On the extended basis shell model and two-nucleon transfer

LD.1 : 2.G

Nrrclear Pht:vics A29S (1978) 345-350 ; © North-Honored Puh/lahing Co., Antatardant Not to be reproduced by photoprint or microfilm without written pamitsion from the publisher

ON THE EXTENDED BASIS SHELL MODEL AND TWO-NUCLEON TRANSFER W. T. PINKSTON

Vanderhilt Universityy", Nashri/le, Tennessee f Received 28 July 1977 Abstract : The enhancement of two-nucleon transfer reactions predicted by the extended basis shellmodel (EBSM) calculations of Feng, Ibarra and Vallieres and other theorists, is shown to be a higher order manifestation of the pairing effects in nuclei . A simple schematic model is introduced which illustrates the constructive interference of the contributions from the highly excited shell-model states included in the EBSM . The model can be used to estimate the effect ofground-state correlations oftheclosed-shell mre. The results indicate that the assumption ofan inert core in EBSM calculations introduces negligible error in cases in which the addition of a nucleon pair to a closed shell is treated. However, in the case of two-nucleon removal from a closed shell, including the effects of correlations may significantly enhance the cross section. A suggestion is made for including these effects in EBSM calculations .

I. IBtrO~1CHOO A complete theory of two-nucleon transfer reactions must take into account a number ofcomplicated factors. Virtual inelastic processes are known to be important. Sequential transfer of the particles may also compete favorably with the direct process. In order to predict either absolute cross sections or the relative importance of the direct and the competing higher order processes, one must be able to compute the direct transfer amplitude accurately. An essential ingredient of the direct process is the overlap of initial and final nuclear states, from which the reaction form factor is computed. In lowest order this overlap can be computed from shell-model wave functions. A number ofcalculations _ have been reported' t') which improve on the lowest order shell model. These calculations have been restricted to nuclei with two valence nucleons and an inert, closed-shell core. They consist of standard configuration mixing calculations in an "extended basis" 9) of hundreds oftwo-particle shell-model configurations. Following ref. 9), we shall refer to this approach as the extended basis shell model (EBSM). The closed-shell nucleus will be referred to as A. The other nucleus will be B. The use of the extended-basis wave functions results in a large enhancement in the direct, two-nucleon transfer amplitude. Although the higher configurations all have very small amplitudes, there are many of them, and they interfere constructively in the r Supported by Vanderbilt University Research Council. 345

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W. T. PINKSTON

reaction form factor in the nuclear surface and exterior regions where the reactions take place. This constructive interference is most easily understood in terms of the zero-range form factor used in distorted wave calculations of light ion reactions such as A(t, p)B. In zero-range approximation, a short-range function of the relative coordinate of the transferred nucleons, representing their internal motion in the triton, is folded into the overlap. This results in a form factor which depends on the c.m . coordinate of the nucleon pair. The highly excited configurations of the EBSM contribute c.m . functions which oscillate rapidly inside the nucleus. Outside, these functions are large and fall oti slowly with distance . The combined effect of many configurations, with small amplitudes, averages to zero inside the nucleus, but in the exterior there is a systematic increase in the form factor because the amplitudes all have the same sign. This constructive interference, sometimes called "coherence", follows from an improved treatment of nucleon correlations and is a higher order manifestation of pairing in nuclei . It is.well known that pairing correlations result in strong enhancements of certain final states in (t, p) and (p, t) reactions. These enhancements have been known for a long time 12); in recent years they have been treated as pairing rotations and vibrations ' a .' a). In the pairing theory calculations, the active nucleons tend to occupy single-particle states, all of which are in the same major shell. A result of this is that the wave functions, describing the c.m. motion of a pair, are the same for all states . Thus, the constructive interference in lower order calculations increases the overall magnitude of the form factor, but does not et%ct its shape. In the present work a simple, schematic model is used to show the origin of the constructive interference of the higher order configurations. If a surface delta interaction (which is essentially the same as a pairing intéraction) is used, all higher order configurations have amplitudes of the correct sign for constructive interference. An apparent shortcoming of the extended-basis calculations is the assumption of an inert core . This leads to the apparent inconsistency that most of the configurations included have smaller amplitudes than core-excited configurations which are excluded. The most important core-excited contributions to the overlap come from ground-state correlations, i .e. configuration mixing, in the A nucleus. Contributions, for example, from 4p-2h configurations in B make a smaller contribution to the overlap. The schematic model of the present work can be used to estimate the eû'ect of the.ground-state correlations . The results indicate that they have very little eûect on the reaction form factor for A(t, p)B. However, in a reaction A(p, tom, in which two neutrons are removed from a closed shell, leaving a two-hole state, ~C), the ground-state correlations are expected to increase the form factor significantly in the surface and exterior. As a result the reaction cross section will be increased. A suggestion is made for the >~asis of quantitative calculations of this latter effect .

TWO-NUCLEON TRANSFER

Let an overlap function, ~,~,H(xi,

347

2. A schematic model CAB,

be defined by

xz) _ !

The ~ are Fock operators. An x represents both position and spin coordinates, r and a. The c are destruction operators for shell-model single-particle states. The subscripts, k and I, represent single-particle quantum number sets, (nijm). The bracket function is an antisymmetric two-particle state, {Y'kY'!}A = Wk(xl)W(lx2)-Y'k(x2)Y'1~x1)~

Since the amplitude of a given excited configuration is small in EBSM, we will assume that it can be calculated in first-0rder perturbation theory, i.e. IB> = I~>+ ~ Ip9> A

P,q

A VPa~

E~,+E -Ep-Eq'

VPq~ = vpgmn - VPq+~'

A single configuration is taken for the lowest order shell-model state ; however, the results do not depend on this simplification . To the same order of approximation, the ground state of A is A ;,;

p>q

Et+Ej -E p -Eq

in which I0~ is the closed shell state. The two-particle, two-hole state is I~l-li-1~ = CpCq c;c110i"

The letters ij represent single-particle core states ; mnpq represent valence-particle states . When eqs. (2) and (3) are substituted into eq. (1), the following first-order result for the overlap is obtained }A A ~ ynuij{Y't~Yj} A ~ V~P4{WpY'q ( ) t>f i f- ~_ n P>9 ~ ~ P 9 Configurations neglected in eqs. (2) and (3) result in contributions of higher order to eq. (4). In order to demonstrate the constructive interference of all the terms in eq. (4), we must relate ~,m to the zero-range form factor and show that the contribution of all tenors has the. same sign as that of the first term: Angular momentum coupling must be considered ; however the transformation of eq. (4) to coupled form is straightforward and trivial. Let IBS be a J = 0* state. Let the lowest order shellmodel state, mn, belong to the configuration, (nj) Z . For a typical two-body wave

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W. T . PINKSTON

function in one of the sums of eq . (4), the cônfiguration will be denoted, nl l'jnzlj". In addition to the zero-range approximation for the reaction process, we also make the Os approximation' 3~' 4 ). This consists of the following: Assume harmonic oscillator single-particle states . Let each jj coupling state in eq . (4) be expanded in LS states . Then, by means of a Brody-Moshinsky 1 s) transformation, expand into states of relative and c.m . motion of the nucleon pair. The Os approximation consists oftaking, as the zero-range form factor, the oscillator c.m . function associated with a Os function of relative motion . This is the c.m . function, Ns, with the largest number of radial nodes; it is the largest in the surface and exterior regions. The phase of the contribution of a given configuration to the zero-range form factor depends upon signs of several factors : (a) the amplitude,
or N = 2n+1.

Thus, the sign of the form factor outside the nucleus is (-1~. For the higher configurations, one has 2N = 2n1 +2nz +21',

or N = nl+nz+1',

and the sign ofthe radial form factor is (-1 p' + "z +`~ . This sign must be combined with the signs of the two-body matrix element and the energy denominator. The energy denominators are all negative . We take a surface delta interaction for the potential, V. It has matrix elements +~+rQiz, (5)
n i ~ nz. viz = ~, The quantity G is positive. It is the product ofa potential strength andthe magnitudes of the four single-particle radial wave functions on some nuclear surface. The signs of these radilil functions, (-1)z"+"'+"', have been included in eq . (5) explicitly . If one combines -1, from the energy denominator, with -(-pl+"=+s+r from the Qiz = 1,

n i = nz ;

TWO-NUCLEON TRANSFER

349

matrix element of V, and (-1)A'+"~+r from the phase of the oscillator function, the result is (-1~. This is the same as the sign of the wroth-order shell-model form factor. Thus, all configurations contribute coherently in this crude model. EBSM calculations neglect the last term in eq. (4). Although these terms also interfere constructively, it is doubtful whether including them would result in significant changes. The core states contribute c.m . functions which are inherently smaller in the surface and exterior than that of the lowest-order shell model. In the reaction, 16at, p)' 80, for example, the Os form factor is a 2s oscillator function . Excited two-particle states result in coherent admixtures of 3s, 4s, . . ., whereas the hole contribution will be ls. Admixing a small amount of is with a 2s form factor will influence the interior region much more than the exterior. On the other hand, correlations of the closed-shell core are important in a case such as A(p, t~. In ls0(p, t)140, for example, the lowest order shell-model configuration, lp -Z, results in a is form factor. If 2p-2h states are included in the 160 wave function, the resulting 2s, 3s, . . . contributions are likely to have a very large effect on the form factor because they are so much larger than is in the surface and exterior. Constructive interference of all states in A(t, p)C can also be demonstrated in lowest order perturbation theory. Let the C-state be, ~C) = c,c~~0~. Then vA J'~~ .I' 1A

p,v Ei+E~-Ep-Eq

Comparing eq. (6) to eq. (4), it is clear that any reasoning about the relative signs of the contributions of various terms in one equation holds equally well for the other. 3. D

oo

The schematic model presented here is of both heuristic and practical value. It makes the connection between (1) recent EBSM calculations, which concentrate on closed-shell-plus-two-neutron nuclei and use fairly realistic nucleon-nucleon potentials, and (2) studies, based on crude pairing forces, of two-nucleon transfer on nuclei far removed from closed shells . It shows how the constructive interference of contributions from many different major shells, observed in EBSM, can be understood, and also how this is related to the "coherence" predicted in BCS calculations, in which almost all orbits are in the same major shell. The practical value of the model is that it indicates rather clearly in which cases the assumption of an inert core for EBSM is valid or invalid. From the model, it can be concluded that, in calculations performed to date, the inert core assumption does not lead to serious defects in the overlap function, at least in the nuclear surface and exterior regions. Clearly, the model is oversimplified and should not be taken too seriously. The

350

W. T . PINKSTON

use ofthe surface delta interaction, in cases in which many major shells are included in the model space, is highly questionable. However, the predictions ofthe model are completely qualitative. The surface delta interaction is used only to determine the signs of the two-body matrix elements. Clearly, it will not predict even the signs correctly for all the matrix elements of a more realistic potential. The purpose of the model, it must be remembered, is to show a tendency for most of the configurations to interfere constructively . Nothing more is intended . If nucleons only interacted on a surface then they would move freely in the exterior region. Thus, the surface delta potential cannot account for pair correlations in the exterior. However, recent studies' 6) of EBSM wave functions indicate that such carrelations are not large. From the model we conclude that including ground-state correlations in overlap calculations should be very important in cases such as ' 60(p, t)'40. The EBSM can be generalized by a particle-particle RPA formalism, which takes into account ground-state conelations. This will be treated in a subsequent publication "). The author acknowledges helpful comments from Professor Norman Austern and Taro Tamura. References

1) R. H. Ibarra and B. F. Bayman, Phys . Rev. Cl (1970) 1786 2) R. H. lbatra, Nucl. Phys. A211 (1973) 317 3) D . H. Feng, R. H. Ibarra and M. Vallieres, Phys . Lett. 46B (1973) 37 4) D. H. Feng et al., Phys. Lett . 47B (1973) 477 Sj F. A. Gareev, ). Bang aad R. M. Janalejcv, Phys . Lett . 49B (1974) 239 6) J. Bang and F. A. Garcev, Nucl. Phys . A232 (1974) 45 7) J. Bang et al., Phys . Lett . S3B (1974) l43 8) P. J. Iano and W. T. Pinkston, Nucl . Phys. A237 (1975) 189 9) R. H. Ibarra, M. Vallieres and D . H. Feng, Nucl . Phys. A241 (1975) 386 10) M. Vallieres, D.~I . Feng aad R. H. bbarra, Nud . Phys. A2S6 (1976) 2i 11) J. Bang et a/., Nucl. Phys. A264 (1976) 157 12) S. Yoshida, Nucl. Phys. 33 (1962) 685 13) R. A. Broglia, O. Hansen and C. Ricdel, Advances in rtuclear phyairs, vol . 6 (Plenum Press, NY, 1473) ch . 3 14) R. A. Broglia, C. Ricdel and T. Udagawa, Nucl . Phys. A169 (1971) 225 15) T. A. Brody and M. Moshinsky, Tables of transformation brackcts (Monograt'ias Del Instituto de Fisica, Mexico, 1960) 16) R. H. Ibarra, N. Austero, M. Vallieres and D . H. Feng, N~id . Phys. A2gg (1977) 397 17) W. T. Pinkston, D. H. Feng and M. Valliers, to be publishedl