particles, where the system has a sharp Fermi surface. 0: The Pairing Interaction After separation of the self-consistent part of the interactions there remains some interaction between the particles; the so called "residual interaction". This residual interaction is rather weak, but it plays an important role in determining various nuclear prOperties. The pairing interaction is one example of such a residual interaction. In this interaction two particles occupy atates related by time reversal, i.e.(a.ma) and (a,-ma) . are coupled to angular momentum J=O, since the J=O state is much more strongly bound than the others, so that the neglect of J¢O states is a good approximation for the pairing interaction. The short range forces play an important role in the formation of bound 15 pairs provided that the interaction is attractive in a sufficiently large neighborhood of the Fermi surface (coherent interaction). To see this, first notice that the energy of a particle close to the Fermi surface is eF=VF(p-pF) where VF and PF are a the velocity and the momentum at the Fermi surface respectivly 6. Now the Schrodinger equation for two interacting particles, (E-H0)w = Vt or w =(E-HO)-1VW where 30:: the kinetic energy, and V the interaction energy, amd E=-A is the binding energy of the system. In the momentum representation the wave function,(pufi for the bound state can be written as +oo < V '> W(P)=- f Eieig‘) ¢(p')dp', (II.16) whereE;Qfi is the eigenvalue of Ho . Let V: -g = constant in a limited range of p. In this case W(p) will be approximately constant, therefore equation (II.16) reduces to d = .17 “FEET 1 . (11 ) For small values of g; A will be small too, and equation (11.17) will have a solution if QL elp) or for three-dimensional case I 16 2a IE??? is very large to compensate for the small values of g. Substituting the value of €(p) near the Fermi surface we get im. pgdp' - 0 VFp' — m ’ pm.:pmax where p' = p - pF. This shows that in the presence of the Fermi sea even a very weak attraction will produce bound pairs near the Fermi surface. In superconductors the interaction transmitted by the phonons is coherent at the Fermi surface, and it produces the phenomenon of superconductivity. In nuclei. however no one has been able to prove the existence of such a coherent interaction. However the following, experimental facts indicate the existence of pairing interaction; 1- All even-even nuclei have J=O ground states. 2- The energy gap in the elementary excitations is equal to the energy required to break a pair of J=O. 3- In the odd nuclei the odd nucleon is weakly bound while it is strongly bound in the neighboring even-even nuclei where it is assumed to form part of a pair. Therefore the experimental data on the odd-even mass difference can provide good information about the pairing strength g. One can choose g to satisfy the gap equations and to fit the odd- even mass difference at the same time. The odd-even mass difference is given by: l7 B(N+1) + B(N-l) - 2B(N) = 2%} , where N is the odd mass no., B(N) is the binding energy of N nucleons and E is the quasi-particle energy. The short-range forces causes two particles which occupy time reversed states to scatter each other twice from their common orbit to another orbit where they still occupy time- reversed states (a.ma) and (a.-m&): in this way they scatter to all possible a-levels and hence cause the particle density distribution to be Spherical, while the long-range part of the two-body interaction tends to align the nucleon orbitals and produce a deformation. This competition between short range and long range forces determines the nuclear shape. the pairing interaction has to be smaller than the field interaction in order for shell structure to exist, but as we saw, only a small residual interaction is needed in order to create bound pairs near the Fermi surface. There exit two approaches to the pairing interaction; the first is the general BCS or Hartree-Fock-Bogoliubov treatment in which the single-particle energies and wave functions must include the effect of the residual interaction for the valence nucleons as well as the core nucleons, and ed is a solution of the general Hartree-Fock(HF) equations. In this case the pairing interaction includes J¥0, and A and A are not constants but rather they change with the 18 excitation energy. The second approach which is ad0pted in this work. is the phenomenological approach in which the single- particle energies, ea . and wave functions are assumed to be known in advance. Where Ed is taken from experiment, and the contibution to the pairing from J=O pairs is taken to be a constant, g; the contributions from Jio are neglected as they are assumed to be small. This method gives good results, since the experimental values of E}, include all the neglected effects; also A» and 1) are taken to be constant and one uses their ground states values which are solution of the BCS equations: 9b 2/g=z—— , Q =b+l/2 I b Eb b (II.16) e —l N = zab{1— g } , b b where the pairing constant g is taken to fit the odd-even mass differences, and N is the number of nucleons outside a closed shell. The above equations are valid for neutrons and protons seperatly. d: The Hamiltonian We can seperate the Hamiltonian to three parts as follow: H = + + H0 Hp HQ-Q 19 where H0 is the single particle Hamiltonian (kinetic energy) and is given by: = Ziegcg+cg , (II.17a) Hp is the pairing interaction and is given by: __ l 5 5+ 5+ 5 E Hp — I ggég SaSBCa C-dc-BCB , (II-17b) Where the matrix element of the short-range interaction is assumed to be larger in the states coupled to J=O i.e. . 2 -. 2 -. . i<(3a) J=0|V|(JB) J=0>|>>|<(JG)ZJ#OIV|(38)2J#0>l . and approximately given by: 2 -g , and Hq_Q is the quadrupole-quadrupole interaction. This part of the Hamiltonian which induces the collective vibrations is assumed to have the following form: HQ-Q = - %ZZZZZZXE(-)qu(aY)qgu(Bd)c§+cg CECE asyégu — %Xpnzzzzz(l+R(np))(-)“q§(a1)qfu(85) aByéu ch+cn+cpcn , (II.17c) a B 6 y where H(n,p). interchanges n and p in all the expressions which follow it in the term where it appears. Following KS and others,* no pairing interaction between neutron and proton is assumed,the effect of p-n pairing is assumed to be important in light nuclei. A<70. although its effect is not very well understood. There exist no reliable method to treat such 20 interactions; their effects can be shown to be incuded in the experimental values of Ed , and hence they are not included in the Hamiltonian. The pairing part of the Hamiltonian is to be diagonalized by means of the BCS approximation after transforming to the quasi-particle scheme by means of the B-V transformation which leads to a new Hamiltonian in terms of the quasi-particle Operators aaand.a; . After the BCS approximations the number of particles is no longer conserved and therefore a chemical potential A has to be introduced as a Langrange multipier. its value to be fixed by requiring that give the correct mean value of the number of particles. The inverse to the B-V transformation is: +_ t c — u a + s v a , a a a a a -a _ t c — u a + s v a . a a a a a -a We define the seniority zero pair operators for quasi-particle as follow 3 _ l i i Aa— saga-a ’ Vfla ma>0 (II.18) _ t i. Aa- (Aa) I a In ad. 0. Their commutation relations are: 21 T _ _ a a t t [N ,A ] = 26 A , a b ab a (II.19) [Na'Ab] =-26abAa . In terms of these Operators the pairing Hamiltonian takes the form: H +H = U + H + H + H = H: P P O l 2 res. where _ 2 _ _ 4 _ 2 U- :[ZQava(€a A) gflava] g( :Qauava) r _ _ 2_ 2 H1— :Hsa l)(ua va)+2guava( Zflauava)]Na, a (11.20) H = 2(2/9 au aav (s a-l) ( za v)/a (u2-V2)](A+ + A ) 2 a g b bu Vb a a a a a ' __ 2A Hres — g :é/aa ab (u: A: -v :A a)(ubA b- vb Ab)+gZZ/% ua va {Na (ubA b 2 t 2 t 2 -vbAb)+(ubAb- vbAb )Na }- ~g ZEu av aubv bN a Nb . ab By minimizing the ground state energy U or equivalently setting H2=O, we find the following results: Qa 68-1 2/9 = X E— , N = ZQa(l- ) a I E a a a where 22 _ _ 2 2 1/2 _ Ea-[(ea A) +A ] , A —g :Qauava , e -A 2_ l _ a 2_ _ 2 va- 7(1 Ea ) , ua—l va . Equations (11.21) are known as the BCS or the gap equations, the first two of which have to be solved samiltaneusly for A and A . In order for a superconducting solution to exist, the pairing constant g must not be very small and the level separations of the single particle energies must not be very large. This does not means that no pairing correlation will exist for small g. but rather that no advantages can be gained by transforming to the quasi-particle picture. The ground state energy of the system is given by: U'= Z[(e -l)29 v2- 9 v4]— 93 +AN (II 22) a a a a g a a g ’ ‘ where N is the number of particles. The new Hamiltonian takes the form: H'-U = 2E N + H aa a . . wnere the term are res. s. is usually neglected in the BCS approximation, as it is small compared to the diagonal part of the transformed H, where the diagonal part represent independent quasi-particles with the excitation energies for seniority zero, J=O+ states given by wa=23a . In this way one can construct n states, where n is the number of levels in the major shell. One of these states is a Spurious state introduced by violating the conservation of particle number. It is well-known that each time a conservation rule is violated a spurious excited state will appear. 23 If the rest of the Hamiltonian is included, the number of particles is again conserved, and the spurious state disappears. The pairing Hamiltonian for both kind of particles is: ._ _ E E E H U — :élsawa + Hres.] . (11.23) After transformation to quasi-particles by the B-V transformation. the quadrupole Hamiltonian can be expressed in terms of the following operators: AiabJM)= 2 (alem m M)a+a+ A(abJM) =[A+(abJM)]+ 1;le a8 (18’ I B (11.24) 0 b+m8 .1. A (abJM)= %a%B(-) (alemamBM)aaa_B . where (alemam M) is the Clebsch-Gordan coefficient. The 8 following recoupling formula will introduce terms in the Hamiltonian which contains Racah coefficients W(ach;LK): L J _ . K J. [(adxa8)uxay]M - :3 VQKQLW(ach,LK)[aaX(aBXay)q]M,(II.25) these terms are neglected as they are small compared to other terms. The final result is E _ - E E _ ml, 5 E 5* _ H— zsawa ZZZXZZ%5 q (bd)q (ac){( ) 4 UdeacA (bd2 u) a abcdug XA€+(ac2u)+ %UE UE A€+(bd2u)A€(ac2u)+ %(-)UUE Ug bd ac bd ac XAg(bd2u)A€(ac2u)+Ug vE (-)“A§+ 50 E 5 bd ac (bd2-u)A (ac2u)+V Uac bd XAEO(bd2u)A€(ac2u)—5(-)d—ad vg vg zz(-)KW(bdac;2K) uo bd ac Kq at E 1 up n p XA (aqu)A (chq)}- I— £Z££2(1+R(pn))x q (bd)q (ac) abcdu xH-H‘lun up An+(bd2-u)Ap+(ac2u)+ in“ up An+(bd2u)Ap(ac2u) 4 bd ac 2 bd ac 24 +(- )“ W4UEaU§ A n(bd2u)Ap(ac2u)+(- -)“Ubdv§CAn (bdZ-u) XApO (ac2u)+VdegcAn W(bd2u)Ap(ac2u)-5(-)d- sovgdvgc X22(-)KW(bdac;2K)An+(aqu)Ap(chq)} , (II.26) Kq Where Ugb= ugvg+ ugvg , V=§b )1qu Viv]? , and X is the strength of the quadrupole force; its value is to be fixed by fiting the energy of the first 2+ state with the experimental value, and a-b q(ab) = : (—) q(ba) , (II.27) is the non-dimentional matrix element, and a0 is given by m is the nucleon mass and mo is the harmonic oscillator frequency, and a,B,6, and y are the harmonic oscillator single-particle wave functions. Applying the Wigner-Eckart theorem to q(aB) yields: (b2almBMmma) q(ae) = (allazr 2y Ilb) , J=2 /2a112“ 2 1 1 2 but (b2almBM ma)=(- )C usl a+ 1 / (baZImB-ma-M), therefore q(aB) =71—s M(ab2|m -m BM)q(ab) . (II.28) 25 The quadrupole P-N interaction has the effect of lowring the energy of the 2+ states; neglecting it will result in increasing the energy due to the increased symmetry energy which is associated with the independent polarization of the neutrons and protons. The exact commutation relation for the pair Operators can be obtained by using the recoupling formula (Eqn.II.25); the result is (Appen. B) i _ _ _ _ a+b-J [A(abJM)’A(chq)] _ 6MqGJKwachd ( ) Gadébc) +p(ab)p(cd)(-)a+d+J+K zz/T2K+1)(2L+1) Lu de o X{aJL}(LKJquM)A(adLu)GCb , (11.29) where P(ab) = (1 - (—)a+b+JR(ab)), and R(ab) is the exchange Operator; R(ab)w(ab)=w(ba)' e: Collective Operators The collective vibrations can be described by means of collective operators BJM and 83M, which obey the following equations of motion [B+ H] = -wB+ [B H] = wB (11.30) JM' JM ’ JM’ JM ’ where w is the energy of the vibrational excited state.The ground state wave function is defined by the set of equations BJMIO> = 0 (11.31) 26 The vacuum IO> just defined is different from the BCS vacuum and is believed to be better than the BCS vacuum. The excited state wave function of angular momentum J is given by: 3+ l0> = lJM> (11 32) JM ' ° This collective Operators can be expanded in terms of A(ab) and A+(ab) with real coefficients; T = l a 5+ _ _ J-M a a BJM 2::§[w(ab)A(abJM) ( ) ¢(ab)A(abJ-M)] (11°33) where E . _ E w(ab) ‘ ' J-M 1‘ ¢§ab) = (-) The amplitudes w and ¢ are defined in Ref. 2. The + eXpression for BJM above is obtained as in Ref.2 with the help of quasi-boson commutation relation; see Eqn.(III.1) If the exact commutation relation had been used (see EPA approximation, Sec. d )the result would be + _ 1 ._1_ g g+ _ _ J-M g g BJM- 2 ii: a5 [w(ab)A(abJM) ( ) ¢(ab)A(abJ-M)] ab I _ T ab - 1- na- nb ’ ni _ (aiai> . (11°34) 27 These quantities will be drived later on when we treat the RPA approximation. f: The Reduced Transition Prgbability B(E2) The ground state reduced transition probability B(E2) is given by: B(E2) = ZZII2 , (11.35) Mu u where Q2“ is the quadrupole operator. In terms of the Operators A(ab) and Af(ab) the quadrupole operator takes the forms ef giab) a 5+ u a Q = 212 u (A +(-) A _ ) , (11.36) 2H abg 2/§ a: ab (aqu) (ab2 u) for U = M = 0 equations 11.35 and 11.36 yeild: _ l E E E a 2 B(EZ) — “‘Z |zzzefq§ab)uab(w(ab)+¢(ab))| , (11.37) 4aO abg where ef is the effective charge (egzl. e:=2). The above result is obtained after appling the Wigner-Eckart theorem to the matrix element q(ab) and using the definition (II.2U) for the operators A(ab) and A+(ab). Terms which contain the scattering Operators A°(ab) make zero contribution. since these operators have non-vanishing matrix elements only between states with definite numbers of particles in contrast to A(ab) and A+(ab) which connect states with different number of particles. 28 III. The Approximation Methods In the equation-of-motion methods, one calculates the dynamical observables by calculating the appropriate relationships, rather than calculating the absolute wave functions. The most interesting quantities in any nuclear system do not depend on a knowledge of the wave functions themselves, but rather on certain relationships between them. Thus the equation-of-motion methods reduce the labor involved in calculating a certain observable, as calculation of the entire complicated wave function is not required by these methods. To linearize the equations of motion one must depend on some kind of approximation scheme. The approximations QBA, TDA, and EPA which are studied in this work all lead to linearized equations of motion. In particular the equations of motion for the Green's functions can be linearized (decoupled) by means of the above approximations. a: The Quasi-Boson ApproximatioanBA In this approximation a pair of particles (Fermions), their angular momenta coupled to integral J, are treated as bosons; this means the Pauli principle is neglected. Therefore one expects QBA overestimates the number of particles which participate in a certain collective states. As a result of this approximation the enhancement predected for the reduced transition probability B(EZ), is larger than 29 the single particle estimate. The effect of the Pauli principle will be demonstrated in (III.d). The commutation relations used in the QBA are: + a+b+J [A(abJM)'A(chq)] = 6JKanMacabd - (-) Gadébc) ’ ”r _ [AalAa] —' 6ab I (111.1) + _ + [Na’Ab] _ 26abAa ’ [Na’Ab] =-26abAa . For the first two relations to be valid to a good approximation. Né/Qa_‘must be small i.e. the number of particles in level a must be smaller than the number of available states. It is clear from the above commutation relations that the different modes of the system are well + + separated, i.e. J20 and J=2 modes are independent. b: The First 0+. Seniority 0 State The exited states for this mode of vibration are usually described by the independent quasi-particle Hamiltonian where the excitation energies of the system are given by u>=2Ea. If only the important part of fires is included. a diSpersion relation for these states can be obtained. A pair is two nucleons with their angular momenta 30 coupled to a resultant angular momentum J=0. The lowest energy state of the system is that in which as many nucleons as possible are paired. The number of unpaired nucleons is referred to as the seniority of the state. Addition of extra pairs to the system will not change the angular momentum or the seniority of the state. Therefore seniority 0 states can be generated by succesive applications of the pair Operator A+ to the vacuum, and are given for the 3n configuration as a follow: where n is the number of particles, and 29 is the maximum number of nucleons that can occupy the J-level. The Hamiltonian for weakly interacting quasi-particles is given by: (see eqn. 11.20) 2 T 2 2 2 + '= - - - . I.‘.2 Hp :EaNa gigVQaQb(uaAa VaAa)(ubAb vbAb) (II ) where other terms have been neglected, as they are assumed to be small compared to the second terms in this Hamiltonian Hg. We define the following Green's functions: 0 _ T _ Gab(w) - ((Aa‘t)|Ab(t')>$ , n - 1 . (111.3) 0+ _ + T . Gab(w) — <5 . 31 The quasi-boson commutation relations lead to the following motion for G and 6+ (see app.G) 6ab 0 — (“'ZEa)Gab(w)"7F’ r—-2 F" 2 O 2 OT - g Rana: Qc(chcb Vchb) '- g/Qavil/QC(VZGO -uZGO+), C 0 ob c cb (III.4) wastes-vies; -g/fi;u§g/5;- Now let Db = :/§;(u:G:b - viGgg) , (III.5a) and Bb = é/S:(V:G:b - ugGgg) . (III.5b) + The equations for GUJD ) and G (w ) then take the form: r—-2 r—-2 Go (w): Gab _ g Qaua D _ g Qava B ab 2n(w-2EaY iw-ZEa) b (w-ZEa) b ’ -g/n‘v2 g/fi—uz (111.6) aaD aa 01 _ _ _______ Gab(w)_ (axisgy‘ b (w+2Ea)Bb ° 32 Using the definitions of D and B we can find two coupled b b equations for Db and B. ,namelyc 2 w(u:-v )+2E :(u:+V:) l/g Db=Cbub— :Qa D 4E wuiv - 29a a —4E SIN!” (III.7) 2 -w(u2-v2)+2E (u4+v4) 1/ B =c v2- 29 ——E———E-D -29 a a a a a B 9 b b b a 2 2 Db a 2 2 b ' w - a a w -4Ea and by using ZQa/ZEa=l/g . a the result can be written 2 an+bBb= Cbub , _ 2 bDb+de-vab , where w +2wEa (u: -va 2)- -8E a=ZQ a 2 a (w - (III.8) 2 u2 v2 a ua va 2 4Ea) 8E WMNDJ 2 b=£9a a a ZEa ( 2 uv2 uva 2 E2 ' w - 4Ea) 2 2 V2 (111.9) )- 8Eaua Va ) -2wE (112 -V d=zna a 2a a 2Ea(w -4E WNQJN /§_ _ b and Cb_ 2n(w-23b)g 33 Figure 2. The first excited (J=0+ and seniority zero) states in Sn isotopes plotted against the active number of neutrons N. 34 N 053“. .2... $0.. JEN . find . NON . bQN >22 35 The poles in the Green's function are determined by the solvability condition for the homogeneous equations; i.e. ad=b (111.10) Substituting the above values of a,b, and d in equation (111.10) we get the following eigenvalue equation: m2-4(A2+nanb) EX QaQb 2 2 2 2 = O , (111.11) ab 4EaEb(w -4Ea)(w -4Eb) where ni= sit A . From the above eigenvalue equation we see that for each w = + wi there exists another solution at -wi , and hence Res G(w) is not zero. (U'=-U). l c: The Collective Motion The 1% Yap force is decisive in producing the collective 2... states. It has two effects: the first is to introduces a dynamical correlation between the effective nucleons, and the second is to produce the effective charges. Since its action is not limited to those effective orbits, therefore neglecting the contributions from far away shells is assumed to be included in the effective charges. If one tries to use the real charges instead one must include the far away orbits; this increases the dimensionality Of the problem. but leads to the same results34 . The following Green's functions are needed 36 b(w)= <> w ' n1 b(w)= < $ , Gpn+ p (w): < n n n n n __ (ab)’ (ab) ‘_ X (w- Eab)§ b(w)— 2“ I—6q(ab)Uab :§q(cd)U; n+ n an n b 22 ( pd)U:d X(ch + ch)- -——q(a )Ug deq c X(G2p+ + an ) Xn n n n Gn+ n (w+Eab) Gab(w)= ——q(ab)U; big q(cd)UCd (G cd + ch) np n n p p + + §Tfiq(ab)uab Z§q(cd)UC d(epn + c3) . 0 Similar equations can be obtained for G§:(w) and Gpn+(w). 37 Now let n_ n n+ n R _ zZqicd)Ucd(ch(w)+ch(w)) , cd Rpm: Zqu(cd)U§d(Ggg(w)+Ggg(w)) . Cd The equations of motion then take the form: n n1 n <[A ,A 1> q?ab)U (w-Egb)ng(w)= (as; (ab) - 10 ab (ann + xanPn) n q?ab)U + (w+E:b)G:b(w)= 10 ab (XnRn + xanpn) , (111.15) p pn qQab)U§b pn n p pn (w-Eab)Gab(w)= - 10 (X R + X R ) , qQab)Up (w+E§b)G§21w)= 10 ab (xann + prpn) These equations of motion are obtained with the help of buscn commutation relations. The first two equations reduce to the following equation n pn n §_fn) _ X R (l - fann - Fn 5 I and the last two equations reduce to the following equation F“ P R“ 55—? - (1 — 1&1")an = o , 5 where Bab = Ea + Eb , 38 52 52 E n2 n q(ab)U E q(ab)U f€=zz g2 ab 3b , F“: nab , 52(p,n) . (111.15) ab (Eab - w ) “(w—Bab) The poles in the Green's function corresponding to the collective motion are given by the condition for the solvability of the homogeneous equations i.e. Xan 25 22’: Xp (1 - 5 fn)(l - g—fp) — fnfp = o . (111.16) Following K8 the strength of the quadrupole-quadrupole forces between neutrons. protons. and proton and neutron are taken to be equal: i.e. xn a xp a xpn s x. If xpn is taken to be different from the other and such that it is much smaller than they are. then the system of neutrons and protons tends to polarize independently causing or increase in the energy of the 2+ state above the observed value: also B(E2) is then not sensitive to xpn. Therefore xpn has to be Of the same size of Xn and Xp. The eigenvalue equation for xpn=0. seperates into two equations each for each kind of nucleons i.e. £2 £2 E 2 = 22 q(ab)UabEab X €2_ 2 ’ ab (Bab w ) and for Xn a Xp = xpn = X. we get 39 £2 EZE E = 222 q(ak‘))UabE ab abE (Biz-u)2 ) xun . (111.17) The collective solution is that value of in which is smaller than the minimum Eab' In the deformed region where X is large no solution exists. To evaluate the amplitudes 1p< b)and ¢( we start as a ab) follows: E 51 = E (A(ab)A(ab)> 2nRes.Gab£:lw. 1 Now ' _ E <0|AE ab) A(ab)|0> _ §§<0|A(abJM)|Kq> (111.18) Applying the Wigner-Eckart theorem we get 5+ _ (OJKIOMq) - (KIIAEJr IIO) . (abJM) m (abJ) (OJKIOMq) = (-)J+M 2K+l (i<.10|--qx(\;10)(S = <-)J‘*M/’—"2K+1[<-)J‘M M 1 , /2EII therefore g+ 2J JK 5M 5+ — (—) ——3(KIIA IIO) , I abJM) /23:T (abJ) hence (OIAE Ag+ |0> = — <[BJ’I‘WabJND]> _ w(ab) ' Therefore we arrive to the following results G§b(w) . (111.22) The extra terms in the equation of motion of a Green's function are similar to the original terms obtained in QBA except for a multiplicative factor 1 and where the average is taken in the ground state. This can be improved by evaluating n in the first 2+ state which leads to the improved random 1 phase approximation (IRPA). In this approximation the effect of the Pauli principle is included; therefore this method will predict a smaller number of particles is participating in any collective motion than is predicted by QBA. This has 43 an effect on the theoretical prediction of the enhancement over the single particle estimate for the reduced transition probability from the 2... state to the ground state. The relations needed here are (see Appendix D ) = n: 6GB (111.23) E £1 _ _ _ a+b €_ 5 <[A(ab)'A(ab)]:>-(6ac6bd ( ) sada bc)(l na n b) The quantity na represents the quasi-particle average which can be calculated (as in Appendix Ea ) from ’ After applying the Hartree-Fock factorization to the product of four Operators the result is E_ 1 52 = 1 111.24 na* E;— g Cp(ab)D ab ’ Dab T_:_€;; ' ( ) The amplitudes ¢(:b) and ¢(§b) for this approximation will be given by £2 52 £2 52 g w(ab)} = q(ab)Uaba ab 2 [ Piw)] l . (111.25) ¢Z£:) 2w(w + E: b) a where £2 £2 5 E .5 q(ab)U E a P(w)=ZZ 2 a: 3b ab I b(w - E ) a ab 5 E E dab = l - na-nb r 44 £2 £2 N0 = :é¢(ab)Dab ' In IRPA we need to evaluate n AC .- _ . obtained from <2IA(ab)A (ab)|2=>.The results are (Appen01x Ed) .5- 5+ E a —<2|aa aa|2> which is 52 £2 _ a :2 g E[W(ab)+¢(ab))(l nb)+¢(ab)]Dab na = .5 £2 £2 ' N0 + 2(w(ab) +¢(ab))Dab (111.26) where .52 _ £2 52 .5 52 No ‘ :§[(¢(ab)+¢(ab))aab W+¢(ab)]D ° The validity of QBA depends on the smallness of na . As an example. the values of n are listed in table 1. for a 202 12h Hg and Sn . The values of na are plotted againest (8 'A) in Figures 3(a.b). For comparison with particle a 2 distribution v , the later is plotted in Figures 3(c,d) a 45 Table l. The values of n; for Sn124 and H9202 isotopes. Hg202 a h9/2 f7/2 l13/2 p3/2 f5/2 p1/2 n; .011 .015 .007 .171 .319 .296 a d5/2 g7/2 ‘31/2 h11/2 d3/2 124 Sn n; .05 .122 .173 .347 .309 46 Figure 3. The values of n; and v: are plotted 124 against <6. - A) for Sn in . This can be done (see appendix E) by applying the Hartree-Fock factorizationa to . (III.27) 2 a a 110 As an example the values of Na/Qa for Sn are listed in table 2 . Table 2 . 110 . The values of Nj and Nj/Qj for Sn lSOtOpe. 3 d5/2 g7/2 S1/2 h11/2 d3/2 Nj 1.50x10'2 1.10><1o"2 0.36x10‘2 0.50x10'2 0.26x10-2 -2 2 2 zr—fiflfl:i Nj/cj 0.50x10 0.28x10 0.36x10 0.08x10 0.13x10 ‘ Similar results are obtained for the rest which indicate that QBA is a very good approximation for J=0 Hamiltonian. This also shows that J=0 pairs are strongly bound, there— fore they behave mor like Bosons. Similar calculations for J=2 pairs show that QBA is less valid. 49 ea Tamm-Dancoff Approximation (TBA) This method of linearizing the equations of motion can be classified as a higher order phase approximation. In this approximation the Hamiltonian is diagonalized in the Space spanned by a limited number of shell model states. It underestimates the reduced transition probabilities for the low-lying collective states. For this approximation the collective operator T1 _ TE 51 BJM " 1/2 222 w(ab)A(ab) 5 (III.28) abg where T refers to TDA and E _ The excited state [J=2 > is therefore given by _ T1 |JM> — BJMI 0> Thus we obtain Tamm-Dancoff results if we set ¢§ab)=o cn: equivalently G+=O in the previous formulas (Eqns. 111.15). since the solution (mngi for the eigenvalue equation is associated with A+ and the negative solution with 2K in the expression for Iliin the EPA. In this approximation one does not expect the resulting eigenvalue equation to admit negative solutions which implies Res. GT=O. Hence setting G+=O we get 50 n n+ n > (w-En )GnT(w)= <[A(ab)’A(ab)] _ qQab)Uab(XnR.+xnpR, ) ab ab 2w l0 n np ' ' qQabmp _ p in _ _ ab np . p . (w Eab)Gab (w)— 10 (X Rn+x Rnp)’ where n n nT I = ‘ I P P DPT Rnp— :§q(0d)Uchcd (w) . From the first equation we get (for Xn: Xp = xpn 2X ) R, _ Fn(1-bnx) n 1-bpx-bnx and from the second equation we get prFn Rnp = n p ’ l-b X-b X where 52 £2 q(ab)U bg = 22 ——E———39- , ab (Bab-w) n n Fn = q(ab)Uab n “(w-Bab) n Solving for G b( w ) we get a 51 n2 n2 GnT(m)— 1+6ab _ Xq(ab)Uab l , ab — _ n _ n 2 _ n _ p 2n(w Bab) 10n(w Eab) l b X b X therefore n2 n2 q(ab)Uab l E _ n 2121310)] . Res.G:§(+w)= “(w-Bab) E T where 52 52 ET q(ab)U P(w)= 22 E a? . ab (w-Eab) Therefore 12 £2 2q(ab)U ET _ — 3b (21mm 1 . _ E (w Eab) E gT2 w(ab) The eXpression for B(E2) is the same as before except now we use (03:10) instead of wfiab) and put ¢€(ab)=0 . The eigenvalue equation correSponding to the collective motions, for this case is given by: 1- (bn + bp )x = 0 , substituting for b we get the following results: E2 E2 10 q(ab)Uab abg Eab- m From this eigenvalue equation we see that the negative values 52 nT of 0) are not solutions. Therefore Bes.G('w)= 0 ' ‘Which implies that ¢Eab)=ojrhe difference between this TDA and EPA is this: the collective operator 3* is expanded in terms of A+ only in TDA, which is equivalent to expanding the wave function of a certain excited state in terms of two quasi- particles states only neglecting the two quasi-hole states, while these two quasi-hole states are included in RPA where the two kinds of states are treated symetrically. Thus the negative solutions are associated with the Operator A(ab) while the positive solutions are associated with the operator A+(ab). 53 Table 3. B(E2+0) values in units of 10-48cm4, calculated for each approximation and compared with the experimental results (the last column). These results are calculated for the parameters given in tables 11-14. wexp. Isotope A QBA RPA IRPA TDA Exp. 1.450 28Ni30 58 .033 .021 .012 .014 .072 1.333 28Ni32 60 .064 .039 .023 .025 .091 1.172 28N134 62 .096 .054 .032 .032 .083 1.340 ‘ 28Ni36 64 .094 .056 .031 .034 .087 0.992 302n34 64 .636 .439 .332 .158 .170 1.039 30Zn36 66 .604 .418 .311 .159 .145 1.078 30Zn38 68 .551 .381 .278 .154 .125 1.040 32Ge38 70 .665 .456 .329 .189 .172 0.835 326e40 72 .864 .529 .377 .199 .230 0.596 32Ge42 74 1.092 .638 .463 .194 .317 0.563 32Ge44 76 1.010* .765 .507 .184 .263 0.635 34Se40 74 1.213 .617 .436 .217 .210 0.559 34Se42 76 1.225 .680 .489 .210 .480 0.614 34Se44 78 .970 .689 .494 .098 .385 0.666 34Se46 80 .804 .613 .465 .188 .283 0.655 34Se48 82 .811 .551 .421 .189 .213 0.450 36Kr42 78 1.440 .750 .492 .202 .510 0.618 36Kr44 80 .900 .666 .468 .189 .340 0.777 36Kr46 82 .638 .499 .398 .177 .180 0.880 36Kr48 84 .545 .408 .331 .178 .150 1.078 388r48 86 .337 .283 .266 .135 1.836 38Sr50 88 .313 .262 .160 .185 .130 2.180 40Zr50 90 .110 .098 .056 .080 0.934 40Zr52 92 .165 .151 .179 .071 .790 0.920 4OZr54 94 .295 .255 .234 .113 .790 0.871 42M052 94 .208 .160 .139 .087 .270 0.778 42M054 96 .379 .273 .214 .129 .300 0.787 42M056 98 .565 .382 .291 .171 .270 0.536 42M058 100 1.125 .653 .495 .212 .610 0.833 44Ru52 96 .250 .157 .106 .105 .250 0.660 44Ru54 98 .476 .262 .195 .146 .480 0.540 44Ru56 100 .807 .397 .327 .183 .570 0.475 44Ru58 102 1.150 .599 .474 .215 .730 0.556 46Pd58 104 .866 .479 .370 .202 .550 0.512 46Pd60 106 1.077 .620 .473 .225 .650 0.434 46Pd62 108 1.400 .794 .586 .244 .740 0.374 46Pd64 110 1.730 .960 .671 .258 .860 Table 3.(Continued) 54 wexp. Isotope A QBA RPA IRPA TDA Exp. 0.633 48Cd58 106 .631 .379 .275 .168 .470 0.633 48Cd60 108 .718 .453 .332 .187 .540 0.658 48Cd62 110 .752 .497 .370 .200 .500 0.617 48Cd64 112 .853 .555 .413 .211 .540 0.558 48Cd66 114 .976 .605 .444 .217 .580 0.513 48Cd68 116 1.060 .622 .452 .219 .600 0.562 52Te68 120 1.162 .752 .564 .269 .550 0.564 52Te70 122 1.080 .702 .530 .264 .650 0.603 52Te72 124 .921 .617 .466 .253 .390 0.667 52Te74 126 .736 .516 .386 .236 .530 0.743 52Te76 128 .554 .407 .295 .211 .410 0.840 52Te78 130 .395 .306 .214 .181 .340 0.441 54Xe74 128 1.610 .929 .730 .339 0.538 54Xe76 130 1.165 .736 .566 .314 .480 0.668 54Xe78 132 .797 .550 .405 .281 .320 0.850 54Xe80 134 .497 .375 .255 .238 .300 1.320 54Xe82 136 .164 .154 .073 .143 0.464 56Ba76 132 1.7507 1.040 .800 .390 .730 0.605 56Ba78 134 1.179 .783 .578 .356 0.818 56Ba80 136 .732 .542 .371 .313 1.426 56Ba82 138 .218 .205 .122 .185 .300 1.257 508n62 112 .328 .249 .168 .135 .180 1.299 SOSn64 114 .351 .267 .179 .145 .200 1.293 SOSn66 116 .364 .275 .181 .149 .210 1.230 SOSn68 118 .357 .268 .178 .145 .230 1.171 SOSn70 120 .333 .250 .169 .138 220 1.140 508n72 122 .294 .225 .152 .127 .250 1.131 508n74 124 .240 .188 .127 .111 .210 0.790 58Ce80 138 .954 .702 .488 .368 1.596 58Ce82 140 .279 .267 .111 .242 360 0.650 58Ce84 142 1.250 .872 .652 .400 590 1.570 60Nd82 142 .170 .166 .193 .150 340 0.695 60Nd84 144 1.283 .965 .761 .426 440 0.453 60Nd86 146 2.300 1.590 1.260 .493 840 0.747 628m84 146 1.270 1.018 .858 .436 0.551 628m86 148 1.990 1.540 1.271 .502 890 0.334 628m88 150 3.810 2.820 1.977 .574 l 320 0.637 64Gd86 150 1.870 1.570 1.304 .515 0.155 7605112 188 10.550 5.210 3.640 .910 2.800 0.187 7605114 190 8.270 4.480 3.270 .885 2.550 0.329 78Ptll6 194 3.000 2.180 1.790 .701 1.940 0.356 78Pt118 196 2.470 1.790 1.430 .656 1.270 0.408 78Pt120 198 1.820 1.360 1.042 .592 1.350 55 Table 3.(C0ntinued) w Isotope A QBA RPA IRPA TDA Exp. exp 0.426 80Hg116 196 1.290 1.047 .952 .469 0.412 80H9118 198 1.270 .983 .820 .458 1.130 0.368 80H9120 200 1.312 .948 .722 .442 .850 0.439 80H9122 202 .854 .657 .447 .376 .590 0.430 80Hg124 204 .621 .482 .297 .297 0.960 82Pb120 202 .267 .239 .151 .186 0.899 82Pb122 204 .193 .175 .087 .148 .170 0.803 82Pb124 206 .081 .077 .025 .072 .130 48 Table 4. B(E2) values in units of 10- cm4, are listed for each one of the approximations QBA, RPA, IRPA and TDA. The experimental B(E2) in units of e210-48cm4, are listed in the last column. These values are calculated for the parameters given in tables 15-20. Isotope A QBA RPA IRPA TDA Exp. 28Ni30 58 .114 .066 .041 .037 .072 28N132 60 .130 .082 .054 .042 .091 28Ni34 62 .146 .087 .057 .044 .383 28Ni36 64 .112 .070 .043 .039 .337 3OZn34 64 .682 .452 .343 .068 .170 30Zn36 66 .622 .414 .308 .165 .145 30Zn38 68 .496 .336 .249 .145 .125 32Ge38 70 .619 .433 .321 .177 .172 3ZGe4O 72 .910 .515 .360 .200 .230 32Ge42 74 .997 .686 .480 .082 .317 32Ge44 76 .872 .946 .562 .169 .263 34Se40 74 1.290 .617 .436 .221 .210 34Se42 76 1.149 .752 .522 .201 .480 34Se44 78 .857 .746 .531 .184 .385 34Se46 80 .698 .592 .467 .174 .283 34Se48 82 .667 ' .457 .351 .173 .213 56 Table 4.(Continued) Isotope A QBA RPA IRPA TDA Exp. 36Kr42 78 1.404 1.010 .571 .202 .510 36Kr44 80 .814 .719 .511 .179 .340 36Kr46 82 .547 .460 .388 .161 .180 36Kr48 84 .444 .347 .294 .151 .150 388r48 86 .302 .265 .288 .125 38Sr50 88 .313 .262 .160 .185 .130 4OZr50 90 .227 .203 .120 .156 40Zr52 92 .256 .245 .312 .103 .790 4OZr54 94 .403 .369 .356 .146 .790 42M052 94 .289 .254 .270 .112 .270 42M054 96 .497 .410 .350 .157 .300 42M056 98 .741 .529 .405 .207 .270 42M058 100 1.351 .802 .600 .242 .610 44Ru52 96 .306 .232 .210 .117 .250 44Ru54 98 .580 .398 .325 .165 .480 44Ru56 100 .992 .568 .456 .206 .570 44Ru58 102 1.340 .734 .578 .236 .730 46Pd58 104 .975 .560 .438 .213 55 46Pd60 106 1.167 .677 .525 .233 .650 46Pd62 108 1.478 .835 .628 .250 .740 46Pd64 110 1.788 .998 .704 .262 .860 48Cd58 106 .673 .408 .300 .170 .470 48Cd60 108 .747 .467 .345 .186 .540 48Cd62 110 .772 .503 .375 .198 .500 48Cd64 112 .866 .562 .417 .208 .540 48Cd66 114 .974 .614 .450 .215 .580 48Cd68 116 1.030 .627 .457 .216 .600 508n62 112 .328 .249 .170 .134 .180 SOSn64 114 .358 .272 .182 .146 .200 508n66 116 .371 .282 .186 .152 .210 508n68 118 .365 .277 .185 .149 .230 SOSn7O 120 .343 .260 .176 .142 .220 508n72 122 .306 .233 .157 .132 .250 508n74 124 .257 .198 .132 .116 .210 52Te68 120 1.210 .800 .598 .280 .550 52Te70 122 1.140 .753 .567 .275 .650 52Te72 124 .975 .663 .500 .264 .390 52Te74 126 .782 .555 .413 .247 .530 52Te76 128 .596 .443 .323 .223 .410 52Te78 130 .405 .323 .228 .189 .340 54Xe74 128 1.720 1.010 .782 .356 54Xe76 130 1.230 .794 .604 .329 .480 54Xe78 132 .788 .569 .426 .284 .320 54Xe80 134 .411 .326 .227 .209 54Xe82 136 .164 .154 .073 .143 57 Table 4.(Continued) Isotope A QBA RPA IRPA TDA Exp. 56Ba76 132 1.830 1.117 .851 .406 .730 56Ba78 134 1.130 .799 .608 .353 56Ba80 136 .561 .447 .344 .260 56Ba82 138 .218 .205 .122 .185 300 58Ce80 138 .682 .557 .467 .292 58Ce82 140 .279 .267 .111 .242 .360 58Ce84 142 .934 .754 .628 .338 :90 60Nd82 142 .170 .166 .193 .150 .340 60Nd84 144 .898 .772 .710 .343 440 60Nd86 146 1.840 1.500 1.192 .452 .840 62Nd88 146 .804 .727 .764 .327 62Nd90 148 1.500 1.300 1.150 .443 890 62Nd92 150 3.195 2.730 1.980 .550 1.320 64Gd84 150 1.270 1.177 1.156 .426 7605112 188 11.360 5.320 3.670 .960 2.800 7605114 190 8.980 4.681 3.340 .938 2.550 78Pt116 194 3.530 2.480 1.960 .761 1.940 78Pt118 196 2.920 2.070 1.630 .714 1.270 78Pt120 198 2.173 1.600 1.230 .648 1.350 80Hg116 196 1.580 1.254 1.115 .521 80H9118 198 1.530 1.166 .975 .502 l 130 80Hg120 200 1.552 1.105 .861 .476 850 80Hg122 202 1.008 .761 .543 .402 590 80Hg124 204 .742 .561 .370 .319 - .. c.-- 58 Table 5 . The average of the absolute deviations from the experimental B(E2) values for the corresponding approx. The Approximations QBA RPA IRPA TDA B(E2)::p. léB|2 , j I O I and JM:> is the wave function for the even-even nucleus ( the wave function of the excited state in the residual nucleus), and |(jJO),JM> is the wave function of the ground state of the odd target coupled to the free neutron wave function, and is given by: ' . n+ |(3JO).JM>’ = 22 (jJOJlmMOM)ij|J M >., o 0 mM 0 where|JOMé:> is the wave function of the target, which c1n be expanded in terms of the seniority one and three states as follows J o O . 1‘ I O O O|0>+§.§,Cj'(ZJ'Jolom'MO)B20aj'm'|O> , Q where |0> is the quasi-particle vacuum. The coeffients CJ J O and Cj? can be obtained by diagonalizing the Hamiltonian H in the Space of one quasi-particles with zero and one phonon. These amplitudes are tabulated in KS . The particle n+ ]m can be expressed in terms of quasi-particle Operator C Operators by means of B-V transformation n+ n n+ n n C. = u. a. + s.v.a. . 3m 3 3m 3 J J‘m Straightforward calculations (see Appendix F ) yeild the following result for SJ = l n Jo ,n J S u.C w . 3 J0 (3J0) J n o 2 — ij. /(2JO+1)/5| . j This formula is similar to that obtained by Yoshida and Sorensen except for the amplitude wng) the value of which depends on the approximatio usgd. The phase of ijJ) is chosen to agree with that of C].0 (see Appendix F). 62 Table 7 . S values, are listed for each one i of the approxim- ations QBA, RPA, IRPA and TDA. The experimental 82’ are listed in column 7. 2 Reaction QBA RPA IRPA TDA Exp. J: 1 Ni6l(d,p)Ni62 0.47 0.32 0.18 0.37 0.26:.04 3/2' 1 Zn67(d,p)Zn68 0.25 0.20 0.19 0.21 0.11:.02 5/2‘ 1 5e77(d,p)5e78 0.07 0.07 0.08 0.08 0.029:.005 1/2“ 2 Zr91(d,p)Zr92 1.64 0.72 0.36 1.44 1.33:.2 5/2+ 2 M095(d,p)M096 0.87 0.27 0.40 0.70 0.30:.05 5/2+ 2 RulOl(d,p)Ru102 0.13 0.20 0.18 0.16 0.032:.008 5/2+ 2 Pd105(d,p)Ple6 0.18 0.20 0.23 0.20 0.068:.03 5/2 2 Snlls(d,p)Snll6 0.18 0.14 0.11 0.16 0.10:.015 1/2+ 2 Snll7(d,p)Sn118 0.19 0.14 0.12 0.17 0.16:.025 1/2+ 2 Sn119(d,p)Sn120 0.16 0.13 0.11 0.14 0.06:.01 1/2' 2 T8125(d,p)T8126 0.07 0.07 0.07 0.07 0.027:.004 1/2 2 Bal35(d,p)Bal36 0.01 0.01 0.02 0.01 0.32:.04 3/2+ 63 Table 8. . . -48 4 B(E2) values in units of 10 cm from KS. Isotope B(E2)theor. B(E2)exp. 28Ni30 0.017 0.072 28N132 0.051 0.091 28Ni34 0.100 0.083 28Ni36 0.092 0.087 302n34 0.264 0.170 30Zn36 0.245 0.145 3OZn38 0.164 0.125 32Ge38 0.458 0.172 326e40 0.476 0.230 326e42 0.609 0.317 3ZGe44 0.729 0.263 326e46 0.451 34Se40 0.696 0.210 34Se42 0.919 0.480 34Se44 0.770 0.385 34Se46 0.594 0.283 34Se48 0.327 0.213 36Kr42 1.784 0.510 36Kr44 0.812 0.340 36Kr46 0.550 0.180 36Kr48 0.313 0.150 38Sr48 0.205 38Sr50 0.143 0.130 4OZr30 0.141 4OZr52 0.080 4OZr54 0.216} 0'790 42M052 0.166 0.270 42M054 0.360 0.300 42M056 0.683 0.270 42M058 0.915 0.610 44Ru52 0.279 0.250 44Ru54 0.563 0.480 44Ru56 0.947 0.570 44Ru58 1.424 0.730 64 Table 8.(Continued) Isotope B(E2)theor. B(E2)exp. 46Pd58 1.006 0.550 46Pd60 1.261 0.650 46Pd62 1.603 0.740 46Pd64 2.009 0.860 48Cd58 0.447 0.470 48Cd60 0.571 0.540 48Cd62 0.687 0.500 48Cd64 0.758 0.540 48Cd66 0.799 0.580 48Cd68 0.809 0.600 508n62 0.350 0.180 508n64 0.381 0.200 505n66 0.399 0.210 508n68 0.414 0.230 508n70 0.416 0.220 508n72 0.365 0.250 508n74 0.273 0.210 52Te68 1.183 0.550 52Te70 1.307 0.650 52Te72 1.080 0.390 52Te74 0.729 0.530 52Te76 0.468 0.410 52Te78 0.289 0.340 54Xe74 1.654 54Xe76 1.174 0.480 54Xe78 0.592 0.320 54Xe80 0.344 54Xe82 0.198 56Ba76 1.814 0.730 56Ba78 0.929 56Ba80 0.509 56Ba82 0.294 0.300 65 Table 8.(Continued) Isotope B(E2)theor. B(E2)exp. 58Ce80 0.631 58Ce82 0.392 0.360 58Ce84 0.828 0.590 60Nd82 0.361 0.340 60Nd84 0.908 0.440 60Nd86 2.101 0.840 62Sm84 0.900 628m86 2.189 0.890 625m88 4.000 1.320 64Gd84 0.974 64Gd86 1.872 7605112 11.800 2.800 7605114 9.300 2.550 78Pt116 5.200 1.940 78Pt118 4.086 1.270 78Pt120 3.060 1.350 80Hg116 1.250 80Hg118 1.355 1.130 80Hg120 0.982 0.850 80Hg122 0.749 0.590 80Hng4 0.461 82Pb118 0.337 82Pb120 0.280 829b122 0.216 0.170 82Pb124 0.101 0.130 The values of S 66 Table 9. 2 from ref. 19. m 2 Reaction Stheor. Sexp. JO 1.18 1 Ni61(d,p)Ni62 0.320 0.26:.04 3/2 1.08 1 Zn67(d,p)Zn68 0.210 0.11:.02 5/2 0.62 1 5e77(d,p)5e78 0.070 0.029:.005 1/2 0.94 2 Zr91(d,p)Zr92 1.500 1.330:.2v 5/2+ 0.81 2 M095(d,p)M096 0.680 0.30:.05 5/2+ 0.48 2 Ru101(d,p)Ru102 0.11 0.032:.008 5/2+ 0.52 2 Pd105(d,p)Pd106 0.14 0.068:.03 5/2+ 1.30 2 Snlls(d,p)8nll6 0.18 0.10:.015 1/2* 1.22 2 Snll7(d,p)Sn118 0.19 0.16:.025 1/2+ 1.17 2 Snllg(d,p)Sn120 0.15 0.062.01 1/2+ 0.69 2 Te125(d,p)Te126 0.06 0.027:.004 1/2‘ 0.83 2 Ba135(d,p)Bal36 0.02 0.32:.04 3/2+ 67 V. Numerical Results In this study we considered the same spherical nuclei studied by KS, the single particle energies are those given in References 1 and 105the values of Ag and Ag are extrapolated from KS (rough estimate) and then used in iterations to satisfy the BCS equations (gap equation). These values of Ag and Ag are the values which minimize the ground state energy, and their dependence on the excitation energies is assumed to be small and is neglected. The values of the theoretical B(E2) are calculated for two sets of single particle energies: the first set are those given in K8, the second set are those given in Reference 10 , Th: following A—dependences are used: a: = 10.74GX1051A-2/3 I 1 1/3 1/3 1/3 €j(A)—€j(AO)(AO/A) +aj(AO/A) [1-(A/AO) ]+A€j(Z,N). If both j=£il/2 are present in the major shell, aj is given by __ _ 2 O‘sz+1/2‘ (81-1/2(Ao) €£+l/2(Ao))2f:1 ' _ 2+1 O‘11-1/2‘(€1-1/2“7"6"€1+1/2“”‘o)’22:?“ ' If only one of them is present, then aj is given by: 68 _ _ 71 O‘2+1/2" X273 ' o a = 7( +1) 1-1/2 ““ng73 ' where A€j(Z,N) is a Special shift in the single particle energy, and A is the mass number. The numerical calculations of B(EZ) and SJ were performed on the CDC 6500 computer at Michigan State University. The energies of the 2+ states are the experimental values taken from the table of isotopes (Ref.48 ). The results for B(E2) are listed in Tables 324 for each approximation. In table 7 the values of 82 are listed, while the averages of the absolute deviations from experimental values [53(Ez)|av for B(Ez)are tabulated along with the corresponding averages of the experimental values of exp . and B(E2)are av B(E2) in tables 5&6 . The quantities |B(E2)1av defined as follows 1 N th. exp. |0B(E2)|av= fi>iil|BiE2) - Bi(E2)l . exp. 1 N exp av i=1 i th exp where B(E2), and B(E2) are the theoretical and experimental values of B(Ez) respectively, and N is the total numbe: of cases. The values of g are those given in KS and in Ref. 10, where those values are taken to fit the eXperimental 69 odd-even mass differences. To convey an idea of the dependence of A and A on the number of particles; these values are plotted against the number of particles for Sn in figure 4. From tables 5 and 6 it is clear that (for this set of parameters) the IRPA gives the best result, while TDA does not differ much from RPA as far as the absolute diviations from experiment are concerned. Indeed the TDA underestimates the B(E2), on the other hand the RPA overestimates it; therefore, including the Pauli principle in TDA will increase the deviations from experimental values. In this work the Pauli principle is not included for TDA. In Figure 4 the quantities A and A are plotted against the neutron number for Sn isotOpes. We have to know the N—dependence of l and A in order to obtain new values by extrapolation. 70 Figure 4. The parameters An and An are plotted against the mass number A for Sn isotOpes. 71 mo.._ 8... so... 00... .... 0.... m...1 h...L m... .N... MN... mu... 3.: as x . / / / mm. .N. m: m: m: h v 050.“. APPEN DI CES 72 Appendix A Useful Relationships éy (abcl mamBmY )(cdelm Hm5m ) = (-)a+b+d+e5%A/(2c+l)(2g+1)u X {32;}(agelmamxmn)(bngmBmamA) , (A1) abc . . . where {d g} 18 the 51x j-symbol. +b-J .. (abJImamBM) = (-)a (abJI-ma-mB—M) (o4) = (-)a+b-J(baJ|mBmaM) (A3; a-ma = /(2J+1)/(2b+17(-) (anIma-M—ms) (A4) b+mB = 712J+1)/72a+1)(-) (Jbal-MmB-ma) (A5) a—m = /(2J+1)/(2b+1)(-) “(JabIM-mams) (A6) b+mB = /12J+1)/{2a+1T(-) (bJaI-mBMma) (A7) %Q%B(abJ|mamBM)(abJ ImamBM )= GJJ'GMM' (A3) § (alemamBM)(abJIma'mB'M ) = Smama'GMM' (A9) In the following commutation relations the collective operator B is given by: JM BJM ‘ 2::20 6b —{w(ab)A (abJM) ( ) ¢(ab) (abJM)} (“1“) 5+ _ g [B JM’A(chq)] ‘ GJKdqu(cd) (”‘~) 5 _ E . . [BJM’A(chq)] - 5JK6Mq( )J M(Med) (”14) i 5+ __ 5 ,~ [BJM'A(chq)] _ 6JK6Mq( )J M43(cd) (AL } 73 I E __ E -- [BJM'A(chq)] ‘ 6JK6Mq w(cd) (A14) g . 0 i 5+ _ _ J—M (ab) 5 _ [BJM’ aB ] — :( ) -;€———(abJ|mamBM)aa (A13) ab + e “’1 b) + _ a ,. [BJM' aB ] - £'—g——-(abJ|mamBM) ad (A16) 0 a ab 1.. .f. 8 .. 1. __ y“ [A(abJM)' ay] — §adcb(achmamyM) a0 %Bdca(ch|m myu,a (A17) Appendix B . T The Exact Evaluation of [A(abJM)’A(chq)] Starting from the definition of the operators A .1. (abJM) these operators as follow (abJM) and A , we can write the commutation relation for + _ . [A(abJM)'A(chq)] — $0§B%Y%0(ale mdeM)(CdKlem5q) X[aaa8,a5ay], by using the fermion commutation relations; i _ f + _ _ {ac'aB} ’6a8 r {aa'aB} - {aaraB} * 0 we arrive to .f. [A(abJM)'A(CdKQ)] = %a%s§y%a(alemam8M)(CdKlemaq){'(3dYSBS lj 74 T f + ) + 6 a a - dcda a 5 a a5 + 6 .1» 97 B 6 B Y BY 9 a a } (El) 86 a Y aE(a,ma). Let us call the first two terms in (B1) by T; T = m 2 -{(abJ|ma m BM)(abKIm m 0 m8 X (baKImBm Bq)0aC6 bd- (abJImamBM) cq)6ad5bc}' and by using relations (A3,A8) we get __ _ )a+b- T — SMqGJK(6ac5bd ( g ad 55c) ' The last four terms can be writen in a compact form by using interchange operator R(ab) which is defined by R(ab)"’(a16)= ll’(ba) . Its effect on the Clebsch-Gordan coefficients is to multiply them by a phase factor i.e. a+b-J R(ab)(alemamBM)=(baJImBmaM)=(-) (abJImamgM). Therefore to keep the terms containing Clebsch—Gordnn + — . . a b JR(aL.-) 111' 9.- f: coefficients unchanged the operator (—) be used. Let the last four terms be T' which can be writen as follow: T =—P(ab)P(cd)Z Z Z Z (abJ m m M)(ch m m q)0 Y8a+a6, ma “8 Y m0 0 where P=<1-(->a+b’JR) 75 Now by applying relations (A3,A5,A1) we get the following d+m % (bdKImBqu)(abJ|mamBM) = (-) 6(-)a+d+J+K/T2x+1)/(2b+1) B d+m 6 d J+K X%B(deIq-m6m8)(baJImBmaM) = (-) (—)a+ + £§7722111x de /12L+1){aJL}(LKJ|qu)(adlea-mdu), l hence T become a+d+J+K de T =-P(ab)P(cd)(-) EE/(2K+1)(2L+1) {aJL}(LKJquM) d+m5 1. X% % (-) (adleOL-m(51.1)a0ta6 . a 6 Now set m5+-m(S , and since d and hence m6 are half integers; F d-m(S d+m(S (’) ='("') I therefore T. will have the following form: T': P(ab)P(cd)(-)a+d+J+K £5/72K+I)72L+17 {§§3}(LKJIVQM) O x A(adKu)6cb f where 0 d+m(S _ _ I A(adLu)-%ama( ) (adLImamau)aaa_5, -6E(d,-m6) . Finally we get: + — _ _ _ a+b-J [A(abJM)'A(chq)] — 6MqGJKwacébd ( ) sadébc) a+d+J+K de +P(ab)P(cd)(-) £§/(2K+1)(2L+1) {aJL}(LKJquM) o a x A(adLu)5cb ' (BL) 76 Appendix C The reduced matrix element q(ab) The reduced matrix element of the nondimensional quadrupole transition operator is given by a-b q(ab) = (N'2'a||pzyz||N2b) = LZL——— /5725IIT .flfi 2+2 1- x (a2bll/2 0 1/2)—il§L——— RmE3 (01) where _°° 4 2 :II Rafi — 0fRa(o)RB(p)p do . a—(Nl) . B—(N.£ ). p= aOr = (mwO/h)l/2r Ra(p) is the radial part of the Harmonic oscilator wave function, m; is the nucleon mass , 1 mo is the oscilator frequency and N is the principal quantum number of the harmonic oscilator wave function, such that the energy is hwo(N+3/2). The radial integrals R are given in Table 10. 08 Table 10. The radial integrals R08 I I N 2 RGB ‘ 3 N 2 . N+§ 1 .1. N12 2 -§[(N+2+2:1)(N-2+1:1)]2 1 N12 2:2 %{(N+2+112)(N+2+312)]2 77 Table 10.(Continued) 1 N 212 -[(N+2+2:1)(N-2+1¥1)]2 1 N22 2:2 %{(N-212)(N-2+212)]2 Appendix D Evaluation of<[A f (abJM)’A(chq)]> Using the following vacum expectation value = n.6 . , (D1) 1 j 1 13 we get I _ <[A(abJM)'A(chq)]>- 2a582Y§5(alemamBM)(CdKleméq) + + + + X{6aY685 - 6056BY -<(acya6aB-Ga6ayaB-6Bya6ad+6B6ayaa)>} and by using relations(A3,A8) we get a+b-J 6 6 ) <[ + _ 6 (5 ad bc A(abJM)'A(chq)]> — GJK Mq acébd-(-) x(1-na-nb) (D2) Appendix E I a: Evaluation of na and na To evaluate na we start with 1. _ 2 _ (A(ab)A(ab)> — ¢(ab) — § § § % (alemlmZM)(abJ|m3m4M) 1 2 3 4 >< . (El) 0 B 8 d 78 where a=(alml) I B=(blm2) I a'=(alm3) I B'=(brm4) 0 Applying the usual factorizations to the product of four operators and using Eqn.(D1),we get T T _ T T T T — - = n.r1 6 '6 - n r1 6 '6 (E2) a b dd 88 a b 08 Ba Substituting(E2)in(El) and using relations(A3,A8) we get T _ _ _ a _ 2a__ (A(ab)A(ab)> _ nanb(l ( ) Gab) ’ ( ) h 1 therefore 6 “ anb = ¢(ab)D ab (E3) where 1 D 2......— ab 1+6ab In RPA the amplitudes ¢(ab) are given by 2 2 2 qfitmuabmab 2 = [2 PM] (ab) 2 2w(m+Eab) 8 a5 q§ab)UabE€ Eab OLab ab 52 w2 2 (Bab-w ) -1 where P§w)=2 where ab=l-na-nb' Since aab appears in both the denominator and nominator G. of ¢(ab)' the later 18 less sensetive to the ch01ce of aab’ therefore it is a very good approximations to set dab=1 in evaluating na. If Gab is included, na can be calculated by iterations. 79 Now let N = Z n = Z n , o a a b b then 2 2 N = 22 n n = 22 0 D , o ab a b ab (ab) ab therefore 1 2 a NO b (ab) ab ' T To evaluate na we need to evaluate <2lA(abJM)A(abJM)|2>’ where |2> = BEIO> , 1. 1 (ab)A(ab) ' ¢(ab)A(ab)) ° B = — 22 (w 2 2.ba Using the following commutation relations: I. _ : [BZV'BZV — avv ' and relations (All,A12,A13,A14), we get 1.. (2| A(abJM)A(abJM) _ 2 2 2 l2) ‘ (w(ab)+¢(ab))aab+¢(ab)) I I Following the same procedure in evaluating na we get 2 2 2 ' §1«w(ab) + ¢(ab)) (Ll-ma) + ¢(ab) )Dab n = I 2 2 I (E5) N0 + Z (w(ab)+¢(ab))Dab b a 80 where na is given by (E4), and N'2= 22 O 2 2 2 ab ((w(ab)+¢(ab))aab+¢(ab))Dab ° b: The Collective Operators The collective operator Bf can be expanded in terms JM + ' o o of A(abJM) and A(abJM) With real coeffeCients T _ T _ _ J-M BJM ‘ XX[XabJA(abJM) ( ) yabJA(abJM)] ' ab From the definetions of w(ab) and ¢(ab) we get .'- w(ab)=<0|A(abJM)|2> = = 2XabJaab ' similarly ¢(ab) =2yabJaab ' 1. Therefore the expreSSion for BJM in RPA takes the following form T 1 1 T BJM= 2 g: 5;; [w(ab)A(abJM)-¢(ab)A(abJM)] ° c: Evaluation of N /9 a a Using similar procedure that used in evaluating na i.e. T _ l T T _ §;.%c£d' Sasc' >0 where c=(a,ma) , c'=(a,ma') , 0a: a+l/2 81 Now applying the factorizations (E2) and (D1) we get T _ 1 2 (AaAa> ‘ 0;— §a>02Na ' .1. N = . a a 0 Using the following % >0l = Qa CT. we get _ ’ T ‘2 Na/Qa — //20a (E6) d: The Factorization In the Hartree-Fock method; it is assumed that the ground state may be represented by an independent particle state vector such that the average in that state of products of single particle operators takes the form =-’ >' (E7) This expression differs from (E2) in the presence of the last term, which is zero unless the particles participate in pairing interaction. In the quasi-particles scheme the interacting particles transformed to independent quasi-particles by means of B—V transformations, where the pairing interaction between the original particles is now absorbed by the quasi—particle energy; Ea . Therefore the product of four quasi—particles 82 Operators takes the following form T T T T T =-, (E8) 8 B 8' a B' B where the third term which describes the pairing sets equal to zero. Appendix F The Sj for (d,p) stripping on odd mass target The quantity Sj is defined by . 2 sj = I| where |JM> = 8+ |0> JM ' I(jJO),JM> = $0 i(on JImMO M)CJ. m|J0M0> , where JOMO is the wave function of the target ngch can be expanded in terms Of seniority one and three states; 0> + z z 2 Cq?(j'2Jo |m 0M0 )aT. B+_i0> ' j'm'V J j "m 20 .1. IJo Mo > = CJoa JO MO where |0> is the quasi-particle vacume~. The coefficients qu can be obtained by diagonalizing the Hamiltonian in the space Of one quasi-particle with zero, one, and two phonons , these coefficients are tabulated by KS. The single particle creation Operator Cgm can be expressed 83 in terms of quasi-particle operators agm and ajm by means of B-V transformations; CT = u.aT + s.v.a. 3m Jlfln 3 J J‘m Now for J=2 we get . T j+Jo <2 2> .<2 . >- — . |(3JO), c u [A(JJO)IO_ ( ) VJC 0 Jo 3 <2|A .)|0> Jo (J03 + (-)J+J0ijgO/7733¥TT7§ + vj/§TZE§:TT j a 2 j' w(ja)w(j' 'a){j' 'Jo 2} X z z(- -)j W+J°C j a (Fl) the last result is obtained with the help of relations (Al,AlO-A13). The first two terms in the last Eqn. reduce to after using the following results CJou jw(on) <2|A 0>== 0 where w(ab)= )a b¢(ab) wZab)= -(-)a-bsgn(a2b|1/2 0 1/2)w(ab) where the phase of w(' ab) is that of Cgo ;'Neglecting the last term in (F1) as it is small compared to the other terms, we get Sj=lufl Ow(on ) - vjcg °/(2Jo+1)/5|2 (F2) 84 Appendix G . o a. Calculation of Gab The Green's function Gib provide the solution to the pairing part of the Hamiltonian with small residual interaction and hence describes the behaviour of the 0f seniority zero states in even-even nucleus. If the exact commutation relations of Aa and A: are used, the following commutation relations are obtained: N N I ._ _ 2 2 ___C. _ 2 2 _ ”-2 T [A0,Hp] — ZECAC g{§/Qcabubuc(l QC)Ab g/Qcabucvb(1 ac)Ab N N -Z/§2'_S2"u2v2A+(l- £)+2/s2""?2"v2v2A (1- —-9-)}. c a a c a Q a c a c a Q a c a‘ c N Let (1- ME) = L We C ’ collecting similar terms we get: [A H'] = 2E A - g/Q u2L Z/Q (u2A -v2 T) c’ p c c c c c.b b b b bAb 2 2 2 + g/chc :Vfla(vaAa uaAa)LC . (61) Similarly 1’ 2 2 2 + ' —- - _ [Ac’Hp] — ZECAC g/QCVCLC é/Qb(ubAb vbAb) 2 2 2 T g/chc i/Qa(vaAa uaAa)Lc (G2) Substituting (G2) in the equations of motion for Go(w) we get equatins (III.4), where LC set equal to one. G =24 A p / 85 Table 11. Proton levels 29 50 Ael/2=-.2,Z=50 Ae3/2=.05(50-Z),Z>50 Isotope A AP AP An An 4OZr54 94 1.893 1.078 -0.041 0.910 42M052 94 2.270 1.137 -0.481 0.713 42M054 96 2.258 1.100 -0.060 0.917 42M056 98 2.248 1.065 0.414 1.063 42M058 100 2.238 1.031 0.839 1.235 44Ru52 96 2.543 1.087 -0.489 0.714 44Ru54 98 2.526 1.056 -0.091 0.929 44Ru56 100 2.510 1.027 0.334 1.078 44Ru58 102 2.494 0.999 0.726 1.221 46Pd58 104 2.698 0.887 0.600 1.205 46Pd60 106 2.677 0.865 0.929 1.287 46Pd62 108 2.658 0.845 1.237 1.334 46Pd64 110 2.639 0.826 1.532 1.355 48Cd58 106 2.869 0.668 0.416 1.187 48Cd60 108 2.844 0.653 0.734 1.231 48Cd62 110 2.820 0.639 1.056 1.245 48Cd64 112 2.797 0.625 1.386 1.246 48Cd66 114 2.775 0.612 1.708 1.249 48Cd68 116 2.753 0.599 2.002 1.247 508n62 112 1.083 1.237 508n64 114 1.395 1.241 508n66 116 1.702 1.240 508n68 118 1.987 1.233 SOSn70 120 2.247 1.209 508n72 122 2.485 1.162 505n74 124 2.705 1.088 91 Table 17. Gp=23/A Gn=23/A Proton levels 50||“(113111)Ml“(lMILlIMNNJHHMMNIH