Oscillator cluster parentage and supermultiplet expansion for states of light nuclei

i

1.D.I: 1.D.3 [

Nuclear Physics A204 (1973) 593--608; (~) North-HollandPublishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

OSCILLATOR CLUSTER PARENTAGE AND SUPERMULTIPLET EXPANSION FOR STATES OF LIGHT NUCLEI P. KRAMER t Instituto Naeional de Energla Nuclear, Mexico

and D. SCHENZLE Institut J'~r Theoretische Physik, Tiibin#en, Germany tt

Received 16 October 1972 Abstract: Cluster parent states are defined in terms of excited oscillator states of high excitation. The expansion in terms of these states is highly selective and keeps the lowest supermultiplets. The matrix elements of the nuclear Hamiltonian are reduced to two-body matrix elements and generalized transformation brackets. Calculations for 6Li levels are reported.

1. Introduction

The a p p r o a c h to states and reactions o f light nuclei presented in this paper brings together three viewpoints on nuclear structure: the supermultiplet scheme, nucleon clustering and h a r m o n i c oscillator states. There is n o w g o o d evidence from several sources that an expansion o f states and reactions o f light nuclei in terms o f supermultiplet q u a n t u m numbers should converge well 1). This evidence comes mainly f r o m supermultiplet expansions within the nuclear shells. I n view o f this evidence it was proposed 2- 4) that a model space for states and reactions o f these nuclei should be based on the supermultiplet scheme. A state corresponding to a given supermultiplet involves a certain complexity o f the orbital wave function as it should transform according to a given orbital Y o u n g diagram or partition f. One would like to find a few degrees o f freedom that could carry the supermultiplet along with the other details o f the state and to develop the dynamics o f these degrees o f freedom. It was pointed out 4) that the nuclear cluster model provides an appropriate scheme: The orbital symmetric internal states o f the clusters determine by Littlewood's rules a selected set o f orbital partitions, the full complexity t Visiting professor of the International Atomic Energy Agency, Vienna. Permanent address: lnstitut f/Jr Theoretische Physik, Tiibingen, Germany. t* Supported in part by the Bundesministerium fiir Bildung und Wissenschaft of the Federal Republic of Germany. 593

594

P. KRAMER AND D. SCHENZLE

of the states being carried by the state of the degrees of freedom associated with the relative motion of the clusters which depends o n f . The framework for treating the cluster model within the supermultiplet scheme was developed in the three papers 2-4) which we refer to as I, II and III and use extensively in the present paper. These figures are also used for quoting equations from these papers. It was shown there that the matrix elements of the Hamiltonian in the scheme may be expressed in terms of interaction integrals classified by double cosets. As a basis for building up a model space we shall use states of the oscillator cluster model as proposed by Wildermuth and Kanellopoulos 5). The group theory of harmonic oscillators has been used elsewhere 6) to study the qualitative features of such states. These states are projected from what we call oscillator cluster parent states. In the cluster parent states the excitation and angular momentum is carried by the degrees of freedom mentioned above. This particular choice is motivated by three reasons. First of all it may be shown that oscillator states are stable under the projection with respect to orbital partitions f. Mathematically this means that the normalization matrix between these states has block diagonal form. Physically it implies that if one wishes to bring two Gaussian clusters as near to each other as is allowed by the Pauli principle one must choose the lowest possible oscillator state for the relative motion and antisymmetrize. Secondly, even for nuclear reactions the oscillator basis has an immediate relation to the normalizing kernel of the resonating group method 7) and may therefore be used to write down the interaction kernels in the reaction region. The third reason is that by the use of group theory one may express all normalization and interaction integrals involving oscillator cluster states in terms of two-body matrix elements. This allows one to probe the proposed model space with the various effective interactions proposed recently. The paper is arranged as follows. In sect. 2 we write down the oscillator cluster parent states and the integrals we want to calculate. Sect. 3 contains the group theory needed for the calculation of normalization and interaction integrals. In sect. 4 we study the transformation properties of relative vectors and corresponding creation operators. The integrals for three-cluster configurations are reduced in sect. 5 to twobody matrix elements. Sect. 5 contains a discussion of the results under the viewpoints mentioned earlier in this introduction. Sect. 6 gives an account of two-cluster configurations with application to excited states of 6Li. 2. Oscillator cluster parent states

The basic assumption in this paper is the restriction to states of the harmonic oscillator cluster model. In this model the full excitation energy of a state is carried by the relative vector between the c.m. of a set of clusters while there is no internal excitation of the clusters. Consider a configuration of three clusters al, a2, a3 with nl, n2, n3 nucleons in each

OSCILLATOR CLUSTER PARENTAGE

595

cluster respectively. The vectors between the c.m. we denote by

,,

=

Eo,, 7,E _

...

_

+x,,)_

]

...

L n I -F n 2 J

n2

,

Ln 1 + n z W n 3 . . I

x

-(x'+...

+

--(x"'*"~+'+...

q-n 2

+ x ''+"2+n~)

.

(2.1)

n3

We introduce momenta pl, p2 conjugate to xl and x2 and define the normalized onevector unexcited oscillator states I0~> = ~Ar exp

[oo] - ~ - (:~)2

X

(m~°l~

= \~-2

s -- 1, 2,

(2.2)

.

The other one-vector states are obtained by applying polynomials in the creation operators [ 1 ½.,, (2.3) \2h ]

#'= (m°J~x'-i t2--m~h)p

to the unexcited states (2.2). If by N s and L~ Ms we denote the total number of quanta and the orbital angular momentum quantum numbers, the normalized one-vector states may be written as IN, L~ M,> = P{:m(#')10'>, where the homogeneous polynomial P is given by 8) p N , ( s', 4 [ ; s :.s~(N~- L,)O~ t.._s'~ L~Ma. ~ ) = ZaNsLskll q ] LsMskll 1, •

ANL...~(__).~(N-L)Ei 4re ]' " N+L+I)!!(N-L)!!

(2.4)

Now we couple the angular momenta L, and L2 to the total orbital angular momentum L and obtain a state of the relative motion XLM(~fl'~2)

t •1 N2 .2 ~-" [P~,(q )PL2(t/ )]LMIO 1 >I0 2 >,

(2.5)

where the square brackets indicate the coupling. The internal states of the clusters depend on n - 3 relative vectors ~', s = 3 . . . n - 1. We write these vectors alternatively as three sets 5c"~'',s I = l . . . n

1-1;

5¢~'2,s2 = 1 . . . n 2 - 1 ;

x,~3, sa = 1 . . . n 3 - 1 ,

(2.6)

-cwhich depend only on the nucleon coordinates within each cluster respectively. A

596

P. K R A M E R A N D D. S C H E N Z L E

cording to our assumption the full internal state is given by a product of three unexcited oscillator states 3

nj- i

q~a,(Xa'~')q~,,2(X"2~2)q~a3(3Ca3~) = I-I [ 1--[ 10aJ~)]' j=l

(2.7)

sj=l

corresponding to the same frequency 09 as used for the relative motion. We define the overall unexcited state 10) by 3

10> = 101>I02>

nj-- 1

I-I 1-[ 10°J~>

j=l

= JV'"-' exp

sj=l

- ~-

.=

•

.

(2.8)

We choose the matrix that gives the relative vectors x~ in terms of the single-particle vectors orthogonal. Then the expression in square brackets appearing in eq. (2.8) and hence the unexcited state are unchanged under permutations. The oscillator cluster parent states are now given by

[~nal a2 a3 NIL1 N2 L2 LM) • alSl .a2s2 • I . 2 )¢pa:(x )cp.s(x. a 3 s 3 )XLM(XX ) = = ¢p~,(x

[P,.,(~ Ni

• 1

N2 • 2" )PL201 )]Lul0>.

(2.9)

For calculation with these states it is necessary to obtain the matrix of the nuclear Hamiltonian. This matrix is obtained in three steps. First of all the states (2.9) are adapted to an orbital partitionf. Then one defines reduced matrix elements both with respect to the orbital and the corresponding spin-isospin states for operators acting on the last pair of particles as outlined in I. The reduced spin-isospin matrix elements are calculated by means of standard techniques. The reduced orbital matrix elements are expressed in terms of basic interaction integrals and algebraic weighting factors as described in II and III. In the third step to be discussed in this paper the interaction integrals are written in terms of two-body matrix elements. As the symmetry-adapted orbital states are no longer orthogonal with respect to each other, a similar technique applies to the normalization matrix which may be written in terms of normalization integrals. The normalization integral may be written according to II and IlI as

(~nal~2a3NxL1N2LzEIZKI~nala2a3N1L1NzL2L),

(2.10)

where the barred quantum numbers belong to a second configuration of three clusters and Z K is a DC generator. The interaction integral we denote by ( ~ " a l (~2 a3

N1 Ll N2 L2 EIIZ~ T~(n - 1 n)Z'x, Z'x',,Zqll~"a~ az a3 N~ L l N 2 L z L),

(2.11)

where 2~Zk, Zk',,Zq are DC generators as explained in III and T~(n - 1 n) is an orbital tensor operator of rank x acting on the last pair of particles. We shall now calculate these integrals.

OSCILLATOR CLUSTER PARENTAGE

597

3. Polynomial expansions and representations of the unitary group Both the application of permutations and the transformation to different sets of relative vectors lead to transformations of the states (2.9) which we shall describe in this section in terms of representations of the unitary group. Consider a transformation n--1

ils ~ ~ iltut~,

(3.1)

t=l

where the matrix u = (ut~) is unitary and of dimension ( n - 1) x ( n - 1). The matrix u is an element of the unitary group U ( n - 1) acting on the upper indices of the vectors Os. Any state of the type (2.9) may be considered as a partner of a basis of a reducible unitary representation of this group. Denote this partner for short by the index p and suppose that the other partners are labeled by a set of indices p'. Then the coefficients in the polynomial expansion n--I

Pp( E iltUt~) = E ep,(Ot)(p'lulp), t=l

(3.2)

p'

are matrix elements of a reducible unitary representation of U ( n - 1). If v = (vst) is another element of U ( n - 1), the unitarity of the representation implies (p'lvl~)* = (~lr*lp'),

(3.3)

and the representation condition gives

~, (~lv*lp'>(p'lulp) p'

= (~l~*ulp>.

(3.4)

This allows us to calculate the scalar product (01[P~(~ #tv~=)J*Pp( ~ #tu,~)i0) = ~ *
t

(3.5)

p'

In general we shall assume that the polynomial P in eq. (3.2) depends on at most n - 1 vectors ~ . For each vector it has a fixed degree w~ given by w~ = N~,

(3.6)

which corresponds to the excitation of the vector ~ . As a consequence it is easy to prove that the representations of U ( n - 1) appearing in eq. (3.2) are given by polynomials in the numbers u~t. It is possible to extend this representation of U ( n - 1) to a representation of the general linear group G L ( n - 1 ) . This can simply be done by inserting into these polynomials the elements of a general complex matrix. We now claim that the following reduction theorem holds. Suppose that we split the labels ~ and p as = ~, p = 2w, (3.7) where w = (w 1 . .. wn-1) denotes the weight and ~. denotes all other row labels of

598

P. K R A M E R A N D D.

SCI-IENZLE

the representation. Assume that both in the weights i~ and w we have s > k-1 :~=

w~=0.

(3.8)

Then the representations of U ( n - 1) appearing in eq. (3.2) may be written in terms of representations of G L ( k - 1) according to (~(~,

..-

~ - 1 0 . . . 0)lul,~(w,

...

wk-10...

= <~(W1"'"

0))u(,-1) Wk-1)Illtl/~(W1""" W k - 1 ) ) G L ( k - I ) ,

(3.9)

where the element u' of G L ( k - 1) is the submatrix of the unitary matrix u with elements ( 3.10 ) u t' s = uts t < k-l, s < k-l. The second line of eq. (3.9) should be understood as explained in the last paragraph. The theorem expressed by eq. (3.9) plays a key role in the present paper since it allows us to treat a k-cluster configuration in terms of extended representations of U(k - 1). In particular we shall apply it to the scalar product of the polynomials Pp = [ P LNI1 ( ~ q..tbtl)PL2( N2 ~ ifbt2)]LM,

(3.11)

f ~ = [ P ~ ( x~~ ~..tc tl)xp~2r L2~ ~ ~tct2)~ LM"

(3.12)

1

t

t

t

to obtain with p = 2w,

2 = L~L2LM,

w = (NIN20~-3~j,

(3.13)

and similarly for/3 the result (01P~ PolO) = ( N I L1 N2 L2 LIe[N1 L1 N2 L2 L ) ,

(3.14)

where e ~

(el,

e12)

e21

e22/

e,~ = (c#b)~s = Z c~b,~.

(3.15) (3.16)

t

The representation matrix (3.14) is a transformation bracket between two-particle oscillator states extended to the group GL(2). Its explicit form is discussed in appendix B. 4. T r a n s f o r m a t i o n o f r e l a t i v e v e c t o r s

For the calculation of the normalization and interaction integrals we must obtain the transformation matrices according to eq. (3.1) that correspond to the permutations of relative vectors appearing in these integrals. The transformation of the operators qs equals the transformation of the relative vectors ~s. The relative vectors xS are linear combinations of the vectors x ~ and hence transform linearly under permutations. We first describe these vectors as bases of irreducible representations of the

OSCILLATOR CLUSTER PARENTAGE

599

p e r m u t a t i o n groups S(n) = S(n I + n 2 ) ~ S(n3) = S(n,) ~ S(.2) ~ S(n3).

(4.1)

U n d e r S(n) the vectors x ' transform according to the I R { n - 1 1 } while the c.m. vector transforms according to the IR{n}. Applying the same arguments to the subgroups one obtains the classification given in table 1. TABLE I

Classification of relative vectors for cluster parent states S(n)

S(ni +n2)

{n-- 1 1}

.~l X-~ .i::,

S(nl) + S(nD + S(nD

{nl + n2 -- 1 1} {n, + n~ } {ha+n2--1 1}

{nx }+ (n, }+ {nl--I (,,~ } + {n, } +

x":~ x~:3

{n2 }+ {n3 } (n2 }+ (n3 } I }+{n2}+{na} (n~- 1 1} + (n, } {n~} + {n3-1 1 }

A similar classification holds for the vectors ~t of the second cluster parent state. The same classifications apply to the creation operators ~t and ~* respectively. N o w we express the application o f the permutation Z r to ~ as a linear combination o f the vectors ~t belonging to the second cluster parent state, n--I

ZK il s = Y'. ~et~(Zr).

(4.2)

t=l

The n u m b e r s e u are matrix elements o f the real orthogonal irreducible representation ( I R ) { n - 1 1} o f S (n) with rows characterized by I R of subgroups according to table 1. By using the results of sect. 3 we shall see that the four numbers

e,~(Zr),

t = 1, 2,

s = 1, 2,

(4.3)

suffice to completely determine the n o r m integrals. These numbers are discussed in a p p e n d i x A. F o r the interaction integrals it is convenient to introduce an intermediate set o f relative vectors in order to separate the matrix elements of T ~ ( n - 1 n). By use of the group chain S(n) ~ S ( n - 2 ) ~) S(2), (4.4) we define n - 1 orthogonal vectors ~ according to table 2. TABLE2 Classification of relative vectors for interaction integrals

~,s = l..n--3 ~.-2 ,~,--1

S(n)

S(n--2)

S(2)

{n--I I}

{n--3 1} {,--2} {n--2}

{ 2} { 2} {11}

600

P. KRAMER AND D. SCHENZLE

The only relative vector of interest is

~n-1 = ~/~(xn-l

x.),

(4.5)

since T ( n - 1 n) acts on it. We shall need the numbers in the linear combinations n--1

z , ¢ = Z ~1 " bts(ZK" '

z,),

t=l

rl--1

= Z

1).

(4.6) (4.7)

t=l

We write these equations for later use as

Z'~'Z~"ZqiI s = Z ' / f b , s + / i " - t b n - , s,

(4.8)

t

Z~l~" = y,, qCts-t.., . . .q. -1 C n - t s ,

(4.9)

t

where the dashed sums rum from t = 1 to t = n - 2 . We shall show in sect. 5 and in appendix A that the only numbers needed for the interaction integrals are

cn_ls,s = 1,2;

bn_ls,s = 1,2.

(4.10)

5. Calculation of norm and interaction integrals

The results of sects. 2-4 enable us to decompose the integrals (2.10) and (2.11). For the normalization we consider the integral (2.10), the polynomial bases (2.9) and the result given in eq. (3.14). We obtain by use ofeq. (3.14) the result (~nal a2 a3 N1 L1

N2 L2 EMIZ~I~"a~a 2 a 3 N1 L1 N2 L2 LM)

= (N1ExN2L2LIeiN1L1 N2L2L>f-LrOT.tMO~I+N2.1vI+t%, (5.1) where e is a 2 x 2 matrix, e = (ell \e2t

e12t, e22/

(5.2)

and where the numbers est = es,(Zr)

(5.3)

are discussed in appendixA. Note that both the cluster structures ata2aa, ata2a3 and the exchange type Z r enter only the matrix elements ea. Any normalization integral which by itself involves a 3 ( n - 1) fold integration is now reduced to a simple general transformation bracket discussed in appendix B. To evaluate the interaction integral (2.11) we first of all apply Z~- 1 to the left-hand and Z k, Zk',, Z~ to the right-hand parent state. Using the linear combinations of eq.

OSCILLATOR CLUSTER P A R E N T A G E

601

(4.8) and eqs. (B.2) and (B.4) we obtain the decompositions N, " PL~M~( E

i~'bts)

=

E

N'sL'sN"sL"s

pN"

{[ L',*( E ' / /

""b xpN",z..n- lxn ,~) 1,,,fl,tl )JL, M.

× A(N'L'sN;'L"NsL )(bn-,s)N"s},

(5.4)

for s = 1, 2. Introducing this in eq. (2.9), recoupling the four angular momenta L'IL'[L'2L'z' and using eq. (B.1) in the form N"I ..n-1 N"2 ..n--1 nN" [..n--l'l.4/ttTtVrltr.lttttvrtlttftt~t [PL", (t/ )PL,', (t/ )]L"M"~- I-L,,M,,~ )_/-I_ll'/1 /.~11~/2/.,21~ .t~ ],

(5.5)

we obtain the expansion

E "tl b,,)PL2( N2 E

il'b,2)-ILM

=

N'IL'IN'2L'2N"IL"IN"2L"2N"L"L"

N', ~, tl..tbt,)Pt.'2(E'tl N'~ ""b,2)]v[ pN,, ..~-1 )]I."]LM {[[PL',( L"(t/

x A(N'~ L', N'~'L'[ N, L,)(b,,_, ,)N"'A(N'2 L'2 N'2'L'~ N2 Lz)(bn-, z) N'2 × A(N;'L'; N';L'; N"L") × <(L'~ L'2)E(E; L'~)E'LI(L', E;)L,(L' 2 t ~ ) t 2 L>}.

(5.6)

The algebraic coefficients in eq. (5.6) are discussed in appendix B. A similar expansion we apply to the left-hand parent state. Now consider the matrix element of T"(n-1 n) between these expanded states. We may split the unexcited state 10> as

(5.7)

10> = 10'>10">,

so that the operators #s, s = 1. . . . . n - 2 act only on the first factor and/~n- ~ acts only on the second factor. The operator T " ( n - 1 n) then acts only on the one-vector state P~;;w,(iT"- ~)10"> = IN"L"M">, (5.8) which appears in eq. (5.6) as the second part of a coupled state. By standard R.acah algebra this leads to expressions involving the reduced one-vector matrix elements .

(5.9)

With respect to the first part of the coupled state we must evaluate in each term of the expansion of bra and ket a scalar product of the type given in eq. (3.14). This gives N' 1 <0 t I[Pz. (Z

t

..t

~'

ff'~ E , tl..,C'2)]Z,~,[PL,, t N,,( Z , tl""bt,)n~.2( tl"' ct')PT.,2( N'2 Z ' tl bt2)]L.u, lO >

= bI, L, 6~, M, 6~, +~'2,N', +N'z,

(ym)

with

e'= [e',, e',2

\e'2~ e'2z]'

(5.I1)

602

P. K R A M E R AND D. SCI-IENZLE

and from eq. (3.16) t ~ e,~

Et

c,,b,s .

(5.12)

t

These numbers are discussed in appendix A. With the results of eq. (5.10) we obtain the final expression for the orbital interaction integral reduced with respect to angular momentum (~"tit a2 d3 N~ L 1N2 £,'2f,[lZo T~Z'K,Z'~',,Z~ll@a~a2 a3 NI L1 N2 Lz L) =

~

(N"£,"IIT~IIN"L'')

N"L"N"L"

x I

E

A(AIT£,'I'N'2'E'z'N"E")A(N'I'L'; N'2'L'~N"L")

t N'lL'lN'2L'21~"IL"IN"2L"2L" N'tL'IN'2L'zN"tL"IN"2L"2

2

× I1 A0V;~; N;'~;% ~0(c.-, 5)"""A(N~E~NsL~N~Ls)(b.-x~) ...... """ s=l

× <(L~L, )LI(Z:~L~)L~ LI(L, L~)C(L, L~)L L> --t

--t~

--

t

--it

--

--

--t

--t

i

~¢t

--it

--tt--

× <(L', L'~)L'(C; C~)L"LI(C, C;)L,(L', C~)L~ L> x (--)L'+L"+L+~[(2E+ I)(2L+ 1)] "~ {~' x ( N ; L1NEL2Ele - ' - ' - ' . . IN1E1N2/~2 . . . . . L'>}.

E

(5.13)

There are several restrictions on the summations in eq. (5.13), most of them coming from the properties of the coefficients A given in eq. (B.2). Eq. (5.13) gives the orbital reduced interaction integral as a sum of one-vector reduced matrix elements of T~(n- 1 n) with algebraic weighting factors. The matrix elements of T"(n- 1 n) may now be evaluated in terms of Talmi integrals or be taken from effective interactions. The expression (5.13) shows the following features which are in contrast to the standard fractional parentage technique: (a) Each weighting factor is by itself a polynomial in the numbers e~s, c,_ 15 and b,_ is. For fixed powers of these numbers, the coefficients of the polynomial depends only on the quantum numbers N1L1N2 L2 NI L~ N2L2 £,LN" L" N"L", not on the numbers ala2aaala2a 3 and the exchange type specified by Z~Zk, Z'x',,Z~. These polynomial coefficients are built from three 9-j coefficients and powers of factorials. They are common to all interaction integrals between three-cluster parent states and their number depends exclusively on the maximum excitation and angular momenta chosen. (b) The numbers e~,, c,_ is and b,_ 1~ in the interaction integral and similarly the numbers e,~ in the normalization integral depend only on the choice of clusters and on the exchange type. These numbers are discussed in appendix A.

OSCILLATOR CLUSTER PARENTAGE

603

(c) No multiplicity appears in any coefficient of eq. (5.13). Rather this is an expression involving combinations of algebraic "building stones" and hence suited to numerical computation. The question arises where all the well-known multiplicity problems of the fractional parentage technique have disappeared. The answer is that these problems appear in the normalization matrix. This matrix is block diagonal, each block being characterized by fixed excitation N, angular momentum L and orbital partition fi The rows and columns of each block are labeled by N1 LIN2L2 axa2a3 and NxL1N2L2ala2a3 respectively. The normalized eigenvectors of each block matrix for non-vanishing eigenvalues provide a set of orthonormal trial states. The matrix of the Hamiltonian may then be diagonalized between this reduced set of states. As the normalization matrix is a purely algebraic quantity we expect that it does not lead to "almost" dependent states. Moreover from properties of overlap integrals discussed in sect. 6 it is easy to show that with high excitation different cluster parent states become orthogonal. (d) We have several ways of controlling the "state explosion" that is so wellknown from the shell model. This is because the oscillator cluster model implies a strong reduction of the degrees of freedom. The two-cluster parent states share precisely the degeneracy of a two-body system. All three-cluster configurations share the structure of a translational invariant three-body system and have the same degeneracy. This means at the same time that the IR [h~ h2 h 3 ] of U(3), although the states are not bases of IR of this group, are restricted by h 3 = 0 for three clusters and to h 2 = h a = 0 for two clusters. It is well known that a restriction of this type is meaningful from the viewpoint of the SU(3) classification of shell-model states due to Elliott 9). (e) By choosing the number and type of clusters we restrict at the same time the orbital partition f and hence the supermultiplet quantum number. Explicitly f is restricted by the condition that it can be contained in the outer or Littlewood product al × a2 x a 3 of partitions. Three-cluster parent states give rise to partitions f of at most three rows, two-cluster parent states to partitions of at most two rows. The expansion in terms of an increasing number of clusters is therefore equivalent to a supermultiplet expansion of states in an oscillator basis as discussed in III.

6. Two-cluster configurations We shall discuss in this section the special case of two-cluster configurations. Given a nuclear Hamiltonian, one may decompose it in terms of coupled orbital and spin tensor operators T~(n- 1 n) and V~(n- 1 n). This includes the internal kinetic energy which may be written as T-

1

E(p,,_p,)2

2nm s<, _

1

2m ~ ( P * ) 2

1

2nm--(~ ¢)'"

(6.1)

604

P. KRAMER AND D. SCHENZLE

From II and standard Racah algebra the matrix element for a fixed tensor rank x is given by

((~)"~ , ~2 "NEf 'gTJII( T~ V")II(~')"a , az N L f STJ)

= (ifllfl)-~ (~)(_)L+~+, IJ sIKL ~} x Z {(~"~1 a2 NEff'f"llT'~Jl~"a, a2 NLff'f")}. (6.2) f'f" The spin-isospin matrix element can be evaluated by standard fractional parentage techniques. In eq. (4.11) of II the orbital matrix elements of eq. (6.2) are expressed in terms of orbital interaction integrals. For the orbital interaction integrals we use eq. (5.13) with the restriction N2 = L 2 = N 2 = L2 = 0 which gives a strong reduction of the sums. The final result for the orbital matrix elements is

(~"al ~2 NE ff'f"llT~llct"al a2 NL f f ' f " ) - Ifllfl ,l!,2,n,,n2, (n) -2 If'J " 2 x ~_, N"L"N'L"

x { ~ A(N'L'N"E,"NE)A(N'EN"L"NL)

x(_:+.+z+.[(2r_.+D(2L+l)]~ {~' r~ × y m-'(K')m-'(K") glK'K"q

x

[: :' :"lV~: ,,'l a:,lF:"/~'1'a'l'(K,,)j a'zl[ilt (o)a2]j a,

x (e', I(~IK'K"q))N'(c, ,,_, (7/))~"(b,,_1 ,(K'K"q))N"},

(6.3)

where for short we have indicated the double coset generators ZqZk, Z'r',,Zq and the corresponding splittings of representations by ~IK'K"q. The normalization matrix of two-cluster configurations is obtained from eq. (3.25) o f l I and eq. (3.14) as

(°t"al az NL flct"al a2 N L f )

[: [:

Ifl nl! fir! nl! n2!

Y m-l(K) al

n!

"~

"

a2

_ Ill ~,! ~! n,! ,~! Z 'n-x(K) ~, n!

r

al a21 (K) J (el l(K))N

a2

01 a z ] [ { n - l l)

(K) JL ~/1 a2

al

a21N

j

(6.4)

OSCILLATOR CLUSTER PARENTAGE

605

From this equation it is possible to estimate the normalization for high excitation, N >> 1. The second 9-fcoefficient appears with the power N. This 9-fcoefficient equals one for the double coset with no exchange and is smaller than one otherwise. For different configurations in bra and ket the second 9-fcoefficient is always smaller than one and the matrix element approach zero as N increases. The expressions eq. (6.2) and eq. (6.3) have been calculated with effective interactions given by Shakin et al. 1o) and Elliott et al. 11). In fig. 1 we show the level sequence 6LI

E(J T)

6.0

10

5.9

10

5.36

21

~.~,

21

'; =J7

20

,;.~

20

3.56

2.1a

(-31.98)

01 3.0

ol

2.4

30

10

6.5

21

5.2

20

3.8

01

2.4

30

30

10

( -9.0 )

10

(-93)

b=I.76 EXP.

74

b=t.5

5HAKIN

10

~.~

10

4.6

21

4.0

2o

I.~

07

1.5

30

(-d.7)

10

b=t5 ELLIOTT

Fig. 1. Energy levels E ( J T ) of 6Li. The states are described in terms of two-cluster parents a l , a2 = 42, 33 with oscillator excitation N < 6 adapted to the orbital p a r t i t i o n f = {42}. The effective interactions are due to Shakin and Elliott. TABLE 3 Probability o f finding the oscillator excitation N = 2, 4, 6 for the calculated states o f 6Li characterized by J T JT

N=2

N=4

N=6

10

0.802 0.784 0.876 0.841 0.803 0.770 0.838 0.800 0.835 0.815 0.699 0.668

0.159 0.176 0.086 0.114 0.155 0.182 0.131 0.161 0.133 0.148 0.253 0.276

0.039 0.04 0.029 0.041 0.042 0.048 0.032 0.039 0.032 0.036 0.048 0.056

30 01 20 21 10"

First row: Shakin interaction (b = 1.4 fm), second row: Elliott interaction (b = 1.5 fm).

606

P. K R A M E R AND D. SCHENZLE

for states of six nucleons and positive parity in a basis including two-cluster parent states with nl, n2 = 42 and 33, 2 < N < 6, L < 2 a n d f = {42}. Note that the part i t i o n f = {33} could come only from n 1 = n2 = 3 and does not give positive-parity states because of the identity of clusters. In table 3 we give numbers in order to show the convergence of the oscillator cluster expansion with respect to increasing oscillator excitation. We emphasize that for these first calculations we did not consider supermultiplet mixing. Both supermultiplet mixing and the inclusion of more than two clusters may be analyzed with the methods presented in this paper. Calculations covering these and other aspects will be presented in future publications.

Appendix A REPRESENTATION MATRICES OF S(n)

In sect. 4 we defined sets of numbers eta, % and bt~. F r o m tables 1 and 2 the labels t, s may be identified with representations of S (n) and corresponding subgroups. O f the coefficients ets we need only those with t = l, 2 and s = 1, 2. We write the I R of S (nl +//2) and S (nl + n2) as .q2 and g2 respectively so that t = 1

02 = { n l + n 2 - 1 1 } ;

t = 2

02 = ( n l + n 2 } ,

s = 1

g2 = {n~+n2 - 1 1 } ;

s=2

02 = {nt+n2}.

(A.1)

Then et~ is given by ets = ( { n - - 1 1}.~2({~}{ff2}){~a}lzKIfn-- 1 1 } 9 2 ( { n ~ } f n 2 } ) f n a } ) = et~(Zr).

(A.2)

This matrix element was discussed in eq. (B.10) of III and was expressed there in terms of four 9-f symbols. For the coefficient c,_ 1~ we get from tables 1 and 2 and from eq. (B.7) of I I I £ n - l s = C n - l s ( Z g 1) = Csn-l(Z~l) = < ( n - 1 1}(n - 2 } [ ( a l + a l } ( ( G

L {11}

F

a'dJ

}(ai})]{uI[(a','}{ai'}{a;'}]l

F

ai' +'d

where the coefficients in the last line are 9-f coefficients. The numbers b , _ t s = b,_l ,(Z~, Z~:',Z~),

(A.4)

depend on the product of three permutations. We could use the representation condition to write them as a sum of products of three matrix elements. But as Zq fixes the

OSCILLATOR CLUSTER PARENTAGE

607

I R { n - 2 } and {11} ofS ( n - 2 ) and S(2) and Z;c,, Z~c',, are members of these groups, the sum reduces to a simple product of matrix elements given by b._, ~(Z~, Z;,',, Zq) = <{n - 2}]Z~,[{n -2}><{11}[Z~,,l{ll}>bn_ t ,(Zq)

F{n-1 I} = (_)z,,,:,,l { n - 2 } L {11}

g2

a3lV

{a'l+a;} F

g2 al a'al|{a'l+a'2 } a', a'~J

L

F

a'l'

a2 l a'2| • a'~J

(A.5)

The matrix elements e~s ofeq. (5.11) are given from eq. (5.12) by

e,s = e,~(Z~, ZK, Zr,, Zq) n-2

= E c,,(Z~)b,,(Z'r, Z'x',,Z,~) t=l r l- - 1

= E c,t(Z~)bt~(Zr' Z'K',,Zq)-- c,~_i b._1,.

(A.6)

t=l

The first sum in the last line is the matrix element of the product Z~Z~:,ZfK',,Zq tween bases of S(n) characterized by { n - 1 1} and r, s respectively. This matrix ment must be equal to the matrix element of some double coset generator Zr. the DC symbols of Zk,, and Z~:',,, Zr, be given by the numbers dij(K'), dij(K") d u (K). Then it can be shown that Z K is determined by

beeleLet and

dii(K ) = dii(K' ) + dii(K").

(A.7)

e,',(Z~, Z'w Z'K',,Zq) = e,~(Zq) -- c,._ ,(Z~)b,_l ~(Z'K,Z'K',,Zq).

(A.8)

Therefore we get All coefficients are now reduced to 9-fcoefficients with partitions of at most two rows. These 9-f coefficients are hence proportional to 9-j coefficients of SU(2) as given in eq. (A.3) of II.

Appendix B TRANSFORMATION BRACKETS FOR OSCILLATOR STATES The general transformation bracket for two particles in a harmonic oscillator basis is defined by the expansion Nl 1 N2 1 [Pc,(~/ u,,+~l 2 U2,)PL~(~I U,2+t/2U22)]LM

=

~

{[f/.,(g )PE2(g )]Lu}.

(B.I)

N1L1N2L2

In ref. 12) the bracket was explicitly given as a polynomial in the elements u u of the matrix u. Defining the coefficients

A(N'L'N"L"NL) = fiN"+lV",N 6(L'E'L)

F(2t'@t(2L + 1)(2E' + 1)] ~, + 1)

x AN,Z, AN"L" A#~ L

(B.2)

608

P. KRAMER AND D. SCHENZLE

with ANL given by eq. (2.4) if L' < N', N ' - L ' even; L" < N", N " - L " even; L < N, N - L even; we o b t a i n the t r a n s f o r m a t i o n b r a c k e t in the explicit form

( N1 L1 N2 L2 LIuIN1 L1 N2 L2 L> =

E (Ull)NII(U12)NI2(U21)N21(U22) N2z NI 1N12N21N22

x

E

{A(NI,Lx,N,zL,zNIL,)A(N2,L2, N22L22N2L2)

LI IL12L21L22

× A(N,, L,, N2, g2,

L,)A(N,2 L, 2 I%2

x <(L,~ L, z ) L , ( L 2 , LzE)L2

N, g2)

LI(L,, L2,)L,(L~2 Lz2)L2 L>}.

(B.3)

T h e s e c o n d s u m is restricted by the p r o p e r t i e s o f the coefficients given in eq. (B.2). T h e last coefficient is a recoupling coefficient o f f o u r a n g u l a r m o m e n t a a n d p r o p o r t i o n a l to a 9-j coefficient. A special case o f eq. (B.3) is

= A(N~E, N2LzNL)(ut2)~'(Uz2) ~2.

(a.4)

T h e expressions (B.2) a n d (B.4) are readily extended to elements o f the full l i n e a r g r o u p G L ( 2 , C ) in two dimensions o n replacing the m a t r i x elements o f the u n i t a r y m a t r i x u b y those o f a general c o m p l e x 2 × 2 matrix.

References 1) P. Kramer, Proc. XV Solvay congress on symmetry properties of nuclei, Brussels, 1970, to be published 2) P. Kramer and T. H. Seligman, Nucl. Phys. A123 (1969) 161 3) P. Kramer and T. I-L Seligman, Nucl. Phys. A136 (1969) 545 4) P. Kramer and T. H. Seligman, Nucl. Phys. A186 (1972) 49 5) K. Wildermuth and Th. Kanellopoulos, Nucl. Phys. 7 (1958) 150 6) P. Kramer and M. Moshinsky, in Group theory and its applications, ed. E. M. Loebl (Academic Press, New York, 1968) 7) K. Wildermuth and W. McClure, Cluster representation of nuclei (Springer, Berlin, 1966) 8) T. A. Brody and M. Moshinsky, Rev. Mex. Fis. 9 (1960) 181 9) J. P. Elliott, Proc. Roy. Soc. A245 (1958) 128 10) C. M. Shakin, J. R. Waghmare and M. H. Hull, Phys. Rev. 161 (1967) 1006 11) J. P. Elliott, A. D. Jackson, H. A. Mavromatis, E. A. Sanderson and B. Singh, Nucl. Phys. A121 (1968) 241 12) P. Kramer, Rev. Mex. Fis. 19 (1970) 241