Shell model calculations on energy levels in the 2s12 1d32 shell (I)

I 1.D.I

Nuclear Physics 56 (1964) 529--547; ( ~ North-HollandPublishino Co., Amsterdam Not to be reproduced by photoprint or m i c r o f i l m without written permission from the publisher

SHELL MODEL CALCULATIONS ON ENERGY LEVELS I N T H E 2s~ ld~ S H E L L (I) P. w. M. GLAUDEMANS, G. WIECHERS t and P. J. BRUSSAARD Fysisch Laboratorium der Rijksuniversiteit, Utrecht, The Netherlands

Received 11 November 1963 Abstract: The shell model is used to investigate quantitatively the nuclei in the range zgSi-4°Ca. An inert ~sSi core is assumed with a two-particle interaction of the outer nucleons in the 2st and ldt shells. Mixing of all possible configurations in these shells is taken into account. The interaction matrix elements are reduced to linear expressions in 17 parameters (i.e. 15 two-particle interactions and 2 binding energies to the core). In this paper the methods, which have been used, are discussed. In a second paper the numerical results will be presented, obtained from fitting the 17 parameters to 50 nuclear levels. The results give (a) the energies, spins and configurations of about 400 nuclear levels, (b) the values of the 17 parameters and (c) the spectroscopic factors for stripping reactions.

1. Introduction I n this paper we shall investigate, using shell model calculations, the level schemes o f the nuclei in the range between 29Si and 4°Ca. Only the outer two shells, i.e., the 2s~ and Id~ shells, will be taken into account and all inner (closed) shells will be considered as an inert 28Si core, providing a central field for the outer nucleons. The nuclear interaction between the extra nucleons will b~ taken as a sum of two-particle interactions. With the latter assumption the matrix elements o f the internucleon interaction between the m a n y possible configurations, can all be reduced to a small n u m b e r o f two-particle matrix elements i). I f only the 2s½ and I d , shells are considered with all the particle configuration mixings, we need 15 two-particle matrix elements. As soon as more shells are taken into account, this n u m b e r increases appreciably; thus, e.g., if also the ldl shell is considered (with consequently a smaller inert core, 160), 63 different two-particle matrix elements are required. Since such a large n u m b e r o f two-particle matrix elements will complicate the calculations considerably and m a k e them much_ less transparent, we have chosen an inert 28Si core, although o f course excitations of the inner shells will not be found this way. As there is no satisfactory theory o f the nuclear forces, the two-particle matrix elements will n o t be evaluated theoretically but we shall consider them as parameters to be fitted to reproduce the experimentally observed energy levels. Thus in the region o f interest to us there are 50 low-lying energy levels which have been identified. There are 17 parameters (i.e., the 15 mentioned above and the binding energies to the core o f a t Present address: Dept. of Physics, University of Cape Town, South Africa. 529

530

P.W.M.

GLAUDEMANS et aL

nucleon in the 2s~ and ld~ shells). Thus the problem of fitting the parameters is three times overdetermined. These calculations differ considerably from Arima's results 9) which do not account for configuration mixing. In the next section we shall discuss the various parts into which the total Hamiltonian is decomposed. In sect. 3 the Coulomb energy term is evaluated. The reduction of the interaction energy of the extra nucleons to a sum of 15 two-particle terms, is treated in sect. 4. In sect. 5 the evaluation o f these two-particle terms is discussed. In a forthcoming pape rl ~) we shall give the numerical results, as well as the spectroscopic factors for stripping reactions that can be evaluated once the nuclear configurations are known.

2. The Energy of a Nucleus with Z Protons and N Neutrons As we consider a nucleus in the range of interest as consisting of a core of 28Si ' surrounded by A' = Z + N - 2 8 nucleons in the 2s½ ld~ shell (hereafter referred to as the sd shell), the total Hamiltonian o f the nucleus can be written as the sum of two parts: H = Hb +He; Hb giving the binding energy of the 2aSi core, and He giving the energy of the A ' nucleons moving in the central field supplied by this core. The latter part can be written as A'

A"

= E r,+ E i=1

i=1

A"

A"

E uF)+ E II,j.

i=1

l<-i
(2.1)

Here T~ and U[ s) denote the kinetic and potential nuclear energies of the ith particle, respectively. The term U[° represents the Coulomb energy of the ith particle in the field of the core and the other A ' - 1 extra nucleons. The nuclear interaction energy of the A' extra nucleons is represented by the sum over two-body interactions H~j. We shall assume that, when filling the sd shell, all particles in the s shell, and similarly all particles in the d shell, are equivalent. That is, the binding of all nucleons in the s shell (or in the d shell) to the 2aSi core is assumed to be independent of the number of extra nucleons. For the light nuclei considered we shall take the isospin T to be a good quantum number. Let the nuclear state with n particles in the s shell and m particles in the d shell be characterized by the mode of coupling of spins and isospins of the particles in their respective shells. Since the spin J and the isospin T of the 28Si core are both equal to zero, the (J, T ) values of the nuclear states are determined by the nucleons outside the core only. The complete specification of a nuclear state may demand the presence of other quantum numbers than spin and isospin (e.g., seniority and parity), but these will be omitted because they are irrelevant for our further discussion. If a nuclear state is described by a pure configuration, i.e., if in thejj-coupling scheme each particle is specified by the values of n~, Ii andj~ (that is, principal quantum number, orbital angular momentum and total angular momentum, respectively), the level energy is

SHELL MODEL CALCULATIONS

(I)

531

given by (in our case of (2s~})"(ld~)" configurations only) n+ra

E = Eb+Ec+nEs+mEd+ ( y' H,.i).

(2.2)

l~_i
Here E b represents the binding energy of the 2Ssi core. Its value can be obtained from a nuclear mass table 2). The term Ec represents the repulsive Coulomb interaction of the protons outside the core as well as their Coulomb interaction with the core (this will be discussed in sect. 3). The terms E s and E d give the nuclear binding energies to the core of an s~r and a d~ particle, respectively. However, if configuration mixing is taken into account, the energy levels with respect to the 2 SSi ground state and with the Coulomb energy of the outer particles subtracted, can be found only by solving the eigenvalue problem of the Hamiltonian H = H e - ~_-' x Ui(c). Therefore we need to evaluate the m a m "x elements of ~1,+m _i< j H~j between the various configurations of the sd shell. In sect. 4 we shall show that these matrix elements can be expressed in terms of 15 two-particle interactions, which could be calculated theoretically, if one knew the effective nuclear two-body forces. These two-particle matrix elements, as well as the binding energies, Es and Ed, will be considered as 17 parameters to be obtained from experimental data.

3. The Coulomb Energy of a System Containing Z Protons Several formulae are available to calculate the Coulomb energy of a system containing Z protons. A semiclassical expression, including exchange energy, is given by Sengupta a); it contains only one parameter r 0 = RA -~, where R is the nuclear radius and A the mass number o f the nucleus concerned. Assuming the charge distribution of the protons to be uniform within a sphere of radius R, Sengupta finds 2

4

Ec(Z ) = [0.60Z - 0 . 4 6 Z r - { 1 - ( - 1)z} x

O.15le2/R,

(3.1)

where e denotes the charge of a proton. The parameter r o has been calculated by Carlson and Talmi 4) for the j j-coupling shell model. They found r o = 1.24_+0.03 fm for the range 29Si to 39K. Another formula * is given by de-Shalit and Talmi. This formula takes into account the orbits of the particles outside the closed shells and may hence be expected to give more accurate results than Sengupta's expression, eq. (3.1). Let C represent the interaction energy of one proton in t h e j orbit with the closed shells. Then, assuming that Z ' protons move independently in the field supplied by the core of closed shells, de-Shalit and Talmi give for the Coulomb energy of these Z ' protons, the following expression: Ec(Z') = CZ'+kZ'(Z't Eq. (30.2) of ref. 1).

1)a+ [½Z']b,

(3.2)

532

P.W.M.

GLAUDEMANS

et al.

where [½Z'] stands for the largest integer not exceeding ½Z'. The values of C, a and b depend on the particular shell in which the protons are moving. They can be derived from the difference in the binding energies of mirror nuclei, which equals the difference in Coulomb energies if one makes the assumption that the nuclear forces are charge independent. According to eq. (3.1) we have A t ( Z ) = E c ( Z + 1 ) - - E c ( Z ) = [0.60(2Z+ 1 ) - 0 . 6 1 3 Z * - ( - 1)Zo.30]e2/R. (3.3) Similarly from eq. (3.2)

AI(Z') = Ec(Z'+I)--Ec(Z')

= C+Z'a+½{1-(-1)Z'}b.

(3.4)

For the d shell there are four pairs ofmirror nuclei, i.e., 363S17_ _ 3 3 1 7 C l 1 6 ' 3 5 1 7 C l 1 8 _ 3518Arl7' 37A 37T~ 39~7 39~ 18Ptr19--191~lS, 191~20--20L,a19. T h e s e are sufficient to determine the values of the parameters ro in eq. (3.3) and C, a, b in eq. (3.4). For the s shell there are only two pairs of mirror nuclei, i.e. 29r, 29e~. and 31 151~14 14~115 16815 -- 31 15P16. We can now employ the differences in Coulomb energy for pairs of nuclei two charge units apart to obtain more independent equations for the unknown parameters. F r o m eq. (3.2) we derive A2(Z' ) = E c ( Z ' + 2 ) - - E c ( Z ' ) = 2 C + ( 2 Z ' + 1 ) a + b .

(3.5)

We can apply this expression to the pairs 30 • 30 14Sh6-16S14 and 32 1 5 P 1 7 - - 32 17Cl15. Binding energies are taken f r o m the tables by Everling et al. 2). Application of formlda (3.3) shows that there is no significant difference in the values of ro for the s and d shells. The value obtained is given by ro = 1.278 fro. Using eqs. (3.4) and (3.5) we find for the s shell: C = +5.659 MeV, a = +0.519 MeV, b = - 0 . 0 1 8 MeV. F r o m e q . (3.4) we obtain for the d shell: C = +6.365 MeV, a = +0.270 MeV, b = +0.110 MeV. TABLE Values Nucleus

Sengupta

de

Shalit-Talmi

of

1

E--Eb--E

c (in MeV) Nucleus

Sengupta

de

Shalit-Talmi

~-sSi

--

0.00

--

0.00

3:tS

--

64.41

--

29Si

--

8.04

--

8.48

3~CI

--

64.35

--

67.12 67.19

2~p

--

8.03

--

8.41

asS

--

70.94

--

74.10

3*'Si

-- 18.23

-- 19.09

3~CI

--

76.50

--

79.86

a0p

-- 18.86

--19.72

3~Ar

--

76.54

--

79.85

a°S

--18.45

--19.17

3~S

--

80.38

--

83.99

31Si

--24.43

--25.68

3nCl

--

84.58

--

88.43

alp

--30.71

--32.04

3~Ar

--

91.24

--

95.10

3~S

--30.72

--31.97

3~CI

--

94.43

--

98.75

32Si

-- 33.35

-- 35.00

37Ar

--

99.49

-- 103.90

3zp

--38.21

--39.97

3~K

--

99.44

-- 103.89

325

--45.30

--47.06

3BAr

-- 110.82

-- 115.74

32C1

-- 37.88

-- 39.64

38K

-- 110.90

-- 115.93

33p

47.88

-- 50.08

39K

-- 123.43

-- 129.0 I

335

-- 53.46

-- 55.70

39Ca

-- 123.47

-- 129.02

3~C1

-- 53.40

-- 55.71

4°Ca

- 138.49

-- 144.64

54.04

-- 56.65

34p

534

P.W.M.

GLAUDEMANS

et aL

Macfarlane and French 5)

~{s"(J'T')dm(J"T")IJT}-- ~ , F where the vector ~ ( = J ' , T') is coupled with the vector fl(= J " , T " ) to give F(--- J, T). The circular arc indicates that ~b is antisymmetric in all n + m particles and the arrow in F denotes the order of coupling ~ and fl to F. If we assume H to be a two-body interaction, it may change the orbits of at most two particles. The matrix elements to be evaluated have the form

=

)-] l~_i
Hi

F

.

(4.4)

F

We can now divide the matrix elements into four different classes giving non-vanishing contributions: class class class class

I :p II : p III: p IV: p

= 0 , a = V and fl = 3; = 0, a -¢: y and/or fl ~ 6; = i; --- 2.

The first class represents all the diagonal elements of the energy matrix . All other classes represent off-diagonal elements. The last two give rise to configuration mixing. We now want to express the matrix elements of many-particle configurations as linear combinations of matrix elements of two-particle configurations. The antisymmetric wave function ff{s"(~)dm(fl)l F} can be written as a linear combination of products of antisymmetric wave functions s) of s"(~) and dm(fl):

~

= L(n+-m~)!J F n!m! l ~ ~' F

(4.5)

(-I)'P, F

In the right-hand member o f e q . (4.5), when composing the two states sn(~) and dm(fl) to a state characterized by F, we assume ascending particle numberings 1. . . . . n for the state s"(~), and n + l . . . . . for the state dm(fl). The natural (ascending) ordering is conventional but important in fixing the phases of the antisyrmnetric

n+m

SHELL MODEL CALCULATIONS

(I)

535

wave functions. The prime in eq. (4.5) indicates that the summation over the permutations P, of the n + m particles is restricted to order-preserving permutations, i.e., the natural ordering within the sets s" and d m separately should be preserved. The wave functions on the right-hand side of eq. (4.5) are orthonormal. 4.1. MATRIX ELEMENTS OF CLASSES I AND II The matrix elements of classes I and II are given by

s.~

<

.+m I ,X<#,,

(4.6) F

r

The left-hand side wave function can be expanded as in eq. (4.5). Since the matrix elements are independent of the particle numbering, we may write

:L n,m, J \

~,

,g< n,, F

r

(4.7) where we have taken one particular natural particle ordering for the states sn(~) and d'(fl), i.e., the numbering 1, 2 . . . . . n in the s" shell, and n + 1, n + 2 . . . . . n + m in the d m shell. Let us consider the energy operator in expression (4.7). We may write: n+m

Z

where

(4.8)

H,j = H # + H b + H # b ,

1___i < j Ha =

l~i
Hij,

n+m Ht, =-- ~ Hiy, n+l<=i
~ H~b = i=1

n+m ~. Hi i • j=n+l

Expand the right-hand side wave function in expression (4.7) as given by eq. (4.5). Consider the effect of the operator Ha. There is just one order-preserving permutation P, that will give a configuration with particles n + 1, n + 2 . . . . . n + rn in the d m shell. Since H a operates on the s~(?) group only, it is this d m configuration solely that can lead to a non-vanishing matrix element of Ha. Carrying out the integration over the d m shell in H a, one obtains (fl = 3)

=(

(,..

/~s~/~

>.

(4.9)

536

P.w.M.

GLAUDEMANS

et al.

The matrix element in eq. (4.9) will vanish unless 0~ = Y. For 0~ = ~ it represents the internal energy of a system of n s particles. Similar considerations hold for the term (Hb), i.e., for /~ = 6 it represents the internal energy of a system of m d particles. The internal energies of both the sn system and the d m system can be expressed in two-particle energies with the use of the recltrsion formula 1):

--n_2~(pnc~]pn-l~)2

n-' O

l<=~i
~.

(4.10)

Here p stands for s or d. The coefficient (pn ~lpn-1/~) is a coefficient of fractional parentage (c.f.p.) and is defined by the relation 5) (see appendix A)

(4.11)

Now consider Hob (eq. (4.8)). Take any term Hij of Hob. From the orthogonality conditions it follows that there are only two order-preserving permutations Pr in the expanded right-hand side wave function in expression (4.7), that will give a nonvanishing contribution to the particular element (Hij). They are Pr = 1 and P~ = Pij (where Pij exchanges the particles i and j). It follows that

(4 F

r

We may use the fact that the sd system has four multiplets to represent the sd interaction by four independent two-body operators. We may write:

Hij(1-Pij) = B o + B j ( j i "Jj)+Bt(ti" tj)+Bj,(j, "jj)(t i " tj),

(4.13)

where Bo, Bj, Bt, Bjt a r e constants and j and t are the spin and isospin operators, respectively. Substituting (4.13) into (4.12) we obtain, using the orthonormality of

SHELL MODEL CALCULATIONS ( I )

537

the wave functions and carrying out the summation over i and j, (4.14)

= {nmB o + Bj(J~ " Ja) + B,(T~ . Ta)}6~cSpa+ B;t(M). Here ( M ) is defined by: .\

/.-/ (4.15)

F

F

where we introduced the operator Otj ~ (Ji "jj)(ti'tj). Using (Jr" Ja) = ½{J(J+ 1 ) - J ~ ( J , + 1 ) - J p ( J z + 1)} and a similar expression for (T~. Tp) we can quite easily write down the coefficients of Bj and Bt in terms of the spins and isospins of the nuclear states. The evaluation of the coefficient of Bit often requires several steps. In the trivial case where n = m = 1, eq. (4.14) gives a simple relation between the two-particle sd energies and the four constants Bo, Bj, Bt and Bit. Noting that the operator O o is symmetric in the m d particles, we may write for eq. (4.15):

(M) = m(

(4.16)

~=lO,,,+t F

F

In evaluating this matrix element we separate the d particle labelled (n + 1) on both sides with the help of c.f.p. (see eq. (4.11)), after changing the order of coupling of the d particles. Recoupling with the help of Racah coefficients and performing the integrations over those parts, that are not operated upon by ~7=t Oi.n+l, we are left with

m ~ ( - 1)2~+2d-P-~ r/

d ( n + 11

0

/'

s"

~d(,+l)

0

(4.17)

Here U is a normalized Racah coefficient with U (ad F ~l; 8 fl) =- U(JJdJrJ~;J~dp) x U(T~Td Tr T~; Ts To); ( - I )" denotes ( - 1)~" + r,. From the relation between the matrix elements of a tensor operator in a hole configuration and in the conjugate particle

538

P.W.M.

GLAUDEMANS e t

al.

configuration t one may derive for the tensor operator 0 u (of even rank)

,?o,. 0

/

')\

<_

/

0

(4.18) 11=,+1

'

~

,3

Thus we see that the matrix element (4.17) vanishes for n = 2 and n = 4 if a = ? (which is therefore always the case for n = 4). The case n = 3, when always a = ?, is immediately reduced to that for n = 1. The only case still to be investigated is n = 2 and ~ # 7. Utilizing the equivalence of the two s particles and the Racah recoupling techniques, we obtain for the s2d configuration

(

1,2 ~ d ( 3 )

2

d(3)

0

--2~U(ssOd;ote)U(ssOd;Ye)~ s~

IO2,3[ s ~ 3 ) ) .

(4.19)

Summarizing we find that the elements of class I can be expressed as a linear combination of two-particle interaction energies. The requirement of antisymmetry of the two-particle wave functions restricts the number of two-particle interaction energies to ten, i.e., of the following configurations (denoted by PP)r): sgt, S12o, doZ~, d21, 812o, d~o , Sdll, sd21 , sdlo, sd2o. The latter four interaction energies m a y be replaced by the four constants B o, Bj, B t and In m a n y cases the coefficients of these constants can be written down without much computational labour. As most of the terms into which (H(o)), eq. (4.6), can be expressed, vanish for the matrix elements of class II, one obtains a much simpler expression (containing the Bjt term only) in class II.

Bit.

* Eq. (22.42) of ref. 1).

539

SHELL MODEL CALCULATIONS ([)

4.2. M A T R I X E L E M E N T S O F CLASS III

The matrix elements of class III are given by

l~"~~~

!.+,,, H

(4.20)

F

F

We rewrite the left-hand side wave function as in expression (4.7). Next we apply eq. (4.5) (with n - 1 and m + 1 instead of n and m) to the right-hand side wave function of expression (4.20) and obtain
F

F = {n(m +

1)} ~

,

,

J"

p

F

(4.21)

For this reduction (see appendix B) we employed the fact that (a) the operator x~_~
// (4.22)

Actually these n ~ n - 2 c.f.p, describe the successive splitting off of two particles.

540

P.w.M.

GLAUDEMANS et al.

Finally the matrix elements and can be expressed in terms of twoparticle matrix elements, to be denoted by t

p(i)~p'(z)

. - / ~

,"

We obtain

",

p"(1),A,p'"(2)

IH,,21

~

./ ~ - - - -~-

\/.

(4.23)

c~

= (-- l)~+o+d-a{n(m+ l)}~{(n-- l)U(~dr~;~6) × ~ U(eso~d;

?v)
Ev

- rn U(~,sFfl; aft) E

(-

1)S< din+ '61d'- 1~/(dZO)>

on

x

U(rld6s;//~)a}.

(4.24)

The only possibility for the pair Jr, T, is given by J~ = 1, Tv = 0, so that the summation over v consists of only one term. Similarly the summation over 0 contains only two terms, i.e., Jo = 1, To = 0 and Jo = 2, To = 1. The phase ( - 1 ) ~ may hence be replaced by ( - 1). Since we restrict our calculations to the d~ shell, the phase ( - 1)a may be replaced by ( + 1). Summarizing we have shown that the matrix elements of class III can be expressed as a linear combination of three two-body interaction energies. 4.3. M A T R I X E L E M E N T S O F CLASS IV

The matrix elements of class IV are given by

l=i
Hi j I

F F Again we rewrite the left-hand side wave function as in eq. (4.7), and we apply eq. (4.5) (with n - 2 and m + 2 instead o f n and m) to the right-hand side wave function of expression (4.25) and obtain (see appendix C) /(rn + 1)(rn +2)/~

= [ //

×\

J

~

sn

\

..... ,.+,

'~

d=

~',

.... +.:

/

n

'--<'
1) r P ,

i "\, . . . , - 2/i/ L o - ,o~

/"

/

•

F

= ½{n(n- 1)(m + 1)(m + 2)}½Z x ( - l)P+'-~u(yerfl; ~6)~, t Only those values o f ~ will occur that yield antisymmetric wave functions.

. . . . +-) \

(4.26)

/

SHELL MODEL CALCULATIONS (I)

541

where the two-particle matrix element (s21HId2), has been defined in eq. (4.23). Since J,, T, can take the values (0,1) and (1,0) only, we have expressed the matrix elements of class IV as a linear combination of two two-body interaction energies. 5. The Evaluation of the Parameters

In the previous sections we have seen that the matrix elements (@jlHId/i) of the Hamiltonian in our case of the 2s½ l d t shell can be expressed in a small number of parameters i.e., 12 for the elements of class I (the diagonal elements) and 6 for the elements of classes II, III and IV (the off-diagonal elements). These two groups have one parameter in common, the Bit term discussed in subsect. 4.1. Hence the total number of unknown parameters amounts to 17. Thus we may write the decomposition 17

< ¢ j l n l ¢ , ) (p) = ~V ~,ii~tP)h",,

(5.1)

r=l

where the h, denote the 17 parameters and t~rji ~(P) represent the coefficients discussed in the previous sections. The superscript p labels the energy matrices, i.e., it distinguishes for each nucleus .4, the various pairs of (J, T) values. Ill our case there are 90 energy matrices, so that p = 1, 2 . . . . 90. Since the order of these matrices may amount to 14, the direct substitution of eq. (5.1) into eq. (4.3), will lead to equations in the unknown parameters hr of too high a degree to be practical. We therefore have chosen a linearization procedure, that involves an iteration process lO). As is well-known 6) the eigenvectors a~k) satisfying eq. (4.3) form an orthogonal matrix A, diagonalizing the energy matrix:

(A-1HA)k, = ~ a~k)(~jlHl~,i)a~t) = E(k)•kt.

(5.2)

ij

For the iteration process we first determine the matrices A (p), starting from a tentative set of the parameters hr, that then determine the numerical values of the matrix elements (~kjlHI~k~)(p). In general the eigenvalues of the tentative matrices (~,jlHl~i) (p) will not be the experimentally observed level energies. If now ill eq. (5.2) we substitute eq. (5.1) and the experimental level energies that we tlave at our disposal, we obtain sets of linear equations in the 17 unknowns h,. (To be sure, we employ the matrices .4(P) that diagonalize the tentative matrices($jlnl$i)(P)). In most cases none or only one or two eigenvalues of a particular energy matrix call be equated with an experimental level energy. Considering all nuclei in the region 29Si-'t°Ca simultaneously, we have 50 linear equations in the 17 unknowns h~. Solving for the h~, we obtain an improved set of parameters and can construct new matrices (~kjlHl~b~)(p). This process may be repeated now, until the solutions h, have become stationary. This was seen to require not more than 5 cycles. All calculations were performed on a computer. The results will be presented in a forthcoming paper 11).

542

P.w.M.

GLAUDEMANS e t al.

We are much indebted to Professor J. B. French for suggesting this problem and the methods to be used. We also would like to thank Professors I. Talmi and P. M. Endt and Messrs. F. C. Ern6 and A. Heyligers for helpful discussions. This investigation was partly supported by the joint program of the "Stichting voor Fundamenteel Onderzoek der Materie" and the "Nederlandse Organisatie voor Zuiver Wetenschappelijk Onderzoek".

Appendix A The c.f.p, for non-identical particles in the 2s~ and Id~ shells can be derived from existing tables of spin-orbit 7) c.f.p, and (iso-)spin 8) c.f.p. The resulting list of c.f.p., applicable only to the lower half of the shells, was checked t and extended with the help of the recursion relation tt for c.f.p., as given by de-Shalit and Talmi. As there is a freedom in the sign of the wave functions for different configurations, the resulting c.f.p, are not always unique as to their sign. This, however, will not influence the physical results (e.g., eigenvalues of the energy matrices), once one has chosen some convention. The calculated c.f.p, are represented in table 2. TABLE 2 Coefficients o f fractional p a r e n t a g e

The 2s½ shell

= + 4 ½ = --~/½ = '11-l The ld~ shell (d 3 J

T I d~ J ' T')

,~.3-~---~ (~i)2 Ut)~

(01)

(21)

(10)

(30)

(1 ½)

+ 4A

+ ,/A

- 4A

- 4¢~

(½ ½)

o

+ ~/½

+ ~/½

(~ ½)

0

+ 4½

-

d-

o

½)

(i

~)

- 4~

~/~o

+ ~/~

+ 4½

o

- 4½

+ ,/-~

o

o

t T h u s we applied two sign c h a n g e s in table 2 o f ref. ~), i.e., ( d * [ 2 1 1 ] ( l l ) J = 2{]da[21](10)J = ~-) = + x/~, ( d ' [ 2 1 1 ] ( l l ) d = 2{]da[21](21)J = ~> = + V ' ~ . tt Eq. (33.16) o f ref. 1).

o

543

SHELL MODEL CALCULATIONS (1)

( d 4 J T I d 3 J' T') ")

(oo)

(~ ½)

(½ ½)

(~ ½)

(~ ½)

(t t)

+ 1

o

o

o

o

v = 2 (20)

- x/¢

v = 4 (20)

0

(40)

0

(11) (31) (21)

+ 4~ - 41 + 4A

(02)

0

+ x/~

+ x/~o

- x/~

0

- x/2-Za

+ x/~

+ x/A

0

+ x/~

-- x/~

0

+ ,/~o + 4& + 4~

0 + J~ -4~

+ 41 -4~ -41

0

+ I

0

- 41 0 + 4~o 0

0

a) Only in the case of (d*20[dSJ'T') is it necessary for a complete description of the state of four d particles coupled to J = 2, T = 0 to specify the value of the seniority v. ( d s J T [ d 4 J' T') (00)

v=2 (20)

v=4 (20)

(40)

(ll)

(31)

(21)

(~ ½) (½ ½)

- 4~ o

- 4~ + ,/1

o + 4~o

o o

+ 4A - 4~o

- 4% o

+ 4~+ 4~

o o

(~ ½)

o

+,/1

- 4A

- 4A

+4/o'o

+ 4~

+ 4A

o

(~)

o

o

o

_,/3

+,/~o

+f¼

o

(02)

+4¼

(d 6 1 T ] d 5 l' T')

(~ ½)

(½ ½)

(~ ½)

(~ ½)

(~ ~)

+ ,/~

o

o

o

+ J~-

- 44

- 4~-

(o 1) (21)

+ 4~

+ 4~

+ 4~

(10)

+ x/~

- x/½

+ x/~3-

0

0

(30)

+ 4~

0

- 4A

- J~

0

544

P. W . M. GLAUDEMANS

el al.

(d7½½ [ d 6 J ' T ' ) (~)6

(~)7

(01 )

½)

(21 )

+ 4A

+

(10)

(30)

- 4A

- 4¼

( d a 0 0 1 d 7 ½ ½ ) = + I. For the reduction of the matrix elements of two-particle interactions it is often convenient to separate off two particles simultaneously. The resulting n ~ n - 2 c.f.p. can be expressed in terms of two n ~ n - 1 c.f.p, and a Racah coefficient. Applying eq. (4.11) twice, one obtains

( pn /~ = " /

(.)

-- ~ (p"ctlp"-16)(p"-16lp"-2fl) E U(otflpp; y6)

Y\\~

The (normalized) Racah coefficient U results from a recoupling. Since in eq. (A.1) only antisymmetric states of the two particles n - l, n will be present, we have replaced /

p(,,-i~/"~p(,,) /'

\

by

// p2 ", ( , - 1 , . ;.

Comparing eq. (A.1) with eq. (4.22) one obtains an expression for the n ~ n - 2 c.f.p.

(p,c~lp.- 2fl(p27)) = )-- (p.0~lp,- 16)(p,,- 161p,,- 2fl) U(oqSpp; ),6).

(A.2)

The resulting n ~ n - 2 c.f.p, should vanish for the symmetric p27 configurations and satisfy the normalization condition

E (PnctIPn~2fl(P2]O)2 = 1. This is a valuable check on the calculations as well as on the n ~ n - 1 c.f.p, one starts with * t T h u s we applied t w o m o r e sign c h a n g e s in table 2 o f ref. 7), i.e., (&[22](ll)J=

2{]da121](21)J=

~)

= + ,v/~,

( d 4 1 2 2 ] ( l l ) J = 2{[da[Zl](lO)J = ] ) = + ~ .

SHELL

MODEL

CALCULATIONS

(I)

545

Appendix B

The order-preserving permutations that will yield a non-vanishing contribution to eq. (4.21) are such that in the right-hand side of the matrix element (H(1)) they produce a wave function having n - 1 of the particles numbered 1, 2 . . . . . n in the s state. All other permutations will lead to vanishing matrix elements for the two-particle interaction ~ 1 __
•"

Inn,.+ll

>

F

F

= Z Z U(esrfl; ~o)(- 1) m-1

/

'

s'-

~

d"

r

r

(B.l)

We now can perform the integration over the particles 1. . . . . n - 1. This yields the delta functions 6~r 66,0, and we obtain * Note that we have restricted the permutations Pro to be such that they will not produce any of the particles n+l . . . . . n + m among the group (sn)a.

P.w.M. GLAUDEMANSe t

546

al.

(n,,.,,+,> = U6,sLa; = a ) ( - 1 ) m-' Y~ (d'nflid'-tr/~)(d"+',Sld"-'r/2(d2,9))

~1~2~'

(n+2 ...~+m~ S(nJ

~.,~

k~(n + 1)

/ - - . 4 3 =

u(~,sr,e;o,,5)(-O=-* ~ (d"flld'-lrh)(dm+'aldm-'rl2(d2oq))

~1t~29

x ~,, U(rh dbs; fl{)(- 1)' * + " ' - p ( - 1)a+"2-a

x<

IH..+,l

) (8.2) a

6

The integration over the coordinates of the particles n + 2 . . . . . n + m then yields the delta functions ~ , , 2 6a~. One thus obtains the second part of the right-hand member of eq. (4.24), that is (Hn, n+l).

Appendix C The reduction of eq. (4.26) can be done with similar methods as discussed in appendix B. The only order-preserving permutations P, that will lead to non-vanishing matrix elements of the two-particle interaction, must produce a wave function with ( n - 2 ) of the particles numbered 1, 2 . . . . . n in the s state. Let again P,o be one of them. Applying the same permutation Pro also to the wave function in the lefthand side of the matrix element, we obtain for the matrix element (because of the symmetry of z.,l<~
References 1) A. de-Shalit and I. Talmi, Nuclear shell theory (Academic Press, New York, 1963) 2) F. Everling, L. A. K6nig, J. H. E. Mattauch and A. H. Wapstra, Nuclear Physics 18 (1960) 529 3) S. Sengupta, Nuclear Physics 21 (1960) 542

SHELL MODEL CALCULATIONS ([)

547

4) B. C. Carlson and. 1. Talmi, Phys. Rev. 96 (1954) 436 5) M. H. Macfarlane and J. B. French, Revs. Mod. Phys. 32 (1960) 567 6) H. Margenau and G. M. Murphy, The mathematics of physics and chemistry (Van Nostrand, New York, 1951) Chapt. 10. 7) A. R. Edmonds and B. H. Flowers, Proc. Roy. Soc. A214 0952) 515 8) H. A. Jahn and H. v. Wieringen, Proc. Roy. Soc. A209 (1951) 502 9) A. Arima, Progr. Theor. Phys. 19 (1958) 421 10) J. B. Sanders, Nuclear Physics 23 ~1961) 305 l l) P. W. M. Glaudemans, G. Wiechers and P. J. Brussaard, Nuclear Physics 56 (1964) 548