A tensor basis approach for pair correlation studies in identical nucleon systems

Nuclear Physics A469 (1987) 273-284 North-Holland, Amsterdam

A TENSOR

BASIS

APPROACH

IN IDENTICAL C.R.

FOR PAIR

CORRELATION

NUCLEON

SYSTEMS

SARMA

and

STUDIES

J. SITA

Department of Physics, Indian Institute of Technology, Powai, Bombay 400076,

India

Received 18 November 1986 (Revised 9 February 1987) Abstract: The unitary group approach has identical nucleon systems. Both oddprinciple and number conservation have A simple algorithm has been developed the 2 = 28 - 50 shell with fixed neutron

been used to study the pair correlated ground states of and even-nucleon systems have been studied. The Pauli been shown to follow in a natural manner in this formalism. based on the method and applied to the protons filling number 50.

1. Introduction

Extensive studies of the nuclear pairing hamiltonian have been carried out in the recent past mainly with a view to obtaining a correlation between shell model and interacting boson model (IBM) ‘,?) ground states. These studies include the zero broken pair approximation (BPA) ‘-‘), the BCS method 6,7), the equations of motion method (EMM) *,‘) and the quasi-boson approach lo). BPA approaches involve recursive use of commutation relations among pair-fermion operators ‘) and lead to relatively slow computer does not yield good ground use and yields

relatively

programmes. The BCS approach, in its simple form, states. The EMM ‘,‘), on the other hand, is simple to

good

ground

state results.

It suffers from the drawback

that in its simple form it neglects summation over intermediate states in handling two-body terms of the hamiltonian. Reported EMM “) results are based on this approximation. Further, elimination of spurious (Pauli principle violating) states requires an auxiliary diagonalization of the density matrix over the full s-pair configuration space and leads finally to a non-hermitian secular matrix. The quasiboson approach leads to simple expressions for the hamiltonian matrix but suffers from the drawback of neglecting number dependent terms in the commutation relations of pair-fermion operators. The procedure cannot, further, be extended readily to odd-nucleon systems. The above considerations led us to consider as an alternative a unitary group approach (UGA), which is relatively simple to implement and well established in electron correlation studies in atoms and molecules i1-i6). In this paper we have attempted the use of UGA to generate and diagonalize the pairing hamiltonian matrix for identical nucleon systems. 0375-9474/87/$03.50 @ Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

C.R. Sarma, J. Sita / Tensor basis approach

274

In sect. 2 we outline the use of tensor basis spanning the Pauli-allowed Nth rank alternating representation of the unitary group U(n) in obtaining the matrix representation handling

of the pairing

hamiltonian

the s-pair configuration

‘). A recursive

approach

is developed

for

space in which the basis states of the N-nucleon

system are expressed in terms of those of the known N -2 nucleon system. The procedure is outlined in sect. 2 and is illustrated using the single-particle levels 1p3,* for protons filling the Z = 28 - 50, N = 50 region. Ground %9/2, lPl/Z, Of,,,, state energies, s-pair amplitudes, occupation probabilities and spectroscopic transfer factors have been determined for both odd and even numbers of protons filling this shell.

2. UGA and the pairing hamiltonian Consider a system of N identical nucleons filling an ordered set of shell-model levels{jJi=1,2,... , k}. Assuming that an ortho-normal set of single-particle basis states ljimi)(-ji s mi cji) are defined over this set of levels, we find that these span a carrier space for the fundamental representation of the unitary group U(n). The dimensionality of this space V,, is n =CF=, 2Q where fii =ji +$. A reduction of V,,(F) using standard procedures yields the irreducible representations of U(n). For identical nucleons, the Pauli principle requires that we need consider only the (i) dimensional alternating representation [ 1 N 0] of the group. The n* generators E&a, b = 1,. . . , n) of the Lie algebra this space and satisfy

J%&&

of U(n)

= %,lj,&

define

a set of shift operators

(1)

,

(2)

[&, JLiI= &xi&x- b&i , E&=&a, These

generators

being

symmetric

on

(3)

in particle

coordinates,

commute

with the

Wigner operators used for obtaining the tensor basis spanning the irreps of U(n) so that the matrix representations of these operators can be obtained easily. Based on the above we now consider the pairing hamiltonian, H = 1

&j.Nj, -G

C A,:Aj, 3

where

with

a,:,_ = (-)j.+m~aJ~_,a .

CR. Sarma, J. Sita / Tensor basis approach

In eqs. (4)-(7),

the a(~+) are shell-model

eja is the single-particle the pairing interaction. operators

of U(n)

fermion

275

destruction

(creation)

operators,

energy corresponding to the level j, and G is a measure of The fermion operators provide a realization of the shift

as, E’,m,:jhm, = aLmaa,,,,

Using eqs. (5)-(g),

the hamiltonian

(8)

of eq. (4) can be reexpressed

as,

We now generate the configuration space of Nth rank tensor products spanned by zero angular momentum coupled s-pair states. As a first step consider two particles in a level j, defining the basis for the representation [ l2 0] of U(n). Adaptation of this basis to yield J = M = 0 leads to the state, (10) where,

Generalizing these considerations to an N-particle system where n, pairs occupy thelevelj,,n2occupyj,,etc.subjecttoni~~i(i=1,...,k)andN=2C~=,niwe obtain the normalized standard Weyl tableau states spanning [ 1 N 0] of U(n) and adapted to J = M = 0 as,

’ ‘-’ jkmkljkekl

where the summation

* * . nk;

(12)

such that 0 < mil<

mi2 <

. * . jkmk,, jktik,,) is a properly

01. As shown in appendix ((n,n2

,

on the right side is over all m-values

. * +< mi,, sji and (j,m,,j,fi,,, of [lN

’ ’ ’ _ikmknk_ikeknk)

00)(Hj(n,n2

normalized SWT basis A, these basis states lead to the matrix elements, * ’ ’ nk; 00)) = i

ni[2&i - G(Ri - ni + I)] ,

(13)

i=l

((n,n~...ni...~j...nk;OO)JH((n,n2...ni-l...~j+l...nk;OO)) =-G[(ni+l)nj(.n,-ni+l)(0j-nj)]“2,

(14)

of the hamiltonian of eq. (9). If, in eq. (14), ni + ni + 1, nj + nj - 1, we interchange i and j. It is interesting to note that eqs. (13) and (14) are identical to the ones obtained by the quasiboson approximation [cf. appendix 1 of ref. ‘“>I. This coincidence will be discussed in sect. 3.

216

C. R. Sarma, J. Situ / Tensor basis approach

The matrix

elements

presented

in eqs. (13) and (14) enable

us to consider

correlated pair states and the matrix representation of the hamiltonian For a two-particle system the pair correlated state is defined as, #b0

=

C

the

using them.

Qoo(j2),

(15)

where Cj are the usual variation coefficients and 4oo((j)2) is defined by eq. (10). We now deal separately with the two cases: (i) N even and (ii) N odd. (i) Consider first the case of two identical nucleons (N = 2). The pair states set spanning the entire configuration +Oo(j’,) (a = I,. . . , k) define an orthonormal space of zero angular momentum states. Using this basis set, the hamiltonian matrix elements of eqs. (13) and (14) can be readily evaluated. Diagonalization of this matrix yields the required eigenvalues and corresponding eigenvectors. The coefficients Cj of 4oo(j2) occurring in eq. (15) are chosen to be those of the ground state $$,) corresponding to the lowest eigenvalue Eg’. Using this known ground eigenstate of the two-particle system we now realize the configuration space of N = 4 particles by multiplying it by each of the pair states +oo(j’,) and antisymmetrizing the result to yield a set of k linearly independent states, l/2

(na+l)p-na) 1

A(2) (,I,,,-n,,(%n2 ' * * nk; 001,

(16)

a

where the summation is over all possible sets of values {n, , n2, . . . , nk} subject to the conditions listed below the summation sign and ( n1n2 * . . nk; 00) are the basis states as defined

+00(h)

by eq. (12). For example,

=

1 1

2(fin, - 1) 1’2 A’,:;...,,(20 .R,

if a = 1, we have the result, * . * 0; 00)

+A(c&,...,,(l!O . - . 0; OO)+- - -+A{;&

.,,,(lOO

. . - 01; 00))

where, in this case of N = 4, Ai:io...o, = C, etc. The states defined by eq. (16) are Schmidt orthogonalized to yield a set of k orthonormal states $~‘((Y)((Y = 1, . . . , k). Using these and eqs. (13) and (14) the hamiltonian eigenvector

matrix for N = 4 can be readily set up and diagonalized. If $2 corresponding to the lowest eigenvalue Er’, it is resolved as,

is the

where Ali),n,...nk, are sums of products of coefficients of (nln2 * * * nk; 00) in cCroo(a) and the corresponding components of the eigenvector. Starting with eq. (17) and proceeding as for N = 4 we can generate the states $g’( j,) for N = 6. Setting up the hamiltonian matrix and diagonalizing it is as before yields Er’ and $g’. This

C.R. Sarma, J. Sita / Tensor basis approach

277

recursive procedure is continued till we reach the required N-value. In general, if 9 $$-“’ is the eigenvector corresponding to the lowest eigenvalue IEhN-*’ we can express it in a manner similar to eq. (17) as, (N-2, ;= rcIo0

A(+Z) (n,nz...?Q)( &312

c

* * * n/&00).

(18)

(n,li=l,...,k; 0c;n,=II,; xq=N/Z-1)

of I&~-” in terms of the basis where Al:;:!. nr, define the resolution (%nz * * * nk; 00) of eq. (12). Multiplying +$F-“’ by 4oo(ji)(a = 1,. . . , k) and antisymmetrizing the result we obtain the k linearly independent vectors,

Thus knowing A{~I?~Lj, the &r’(j,) can be determined and, using the procedure outlined before, the lowest eigenvalue ELM’and the corresponding eigenvector can be determined. We have used this recursive procedure to carry out calculations for variable numbers of protons filling the 2 = 28 - 50 shell with a fixed neutron number 50. The active single-particle levels used are OggLZ,IP,,~, Of,,, and lp,,,. The single-particle energies and pairing strength G used are “) (in MeV): E(fg,*) =0, &(p& =0.6, ~(p,,~) = 1.8, E(gg12)= 3.4, G =0.291. The results for EiN’, where N is the number of protons, are summarized in table 1 and compared with those of exact shell-model calculations. Using $oo (N) the s-pair amplitudes have also been extracted and their variation with even proton number is presented in fig. 1. Further, the occupation probabilities follow as,

Cm

TABLE

Ground-state

exact *) tensor

1

energy (MeV) for even number of protons filling the Z = 28-50 fixed neutron number 50

shell for

4

6

8

10

12

14

-2.41 -2.40

-2.48 -2.47

-1.71 -1.68

-0.02 0.04

3.32 3.38

8.48 8.51

The row labelled tensor is the value from the present approach.

278

C.R. Sarma, J. Sita / Tensor basis approach

Fig. 1. s-pair

amplitudes

I

I

4

6

I 8

N

I 10

I

12

I

14

for even nuclei in the shell A = 28 - 50 for fixed neutron

number

50.

These values are listed and compared with shell-model values in table 2. Another important quantity that can be determined using $bF’ is the two-particle transfer amplitude XCN’(j,) for these nuclei. These quantities can be defined as

(21) TABLE

Occupation

numbers

pj( N) for even number fixed neutron

2 of protons filling the 2 = 28 - 50 shell with number 50

4

6

8

10

12

14

3

(

exact “) tensor

0.497 0.495

0.706 0.698

0.840 0.828

0.942 0.941

0.968 0.964

0.970 0.972

t

exact ‘) I tensor

0.187 0.189

0.337 0.347

0.594 0.605

0.898 0.887

0.95 1 0.947

0.956 0.960

t

exact *) ( tensor

0.049 0.050

0.080 0.082

0.122 0.127

0.179 0.204

0.845 0.838

0.883 0.890

exact *)

0.018 0.018

0.026 0.026

0.034 0.035

0.040 0.044

0.070 0.074

0.259 0.255

z’

1 tensor The row labelled

tensor

is the value from the present

approach.

279

C.R. Sarma, J. Sita / Tensor basis approach

These have been listed and compared

with shell-model values in table 3. (ii) Odd IV. Consider the tensor basis state (nlnZ * * * nk; 00) defined by eq. (12). Choosing a single-particle state (jama) and multiplying both sides of eq. (12) with it, antisymmetrizing and normalizing the result we obtain, (n,n* . * * nk;_h%)=[

5, (~)]m”‘(“~,1)-1’2 (i#o)

TABLET

The two-particle

transfer

amplitudes

_,x

for even number of protons fixed neutron number 50

in the Z = 28 - 50 shell with

4

6

8

10

12

14

f

exact “) tensor

1.822 1.819

1.782 1.779

1.477 1.494

1.127 1.170

0.657 0.711

0.530 0.584

t

I exact “) tensor

0.832 0.832

1.046 1.060

1.253 1.252

1.196 1.175

0.588 0.618

0.432 0.445

t

[ exact a) tensor

0.219 0.220

0.275 0.278

0.333 0.340

0.394 0.417

0.833 0.816

0.369 0.377

exact a) tensor

0.659 0.659

0.794 0.800

0.902 0.915

0.965 1.004

1.213 1.262

2.408 2.407

The row labelled

tensor

is the value from the present

approach.

where the prime on the summation sign on the right indicates omission from it of all possible values of m,. Using these basis states and a slight modification of the method outlined in appendix A leads to the following non-trivial matrix elements of the hamiltonian H:

= i

ni[2~i-G(~i-ni+1)]+&,(2n,+l)-Gn,(’,-’,)

3

(23)

i=l ifo

and, ((nl?tz*

‘.

ni’

**

nj*

*.

nk;jama)(H((nln2'-

=-G[ni(nj+l)(ni-ni+l)(~j-nj)]“2,

*

ni-1

*’

*

nj+l

.

”

nk;jama))

(24)

280

C.R. Sarma,

J. Situ / Tensor basis approach

if i, j Z a, and ((n, * * * n, * * * n,. * * nk; j,m,)lH1(

n, * * . n, - 1 - * * nj + 1 . . * nk; jarno))

=-G[n,(nj+1)(R,-n,)(0j-nj)]1’2,

(25) if i = a fj. If n, + n, + 1 and nj + nj - 1 in eq. (25), we interchange the bra and ket vectors and use it. Thus the matrix representation of H over the tensor basis for odd-N follows readily on using eqs. (23), (24) and (25). Just as in the case of even-N, we start the consideration of odd-N with the lowest value, N = 1, the odd particle being in the single-particle state Ijllm,). Multiplying this by the pair state &-,O( j2) of eq. (10) and antisymmetrizing leads to the result,

(26) n,n2 * * . nk; jam,)

if i#a,

where all nj = 0 except the one for which ni = 1. The k states defined by eq. (26) form an orthogonnal set. Normalizing them, the hamiltonian matrix can be readily set up using eqs. (23), (24) and (25). Diagonalization leads to a set of eigenvalues and eigenvectors from which we select the lowest Eh3’ as the ground state energy and the corresponding eigenvector $j$.. A resolution of $;:A, in terms of the basis defined by eq. (22) and a recursive procedure similar to the one followed for even-nucleon systems leads to the determination of the ground states of N = 5, 7, 9 etc. nucleons. The only difference is that we now use eq. (22) for the primitive basis states and eqs. (23), (24) and (25) for matrix elements of H. The calculated ground state energies for the proton system considered earlier are presented in table TABLET Grround state energies (MeV) for odd number 28 - 50 shell for fixed neutron number 50

$

2

exact * tensor

1.91 1.91

5 i exact *) tensor

of protons

filling the Z=

;

f

-1.15 -1.15

-0.75 -0.75

0.36 0.35

1.13 1.13

-1.51 -1.50

-1.37 -1.39

-0.39 -0.38

7 1 exact *) tensor

1.09 1.11

-0.95 -0.93

-1.20 -1.11

-0.37 -0.37

g I exacts) tensor

1.90 2.06

0.52 0.56

-0.01 0.06

0.48 0.49

11 ( exacts) tensor

3.58 3.65

3.80 3.97

3.23 3.26

2.20 2.24

13 ( exact s) tensor

6.99 7.07

8.97 9.04

8.41 8.43

7.36 7.40

3

The row labelled

tensor

is the value from the present

approach.

CR. Sarma, J. Situ / Tensor basis approach

281

TABLE 5 Occupation

numbers

pj( N) for odd number of protons filling the Z = 28 - 50 shell with fixed neutron number 50

N

PP/Z

3 ( exact *)

PS/Z

P3/2

PI/Z

tensor

0.107 0.107

0.419 0.419

0.290 0.290

0.500 0.500

5 t exact *) tensor

0.113 0.113

0.643 0.637

0.342 0.343

0.500 0.500

7 1 exact ‘) tensor

0.119 0.119

0.734 0.726

0.423 0.425

0.500 0.500

9 { exact *) tensor

0.124 0.126

0.801 0.797

0.709 0.708

0.500 0.500

11 1 exact *) tensor

0.127 0.131

0.815 0.816

0.730 0.728

0.500 0.500

13

0.146 0.153

0.815 0.817

0.730 0.732

0.500 0.500

exact *) tensor The row labelled

tensor

is the value from the present

approach.

TABLE 6 Single particle spectroscopic factors Sj( N + 1) for odd number of protons filling the Z = 28 - 50 shell with fixed neutron number 50 N

s&z

s:,,

$2

s:,2

exact s) ( tensor

0.991 0.993

0.739 0.747

0.917 0.918

0.976 0.976

exacts) I tensor

0.982 0.982

0.492 0.496

0.810 0.808

0.950 0.949

7

exact ‘) ( tensor

0.974 0.973

0.285 0.296

0.654 0.636

0.918 0.916

9

exact *) ( tensor

0.965 0.941

0.146 0.160

0.390 0.378

0.876 0.870

11

exact *) I tensor

0.959 0.948

0.048 0.052

0.088 0.099

0.816 0.784

13

exacts) ( tensor

0.927 0.920

0.815 0.817

0.730 0.732

0.154 0.162

3 5

The row labelled

tensor

is the value from the present

approach.

CR.

282

Sat-ma, J. Sira / Tensor basis approach

4 and compared with shell-model results. The occupation before are listed in table 5. The single-particle spectroscopic

numbers calculated as factors are determined

using (27) where *;;+I)

These quantities

=

4 ?jm(ji)

.

are listed in table 6 and compared

with the shell model

results.

3. Discussion There are a number of interesting features of the present approach that are noteworthy. The procedure is extremely simple to use and only minor modifications are required in going from even to odd nucleon systems. Particle number and Pauli principle are automatically conserved by generators of U(n). The results presented in tables l-6 compare quite favourably with the ‘exact’ ones. An interesting feature is the variation observed in the s-pair amplitudes as shown in fig. 1. We attribute this to the subshell closure 17) at **Sr,,. Apart from these aspects, one of the most interesting results is the coincidence of eqs. (13) and (14) and those obtained using the quasiboson approximation lo). This result implies that, though pair-fermion operators are generators of the covering SO(2n) algebra, whenever they appear in the bilinear form, Crr,b GabA,:Aj,, and act on an antisymmetrized ket of pair operators they behave as boson operators. A similar result was established for uncoupled Ijm) shell model basis recently by Gambhir et al. ‘*). Since zero angular momentum projected

kets used here are just linear

combinations

of uncoupled

antisymmetric

pair fermion states, the present identity follows. The generalization of the present work to excited states using a more realistic two-body potential is in progress. The authors are thankful to Professor Y.K. Gambhir and helped us with many useful discussions.

who suggested

the problem

Appendix A MATRIX

ELEMENTS

OF THE HAMILTONIAN

OF EQ. (9) FOR EVEN-NUCLEON

SYSTEMS

Consider first the diagonal block of matrix elements in eq. (13), for even-nucleon systems. We find that the first two summations on the right side of eq. (9) contain only the weight generators, Ej3,,;j,,L of U(n). Hence each of the ni pairs, occupying a given ji in either the bra or ket vector contribute a factor (2.5 -G) to eq. (13) leading to a total contribution ni(2.si - G) to it. The last summation on the right of eq. (9) contains non-trivial shift operators, Ejm;j~A~Ejfi;j~m~, of U(n). These operators

C. R. Sarma, J. Sita / Tensor basis approach

cause a shift from the occupied

pair (jam,, jar?&) to the pair (j@,,

283

j,m,).

Further,

since we are dealing with the diagonal block, only the j, + j, excitations need be considered. Since the tensor basis of eq. (12) is antisymmetric the transition (jam,, j,fi,)

jam,) yields zero contribution to eq. (13). We need to consider, only those transitions for which m, # mb for a given j,. Further, for the + (j,rB,,

therefore, tensor basis not to vanish, the pair state jamb, j,mb should be initially unoccupied in the ket vector. There are only 0, - n, such vacant pair states for a given j, with pair occupancy n, so that this term of eq. (9) contributes -G(L$ - n,) to eq. (13). The negative sign occurs because of the reordering permutation necessary to restore the pair j,fib, j,mb to the standard ordering jamb, j@t,. The total contribution to eq. (13) from n, occupied pairs of given j, is thus na(2&, - G(& -n, + 1)) leading to the required result. The off-diagonal block of matrix elements given by eq. (24) may now be considered. The first two summations of operators on the right side of eq. (9) do not contribute to this matrix element. Further the form of the product operators Ejhm,;j~m,Ejhmb;j~,m,,occurring in the last SUmmation implieS that single excitationS Of a pair (j,m,,j&) to (jbfib, jbmb) (a # b) are the only ones yielding non-zero contributions to eq. (24). Thus only n, + n, f 1, nb + nb T 1 need be considered. Choosing the lower signs given above, these arguments imply that any one of the occupied n, pair states of the ket could only be excited to one of the vacant (& - fib) states of the bra vector. Thus, the net contribution to eq. (24) is -Gn,(&, - nb) where the negative Sign again follows as before. Further, the excitation changes the normalization due to changes in occupancy so that we need a renormalization factor,

Thus the total contribution to eq. (24) follows as a product of -Gn,(Gb - nb) and the factor occurring on the right side of eq. (A.l). Similar arguments, after including the odd-particle state [jam,) as in eq. (22) leads to the matrix elements presented

in eqs. (23) and (25) for odd N.

References 1) 2) 3) 4) 5) 6) 7) 8)

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284 11) 12) 13) 14) 15) 16)

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ed.