Self-consistent calculation of the optical potential and ground-state properties of nuclear matter

ANNALS

OF PHYSICS

86, 233-261 (1974)

Self-Consistent Calculation of the Optical Ground-State Properties of Nuclear

Potential Matter

and

Q. HO-KIM* Department of Physics, Lava1 University, Quebec, Canada GIK 7P4 AND F. C. KHANNA Atomic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada KOJ IJO Received August 1, 1973

On the basis of the Green-function formalism, we performed a self-consistent calculation of the self-energy Z(k, W) of a particle interacting with the infinite nuclear medium. The function Z(k, w) was mapped out in the energy-momentum plane, and the single-particle energy w(k), momentum distribution p(k) and the “on-shell” part of the self-energy, Z(k, w(k)), were defined, from which all physical properties followed. In particular we investigated the ground-state properties of nuclear matter in two .4approximations of the T-matrix. In one, the intermediate two-particle propagator, A 00, represented free-particle propagation; in the other, called A,, , intermediate states included both interacting particles and holes. Pauli principle effects were included in both approximations. The second approximation was expected to be conserving because it included a large part of the rearrangement effects which, we found, contributed ~6 MeV per particle to the average energy and -28 MeV to the singleparticle energy at zero momentum. The Hugenholtz-van Hove theorem was nearly satisfied, with only 1 MeV separating the chemical potential from the average energy. We also studied, in the &,-approximation, the optical potential for the scattering of a particle by a large nucleus; it was directly related to the “on-shell” part of the selfenergy. It was found that, below 100 MeV, the real part varied as (-90 + 0.584E) [MeV], and the imaginary part as (2.4 + 0.009 E) [MeV].

1. INTRODUCTION ;31 recent years the theory of nuclear matter has been developed with considerable success along a program proposed by Brueckner [l]. The method on which it is based features a linked-clustered expansion [2] of the nuclear ground-state energy * Supported in part by the National Research Council of Canada. Copyright 0 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.

233

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expressly desiged to treat the many-body problem encountered in nuclear matter for arbitrary interaction between nucleons. The theory and its applications have recently been reviewed in several excellent papers [3-51. There exists a less well-known and quite different method based on the Martin-Schwinger hierarchy of the Green function equations of motion [6] which attempts to solve the same problem with, however, an essential difference of not resorting to an expansion in powers of the nucleon-nucleon interaction. Further, in simple unbound systems, it offers the additional advantage of treating particles above and below the Fermi sea in a unified way. The work reported here follows this second approach. In practice the infinite hierarchy of the Green-functions is always truncated to yield a closed set of equations. The ambiguity in the choice of a proper truncating procedure works to our advantage in letting us assign an important role to the strong correlations governing the motion of particles in nuclear medium. This leads to the independent-pair model, several variants of which exist treating interactions in the intermediate states differently and satisfying the Pauli principle to varying degrees of accuracy [7-121. Although one may require that any acceptable approximation satisfy physical conservation laws, such a criterion is not always the most practical or advantageous. One of our purposes is to investigate numerically the effects of these approximations on the ground-state properties of nuclear matter, using as our basic input a number of separable interactions in S, P and D states which fit accurately two-body data [13-151. In most previous calculations S-wave separable interactions were used [7-II]; only recently [12] was a more realistic hard-core potential considered. As was first pointed out by Bell and Squires [16], the Green-function method, apart from its usefulness in investigating bound states, provides an accurate procedure for calculating the optical potential, which can be shown to be related to the self-energy of a particle interacting with the nuclear medium. Being based on the quantum field theory, it is ideally suited for treating correctly the effect of quantum statistics known to be pronounced at low energies. The theory, however, can accommodate the high-energy region as well, and may even be simpler because of the possibility of neglecting the Pauli principle [17]. In spite of our theoretical understanding of the problem and the experimental success of the model, there exist very few attempts to calculate the optical potential from basic principles. Shaw [I81 seemed to be among the first to apply the Brueckner method of nuclear matter to calculate the lowest-order contributions to the imaginary part of the optical potential. Reiner [19], on the other hand, on the basis of a simple version of the independent-pair model investigated both the real and imaginary parts using a separable S-wave interaction. These two calculations were intended for nuclear matter and gave analog of the real and volume absorption parts of one body optical potential in finite nuclei. In a finite system we have to consider surface effects in detail, and at low incident energies, the long-range correlations

SELF-CONSISTENT OPTICAL

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POTENTIAL

(or collective effects) play a dominant role [20]. Vinh Mau [20] studied the effect of random-phase correlations on scattering of 20 MeV nucleons on “OCa and found them to give a substantial contribution to the absorptive part of the potential. On the theoretical side, besides a critical review by Fetter and Watson [17], Hiifner and Mahaux [21] recently gave a systematic expansion of the self-energy in powers of the density, similar to the Bethe-Brueckner expansion for the binding energy of nuclear matter and showed that the same expansion parameter appeared in both. Thus our interest in the problem stemmed from our desire to obtain a more concrete understanding and to see how successful would be a unified treatment of the binding energy and the optical potential. Such a study, which would give a fairly stringent test of the theory, seems not to have been attempted before. Preliminary versions of this work have been reported earlier [22]. In the following section we outline the general Green-function formalism, then specialize it to the case of the infinite nuclear matter. Different approximations introduced in the formalism are discussed at length. Results of our calculation, which is fully self-consistent, are presented and discussed in Section 3. Finally Section 4 presents the main conclusions that can be drawn from this study.

2. THE GREEN FUNCTION

FORMALISM

A. General Theory Let a system of N identical fermions be described by the Hamiltonian

[6]:

ff = j d-3 11/'(l)(- g w) W) + j dx, --- dx, #+(l) #+(2) (12 I v I 34) 3(4) #(3),

(1)

where n stands for the space-time variables (x n , t,), and x, refers to spin, isospin and position (or momentum) coordinates. The operator u is an instantaneous interaction, and in coordinate space, is nonlocal and contains an implicit factor 6(x, + x2 - xa - x4) as a reflection of the translational invariance. The n-particle Green function, which forms the basis of our discussion, is defined by

GO -*- n, I’ a**n’) = (-Qn(T[#(l)

e-0#(n) #+(n) =*- I/J+(~)]),

(2)

where T is the Wick time-ordering operator. The ground state expectation value (X) is the limit approached by the grand canonical average Tr Gf exp[-/W

- pN)I)/Tr

exp[---p(H - PN)],

(3)

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AND

KHANNA

as the temperature /3-l tends to zero. The symbol p stands for the chemical potential (2EO(N)/2N), . From the Heisenberg equation of motion for the field operator #(x), a system of N-coupled differential equations can be derived for the Green functions [6]. Alternatively one can introduce integral forms of the equations with the help of a free-particle Green function Gi”) and appropriate boundary conditions in time. The first two equations of the system then read: G,(ll’)

= Gt’(ll’)

G,(12, 1’2’) = G,(ll’)

+ iGp’(14) (45 1v ( 23) G,(23, 5+1’),

(4)

G1(22’) - G,(12’) G,(21’) + L(12, 1’2’).

(5)

The notation n+ = (x, , t n + 0+) and the summation convention on repeated indices have been used. The operator L which can be expressed in terms of G3 , ties these two equations to the remaining chain of equations. The object of our study is the self-energy operator defined by z(11’) = Gp’(ll’)-l

- G,(ll’)-‘,

(6)

or, using (4), Z(11’) = i(14 1a j 56) G,(56; 4+7) G;r(7, 1’).

(7)

In terms of Z the one-particle Green’s function G, assumes the form: G,(ll’)

= Gp’(l1’)

+ Gp)(14) z(44’) G,(4’1’).

(8)

The propagation of a particle through the medium is thus described by a scattering equation in which the one-body operator Z plays the role of an effective field produced by all other particles in the system. Clearly the self-energy 2 vanishes for a noninteracting system, is nonlocal in the coordinate system, complex and energy-dependent. Its imaginary part, which is nonvanishing only in the positive energy region, results from the finite lifetime of the initial state and corresponds effectively to the width of its decay spectrum. Thus the optical potential for elastic scattering of a particle can be identified with its self-energy [ 161. B. The (l-Approximations All physical quantities of interest, such as the binding energy, density and selfenergy, can be expressed in terms of G,; however the entire hierarchy of the Green functions must still be solved. Obviously some approximation must be introduced. The simplest is to set L = 0 in Eq. (5), which results in an independent-particle picture characterized by the absence of any form of particle correlations. A better

SELF-CONSISTENT

OPTICAL

POTENTIAL

237

approximation which takes account of the particle correlations but still keeps Eqs. (4)-(5) self-contained, is to factorize the operator L in terms of G, and Gz only, neglecting its dependence in G, . For a system of low-density but governed by short-range forces it is advantageous to keep interacting pairs correlated; longrange correlations are effectively deemphasized and can come only as corrections in higher order terms. We may then have several possibilities for L, all written in the form: L(12, 1’2’) = r1(12, 34)(34 j v 1 56) G2(56, 1’2’),

(9)

where the two-particle propagator /l assumes different forms depending on how accurately we want to treat the intermediate states. Thus we may have fl,,(l2,

1’2’) = iG1(ll’)

~l,,(12, 1’2’) = $[Gt)(ll’) /1,,(12, 1’2’) = iGp)(ll’)

G,(22’),

(10)

G,(22’) - Gp’(12’) G,(21’)], G$)(22’).

(11) (12)

An exact calculation necessarily leads to quantities that satisfy all conservation laws. In a self-bound system, for example, the removal energy of a particle at the Fermi level should be equal to the average energy per particle in the ground state [23,24]. To take another example, values of the self-energy obtained from either Eq. (6) or (7) should be the same at any momentum. Obviously it is desirable to have a conserving approximation; that it is indeed the case for the -&,-prescription has been shown by Baym [25]. The cl,,-propagator has the additional advantage of including a large number of diagrams in the perturbation expansion and does not lead to double-counting. Its drawback lies in its complexity. On the other hand, the fl,, and ./lo0 approximations, though nonconserving and less satisfactory from the formal point of view, are much easier to handle and have been more often used in practice [7-10, 121. Whatever model we choose for Ll, Eqs. (7)-(g) yield the following relations which we write in the operator form: 2’ = iTG, , G 1 = G(O) + G’O’zG 1 1

(13) 12

T = v - v,,,~ + VAT.

(14) (15)

The form of Eq. (15) shows that T is a generalized two-particle T-matrix which incorporates the effects of the interacting medium through which the particles propagate. Since in the three models we have adopted, (1 is given in terms of the

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one-body Green function, Eqs. (13)-(15) form a closed system and can be solved self-consistently for a given two-body potential v. C. Injinite

Nuclear Matter

The structure of Eqs. (13)-(15) can be considerably simplified if the system of interest is completely homogeneous as in infinite nuclear matter. It is then advantageous to use the momentum-energy representation. Let w be the energy variable and k, be the laboratory momentum of the ith particle, and K = a(kl - k,) and K = k, + k, the relative and center-of-mass momenta of a two-particle system. Knowing the free-particle Green function Gp’(k, co) = [w - k2 + ,u]-I,

(16)

we write the interacting particle Green function as G,(k, o) = [w - k2 + p - Z(k, co)]-‘.

(17)

To make the calculations practical, further approximations are needed. While the quasiparticle assumption [26]-according to which particle states live long enough to have a well-defined energy o(k) and a small spreading width y(k)is sufficient to express G, as a relatively simple function of k and w, the two-particle propagator /1 still appears too unwieldy to be of much practical use. A further assumption, that the damping width y(k) be small, is then introduced. Thus given the definition [k2 - p + Re Z(k, w)] [l - 2 “FL w(k) =

w, 1-l 1 , 4k) w=--w(k)

k2 - p + Re z(k, w(k)),

43

> 0, (18)

< 0,

/

y(k) = p(k) Im W, 4k)>,

(19)

and

we have, for the one particle Green function,

G&G4 = p(k)Iw - 4W - MW,

(21)

SELF-CONSISTENT

and for the two-body

OPTICAL

239

POTENTIAL

propagator

&,,(k,

, k, , w) = [w - k12 - kz2 + 2p]-l,

A,,,(k,

, k, , w) = ${[a~ - k12 - k22 + 2p - Re Z(k, , w - k22 + ,u)]-l

- dk,) F-w(k,))b fldkl>

k, 2 ~1 = p(k)

p(k,W(4.k,N

(22)

- h2 + EL- 4kW ‘G(k,))

x by - 4,) - 4k,)l-1,

- e(--w(W)

+ (1 ++ 2)},

(23)

&--w(Q)] (24)

where e(w) = 1 for w >, 0 and e(w) = 0 otherwise. The 4& , k2 , w) propagator implies that the two particles in the intermediate state are treated as free particles and their energy spectrum is given by the kinetic energy of the particles. In this approximation the T-matrix has no singularities except on the positive real axis. A,,(k, , k, W) is quite similar to &, approximation with regard to the analytic properties of the T-matrix in the complex energy plane. Though this approximation treats one of the particles as dressed and the other as a free particle, the computational details are quite similar to that of A,,, approximation. Even though both Ll,, and A,,, approximations are not conserving, a great deal of effort has been put in to explore them [7-121. The 4dkl , k2 , w) is the exact two particle propagator that treats the intermediate particle spectrum properly and takes into account the occupation probability [p(k,) for states of momenta k,]. This is the analog of the renormalized Brueckner-Hartree-Fock calculations [27] that have been done in some closed shell finite nuclei. In detail this propagator is different from the usual propagator used in the calculation of Brueckner G-matrix for nuclear matter. The first term B(w(k,)) O(w(k,)) in Eq. 24 expresses the propagation of particles in the intermediate state while the second term O(--w(kl)) O(--w(k,)) implies the propagation of two holes. If the second term (hole propagation) is neglected, then &-propagator is similar in structure to the propagator suggested by Thouless [28]. Further if we ignore the factors of p(k,) and p(k,) which expresses the renormalization due to the occupation probability and assumes that the single particle spectrum in the intermediate state is expressed as w(k,) = A + Bk12, with A and B determined by a self-consistency procedure [29], then the propagator L’!,, is similar to the reference spectrum method. In the usual linked cluster expansion [30] the ordering of the perturbation theory is done by the number of hole-lines in a particular diagram. An inclusion of the second term in Eq. (24) will not be quite consistent with an expansion in terms of hole lines. In addition Brandow [30] has estimated that such an inclusion of hole line propagation does not necessarily lead to a more convergent perturbation expansion (see Iwamoto [31]). Actual contribution to the potential energy may be -1 MeV per particle. In our case we have retained the

240

HO-KIM

AND

KHANNA

second term due to hole line propagation since it is only the fl,,(k, , k, , W) propagator as given in Eq. (24) that leads to a conserving approximation. It should be emphasized that the expressions for A,,(k, , k, , w), A,,(k, , k, , w) and All(kl , k, , w) have been obtained in the quasiparticle approximation [6] for the spectral function S(k, w)(= (1/2ni)[G(k, w - iy) - G(k, w + iv)] which is written as S(k, w) =

dk)

b - 4W

y(k)

+ tMW

.

However for low-energy-excitation spectra it is physically reasonable to assume that the single-particle states are relatively stable and they have very small width. Specifically if we assume that y(k) Q w(k), then the spectral function can be written as S(k, w) = 2np(k) S(w - w(k)).

This form implying that the width y(k) is zero, is not correct since Z(k, w) for w > 0 will have a finite imaginary part. This suggests that a complete self-consistency in which both the Re C(k, w) and Im Z(k, w) are varied at each stage of iteration is not possible. The procedure we follow is partially self-consistent in that Re Z(k, w) is done properly. This implies that the ground-state energy and other ground-state properties of nuclear matter are calculated in a self-consistent manner while the optical potential is only partially self-consistent. We should expect some small deviations from exact conservation laws due to the quasiparticle approximation. The definitions of w(k) and p(k) given in Eqs. (18) and (20) when w(k) < 0 are obtained by noting that G,,(k, w) has the analytic structure with a pole at w(k). If this resonance structure is well-defined, i.e., y(k) < w(k), it is useful to expand Z(k, W) in a Taylor series about w(k) and then retain only the term linear in [w - w(k)]. Physically w(k), y(k) and p(k) represent the energy, width and the occupation probability for a hole with momentum k. The propagation of the hole in the medium is described by these quantities completely. The definitions of w(k) and p(k) for w(k) > 0 as given in Eqs. (18) and (20) are just a conjecture and were suggested as prescription by Koltun et al. [32]. All the three propagators have a branch cut on the positive real energy axis in the complex energy plane. This makes it hard to give a good definition of the single-particle energy for w(k) > 0 since Z(k, w) is a rapidly varying function of w. The prescription, therefore, relates w(k) and p(k) for w(k) > 0 to the values of the self-energy function Z(k, w) for negative values of w and Z(k, w) is analytic in this region of the complex w-plane. Since we calculate the optical potential only in fl 0,, approximation, this prescription may be reasonable though we have made no attempt to investigate alternatives. It may be pointed out that our

SELF-CONSISTENT

OPTICAL

241

POTENTIAL

failure to compute the optical potential in (1,, approximation may be related to this prescription to some extent since the propagation fl,, has singularities (pole as well as branch cut) on the negative real energy axis and the prescription of Eq. (19) will not provide any simplification. We have not been able to obtain a useful prescription for the definition of the single particle energies in A,,-approximation. In the same approximations the self-energy has the form

w, 44 - id = j & p(k’)

lO(-c&Y))

(kk’

I 7@(k)

-j s _

(kk’

when written

in relative coordinates,

’ dw’ --m 2rr

where the T-matrix,


+ w(P)

- iv) 1kk’)

/ T(w’ + iv) - T(w’ - iq) 1kk’) w(k) + w(k’) - w’ - iq

(x I v I H”) z&(x”,

[ !’

(25)

is given by

w) (K” j TK(w) 1H’).

(26)

Since our T-matrix is similar in structure to that found in the two-body scattering problem, it is natural to attempt a solution by similar techniques, e.g., by a partial wave expansion. However because of the dependence of Ton the relative direction of the CM momentum vector and the relative momentum vector, an approximation must be introduced to remove this coupling in the fl,, and /I,, propagators. (A procedure to take into account the motion of the center of mass exactly has been given by Werner [33]). Thus we replace A,(x) in Eq. (18) by the angle-averaged expression A,(K)

=

&

j-

dri- &(K).

This procedure of angle averaging is exact for the important introduce some errors for other partial waves. The T-matrix then may be expanded in partial waves

(27) S-states but does

The operator P, projects onto states of isospin T, the symbol 01stands for S, J, T, and G$‘(i?) are the vector spherical harmonics. In terms of trLP, which can be calculated exactly if o is separable, the self-energy becomes

z@,Z>= cal&

(2J

+ 1)(2T + 1)j $$ @')&-+'>)t:(KKK, Z+ w(k'>). (29)

242

HO-KIM

AND KHANNA

In obtaining this result we have ignored the second term in Eq. (23). In the &,-approximation the only pole in the T-matrix occurs in the triplet interaction at a positive frequency associated with the deuteron binding energy. Similarly any poles in T(fl,,) are expected to remain on the positive w-axis. Consequently in these two approximations the second term on the RHS of Eq. (25) vanishes. It subsistshowever in the (I,,-approximation becausethe propagator has singularities all along the real w-axis and T(W) is discontinuous on the negative real w axis. However the second term in Eq. (25) is nonzero in a small finite interval. Estimates suggestedthat the second term may be small compared to the first term and for computational simplicity it was dropped. The step function in Eq. (29) divides the momentum spaceinto occupied states and unoccupied statesat the Fermi momentum k, defined by w(kF) = 0.

(30)

Then Eq. (18) shows that the chemical potential p is identical to the separation energy: p = kp2 + Re Z(kF , 0). (31) It is appropriate at this stage to compare the Green’s functions approach that is followed in this paper and the conventional Brueckner approach for the calculation of the separation energy. As mentioned earlier the Brueckner theory is a hole line perturbation expansion. This implies that the lowest order ground-state energy

:h)

FIG. 1. The line goingup (down) is a particle(hole) line. The dashed (curly) line represents bare potential u (G-matrix). (a) gives the definition of reaction matrix, (b) lowest order groundstate energy of nuclear matter, (c) one of the third-order corrections to the ground-state energy, (d) Hartree-Fock definition of single particle energy, (e) and (f) the second order corrections (rearrangement energy) of the single-particle state.

SELF-CONSISTENT

OPTICAL

POTENTIAL

243

(Fig. lb) is just the sum of ladders that yields the Brueckner G-matrix (Fig. la). The lowest order correction to the binding energy of nuclear matter is of third order in G and of the diagram that contributes in this order is shown in Fig. Ic. On the other hand, the single particle (or hole) energy in the lowest order is given by Fig. Id and has corrections that are second order in G (Fig. le). These secondorder corrections to the single particle energy can be incorporated in the Brueckner reaction matrix if we include the repeated scattering of holes as well as particles in the intermediate states (see Iwamoto [31] and Young [34]). This implies that the Brueckner reaction matrix should be changed to an analog of the T-matrix defined in this paper. In the self-consistent Green’s function approach the contributions due to diagrams in Figs. (le and lf) are included through the definition of the T-matrix. An alternative approach for including these diagrams has been used by Brueckner and Goldman [35] and follows a suggestion by Landau [36]. This method considers the scattering of two quasiparticles (i.e., one particle and one hole) and the diagram of Fig. If appears as the scattering of a particle-hole pair. A detailed comparison of the Green’s-function approach and the conventional Brueckner theory has been given by Brandow [30]. The splitting of the self-energy and an “off-shell” part is considered so that operator Z(k, w) into an “on-shell” it is convenient to calculate in the hole-line expansion scheme. However if this approach is followed, the renormalization of the single particle energy due to rearrangement effects (Figs. le and If) have to be calculated explicitly. For a selfbound system it has been shown by Bethe [37] and by Hugenholtz and Van Hove [24] that the average separation energy of a particle from the top of the Fermi-sea should be equal to the chemical potential and to the average energy per particle, i.e., aE, xp=E” ( aN 1 v

N’

This is an exact result. It appears to us that a calculation scheme that satisfies this relationship will be most desirable and this has been the objective of this present work. It may be appropriate at this stage to remark on the relevance of this discussion about the definition of the single particle energy and the effects of rearrangement energy on finite nuclei. In recent years Hartree-Fock calculations have been done on finite nuclei and it is found that the energies of the single particle orbitals are not the same as that obtained in experiments like (p, 2~) and (e, e’p). Engelbrecht and Weidenmtiller [38] have proposed an extension of the Green’s function methods to the study of single particle energies in light nuclei. Padjen et al. [39] have made some calculations based on this proposal and have obtained large values for the rearrangement energies (i.e., calculated diagram of Fig. le with proper renormalization due to the occupation probability taken into account). However the positions and widths of the resonances observed in such reactions 595/W2-4

244

are identified function,

HO-KIM

AND

KHANNA

with the parameters w(k) and y(k) of the maxima S(k, co) = (1/2xi)[G(k,

w - iv) - G(k, w + iv)],

of the spectral (32)

as a function of w. A detailed mapping of G(k, w) or Z(k, w) in the energymomentum plane can have some relevance to the physical observed quantity, the spectral function. There are many other definitions and prescriptions for the definition of the singleparticle energies in finite nuclei and these are summarized quite well in the paper by Koltun [40]. The relevance and detailed comparisons of the methods of computing single particle energies using Brueckner reaction matrix are given by Brandow [30]. Turning our attention now to the intermediate states of two interacting particles, we notice two important factors: the Pauli principle and the interaction with the medium. While the L!&,, ignores them both and the fl,, satisfies them half-way, the II,, gives a correct treatment on both accounts. Another point which is of special interest is the presence in the fl,, of a term describing hole propagation in addition to the usual term associated with particle propagation. Hole propagation is nonexistent in both the traditional G-matrix (p(k) = 1) and the renormalized [41] G-matrix (p(k) # 1). We may recall that in Brueckner’s theory such hole-hole interactions give rise to rearrangement effects [35], which indeed are small in the ground-state energy (about 1.5 MeV per particle) being a third-order correction and are rightly neglected; but in single-particle energies they appear in secondorder and are not necessarily negligible (about 26.8 MeV at k = 0). Hole propagation appears in a natural way in the Green-function formalism and ensures that a self-consistent calculation with the &prescription satisfies the HugenholtzVan Hove theorem [24]. Finally once the calculation yields Z(k, w), all physical quantities can be obtained: single-particle energy w(k), damping width y(k), momentum probability p(k) and chemical potential p from Eqs. (18), (19), (20) and (31), respectively. Other properties of nuclear matter of special interest are: the energy density, <=-- 2 kFdk k2p(k) W 5-P s0

+ 4k) + ~1,

(33)

the energy per particle, E =

EIP,

the average particle density, p = -$ Jo”’ dk k2p(k),

(34)

SELF-CONSISTENT OPTICAL

POTENTIAL

245

the effective mass, (36)

and finally the symmetry energy, ES,. , in first-order perturbation theory [42]. The introduction of a single-particle energy, w(k) + TV,motivates us to define the “on-shell” part of the’ self-energy as Z(k, w(k)), which in turn determines the optical potential for the scattering of a particle from a large nucleus as follows:

V,,,,(k) = Re -W, 4k)) + ip(k) Im z(k, 4W. 3. CALCULATIONS

(37)

AND RESULTS

The preceding discussion shows that the self-energy contains a great wealth of information, some of which is exploited in the present work. Of the three versions of the independent-pair model mentioned previously, only two are retained: the LI,, and (I,,-versions. In using the /&,-version we incorporated the Pauli requirement that the intermediate states be restricted to the unoccupied subspace (k > kF). Bound states properties of nuclear matter were calculated in both versions, but for lack of computer memory, we were not successful in our calculation of the optical potential using the A,,-approximation. The time involved in each iteration becomes so large that a self-consistent calculation for the optical potential in /lrr- or /I,,approximation becomes nearly impossible. A. Two-Body Interactions The common feature of the interactions considered for use in this work is that they are nonlocal and separable; this may be objected to as not being very realistic; however the calculation of the T-matrix is certainly made much simpler. In the relative momentum coordinates one may write the radial form of a rank-2 separable potential as follows: z&(k, k’) = i”-‘[cl(:! g,(l)(k) gi!)(k’) - c,‘$g:‘(k) gf?)(k’)],

(38)

where gti)(k) are the fourier transforms of the radial form factors containing parameters chosen to give a good fit to two-body data. We used Hammann and Ho-Kim [I 31 and Mongan’s Case II [ 141 potentials in our analysis. The radial form factors for the first are given by g,(i)(k) = kz/(kz

+ a(i)z)(z+l) 1 3

i=

1,2,

(39)

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HO-KIM

AND

KHANNA

and for the second #(k)

=

k’/(k2

a9(l+2)12

+

2

i=

7

1,2.

The parameters are listed in Table I. TABLE

I

Parameters of Hammann’s Potential, Eqs. (38) and (39) and Mongan’s Case II Potential, Eqs. (38) and (40). The Strengths Are Given by -(1/C)* or +(4/c)* Depending Whether a Star Is Indicated or Not. In the ?$ - 3D, Coupled Channel the Strengths Are: CA:’ = -7391.613, C$ = -42.299 MeV fm-3 for Hammann’s Potential, and CA:) = 4628.88, C$ = 1382.753 MeV fm-1 for Mongan’s Potential.

(C(l))l/Z [MeV/fm’2”+1~]1/2

‘SO % lP1 3PLl sPl

3P, ‘D, ‘JA =JA SD,

Hammann-Ho-Kim (CCZl)lP

all,

[MeV/fm(2z+1’

56.266 57.219 252.119 1878.142 437.991 132.483* 1322.917* 870.076 534.391* 160.253*

] II2 ( fm -I)

18.277 22.514 30.641 55.667 22.327* 29.985 20.427 41.132* 27.576 10.072

3.50 3.50 2.55 4.60 3.50 3.0 3.45 2.65 2.48 2.28

al2l

Mongan-Case (p,)l/2

(Ccl,)l,a

II &’

al2)

(fm-I)

[MeV/fm1112

[MeV/fm]‘12

(fm-I)

(fm-‘)

1.50 1.50 1.54 1.75 1.50 2.0 1.33 1.32 1.22 1.30

302.0 93.74 40.88 988.10 45.63 0.0 0.0 49.38 0.0 493.80

27.33 41.08 30.21 13.94 0.0 24.14 21.09 33.66 20.60 7.716

6.157 3.612 1.41 4.46 2.178 0.0 0.0 1.264 0.0 6.558

1.786 1.994 1.258 1.313 0.0 2.192 I .944 1.161 1.468 1.451

A common weakness of separable potentials is their ineffective tensor force [43], which leads for example to a small D-state mixing in the deuteron and an overbinding in nuclear matter. Afnan et al. [ 151 have proposed several simple one-term separable potentials in the triplet-even channels which fit low-energy two-nucleon data almost equally well but which differ from each other in the amount of the deuteron D-state probability; the radial form factor suggested is the same as given in Eq. (40), with the parameters listed in Table II. One-term

Separable

TABLE Potentials

II in aS-3D

Channel,

Eq. (40)

Potential

PDl PD3 PD4

(Me?yfm)

(Me?;fm)

-188.77 -131.94 -80.12

-27.53 -246.31 -468.17

(Me?;

fm)

23.74 373.02 964.12

(frz1,

(fZ-1)

1.39 1.338 1.31

0.696 1.35 1.56

1.0 3.0 4.0

247

SELF-CONSISTENT OPTICAL POTENTIAL

B. Numerical Procedure

The calculational scheme started with the computation of the elements of the T-matrix for a number of energy arguments. The iterative cycle was initiated by some trial function w(k) and p(k) which allowed the calculation of the self-energy Z(k, o) via Eq. (29). The new Z(k, w) in turn yielded improved function values of w(k) and p(k), which brought us back to the next iteration. In the zero-width approximation only an iteration of the real part of the self-energy is required, and, consequently, a convergence criterion was imposed on the binding energy and the separation energy. The imaginary part of the self-energy was calculated at the end

(Mev)

I

Re

-4O-

Z(k,w) r

-80

,p

2 3

-

-120-

-160-

-100

-60

-20

20 w+p

60 (MeV)

100

140

160

FIG. 2. Re X(k, UI) and Im Z(k, w) are plotted as a function of w + ~1 for the fist three iterations in &,-approximation (k~ = 1.4fm-l and k/kp .= 0.75).

of the iterative process for the binding energy. The calculation was repeated for each desired value of k, . It was found that the convergence rate could be speeded up by using as starting solution the values of w(k) and p(k) obtained either for a nearby k, , or for the same kF in the A,,, approximation if a A,,-calculation was being performed. The self-energy was first mapped in the energy-momentum plane. Figure 2 illustrates the fast convergence rate of the solution at k, = 1.4fm-l; the selfenergy varies with k but stabilizes rapidly as the convergent solution is approached.

HO-KIM AND KHANNA

248

C. Properties

of Nuclear Matter

1. The A,, Approximation This approximation has been most often used in the past. In the matrix elements of T(z) the energies of the intermediate states are treated as free-particle energies; the starting energy argument of the T-matrix is always treated correctly in a selfconsistent manner. This part of our calculation follows the same pattern as in previous works [7-IO], from which it differs however on two points. First, our &,-propagator contains an explicit Pauli operator which limits intermediate states to the unoccupied subspace; this makes it quite similar to the Brueckner propagator. Second, all S, P and D interactions are included, which, of course, affects all self-consistently calculated properties.

k/k,

-f

/’

= 0.987

k/kF=0.717

-,

FIG. 3. Particleenergy~(k,w) in the -&-approximationat kF = 1.4fm-’ for variousvalues of k and W.

Since it turned out that Hammann and Mongan’s potentials yielded quite similar results, we shall restrict our discussion mostly to the former. As already mentioned Z(k, w) was first mapped out in the energy-momentum plane (Fig. 2); a more complete picture is given in Fig. 3 where we plot

l (k, co) =

k2 + Re Z(k, w)

(41)

as a function of w + p for several values of k at a tied Fermi momentum kF = 1.4 fm-l. Re Z(k, w) varies slowly with w at fixed k and kF . The intersections of E(k, w) with the line E = w + p correspond to the values of the single-particle

249

SELF-CONSISTENT OPTICAL POTENTIAL

FIG. 4. Single-particle energy spectrum w(k) + p is plotted against k,!kp for various values of kF . k/kp = 1 gives the value of the chemical potential p.

5 300k

FIG.

=I.4

5. Single-particle energy w(k) + p for k > kf in A,, approximation.

250

HO-KIM

AND

KHANNA

energy w(k) + p. These are plotted in Figs. 4 and 5 for k < k, and k > k, , respectively, for several values of k, . At k = k, the curves give the chemical potential CL. They exhibit no apparent energy gap at the Fermi momentum, in contrast to the usual Brueckner calculation which prescribes a gap separating the hole energy from the particle energy at the Fermi sea. Figure 6 gives the energy per particle predicted by various potentials considered in our work. Hammann’s potential (H) predicts saturation at k, = 1.85 fm-l for

-SOL 1.2

1

1.4

I

1.6 kF

I

I.3 IN

I

2.0 FERMIS-’

I

2.2

I

2.4

i 6

FIG. 6. Binding energy per particle and chemical potential in nuclear matter in A,,, approximation. Curves labeled Hand M refer to Hamman [13], Mongan [14] potentials respectively. Afnan et al. [15] potentials with 1, 3 and 4 % D-state probability in deuteron are labeled as PDl, PD3 and PD4, respectively. p(H) and p(M) give the chemical potential for Hamman and Mongan potential, respectively.

an energy of -30.5 MeV per particle while Mongan’s potential (M) yields stability at k, = 1.75 fm-l for an energy of -26.5 MeV. Thus our &,-calculation gives results in close agreement with those obtained by the Brueckner method [43] but clearly in disagreement with empirical values. The overbinding so obtained is caused by a weak tensor force and a concomitant strong “S, force. This point has been tested by modifying the parameters of the triplet-even components to give a correct D-state mixing in the deuteron (the PD forces of Afnan et al. [15]) while keeping Hammann’s parameters for all other components. The results are shown also in Fig. 6 by the curves labeled PDl, PD3 and PD4. The PD4 potential, for

SELF-CONSISTENT

OPTICAL

251

POTENTIAL

example, leads to saturation at kF = 1.75 fm-l for an energy of - 14 MeV per particle. Table III summarizes the bound-state properties of nuclear matter as predicted by Hammann’s potential. Of course no rigorous agreement with their empirical values is to be expected; however the results are reasonable in the range of k, = 1.4 - 1.8 fm-l. TABLE Bound-State Properties of Nuclear Approximation. The Numbers

III

Matter Calculated with Hammann’s Potential in the A,,Given in Brackets Are Calculated in &-Approximation.

kF (fm-I)

m*/m

ESYM

(A)

(i&)

(MeV)

(f&

1.2 1.4 1.5 1.6 1.7 1.8 1.9 2.4

- 17.25 -23.61 -25.74 -28.16 -29.45 -30.16 -29.49 -11.78

(-16.5) (-20.5) (-23.3) (-24.0) (-23.0) (-20.2)

-34.82 -44.44 -45.05 -47.04 -46.16 -40.0 -35.58 -4.4

(-32.0) (-37.0) (-38.5) (-35.0) (-24.2)

13.20 21.04 27.37 31.69 36.21 41.83 49.36 64.50

0.11 0.17 0.21 0.25 0.30 0.35 0.41 0.75

0.75 0.64 0.57 0.56 0.55 0.55 0.51 0.61

2. The A,,-Approximation As discussed in the last chapter, it would be of great interest to investigate the degree to which different versions of the independent-pair model satisfy conservation laws, an expression of which is embodied in the Hugenholtz-Van Hove theorem on the equality of the chemical potential and the average energy. The expectation is that, because the Ll,,-prescription incorporates an important class of rearrangement effects, it must be conserving if the calculation is performed selfconsistently and correctly. The results we are now reporting refer to Hammann’s potential. A comparison of the hole energies obtained in the A,,, and &-approximations shows (Fig. 7) that in the latter the particle is less bound; in the momentum range k/k, = 0.0 to 1.0, for k, = 1.4 fm-I, the difference amounts to 28.0 to 9.0 MeV, a large part of which results from rearrangement effects. This is in remarkable agreement with the values 26.8 and 4.4 MeV obtained by Brueckner et al. [35], and 25.5 and 4.8 MeV given by Kijhler [44], although the interactions are quite different in the three calculations. Figure 8 illustrates the important role played by such effects in the conserving

HO-KIM AND KHANNA

252

-*O/ I

kF = 1.4

fm-’

.2

0

.4

.6

3

.s

k/k,

FIG. 7. Comparison of the single particle energies in no0 (solid line)- and rl,, (dashed line)approximations for kp = 1.4 fm-‘.

-50 1.2

7 1.4

, 1.6

I 1.8 kF

t 2.0

I 2.2

I 2.4

6

(fm-‘1

FIG. 8. Comparison of the binding energy per particle (E/A) and chemical potential Q in the d,, (solid lines)- and II,, (dashed line)-approximations.

SELF-CONSISTENT

OPTICAL

POTENTIAL

253

properties of the theory. The crossing point of the average energy and chemical potential curves is to be compared with the energy saturation point. They are, respectively, (-28.5 MeV, kF = 1.98 fm-l) and (-30.5 MeV, kF = 1.83 fm-I) in the &,-approximation, and (-23 MeV, 1.81 fm-l) and (-26 MeV, 1.70 fm-l) in the (I,,-approximation. On the one hand, the rearrangement effects included in the A,,-propagator lower the binding energy by 6.5 MeV per particle. If all of this energy comes from rearrangement effects, it is somewhat larger than the previously quoted result, 1.5 MeV per particle [35]; we believe, however, at least part of the difference can be explained by the particular features of the potentials used in the two calculations. On the other hand, from the viewpoint of conservation laws, the use of the A,,-propagator is no doubt an improvement over that of the A,,-propagator, as witnessed by our results; however it is not obvious that the introduction of the quasi particle assumption, or even less so, of the zero-width assumption, allows the formalism to remain conserving. Our results prove that this certainly has not been the case, but also that any violation of the conservation laws has been mild. resulting in a deviation of --I MeV between the separation energy and the average energy; this number however is subject to change with the interaction, roughly in proportion of the rearrangement contribution. The discrepancy of 1 MeV between the average energy per particle and the chemical potential can be attributed in decreasing order to the use of zero-width approximation (i.e., using S(k, w) = 2np(k) S(w - w(k)) and to the angle averaging procedure for the propagator. A detailed numerical investigation of these aspects will involve enormous amounts of computer time. Considering the fact that a discrepancy of 1 MeV implies an error of 1-2 y0 in the evaluation of the potential energy, the estimate presented here is reasonable and is in accord with the conservation laws. To summarize, we compile in Table IV results of the more recent calculations of nuclear matter energies in the A,,, , /I,, and fl,, approximations. For each calculation we give the energy-Fermi momentum coordinates of the minimum point of the binding energy-curve and the point where this curve meets the chemical potential curve. Foster and Fiset [lo], Wegman and Weigel [l l] use the same s-wave separable Yamaguchi potential and make the same approximations. Contrary to expectations [8], the binding energy from the rl,, calculation is not the smallest; also, somewhat surprisingly, the Hugenholtz-Van Hove theorem is very well satisfied in their calculations. However the Fermi momentum is always too large. Fiset and Foster calculation [12] is the only one existing to use a local hard-core (Hamada-Johnston) potential [45] in the A,,, and rl,, approximations; they obtain a very small binding in contrast to the standard Brueckner type calculation, which give -11 MeV binding at k, == 1.35 fm-‘. In order to appreciate the results for the energy per particle in fl,, approximation, we would like to mention the results obtained with conventional Brueckner

254

HO-KIM

AND TABLE

KHANNA IV

Comparison of Nuclear Matter Energies Calculated in Different d-Approximations. For Each Entry, the First Line Gives Minimum Energy per Particle and the Second Line Gives the Crossing Point of p and E/A. Calculations

f4m

Foster and Fiset”

&I

E(MeV)

kF(fm-I)

E(MeV)

-35.8 -35.3

2.30 2.43

-31.4 -30.9

4,

kF(fm-I) E(MeV) kF(fm-I) 2.24 2.36

Wegman and Weigel”

-6.4 -30.5 -28.5 -26.0 -19.0

Fiset and Fosterc Present workd

tT

1.25 -

-4.6 -

1.83 1.98 1.75 2.1

-34 -34

2.5 2.5

-24 -23

1.70 1.81

1.03 -

0 Reference 10; s-wave Yamaguchi potential; p(k) = 1 in the calculation of Z(k, w). b Reference 11; s-wave Yamaguchi potential; p(k) = 1 in the calculation of B(k, w). c Reference 12; Hamada-Johnston interaction; p(k) + 1 in the calculation of X(/c, w). d Hammann potential. Conventional Brueckner calculation yields a B.E. of ~-31 MeV at kf = 1.8 fm-I. e Mongan potential. Conventional Brueckner calculation yields a B.E. of w-26.0 MeV at kf = 1.75 fm-I. [Both (d) and (e) use p(k) obtained from Eq. (20)).

reaction matrix. Clement reaction matrix given by

et

al.

[43] used an angle averaged propagator

in the

where Q(K,p) is the Pauli operator that restricts the intermediate states of two particles of relative and centre of mass momentap and K, respectively. The single particle energies c(q) are given as 44) = 2M q2 + U(q),

h2 = 2M q2,

q
(43)

SELF-CONSISTENT

where U(q) is a single-particle

OPTICAL

POTENTIAL

255

potential

u(q) = C (w I GKW = 4) + 40) I w’ - q’q), P’
(45)

where A and M* are parameters that are fitted to single-particle energies given by Eq. (43). Clement et al. find that Mongan potential yields a minimum for E/A at -25.0 MeV for kf = 1.75 fm-l while Hamman-Ho-Kim potential yields E/A at ~3 1 MeV for k, N 1.8 fm-l. These numbers are quite similar to the values we have obtained in /I,, approximation. It is found that at k, = 1.36 fm-l the HammanHo-Kim and the Mongan potentials yield a binding energy per particle of -27.26 and -19.86 MeV, respectively. This large binding of nuclear matter with either of these separable nonlocal potential is attributed to the weakness of the tensor force. For example the D-state probability in deuteron is found to be 0.05 % and 1.1 % with Hamman-Ho-Kim and Mongan potential, respectively. Our numbers for /l,, approximation are definitely an improvement over the values obtained with either the II, or the conventional reaction matrix techniques. Before leaving this topic about the binding energy per particle in nuclear matter, some comment should be made about corrections due to the three and higher particle clusters. It was noticed by Bethe [46] that in the conventional theory the three hole line diagrams should not be treated perturbatively but the technique due to Faddeev [47] ought to be used to sum all the third order diagrams. Solving Faddeev equations has yielded that the binding energy per particle is increased by l-2 MeV depending on the value of k, and the type of potential used (i.e., hardcore or soft-core). A detailed discussion of these numbers is given by Bethe [4]. It appears that third and fourth order (Day [48]) terms contribute 2-3 MeV per particle and are quite significant as far as attaining the empirical volume energy of -15 MeV per particle. Using the Green’s function technique, Fiset [49] has estimated the effect on the binding energy per particle if three particle Green’s function (G3) is not approximated by products of G1 and G, but additional effects due to the correlation among three particles are retained. Fiset finds that the change in E/A using a Yamaguchi potential in s-states only is of the order of 1-2 MeV in the region of the minimum of the energy. These numbers are consistent with the results obtained by Bethe [46] and Day [48]. We estimate that the change in the binding energy of nuclear matter using Hamman’s or Mongan’s potential will be

256

HO-KIM

AND

KHANNA

comparable in magnitude, perhaps somewhat larger. A detailed investigation of these contributions to the binding energy of nuclear matter constitutes a separate study. D. The Optical Potential The optical potential has been calculated in the A,,-approximation using Eq. (37). The results, based on a calculation at k, = 1.4 fm-l with Hammann’s potential, are presented in Fig. 9 as function of the scattering energy E = w(k). --

o-

l

Re

VOPT

+

Im

VOPT

.

.

> 2

. -2o.

z : sp

-6

-4o.

-5 .

cr" -6O-

+

+ +

+

+

. 0

I 50 E

I 100 (Me’/1

-2

2 I z -L “0 >

-I

E

-4 -3

-8Ok -100

+

I 150

2000

FIG. 9. The depth of the real and the imaginary part of the optical potential as a function of the energy (E) of the incident particle for kp = 1,4fm-’ using Hamman potential and A,,approximation.

The real part of the potential is about 90 MeV deep at zero energy and becomes more shallow as energy increases at an almost linear rate. Below 100 MeV it may be fitted with the function Re I’,,, ~1’ (-90

+ 0.584 E) [MeV].

(46)

In comparing our results with experimental data we must keep in mind two important factors already discussed in connection with nuclear matter. The first is the strong triplet-even component of Hammann’s force, which may overshoot the binding energy and the chemical potential by as much as 20 MeV; the second is the rearrangement contribution of the order of 9 MeV at the top of the Fermi sea for k, = 1.4fm-I. Both factors work to decrease the potential depth to about 60 MeV at zero energy, but to an extent unknown at higher energies. With these reservations in mind we can state that our calculation has some

SELF-CONSISTENT OPTICAL

POTENTIAL

251

degree of success if compared to Perey’s analysis [50], which leads to the following empirical value for the real part of the potential inside a heavy nucleus: (-53.3

+ 0.55 E) [MeV].

(47)

The depth at zero-energy can also be compared, although with less significance because of different methods and potentials, with the theoretical estimates, 84 MeV by Reiner [19] and 63 MeV by Kidwai and Rook [51]. The imaginary part of the potential, obtained in the present work, varies very slowly with energy and below 100 MeV it can be represented as (2.4 + 0.009 E).

(48)

This is to be compared with the calculation of Shaw [18] which in second order gave a value of 2 MeV at 16 MeV incident energy. Reiner obtained similar magnitude of the imaginary part of the optical potential below 100 MeV incident energy but above 100 MeV there appears an abrupt change. The empirical values used to fit the elastic neutron scattering from various nuclei cannot be compared directly with the values given by Eq. (48) since the nuclear surface plays an important role in the case of finite nuclei. In addition it is usual to use a surface absorption rather than a volume absorption optical potential. Using a Woods-Saxon shape, Seth [52] fitted the experimental strength functions with a magnitude of 3.2 MeV for the volume absorption optical potential. A more recent collection of the optical potential parameters [54] indicates that the strength at low energies is - 5 MeV and increases slowly with energy. Our results are in qualitative agreement with the empirical values and the difference can be attributed to the presence of collective effects in finite nuclei. It has been shown by Vinh Mau [20] that the collective effects can contribute approximately 2 MeV to the imaginary part of the optical potential while they have a small effect on the real part of the optical potential. This explains why our results (Eq. (46)) after correction for the anomalous strength of triplet-even component of Hamman’s potential yield a good estimate of the depth of the real part of the optical potential and an excellent energy dependence which can be compared quite well with the values used in heavy finite nuclei [55, 571. 4. CONCLUSIONS On the basis of the Green-function formalism, properties of the particles below and above the Fermi level were calculated self-consistently and on the same footing. The main object was to obtain the self-energy of a particle interacting with an infinite nuclear system as a function of energy and momentum. From this general function Z(:(k, w), a single-particle energy, w(k), and a momentum distribution,

258

HO-KIM

AND

KHANNA

p(k), were defined, and the “on-shell” part of the self-energy was introduced, Z(k, w(k)), leading to all the physical quantities of interest. The energy per particle in nuclear matter is found to be too large for the case of Hamman’s [13] and Mongan’s [14] potential. But this can be traced to the fact that both of these potentials give a D-state probability in the deuteron of -1 %. When the tensor force is strengthened by using the potentials due to Afnan et al [15] the energy per particle decreases and for a D-state probability of 4 % the minimum for the energy appears at kF - 1.75 fm-l with a magnitude of -14 MeV per particle. This is further evidence for the essential role of the tensor force in bringing about saturation in nuclear matter [4, 51. It is interesting that the T-matrix approach that includes effects due to hole-hole scattering in the intermediate state leads to results for the binding energy of nuclear matter that are quite similar to those of the conventional reaction matrix techniques using reference spectrum method. The hole-hole scattering adds a small amount -1 MeV to the energy of the ground state. This is quite consistent with the conclusion by Day (48) that hole-hole scattering in perturbation theory yields only -0.34 MeV additional binding. Our results substantiate the perturbative conclusion of Day and support the linked cluster expansions of Brueckner-Bethe-Goldstone based on hole-line expansion type that have been advocated by Brandow [30]. The use of Green’s functions seems very attractive but the complete mapping of functions like Z(k, 0) involve a great deal of work that may be unwarranted due to the development of other techniques. Furthermore the study of Fiset allows estimates of three particle clusters and it appears that the magnitude of these numbers will be comparable to those obtaine by Bethe [46] and Day [48]. The single particle spectrum given by w(k) indicates that there is neither a discontinuity nor a gap at k = k, . The single particle energy for k < kF is a complicated function of k/kF but it approaches k2/2A4 asymptotically for large values of k. For k < kF , w(k), the single-particle energy, contains a contribution due to rearrangement effects. The momentum distribution function, p(k), deviates from unity and has a magnitude in the range 0.854.95 for k/kF < 1 for several values of kF near the minimum of the energy. Inclusion of the calculated value of p(k) rather than unity changes the binding energy per particle by a few MeV. Of the two two-particle propagators investigated, one, the -&-propagator, is expected to be conserving, and indeed our numerical results on nuclear matter prove that, in this respect, the class of rearrangement effects which it includes played a deciding role. With a contribution of 6 MeV per particle to the average energy, and 9-28 MeV to the single-particle energy from such effects, the Hugenholtz-Van Hove theorem comes close to being satisfied. The small deviation, normally tolerable considering errors from other sources, come from several approximations introduced for practical reasons.

259

SELF-CONSISTENT OPTICAL POTENTIAL

The “on-shell” part of the self-energy, as we define it, is essentially the optical potential for a particle scattered by a large nucleus. The optical potential we present is obtained in a nonconserving approximation, and consequently without the benefit of rearrangement effects. These may not seriously affect the real part of the potential, but may modify the imaginary part by as much as 1 MeV. We know that the same effects appear in the experimentally observed width of the IS hole states in nuclei with A < 40; the prediction, basedon a calculation [53] using Mongan’s potential [14], is that the width increaseswith A, being 14, l&6,20.15 and 20.8 MeV for Y, 160, 28Siand 40Ca, respectively. Finally mention must be made of the long-range correlations, which are ignored in our truncating procedures and are unimportant in infinite nuclear matter, but which become an essentialpart in the correct treatment of finite nuclei at low energies,ashas been emphasized by VinhMau [20]. At higher energies (E > 50 MeV), our formalism, which treats shortrange correlations correctly, should describe the experiments quite well. The depth of the real part and the volume absorption imaginary part of the optical potential appear to have very reasonable magnitudes and the energy dependence of the potential is in excellent agreement with the empirical values [46]. The successof the present investigation seemsto justify renewed efforts to overcome the technical difficulties and give a firmer foundation to the calculation of the optical potential in nuclear matter. While the use of a more realistic local interaction poses no obstacle, the inclusion of the rearrangement effects asin the A,,-approximation and the removal of the zero-width assumption are more difficult but certainly not beyond our means. Because of the analytic properties [56] of fl,, propagator, it has not been possible to define the single-particle energies for k > kF in a form that is easily tractable on the computer. Even a prescription for single-particle energies will be most useful.

ACKNOWLEDGMENT One of us (F.C.K.) L. Wilets.

would

like to acknowledge

some very

useful

conversations

with

Professor

REFERENCES 1. K. A. BRUECKNER,

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AND

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260

HO-KIM

AND KHANNA

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