Resonating group kernels for complex cluster systems

Nuclear Physics @ North-Holland

A405 (1983) 40-54 Publishing Company

RESONATING

GROUP

KERNELS FOR COMPLEX CLUSTER SYSTEMS Y. SUZUKI

Physics Department, Niigata University, Niigata 950-21, Japan Received

3 January

1983

Abstract: The Bargmann transform technique has been generalized to evaluate the resonating group kernels needed for the microscopic cluster model in which constituent fragments are described by harmonic oscillator functions with unequal width parameters. All the techniques developed for the case of equal width parameters are applicable without any change except for a slight modification of the transformation to the Bargmann space cluster-internal and cluster-relative motion vectors. The inversion of the Bargmann transformation has been carried out for those Bargmann transforms which appear in the cluster model, making it possible to gain the kernels in a closed algebraic form. The kernels for the 01+d and g-quark 3-nucleon systems have been evaluated to illustrate the technique.

1. Introduction The microscopic cluster model ‘> provides a basis for a number of interesting nuclear problems, including not only the study of possible cluster structure within nuclei, but also a detailed treatment of nuclear reactions in which the Pauli principle is properly incorporated into the theory. The basic idea of the cluster model is the decomposition of wave functions into the constituent clusters. Their internal motion is described by simple wave functions, usually harmonic oscillator functions. This assumption allows integration of the Schriidinger equation over the internal variables of the clusters to obtain a nonlocal resonating group (RGM) equation for the relative motion of the clusters. Practical calculations in the framework of a microscopic cluster model, however, are dependent on the availability of techniques for the evaluation of challenging overlap and interaction kernels. Powerful techniques have recently been developed for the evaluation of norm ‘,‘) and overlap 3, kernels for a fully antisymmetrized cluster basis. Much of the recent progress in the calculation of norm and overlap kernels has been aided by the introduction of the Bargmann-Segal transform “). Suzuki, Reske and Hecht ‘) have continued to show that the Bargmann transform technique can also be generalized to the calculation of interaction kernels, tabulating the single-column Bargmann transforms needed for the RGM kernels of SU(4) scalar nuclei with A = 12-24 to all the possible SU(4) scalar fragment decompositions. It has been assumed in refs. 335)that the internal functions of the cluster fragments are built from harmonic oscillator functions with equal width parameters for all fragments. The generalization to the case of unequal width parameters requires special treatment of the 40

Y. Suzuki / Resonating group kernels

41

cross terms between relative motion and c.m. excitations. It is one of the purposes of the present paper to show that the Bargmann transform technique can also be used to advantage in the calculation of the RGM kernels for the case of unequal width parameters. In a previous investigation 5, the emphasis has been the evaluation of the needed Bargmann transforms in general algebraic form. RGM matrix elements of the kernels can be extracted from appropriate expansions of the Bargmann transform of the RGM kernels. Since the expansions, particularly in the harmonic oscillator basis, may be slowly convergent, it would be advantageous to evaluate the RGM kernels in completely closed algebraic form. The evaluation of the RGM kernels in closed form is essentially needed in the case of unequal width parameters to get rid of spurious c.m. excitations. This problem was left for future investigation. It is another purpose of the present paper to discuss the inverse of the Bargmann transformation to gain such a closed form. The Bargmann transformation is given in sect. 2. The Bargmann transforms are first expressed in single-particle Bargmann space variables. In the case of equal width parameters the transformation to Bargmann space cluster-internal and cluster-relative motion coordinates is achieved directly by the same orthogonal transformation which effects this transformation in real space. This is not the case in the unequal width assumption. It is necessary to transform the single-particle Bargmann space variables to the Bargmann space cluster-internal and cluster-c.m. coordinates. This intermediate step was employed by Seligman and Zahn 6, in their dilatation technique. The cross terms between the relative and c.m. motion are dealt with more simply by the method presented in this paper which makes use of delta functions in the RGM kernel. The construction of the RGM kernels from the Bargmann transforms is made in closed algebraic form in sect. 3. The procedure explained in sects. 2 and 3 is illustrated in some detail with simple examples, the (Y+ d cluster system and the 9-quark 3-nucleon system, in sect. 4. Finally sect. 5 gives a summary. 2. The Bargmann transform in cluster coordinates The RGM kernels of the operator 6’ for the cluster decomposition into fragments f+ (A -f) in the bra side and f’+ (A -f')in the ket side are defined by K(R’, R’) = (~~f~_fS(R-R’)I~~J~f,~*-pS(R

-A!‘)).

(1)

The Dirac bracket notation implies integration over all spatial coordinates other than the c.m. coordinate R,.,. and a summation over all spin and isospin coordinates. 8 and R are the relative coordinates corresponding to the fragment decomposition f, A -f for R, and f’, A -f'for R. For example, J? is given by

42

Y. Suzuki / Resonating group kernels

where ri is the physical single-particle coordinate. [Although the notation of this paper will follow that of refs. 3*5)as much as possible, note the different definition Of ri from refs. 3’5) in which ri is equal to the physical single-particle coordinate divided by a common oscillator width parameter, b = [It/m~]~‘~.] The internal wave function C&is a function of f - 1 internal coordinates and includes the spin and isospin dependence of the fragment f. The calculation of the complicated multi-dimensional integrals implied by eq. (1) can be reduced to one of evaluating orbital integrals after carrying out the spin and isospin integrations. It is not the purpose of the present investigation to discuss a method of separating the spin-isospin and orbital integrations. See e.g. refs. 377) for this subject. In the orbital integrals the antisymmetrizer, ,QZ,is replaced by an appropriate operator built from a sum of permutations, 9. It is sufficient for the present purposes to show how the RGM kernels for the operator 0’!? are derived from the Bargmann transforms. The Bargmann transform of the operator ~29 is defined by

H(& k”) =

I” drl ***Irn drA fi A&(& -cc -CC

r#XP

fi i=l

i=l

Abi(k?, ri>,

(3)

where Ab (k, r) = [rb2]-3’4

* (r/b) -$(r/b)‘]

exp [-$*+&

is the kernel function which generates the Bargmann dimensional single-particle space. It is easy to show that

drAb(k, hr)A,Jk’,

pr) =

2bb’ p2b2+A2br2 p2b2_A2b~2 -2(p2b2+h2br2)

312 1 [ exp

(4)

transform

2A,ubb’ $b2+h*b,2

in the three-

k.k’

(k*- k’*)]

for real A, p except A = p = 0. A useful property of A is

s

dg(k)&(Ak*, r)A&k,

r’)

(5)

Y. Suzuki / Resonating group kernels

with the Bargmann

measure dg (k) =

where

43

the integration

n 6’ (I-2X.“.I

of k, ranges

em kWkzd Re (k,) d Im (k,), over the whole complex

plane.

(7) For other

of A-, p-values the integral does not exist. A real space square integrable f(r) and its Bargmann transform F(k) are related by the equations

cases

function

00 F(k) =

r)f(r) ,

drA/,(k,

I --Jo

f(r) = j dg(k)Ab(k*,

The harmonic in the nuclear

oscillator generating cluster model:

function

(8)

r)F(k) .

property

Ab(k, r) = 1 P:““‘(k)+ho”‘(b, Qa

of A is of prime

importance

r)* .

(9)

harmonic oscillator function with the oscilHere ~$hO”‘(h, r ) is a three-dimensional lator parameter b, PxO” (k) is its Bargmann transform, and (Yis any set of convenient subgroup labels of the SU(3) representation (QO). The k;: and ki in eq. (3) are the Bargmann space single-particle vectors. The Bargmann transforms of 09 are easy to evaluate in these single-particle coordinates as was shown in ref. ‘), even though unequal width parameters are assumed for the fragments. The width parameters for the fragmentsf, A -f,f, A -f’ are denoted by 6; g, h, b’, respectively. (This means 6’ = . . * = 6; = J, &+, = * - . = iA = &‘, etc. in eq. (3).) In this case of unequal width parameters the transformation from the single-particle k;:(ki) to Bargmann space cluster-internal and cluster-relative motion variables is not achieved directly by the same A XA orthogonal transformation matrix which effects this transformation in real space. Instead of this direct transformation, Bargmann space cluster-internal and cluster-c.m. vectors are introduced by the fxf

and A -fxA

-f

(f’xf’

and A -f’ XA -f’)

orthogonal

transformations

in each fragment. Their vectors are denoted by Kc, &, EA. /(Ki, Kf, Kn .r), where e.g.& (i=l,..., f-1 andf+l,... , A - 1) stand for A - 2 internal variables, K, and KA , denote the cluster-c.m. vectors of the fragments f and A -f. This transformation leads to

A--l

f- 1 X

JJ i=l

Ac(Ki, Ri)

n i=f+l

As(1E,y Hi),

(10)

where Ei are the cluster-internal coordinates and & RA-, are the cluster-c.m. coordinates of the fragments f, A -f, e.g. & = [rl + . . * + rf]/f”‘.

Y. Suzuki / Resonating group kernels

44

The simplest replace

way to relate

& and I&-f

the Bargmann

with Bargmann

transform

to the RGM

space cluster-relative

kernel

is to

vector K by

where 6a2 = [(f/6”) + (A -f/p2”2)]/A. Similarly Kp and KAef are given by K and b. with b02 - [(f’/b2) + (A -f’/bf2)]/A. This replacement of the Bargmann space cluster-c.m. vectors implies the following transformations of the Bargmann space single-particle &i and ki. For the fragments f and A -f, the single-particle i?i transform into

1

1’26o _

-- A-f [ Af

FK+*..,

fori=l,...,f

[A(L_fJ1’22K+..., whilst the single-particle k:, ments f’ and A -f’, transform

[

-- A-f’

Af

k” =

1

fori=f+l,...,A,

corresponding into %q, FK*+.

to a cluster

..,

[Ac~_f,]“2~K*+*~*,

where

the terms

cluster-internal

A&f,

@/I-& =Aw&,

with

0

The combination

into frag-

fori=l,...,f

fori=f’+l,...,A,

(13)

associated

&)A,@&,

x CR,JLn.) = [Tb-2 3-3/d

decomposition

by + - . * stand for the respective Bargmann space with each fragment. It is easy to show that

abbreviated variables

(12)

exp

[

-;(!L$)2

of eqs. (6), (10) and (14) yields

@17(J% &.,.I

,

(14)

Y. Suzuki / Resonating group kernels

45

Using eq. (16) and its analogue in the ket side, eqs. (1) and (3) are combined as {~~~~-~~(~ -~‘)l~~l#~~~-~~(~

-R’))

with

where H(R, K”) is the Bargmann transform of 0~2 expressed in terms of the Bargmann space cluster-relative variables K, K* and cluster-internal variables Ki, KT. The relationship between H(&?, K”) and H(k; 12”)of eq. (3) isfully explained in refs. 3*5).Since oscillator excitations associated with most of the internal degrees of freedom of the fragments are restricted to OSstates, most of the A(& @iii,and A(Kf, &) expansions can be frozen in their zeroth-order terms (see eq. (9)). Only a few internal degrees of freedom can carry oscillator quanta. This property simplifies the calculation of H(&?,iK*)_ H@,K*) is obtained by applying three basic operations on H(k; k*): (i) di~erentiation with respect to the single-particle &, k: which carry oscillator quanta, (ii) subsequent transformation from the Ei and k: to E, i%: and K”, &CT as is given by eqs. (12) and (13), and (iii) setting all but the cluster-relative motion variables .&?,K” equal to zero. Eqs. (17) and (18) demonstrate that the RGM kernels for the case of unequal width parameters can also be obtained by the completely same technique as has been developed in the case of equal width parameters except that the transformations of the Bargmann space single-particle variables to the cluster-internal and cluster-relative variables are slightly modified. It is noted that eq. (17) reduces to a familiar relationship between the RGM kernels and Hr(g, K”) in the case of equal width parameters and G(R’, R’) collapses to unity. The method developed in this paper to deal with the cross terms between relative motion and c.m. excitations is much simpler than the dilatation technique 6). The simplicity of the method arises from the replacement of the dynamical coordinate 2 with the RGM parameter J?’ in eq. (16), which is permissible by the presence of the Dirac delta function. The method is akin in philosophy to that employed by Horiuchi ‘) insofar

Y. Suzuki / Resonating group kernels

46

as it uses the delta function property. Tohsaki-Suzuki ‘) investigated a derivation of the RGM kernels from appropriate generator coordinate kernels through double Fourier transformation which requires repeated multiple integrations. It appears that the method presented here is more compact since the inverse of the Bargmann transformations needed in eq. (17) is easily calculated as will be shown in sect. 3.

3. The construction

of the kernel

With a slight change of notation in eq. (17), a closed form of the RGM kernels K(r, r’) is given as the inverse transformation of the Bargmann transforms H(k; k*): K(r, r’) =

J dg(6) J dg(k)Ar,(k*,

r)H(& k*)Adk,

r’) ,

(19)

where a and k” in this section denote the Bargmann space cluster-relative motion variables. Since a straightforward integration for a given H(k; k”) is tedious, the inversion of eq. (19) becomes more convenient: H(E, k*)=

Jm dr Jmdr’Ab(k; -m --m

r)K(r, r’)AbJk*, r’) .

(20)

It is easy to show that this inversion is unique; that is, if two kernels K1 and Kz give the same H(k; k”), then K1 and Kz must be identical. This trivial statement proves useful in determining K(r, r’) when a functional form of K(r, r’) is inferred from the form of H(k; k”). In general the Bargmann transforms H(k; k”) for norm and those interaction kernels that are derived from a gaussian interaction are of the form ‘) H(k; k*) =P(/V, k*‘, (E* k”), (Ki *Z), (Kj. k”)) xexp[-~(W~-~pk~‘+yB.k*],

(21)

where Ki stands for the Bargmann space cluster-interval variables and P is a polynomial of k2, k*‘, (Emk”), (Ki +I%) and (Kj - k*). The kernel corresponding to eq. (21) is obtained by K(r,r’)=[P(-2~,-2~,~,(K..~),(~.~))Ko(r,r’)]”=o,

(22) u=o

where Ko(r, r’) is a kernel corresponding to the Bargmann transform of the exponential form, Ho& k*), Ho(k;k*)=exp[-$.x~-$?k*2+y~~k*+u~~+u~k*].

(23)

The calculation of the kernels is thus reduced to one of evaluating Ko(r, r’) and differentiating it as implied by eq. (22).

47

Y. Smuki f Resonating group kernels

A functional form of &(r, J> is easily inferred as follows. Since the exponent of H,(z, k”) is quadratic in I!, Ho& k”) is considered proportional to A&&$, yk*+u). The integration of eq. (19) over a yields terms r2, (yk*+u)‘, r - (yk”+u) in the, exponent according to eq. (6). These terms and the rest of Ho&, k”), -&k*‘+ 0 Sk”, are again quadratic in k”. The same argument can be appliedto infer afinalfunctionalformof &(r, r’) as co exp [-cl(r2/b2) -c~(r”/b’~) -tc&/b) ’ @‘lb’) +c4u ’ (r/b) + c5u ’ (8/b’) +&ju ’ (f/b’) +c7u ’ (r/b)-c8a”-c9v2+ clou -01. The unknown coefficients co-cl0 are determined by substituting this expression for Ko(r, s’) into eq. (20) and equating the resulting integral to eq. (23). Table 1 lists some useful formulas for the kernels K(r; r’) corresponding to the Bargmann transforms H(& k”) that appear frequently. The gener~izatio~ to the case of a m~ticluster system is straightforw~d. Suppose that one wants to gain the kernel for the Bargmann transform H(k; - - - Em, k? - * +kz) through

X

fi dg(kj)A~~(kj~ r,)H(k; * * . Em, k: . * . kz) ,

(24)

i=l

where & and k” are the Bargmann space cluster-relative inverse of eq. (24) leads to

X

I

m fi drjAq(kT, -co j=l

rj)K(rl

motion variables, The

. . - r,, ri ..*rlr).

(25) As an example let us find a kernel corresponding H(E~*4&T.*

- k:) = exp [-$(&

to the Bargmann transform:

A&)--&K*,

- &fil*, “CYiyj ,

BK”) -@,

CK*) (261

where &(K*) is a column vector of dimension m(n) with ith component $(k? ). A and S are M x m and n. x it real symmetric matrices and C is an m x IZ real matrix. The scalar product (g, Ah?) stands for xiiA,& - 5. The matrix T, (27) is assumed as positive definite. E, is an m x m unit matrix. The appIication of the same argument as used in the paragraph below eq. (23) suggests a functional form

48

Y. Suzuki / Resonating group kernels

of the kernel R(rl * * * r,, r: - * * rk) = [pw1’2imtn) detD det D’]-3’2 x

exp [-$(Lt-‘r,

LD-‘r) --$(D’-‘rr, A4D-1~3)

-$(D-“r, ND’-lr’)-$(D’-lr’,

'ND-'r)]

,

(28)

where, e.g. r is a column vector defined by % = (rl, . . . , r,) and D an w1x m diagonal matrix with (i, j) element b$ip The matrices L and A4 are m X m and y1X n real symmetric and N is an m X n real matrix. The substitution of eq. (28) into eq. (25) leads to H(l&

..k-,,kf

xP

-3/2

. ..k~)=~xp[-l(~~~~+~~~k.‘)]

En 1’2(m+n) det D det D’]-3 1 m mc dRi exp [-(R, XR) + (S, Z?)] , -00 i=l

(2%

with

* )( “)

fl)t-1

g*

L+E, ‘Iv

>(

’

N M+E,

(30)

where R is a column vector defined by ‘R = (rl . - . rm, r; . . . P;). The integral of eq. (29) for a real symmetric and positive definite matrix X is easily carried out by diagonalizing X. The result is * m+n I_[ dffi exp [-(RF XR) + (S WI I -Cc i=1 m+n 3/2 7r ZZCZF exp [Z(S, X-IS)] . [ det X I The combination of eqs. (26), (29) and (31) leads to a determination unknown matrices L, M, N of eq. (28): p=detT,

L+Em ‘N

N = 2T-I. M+E, >

(31) of p and the

02)

4. Specific examples It has been shown in the previous sections that the Bargmann transform technique can be used to advantage in the calculation of the RGM kernels for the case of unequal width parameters and the Bargmann transforms of large class are readily converted to the corresponding RGM kernels in closed algebraic form. The method

I’. Suzuki /’ Resonating group kernels

49

developed is best understood with a few simple examples. The norm kernel for the cy+ d system and a part of the interaction kernel for the g-quark 3-nucleon system are calculated to illustrate the technique.

The deuteron is assumed to have spin S = 1 and isospin T = 0. The spatia1 parts of the ru-particle and deuteron wave functions are assumed to be described by the OS harmonic oscillator fu~etions with width parameters b and 6’. The calc~atjo~ of the norm kernel,

K (K, K’) = (&&is (K - K)I~I#&&

(K -K’)) ,

(33)

is simplified by the use of the double coset expansion I’) for &: d = 1 - 8Ps+s+ 6P35P46 ,

(34)

The spin-isospin matrix elements of the permutation operators are easily evaluated by a standard shell-mode1 technique. The results are (ff zT’eS&““’iP$%T’ j4j~~~~~T~)= 2 and (~~~‘~,~‘IP~~T~P~~T’j~~~~#~sT’) =b The calculation of the norm kernei is thus reduced to a pure orbital integration:

where the permutation B act on the spatial parts only. Using eqs. (3) and (5), the Bargmarm transform of the operator, F&+&S, is given by H(k,k*)=exp

I -2Y~,+

i & k+ 11-20Jexp[-~~.(k~-CTk~) [isI i’ ‘1 -k&k:

-f+k~)-7fk~-k$Z-~~kK22)]

+a6expt-~~~jk~-ak~)-~~~(k:‘-trk$)-~~~(k~-ok~) -i&j~(k~-[+k~)-&-k~2-~+kf2+&--k:2-&bk~2)]},

(36)

where cr = Z~~‘/~~*~~“~ and 7 = (62-6’2)/2(b2+b12). The transformation from the si~g~e~parti~le k;; kf to the Bargmann space cluster-relative motion variables &, K” is given by eqs. (12) and (13) and converts N(c k”) to H(K, K”) =e”‘K*[l --2e’i-e”],

(37)

where 8 is a generalization of the quantity introduced in ref, 3, to indicate p pairs of n&eons exchanged and is defined by (f = 4, A -f = 2) e~=~3~expr-~A(~2+K~z~-~~~*K*],

(38)

Y. Suzuki f Re~onu~ing group kernels

50

with

b:

f

I ’

---A(A-f)p A-f b: EL=Afp+

b;

f A(A-f)

2u b;

(39)

ii%bb”

Making use of the formulas of table 1, the combination (38) yields an expression for the norm kernel:

of eqs. (17), (1% (37) and

+cr6[p2ii(~)z]~3’2exp.[-~(l-16h’+(l-2~)2)

x

.R’1} (2 >2(R2+R’3+2~1~2z~)(~)2~ )

TABLE

1

The Bargrnann transforms I$(&, k*) and corresponding

(1) exp [*E*k*]

[bb’]-3”S(x

kernels K(r, I’) related by egs. (19) and (20)

TX’)

(2) for y2 < 1 exp [#uk*]

exp

[(l - y2)db’]-3’2

(3) fora<~,~<1,r2<(l--)(f-~) exp[-~~~-~~pk*‘+~~.k*]

K&i; 8) = [p7ibb’]-3’2 exp

-

(lf4(~-_pWY2 2P

-- (1--(y)(lfS)+Y2X,2+1LYX.X, 2P P

33

I

withp=(l-*)(I-j3)-y2

_

exp [-fah’-&k*’ +&.k*+&.u+k*,v]

K&,r’)exp

i

42 -[(l-p),.x-yu.n’+(l-cu)v.n’--yu.x]

?-,Y l-8 ~~~~~~*~+Y~ 2P 2P

.v P

1

The dimensionless variables x, x’ defined by x = r/b, x’ = r’f b’ are introduced to simplify the expression.

51

Y. Suzuki / Resonating group kernels

with P2=(1-4A)*-(1-2~)*.

pr=(1-2A)*-(l-j_L)*, 4.2. THE 9-QUARK

3-NUCLEON

(41)

SYSTEM

An RGM study 11,12)for a three-nucleon system of nine quarks is motivated by an experimental finding that a three-particle “point” density of 3He exhibits a pronounced central depression. The system is taken up in this section to illustrate how to calculate interaction kernels through their Bargmann transforms in multicluster systems. The RGM interaction kernel is defined through the equation K(R12R12-3, x C

i
R;,R;*-,)

=~~~~N~N~NI~~~~12-~12~~~~12-3-~12-3~l

2

~ij~l[~~~,~,l::~(~,*-R~2)~(~12-3-R;-3))

(42)

where RI2 = v@, -R2), &12_3 = &(@r +Pz - 2@,) with e.g. I@,= X&I + r2 + r3). Each nucleon wave function constructed from three quarks is a color singlet of color symmetry [13], spin-isospin symmetric with SU(4) symmetry [3] and includes the OS oscillator functions with the width parameter 6. The spins and isospins are coupled to total S =$, T =& .d is expanded in terms of the double coset generators lo): d =

[l-w36+p39+p69)

+w25P39+P35P69+P38P69)

+27(p369+p396)l(pf!1

(-1)“‘“‘p)

-216p25p3869,

(43)

where P acts on the full space, color, spin and isospin degrees of freedom of the quarks and six 8 include those quark exchanges which are equivalent to nucleon exchanges. For the sake of simplicity only a part of the interaction kernel corresponding to the second term of & -9(P36+P39+P69)(~&1 (-l)“‘P’P), is evaluated here by assuming a simple-minded quark-quark interaction oii = uo(Af*Ai) exp [-&((ri

-rj)/b)*]

,

(44)

where hi is the color SU(3) generator

for the ith quark normalized such that kernel for more sophisticated interactions will be given in a future publication. Using 3-particle fractional parentage coefficients and SU(2) recoupling transformations as explained in detail in ref. ‘*) the color and spin-isospin matrix elements are easily evaluated to reduce the calculation of the kernel to a pure orbital integration:

P$‘=‘lor)= $(Ai -Ai) + 4. The full RGMinteraction

K(2)(Rr*R1*_3,

Ri2Ri*_3)

X~~b)~~b)~~b)~(Rl*-R;*)6(Rl*-3-R12_3)),

=$(-9)(-&)+

Ug (~~b’C$~b’+~b’

(45)

Y. Suzuki / Resonating group kernels

52

with 0 = 5(-&h)

+4(-im23)

+ 4(3%5)

-tf(&)l

+ 1 (YY(r36) ,

(46)

where the permutation P acts on the spatial parts only and exp L-$3 C(ri-rj)/b121. The Bargmann transform of the basic operator f(rij)kP is given by ‘)

= [l +p]-3’2 exp

-(p/4(1

-t/3))& -$

+8-l@

f(r:j) =

-.F1k~)z + jI (k;:.P_‘RT)]

.

(47)

The substitution of eq. (47) into eq. (45) gives the Bargmarm transform of the kernel, H(k; k*), expressed in terms of the Bargmann space single-particle &, kT. The transformation from k;: to the internal-motion and nucleon-relative motion Bargmann space variables is achieved by the same orthogonal transformation which effects this transformation in real space. The single-particle Ei transform into 12) yf&@& + J-r&&] + * * - ) pi = J~[-J~~~,

+ J~13b,] + ’ ’ . ,

1&-&&]+.

-.)

for i = 1,2,3 fori=4,5,6 fori=7,8,9,

(48)

where the terms indicated by + * - * can be set equal to zero since they stand for the inter& motion and cm. motion variables frozen to the OSoscillator excitations. Using eq. (48),and its analogue for k? and introducing auxiliary Bargmann space

(and similarly K& A$) H(k; k”) is transformed to the appropriate transform H(K, K”) which is compactly expressed as H(K, K”) = $(-9)(-&)(-$)2?0[1

Bargmann

+8]-3’2

X i: 5 (5+4(expE-~(PI(1+P))k2,if+cxp[-~(P/(1+P))K~~~l) i=l j-1

-2exp[-~(~/(1~~))(~~,+K~j)2l-2exp[-~(Pl~~+~~)(~~i-~$~2l~ x{exp [$& .K$ -t-K,{ .K$]+exp The symmetry of the three-nucleon

[-$A& *KZi +IGi *Kg]}.

(50)

wave function makes it convenient to express the kernel in terms of the dimensionless full set ai, bi(a:, b:) where al = Rw’~, bI =R12-3/b and ai, bi(i = 2,3) are related to al, bl in the same way as &, & are

53

11.Suzuki / Resonating group kernels

given by &,, &, in eq. (49). The Bargmann transform of eq. (SO) is converted to the kernel with the aid of table 1. Note that the variables I&,,, Kg are factored out the form exp [&, *K$] yielding the kernel S(& -6;). The kernel is thus obtained in a closed form: K(2)(R~zR~2-3, RLRL-3)

=-&[$~b~]-~‘~~g

i

C S(bi -6:)

exp [-Z(U~ SU;~)]

-

I

al +exp

-2[1 +$fi]-312 exp 3 4+28u:.ai

6+6p

E

-------uj t3+5p

JII

.a!

- 2(&&:p4@) (uf +a:?]

I

texp

I

3 --------ai 4+2@

.a!

’II

The calculation of the norm and interaction kernels is of vital importance in microscopic nuclear cluster models. The complicated multidimension~ coordinate space integrals needed for the evaluation of the kernels have been aided by the introduction of the Bargmann transform. The Bargmann transform of appropriate operators is first calculated in a single-particle basis and then transformed into a cluster-internal and cluster-relative coordinate basis by applying the three basic operations on the B~gmann transform. A problem arises when constituent fragments are described by harmonic oscillator functions with unequal width parameters, because the transformation to the Bargmann space cluster-internal and cluster-relative motion vectors is not achieved by the same orthogonal transformation which effects this ~ansformation in real space. The present investigation has shown that the problem is solved by transforming from the single-particle Bargmann space variables to the cluster-internal and cluster-c.m. coordinates and by introducing a slightly modified transformation of the cluster-c.m. vectors to the clusterrelative vectors. The cross terms between the relative motion and cm. excitations are most simply dealt with by the transformation through the Dirac delta function.

54

Y. Suzuki / Resonating group kernels

The present investigation has thus shown that the Bargmann transform technique can be generalized to the case of unequal width parameters without any trouble. To gain .a completely closed algebraic form of the kernels the inverse of the Bargmann transformation must be done. It has been discussed in the present investigation that those Bargmann transforms which frequently appear in nuclear cluster models are readily converted to their corresponding kernels in closed form. The technique developed in this paper has been applied to the calculation of the kernels for the (Y+ d system and the 9-quark 3-nucleon system to demonstrate both the feasibility and powerfulness of the technique. References 1) K. Wildermuth and Y.C. Tang, A unified theory of the nucleus (Vieweg, Braunschweig, 1977); P. Kramer, G. John and D. Schenzle, Group theory and the interaction of composite nucleon systems (Vieweg, Braunschweig, 1981); K. Ikeda et al., Progr. Theor. Phys. Suppl. 68 (1980) 1; Y.C. Tang, Microscopic description of the nuclear cluster theory, Lecture notes in physics, vol. 145 (Springer, Berlin, 1981) 2) Y. Fujiwara and H. Horiuchi, Progr. Theor. Phys. 63 (1980) 895; 65 (1981) 1632, 1901 3) K.T. Hecht, E.J. Reske, T.H. Seligman and W. Zahn, Nucl. Phys. A356 (1981) 146 4) V. Bargmann, Commun. Pure and Appl. Math. 14 (1961) 187 5) Y. Suzuki, E.J. Reske and K.T. Hecht, Nucl. Phys. A381 (1982) 77 6) T.H. Seligman and W. Zahn, J. of Phys. 62 (1976) 79; H.H. Hackenbroich, T.H. Seligman and W. Zahn, Helv. Phys. Acta 50 (1977) 723 7) K.T. Hecht and SC. Pang, J. Math. Phys. 10 (1969) 1571 8) H. Horiuchi, Progr. Theor. Phys. 47 (1972) 1058; ibid. Suppl. 62 (1977) 90 9) A. Tohsaki-Suzuki, Progr. Theor. Phys. Suppl. 62 (1977) 191 10) P. Kramer and T.H. Seligman, Nucl. Phys. Al36 (1969) 545; Al86 (1972) 49 11) H. Toki, Y. Suzuki and K.T. Hecht, Phys. Rev. C26j1982) 736 12) Y. Suzuki, K.T. Hecht and H. Toki, Kinam 4 (1982) 99