generator-coordinate wave functions suitable for a many-body description of nuclear reactions

Nuclear Physics A197 (1972) 193--210; ( ~ North-Holland Publishin 0 Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

GENERATOR-COORDINATE WAVE FUNCTIONS SUITABLE

FOR A MANY-BODY DESCRIPTION OF NUCLEAR REACTIONS

CHUN WA WONG t

Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts

and Department of Physics t*, University of California, Los Anyeles, California, 90024 tt*

Received 24 January 1972

(Revised 14 August 1972)

Abstract: Generator-coordinate wave functions suitable for nuclear reactions are constructed from Peierls-Thouless wave functions for separate clusters of nucleons. Antisymmetrization of the total wave function is achieved by ensuring that the basic many-body wave function in the Pei- erls-Thouless linear combination is a Slater determinant. A prescription is given for the calculation of operator matrix elements such that only smooth functions, and not Dirac &functions, are explicitly calculated. For certain oscillator single-particle wave functions the present method is shown to give results identical to those of previous formulations of the generator-coordinate method. The generator-coordinate wave functions used here are wave functions of good cluster momenta: momenta conjugate to the actual cluster c.m. coordinates, not their average values. Consequently the interaction kernel in coordinate representation is identically the kernel of the resonating-group theory. Finally channel-channel coupling is formally included by considering the Griffin-Hill-Wheeler variational equations as coupled-channel equations.

1. Introduction

We would like to report here a study of a class of generator-coordinate wave functions (w.f.) which are suitable for a many-body treatment of nuclear reactions. By a generator-coordinate w.f. we mean a fully antisymmetrized many-body w.f. containing certain parameters called generator coordinates 1, 2). We are interested in those w.f. in which antisymmetrization is to be realized by means of Slater determinants 3).

This type of w.f. has been used in both nuclear structure 3) and nuclear reactions [refs. 4-12)]. In the study of nuclear reactions many important results have been obtained, especially with the help of simple analytic single-particle (s.p.) w.f. For example, the differential form of the kinetic-energy operator of relative motion has been isolated 7-10), and the relation between these w.f. for nuclear reactions and the familiar resonating group w.f. x3) has been clarified 7-9, 12).

It is nevertheless useful to have a formalism in which no knowledge of the s.p. wave functions is assumed at the beginning. Perhaps in such a general treatment the struc-

* Alfred P. Sloan Research Fellow, 1970-72. +* Permanent address.

*~t This work supported in part through funds provided by the Atomic Energy Commission under Contract AT(11-1)-3069 and in part by the National Science Foundation.

193

194 c .w . WONG

ture of these generator-coordinate w.f. and the practical problem of numerical solution receive greater emphasis.

We start, in sect. 2, by writing down the Griffin-Hill-Wheeler (GHW) variational equation 1.2) which determines the linear combination of generator-coordinate w.f. (to be called generating w.f.) which makes up an (approximate) eigenstate of the nuclear system. The well-known Peierls-Thouless generating w.f. 14), based on an "underlying" Slater determinant, is introduced.

In sect. 3, a generating w.f. for a two-cluster nuclear system is constructed by using the fact 14) that the Peierls-Thouless generating w.f. for each cluster is an eigenfunction of the c.m. momentum of the cluster. Antisymmetrization of the total wave function is achieved by building a super underlying Slater determinant from those of the separated clusters.

Operator matrix elements between such two-cluster generating w.f. are studied in sect. 4. Besides an overall f-function in the c.m. momentum of the entire system, each such matrix element is shown to consist of a term proportional to a Dirac 6-function in the intercluster momenta, and a term which is a smooth function of these momenta. An explicit expression is obtained for the f-function term of w.f. overlaps.

In order to isolate the smooth part of an operator matrix element it is necessary to subtract out the f-function term. This is done in sect. 5 with the help of another determinantal w.f. (not a Slater determinant) with which one can construct just the 6- function term to be subtracted. A generalization of the subtraction procedure for m-body reaction channels is given in the appendix.

Sect. 6 shows how a knowledge of the 6-function part of an operator matrix element permits the isolation of the smooth part in such a way that no numerical construction of a Dirac f-function is required. In sect. 7 the formalism is applied to Gaussian s.p. wave functions. The results are found to be identical to those of earlier approaches [refs. 7- 9)]. The condition under which all these methods give the same results is given.

The Peierls-Thouless generator-coordinate w.f. considered here are labeled by the actual (asymptotic) relative cluster momenta. It is shown in sect. 8 that the coordinates conjugate to these momenta are the dynamical variables of the resonating-group method 13). This means that Fourier transformation to a coordinate representation will automatically reproduce the resonating-group formalism.

A qualitative discussion of the types of states appearing in the many-body description of the nuclear reactions is given in sect. 9. In sect. 10, the G H W equation is discussed in terms of coupled-channel equations. Finally, brief concluding remarks are made in sect. 11.

2. GrilFm-Hill-Wheeler equation and Peierls-Thouless wave function

We are interested in constructing many-body wave functions of the type 1, 2)

;

1

(2.1)

G E N E R A T O R - C O O R D I N A T E WAVE F U N C T I O N S 195

suitable for the description of scattering as well as bound-state problems. Here • (x, ~) is itself a many-body w.f. in the dynamical variables {x}, carrying parameters

commonly called generator coordinates. This #eneratin9 wave function #(x, a) is usually related to determinantal w.f., but must always be simple enough to be handled.

The unknown function f (~ ) in the linear combination (2.1) is t o be determined by the variational principle 1.2)

6 F(~[ ' J t° ]~) l = 0, (2.2) L <~1~> J

where ~ is the many-body Hamiltonian for A nucleons, usually assumed to be of the form

A

~'~(A) = • ti+ ½ ~ vii. (2.3) i= 1 i j

Here ti is the s.p. kinetic-energy operator and vi~ the two-nucleon interaction. The variational principle leads to the Griffin-Hill-Wheeler (GHW) variational equation [refs. 1, 2)]

where

f~iE U(~, ~')-EN(ot, cd)]fi(E)d~' = O,

n(~, ~') = f ¢*(x, ~)<xl~lx'>~(x', ¢)dx dx;

= f The normalization is

(2.4)

(2.5)

(2.6)

f f * i t i d~dg (2.7) i (~)N(~, ~ )fj(~ ) = 6ij.

A recent review of this method has been given by Brink s). In a scattering situation there are at least two clusters moving away from, or to-

wards, each other. A natural choice of the generator coordinate to be used is the relative separation between the mean positions of the two clusters 6-12). However, the formal description of this situation is relatively simple if the generating w.f. for each cluster is already in the form

,t,k(,, ) = e'k'"~i"t(, , --X), (2.8)

where

X -1 ~ x i A i

196 C . W . W O N G

is the c.m. coordinate of the cluster and x - X represents the set of s.p. vectors mea- sured from X. That is, the c.m. of the cluster is in a known momentum eigenstate, and the internal w.f. ~i,t is independent of k, i.e. Galilean-invariant. We are particularly interested in (j~int of the type 14)

• int(x- X) = f d3~t G(~¢- X)tJ~D(X - - ~), (2.9)

which can be generated from a Slater determinant ~o. (The subscript D will be used in this paper to denote a determinantal w.f., not only a Slater determinant.) We note that since the original w.f. ~D is not a momentum eigenstate of the c.m., all dependence on X is first removed from tb D by simply integrating over X with the help of the generator coordinate • which is its average value. The correct c.m. state exp(ik • X) is then simply put in.

Although any function G ( ~ - X ) will do in eq. (2.9) we can normally and generally handle only determinantal w.f. The integrand on the r.h.s, of eq. (2.9) can be ex- pressed as a determinantal w.f. only if

A

G(~t- X) = I-I 9,(a¢- x,), (2.10) , = 1

so that the s.p. factors can be absorbed into ~o. This calls for an exponential function:

G(at- X) = e i'' (*-x' (2.11)

The corresponding generating w.f. of the type (2.8) is:

• q ) = a~)d3~. (2.12)

There are two parameters in the w.f. (2.12): the c.m. momentum k and the parameter q. The parameter q either describes internal degrees of freedom or is redundant. (Redundancy appears if the c.m. wave function already factors out in ~o, so that the internal w.f. in (2.12) is the same for any choice of q.) In this paper we suppress internal degrees of freedom as much as possible, and concentrate on k. The generating w.f. (2.12) was first introduced by Peierls and Thouless x~). We call it the Peierls- Thouless generating w.f.

To emphasize the underlying Slater determinantal structure of the Peierls-Thouless generating w.f., we shall often write

where

~k(x, q) = f ei~'*~(x--~, k - q ) d a ~ ,

A

• k - q) = I7 e j = l

is also a Slater determinant.

(2.13)

(2.14)

GENERATOR-COORDINATE WAVE FUNCTIONS 197

3. Intercluster and scattering wave functions

Given a Peierls-Thouless generating w.f. for each cluster we would like to construct a generating w.f. for a nuclear system of two or more clusters which has the same Peierls-Thouless structure.

Consider the product of Peierls-Thouless generating w.f.

~k,(X,, q,)~k~(X2, q2) eit~.x + ~.~)~?t(x _ X , i,,t = , q,)~2 ( x 2 - X 2 , q2), (3.1)

where @i "t is the internal w.f. of cluster i of As nucleons appearing on the r.h.s, of eq. (2.12). The momenta

k = (M 2 k 1 - M, k2)(M 1 q- M2)- 1,

K = kl +k2 , (3.2)

are the intercluster and c.m. momenta respectively. The coordinates

X = ( M I X , + M 2 X 2 ) ( M I + M 2 ) - ' ,

X = X l - - X 2 , (3.3)

are the c.m. and relative or intercluster coordinates, respectively. We have used x~ to denote the set of s.p. coordinates in cluster i. (Note that in this paper we use x to denote either the relative dynamical variable of eq. (3.3) or the set of s.p. coordinates (2.8) in each cluster.)

Eq. (2.13) shows that the product (3.1) can be written in the alternative form

= ff e '(~' "~'~+q~" ~2)daat 1 d3at2 ~ l ( X l - O i l , k 1 - - q l ) q ~ 2 ( X 2 - a~2, k 2 - - q 2 ). (3.4) 4) k, 4~k:

In this expression the dynamical variables xt are in the appropriate Slater determinant, and nowhere else. This structure immediately suggests that a fully antisymmetrized generating w.f. is simply

~k~({x,; q,}) = d [ ~ , ~k2] = f/d3~ld3~ e"~''''+'2 Here ~D is the super Slater determinant

t , t

k , - q , } ) = =

k,-q,}). (3.5)

~D1 exch. . . . . . . . . . . . . . . : . . . . . t . . . . . '

exch. ~D2 (3.6)

containing ~ on its upper left (A 1 x A 1 elements), 1~2 on its lower right (A 2 × .4 2 elements), and in the remaining two blocks the exchange s.p. wave functions appropriate to a Slater determinantal w.f. The complete determinant ~ is such that the set of dynamical variables xl appear only in the first A1 rows, x2 only in the last A2 rows, while the parameters k l - q l , el , together with the s.p. state labels of nucleons in cluster 1, appear only in the first A 1 columns, and k 2 - q 2 , e2 and s.p. labels of cluster 2 only in the last A2 columns.

198 C . W . W O N G

The method can be extended easily to a system of m clusters. The super Slater determinant to be constructed has the known cluster determinants ~ i on its diagonal blocks, and suitable exchange terms elsewhere in an obvious generalization of the form (3.6).

4. Operator matrix elements

Operator matrix elements of the type (2.5) between two-cluster Peierls-Thouless generating w.f. have a simple structure. We note first that the antisymmetrizer d of eq. (3.5) has no effect on the c.m. factor of the w.f. (3.1). Therefore all matrix elements of c.m. operators are the usual ones between plane-wave states. For example, the Dirac 6-function (27036(K-K ') appears for simple w.f. overlaps.

The antisymmetrizer has an effect on the factor exp(ik • x). The effect depends on whether the clusters are identical.

If the clusters are not identical, the w.f. ~ ) contains a unique direct part, ~D1~2, and exchange parts corresponding to the exchange of one, two . . . . . min[A1, A2] nucleons between the clusters 7). The exchange terms decrease very rapidly whenever the cluster separation x becomes much larger than the contact distance between the clusters. On the other hand, the momentum eigenfunction exp(ik • x) in the direct part fills all intercluster space, and gives rise to the Dirac 6-function (27Q36(k-k ') in w.f. overlaps.

We therefore expect that a general operator matrix element for non-identical clusters has the form

<~l~l~k,> = F~(k)(27z)a~(k-k')+ <~l~l~,>s, (4.1)

where we have suppressed all state labels except k. That is, there is a term proportional to a &function of the intercluster momenta, and a remaining part which is a smooth function (denoted by the subscript s) of the intercluster momenta.

For w.f. overlaps we have already noted that the direct parts of the w.f. give rise to the 6-function term. This is an important result because the generating w.f. are now asymptotically orthogonal. We say that k is an asymptotic quantum number. The smooth part of the overlap has the effect 7) of changing the eigenvalues of the overlap function (2.6) from the values of one expected for truly orthogonal states. This situation is exactly identical to that for resonating-group w.f. 15).

For matrix elements of the Hamiltonian (2.3) we see that if ~k is already an eigenfunction of internal energies E~ "t, then it must follow that

F~r(k) = E k f l(k), (4.2)

where

h 2 h 2 int int

E k ~- - - k 2 d - K 2 + E 1 - k E 2 , (4.3) 2/~ 2M a

# = MI ME/MA, M a = M 1 + M 2 , (4.4)


Mi being the mass of cluster i. Consequently

( ¢ , ~ I ~ P - E I ~ , ) = (Ek-E)F~(k)(2~)aJ(k-k')q-(~,l.;~F-El~k,>s. (4.5)

As in the case of resonating-group w.f. 13.16), the smooth part of the operator matrix element can be considered an effective energy-dependent interaction kernel in the GHW equation (2.4).

We note finally that F~ (k) can be written in closed forms. If the subscript d denotes the overlap of just the direct parts of the Peierls-Thouless generating w.f., then

Fl(k)(27T)6j(k - k ')J(K-K') ~ ((I)kK((X i, qi})l~k'K'((Xi, q~})>d

2 = (2•)6J(k - k')J(K-K') ~I (~iint(xi--Xi, qi)lcrpiint(xi-Xi, qi)), (4.6)

i=1

use having been made of eq. (3.1). Each internal overlap factor can be calculated by using the Peierls-Thouless generating w.f. (2.12) for one cluster.

In the following we concentrate on only one of the clusters, dropping the subscript i from x~, X~, 0t i and q~ for simplicity. We need

(¢~k(X, q)l~,(X, q')>d = ((~int( X - X , q)l~pint( x - X , q')) Ifei~k'-k)'xd3X]. (4.7)

According to eq. (2.12), eq. (4.7) has an alternative determinantal form:

= (4.8)

A change of variables to x' = x - ½ ( ~ + ~') permits the extraction of the c.m. J-function from eq. (4.8) by integration over ~t+ ~'. The remaining factor when compared with eq. (4.7) gives

q')> = f d pe P

x (~o(x'-½[J)le -i~a'-a)'x'l~D(x'+½/I)), (4.9)

where/~ = at-~t'. If the clusters are identical, the exchange part involving the exchange of all the

nucleons of one cluster with all those of the other differs from the direct part by at most a sign 7-9). This spatial symmetry can be built into the two-cluster generating w.f. by the replacement 7 - 9 ) in eq. (3.1) of

e i~" = --* (1/x/2)(e lk" =-t- e-ik .=), (4.10)

where x is now the cluster separation. If each cluster has spin s, isospin t and ½A nucleons, and the total system has spin S and isospin T, then the sign in (4.10) is

spatial symmetry = (_)~a+2,-s+2,-r , (4.11)

200 C.W. WONG

because of Fermi statistics and the symmetry properties of vector coupling coefficients. Incidentally, the symmetry (4.11) implies that the parts of ~b D involving the exchange

of

n = integer [¼A]+ 1 (4.12)

nucleons or more are identical to those involving the exchange of fewer nucleons. The number of terms in the expansion of the determinantal w.f. can therefore be reduced. This symmetry has already appeared in previous works 7-9).

5. The 0-function part of an operator matrix element

The smooth parts of matrix elements in eq. (4.5) should not be computed numerically from the Slater determinant ~ directly. To do this will require a computer construction of the Dirac &function 6 ( k - k ' ) . Such a construction is difficult, because any computer noise will spread out the 6-function, making the numerical problem difficult and time-consuming, if not downright unstable. What is needed is a formula for just the f-function term:

(~(~d)l(Pl~(ka,))' = Fe(k)(27r)a6(k - k'), (5.1)

having a basic structure as close as possible to the 1.h.s. of (4.1). The smooth part of (4.5) is then simply

(4~kl~--El~k,)s -- (qod.; '~--El~k ' ) - - (~kd) lyY--El~ka' ) ) '. (5.2)

We shall show in sect. 6 that the numerical evaluation of the right-hand side of eq. (5.2) does not require the construction of a Dirac 6-function.

We first demonstrate that the desired generating w.f. O~e) of eq. (5.1)is a function again of the type (3.5), but involving the underlying determinantal w.E

~'D ,t--i, O~i, k i - -q ,}) = • (5.3) l O i~bz[

This differs from (3.6) by having zeros in the two remaining "blocks". We call it a direct determinant and distinguish it from the Slater determinant (3.6) by the superscript (d). (This direct determinant is just the product w.f. in eq. (3.4).)

First of all, it is clear that

<4'L~)I~,)> - <4~klO~,>d (5.4)

of sect. 4. Therefore

(0,~")1~",)>' = <~)1~,~,)>. (5.5)

Secondly, (4~)1CI4,~,)) can be calculated in exactly the same way as (~kl~l~k') because both involve matrix elements of g) between the underlying determinantal w.f., • ~) and ~ respectively. The same standard method 3) for determinantal w.f. applies in both cases.


For the determinantal w.f. @~a) the overlap matrix B [see Brink's eq. (A.9), ref. 3)], whose element is a scalar product of two s.p. wave functions and whose determinant gives the overlap of the many-body determinantal w.f., has a block-diagonal structure, like the r.h.s, of eq. (5.3). The cofactor matrix, being related to B-1, also has a block- diagonal structure. Both features arise from the absence of nucleon exchange between clusters in @~).

Consider now the matrix elements of the Hamiltonian (2.3):

~'~(A) = Z ti+½ Z vii = 3 - + V (5.6) i t j

respectively. It is immediately clear, especially with reference to Brink's eq. (A.9) and the block-diagonal structure of B -1, that

(q~d)l~q-l@~d,)) = k 2 + g 2 +Z~i"t +T~,t (q~(kd)l~(k~,)), (5.7) 2Ma

where T~ nt is the internal kinetic energy of cluster i. We choose

(@[d)l~'l@~d,))' = (@~d)l~'-]@~q)). (5.8)

For the two-body operator 3 ¢r we use Brink's equation (A.13) in the form

(@~)(. . . k . . . ) l ~ l ~ d ) ( . . , k ' . . . ) ) = ½(~d)( . . . k . . . ) 1 ~ ) ( . . , k ' . . . ) )

× E <~o, ~jlvl~k ~,>[(B- ~)k,(B- x),j-- (n- 1)kj(n- 1),, 3. (5.9) ijkl

Here q5 i and ~k k are the s.p. wave functions in @~d)(... k . . . ) and @~Dd)(... k ' . . . ) , respectively. Because of the block-diagonal structure of B- 1 they are identically zero whenever the index i or k falls among the wrong block. As a result, the terms on the r.h.s, of eq. (5.9) are of only two types: (i) all four indices i,j, k, l belong to the same block, or (ii) two indices i, k or i, I belong to one block and the remaining two belong to the other block. The terms of type (i) contribute to the internal potential energies of the clusters; those of type (ii) to the interaction energy between the clusters. Since we want to isolate the internal energies in eq. (4.3) we define (@~od)(... k . . . ) 1 ~ 1 ~Dd)(... k ' . . . ) ) by an equation similar to eq. (5.9), except that the summation on the r.h.s, is replaced by ~ ' over type (i) terms only. The prescription then defines

The prescription can easily be extended to n-body operators n < min(A1, A2) by requiring that the sum ~ ' extends over terms in which all indices belong to the same block.

If the clusters are identical, a knowledge of the spatial symmetry (4.10), (4.11) might reduce the computational requirement for calculating operator matrix elements. On the other hand, standard formulas like (5.9) are so simple in appearance that one hesitates to modify them. It is probable that in applications involving heavy nuclei it is necessary to determine the best formulas to be used.

202 C.W. WONG

6. The smooth part of an operator matrix element

For simplicity of notation we consider two distinguishable clusters of equal masses, and operators which commute with the total c.m. coordinate X = ½(X 1 +X2) of the system. The form of the w.f. (3.5) shows that the matrix element (~kKI01~k'X') involves integrations over four generator coordinates ~. These variables can be written in terms of

R = og 1 - - 0 g 2 ,

S = ½(~1 +~'2), (6.1)

for the bra, and R', S ' for the ket. The steps similar to those leading to eq. (4.9) can now be taken, using the antisymmetrized w.f. (3.5) for the complete matrix element and the "direct" w.f. (5.3) for the ~-function term. In each case the fi-function (27r)3j (K-K ') is factored out by integration over ½(S+S'). The final result for the smooth part is

where

(~k[~[~k,)s = JJJda(S-S ' )daRd3R ' exp [ i ( q ' . R ' - q . R ) + i ½ ( Q ' + Q) • ( S ' - S ) ]

! t . t t . ! × [(~D({x,, .8,, Ks-q,})l¢l~d{x,, ~,, K~-q;}))-(~(same)l~l~(same))']. (6.3)

In this equation,

q = ½(ql--q2), Q = (q l+q2) ,

~gl = ½(S-S '+R) , f12 = ½ ( S - S ' - R ) ,

~'~ = ½ ( S ' - S + R ' ) , IJ'~ = ½ ( S ' - S - R ' ) ,

K1 = - g 2 = ½ k , !

K~ = - g 2 = ½k',

x~ = x i - ½(S'+ S), (6.4)

and "same" means the same arguments as those in the corresponding position in the preceding matrix element.

In eq. (6.3), the direction of one of the three integration variables (R, R' or S - S ' ) may be chosen arbitrarily. A seven-dimensional integral is left. We note that there is little contribution to the integral when [S-S '[ is large. The subtraction between the two determinantal matrix elements is also such that the integrand falls off rapidly with increasing R or R'. Nevertheless, the evaluation of this multiple integral is formidable.


The computational problem is reduced if the determinantal wave functions ~ and ~toa) are such that the dependence on S - S ' in the integrand of eq. (6.3) factors out. Then the S - S ' integration can be performed separately, leaving a four-dimensional integral. If in addition the S - S ' factor does not depend on the momenta ki, we may simply set S - S ' = 0 instead of integrating over it, because the overall normalization is unimportant in the equation of motion. It can be shown that this factor- ization property holds for those oscillator wave functions 17) for which the c.m. dependence of each cluster w.f. factors out as a Gaussian. In general, the prescription of using S - S ' = 0 could be a reasonable approximation.

7. Oscillator wave functions

Analytic expressions for certain operator matrix elements can sometimes be obtained for simple s.p. wave functions, e.g. Gaussian functions. Gaussian functions have already been used in previous generator-coordinate formulations 5- 9. 11, 12) of the scattering problem.

For the Gaussian s.p. wave function

q~(r, b) = (bn½) -k exp (-rE/2b2), (7.1)

where b 2 - - h/Mw, the Brink overlap matrix 3) B; i for the Slater determinant (3.6) with all q l = 0 is

Bij = <e'k"'tP( r - ~i, b)leikJ'rtP(r--flj, b)>

= <tpil~0j> exp [ -¼b2(k j - k , )2+i (k j - k , ) " Y/j], (7.2)

where

tp, ------ ~p(r-fli, b),

Y/j = ½(fl,+flj). (7.3)

Similarly, for the s.p. kinetic energy operator t,

(e ik'''tpiltl ei~''~0j>

hE 1 I 3 1 (/~i- 'j)E+½(k, +kj ) 2 + i k ' + k j " 1 = B,J2m 2 52 2b 4 ~ (fli-fl.i) • (7.4)

The total overlap and kinetic-energy matrix elements can now be calculated by using Brink's formulas 3). The final results for ~-ct scattering are found to be identical to those of Zaikin 7).

The general condition under which these two methods - the method of Giraud, Zaikin and others 7-9) on the one hand, and ours on the other - give identical results can be derived as follows. We note first that the former method 7-9) can be

204 C . W . W O N G

formulated in terms of the Peierls-Yoccoz ~s) generating w.f.: J" eik "~t~D(X-- 0t)d30t This w.f. can be obtained from eq. (2.12) by setting q = k. With this w.f. in eq. (6.3) we find that the method of refs. 7-9) is the S - S ' = 0 prescription of the Peierls- Yoccoz method.

Now it is known ,s) that if the cluster determinantal w.f. ~D is such that its c.m. dependence already factors out, the generating w.f. (2.12) for different choices of q are equivalent to one another, except for an overall normalization which depends on q. Examples of factorable determinantal w.f. are the oscillator w.f. discussed by Elliott and Skyrme 17). For such factorable w.f. then, the final results must also be the same, provided that when q is not a constant, but depends on k, the k-dependent normalization must be corrected out. This is the reason why there is an additional "F-transformation" in refs. 7- 9). In contrast, the q used in our method (only one q is needed for factorable ~i~) is a constant, and no F-transformation is required.

In addition, the S - S ' = 0 prescription is exact only if the S - S ' dependence in eq. (6.3) factors out. This happens if the c.m. motion for the entire system is factorable - a condition more restrictive than that for the validity of the Peierls-Yoccoz method.

I f the c.m. dependence does not factor out in the cluster w.f. ~o, the internal w.f. of the Peierls-Yoccoz method depends on k. The Peierls-Thouless method avoids this difficulty. On the other hand, the ovexlap matrix o r b eq. (7.2) is now more compli- cated because of its momentum dependence.

Finally, the plane waves in the Peierls-Yoccoz generating w.f. are those for the generator coordinates ~t. Consequently these generator coordinates are the coordinates conjugate to the cluster momenta, as pointed out in refs. 7-9). The situation with respect to the Peierls-Thouless w.f. is different, and will be discussed in sect. 8.

8. Coordinate representation: equivalence to the resonating-group formulation

Giraud and others s, 9, 12) have shown how the scattering kernel in the resonating- group formulation can be derived from the GHW generator-coordinate kernel in the case of Gaussian w.f. Now the coordinates of the resonating-group method are the actualdynamical variables constructed from the cluster c.m. coordinates Xi; whereas the G H W generator coordinates ~i are only their average values. It is therefore interesting to show, for arbitrary s.p. wave functions, that the coordinates conjugate to the cluster momenta of eq. (2.8) are the actual dynamical variables, not the G H W generator coordinates. Consequently, the Fourier transform of the momentum-space scattering kernel of previous sections is identically the resonating-group kernel.

We start from eqs. (3.1) and (3.5). Ignoring the unimportant c.m. factor exp(iK • X ) we have for the channel w.f. of a plane wave

. o r - ik . x = i n t ~ i n t l 4'k = ~;zke (1)1 (1)2 .]' (8.1)

G E N E R A T O R - C O O R D I N A T E W A V E F U N C T I O N S 205

where x is the relative dynamical variable of eq. (3.3). In the Fourier transform

t" cb, = (2re) - 3 je-'~"~d3k, ( 8 . 2 )

both k and r are c-numbers which commute with the antisymmetrizer ~¢. Conse- quently

int int q~, = d [ 6 ( r - x ) ~ l ~2 ]. (8.3)

This shows that the Fourier transform leads to the actual dynamical variable x without affecting antisymmetrization. Similarly, functions of r commute with d , but functions of x do not. Therefore the coordinate-space scattering w.f.,

ff(r)q~rd3r i,t int = ¢ 2 ], (8.4)

is just the resonating-group scattering w.f. Fourier transformation to the coordinate representation is desirable in practice be-

cause scattering w.f. are usually studied in coordinate space. But unfortunately we have not been able to find a short cut to the coordinate representation for arbitrary (e.g. numerical) s.p. wave functions.

The attempt sheds sufficient light on the structure of the w.f. to merit a short discussion. The problem involves integrations over (i) the s.p. coordinates in the s.p. wave functions; (ii) the generator coordinates ~i, which are the average cluster positions; and (iii) the momenta k~. In the formalism presented here these integrations can be taken in any order.

If the generator coordinates are integrated first, the dependence on the s.p. coordinates becomes mixed up. The momentum dependence, being factorable in eq. (2.8), should next be integrated. The result is just the resonating-group ansatz, although one in which the internal w.f. (there might be more than one for each cluster) are related to known Slater determinants.

If the momenta are integrated first, 6-functions like that in eq. (8.3) appear. They again mix up the s.p. dependences so that antisymmetrization cannot be performed as easily as in a simpler Slater determinant. The overall result is equivalent to the method of Tabakin 12).

The simplest procedure seems to be the integration of s.p. coordinates first, as advocated here [also in refs. 4-1x)]. After this the order of integration of ~ and ki does not appear to be too important. We should note however that when the momenta are integrated first, integral relations between the generator coordinates and the dynamical (or resonating-group) variables are obtained. These transformations are equivalent to the inverse F-transformation of Giraud et al. 8.9), but without its singu- larity.

In the remaining sections we shall discuss only the momentum representation.

206 C.W. WONG

9. Reaction channels

We first consider two-body reaction channels. These channels can be labeled by the neutron number Ni, the proton number Zi, the internal-state label n~ (including spin- isospin labels) of cluster i (i = 1, 2), and the (asymptotic intercluster momentum k (or k,/, m)). We again suppress the c.m. momentum K because the 6-function appears in all terms and can be factored out.

In the spirit of the generator-coordinate theory, the internal w.f. of cluster i is assumed to be expressible in the form

~i,"~t(x,-X,)= fff,,(qi,fli)e'~"(~'-x')~Di(X,-~ti, fll)d3~t, d3qidfli, (9.1)

where fli represents the set of generator coordinates (in addition to qi) necessary for the complete description of internal states. To be sufficiently flexible, we must include as generator coordinates not only parameters appearing in the w.f., but also shell- model configuration and other state labels. We assume all these can be done without causing irreconcilable difficulties.

The development of sects. 4-6 can now be repeated through the construction of super determinantal w.f. cb~, q~), and of the generatingw.f. ~ak and q~a d), i.e. including the internal degrees of freedom. Here 2 = 2~, 4 2 (~'i = Ni, Zi , ni) is the set of quantum numbers for the internal states of the clusters.

Now i f2 ' 4 :2 the overlap (q~k[~Z,k,) can never build up the 6-function 6(k-k') because the orthogonality of the internal states ensures that the integrand of the scalar product does not fill up all space. The same holds for matrix elements of j~,o, because q~Zk is by definition an eigenstate in the degrees of freedom internal to the clusters. Therefore calculations of overlaps and Hamiltonian matrix elements need not involve the subtraction procedure of sect. 5. But numerically, we might be better off using it. If 2' = 2, we should certainly use the subtraction procedure.

A general m-body reaction channel can be labeled by 2 = 2~ . . . . . 4,, and the set o f (asymptotic) intercluster momenta k = kl . . . . . kin-1. A general operator matrix element involves terms proportional to m - 1 6-functions: 6(kl-k'1).. . 6(k,,_ 1- k~,_~), m - 2 6-functions . . . . . and finally terms which are smooth functions of the channel momenta. The general procedure for the isolation and construction of these terms is given in the appendix.

In addition to reaction channels which can become open at sufficiently high energies there are also shell-model states which are always localized if left entirely to themselves. These are many-body states of A nucleons, each in a bound s.p. state. They can be described by a simple Peierls-Thouless w.f. of the type (9.1), but for the entire nuclear system.

Operator matrix elements among shell-model states or between shell-model states and reaction channels are the simplest, because they are " smoo th" and contain no term proportional to a 6-function.


10. The coupled-channel problem

The coupled-channel problem can now be formulated. If ~ denotes the set of all necessary state labels, including the cluster label ~ and the set of asymptotic intercluster momenta k, then the variational principle (2.2) will give the coupled-channel G HW equation (2.4). This is the same procedure as the resonating-group method [refs. 13. t6)], as far as the intercluster degrees of freedom are concerned.

A more vigorous derivation of the coupled-channel G H W equations has been given by Ohmura et al. 19) in terms of a variational principle on matrix elements of the scattering matrix.

The structure of eq. (2.4) also suggests that, like shell-model problems, we are actually solving a truncated many-body Schroedinger equation. But unlike usual shell-model problems we are using generally non-orthogonal w.f. (Our scattering w.f. are only asymptotically orthogonal, while the generating w.f. for shell-model states may or may not be orthogonal.)

The type of solution to eq. (2.4) depends on whether the channel is a reaction channel or a shell-model state. For shell-model states an eigenvalue problem is to be solved. Coupling to open reaction channels, if any, gives the energy eigenvalue an imaginary part which is the width of the quasi-stationary state.

In open reaction channels the energy E is fixed while the scattering w.f. is to be calculated. Angular-momentum decompositions needed are the standard ones 16, 2 o). The fact that generator coordinates k are themselves the asymptotic intercluster momenta considerably simplifies the calculation of scattering w.f. This is because there is no zero-point oscillation in these degrees of freedom. Thus in two-body channels and in the absence of Coulomb interaction, the matrix element (5.2) can be used directly to calculate phase shifts by way of principal-value integrals 2 i). If a Coulomb interaction is present, an additional manipulation is needed before the application of the principal-value method: either a transformation to Coulomb scattering states or use of the Coulomb Green function 22). The situation with three-body channels is not so clear, but presumably one can solve Faddeev equations.

One nice, albeit only formal, feature of the coupled-channel problem is that it automatically includes dynamical distortion effects through channel-channel coupling. According to the matrix elements (5.2), this channel-channel coupling is important only when the clusters are close together. It is also to be expected that the type of internal excitations of importance depends on the nature of the reaction under con- sideration.

In reality the number of channels which can be coupled together is limited. One might be forced, on the one hand, to use a residual optical potential. On the other hand, one might work in terms of effective nucleon-nucleon interactions related to the original bare interaction through diagrammatic perturbation theory 23). The numerical problem is similar to, but more formidable than, the corresponding shell-model problem.

208 C . W . W O N G

11. Concluding remarks

We have discussed in rather general terms a generator-coordinate method for nuclear reactions which is the momentum-space analog of the well-known resonating- group method 13). It differs from the conventional resonating-group formulation by explicitly requiring that antisymmetrization is to be achieved by means of determinantal wave functions. In this and many other aspects the formalism is very similar to the generator-coordinate method of Giraud and others 7- ~ 0). But unlike the latter it can be used with arbitrary s.p. wave functions in a more direct way. In particular, the general structure of generator-'coordinate approaches for nuclear reactions tends to stand out more clearly in the present formulation. However, these advantages are obtained at the expense of making the integral transformations involved more compli- cated and harder to apply. Indeed, from a practical point of view one would, if one could, solve the original, i.e. untransformed, Griffin-Hill-Wheeler (GHW) integral equation 1, a) directly.

There are now at least two direct numerical solutions 6, ~ ) of the original G H W equation. They both give reasonable numerical results. It would be interesting to find out to what extent these direct methods are exact. A knowledge of the general structure of the problem gained in indirect approaches using integral transformations could be of help in this connection.

The author would like to thank Professor H. Feshbach for the hospitality of the Center for Theoretical Physics, MIT, and the Alfred P. Sloan Foundation for financial support.

Appendix S U B T R A C T I O N P R O C E D U R E F O R O P E R A T O R MATRI X; ELEMENTS BETWEEN G E N E R A L m-BODY C H A N N E L S

A general operator matrix element between two m-body channels has terms proportional to m - 1 ~5-functions, m - 2 6-functions . . . . . and finally terms which are smooth functions of the channel momenta k = k I . . . . . k,~-l. The subtraction procedure of sect. 5 used with a "direct" determinantal w.f. which in matrix form is block-diagonal with m diagonal blocks will give the term proportional to m - 1 &functions. This direct determinantal w.f. will be denoted by ~ ' m), where ). is the cluster label (Ni, Zi, n~} of sect. 9, and the superscript stands for "m clusters and m diagonal blocks". The multiplicative factor in this term of the operator matrix element is calculated by a standard formula 3) of the type (5.9) for determinantal w.f., but with the summation further restricted by the requirement that all s.p. labels i,j, k, l must belong to the same block, as discussed in sect. 5. We denote this restricted sum ~ ' and the resulting matrix elements(4~(~ ' ~ ) 1 0 1 ~ , ~ , ~ ) ) '

For d~ = 1, ~;¢¢', these matrix elements are already known: ra-1

(~-~' ~'~[q~Tk''))' = (2n)36(K-K') [I [(2n)36(ki-k~)]dzz'( int (2)1 int (2)), (A.1) i=1

G E N E R A T O R - C O O R D I N A T E W A V E F U N C T I O N S 209

where K is the c.m. momentum of the entire nuclear system. Also

(int (2)lint (~.)) = f i (int (2~)lint (2~)), i = 1

where

(A.2)

t*t • (int (2i)lint (2/)) j j . int int t t t = f~,(fl,)(* (x-X, fl,)[* (x-X, fl,))f,,(fl,)dfl, dfl,, (A.3)

the scalar product on the right being those of the type (4.9). Similarly,

( t i~ (m'm) l~Ol~) (re 'm) \ ' E ( m ' m ) ( ~ n k 'm) I~ (m' m ) \ , (A.4) "~ ;,.k I I 2"k" I = 3, k" I

where

E el °', (A.5) j = l i = 1

T i being the kinetic energy associated with the intercluster degrees of freedom described by k i.

Before proceeding to terms with m - 2 6-functions, we should note that the (m, m) overlap (A.2) is factorable into m factors, each referring to one cluster. This is a con- sequence of the block-diagonal structure of ~ ' "). Similarly, the nature of ~ and of the restricted summation ~ ' accounts for the additivity of the energies in (A.5).

Next, there are ½m(m - 1) distinct terms each involving a distinct product of m - 2 6-functions. The disappearance of one 6-function is related to exchanges of nucleons between two clusters, say 1 and 2, whose intercluster momenta no longer appear in the 6-functions. The determinantal w.f. of interest is 4~]' " - i ) , which in matrix form is zero everywhere except the block diagonal and the 12, 21 block positions. The blocks 1~2 can be treated as one super block, but unlike the other blocks we want the smooth part, not the 8-function part, of their operator matrix elements. This means that the desired terms in the (m, m - 1) overlap are of the type

~ d s f m , m - 1) {d~(m,m- I) N ~,z2~. ~,~, ~-~.' ,z'~,'. k',=~'/ = (~z ,~ , , ~l~bz ' ,~'~. k'~ 2)~<~cu~ - z.,, - 2) -,'~'~¢'- 2. ,, - z).~,,, (A.6)

in the notation of sect. 5. Similarly

( ¢ / ~ ( m , m - 1 ) i 2KOidS(m,m-l) N ~ ( ~ ( m - 2 , m - - 2 ) ( ~ ( m - - 2 , m - 2 ) ~ e

In this equation

E(m,m-2) 2, Ti+ ~ ~,,t 1 , 2 = El ,

j i = 3

where ~ ' j excludes the contribution from k~2 and j-(fo. 2) is that part of the Hamil- tonian which involves degrees of freedom internal to clusters 1 and 2 and for the intercluster momentum k~2.

210 c . w . WONG

Next there are m(m- 1 ) ( m - 2 ) / 3 ! distinct terms, each involving a distinct p roduct

o f m - 3 6-functions. Here we need the smooth par t o f the over lap and Hami l ton ian

matr ix elements for three clusters. They are just the ordinary three-cluster results less

those terms p ropor t iona l to two 6-functions and to one h-function.

In this way the decompos i t ion o f opera to r matr ix elements into terms involving a

different number o f 6-functions can be made wi thout depar t ing f rom the underlying

de te rminanta l s tructure for the many-body wave function.

This t rea tment can be extended easily to cases where the left and right channels

have different numbers o f bodies or clusters. In such an opera tor matr ix e lement there

is certainly a smooth par t (no f - funct ion) . A n d there are terms with 6-functions when-

ever there are separated clusters with 2' i = 21.

References

1) D. L. Hill and J. A. Wheeler, Phys. Rev. 89 (1953) 1102 2) J. J. Griffin and J. A. Wheeler, Phys. Rev. 108 (1957) 311 3) D. M. Brink, The alpha-particle model of light nuclei, Int. School Phys. Enrico Fermi, course

36 (1965) 4) B. Giraud, Proc. Herceg Novi Summer School Nucl. Phys. (1966);

B. Giraud, Hartree-Fock techniques for a practical microscopic model of nuclear reactions (1967) unpublished

5) D. M. Brink, lectures given at Orsay, 1969, unpublished 6) H. Horiuchi, Prog. Theor. Phys. Jap. 43 (•970) 375 7) D. Zaikin, Nucl. Phys. A170 (1971) 584 8) B. Giraud, J.-C. Hocquenhem and A. Lumbroso, invited lecture C6-1, Colloque de La Tous-

suire, Lycen 7104 (1971) 9) B. Giraud and D. Zaikin, Phys. Lett. 37B (1971) 25

10) T. Yukawa, Phys. Lett. 38B (1972) 1; Nucl. Phys. A186 (1972) 127 11) N. B. de Takacsy, Phys. Rev. C5 (1972) 1883 12) F. Tabakin, Nucl. Phys. A182 (1972) 497 13) K. Wildermuth and W. McClure, Cluster representation of nuclei (Springer, Berlin, 1966) 14) R. E. Peierls and D. J. Thouless, Nucl. Phys. 38 (1962) 154 15) S. Saito, Prog. Theor. Phys. Jap. 41 (1969) 705 16) E. van der Spuy, Nucl. Phys. 11 (1959) 615;

A. C. Butcher and J. M. McNamee, Proc. Phys. Soc. 74 (1959) 529 17) J. P. Elliott and T. H. R. Skyrme, Proc. Roy. Soc..4.232 (1955) 561 18) R. E. Peierls and J. Yoccoz, Proc. Phys. Soc. A70 (•957) 381 19) T. Ohmuras B. Imanishi, M. Ichimura and M. Kawai, Prog. Theor. Phys. Jap. 41 (1969) 391 20) R. Balian and E. Brezin, Nuovo Cim. 61 (1969) 403 21) F. Tabakin, Ann. of Phys. 30 (1964) 51;

G. E. Brown, A. D. Jackson and T. T. S. Kuo, Nucl. Phys. A133 (1969) 481 22) J. Schwinger, J. Math. Phys. 5 (1964) 1606 23) B. H. Brandow, Rev. Mod. Phys. 39 (1967) 771

generator-coordinate wave functions suitable for a many-body description of nuclear reactions

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