Nuclear Physics A548 (1992) 39-48North-Holland

Elimination of Pauli resonances in the generator-coordinatedescription of scattering

M. Kruglanski' and D. BayePhysique Nucléaire Théorique et Physique Mathématique, C.P. 229, Université Libre de Bruxelles,

B-1050 Brussels, Belgium

Received 30 March 1992

UCLEPHYSICS A

Abstract : Microscopic descriptions of elastic scattering in a cluster model with different oscillatorparameters lead to the occurrence of resonances without definite physical meaning, related to thealmost forbidden states, known as Pauli resonances . In the generator-coordinate method, a simplebasis change allows one to obtain a microscopic model free of such resonances. This technique isillustrated on a +'60 scattering . Beyond the resonance region, phase shifts change very weakly .On the contrary, bound-state and physical-resonance energies may significantly be modified . Themodel seems to be physically more consistent when the almost forbidden states are eliminated .

1 . Introduction

The microscopic cluster model 1,2) aims at providing a unified description of lightnuclei and of their reactions. Each improvement of the cluster description enlargesthe model space and should lead to an improvement of the physical properties ofthe global system . This expectation is, however, not fully encountered when themodel is applied to elastic scattering . A remarkable agreement with experiment isobtained in the description of a-scattering on `60 and 4°Ca [refs. 3,4)]. The micro-scopic phase shifts are very similar to those given by the real part of accuratephenomenological potentials 5,6) . However, these phase shifts are obtained in thesimplest version of the model, in which all clusters are described by harmonic-oscillator wave functions with the same oscillator parameter. If different parametersare employed for the clusters, the microscopic results are contaminated by theoccurrence of many resonances, without obvious physical counterpart, known asPauli resonances'' ).The existence of narrow resonances above the Coulomb barrier in microscopic

calculations with different oscillator parameters is known for a long time g), as wellas their extreme sensitivity to numerical accuracy. Although their physical interpreta-tion remained controversial 9- " ), their relation with the forbidden states appearing

Correspondence to : Dr. D . Baye, Physique Nucléaire Theorique et Mathématique, Campus de LaPlaine, CP 229, B-1050 Bruxelles, Belgium .

' Boursier IRSIA .

0375-9474/92/$05 .00 © 1992 - Elsevier Science Publishers B.V . All rights reserved

40

o ic-oscillator cluster model with a single parameter was soon estab-e attitude towards these resonances ranged from ignoring them as they

ear when cannel coupling or absorption are introduced s) to analyzing theirin details 9,") and, to this end, trying to describe them accurately 7,12).

ow, the picture about

auli resonances is clear. These resonances arise fromst forbidden by the Pauli principle which would exactly vanish in the

ara eter model.

eappearance ofthese states as resonances (or, sometimes,states) is due to a limitation of the model space to a single or a few

rations. They are poor approximations of cluster configurations which arenot included in the model space. They play a crucial role in providing a realisticehaviour o the phase shifts with respect to the Levinson theorem 9.12) . The fact

uli resonances are well understood does not prevent them from polluting thescopic results.

e accurate phase shifts of ref. 7) for the a +'60 system showlicate pattern whose physical signification is hard to deduce. Any com-wit

phase shifts from another model is precluded above 8 MeV becauseccurrence o

any resonances without real physical signification .e aim of the present work is to develop a practical method for removing Pauli

resonan,Les from the microscopic model. In the generator-coordinate method) 0,'4), the removal is very simple and provides smooth and easy-to-interpret

ase shifts.

e technique of elimination of Pauli resonances is defined in sect . 2.n application to a + °®

scattering is described in sect. 3. Concluding remarks areresented in sect.

stsinasc

tes alle

n

thaticrCo

pariso the

a

glanski and bye / Elimination of Pauli resonances

au i resonances

et us denote as

; ( ;) translation-invariant antisymmetric cluster functionsinvolving

; nucleons, defined in the harmonic-oscillator model with oscillatorparameter ,. Ifwe assume for simplicity that the clusters have zero spin, a two-clusterwave function can be written in the notations of the resonating--'1p method ''21

) asrm=

Ol(bl)02(b2'Y7'(~~igl(U)

(1)

eq . (),

is the antisymmetrization projector ofA= A, +A, nucleons, p = (P, p1P )is the relative coordinate between the cluster centres of mass and 1 is the oïtiialmomentum .

e relative function g' must be derived from a microscoj_-4c hamil-tonian. This is obtained by expanding

'"' on the basis functions"(b,

) = .W

l(bl)4b2(b2)

i'(

p )ri(lu, bq P,

)(2)

were P, is the reduced mass number and l', is a partial wave of a displaced gaussianfunction with parameter IA -'/ Zb [see eq. (7) of ref. "j]. The parameter R in eq. (2)is called generator coordinate in the following. As shown in refs . '5,'2), 0'' can beobtained after several integral transforms from a Slater determinant O(S) defined

the two-centre harmonic-oscillator model with different oscillator parameters for

M. Kruglanski and D. Baye / Elimination of Pauli resonances

41

each cluster (S is the distance between the centres). The successive transformationscorrespond respectively to eliminating the c.m. motion of the Slater determinant,modifying the width of thedisplaced gaussian and projecting on angular momentum.Let us recall why and when a width transformation is necessary. In calculationsinvolving distortion, it is essential that the parameter b be decoupled from b, andb2 , because different sets of b, and b2 values occur. This decoupling is obtained atthe cost ofadouble integral transform in the matrix elements ' 5,12) (see theappendix).However, since the c.m. projection already requires an analytical treatment of theGCM kernels, this decoupling does not introduce much additional complication .Here, distortion is not considered but the width transformation offers the possibilityof choosing the parameter b in an optimal way for numerical calculations (seesect. 3) .

Let us calculate the eigenvalues of the RGMnorm kernel in the GCM framework,as in ref. ' 6 ) . The definitions

with

The eigenvalues of eq. (5) approximate the eigenvalues of the RGM norm kernelto an excellent accuracy"2). In the equal-parameter case, m, eigenvalues are exactlyzero and correspond to forbidden states . For different oscillator parameters, the m,first eigenvalues are much smaller than the other ones, but larger than zero . Theycorrespond to almost forbidden states, which are responsible for the occurrence ofPauli resonances in the phase shifts .Now, we can define the orthonormal basis 9' 12 )

4 nm - (11

n) -'/2

Cnk0 ,m (Rk) - (lun)-1/2~0102Yin (f2p)Xn(P) ,k

Xn(P) - 1 cnkr!(l1 -1/2b,P, Rk).

(7)k

The elimination of the Pauli resonances is easily obtained by working with the basisnstates ~n

restricted to n > mi. In the resulting calculation, only states which arephysically significant for a scattering process are taken into account and serve asbasis states . In ref.' 2), we have shown that the ~nm with n--<m, are useful forlocal zing the Pauli resonances . Here we observe that the other states are even muchmore interesting since they constitute a simple and convenient basis for GCMcalculations without almost forbidden states .The new basis is composed of states which are exactly orthogonal to the almost

forbidden states . The Pauli resonances are therefore avoided. In some sense, the

N1(Ri, Rj ) =(0"(Ri)10"(Rj)i , (3)

n'(Ri, R; ) _ (ri(M--1/2b, P, Ri)!ri( . -1/2b,P,

R1»,(4)

lead to the generalized eigenvalue problem

(N`(Ri, Rj ) ju-'nn'(Ri, Rj »cl; =0 . (5)

in lanski and

Baye / Elimination of Pauli resonances

et with the restricted basis contains exactly forbidden states and obeys thereforethe extended version of the Levinson theorem "). In practice, the restriction to the>

9 states is easy to introduce and reduces the size of the matrices to be handledin the

The technique of Pauli-resonance elimination is applied to a +'60 scatteringunder exactly the same conditions of calculation as in ref. 7 ). This choice allows usto check the C calculation and to show the usefulness of phase shifts which arefree of Pauli resonances.

e hamiltonian and overlap GC

kernels are calculated as explained in theappendix for b,, =1 .395 fm and b® =1 .766 fm, and for selected values Ri of thegenerator coordinate appearing in eq. (2). The two-gaussian Minnesota interactionis employed as in ref. 7 ) where u = 0.812 is chosen in order to reproduce the 20Nebinding energy. The microscopic R-matrix method "") then provides the phaseshifts . e location and width of narrow resonances are obtained with an R-matrixiteration technique ")

First, let us discuss the choice of the generator-coordinate values Ri and of therelative-motion parameter b. To this end, let us recall the situation in the simplestversion of the single-parameter model which does not require any c.m. eliminationand can therefore be treated in a purely numerical way '4). In that case, therelative-motion parameter takes the same value as the common internal parameter.ne spacing AR between the generator coordinates must be small enough so that

the basis can follow the oscillations of the wave function up to the highest energyof interest 20). It is therefore proportional to the highest wavenumber considered.In general, a constant spacing is employed but a small spacing is not so useful forsmall values of the generator coordinate because the existence of exactly forbiddenstates reduces the importance of the corresponding basis states. Let us illustrate thespacing choice by a realistic example. For b, = 62=1 .62 fm, a spacing of 0.8 fmprovides fair phase shifts up to about 40 MeV. If ®R is reduced to 0.6 fm, thevalidity domain is extended to about 70 MeV. An optimization of the relative-motionparameter b (which is 1 .62 fm here) might enlarge the domain where the phaseshifts are accurate, for a given spacing ®R, but would significantly complicate thesingle-parameter model.

n the model with different oscillator parameters, the simplicity of the equal-parameter case is completely lost but the parameter b is available for optimization .

e choice ofthe spacing dR is now constrained by several requirements . Therefore,it is convenient to employ different spacings in different regions. Let us considerthem separately . (i) As before the spacing AR, for large values of the generatorcoordinates depends on the highest energy of interest 2°). (ii) The spacing ®R, forsmall

relies on the accuracy required in the description of Pauli resonances . To

C

Application to a+'6scattering

0

M. Kruglanski and D. Baye / Elimination of Pauli resonances

43

illustrate the difficulties of this choice, several cases are exemplified in fig. 1 . Thedifferent curves are labelled by the spacing couple (4R, 4R,), the transitionbetween 4R, and 4R, being in all cases located at 3.8 fm. The smallest and largestvalues of the generator coordinate are respectively 0.2 and 11 fm. In order to avoidany problems with Pauli resonances at this step, we only consider energies largerthan 50 MeV, i.e . higher than the Pauli resonance domain for a +'6O. For the sakeof simplicity, we employ the same parameter b for all generator-coordinate values.For 4R, = 0.8 fm, eq. (23) of ref. '2) favours b =1.4 fm. However, the case (0.8, 0.8)provides very poor results beyond 50 MeV. These b and ®R, values are too large.For 4R, = 0.6 but still with b =1 .4, the phase shifts are underestimated by about67r (see fig. 1) . The situation improves for b =1.1 fm. Then 4R, = 0.6 or 0.4 fmprovides phase shifts which remain satisfactory up to 60 MeV. The larger validitydomain than in the equal-parameter model is due to the use of an optimal b. Theinner spacing ®R, must be small enough (-0.6 fm) in order to reproduce theproperties of the wave functions at small distances, i .e . correctly describe the Pauliresonances to which they are orthogonal. However, reducing AR< from 0.6 to 0.4 fmdoes not provide a significant gain . Employing a smaller 4R, should provideaccurate phase shifts up to much higher energies . The case (0.6, 0.6) shows theopposite . noimprovement is obtained becausethe description ofthe Pauli resonancesis not accurate enough. This is shown by the (0.4, 0.6) curve which remains accurateup to 140 MeV. Changing the relative parameter b does not modify much the resultsbelow that energy. Fig. 1 shows that a good balance must be kept between ®R{ ,

E (MeV)

Fig. 1 . 1= 0 and l =1 phase shifts beyond the Pauli-resonance region for different choices of the spacingcouple (AR,, 4R,). The relative oscillator parameter b is 1 .1 (full lines), 1.2 (dotted line) or 1 .4 fm

(dashed lines) . For the sake of clarity, the I =1 phase shifts are shifted downwards by 7r.

44 Kruglanski and D. Baye / Elimination ofPauli resonances

eaching higher energies requires a smaller AR, to follow the outerscillations of the wave function and a small enough AR.-- to describe accurately

its inner oscilations. In the following, we choose (U.4, 0.6) with 1) =1.2 fm.he GCM phase shifts of the partial waves which display Pauli resonances areresented by full lines in fig. 2. The zero-energy phase shifts are chosen equal to

es the number of bound states, in agreement with the usual Levinson theorem.e phase shifts exhibit a complicated pattern whose detail is physically unimportant

except for three aspects. (i) The accuracy of the results is confirmed by the closeagreement with fig. I of Walliser et al.) obtained with the same interaction . Noticehowever the occurrence near 26 MeV of a narrow 1=1 Pauli resonance which ismissing in ref.'). (ii) From I = 0 to 7, each phase shift displays one or several Pauliresonances beyond the barrier resonance. As expected, their number is equal to thenumber m, of forbidden states in the equal-parameter case, i.e . 48 -1) for even 12

or ;(9 - 1) for odd L (iii) Above the last 0' Pauli resonance near 41 MeV, the phaseshifts are regularly ordered as they would be in potential scattering.

In order to suppress the Pauli resonances, we now turn to a basis where the statescorresponding to small eigenvalues are eliminated. The eliminated norm eigenvaluesam displayed in the first four lines of table 1. Larger eigenvalues resemble, for eachparity, the corresponding eigenvalues of the equal-parameter case 2 ' ) . The dashedlines in fig. 2 represent phase shifts after elimination of the Pauli resonances (I -,-- 7)or without such resonances (I >-- 8) . At 50 MeV, they change by 0.4" for I = 0 andless than 0.2" for the other /-values . At higher energies, the differences soon become

reV ti

10 20 30 40 50

E (Mev)

Fig . 2. Accurate GCM phase shifts of a +'60 scattering with (full curves) or without (dashed curves)Pauli resonances, calculated with the Minnesota interaction for u = 0.812 .

M. Kruglanski and D. Baye / Elimination of Pauli resonances

TABLE 1

Lowest eigenvalues W � of the a +'60 RGM norm kernel calculated with eq . (5) for bQ =1 .395 fm andbo=1.766 fm

45

negligible . Above the resonances the phase shifts are almost not modified by theelimination process. However, we have seen in fig. 1 that their accuracy relies on agood description of the wave functions of the Pauli resonances .The phase shifts now show a simple and regular pattern, with broad barrier

resonances for even waves and narrow ones forodd waves. They follow the modifiedversion of the Levinson theorem ") . They can be employed for a comparison withthe potential model 5). The broad barrier resonances qualitatively resemble thecorresponding resonances before elimination. However, a detailed comparison for1= 0, 2 or 4 shows that their shape is modified by the suppression of the Pauliresonances . More surprisingly, the location of the 4+ resonance below the barrieris modified by more than one MeV. This shift deserves more attention .

In table 2, we display the energies of the 20Ne bound states and narrow resonancescalculated in a bound-state approximation. The ground-state energy does not per-fectly reproduce the experimental energy here [in ref. 7), a finite-size Coulombpotential is employed]. The energies of the positive-parity states are all significantlyaffected by the elimination process. This is an effect of the variational principle.The almost forbidden states simulate configurations which are missing in the model

TABLE 2Bound-state and narrow-resonance energies calculated (a) with

and (b) without almost forbidden states

1u=0.812

(a)u=0.812

(b)u=0.836

(b) EXP.

0 -4.62 -3.09 -4.74 -4.731 1.43 1 .72 0.93 1 .062 -2.87 -1 .71 -3.38 -3.103 3.36 3.57 2.86 2.434 0.32 1 .41 -0.31 -0.48

1=0 1=1 1=2 1=3 1=4 1=5 1=6 1=7

2.022 (-6) 6.395 (-6)4.283(-Z) 1 .515(-4) 2.328(-5) 8.981(-5)1.463(-3) 3.039(-3) 7.045(-4) 2.372(-3) 4.339(-4) 1.739(-3)2.020(-2) 3.970(-2) 1.854(-2) 3.781(-2) 1.501(-2) 3.450(-2) 9.676(-3) 2.973(-2)0.2441 0.3557 0.2430 0.3550 0.2404 0.3538 0.2365 0.35210.5144 0.6173 0.5139 0.6171 0.5127 0.6166 0.5108 0.61590.7153 0.7843 0.7151 0.7842 0.7147 0.7840 0.7140 0.78370.8407 0.8817 0.8407 0.8817 0.8405 0.8816 0.8402 0.88140.9132 0.9362 0.9131 0.9362 0.9131 0.9361 0.9130 0.93610.9534 0.9660 0.9534 0.9660 0.9534 0.9660 0.9533 0.9660

46

since they correspond to closed channels . Although only small components of suchcon6gorations are present, they are favoured by the variational principle becausethey enlarge the configuration space. However, in a single-channel calculation, these

ponents, arenot very consistent with the model. They appear in an uncontrollableway in some NMI waves anddo not appear in otherwaves. An explicit introductionfadditional configurations would be far more physical . Hence, we think that the

et is more consistent without the Pauli resonances . However, in that case, theinteraction underbindS '°" e. We have readjusted the parameter u of the interaction

C

t

Co

0

.836 in order to recover the correct binding energy. The results are shown intable 2. The comparison with the energies obtained for u =0.812 without eliminationshows that the spectrum is more regular, with an almost perfect rotational band inositive parity . The 4' state is now bound as expected. The results are more

parable to those of the single-parameter case or of the potential model.

MWNandR"/ Elimination of Pauli resonanm

Conclusion

In the GCM, almost forbidden states can be eliminated very easily, just by asimple basis change followed by a restriction of the new basis. In the restrictedbasis, these states are exactly forbidden. The code modification performing the basischange is simple and does not cost any significant computer time.

In the new model, Pauli resonances do not appear in the phase shifts whichdisplay a regular shape beyond the barrier-resonance energy . The results are muchbetter suited for physical comparison with experiment or with other models .

e a + 160 example shows that the bound states and resonances are affected bythe elimination process. The shift of their energy location is often not small andrepresents a loss with respect to the variational principle. Indeed, the almostforbidden states enlarge the configuration space, but in an uncontrollable way. Littlecan be learned about the physical reasons of the energy improvement. We thinkthat the model without almost forbidden states is physically more consistent andbetter suited for detailed comparisons of both the bound spectrum and the scatteringproperties with other approaches.

Ithough ourgoal is to eliminate Pauli resonances, we observe that this eliminationrequires an accurate description of these resonances. Even absent, the Pauli reson-ances must be treated carefully. The generator-coordinate spacing has to be chosenaccording to different rules for small or large 1Z-values. These findings will be usefulfor our forthcoming study of the importance of the cluster description in low-energyradiative capture.

P enuix

In this appendix, we explain how the GCM matrix elements appearing in thea +'60 scattering calculation are derived. Let us consider a translation and rotation-

M. Kruglanski and D. Baye / Elimination of Pauli resonances

invariant operator O. The most general matrix element in the basis (2) is expressedas a function of Slater determinants (D(S) [refs.'"-'2)] as

(tb'(b', R')101 ç5(b, R)) = (b2b2b126iß21a'2/A2A2b2bY2)-3/4

x (41r)-3 I r(u-'/2)r3', R'-S') dS'

x f T(iu,-'/2p,

x f (0'(S')1O exp (-ia - P~,.m./ b)10(S)) da,

(AA)

S) dS

47

where P,.� , . is the total kinetic momentum . The primed functions correspond to theprimed internal parameters b, and b2 . The parameter

132 =(Ab 2-Al b; -A2b~)/A (A.2)

may be positive or negative according to the choice of b [see ref. 12)] .The matrix element appearing in the r.h.s . of eq. (A.1) is calculated analytically

on a computer with a symbolic code based on the usual cofactor method 22,.3,13) .

For closed-shell nuclei such as a and 160, the resulting expression is a sum ofpolynomials involving scalar products of S, S' and a multiplied by the exponentialof a scalar quadratic form of the same vectors . The triple hector integration corres-ponds at each step to integrals of the type

In l n,n 3 = 1 (U . U)n,(U . W)n2U2n3 eXp ( - a,U' v - a2U' W- a3U2 ) du,

(A.3)

where the n; are non-negative integers and the a; are real parameters, with a3positive . The integral (A.3) must be performed analytically but the constants a,,a2 and a3 may be evaluated numerically at each step . In close analogy with ref. 21

),

the integral (A.3) is given by the recurrence relations

where

with

_ A, h2 A3 A, À2 A3In,nzn3

- ~

Fn,-l,,n,-1~,n3-13h,1~13 ,11-0 12-o 13 -0

1 11 2 )( 13A.4

IWO = ff3/2 exp (F),

(A.5)

F = -1 1n a3 +X2/4a3(A .6)

x= cr,v+a2W .

(A.7)

The triplet (A,, A2 , A3) of positive integers in eq. (A.4) can be any of the triplets(n,, n,, n3 -1 ), (n,, n2 -1, n3 ) or (n, -1, n2 , n3) . In practice, one choice may be

the sum.

where s;, ;, is given by

glanski and D Eaye / Elimination ofPauli resonances

convenient than the other two, for example to reduce the number of terms in

e functions

®,®` i,(i, + i2 + i3 > 0) appearing in eq. (A.4) are defined as

Fili®®i

iaa,' aa~- aa

-1 3' '

+ i Si

(Ä.ô)

S

-?X-,

Sl0-x . U'

Sol -x . W,

or vanishes for i, + i, > 2.After the triple integration, the partial-wave expansion is easily obtained as a

function of spherical Hankel functions.

eferences

1) K. Wildermuth and Y.C . Tang, A unified theory of the nucleus (Vieweg, Braunschweig, 1977)2) Y.C . Tang, in : Topics in nuclear physics 11, Lecture Notes in Physics, vol . 145 (Springer, Berlin,

1981) p . 5723) T. Wada and H. Horiuchi, Phys. Rev. Lett. 58 (1987) 21904) T. Wada and H. Horiuchi, Phys . Rev. C38 (1988) 20635) F. Michel, J. Albinski, P. Belery, T. Delbar, G. Grégoire, B . Tasiaux and G. Reidemeister, Phys .

Rev. C

(1983) 19046) F. Michel and R. Vanderpoorten, Phys. Lett. B82 (1979) 1837) H. Walliser, T. Fliessbach and Y.C . Tang, Nucl . Phys . A437 (1985) 3678) D.R . TI®ompson and Y.C . Tang, Phys . Rev . C8 (1973) 16499) S. Saito, S . Okai, R . Tamagaki and M. Yasuno, Prog . Theor. Phys . 50 (1973) 156110) D. Clement, E.W. Schmid and A.G . Teufel, Phys . Lett . B49 (1974) 30811) T. Fliessbach and H. Walliser, Nucl . Phys . A377 (1982) 8412) D . Baye and M. Kruglanski, Phys. Rev . C45 (1992) 132113) H . Horiuchi, Prog . Theor. Phys. Suppl . 62 (1977) 9014) D . Baye and P. Descouvemont, Proc . 5th Int . Conf. on clustering aspects in nuclear and subnuclear

systems, Kyoto, Japan, 1988, J. Phys . Soc. Jpn. Suppl. 58 (1989) p.10315) M. Hanck, Nucl . Phys . A439 (1985) 116) K. Varga and R.G . Lovas, Phys . Rev. C37 (1988) 290617) P. Swan, Proc . Roy. Soc. A229 (1955) 1018) D. Baye, P.-H . Heenen and M. Libert-Heinemann, Nucl. Phys . A291 (1977) 23019) P. Descouvemont and M. Vincke, Phys . Rev . A42 (1990) 383520) D . Baye and Y. Salmon, Nucl. Phys. A323 (1979) 5212;) A. Tohsaki-Suzuki, Prog . Theor. Phys . 59 (1978) 126122) P.O. L6wdin, Phys . Rev . 9', (1955) 147423) D. Brink, Proc . Int. School "Enrico Fermi" 36, Varenna 1965 (Academic Press, New York, 1966) p. 247