Nucleon-pair shell model: Formalism and special cases

NUCLEAR PHYSICS A ELSEVIER

Nuclear Physics A 626 (1997) 686-714

Nucleon-pair shell model: Formalism and special cases Jin-Quan Chen 1 Department of Physics, Nanjing University, Nanjing, 210008, PR. China 2 Department of Physics, The University of Pennsylvania, Philadelphia, PA 19104, USA

Received 30 May 1997; revised 12 September 1997; accepted 16 September 1997

Abstract

A formalism is described for a nucleon-pair shell model (NPSM) for even-even and even-odd nuclei. The building blocks of the model space are collective nucleon pairs of angular momenta J = 0, 2 . . . . for an even system, and nucleon pairs plus one unpaired nucleon for an odd system. Analytical formulas are given for the matrix elements of a general Hamiltonian. Without any space truncation, the NPSM is equivalent to the shell model in full space, while with a special choice for the building blocks and Hamiltonian it reduces to the broken pair model, the favored-pair model of Hecht and the fermion dynamical symmetry model. (~) 1997 Elsevier Science B.V. PACS." 21.60.Cs

1. I n t r o d u c t i o n

The discovery o f collective motions, such as collective vibration, collective rotation, giant resonances, etc., in nuclei has great significance. How to describe these collective motions in terms o f the shell model is a central task in nuclear structure theory. The shell model contains all the nucleonic degrees of freedom and can in principle account for any of these collective phenomena. With the explosive growth of computational power, shell-model calculations have undergone a tremendous development, as documented in Ref. [ 1 ]. Effective diagonalization o f the shell-model Hamiltonian in the model space I E-mail [email protected]. 2 Present address.

0375-9474/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved. PI1 S0375-9474(97) 00502-2

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with dimensions in the millions becomes feasible. However, even in relatively simple cases of the medium weight and heavy nuclei astronomical numbers, around 1014-1018, of shell-model configurations are involved [2] and modern computers fail for all these cases. Moreover, even if we could perform such calculations, the results would only have a restricted value, since the simple and regular features observed in actual nuclei would emerge merely as numerical results. It is the fundamental task of nuclear physics to select out of these 1014-1018 states the ones which are "simple" and collective, and are brought down to the low end of the energy spectrum due the coherent interaction among the nucleons. In other words, we need to seek a judicious truncation scheme. There are mainly three approaches for truncating the shell-model space. (1) Truncations based on guidance from mean field method. The Hartree-Fock-Bogoliubov method incorporates the pairing force and the deformation-driving force in a variational calculation of the optimum quasi-particle wave function. It provides the optimal mean field for the nucleons. The ground state wave function is given by a single Slater determinant. The price one has to pay for the simple structure of the wave function is that it does not have a definite particle number and angular momentum. These symmetries may be restored numerically by projection. The most ambitious undertaking of this project is the EXCITED VAMPIRE code [3]. Recently, impressive developments have been made in the quantum Monte Carlo method for the shell model [4,5]. The truncation of the space is achieved by choosing the many-body basis states in the form of intrinsic states (mean field solutions) by using the auxiliary field Monte Carlo technique. (2) Another convenient truncation scheme is to use collective fermion pairs with the angular momentum J~ = 0 +, 2 + . . . . as building blocks for many-body wave functions. The properties of the residual effective interaction are such that it is often sufficient to consider only correlated pairs with few possible angular momenta. The work along this line includes (a) The favored pair model or pseudo-SU(2) model due to Hecht et al. [6] and McGrory [7]. The single-particle levels of normal parity are re-classified in terms of pseudo-orbital angular momentum and pseudo-spin, j = 1 + ~ with pseudospin -- 1/2. The truncation is achieved by coupling the total pseudospin of two nucleons to zero, S = 0. A calculation of an idealized model has been done for two neutron pairs and three proton pairs. Similar work has been done by Otsuka [8], but using the so called m-scheme method. He considered up to three non-S pairs for like nucleons. In these works [6-8], the splitting of the single-particle levels as well as the abnormal level are ignored. (b) The broken pair approximation (BPA) [9,10], in which all like nucleons except a few are pairwise coupled to the angular momentum and parity j r = 0 +. (c) Through the success of the interacting boson model (IBM) [ I ! ], it was recognized that the S and D pairs play a dominant role in low-lying nuclear spectroscopy [7,8,12]. The ( S - D ) subspace is only 102-103 in size and it can be easily handled but for the extreme difficulty associated with the computing of matrix elements in the fermion space. To circumvent the difficulty, the collective S and D fermion

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pairs are approximated as s and d bosons in the IBM [ 11], and the Pauli effect in the IBM is taken into account partially by various mapping procedures [2,13]. (3) The group-theoretical truncation methods. (d) In the fermion dynamical symmetry model (FDSM) [14,15], a collective S - D subspace is constructed with the SO(8) or Sp (6) symmetry-dictated S and D pairs as the building blocks. The FDSM S - D subspace is simple in that it is closed under commutation, but the price one has to pay is that all the single-particle levels of normal parity in a major shell have to be degenerate. Similar to the IBM, the FDSM has analytical solutions for several symmetry limits. For general cases, Wu and Vallieres [ 16] have developed a code for diagonalizing a general FDSM Hamiltonian in the FDSM S - D subspace. (e) The generalized pseudo-SU(3) model [17]. The harmonic oscillator structure of the Hamiltonian is restored in the pseudo-space scheme, and the normal ~-~ pseudotransformation makes it possible to invoke powerful group-theoretical method so that shell-model calculations can be carried out even for nuclei with A >/ 100. In Ref. [ 17] the s.p. energy term is simulated by a many-particle extension of the s.p. Nilssen Hamiltonian and a truncation to the total pseudo-spin zero (S~ = S~ = 0) subspace is made. As is seen, the number N of pairs in previous shell-model calculations in (a) for medium-heavy nuclei are restricted to less than or equal to three. The main reason for this restriction is associated with the difficulty of calculating the commutator between many coupled operators. Though there is no restriction on the number of pairs in the schemes (d) and (e), it suffers the shortcoming that the s.p. energy term has not been taken into account. The s.p. energy term H0 comes from the mean field which is the main part in a shell-model Hamiltonian, while the residual interaction is secondary. As is known, H0 favors independent particle motion, and is detriment to the collective motion. Therefore, for a microscopic description of nuclear collectivity, it is vital to develop a formalism which takes the s.p. energy splitting into account and is valid for any number of pairs. In 1993 a new technique [18] was found for computing the commutators involving angular momentum coupled operators, and with it the Wick theorem for a product of fermions has been generalized to a product of coupled clusters by extending the contraction between two fermions to the contraction between two clusters, and analytic formulae are found for the commutators between many coupled fermion clusters [19] (referred to as 'I' henceforth). Right after the discovery of the generalized Wick theorem, a new formalism for the nuclear shell model, called the nucleon-pair shell model (NPSM), was proposed in 1993 for even nuclei [20], in which the building blocks are collective nucleon pairs with various angular momenta and a code for the NPSM was also written. Following the same lines as in Ref. [19], Gringberg et al. [21] developed a lbrmalism and a code for multi-phonon states in even-even spherical nuclei. In the present paper, we give the detailed formalism of the NPSM for both even and odd nuclei, while leaving its numerical results for a subsequent paper [22]. We use a Hamiltonian which contains the single-particle energy term, multipole-pairings between like nucleons

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and multipole-multipole interaction between all nucleons. All relevant expressions for calculating the overlap matrices, the energy matrices, the B(E2) transition rate, etc., are presented in a coherent form suitable for numerical computation. The advantages of the NPSM are that it accommodates various truncations, ranging from the truncation to only the S subspace, the S-D subspace, up to the full shell-model space, and that it is flexible enough to include the broken pair approximation [10], the pseudo-SU(2) or the favored pair model [6] and the fermion dynamical symmetry model [15] as its special cases. The paper is organized as follows. The commutators for the coupled clusters found in Ref. [19] are summarized in Section 2, the multi-pair basis and its overlap are constructed and evaluated in Section 3, the matrix elements of the Hamiltonian and transition operators are calculated in Sections 4-7, the formalism for even-odd nuclei is outlined in Section 8, and the relations of the NPSM to the other models are the subjects of Sections 9 and I0.

2. The c o m m u t a t o r s

The commutators which are essential for computing the overlaps and matrix elements in a basis with N pairs have been derived in Ref. [ 19]. In order to avoid the frequent citing of the formulae in it, the main results of Ref. [ 19] are summarized in this section. A collective pair of angular momentum s with projection u, designated as A'J, is built from many non-collective pairs A~(cd) t in the single-particle orbits c and d in one major shell,

Ast = Z

y( cds) A;( cd) t,

(2.1a)

cd

AS(cd)t=(Ctc x C~)I,

(2.1b)

where y (cds) are structure or distribution coefficients satisfying the symmetry

y(cds) = -O(cds)y(dcs),

O(cds) = (_)c+d+,.

(2.1C)

The time reversal pair-operator of angular momentum r is

A~r~ = Z

y( abr )'4ru ( ab ) '

(2.2a)

ab

A~(ab) : -

(C a

X

Cb) ~ .

(2.2b)

It is assumed that for each angular momentum s or r, there is only one collective pair. The multipole operator is denoted by

Qt = Z q ( c d t ) P t ( c d ) cd

'

Pt(cd) = (C~ x Cd) t ,

(2.3a)

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and it is assumed that (Q~r) t = ( - 1 ),~Qt_,,.

(2.3b)

The creation operator for N pairs rl . . . . . rN, coupled successively to the total angular momentum J u and with Ji as the angular momentum for the first i pairs, is designated by MN

= AMN

...

X A r3t

x . . . × A rut MN

=-- A ( rlr2 . . .rN; Jl J2 . . . JNMN)

t.

(2.4a)

The time reversal operator for N pairs is ~J~

A MN ( Si .

J[) .=

.

.

/

/t~" .x AS2) J2 .x ,,i.~, .

3x

x A s~ MN

--= /~(SlS2... SN; J~ J; . . . J N M N ) .

(2.4b)

~JN Notice that AMu(Si, ~ t ) and --Jut[r" eaMN~ ,, Ji) have the same coupling order, resulting in a great simplification of the phase problem. From I-(4.5) we have 3 M,ut~ ~ ~JN [fi'J~(ri, J i ) , Q rJLMs ~ = Z CLN J~MN, M'N~

~[AMNO 'J)i'r(~]

1

=~-'~ Z QN(t)...Qk+l(t) k=N r~Lk...Lu 1

×,~(rl . . . rtk . . . rN; Jl . . . J k - I Lk . . . L N M N ),

(2.5a)

where Q i ( t ) 4 is related to the unitary Racah coefficient by Qi( t) = O( J i J i - l L i L i - i

) U ( r i L i J i - l t; L i - l Ji) ,

(2.5b)

We also use the convention that 1, QN(t)...Qk+j(t)

=

Qu(t),

for k = N, fork=N-l,

(2.5c)

and A(r~) is a new pair _

F /

A(rlk)=_a~=ffllc(tr2)

t

[Ark,ot]rk = Z y t ( d a k r't ) A ~r'k ( d a k ) , dak

lf/l~( tr'~ ) = U ( r k t J k - l Lk;r~Jk ),

(2.6a) (2.6b)

" 3 There is a misprint in Eq. (2c) of Ref. I18] and Eq. (2.1c) of Ref. I191, which should read (A,B)~ = (IA, B]±): = 2 a f l c c J , bB(Aa,[~fl). 4 It should not be confused with the multipole operator Qt in (2.3a).

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with the new structure coefficients y~(dakr~) given by

y' ( dakr~k ) = Z ( dakr~k ) -- O( dakr~k ) Z ( akdr~k ) ,

(2.6c)

z(dakr~k)=fkt^ff4k(tr~) Z y ( a k b k r k ) q ( b ~ d t )

{ r' t r~k } .

bk

(2.6d)

d ak bk

Notice that the coefficients in (2.6c) differ from the expression I-(4.6b) in that here the coefficients y ( d a k r ~ ) have been symmetrized so that they obey the condition (2.1c). For a recursive calculation this symmetrization is indispensable. Eq. (2.5) has a simple physical interpretation [19]: For the ! particle-hole pair Qt to "interact" with the pair A rk and change it to the new pair A rk, it has to cross over the pairs A rN. . . . . A r~`', associated with the occurrence of a product of the propagators,

Q N ( t ) . . . Qk+l (t). According to I-(4.18b), 1

IQt'AJu(ri'Ji)t] LN=(-)JN-LN Z

Z

QN(t)...Qk+,(t)

k=N r~kLk...LN_l

×A( r~ . . .r~k . . .rN; JI ... Jk-~Lk . . . LN) t,

(2.7)

where the magnetic quantum numbers are suppressed for simplicity. From 1-(4.19) we have 1

[,~s, zJu(ri, Ji)t]Lu-t = (_)Ju+s-LN 1 Z

Z

HN(S)...Hk+I(S)

k=N Lk- I...LN--2

× [~k 6s,rk •Z,k ~,Jk ~A (rl • • • rk- l, rk+l • .. rN; J1 ... Jk- 1Lk... LN- 1) t 1

+ Z

Z

A(rl'"r~'"rk-l'rk+l'"rN;Jl"'Ji-lLi'"LN-I)t]

i=k-- 1 r~Li...Lk-2

+ T ( L N - I ) t,

(2.8)

where the term T(LN_t) t has no contribution to the overlap in (3.3) and thus is irrelevant, see I-(4.20), and ~'k is a constant depending on the structure of the pairs rk and s,

~Pk = 2~OkZ y ( a k b ~ r k ) y ( a k b ~ s ) ,

q~k "~ (--)Jk--J'-l--rk(L/L--I),

(2.9)

akb~:

with J = x / ~ +

l. The factor Hk(s) is

H~( s) = O( J k J k - l L k - l L k - 2 ) U ( r k L k - l J k - l S ;

Lk-2Jk).

(2.10)

The pair fi,(r~) in (2.8) is a new collective pair

A(r~) = - Z G ~ ( s t ) Q k _ l ( t ) . . . Q ~ + l ( t ) ~ 4 i ( t r ~ ) t

[fi r', [,4~, A~t]t] r' ,

(2.1 la)

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Gk (st) = -U(rksJk-i Lk-1; tJk).

(2.1 lb)

The corresponding pair creation operator is denoted by B r;t, A(r~)t --

Br;t = Z y t (ata~ri)A . t r; (atai) t,

(2.12a)

~lkt'l i

and its structure coefficients are given by a rather complicated expression,

Y(atair~) = z (atair~) - 0 ( atair i' ) z (aiakr i' ), Z (atairl) = 4~i'rt~-'~ t'C-,k(st) Qt-1 ( t)... Qi+ l (t) ~ti(try) t

x

~-'y(btbis)y(atbtrt)y(aibirs) bkbs

{ ){ / rt

S

bi a t

t

bt

ri

t ri

,

(2.12b)

a t ai bi

Notice that due to I-(4.11c), Y(atair~) = 0 when i = 0 . For N = 1, (2.5) and (2.7) reduce to the basic commutators in Ref. [ 19] 5

['4r'QSlt=2UsZy(abr)q(bds)

{ dr a bt } ~t(da)'

[Q~"Artlt=20(rt)~y(abr)q(bds)

{ dr a bt } At(da)t

(2.13a)

(2.13b)

3. The overlap of N-pair wave functions A many-pair state is designated as

17-, J N M N )

= [rlr2...

rN; J I J 2 . . . J N M N ) ,

(3.1a)

with 7- representing the 2N - 2 additional quantum numbers 7"= ( r l . . . r N ;

Jl . . . J N - I ) ,

rl = Jr,

(3.1b)

under the convention

r l >>.r2 >~ ... >~r N. For a given J N M N in general there are several possible values for each of the intermediate quantum numbers Ji. Those functions with same ri, JN, MN but different Ji, i = 2 . . . . . N - 1 are neither linear independent nor orthogonal. They form an overcomplete basis. There are two ways to handle the overcompleteness. One way is using 5 Misprints in Ref. [ 19]: In the second term of (2.4) and (2.5a) the factor Obc and (_)n,ui are missing, respectively; in (3A0d) the factor q(bds) should be q(dbs); in (3.10e) the factor O(rs) should be O(rt), and in (4.9a) a minus sign should be inserted in front of ~bk.

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the overcomplete basis for diagonalizing the Hamiltonian, as is done in Ref. [ 21 ], however a much better way is to choose a complete set of non-orthonormal but linearly independent states. A method for this will be discussed in Ref. [22]. In the following we assume that IT, JNMN) are linearly independent. For neutron-proton systems, a complete set of states for given numbers of valence protons and neutrons is obtained by coupling the neutron and proton states to the states with total angular momentum JM, I'rvJvr~rJ~r" JM) = [Ir~J,,) x IT~J~)]~. The overlap between two states with N fermion-pair is the key quantity, since all the one- and two-body matrix elements, or the matrix elements of the pair creation operator and the multipole operator can be expressed in terms of the overlap. From (2.4b) we have,

(OlAMu(Si, J'i)AMu(ri, Ji)*[O)= JN ' JN

(01 (.4J; x AJUt) °

Io)

= ~NN(0I(/~Ju ' X [,~SNxA/Nt]/N--') 0 10) A

J~v- ~

-

JN

J;,_, (OIAM~_, ×

J;,_, [ASN,AJNtlMN_,IO ).

(3.2)

From (2.8) and (3.2) we get

( O I A ) N ( S i ,a' i' )AZJaN M u kt r i, j ~il t O\/ (sl s2 . . . SN; Ji " " fN-I JNlrlrz ... ru; J1. . . JN-l JN) 1

zk=N Lk I...LN-I × ( s i s 2 . . . SN-1; J l . . . J'N-1 ]rt... rk-l, rk+z.., rN; J 1 . . . J ~ - l Z k . . . LN-1) [

+ZZ i=k--I r~Li...Lk_2

×(s~se. . .SN-1;JI . . .

JN-l' Irl...ri...rk-l,rk+l:

. rN;Jl. . . . .Ji-lLi .

.LN_I) ], (3.3)

where ~/'k is given by Eq. (2.9) with s replaced by su, while r~ represents the new collective pair B ~;t defined in Eq. (2.12) but also with s replaced by SN. Notice that the summation over L u - I in Eq. (3.3) is redundant, since it has to be equal to J~-lThe overlaps for N = 1 and 2 are given in Appendix A. It is interesting to note that when all the angular momenta are zero, ri = J i = L i = O, the overlap for N coupled fermion pairs, Eq. (3.3), reduces to the recursive formula for the overlap of N phonons given in Eq. (2.24) of Ref. [23].

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Eqs. (3.3) and (2.1 1 ) have a simple physical interpretation [ 19] in that the first term inside the square brackets of (3.3) represents the annihilation of the pair A rkt by the pair A sN, while the second term implies that the pair A sN first transforms the pair A rkt into a collective particle-hole pair 7~t = [,~rk, ASut] t, with the probability amplitude Gk (SNt), which propagates forward, crosses over the pairs rk-l . . . . . ri+l, contributes a product of the propagators, Q k - l ( t ) . . . Qi+l ( t ) , and finally transforms the pair A rit into the new pair B r~t = [/(rl,7~t]r; with the probability amplitude 19Ii(tr~). The new pair B r~t differs from the original pair A ~'t not only in angular momentum (e.g. if originally the basis is in the S - D subspace, rk = 0,2, r~ could be 0, 1 , 2 , 3 , 4 . . . . ) but also in the pair structure. Therefore the intermediate states are very complicated due to the appearance of all kinds of new pairs, in contrast to the FDSM S - D subspace which is closed under commutation. The "openness" of our subspace prevents us from using the standard coefficients of fractional parentage (CFP) technique for computing the matrix elements for a system with N pairs by inserting a complete set of intermediate states with N - 1 pairs. Consequently, the matrix elements for a system with N pairs can not be expressed in terms of those for the system with ( N - l) pairs and have to be calculated right from the beginning, and this is why the computing time increases very rapidly with the increase of the number of pairs. The right-hand side of Eq. (3.3) is a linear combination of the overlap for N - 1 collective pairs, and thus a recursive formula is obtained. The initial values of some angular momenta which are needed for recursive calculation are listed below: rN+l = r0 = J0 = L0 = 0,

J1 = rl,

Jt1 ----

(3.4)

S 1.

The formula (3.3) involves multiple summations. The order of summation and a precise determination of the summation ranges for the intermediate quantum numbers, especially for the key indices, is crucial for saving computing time. It turns out that the summations over t and r~ are the most time-consuming ones, and their ranges should be determined as precisely as possible. The order of summation and the order for determining the summation ranges of the angular momentum are as follows: ( 1 ) Given the index k. (2) Decide t from t = rk + SN. (3) Given the index i. (4) The range of r~ is decided from r~ = ri + rk + SN. (5) After r~ is given, we can further use t = r~ + ri t O restrict the range of t. (6) Given r~ and ai, the range of at, is decided by ak = r~ + ai. The quantum numbers L i . • • L N - 1 , L N - 2 , are decided by Li = J i - i + rl = Ji + t,

Li+l = Li + ri+l = Ji+l + t . . . . .

(3.5a)

L k - i = L k - 2 -+- r k - I = J k - I -I-t; L~ = L k - I -k- rk+l = Jk+l -k- SN . . . . . t

L N - 1 = J N - 1 = L N - 2 q- rN = J N + SO.

L N - 2 = L N - 3 q- rN--I = J N - I q- $N,

(3.5b)

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The range of the angular momenta Li, Li_ 1 . . . . are decided from the beginning of the angular momentum chain where J/-1 and r~ are given, while the angular momenta LN-2, LN-3 . . . . are decided from the end where rN and LN-1 ( = J~v-1 ) are given, and they are matched in the middle. It should be emphasized that the intermediate quantum numbers r~, t, and Li,,i ~ = i . . . . . k - l, can take both even and odd numbers.

4. The matrix elements of multipole pairing interaction We use a Hamiltonian consisting of the single-particle energy term H0, and a residual interaction containing the multipole pairing between like nucleons and the multipolemultipole interaction between all nucleons, H= Z

(Ho(cr) + V ( o ' ) ) + Z K t O t .

cr=-rrd,

Ho=Zeaha,

V=ZgsASt.A"+ZktQt.

a II Qt= Z

Ot~,

(4.1a)

t

S

O t,

(4.1b)

t

( r i ) t y t ( oiq~i),

(4.1c)

i=1

where ea and h,, are the single-particle energy and the number operator respectively, and the pair creation operator

A~t = Zyo(cds) (C~ × CJ) I.

(4.2)

cd

Notice that the structure coefficients yo(cds) depend on the Hamiltonian to be used and are in general different from those in the building blocks, y(cds), in Eq. (2.1a). Exceptions are the FDSM [15] and pseudo-SU(2) case [6], where the pairs appearing in the Hamiltonian are at the same time the building blocks (see Section 9). The second quantized form of a t is given by (2.3a) with the coefficients q(cdt) equal to

q(cdt) = (_)c-½ Zc , =

~2X/~-~Cc, t od_, A~dt(Nlclr*lNld),

(4.3)

[1 +

where N is the principal quantum number of the harmonic oscillator wave function, such that the energy is ( N + 3/2)hw0 and Ic and ld are the orbital angular momentum of the s.p. levels c and d, respectively. It is easy to show that the reduced matrix elements of a pair creation operator A~t between two states differing by one pair is equal to an overlap,

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(0[A J~ ( s i, fi ) [[ASt[ [A Ju-' ( r i , J/) t [0) J'~

,

J'~

= ( O I A M , ( S i, g~) × [ A J~-~(ri, gi) t × ASt]M~lO ) . . . .(sl

SN,'' J I . . .

J'N _ I J N' I r l . . . r N - I S ; J 1 . . . J N - I J N ) ,

(4.4a)

where the definition for reduced matrix element in Ref. [24] is used. If Ast is a non-collective pair, A ~t = AS ( cd) t then (4.4a) becomes

(OIAJ~ ( si, J[) [[aS( cd) tl[aJN-t(ri, Ji) t[O) = ( s j . . . sN; J[... J~_ i f NIrZ... rN-I (cd)s; J , . . . JN-1 fN),

(4.4b)

Using elementary decoupling technique, we have

(olaJ a.~t • A sA M st 10) = ~--~ U(ssJJ;OL)(O[AJM × (a st × [/~S, AJt]L)J [0) L

= ~-~(L/)')(OIA ~ x [ ( [ A s , a J t ] L L

J T(L) t) x a s t]M]0 ).

(4.5)

From (4.5) and (2.8) we get the matrix elements of a general pairing interaction

Ju + st • asaJu (O[AMN(Si, J~)A MN(ri, Ji)t[O) 1 k--N

Lk-i ...Lu-I

[

,

× Ok~s,rk~Lk 1,Jk-, (Sl S 2 . . . SN; J~ . . . J N - 1 J N I r l " . r k - l , rk+l ... rNS; J1. . . J k - I L k . .

.LN-IJN)

l

+ Z

Z

(s''''sN;Jl'''JN-'JNIrl""r:'''rk-l'rk+l""rNs;

i=k--1 r~Li.,.Lk-2 Jl . . . J i - 1 L i . . . L N _ 1JN)J,

(4.6)

where ~Pk is again given by Eq. (2.9) with y(akbkS) replaced by yo(akb~s), while r~ represents a new collective pair B~' defined in Eq. (2,12) but with y(bkbis) replaced by y o ( b k b i s ) . The matrix elements of A st . A s for N = 1 and 2 are given in Appendix A.

5. Matrix elements of the Q t . Qt, It has been recognized that the proton-neutron quadrupole interaction is the driving force for nuclear quadruple deformation [2]. The matrix elements of the n-p multipole interaction in the n-p coupled basis can be expressed in terms of the matrix elements of the multipole operators for protons and neutrons:

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J.-Q. Chen/Nuclear Physics A 626 (1997) 686-714

(%J,,r~J~ : jMllOt. Q~rllr~Jv1"crJ~ " JM) ,,,, , (_)j;+j,,+j+t{J'~

J~ J } ~ .

J~r J~ t

, , t ~, , , J~{r,,J,,llQ~llr, d~)J~r

t

(5.1)

The matrix elements of the multipole operators for neutrons or protons in the nonorthonormal basis can be calculated in the following way. First we have

(OIAJ'IIQtIIAJtlo)

= ( - )J-J'-t(OIZ ~ × [O',ZJt]~tlO ).

(5.2)

From (2.7) and (5.2) we get

(oI a J'u( si, J[ ) I[atl lZ JU( r i, Ji ) t lO) 1

= (-)'~

~

eN(')...ek+,(')~LNJ;

k=N r~Lk...LN

X(SI • ..SN, . J~, t

..

t . r tk. . .ru;J~. . Julrl..

. .Jk-lLk. .

.LN)

(5.3)

where r~ represents the new collective pair A r; defined in Eq. (2.5). The following property of the reduced matrix elements of the multipole operator can be used not only for testing the NPSM code but also for saving computing time, fN(OIZJu(ri,

Ji)llatllAJ'u(si, J/) tl0 )

= (--)JN-J;YN(OIAJ;(si,

J[)IIQtlIAJN(ri, Ji)tl0)

(5.4)

The single-particle energy can be identified with a collective monopole operator E ~ , _= Q0 = a

q( abO ) P° ( ab ) ,

(5.5)

ab

q( abO )

(5.6)

= - - ~ a b " d e a.

Notice that the single-particle energy term is diagonal in the full space (its eigenfunction is a product of single-fermion states), but is not diagonal in the truncated subspace with collective pairs, and it will be rather difficult to calculate its matrix elements in the multi-pair basis but for the general expression, Eq. (5.3). For the matrix elements of the Q-Q interaction between like nucleons we have

(OiZJMOt . a tA M Jt I O ) = Z ( - - )

J+'-c 2 j + 1 (01 ( [ , ~ J ,

Qt]L ×

[Qt, AJt]L)O[o),

L

(5.7) Using (2.5) leads to

(OlA~,(si, J[)(_)tQt • Q tAM~(r,, J~ J,)tlO) 2LN + 1

-- ZL~

1

+1 E Z k=N r~Lk,..Ltc-t

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698

1 • " •

i+1

i=N s;L;...L~_ I X (S,...

I Si...

I SN; J , . . .

I J;_lLi...

L~N_,LNIr,...

i rk...

rN;

J1 ... Jk-lLk... LN).

(5.8)

where the pair A ( ~ ) t is given by the Hermitian conjugate of Eq. (2.6a), and the pair fi,(sl) is given by

A(sl) -- A~~ = M~(ts~)[~s,, Qt] s~ = Z Y'(da~s~)AS~ (dai),

(5.9a)

dai

ffli(ts~) = U(sitJ[_ 1L~; s~J[), and the structure coefficients y(dais

(5.9b)

I) are given by an equation similar to (2.6c) with

z(dais,)=sit'l(4i(ts,)Zy(aibisi)q(bidt){ b~ The propagator

sits'

}.

(5.9c)

d ai bi

Q[(t) is

Q[( t) = O( J[ J[_ILIL~_ , )U( siL~J[_,t; LI_,J[). The matrix elements of

(5.9d)

Qt. Qt for N = 1 and 2 are given in Appendix A.

6. Diagonalization of the Hamiltonian The Hamiltonian can be diagonalized either in the non-orthonormal basis [rpJ~7"~J~ : JM), or the orthonormal basis I~pJ,,~rJ~r " JM). For either neutron or proton section, the orthonormal basis Isc, JNMN) can be expanded in terms of the non-orthonormal basis

It, JNMN), JN Zi,7-lT", JNMN).

]sc, JNMN) = ~

(6.1)

7" -- JN

The coefficients Z~J,~ are found in the following way. Suppose that Z¢ is an orthonormalized eigenvector of the overlap matrix AJ',

AT-,

=

(T, JNMNI.rJNMN) '

corresponding to the eigenvalue ZJN = ~ J N /

(,~

¢,7-/V/ ~a~,JN.

(6.2)

A(.jN. Then (6.3)

As a matter of fact, it is more convenient to use the orthonormal basis. In this basis the eigenvalue equation of the Hamiltonian for a given angular momentum J can be written by the matrix equation

J.-Q. Chen/Nuclear Physics A 626 (1997) 686-714 (H J -

EjI) YJ = 0,

699

(6.4a)

where I is the unit matrix and YJ is a column vector. The matrix elements of H J are given by

((~g,.~g~ ' ' ' ' : JMIHI(~J~(c~J~: JM) 6

,j

o s

3

!

+ ~-~ xt(¢',.f,,¢'9'~ : JIIQ'~" Q'II¢,,J~(~.J,~ J) •

(6.4b)

t

Suppose that ¥~ = {Y~(s%J,, s%&.)} is the eigenvector of the matrix H J corresponding to the eigenvalue Ej~, /3 = 1,2 . . . . Then the eigenfunction of the Hamiltonian is

I~,JM) = ~

Y/~(~.Jv,(~Jcr)l(vJ.(~rJcr : JM).

(6.5)

Using Eq. (6.1), ]/3, JM I can be expressed in terms of the non-orthonormal basis

Ir~J,r,~J,~ : JM), X~(r~J,,r,~J,~)[r~J,r~J,~:

I/3, JM) = ~

JM),

(6.6a)

r~J~r~J~

XJ(r.J., r~J.) = ~

Y~(e.J.,e~rJ.)ZJ:,rZ~:,r ..

(6.6b)

To analyze the composition of the various states I/3, JM), it is advantageous to use the normalized but non-orthogonal basis Ir, J, rcrJ~ : JM)N,

I/3, JM) = ~

XJ~(r~J~,r~rJ,~)NIr, J,r,~J,~ : JM)N.

(6.6c)

r.J~r~J~

The new linear combination coefficients are given by

XJ (r~j~ , r,,j,~)N = (A..,rA j. j. ) l/e Xlj(r.J,., j ..... r~J~),

(6.6d)

7. The E2 and M1 transition rate

The M1 transition operators are MI = M l ( r r ) + MI(~,),

M1 = g t L + g ~ S ,

(7.1a)

where gt and gs are the orbital and spin gyromagnetic ratios. The total orbital angular momentum operator L and total spin S can be identified with collective dipole operators.

J.-Q. Chen/Nuclear Physics A 626 (1997) 686-714

700

L,, =- Q~ = E q(cdl )P~(cd), cd

S,, =- Q[) = E q'(cdl)P,~(cd)

(7.1b)

cd

with

ll½

'

)t+U2+c 1 e d f c d 1 }

(7.1c)

The E2 transition operator is

E2 = e~rQ2 + evQ 2,

(7.2)

where e~ and e~ are effective charges of a neutron and proton, respectively. The B(E2) and B(M1) values are given by B(E2) = 2Jf + 1 2 2Ji + 1 T ~ i '

2Jf + 1M2fi '

(7.3)

B(M1) = 2Ji + 1

Tfi = e~(/3', &IIQ2II/3, Ji) q- e~ (/3', JfllQ~ll~, Ji), Mfi = (i ~t, Jfl IM1 (Tr) II/3, Ji) + (/3', Jfl IM1 (v) [t/3, Ji).

(7.4)

To facilitate the computation of Eq. (7.4) by computer, it is convenient to introduce the matrices X J and Q~, p = u, 7r, by their matrix elements

(/3lX+lr~J~, r~-&-} = xe#(r~J~, r~J~),

(7.5a)

(7'uJ~r;J; : JflIQtllrvJvT~rJ~r : Ji} ' t t = 6j, j AJf r U(JtvJ~rJft; JiJ~) (%J'~llQ~llr~J~),

(7.5b)

v,

v

v,

v

(r'~J;r~J~ : JflIQtvll.rvJvrrrJ~r : Ji) = 6J',J, AJT~,r~(-)J~-J~+J+-J'u(J'rrJvJft;'

,

jiJtv)

t t t (r~J'~llQ~llr~J~).

The reduced matrix elements (/3', Jfl IQ~[[/L Ji) and (fl', Jfl by the (/Y/3)th entry of the matrix XJIQtpXJ',

(/3', J fI IQtalI/8, Ji) = ( ~ t t x J f atpxJ'llo).

lOLlI/L J/} are then

(7.5c) given

(7.6)

The quadrupole moment Q and the magnetic dipole moment of a nucleus are given by the diagonal matrix elements T/i and M i i , respectively, /-'~q'F

j j

Q = V - ~ CJJ'2°Tii'

~l",J

J ..

tx = V -3 -wJJ''°'v'ii"

(7.7)

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701

8. The NPSM for odd A nuelei

It is assumed that in an odd system with 2N + 1 nucleons, all nucleons are coupled to collective pairs except one which may occupies any single-particle level a in a major shell. The wave function is denoted by

I(r, r 2 . . . r N ; J i J 2 . . . J N ) a " IM)= (ZJNt(ri, Ji) × C,~)IMtO).

(8.1a)

For simplicity the Hermitian conjugate of (A Jut × C~t ) ~ is denoted as ( AJu × Ca)lM ~

,tiM] " ( ( z J u t × c~/~t

(8.1b)

The possible values of the intermediate angular momenta J1 . . . . . JN are the same as for the even system, and the total angular momentum 1 = ]JN -- a[ . . . . . JN + a. In the following we are going to show that the overlap, the matrix elements of the operators A 't • A ~ and Qt for an odd system can all be expressed in terms of the known matrix elements for an even system. Using the standard recoupling technique, we can show that the overlap for the basis (8.1a) can be expressed as the matrix elements of a one-body operator in an even system, ( ( A J~' × Ca ) ' M ( A Jut × C ~ ) M ) (0[

(

a J'u × C a

)'(

,)'

a Jut × C b

M

IO)

M

J'lN , , , = -T-~-'~(--)su-z+b-'U(JNaJNb;lt)(AJNllfJ(ab)l

iaju, ),

(8.2)

1 t

where the operator pt(ab) is related to the operator pt(ab) in (2.3a) as

(

,)' = -O(abt)Pt(ba)

j ( a b ) = Ca × Cb

+ b6abStO.

(8.3)

In order to express the matrix elements of the Hamiltonian for an odd system in terms of those for an even system, it is convenient to recast the Hamiltonian (4.1) in a form in which the pair creation operators appear to the right,

H=Eo + Z (7%(~r) + V(cr)) + Z K t Q ~ . Q', (r=-~7",p

7-(0 = ~

e:,ha, a

Qt = ~

(8.4a)

t

"13= Z

g'A'* " Ast + Z S

ktQt " Qt, t

O(cdt)q(dct) gJ(cd),

cd

where E0 is an irrelevant constant term. Using (8.3) and

(8.4b)

702

J.-Q. Chen/Nuclear Physics A 626 (1997) 686-714 A "t • A ~ = A s. A st - 2 ( 2 s + 1) Z y o ( a a ' s )

2 1-

ClO t

we can show that the coefficients kt and Kt in (8.4) are the same as in (4.1) and

e~, = ea + 2/gla Z ( 2 s

+ 1)yo(aa' s) 2

(8.5)

s,a t

The matrix element of A s • A st ((AJ'NxCb)~AS.ASt

-

(AJNt x C ~ ) I M )

Z_~X--" (_)J'~,I+J'~-Ju+I-JNU(sJNLa;JN+II)U(sfNLb;

2_ L_+ 1 21-t- 1

JN~1JN+1L X ( [ ( A) J'N x A ~ JN~I X Cb jL [(AJNtX ASt)"~Ju~l X C ~a]L )

JN+jl)

t

M

,

(8.6)

M

where JN+l = IJN -- sl . . . . . JN + s, since the pair A st is different from the building blocks A r't. Eq. (8.6) is the overlap for the 2 ( N + 1) + ! odd system, which can be calculated by (8.2).

The matrix element of ~ot(dc )

< ( A J'N XCb) 1' f J ( d c ) ( a =-- Z

Jui

XCbi)/)

V((bd)r''(ca)r''JtNJNJN+ll'l)

rrt JN ~ 1

x

(Iz"

'

x A r (bd)

, ),

x Art(ca)j Js~

(8.7)

V ((bd)r', (ca)r,' J~JNJN+II'I) = (_)1'+r-a-t-Ju~, Ju+l^ U(J~bJN+jd;I'r') II x Z

U(taI'JN; kl) U(dcka; tr) U(drl'JN; kJN+l ).

k The matrix element of C a ((AJ'NxCb) '

Ct

AJNt)=((AJkxCb)IM(AJutxC,,)tM).

(8.8)

It is to be noted that in the expressions (8.2)-(8.7) the dependence on the total angular momentum 1 is only through the Racah coefficients, while the matrix elements for the even system is independent of 1. This has great advantages in practice, since the calculation of those matrix elements are time consuming.

703

J.-Q. Chen/Nuclear Physics A 626 (1997) 686-714

9. The special cases of NPSM In this section it will be shown that the NPSM includes the SO(8) or Sp(6) FDSM and the pseudo-SU(2) scheme as special cases.

9.1. The N P S M in single j case Before proceeding to our main goal in this section, let us first discuss the single-j case of the NPSM. The importance of the single-j case is twofold; first it is closely related to the S O ( 8 ) or Sp(6) FDSM case, and secondly the code for single j serves a prototype for the rather complicated NPSM code. By letting all the structure coefficients y ( a b r ) and q ( c d t ) be equal to one, a = b = c = d, and deleting all the summations over a, b, c, d, the formulae for the collective pairs are reduced to the formulae for the single-j case. For example, Eqs. (2.6a) and (2.12) are simplified as t

!

~

t

A rk = l(4k(try) F2(rktrk)A rk,

(9.1a)

t r.,t r ti ) A G k ( s t ) F l (r~st)Qk-l (t) . . . Qi+l (t)lf4i(tri)F2(

Brft = Z

rft

,

(9.1b)

t

where Fl (rst) and F2 (rst) are the coefficients in the basic commutators [/T,r, A.,'t ] t = 2P6rsStO - FI (rst) pt, [,~r, p., ]r = F2(rst),4'.

(9.2b)

From I-(3.5) we have

Fl (rst) = 2Fz(rst) = 4P2~

(9.2a)

r s

aaa

t}

(9.2c)

9.2. The F D S M The SP(6) and S O ( 8 ) FDSM Hamiltonians for an n-p system are of the form [ 16] HFDSM = Z Z gr(°')Art(cr) " Ar(°') "q- Z Krpr(q'r) " p r ( p ) , o'=¢r,vr--O.2 r

(9.3)

where S t = A °t and D t = A 2t are the building blocks, and p r the multipole operators in the model. The truncation to the S - D subspace in the FDSM is achieved by introducing the k-i basis [ 14] with j = k + i, k = 1 or i = 3/2. The creation operator in the k-i basis and that in the j basis are related by a Clebsch-Gordan coefficient

C]m"

btkmk,im, ~ Z

CJkmnktim,

J By freezing either the k or i degrees of freedom (i.e. couple either k or i of two nucleons to zero), the fermion space is truncated into an S - D subspace with the SO(8)

J.-Q. Chen/Nuclear Physics A 626 (1997) 686-714

704

dynamical symmetry for the i-active (i.e. k-frozen) case, or the Sp(6) symmetry for the k-active (i.e. i-frozen) case [ 15]. The pair operators and multipole operator in (9.3) can be defined either in the k-i basis or the j - j coupling basis.

9.2.1. The k-i basis (1) For SO(8) (i-active), i = 3, k = 2, S2 = S2ki = 1 ( 2 k + 1 ) ( 2 i + A~t

r e~-~

[b~i×bkijo ~ ,

=

1) =10,

[b~iXbki]Or

(9.4a)

0/~

(2) For Sp(6) (k-active), i= ½, 7, k = 1, O = ~i=l/2,7/2S2ki = 15,

b~i × b~i I r° '

A~t = ~ i=1~.7,/2

Pu = Z

#0

b~i x [~ki

•

(9.4b)

/.tO

i=1/2,7/2

The definition of all the operators in Eq. (9.4) are the same as used in Ref. [15], except that the multipole operator P~ in the SO(8) case differs in sign, which is due to the definition of the time reversal operator 6",~,~= ( - 1 ) " - " C , , _ , used here instead of (~,,,~ = ( - 1 ) ~ + ' ~ C , _ , , From Ref. [ 15], or from Eq. (8.6) of Ref. [ 18] we have the commutators [ ,~, Ast ] t

=

(9.5a)

f2~t~rst~t0 _ FI ( r s t ) p t ,

[~r, ps]t = F2(rst)~t,

(9.5b)

Fl(rst)=-2Fz(rst)=2(-)zi"+lts)a{

r s t

'

(9.5c)

where j,, is the active part of the single-particle angular momentum [ 15], i.e. j,, = 3/2 for/-active, i.e. the SO(8) case, and j,, = 1 for k-active, i.e. the Sp(6) case. A comparison of (9.5) with (9.2) shows that if we let the s.p. angular momentum a be equal to j~, the factor ~k (see Eq. (2.12)) ---+(O/2)~ok, and Fi(rst) are replaced by (9.5c), the NPSM for the single-j case will go over to the SO(8) or Sp(6) FDSM depending on whether ja = 3/2 or 1, respectively.

9.2.2. The j - j coupling basis The S and D pairs and multipole operators in (9.4) can be expressed in terms of the

j - j coupling basis, A~t = ~-'~y(abr)[CJ × Ctb]u, r

P• = Z y ( a b r ) [ C t a x Ca] u , r

ab

~

r

(9.6)

ab

where the distribution functions are given by (1) For SO(8), k = 2 ,

.v(abr)=(-)r+k+a+3/2g~b{ a3 b3 kr } '

(9.7a)

J.-Q. Chen/Nuclear Physics A 626 (1997) 686-714

y(abO) = ½Sab&, (2) F o r S p ( 6 )

'

i=l

2'2'

1 3 5 7 a , b = 7,7,~,-~.

705 (9.7b)

7

i=I/2,7/2 y(abO) =~ab~a, l^

l 1 i

a , b = 71 , ~3, ~5, ~7, 7 9.

(9.8b)

The S and D pairs in the FDSM are related by 2t

2

0t

AVDSM = -- [ P , AFDSM].

(9.9)

From Eq. (9.5) we see that the SO(8) and Sp(6) cases in the k-i basis can be treated as a single-j case with j = ja, while according to (9.7) and (9.8) in the j-basis they can also be treated as a multi-j case. This duality provides an invaluable check for our NPSM code. We first check our single-j NPSM code with the analytic expressions in the FDSM, and then put the structure coefficients of (9.7) and (9.8) into our multi-j NPSM code. These two results should coincide if the code is correct. Since the multi-j NPSM code is rather complicated, this check is of vital importance. We checked our code with the analytical expression [14] for the norm in the SO(8) D SU(2) ® SO(3) symmetry limit of the FDSM, 6 and with Eqs. (3.20) and (3.18) of Ref. [ 14] for the matrix elements of pairing interaction S t • S and the multipole operator p2. We also checked our code with some analytic expressions in the Sp(6) D SU(2) × SO(3) limit i n R e f . [15]. It is seen that if we use the ( S , D ) pairs in (9.7) as the building blocks for the normal parity levels but neglect the D pair of the abnormal parity level, the NPSM will be reduced to the FDSM.

9.3.

The pseudo-SU(2) model

The pseudo-SU(2) scheme [ 6 - 8 ] can be regarded as a special case of the k-i basis, with k and i ( = 1 / 2 ) as the pseudo-orbital and pseudo-spin, respectively, and it belongs to the k-active case. In the pseudo-SU(2) model the three levels PU2,P3/2, f5/2 in the 2 8 - 5 0 shell are assigned to the pseudo-orbital angular momentum k = 0,2; while the four levels sl/2, d3/2,d5/z, gT/2 in the 50-82 shell are assigned to k = 1,3. The Hamiltonians in the pseudo-SU(2) model consist of surface delta interactions (SDI) [25] for like nucleons and quadrupole-quadrupole interaction between neutrons and protons [6], 1

Hesu2 = ~ ~ (r=Tr,/.,

ZG(cr)AstDI(o').A~D,(cr)+Su:rQr(Tr).Qr(~), r

(9.10)

r

6 Notice that there are two misprints in Eq. (3.12) of Ref. 1141 for the coefficients o~p(Kr) of the SO(8) • SU(2) ® SO(3) wave function.

J.-Q. Chen/Nuclear Physics A 626 (1997) 686-714

706

where the building blocks A rt are called the favored pairs of the SDI in Ref. [6] (see Eqs. (9.11) and (9.13)).

9.3.1. The k-i basis In the k-i basis, the favored pairs and the multipole operator are expressed as A~tDl=Zy(kakbr)Ar(kakb)t,

Ar~(kakb) t

=

[b~,,, × bk,,iJ~zO, t lrO

(9.1 la)

ab r jr0 P~(kckd) = [b~ic x b kdijuo,

Qr=Z(-)key(kckdr)pr(kckd),

(9.11b)

cd

where

y( kk'r ) = x/2 ( IJ¢'/ ? )C~o°~,o.

(9.11c)

rt I and Qr differ from Hecht et al. [6] by a factor of v ~ and The pair operators ASD -1, respectively. Notice that Qr is the second quantized form of the operator Qr of Eq. (8.4b) in the k-i basis with r being the pseudo-orbital angular momentum. As #f pointed out in Ref. [6], the set of operators Asp I and Qr do not form a closed algebra, see Eq. (7.6) in Ref. [19]. According to the prescription (7.1)-(7.4) in Ref. [ 19], all the formulae in Sections 27 remain valid for the pseudo-SU(2) case in the k-i basis if the indices a,b,c,d are understood as the indices for the pseudo-orbital angular momentum, together with the replacements O( abr) --~ O(½½0)O(abr) = -O( abr),

(rsr I {O00}{rst) m{rst} 31½ 1

dab

dab

=--~

dab

"

(9.12a) (9.12b)

9.3.2. The j - j coupling basis Eq. (9.11 ) can be expressed as

rt = Z Y s D , ( a b r ) [ C ~ x C~] r, AsoI ab

Qr= Z (__ )kOySDl(abr) [ Cta x cb ]r

(9.13)

ab

Ysol(abs)

=

( _ ) l o l [1 + (_)t,,+t,,+s] ~C",-½ b- I/2,sO

From (9.13) and (2.13) we get a relation between the favored pairs of the SDI, rt 1 ASD I = --

0~ ] . [ Qr, AsDI

(9.14)

It is interesting to observe the similarity between the relations (9.9) and (9.14) for the pairs in the FDSM and the pseudo-SU(2), respectively.

J.-Q. Chen/Nuclear Physics A 626 (1997) 686-714

707

If the favored pairs in (9.13) are used as the building blocks and the abnormal level is ignored, the NPSM goes over to the pseudo-SU(2) scheme of Ref. [6].

9.4. The s.p. energy term in the FDSM and pseudo-SU(2) A common feature of the FDSM and pseudo-SU(2) model is that there is no s.p. energy term in their Hamiltonians. In the following we will show that even if a s.p. energy term is added, it will have no effect at all on the spectra and B ( E 2 ) values, etc. In other words, the shortcoming of the neglect of the s.p. energy splitting in these two models can not be remedied unless the truncation to the total pseudo-spin zero space or total pseudo-orbital-angular momentum zero space is released. In the k-i basis the single-particle energy term takes the form

HO = Z e ( k i ) n ( k i ) ki

- Z

~

er

[b~i × [)k'i'] (rr)O

(9.15)

0

kiU i' r=1,2 ....

where the superscripts denote the coupling k + k t = r, i + i t = r and r + r = O, while e(ki) and n(ki) are the average s.p. energy and the total number of the particles in the "subshell" a = [ k - i l . . . k + i, k+i

e(ki)=

Z a=lk-i I

2a+l (2k+1)(2i+1)

k+i

ca'

n(ki)=

Z

n,,

(9.16)

a=lk-i]

while e r is given by k k/ r }

er = F Z

O(aUir)(2a+ l)

0

it i a

Ea"

(9.17)

In the Sp(6) S-D subspace of the FDSM, or the favored pair space in the pseudoSU(2) model, the total pseudo-spin of all states are zero, 1 = 0, therefore the second term in Eq. (9.15) which has I = r ( = 1,2 . . . . ) has no contribution to the expectation value of H0. Similarly, in the SO (8) S-D subspace of the FDSM, the total pseudo-orbital angular momentum K of each state is zero, and thus the second term in Eq. (9.15), which has K = r ( = 1,2 . . . . ), has no contribution to the expectation value of H0. Furthermore, since the nucleons in these two models are distributed evenly over all s.p. levels, the first term ~ki e(ki)n(ki) is a constant for a given nucleus.

10. Discussions In the above, the NPSM is cast in the most general form for both even-even and even-odd nuclei under the assumptions that all nucleon pairs are collective and for a given angular momentum r there is only one collective pair Art. If there is more than one type of collective pairs with a given angular momentum r, we only need to introduce an

708

J.-Q. Chen/Nuclear Physics A 626 (1997) 686-714

additional label to distinguish them. The NPSM can also be easily extended to include non-collective pairs. For example, suppose the Nth pair is non-collective. Then we only need to let the structure coefficients of the Nth pair be y ( d b t rN) = ~1 p(aNbNrN)~a,,aN~Sb,,b~,

(10.1a)

p( abr) = 1 - O( abr)p~ b, where Pab is the permutation operator a ~ b. Similarly, by letting the structure coefficients of the p - h pair Qt be a Kronecker symbol, q ( c ' d ' t) = ~cc,6da, ,

(10.1b)

the pairs Qt will become non-collective pairs. Therefore, all the formulae for the collective pairs apply to the cases for non-collective pairs, or a mixture of collective and non-collective pairs. When the building blocks consist of all possible non-collective pairs, the NPSM is equivalent to the shell model in the full space. The formulae for this case are compiled in Appendix B. For applying the NPSM to medium and heavy nuclei, we have to truncate the shellmodel space to the collective SD, or SDG, subspace, etc., which will be the subject of our next papers. For convenience the NPSM with SD truncation will be termed the SD nucleon-pair shell model, or SD pair model (SDPM) for short. rst From (9.2a) or (9.5a), we see that the 6j symbol {aa,,} can be interpreted as the Pauli correction term. Since

{rs ) a aa

for : v n "~----'~ 1 / ~ / 2 ,

fort=odd

with r,s = 0,2, and /2a = a + ½, the Pauli correction term will vanish when the shell degeneracy goes to infinity. It is interesting to note that if the shell degeneracy ,(2 goes to infinity, Eq. (3.3) will go over to (sl [rl) = 6~.1r, , < s l s 2 : J 2 l r l r 2 : J 2 ) = (1 +O(rlr2J2)prtr~)(6slrlSs2r2),

(10.3a)

under the normalization 2Zy(abr)2=

1.

(10.4)

ab

They are precisely the overlaps between coupled bosons. In other words, when the Pauli terms are neglected, the collective fermion pair becomes a boson, A st ---, b 't, which is the foundation of the physical interpretation of the IBM. Naturally, the NPSM can used as an alternative to the mapping method for studying the microscopic foundation of the IBM.

J.-Q. Chen/Nuclear Physics A 626 (1997) 686-714

709

Among the several models the NPSM involves the least approximations and its relation to the other models can be seen from the following hierarchy relations: The hierarchy o f various nuclear models

Pseudo SU(2) ~,i,h i ...... d

l NPSM

pairs

I with 1, or 2 non-S pair(s)

l

BPA

Pairing Model l,,o v paiT~ at all

, SDPM

mapping

I IBM

N~¢<,

, BMM

with symmetry[dictatcd ,I.S and D pairs

FDSM

The formalism of the NPSM is much simpler than that of the BPA [9], which applies only to semi-magic and near-magic nuclei. A broken pair (or a non-collective pair) in the FDSM is hard to treat, since it breaks the dynamical symmetry, but it is easy in the NPSM, where the collective pair and the broken pair are treated on the same footing. Besides, many of the broken pairs in the FDSM are no longer broken pairs in the NPSM, since the sense of pair in the FDSM is too restricted. Compared to Ref. [21 ] our formalism is more concise with the introduction of the factors Hi(s), Qi(t), interpreted as the propagator of the p-p pair A d and p-t pair pt, respectively, and the factors Gk ( s t ) and IQ~(try), interpreted as the strengths of the vertices for pair transformation Ar, ~ p t and pair scattering A rk ~ B r'k, respectively. Consequently, the expression (2.1 la) for the new pair suggests a clear physical picture for the formation of the new pair.

Acknowledgements The author would like to express his gratitude for many stimulating discussions with Drs. B.Q. Chen, X.W. Pan, A. Klein, F. Iachello, K.T. Hecht, H. Wu, J. Ginocchio, S. Koonin, and especially for the hospitality of A. Klein and S. Koonin extended to him in 1992 when most of the present work was done at the University of Pennsylvania and Caltech. The author also thanks Y.M. Zhao and Y.A. Luo for checking all the formulas. This work was supported in part by the State Science and Technology Commission of China, the Research Fund of the State Educational Commission of China, the Natural Science Foundation of China and U.S. Department of Energy under the grant number 40264-5-25351.

Appendix A. Some special cases of the overlap and matrix elements To illustrate the application of the general formulae (3.3), (4.6) and (5.8) for the overlap, the matrix elements of A st • A s and Q~ • Q', in the following we give explicit

J.-Q. Chen/Nuclear Physics A 626 (1997) 686-714

710

formulae for the cases of N = 1 and 2.

Overlaps: (1)

(s, lrl) =2as, r, Z y ( a b r l ) y ( a b s , ) . ab

(2)

(SlS2;J2lrlr2;J2) = ~_, 4yl(alblrl)Y2(a2b2r2)Y3(atblrt)

(A.I)

albla2b2

×y4(a2bir2) ( 1 + O(rlr2J2)Prlr2) ( a s l r I (~s2r2)

al bl rl -16

yl(ala2sl)Y2(blb2S2)Y3(alblrl)y4(a2b2r2)

Z

a2 b2 r2

al bj a2b2

SI $2 J2 (A.2)

The matrix elements of A st • A s (1)

(rlA st. A'lr ) = 4 & s ~ _ , y ( a b r ) 2 y ( c d r ) 2 abcd

(2)

(sis2 " J2IA st. ASlrlr2 : J2) = 2(1 + O(rlr2J2)Pr~r2)

(A.3)

x { ~sr2Zy(abr2)2(sls2;J2lrls;J2)}ab -}-J2 -1

(-)J2+s-r'l(SlS2; J2lrt, s; J2),

Z

(A.4)

i r1

where the pair r~ is determined by (2.12a) with

z' ( aa'rll ) = 4F1 ~2"~Z t'G2 (st) 101 ( tr'3) t

Z

x

y(bb's)y(abr2)y(dblrl) bb'

{ /{ } r2 s t

rl

bI a b

a a~ bI

t r1

"

The matrix elements of Qt . Qt

(1)

2/+ 1 (slQ' " Qtlr) = 6rs Z 2 r + 1 (r'lr')' r,

(A.5)

with the pair r ~ determined by t

~r = [~r, Qt]r'. (2)

(sis2 : J2]Q" Qtlrlr2 : J2} = L2

2L2 + 1 2J2 + 1

x ( 1 + O(sls2J2)ps, s:) ( 1 + O(rl r2J2)Pr, rz)(Sl st; L2lrlr~; L2), (A.6)

J.-Q. Chen/Nuclear Physics A 626 (1997) 686-714

711

where the pair r~ and s~ are determined by (2.6a) and (5.9a), respectively.

Appendix B. The N P S M in the full space When the building blocks of the NPSM consist of all possible non-collective pairs A '~t

A rkt = A r ~ ( a ~ b k ) t,

(B.1)

= AS'(cidi) t

the NPSM is equivalent to the nuclear shell model in the full space of one major shell. In the following the quantum numbers rk and si in the argument of a bra or ket vector are to be understood as (akbk)rk and (¢idi)si, respectively. A general Hamiltonian can be written either as 1

H = ~ ean,+ ~ Z a

G(abcd, s)AS(ab) t. A'~(cd),

(B.2a)

F(abcd, t)O(cdt)pt(ab), pt(dc),

(B.2b)

abcds

or as

1

H = ~ e~,na- ~ ~ a

where %' and

abcdt

F(abcd, t) are related to ea and G(abcd, s) by 1 e;,=e. + ~ Z

t;s

F(abcd, t) = Z ( 2 s .~

(2s + 1 ) (2a+ B

+ 1 )O(bcs)

G(abab, s)

{a s} cb t

(B.3a)

G(adbc, s).

(B.3b)

Eq. (B.3) was obtained from (B.2) by using Eq. (25) of Ref. [ 18]. The matrix of the single-particle energy is diagonal in the basis with non-collective pairs: N

Z eanalr'"" rN; J1... JN> = ~-~(e,,, + e,,,)[r~.., rN; J , . . . JN). a

(B.4)

k=l

By using (10.1) the following formulae can be obtained from their counterparts for the collective ones, (3.3), (4.6), (5.3), and (5.8):

The overlap matrix (S1S2...

SN; J~ . . . JtN_ 1 J N l r l r 2 . . .

rN; JI . . . J N - I JAr} 1

HN(,N)

Hk+,(sN)

k=N Lk-i ...LN-.I X I~Ok6Lt ,.Jk lt$sN.rk~(akbk, CNNN) rk

× (sl • • • SN-I ; Jl""

fU-, [rl • • • rk-1, r~+l • • • r u;

Sl-..

Jk-1 tk...

LN -1

)

J.-Q. Chen/Nuclear Physics A 626 (1997) 686-714

712

1

--

Z

Z

rkrisutGk(st)Qk-l(t) ""Qi+l(t)iVli(tr~)

i=k-I r~tL~...Lk 2

×(S1.

. .SN-I;J~ . . .

t JN_l]rl.

. . . (aeai)rl.. rk-l,rk+l. .rN;

.

(B.5)

J1...Ji-ILi...LN-1) }].

where ~5( ab, cd ) r = ~ac ~bd

-- 0 ( a b r )

6adaOc.

Special cases of the overlap ( ( Cldl ) sl l ( albl )rl)

= ~rlsl

6( albl, cl dl )r,,

(B.6)

((cldl)Sl (c2d2)s2 : J2l (albl)rl (a2b2) r2 : J2) = (l + O(rlr2J2)Pala=Pb,b2Pr,re) [~slr, as=r=~(albl,Cldl)r'a(a2b2,c2d2) r2] al bj rl

|

-p(a2b2r2)p(alblrl ) {6(ala2, Cldl )s*a(blb2, c2d2) s2 a2 b2 r2 i SI $2 J2

(B.7) where [ ] represents the unitary 9j coefficient. Matrix element of the pairing interaction JN (OIAMN(Si, J'it )aS(e'd') t . aS(ed)a~N(ri, ./1.)1-10) 1

k=N Lk ~...Lu-i

x [~Ok~S,~k6Lk_l,Jk ltS(akbk, cd) ~k X

(S1$2...

SN; J ' l ' " JtN-1 JNI r'''"

rk-1, rk+l.., ru, (c'd') s;

J1 ... Jk-lLk... LN-1JN) 1 -]- Z Z ~k~iSNtGk(st)Qk-l(t)...Qi+l(t)lVli(tr~)p(akbkrk)p(aibiri) i=k--1 r~tLi_l...Lk

x a(cd, bkbi) s

{1{ rk s t bi ak bk

I

ri t r i ai bi

ak

J.-Q. Chen/Nuclear Physics A 626 (1997) 686-714 × (SI...

! JN_lJulr,...

SN; J ~ . . .

(akai)rl.. • r k - l ,

rk+j..,

713 rN, (c'd')s;

JI . . . J i - I L i . . . L N - 1 J N ) } ] .

The matrix elements of the multipole operator (OlAJ'~(si, J[)lle'( cd)l AJN(ri, Ji)tlO } 1

=Z

Z

QN(,

•Qk+l (t) 6LuJ'uMk (cdt, dak r~)

k=N r~Lk...LN

× (SI...

SN; J~... fulrl... (da~)r~... rN; J1... J k - l L k . . . LN).

(B.8)

The matrix elements of the multipole-multipole interactions

(oIa~ (s. ~)Pt(c'd').

pt(dc)A~N(ri, Ji)t [0)

1

2LN + 1 2JN+ l E

=Z LN

E

QN(t)'"Ok+'(t)Mk(cdt'da~rk)

k=N r~Lk...LN-i 1

xZ

E

Q~u(t)'"Q;+l(t)M~(c'd't'd'cis~)

i=N r;L;...L N I I I I × (s~... (d'ci)s~... SN; J~... Ji-t L Ii ' " L~u-1LNIr~ ... (daDrk'" rN;

J1 • .. Jk-I Lk... LN-I LN}, where × M~(cdt, dakr~) =U(rktJk_lLk;rkJk)rktp(akbkrk) ' ^ ^

(~bkc

Mi(ctd't, dtcis~) = U(sitJ[_ 1LI; slJ[)~itp(cidisi) × t~d,c,

t 4},

d ak bk

(B.9)

d t ci di

The above formalism provides an alternative approach to the multi-shell and multi-spin shell-model formalisms [26,27]. References

1 ] X.W. Pan et al., Eds., Contemporary Nuclear Shell Model, Lecture Notes in Physics (Springer, Berlin, 1996). 21 E lachello and I. Tahni, Rev. Mod. Phys. 59 (1987) 339. 31 E. Bender, K.W. Schmid and A. Fassler, Nucl. Phys. A 596 (1996) 1; K.W. Schmid, F. Grummer, M. Kyotoku and A. Faessler, Nucl. Phys. A 452 (1986) 493; K.W. Schmid, R.R. Zheng, E Grummer and A. Faessler, Nucl. Phys. 499 (1989) 63. 4] C.W. Johnson, S.E. Koonin, C.W. Johnson, S.E. Koonin and W.E. Ormand, Phys. Rev. Lett. 69 (1994) 3157; W.E. Ormand, D.J. Dean, C.W. Johnson, G.H. Lang and S.E. Koonin, Phys. Rev. C 49 (1994) 1422.

714

J.-Q. Chen/Nuclear Physics A 626 (1997) 686-714

151 M. Honma and T. Mizusaki and T. Otsuka, Phys. Rev. Lett. 75 (1995) 1284; 77 (1996) 3315. [6] K.T. Hecht, J.B. McGrory and J.P. Draayer, Nucl. Phys. A 197 (1972) 369. I71 J.B. McGrory, Phys. Rev. Lett. 41 (1978) 533. 181 T. Otsuka, Nucl. Phys. 368 (1981) 244. 191 Y.K. Gambhir, S. Haq and J.K. Suri, Ann. Phys. 133 (1981) 154. 10] K. Allaart, E. Boeker, G. Bonsignori, M. Savoia and Y.K. Gambhir, Phys. Rep. 169 (1988) 209. 111 A. Arima and I. Iachello, Adv. Nucl. Phys. 13 (1983) 139. 12] P. Halse, L. Jaqua and B.R. Barret, Phys. Rev. C 40 (1989) 968. 131 A. Klein and E.R. Marshalek, Rev. Mod. Phys. 63 (1991) 375. 141 J.N. Ginocchio, Ann. Phys. 126 (1980) 234. 151 C.L. Wu, D.H. Feng, X.G. Chen, J.Q. Chen and M. Guidry, Phys. Rev. C 36 (1987) 1157. 161 H. Wu and M. Vallieres, Phys. Rev. C 39 (1989) 1066. 17] Troltenier, C. Bahri and J.P. Draayer, Nucl. Phys. A 589 (1995) 76; D. Troltenier, C. Bahri, J. Escher and J.P. Draayer, Z. Phys. A 354 (1996) 125. [181 J.Q. Chen, B.Q. Chen and A. Klein, Nucl. Phys. A 554 (1993) 61. 1191 J.Q. Chen, Nucl. Phys. A 562 (1993) 218. [201 J.Q. Chen et al., Nucleon-Pair Shell Model: Even system, U. Pennsylvania Preprint, UPR-0085 NT 1993. 1211 M. Grinberg, R. Piepenbring, K.V. Protasov and B. Siverstre-Brac, Nucl. Phys. A 597 (1996) 355; R. Piepenbring, K.V. Protasov and B. Siverstre-Brac, Nucl. Phys. A 586 (1995) 396; A 586 (1995) 413. [22] J.Q. Chen, Y.A. Luo and X.W. Pan, Nucleon-Pair Shell Model: Numerical calculation in the S-D subspace for 132Ba, to be submitted to Nucl. Phys. A. t23] B. Silverstre-Brac and R. Piepenbring, Phys. Rev. C 26 (1982) 2640. 124] M.E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957). [ 25] R. Arvieu and S.A. Moszkowski, Phys. Rev. 145 (1966) 830. [261 P.J. Brussaard and P.W.M. Glaudemans, Shell Model Applications in Nuclear Spectroscopy (NorthHolland, Amsterdam, 1977). 1271 J.Q. Chen, Phys. Rev. 43 (1991) 152.