THE GENERATOR COORDINATE METHOD FOR ... - Science Direct

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l.D.2: 2.L

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Nuclear Physics

A182 (1972) 497-521;

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THE GENERATOR

COORDINATE

@ North-Holland

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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 &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’), (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)]. 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,,,) = &,( 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(~ 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: 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,. =

[+ - vC@, iJ1

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-

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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 ~~~= , % = (+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

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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)

, 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

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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

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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).

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