The generator coordinate method for nuclear scattering

I

l.D.2: 2.L

I

Nuclear Physics

A182 (1972) 497-521;

Not to be reproduced by photoprint

THE GENERATOR

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@ North-Holland

Publishing

or microfilm without written permission

METHOD

FOR NUCLEAR

Co., Amsterdam

from the

publisher

SCATTERING

F. TABAKIN Physics Department,

University

of Pittsburgh,

Pittsburgh,

Pa. 15213 t

Received 5 October 1971 Abstract: A generator coordinate description of nucleus-nucleus scattering is formulated. By imposing proper scattering boundary conditions on the Hill-Wheeler integral equation, one can derive an effective nucleus-nucleus potential directly from the basic two-nucleon Hamiltonian. The effective potential is a rather complicated non-local and energy-dependent operator, but it does incorporate both Pauli exchange effects and the occurrence of nuclear shape, size and orientation changes during the collision. The generator coordinate and resonating group descriptions are compared and it is concluded that they are essentially equivalent. However, the generator coordinate approach is a special truncation which is based on semi-classical ideas of collective nuclear motion. Use of generator coordinates leads one to a set of continuous, coupled Lippmann-Schwinger equations in a biorthogonal basis. As a step toward practical solution of these equations, an effective z-cc potential including monopole deformation effects is derived using the independent particle model and a simplified two-nucleon potential.

1. Introduction The generator coordinate method (GCM) provides a practical and flexible microscopic theory that enables one to visualize the collective motion of nuclei ‘). It has been used in various ways to describe the shape, orientation and size changes of nuclei as they vibrate and rotate ‘). The basic idea of the GCM is to use a set of many-body wave function with appropriate parameters that can be identified as collective coordinates. One then forms a superposition of these model wave functions using a weighted average over the collective coordinates. A great advantage of using a GCM approximation for the full many-body wave function is that one avoids the more rigorous but extremely difficult task of finding and using canonically conjugate collective operators “). Instead, the generator coordinates are simply dummy variables used to generate approximate but fully antisymmetric many-body wave functions. Numerous applications of the GCM to describe nuclear collective motion, such as vibrations, rotations and pairing vibrations, have been published “). Also a-cluster wave functions with generator coordinates have been successfully used to describe the collective motions of “Ne [ref. “)I. In this paper another important type of collective motion is considered, namely, the scattering of two nuclei. The natural collective coordinates for this scattering probi Supported in part by the US National Science Foundation. 497

498

I=. TABAKIN

lem are not only a relative separation, but also coordinates that describe the nuclear shape, orientation and size changes that occur during nuclear scattering, particularly at close range. By including such effects, it is possible that the generator coordinate method can provide a microscopic theory of nuclear scattering applicable to heavyion collisions and nuclear fission “). To formulate such a theory, one must first consider how to describe and suitably parametrize a many-body wave function. It is required that this many-body wave function should, of course, be totally antisymmetric and also should have proper behavior when the nuclei are close together. In sect. 2, a suitable wave function having these properties is introduced and the GCM approach for scattering is presented. After appropriately defining the elastic scattering amplitude (sect. 3), one finds that an effective interaction between the nuclei can be obtained from the basic Hamiltonian. This effective interaction, which includes Pauli principle and nuclear deformation effects, is a rather complicated nonlocal and energy-dependent potential. Nevertheless, with suitable approximations it is possible to use the effective interaction to calculate the phase shifts for nucleus-nucleus scattering. The significance of superfluous solutions ‘) as they appear in the GCM is also discussed in sect. 3. The GCM analysis of nuclear scattering “) is closely related to the open channels or resonating group method (RGM) “). Th e exact relationship is discussed in sect. 4, where it is shown that the two methods are essentially equivalent. The equivalence is not really surprising since the RGM is based on a complete set of states; one can always turn away from the classical picture of size and shape changes provided by the GCM and use a large but complete set as in the RGM. After establishing the mathematical relation between these two methods, it is emphasized in sect. 4 that the GCM provides significant advantages if one is willing to make use of intuitive, classical ideas about nuclear motion. Then one can avoid confronting the RG problem of solving an overwhelmingly large set of coupled equations and instead deal with a truncated set of coupled equations that are a consequence of our choice of generator coordinates. The generator coordinate description is thus shown to lead to a Lippmann-Schwinger equation in a biorthogonal basis ‘). Various approximations for the Green function are suggested to simplify this integral equation (sect. 4). Finally, the independent particle model is used to evaluate the effective nucleus-nucleus potential, and the procedures needed to calculate phase shifts including deformation effects are discussed in sect. 5 for the case of cc-a scattering.

2. The generator coordinate wave function Two types of collective coordinates are needed to describe the collective motion involved in nucleus-nucleus scattering. The first is a coordinate R that generates changes in the relative separation of the nuclei. Another class of generator coordinate, called 1, is used to describe the size, orientation and shape of the interacting nuclei,

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e.g., A denotes the standard Bohr-Mottelson collective coordinates R,, /I, y and Q. Let us now introduce a totally antisymmetric model wave function Q,,(r) which is parametrized by our choice of collective coordinates R and 1. Here r denotes the spat:, spin and isospin coordinates of all of the nucleons. The GCM many-body wave function is then defined by forming a continuous superposition of Q&(r) Y(r) = Yfr, . - * rA) =

s

dRdIf,(R)@,,(r),

(24

wheref,(R) is a weighting function and the integral extends over all possible values of R and I. The exact antisymmetric wave function for the many-nucleon system satisfies the Schrodinger equation in the form <%.IHl Using the GCM approximation Wheeler ‘) integral equation

V

to the exact wave function,

s

[H(RAJR’X)-EN(RAIR’l’)]f,@‘)dR’d~’

To solve (2.3) one must first obtain Hamiltonian

which includes defined by

both nuclear

H(RAlR’X)

and Coulomb H(RI(R’I’)

and the overlap

integral

(2.2)

= R<@,,lV.

from

interactions.

one obtains

the Hill-

= 0.

(2.3)

the microscopic

The matrix

A-nucleon

element

of H is

= (~RIIHI~R’j,‘),

(2.5)

= (@,&DR,rl,).

(24

is N(RAlR’A’)

The Hill-Wheeler equation describes the nucleus-nucleus problem in terms of the collective coordinates R and ;1; the nucleon coordinates appear only in evaluating (2.5) and (2.6). 0 nce these matrix elements (operators in the Ri representation) are completely defined, one can consider solutions of the Hill-Wheeler equation with scattering boundary conditions (see sect. 3). To evaluate (2.5) and (2.6) an explicit choice for 4 R1 is needed. A natural choice is to use an independent-particle model wave function, i.e. a Slater determinant of the orbitals for a two-centered oscillator well “). We shall return to that case later (sect. 5). Let us first consider another form + @k(r)

+

= C,4RP’[4,(SI,

JF,@, z- RI],

(2.7)

Although (2.7) does separate the relative and internal motion, it is not the most general ansatz. Later special, useful properties are assigned to the parametrized functions Fn and CA. Eq. (2.7) does not include break-up and other channels.

500

F. TABAKIN

which is made antisymmetric by means of the standard operator JZI = (LI!)-~& (- l)PP, where P denotes all permutations of the nucleon coordinates r. This form for @,, enables one to examine the general properties of the GCM free from c.m. and shell-model configuration-mixing difficulties. In defining GRn, we have replaced the nucleon coordinates r by the corresponding internal coordinates (cl, 2 : t1 and c,) and the relative coordinate (R,,) of the two nuclei. For example, one choice of internal coordinates for the cl-a scattering problem is
for

i=l-4,

& = ri-R2

for

i = 5-8,

(2.8)

4R, = r,+r,+r,+r,.

4R, = r,+r,+r,+r,,

For the c.m. system used in (2.7), the c.m. coordinate R,,,. = 0 is used. Furthermore, the function 4n(rI, 2) is taken to be a product of the two nuclear wave functions parametrized by their respective size, shape and orientation collective coordinates (A : A,, A,) WI,

2) = hW~~,&).

(29

The nuclear wave functions in (2.9) are understood to be parametrizations of the nuclear ground state wave function. One simply identifies suitable shape-size and orientation quantities in the ground state wave function and assigns them the role of collective coordinates. To make this parametrization clearer, suppose the nuclear wave function to be of Gaussian form &AC;) = e-

(aXL~+~,ty*+~ZL2) 3

(2.10)

with the known values of a,, a,, and a, fixing the actual size and shape of the nucleus. Then one could assign these quantities the role of generator coordinates and use 4,(t) = exp( -A,<: -A,I$ -A,(Z) in (2.9) +. More generally, the exact nuclear ground state is an eigenstate of the internal Hamiltonian, which is obtained by separating (2.4) into center of mass, relative, and internal terms H = T,.,.+H,(5,)+H2(52)+H12(R12 The relative

Hamiltonian H,z

(2.11)

2 51 a*

is

= -2;V:2+

c kl,

[Oi,j+ je2

~,:(l+i,(i))(l+r,(i))]

(2.12)

3

1.l

where /,t is the reduced mass for the two nuclei. Note that (2.7) is an eigenstate c.m. energy, T,.,. Y = 0. t For the case of a deformed be an Euler angle.

nucleus

a similar

parametrization

can

be adopted

where

of the

1 would

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Now consider the complete set of exact eigenstates of the internal nuclear Hamiltonian H,(5,) (or HZ(&))

Hi(t,)$“(ti> =

hi +n(ri>*

(2.13)

Throughout this paper #,(c) and 4,(t) are used to designate the exact and parametrized wave functions, respectively. One can always identify some shape-size and orientation parameters in the ground state wave functions Ibe and use them as the generator coordinates as in (2.10). Of course, #2,(5i) is not an eigenstate of wi(ei); it is only an eigenstate when 3. takes on the true ground state value denoted by lo +. Hence for L -+ lo one has (PA,= $e. For general values of 1, (Pnis a mixture of the exact eigenstates (i = 1,2) #&(ti>

=

T


(2,14)

and the product (2.9) becomes

The coefficients in (2.15) determine the mixture of eigenstates corresponding to a particular nuclear size and shape labelled by Iz : A,, A,. The generator coordinate Iz refers to a classical picture of the size and shape of the interacting nuclei. The classical picture, and hence #,, might be a very simple one whereas the corresponding expansion (2.15) could require an overwhelmingly large number of terms - including continuum states. In that case, the GCM provides a useful intuitive approximation based on simple reasonable choices for (Pt. Although the choice given in (2.7) for QtRihas been partly defined, the functions CA(R) and P(& -R) have not yet been selected. The function C,(R) is a normafization factor determined by the requirement iV(RA.iRA) = 1. With (pK&so normalized and antisymmetrized, the many-body wave function is meaningful even when the nuclei are close together. Explicit examples of CL(R) and its role will be discussed later (sect. 5). where The function FA(R1, -R) is taken to be some peaked function of RI,-R, R is used to generate changes in the distance between the nuclei. The width and shape of FA is characterized by iz : Al, &. The reason for taking FA to be a peaked function is made clear by considering the evaluation of N(RLfR’A’) and H(RnlR’X), eqs. (2.5) and (2.6). The overlap integral, which needs to be specified before one can solve the t Xn the case of a deformed nucleus the true ground state involves an average over the Euler angles AO. Here we visualize nuclei with definite orientation in the asymptotic region stipulated by Lo, The average over & should then be restored after one formulates the scattering problem.

502

F. TABAKIN

fill-wheeler

equation,

N(Ra/R’n’)c;

“(R)c;

is given by ‘(I?‘)

= nAx(R -R’) -i-n,,;(R in (2.16), the long-range

+ R’) + &(RIR’).

terms are defined

by

with 2; corresponding to switching /2; and 2; in (2.17). These quantities are functions of R_+ R’ because Fk is a function of R - Rlz. Now, by virtue of our decision to take FL as a peaked function, the remaining term EL,, is a short-ranged operator. For R and/or R’ larger than the range of Fi, -kLd, rapidly vanishes. Thus picking FA to be a peaked function of R,, -R, permits a separation of N(RIJR’A’) into long- and short-range terms. Similarly, H(RA,IRd’) is separated into long- and short-range terms .Fjr(Rr.,rR’n’)C,‘(R)C,‘(R’)

Note that the special choice A = /2, has been used in (2.18) to extract the eigenvalues eO1 and co2. For the matrix elements in (2.2)-(2.6) the same choice n = Jo is now made, where a, denotes the shape-size-orientation parameters of the exact ground state, Le.

The long-range

terms in hn,+ are

h ~,&W’) where TI, includes

= (i.~~~~)(CJi;>SF1,(R+R;2)41F~~(R’-~~~~dR12 both the relative

kinetic

3

(2.20)

energy and the relat.ive Coulomb

inter-

action h2 qz=--

v2

+

ZJ2e2

12 2/J

(2X)

-ET’

&ye C-I denotes the reduced mass and Z,Z, the atomic numbers of the colliding nuclei. AlI other terms occurring in (2.18) are of short range and are included in the Mini(RJR’). Again ,I& means switching 2; 3 Ah. tion of u”;cOL

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The separation of H(R&lR’n’) and N(Rl,lR'A') into long- and short-range parts is analogous to the usual step of separating a Hamiltonian into a kinetic-energy operator plus a short-range potential. Correspondingly, the separations made in (2.16) and (2.18) will enable us to define asymptotic incident wave solutions of the HillWheeler equation “). Using (2.16) and (2.18) the Hill-Wheeler equation with i --f 1, + becomes

I

&,,(RIR’)+

h,,,,(RIR’)

-G,[nA,,,(R

=-

- R’) +

K,,,(RIR’)}C,(R’)fn(R’)dR’dl,

&,,(RIR’)-J%

s

nnoA,(R+ R’)]}C,(R’)f,(R’)dR’dil (2.22)

where the relative energy E, = E- (co1 +coz) now appears in (2.22). For the special case of scattering of identical nuclei (~-a, 0i6-01’j, etc.), some useful symmetries follow from (2.13)-(2.18)

b,(W’) n&R-R') Consequently, fies to

the Hill-Wheeler

s

equation

= h,,(W)> = q,,(R-R'). for the scattering

(2.23) of identical

nuclei simpli-

[hAol~(R~R~)-EnA,~~(R-R')]C,(R')f~~(R')dR'~X =-

s

[9,,,,(RIR')-EIZ,,,.(RIR')]C,.(R')~~,(R')dR'dl'. (2.24)

In (2.24) 6 and K have been defined identical nuclei scattering f,(R)

using

the following

properties

for the case of

= fJ-R),

6>&RIR')= O,,,,(RIR')+D,,,(Rl -R'),

(2.25)

R,,,(RIR') = R,oA~(~~~~)+IZ,o,~(R~ -R'). An important consequence of (2.25) is that only even partial waves appear for identical nuclei scattering. Throughout this paper the discussion is restricted to the special case of the scattering of identical nuclei; however, it should be possible to use (2.22) for other cases such as nucleon-nucleus scattering and to obtain a corresponding generator coordinate description. So far the Hill-Wheeler equation has simply been separated into long- and shortrange terms. In the absence of a Coulomb interaction the long-range terms (1.h.s. of (2.24)) are translation invariant and permit plane-wave solutions; in general Coulomb + Eqs. (2.22) and (2.24) use only the ground state (A,,) component of the Hill-Wheeler equation. The remaining components are introduced in sect. 4, where inelastic scattering is discussed.

504

F. TABAKIN

waves are needed. The r.h.s. of (2.24) is just a short-range non-local and energydependent interaction. Not only is the weighting factorf>(JZ) unknown in (2.24), but the function F,(R-R,,)has not really been specified aside from its being some peaked function of R-R,,. In order to clarify the role of these functions and to facilitate solution of the Hi&Wheeler equation for scattering boundary conditions, let us now consider the definition of the scattering amplitude. 3. The scattering amplitude and the effective potential The goal is to solve the Hill-WheeIer equation (2.24) with proper scattering boundary conditions. Instead of solving (2.24) directly for the weighting functionf,(R), let us first define the boundary conditions by forming the projection of the GCM function (2.7) onto the internal wave functions of the two nuclei “) djAo= 4e = ~~~(~~~~~~(~*) (a fixed or d er of the nucleon coordinates is assumed) +

The relative wave function I,!I~~(R,,) gives the probability amplitude for finding the two nuclei in their ground states when they are separated by a distance R,,.Here RI2 is the separation between nuclear center of masses and should not be confused with the generator coordinate R. It is ~,,(R,,) that is required to satisfy scattering boundary conditions. For large internuclear separation, Il/no(R,z) approaches an incident Coulomb wave plus a scattered wave. From the amplitude of the scattered wave in $ lO, one obtains the elastic scattering cross section. Generalization of these steps to include inelastic collisions will be discussed in sect. 4. The wave function for the relative motion of the nuclei (3.1) can be separated into long- and short-range functions of R,,;the long-range part is xi0

and the function ~~~~(R~~R~~}, whose properties will be discussed later, is a shortrange function of RI2and R;,.Therefore, $,,(R,,) approaches &(R~z) asymptotically and the amplitude of the scattered wave can be obtained directly from xn,(Rrz). The problem is to find x~,(R,,) which is related to xt,fA,CAand FAby (3.3) where

f In (3.1) a six-fold integration over the independent R12 and IL,,. are not integration variables.

variables

(& and &)

is performed, but

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It is this combination (3.4) off, and Fn that is needed to find the scattering amplitude t Eq. (3.3) can be viewed as a change in basis (sect. 4) or as a projection operation on fn. The basic question is: can one derive a unique equation for xi0 using scattering boundary conditions and the Hill-Wheeler eq. (2.24)? Eq. (2.24) does not contain the function xlo(R12) explicitly. However, one can rewrite the Hill-Wheeler equation in a simple, useful form by introducing the inverse of f;,(R-R,,). Let us define FL1 by

s

F,‘(S-R)F,(R-S’)dR

= 6(S-S’).

(3.5)

As a consequence of (3.5), one has F,‘(S-R)dRh,,(RIR’)dR’F,‘(R’-S)

s

=

(

- ++

Y)

@-S’)<&$#Jn>, (3.6)

s

F;O1(S-R)dRn~,,(R-R)dR’F;“(R’-S’)

= 6(S-S’)(~no~~i>.

(3.7)

Although the inverse FL ’ of a peaked function FA is a singular operator, FL 1 shall be applied only to functions that lead to bound quantities ++.Equivalently one can always make Fi ’ bounded by using box normalization in momentum space; for the infinite volume limit all of the results would then remain bounded. Applying the inverse Fi’ defined by (3.5) to the Hill-Wheeler equation (2.23) one obtains

- $*+

--R

2, Z2 e2

-En x&d

12

= -

s

U,1(R,zlR;,)jil(R;2)dldR;,.

(3.8)

Using the symmetries given in (2.24), one finds the effective interaction in (3.8) to be a non-local, ener~-dependent potential h(~,2lR;2)

=

~,~(R~~R;~)-E,K,~JRI~IR;~),

(3.9)

where vnl and KnB are ~td&2lR;2)

K’&2-

=

R)u,JR(R’)F;

‘(R’- R;,)dRdR’

J’

=

J~~~,(r,,2)~(R1~-Rt2)HI2(~~2,

S1r,)~E#,(5,,2)6(R;,-K,,lld51d42dR,,

[6(R,,-R;,)+6(R,,+R;,II, t A similar observation has been made by Horiuchi [ref. 12)]. tr See appendix 2 for an explicit example of how to construct the inverse operator Fi-‘.

(3.10)

506

F. TABAKIN

=JF,-‘( R,,-R)i?,I(RJR’)F,l(R’-R;,)dRdR’

Kni(RlZIRi2)

-

-R;,)f6(R,,+R;z)]. <#d(Pn>lI~(R12

(3.11)

In eqs. (3.X)-(3.1 1) the subscript A0 has been replaced by n in preparation for a later generalization of the discussion. The energy is thus also written as E,, = I!?-(&,~ ++). The above expressions for the effective potential are exact consequences of the assumed GCM wave function (2.1). From the microscopic Hamiltonian, the exact nuclear wave function $I~~, and the GCM wave function #J, one obtains an effective potential using (3.9~(3.11). Evaluation of (3.9)-(3.11) is clearly a difficult task and approximations are needed. Nevertheless, these expressions serve as a useful starting point for approximations; as derived they are free from c.m. and configuration mixing problems and include the expected reduced mass form for the kinetic-energy operator. Note that F,disappears from (3.10)-(3.11). H owever FJ is stiI1 contained in x2(3.4), but only in the combination f, FA. Any peaked function can be used for FA, which provides a freedom that is used later to simplify the evaluation of (3.10) and (3.11). Assuming U,,i’ is known, the scattering wave solution of (3.8) is given by the formal expression (3.12) where E, = (h2/2p)ki and ,u is the reduced mass of the two incident nuclei. A Coulomb potential is included in the definition (2.21) of r,, and Ikbf’) thus denotes an outgoing wave solution of [T,, -E,]fkb+‘> = 0. Eq. (3.12) is essentially the Lippmann-Schwinger equation for two-body scattering except for the presence of two wave functions xn, and x1 which are however related by (3.3). Eq. (3.12), together with the relation (3.3) between x1, and &, is called the Hill-Wheeler equation for scattering. Now the problem is to find the scattering amplitude from (3.12). A basic mathematical problem that is not answered in this paper is to prove that (3.12) has a unique solution. The appearance of both xnO and xn in (3.12), along with their projection relationship (3.3), does not obviously assure a unique solution. This mathematical difficulty is in fact a most significant one since it is a reflection of our desire to use generator coordinates to project our complicated problem into one of smaller dimensions. At this stage, we can onIy point out this basic dif%ulty and shall proceed on the arbitrary assumption that a unique solution exists. Another related aspect of (3.12) is the existence of superfluous solutions, wa(R,,), which have been discussed in the literature *r ‘). These solutions are called superfluous because they are solutions to (3.3) and (3.8), but lead to no observable effect after antisymmetrization. In the present case, it is the role of I& to project out super-

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the fluous solutions when one constructs $Ao(RIZ) using (3.2). Correspondingly, superfluous solutions are the eigensolutions of Knon of unit eigenvalue ‘) (q = 1) Y

s

K,,,(R,,IR;,)o,(R;,)dR;, = -(~,l~,>{w,(R,2)+0i.(-R12)}. t3.13)

Using (3.13) in (3.2) it follows that superfluous solutions wA lead to null GCM wave functions. The construction of mn(R,,) using (3.13) has recently been examined by Zaikine ‘). Practical reasons for projecting out superfluous solutions are discussed in ref. “); for example, one can minimize the contribution of difficult exchange terms. The role of o1 in establishing a short-range repulsive effect in LX-CI scattering has been discussed by Saito “). The similarity of this projection procedure to that discussed in the open-channels method “) suggests that the generator coordinate method is closely related to the resonating group method. That relationship is discussed in sect. 4.

4. The Lippmann-Schwinger 4.1. THE COUPLED-CHANNELS

equation in a biorthogonal basis

EQUATION

Clearly, the generator coordinate (GCM) wave function (2.1) and (2.7) is quite similar to the usual open channels or resonating group (RGM) wave function

(4-l) However, use of generator coordinates to describe changes in nuclear size, shape and orientation introduces a significant difference between these two approaches to nuclear scattering. Before discussing the significance of that difference, let us first illustrate how the RG and GC methods are related. The previous discussion can now be generalized by projecting the GCM wave function (2.1) onto excited internal nuclear states &(t,,,) = &,(<,)&,(g,). In addition to projecting with a fixed order of the nucleon coordinates r, let us also assume that the nuclear excitations are ordered by ~1~2 Q. In evaluation of the cross section one must then include the proper Pauli principle weights “). The discussion in sect. 3 can now be applied to inelastic scattering by simply replacing the ground state label 1, by the general nuclear state label n. Correspondingly, one finds the overlap xn(R12), which is defined by (3.1) with A,, -+ n, satisfies xn = Ikbf’)G,,+ Here En = E- (E,* +Q)

I

(En-T,,)-‘U&dL

(4.2)

and xn and Xn are related by

where xn is again given by (3.4). The elastic and inelastic

amplitudes

are obtained

by

508

F. TABAKIN

x0 and x,,> o, respectively. Of course x ,,> o gives only part of the total inelastic amplitude since breakup and particle exchange channels are omitted in the many-body wave functions (2.7) and (4.1). Using the expansion (2.15) of&r in a complete set, one obtains a large set of coupled integral equations

(4.4) where U,,,, is defined by u,,

= C U,,.

(4.5)

The coupled channels potential U,,, is thus given by (3.9)-(3.11) with 4I replaced by the exact nuclear wave functions &,1(51)&,2(<2) E &,,(<,,,). This expression for U,,, and the corresponding one for K,,, agree with those found in the cluster model “). Although (4.4) is a coupled-channels problem, it is subject to the constraint (4.3) where X(RIZ) satisfies (3.4). Thus the GCM leads to a constrained set of coupledchannels equations where the constraint is an expression of our ideas about nuclear shape, size and orientation during collision. The generator coordinate method serves therefore to select a subset of the complete coupled-channels problem, that subset corresponding to our prejudices about nuclear behaviour. That rather arbitrary procedure parallels the steps taken for bound state problems where one selects a suitable deformed basis for describing nuclear motion and hopes that it properly truncates the full Hilbert space problem. Both in the bound state and continuum truncations one must heed the Griffin-Wheeler warning ‘) that “In such problems, the method of generator coordinates provides no magic machinery that can be operated without the exercise of judgment”. Thus we see that the GCM is in a sense equivalent to a new choice of basis and a special truncation of the full open channels method. If a good choice of basis is made, perhaps corresponding to the actual nuclear motion, then one might hope for great simplifications. Our next step is to show that the GCM scattering problem is a COUpled-channels problem in a biorthogonal basis. 4.2. THE BIORTHOGONAL

BASIS

Let us assume without proof that the expansion of 4i(S,,2) in terms of the exact nuclear eigenstates (2.15) can be inverted; i.e. assume that &n defined by 4~ = C Im>(mP>~ m can be used to construct

the exact eigenstates q5, = /{,:,n)&dl

(4.6)

4, t.

(4.7)

t This is at least reasonable for the lowest few states which correspond to simple modes of motion.

However, a general proof of (4.7) is lacking.

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Similarly, assume that (4.3) can be inverted to express & in terms of xa

One can prove that the same transformation bracket {I.[m) is needed in (4.7) and (4.8). From the orthonormality of the exact eigenstates (nltn) = S,, and from (4.6~(4.8), it follows that fI.1 and In> form a biorthogonal set defined by the properties

s

IA){L/d,I = 1.

(4.9)

Using these properties, one finds an integral equation for &(R12) alone (4.10) Both the Green function Gdrl, and the effective potential Uin, in (4.10) are rather complicated operators G12, = (~f(T,,+H,fHz-E)-‘l~) = c {~ln>(nl(~z-En)-lln)(nll’},

(4.11)

= t: <4n>u?sn

(4.12)

n

u,,

Using the definition of U,,, (3.9~(3.11) along with (4.12), (4.6) and (4.7), one finds that UAi, is given by the short-range operator u.I,. =

[<+A c2~n~>+ - vC@, iJ<#44~~>1

xf~(R,,-R;*)+~(R,,+R;,)l

-(nl(H-E)la’>Cs(R,,-R;,)+S(R,,+R;,)].

(4.13)

The first term of U,,, is the direct, local contribution of the nuclear VN and Coulomb Y’(R,,, tI,Z) interactions in (2.12). The remaining exchange terms are non-local as a result of the antisymmetry requirement. The main part of the Coulomb interaction has already been included in r,, and hence (4.13) includes only short-range Coulomb changes, such as correcting for nuclear size and Pauli principle exchange effects. Clearly, to solve (4.10) and to calculate the scattering amplitude one needs to use approximate forms for G,,, and V,,.. An integral equation for the scattering amplitude will now be derived and then some useful approximations for GAr and Uhr, will be proposed.

510

4.3.

F. TABAKIN THE

Instead Coulomb

LIPPMANN-SCHWINGER

EQUATION

of solving (4.10) directly distorted waves

=

for xi, let us introduce

s

(n,n}(kr-),T,,,lkb+))d~.

a T-matrix

defined using

(4.14)

The distorted wave (ki-‘1 satisfies an incoming spherical wave boundary condition. Here (4.7) and (4.12) have been used to introduce the potential U,,,. From the onshell value of YnO, one obtains the amplitude of the elastic (n = 0) and inelastic (n > 0) outgoing waves of (4.2). Knowledge of Y,,,, thus enables one to calculate the scattering cross sections. For elastic scattering (n = 0), it is known that (O/n} = dlj,, and (4.14) simplifies to (k;-‘J&,lk~‘)

= (k;-‘JT,,,Jkb+‘).

Use of the integral equation for X2(4.10) and the definition an integral equation for (ki-‘jT,,,Jkb+‘)

(4.15) (4.14) of Tnj,O, leads to

This equation is simply the Lippmann-Schwinger integral equation for coupled-channels in the biorthogonal basis In), In}. The channels are here labeled by the generator coordinate variables 1. Note that (4.16) is an operator equation in a Hilbert space of Coulomb wave functions jk(+)) and (kc-‘1; TnA, thus gives the nuclear part of the scattering amplitude to which one must add the purely Coulomb amplitude

@,I WW)~,,. The main result is that the generator coordinate method is equivalent to a continuous set of coupled channels Lippmann-Schwinger equations formulated in a biorthogonal basis. That conclusion has been previously emphasized by Wong ‘) - the same result follows here for scattering. The expressions (4.14)-(4.16) for the elastic and inelastic scattering T-matrices are the main result of this paper. To solve (4.16) for Tnn, both the effective potential U,,, and the Green function GAi, are needed. The rather complicated energy-dependent, non-local interaction U,,. can perhaps be evaluated if simple, reasonable guesses for the parametrized wave functions 4A are used. Some guesses based on the independentparticle model are presented later. Also, to evaluate the inelastic amplitude, the biorthogonal states (Al must be constructed; here Wong’s suggestion ‘) of using the Schmidt orthogonalization procedure is probably of help. The integral equation for Tile, and the expressions for Grin,, U,,. and In} are all exact consequences of our generator coordinate formulation of the scattering of iden-

GENERATOR

COORDINATE

METHOD

511

tical nuclei. Consequently, they are as complicated as the original many-body problem. The difficulties have simply been isolated into the explicit evaluation of GznP, U,, and (_&I. Despite these awesome difficulties there is one important feature of the generator coordinate formulation that provides some hope. Namely, if the parametrized nuclear wave functions 42. are chosen to include the actual nuclear deformation effects, then soIving (4.16) n~erically might require only a small grid of J.-values. In that case the GCM would serve to reduce a large scale coupled-channels problem to a very much smaller coupled-channels problem. Also one might hope that simple approximations for Gnn, and U,,. suffice to obtain reasonably reliable estimates of the effects of deformations during collision. Finally it is hoped that practical computer codes for solving (4.16) using Coulomb distorted waves can be developed. 4.4. SOME APPROXIMATIONS

So far the discussion has been exact and consequently the scattering problem has only been redescribed. However, several approximations suggest themselves on the basis of the following physical interpretation of the Lippmann-Schwinger equation. One sees from iteration of (4.16) that the scattering is a sequence of transitions, generated by O;IAo,from an initial set of collective coordinates 2, to another set R. Higher terms, such as UGU, also describe the propagation of the nuclei by the Green function G,,,. As the nuclei propagated according to GAn,,they also undergo a change in collective coordinates from 1 to II’. Of course, for each value of /2 the shape, size, and orien~tion of both nuclei at that stage of the collision are described. One natural approximation is to assume that the nuclei are rigid and experience no alteration in shape, size, or orientation during the collision. The initial value 1, is then maintained and the rigid-body approximation is G11 = &“J911

(4.17)

with gn = &(TIz-E,?

(4.18)

Then the T-matrix equation simplifies to TAi, = Silo t,, with tao = LJlolo+ Ulolo g,l*t3.0*

(4.19)

The rigid-body or no polarization approximation (4.17)-(4.19) has often been used, especially for ct-crscattering I*). Another approximation suggested by (4.16) is to assume that only U,,, generates changes in il, whereas no shape, size or orientation changes occur during propagation. That assumption is expressed by taking (4.20) along with (4.17). Here dalo is the energy gain (or loss) acquired by changing the collective coordinates from il, to ,I; it is approximately given by Allo z EIZ--eO, where

F. TABAKIN

512

EAcan be evaluated from En = (LIH, +H,IA)

(4.21)

B E&+E&,

and ~~~= <91lHll+n>, % = (+121HzlQ,). Knowledge of the nuclear compressibility and/or of the nuclear energy as a function of the collective variables, can be used to evaluate An*,. A shell model estimate for A lrlo will be presented later (5.20). Using (4.17) and (4.20), our approximate T-matrix is (4.22) The difference Tnn, -Bllo tl, is a measure of how deformations during the collision effect the scattering cross sections. In sect. 5, further approximations are suggested that might be used to evaluate these deformation effects. 5. The independent-particle model In sect. 4 it was shown that the generator coordinate method for scattering is equivalent to solving a Lippmann-Schwinger equation in a biorthogonal basis. To make the GCM a practical scheme for calculating cross sections, some approximate forms for the Green function GAA,where proposed. Now the independent particle model (IPM) is introduced as a practical way of evaluating the effective potential U,,. from (4.13). The IPM wave function for the scattering of two nuclei is constructed from the orbitals of a double-centered single-particle potential “). For simplicity, this potential is taken to be harmonic oscillators centered at ++R. Also the present discussion is restricted to a simple model for a-u scattering. The corresponding IPM choice for QiRlis then Q&r) = C,(R)cQZC&,(~, -3R). = WW”(Rcb+h(t~

+ . A,@, +3R) . . .I 2 ~JF,@Iz

(5.1)

- R)l.

Here 1s orbitals centered at ++R are used with 4,Jri) = r:)fe-iirl’lspin-isospin)i, and I, = 1, = (2b2)-l = Mo/2ti. The generator coordinates are thus related to the oscillator size parameter b and are here being used to induce nuclear size changes only. To include shape and orientation changes one could use a non-spherically symmetric oscillator and take 1, # i, # 1, as the corresponding generator coordinates in (5.2). Assuming a (1~)~ configuration for the a-particle, the rms radius of the x-particle is used to fix the asymptotic value of the generator coordinate 2, = 9/16(r2>) [ref. ‘“)I.

(t’(r2)

= zb and

GENERATOR

COORDINATE

Introduction of internal (5r, t,), relative (RJ to the identification

513

METHOD

and cm. (R,) coordinates leads one

For the IPM case the internal wave function is simply

where Fi are the internal nucleon coordinates (2.8). The corresponding normalization factor is given by CA(R) = (1 -e-aRZ)-2. The above forms apply to the case of two a-particles in a (l~)‘fR(ls)?*R configuration. As R approaches zero, however, the normalization factor CA(R) approaches infinity, whereas the antisymmetric product of orbitals in (5.1) simultaneously vanishes. The proper limit leads to the expected development of extra nodes as R -+0 and to the evolution of a (l~)~(lp)~ configuration for eight nucleons in a common oscillator “). Thus the normali~tion factor C,(R) plays the important role of assuring satisfaction of the Pauli principle for all values of R.Note that the weighting function f,(R)must vanish faster than R2 as R + 0 in order to yield a well-defined fl(R,,) in (3.4). To use the IPM wave function, it is convenient to express U,, directly in terms of the many-body matrix elements (@PLRIHl@lSR,)and (@,,/G1,,,). lnserting the c.m. function P”(R,) into (4.13), one finds that Udl, is given by +

uiliz’

=

s

F, ‘(R,

2

-

R}C, '(R)

<@%RIH-Ed@?.‘R’) ,-l(R’)F-1(RI_R 1’

)dRdR’

I’

12

i-W'1

Here we have defined [,%I,%‘] by [WI

= jkR.)F”(K)dR,

2&W = (=)

*

3

(5.7)

and Ear = Ef [AlA’]- ’ [AlT_,, [A’]. Eq. (5.6) IS . exact and can be used with any manybody wave function GnR. Eq. (5.4) is equivalent to the earlier expression (4.13) for UA‘X9but (5.6) is expressed directly in terms of many-body matrix elements of the microscopic Hamiltonian H. Fortunately, using an IPM wave function for $rlR, the many-body matrix elements of H can be readily evaluated. Expressions for the matrix + The matrix element in (5.6) involves an eight-fold and R',,.

integration.

Also tJax is a function of RI2

514

F. TABAKIN

elements

of one- and two-body

operators

orbitals are given in ref. “). A two-nucleon central potential

using Slater determinants

of Gaussian

form is adopted

of non-orthogonal for convenience

V = u(r)( W + BP, - HP, - MP, P,),

(5.8)

u(r) = ~~e-~*‘~~, with the usual

exchange

operators.

and C~‘(R)(~PIRI~~,R,)C~l(R’)

The result

for

= h,,@IR’)

+

C,-l(R)(~~,IHIQj,,,,)C~‘(R’)

is

c,; ‘(R)(~~,IH~~~,,)C,‘(R’) c,‘(R)(~,,I~~,,,)C,‘(R’)

h,:@IR’) + t&W’),

= n,,(R-R’)+&(R+R’)+

(5.9)

&,(RIR’)

= [n~~~(R-R’)-n~x(R+R’)]4, where ij and R are given in appendix

1. The long-range

+12A,,,(W+M)+2J(rl+d’)e2J2/n+

terms hlns and nAl, are

zll~~~~~:“erf

In (5.11) and (5.12) erf(x) is the error function defined

(5.10)

(l~~p,‘3).

and the following

quantities

(5.12) have been

A, = 2nn’/n+n’, A II = uoa;l,, a;,. It is possible to apply the inverse result of that inversion one finds v’,, = vA1,- EK,,.

= pz/(pz + k/M’). operators

u~,,(R,~IR;~)-EK~~~,(R,~IR;~)+ =

(5.13)

Fy ‘Fil

in (5.6) to IZ~~,and hAA,; as a

T/,“,,(R,,)CG(R,,-R;,)+6(R,,+R;,)l,

1

F,‘(R,,--R)

F-‘(R’-R;,)dRdR’.

(5.14)

As part of the independent-particle model used to derive (5.14), the following mate for (J&Y, + H21A’) has been adopted <~(H,+If,lI’)

= (AIL’)

9$i,+l2A,,,(W+M)+Z,Z,e’

esti-

. (5.15)

GENERATOR

COORDINATE

515

METHOD

The term V&(.ZZ,,) in (5.14) is a short-range ~nodification of the Coulomb potential which accounts for the finite size of the nucleus (5.16)

(AlA’)

When the g-particles overlap, a considerable decrease in the Coulomb potentials occurs as a consequence of the short-range Coulomb correction V&. Eq. (5.16) also includes Coulomb distortion effects since iJ,,, induces transitions from R to X. For the case of rigid cr-particles 2 = I’ = do, eq. (5.15) has a simple interpretation; namely, it gives twice the a-particle energy

Here the average kinetic energy (3hw) is corrected for the c.m. kinetic energy (jho), and the average potential energy consists of 6 nuclear bonds plus the average Coulomb repulsion. To complete the evaluation of UnX, one must apply FI’Fi’ to the functions Gnz and KA1*(see appendix 1). Th at t as k IS . st raightforward but tedious and therefore only an outline of the inversion procedure and some typical results will be presented. For example, application of FL ‘FG ’ to R,,. 12

(6e- ~,(RZ+(R’Z)_4e-fl,(RZ+R’2)[e-~I,(R+R’~Z+e-i~.(R--R’)Z]~, (5.18)

can be completed immediately using the inversion relations derived in appendix 2. The corresponding term in (5.14) is found to be {6ex

e-(10/3)(R,z++R~z’Zfdl’jZ,

2(A’Rd+1R12’2)

fe-

(16?.1’/3A,)R,2

-4($)+e * R,z’

-(RIz*-R,2’*)(I’--I)

+e

+(16A1’/3&.)R~z~

Riz’ 11.

1w

This function is of particular interest because of (3.13), where it was pointed out that 1 -K,, projects out superfluous solutions. In fact, for 1 = I’ (5.19) reduces exactly to Saito’s result and therefore his discussion of superfluous solutions applies here “). The application of the inverse operators Fi IF;;’ to all of the terms in ij,, can be handled by the procedure described in appendix 2. The full result for VAXis rather complicated and will be omitted. It is clear, however, that U,, can be evaluated using the GCM inversion procedure and the IPM. Knowledge of the effective potential is significant since U,,,provides one with some idea of the nuclear size changes that are induced by the basic two-body nuclear and Coulomb interactions. For the special case of rigid-body scattering, the effective potential U;lczOwas evaluated using the above inversion procedure. The result is given in appendix 3; it proves to be exactly the result derived by van der Spuy ‘l), which has often been used in RG calculations of U-M.scattering ‘* ’ “). Thus the GCM proves to be identical

516

F. TABAKIN

to the RGM for the special case of rigid a-particles that are represented by a single Slater determinant. That result is not surprising in view of the discussion in sect. 4. Therefore it is only for non-rigid-body scattering that one makes full use of the special feature provided by the GCM. Namely, in the GCM a semiclassical description of collective nuclear motion is introduced by defining suitable generator coordinates. Then, instead of solving a large set of coupled channels eq. (4.4) it is necessary to solve the Lippmann-Schwinger (4.16) in a smaller biorthogonal basis. Of course, bold approximations are needed. In this section the IPM has been used to evaluate the effective potential U,,. Solving the integral equation (4.16) can be further facilitated by using the approximations (4.20)-(4.21) along with an IPM estimate for the compressibility factor Allo

6. Conclusion

A generator coordinate description of the collective motion of colliding nuclei has been presented. Proper scattering boundary conditions for the Hill-Wheeler integral equation have b:en defined and used to derive a set of coupled eqs. (4.2). The coupled equations are similar to those found in the open channels or resonating group method, but differ in one important respect. Namely, the GCM coupled channels are subject to a constraint (4.3); that constraint is a reflection of our prejudices concerning changes in nuclear size, shape and orientation during the collision. Thus the GCM is a means of inserting classical ideas about nuclear collective motion into a microscopic theory. In a sense, a special truncation of the full coupled-channels problem is made by defining generator coordinates. If that truncation provides an adequate description of the actual nuclear motion, then the GCM provides an important reduction in the number of coupled equations. In sect. 4, the constrained, coupled equations have also been described as coupled Lippmann-Schwinger equations in a biorthogonal basis - in a manner similar to Wong’s discussion ‘). Proof of the existence of that biorthogonal basis has not been given in our paper. Despite this serious omission, the assumption that one can construct a biorthogonal basis (4.9) has suggested some possibly useful approximations. The collision is viewed as a sequence of siz e, shape and orientation charges induced by an effective potential U,,. The effective potential is a complicated, non-local, energy-dependent function that can be exactly expressed in terms of the microscopic Hamiltonian and the nuclear eigenstates (4.13) and (5.6). An exact expression for the nuclear propagator in a biorthogonal basis is also found (4.11). The corresponding kinetic energy is simply - (h2/2p)V:2 where ~6is the reduced mass. The cm. motion is also treated properly in the exact version. Approximations are made for the nuclear

GENERATOR

COORDINATE

METHOD

517

propagator GnZ based on the ideas of rigid and non-rigid nuclei. The effective potential has been evaluated using a single Slater determinant to represent a-& scattering. For rigid tl-particles, it is explicitly shown (in agreement with the conclusions of Giraud “) and Horiuchi ’ 2) that the GCM reduces to the often-used RGM. However, for non-rigid a-particles the effects of size changes during the scattering can be evaluated in the GCM since the corresponding effective potential U,,. can be constructed. That construction procedure is described in sect. 5 using the independent particle model. A generalization of this paper to include shape and orientation charges can be made by considering the IPM in a non-spherically symmetric well with J., # ;1, # 1,. The effective potential UA,, can then be used for a-a scattering. However, it is more important to consider other generalizations. First, one should extend the original ansatz (2.1) to include break-up and other possible channels. Also, the basic question of constructing a GCM effective potential for heavy, non-rigid and deformed nuclei has not been treated fully. Numeri~l methods for sofving (4.22) with Coulomb distorted waves are required, along with methods for constructing the biorthogonal basis ]A}. Finally, the evaluation of the nuclear eigenstates 4, and, correspondingly, of U,,, could perhaps be based on microscopic Brueckner calculations [a GCM Brueckner approach has been recently suggested by Bando, Nagata and Yamamoto ‘“)]_ Indeed the generator coordinate method does not provide magic machinery for solving the complicated problem of interacting many-body systems. However, if generator coordinates are defined which approximate the actual nuclear motion then one might hope for considerable simplifications. The author wishes to thank Professor Sir R. E. Peierls for his hospitality at the Department of Theoretical Physics, Oxford University, where this work was initiated. Stimulating and encouraging conversations with Dr. D. M. Brink are gratefully acknowledged. The author also appreciates the valuable insights provided by Professors N. Austern, R. Drisko, M. Vincent and I. Unna.

Appendix 1

For CI-c(scattering the matrix elements for one- and two-body operators. Knowing these many-body matrix elements, one simply uses (5.9) and (5.10) to determine the functions pAit and r?,, . The results are

+same with R

4

-R,

(A.l)

F. TABAKIN

518

-2(X,+

&)A&‘;)

-t2&,(n? +same

- ntj2

- #‘+>%I”;nP ( exp (-S exp (_$

with R --, -I?+

The quantities

~~~~j2j

Coulomb

(g$j’) (xgi

+exP (-s +x&

($j2j]

Izi;)

term,

(A-2)

A,, Anlp and al.ns are defined in (5.13) and s, Xo and _I’, are s s (anL#)“, x1, = 8W+4B-4ff--2M,

(A.9

XE = 8Mi-4H-4B-2W In (A.2)+ denotes PzJR+R’), which is given by (5.11). Coulomb contribution to (A.2) is

Finally,

the short-range

-I-same with R -+ -R, for ~-LXscattering where I(p) = Iplerf((pl) and erf(p) is the error function; 2, = 2, Both ELI, and p’LT are short-ranged functions of R and R’,

Appendix The inversion known function

procedure di,,, by

consists

of finding

2,

=

2 a function

sZAz,,that is related

ta a

Here b-1z, refers to either ;j,.A,or I?,, (5.9) and (S.lO>, which are given in appendix 1. For X-J scattering, the function Fi. is given by (5.4). Eq. (B. I), which has the formal solution tinnS = Fc’fi,,>FG I, can also be expressed in momentum space

GENERATOR

COORDINATE

METHOD

519

where (B.3) Note

that

FL(k) cc JnAl(k),

where

n,,(k)

is the

corresponding

transform

of

%.L(R - R’). The steps needed to find a,,, are first to find fi,,,(klk’), then divide it by F#c)F&‘) and finally transform Q(k1.k’) to configuration space. For the nuclear terms given in appendix 1, these steps can be completed analytically. For example, a typical term in B,,, is d,,, The result is found

= e -.A.(R-R’)2

e-bA,(R+R’)z (B.4)

to be

_

(R+R’)2A,b_ (R-R’)21,a _ (I-

b)t

(l-a)LE

(R2-R’2)I,(I’-1) U’(l -a)(1

ub 5 = l-

- b)

“b

,

(

tw

(l-u)(l-b)

which is valid for 5 > 0. Other terms in i? and K can be treated by taking derivatives of (B. 5) or simply repeating the steps discussed. In general, one might need to solve (B.l) numerically. Fortunately, for the IPM wave functions used here an explicit solution is found and one can construct the effective potential UAn, directly from the many-body matrix elements.

Appendix 3 For the special case of rigid a-particles (2 = 1’ = A,), the effective potential ULo10 has been constructed using the procedures described in appendix 2. The result is Uloj,o = [VD(R,2)+V~(R,2)]G(R,2-R~2)+vo-~Ko+sa~e

terms with RI, --t -RI,,

where the local potentials

(C.1)

are

(9 (C-3) Here A0 = ~~/3~(5.13); In

and ZE are defined

in (A.3).

F. TABAKIN

520

The functions

K, and vO are

- -

(10/3)lo(R122+R1z’2)e-(16/3)~.~R1z~

4($)+e

RIZ’ >,

(C.4)

~~~We-~~013~~olR~~'+R~~~z~(~~~~~~o(~12_R;o~1z~~ +[~RO(R12-R~2)2-~~]e~‘16’3’RoR12’R’2’)+(~D+~~)~l+~T;D~2+~~u3.

The functions

X

(

exP

G 5)

ul, ut and v3 are found to be

-4A pR:2 ( 3/T2A+2

The effective potential

+exp 1

-41 r2 -~ ( 3/12A+2 (R ‘* 9

given by (C. I)-(C,

(C.6)

’

7) is identical

to the x-01 potential

given by

GENERATOR

COORDINATE

METHOD

521

van der Spuy ‘I), where short range Coulomb terms have been ignored. Therefore the GCM for rigid cc-particles should produce the same phase shifts as in the RGM; that conclusion is confirmed by the numerical results of Horiuchi 12).

References 1) D. L. Hill and J. A. Wheeler, Phys. Rev. 89 (1952) 1102 2) J. J. Griffin and J. A. Wheeler, Phys. Rev. 108 (1957) 312; R. E. Peierls and J. Yoccoz, Proc. Phys. Sot. A70 (1957) 381; R. E. Peierls and D. J. Thouless, Nucl. Phys. 38 (1962) 154; F. Villars, Int. School of Physics Enrico Fermi, course 36, ed. C. Bloch (Academic Press, New York, 1966) 3) D. M. Brink, Orsay Lectures, 1969; Int. School of Physics Enrico Fermi, course 36, ed. C. Bloch (Academic Press, New York, 1966) 4) D. M. Brink and A. Weigung, Nucl. Phys. A120 (1968) 59; B. Giraud, J. LeTourneux and S. K. M. Wong, Phys. Lett. 32B (1970) 23; M. V. Mihailovic, E. Kujawski and J. Lesjak, Nucl. Phys. A161 (1971) 252 5) B. Giraud, private communication and unpublished manuscript, 1967; P. Bonche and B. Giraud, Microscopic form factors for transfer between heavy ions, Symp. on nuclear physics, Novosibirsk, USSR, 1970; B. Giraud, J. C. Hocquenghen et A. Lumbroso, Coordonnees gentratrices dans les &tats du continu, Nuclear Reaction Conf., La Toussuire, France, fev. 1967; D. A. Zaikine, Principe de Pauli dans la theorie des reactions avec les noyaux l&gem, Nuclear Reaction Conf., La Toussuire, France, f&v. 1971 6) J. A. Wheeler, Phys. Rev. 52 (1937) 1083 and 1107; K. Wildermuth and W. McClure, Springer Tracts in Modern Physics, vol. 41, Cluster representations of nuclei (Springer Verlag, Berlin, 1966) 7) C. W. Wong, Nucl. Phys. Al47 (1970) 545 and 563 8) N. Austern, Direct nuclear reaction theories (Wiley-Interscience, New York, 1970) 9) S. Saito, Prog. Theor. Phys. 41 (1969) 70.5; H. Feshbach, Ann. of Phys. 19 (1962) 287; R. Tamagaki, Prog. Theor. Phys. Suppl. extra number (1968) 242 10) D. R. Thompson and Y. C. Tang, Nucl. Phys. A106 (1968) 591; Y. C. Tang, Few-nucleon systems and effective interaction of clusters, in Int. Conf. on clustering phenomena in nuclei, Bochum, IAEA, Vienna, 1969; R. E. Brown and Y. C. Tang, Phys. Rev. 176 (1968) 1235 11) E. van der Spuy, Nucl. Phys. 11 (1959) 615 12) H. Horiuchi, Prog. Theor. Phys. 43 (1970) 375 13) H. Bando, S. Nagata and Y. Yamamoto, preprint, Kyoto University, 1971