Binding energy of nuclear matter from a physical particle spectrum

Nuclear Physics A245 (1975) .41.1~t28; ~ ) North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or mierofilm without written permission from the publisher

B I N D I N G E N E R G Y OF N U C L E A R M A T T E R F R O M A PHYSICAL PARTICLE S P E C T R U M J.-P. J E U K E N N E , A. L E J E U N E and C. M A H A U X

Institut de Physique, Universit~ de Likge au Sart Tilman, B-4000 Lidge 1, Belgium Received 18 July 1974 (Revised 24 January 1975) Alrstraet: We use a non-vanishing potential energy for the intermediate particle states in the BetheBrueckner expansion for the binding energy of nuclear matter. We identify this potential energy with the depth of the optical-model potential. It turns out that, in addition to this physical meaning, the potential spectrum shares the properties previously required by several authors: it presents no gap at the Fermi m o m e n t u m kF; it is small and possesses the proper energy and density dependence at high momenta. Detailed calculations are performed with Reid's hard core interaction. The twohole contribution B 2 is found to be - 1 2 . 2 MeV at the saturation m o m e n t u m k F = 1.30 f m - 1 . A crude estimate o f the three- a n d four-hole line contributions gives B = - 15.60 MeV, k F = 1.34 fin- 1. These results lie closer to the empirical values t h a n those obtained in the conventional calculations where the potential energy in particle states is set to zero : In the latter case one finds, in the two-hole line approximation, B 2 = - 9 . 0 5 MeV, k v = 1.30 fro- 1 ; by estimating the three- and four-hole line contribution, one obtains B = - 9 . 7 0 MeV, k~ = 1.36 f m - 1 .

1. Introduction

The rate of convergence of the Bethe-Brueckner expansion for the binding energy of nuclear matter 1) depends upon the choice of an auxiliary single-particle potential energy U k. Henceforth, we call U h the potential energy for the hole states (k < kF), and Up that for particle states (k > kF). Here, k F denotes the Fermi momentum. The general consensus is that U h should be taken self-consistently according to the Brueckner-Hartre~-Fock prescription. Then, the contributions of graphs lb and lc cancel each other. Since the middle reaction matrix in fig. 1d is off-shell, no similar and simple prescription exists for particle states. Brueckner and Gammel 2) and Coon and Dabrowski 3) define forp > k F a potential Up(z) which depends upon an off-shell parameter z and is such that l d + le = 0. It was, however, emphasized by Bethe 4) and Brown 5) that for large values of p (p >__3 fm-1) other three-hole line graphs should be included on the same footing as 1d. Their total contribution can be cancelled by taking, for k F = 1.4 f m - 1, a potential Up which is slightly positive but small for p ~ 3 f m - 1. Therefore, it appears convenient to take Up = 0 [refs. 4, s)]. These arguments were systematized by Brandow 6, 7) who, however, emphasizes that the choice Up = 0 is probably not optimal for low intermediate states, i.e. below 2 or 3 fm- 1. There, it rather appears more natural to adopt a shell-model type prescription, i.e. some suitable extrapolation of the hole-state definition 7, 8). In particular, K6hler 411

412

J.-P, JEUKENNE et Ow~O

(a)

~

al.

~--×U

(b)

(c)

{e)

(f)

(hl

(il

C3 (d)

(g)

Fig. 1. Severalgraphs appearingin the hole-lineexpansionfor the bindingenergy. and Baranger 9-12) proposed that Up should be taken according to Brueckner's original prescription, i.e. in such a way that ld + le = 0 for low intermediate states only (kr < p < kL), while one sets Up = 0 for p > k L. They suggest the value k L ~ 2k F. Calculations of this type were recently performed by Kao 10), K6hler 11,12) anJ Dahlblom and Kouki 13). They find that the use of a non-vanishing potential between k~ and k L increases the value of graph 1a. However, K6hler 12) also observes that this value, and even more so the sum 1a + 1f, critically depends upon the choice of k L. He also shows that the value of graph If is increased. Here, we return to the intuitive feeling, expressed for instance in ref. 9), that one should take into account the fact that a nucleon slightly above the Fermi surface feels an attractive potential. This corresponds to the introduction of a model space 7,14-is) or of shifted particle energies 19,20) in finite nuclei. More precisely, we investigate whether the use of physical (shell-model) energies for all momenta would lead to sensible results for the binding energy. By physical energies, we refer to the on-shell self-energy insertions, which correspond to the depth of the opticalmodel potential 25). In lowest order, this is the standard choice for hole states; for particle states, our value for Up would cancel a graph analogous to fig. ld, but with the middle g-matrix put on-shell. Such a graph does not occur in the Bethe-Brueckner energy expansion, but this is irrelevant for the practicability of our prescription. We emphasize that our choice differs from that proposed in refs. s- 12), which involves off-shell parameters. In particular, our values of Uk are in keeping with Brandow's suggestion 7) that "a general requirement is that the intermediate spectrum should be continuous at kF". Besides its attractive simplicity in terms of physical intuition, there exist two main reasons to expect that our prescription will yield reasonable results for the

BINDING ENERGY

413

binding energy. Firstly, our choice for Up leads to a near-cancellation of graphs ld and le for low intermediate states p [refs. 7,14)]. Secondly, the on-shell (physical) potential energy is small and positive for the momenta (3-5 fm- 1) which are relevant for the short-range three-body correlations discussed in refs. 4- 7); this is shown by the systematics of the empirical optical-model potential 21). These properties are corroborated by the calculations described below. Hence, our on-shell prescription shares the properties required in refs. a-12) and in refs. 4-7) in the relevant energy ranges, namely p < 3 fm-1 and 3 f m - 1 < p < 5 fm -1, respectively, while also carrying a physical meaning 3 5). However, we note that our choice for Up is not based on a systematic cancellation of graphs. As we just indicated, and discuss further in subsect. 3.3., we believe, however, that it keeps the three-hole line corrections small. In sect. 2, we give a more detailed description of our prescription for Up, of the corresponding energy expansion and of the relationship with other approaches. Our numerical results for the potential U k at various kF are presented in sect. 3, for Reid's hard core nucleon-nucleon interaction 23). The values of the binding energy corresponding to graph la and to l a + If, respectively, are calculated in sect. 3, where we also discuss the magnitude of the higher cluster terms. Finally, sect. 4 contains a brief discussion.

2. Single-particle potential A low density expansion of the physical ("true" 24)) potential energy in nuclear matter is given in ref. 25) and its first few terms are shown in fig. 2. In figs. 2a-2c, all g-matrices are on-shell; in fig. 2d, the middle g-matrix is off-shell. We briefly recall a few definitions. We denote by v the nucleon-nucleon interaction and introduce the notations (h = 1) n>(k) = 1-n<(k) =

{10 if k > k F if k < kv,

(2)

e(k) = kE/2m+ Uk,

(a)

(b)

(dl

(1)

(c)

(el

Fig. 2. Several graphs appearing in the hole-line expansion for the physical single-particle potential.

414

J.-P. JEUKENNE

et al.

where Uk is the auxiliary ("model" 24.)) potential energy of sect. 1. We define a nonhermitian reaction matrix by the Bethe-Goldstone equation [Pq)(Pql # ( W ) = v + v ~ n >(p)n >(q) W - e(p) - e(q) + i6 g(W)'

(3)

P,q

where for simplicity we use the same notation for a vector and its length. An infinitesimal i6 appears in the denominator because values such that W = e(p) + e(q) will appear below. In lowest order, graph 1a, the binding energy of nuclear matter is given by B2 = ~ k~/m + WE,

(4a)

W2 = ~ n < (j)n < (k)( k jlg(e(k) + e(j))lkj)A,

(4b)

where j,k

where the index A refers to antisymmetrization. In first order, graph 2a, our prescription for U k reads Uk = Re ~ n <(j)(kjlg(e(k) + e(j))lkj)A.

(5)

J

For hole states, definition (5) is formally identical to the standard self-consistent choice 1.2); it corresponds to the "on-shell" part of the mass operator (M °n) in Brandow's terminology 7). The potential (5) is continuous at k F. For particle states, expression (5) is the leading contribution to M °ff in the language o f ref. 7). However, it is here evaluated on the energy shell. In the remaining of this section, we describe the difference between our prescription and previous choices for Uk, and we discuss the higher-order contributions. In most calculations performed since 1967, U v is set equal to zero. The resulting potential U k has a gap of about 50 MeV at k F . The open dots in fig. 3 show the general trend of the corresponding results for the binding energy and saturation momentum. Most calculated values are taken from the work of Wong and Sawada 26) and the attached letters refer to the initials of the authors who developed the corresponding nucleon-nucleon interactions 23, 2v- 30). The dot labelled R H C (Reid's hard core 23)) has been calculated by us, as described in sect. 3 ; it only includes l __<2 partial waves, except 3D 3 , while the other points include all l < 10 partial waves 26). According to Siemens 45), the omitted partial waves would modify our R H C value by a negligible amount, namely - 0 . 0 2 MeV. In the case of the RSC interaction, these higher partial waves give a contribution of about - 0 . 4 0 MeV at k F = 1.36 fm-1 [-ref. 43)]. It is a general feature that an increased binding is systematically associated with an increased saturation density. Consequently, all the open dots in fig. 3 are located within one band 26), delimited by the two dotted lines drawn to help the eye. This band does not contain the empirical region (B = - 16 MeV, k F = t.36 f m - 1), which

BINDING ENERGY

-5

415

% \ R~\

BKR

-10

>~ -15

\\

-20

\ -25 1.1

I

I

1.3

I

I

1.5 k F {fro-1)

I

I

1.7

Fig. 3. Compilation of binding energy calculations for several nucleon-nucleon interactions. The open circles refer to the two-hole line contributions (B2) with Up = 0 (sect. 2), the triangle and the cross to the values o f B 2 obtained from our choice o f Up # 0 (subsects. 3.2 and 3.4, respectively). The arrows take higher-order graphs into account (subsect. 3.3): the full arrows point towards B 2 ÷ W 3 + if'4, the dotted ones to B 2 + if'4 and the dash-dot ones to B 4. The square represents the empirical values of B and k F.

is represented by a square. Hence, the major problem is to move the calculated saturation point out of this narrow band. Day 31) mentions three main ways of doing this. (i) Including higher-order terms of the expansion may be helpful. The arrows in fig. 3 indicate the effect of the inclusion of the three- and four-hole line contributions, as evaluated in ref. 26) and as discussed in subsect. 3.3. We see that these two corrections do not move the saturation points outside the narrow band. (ii) A second possibility is that long-range correlations should be better taken into account. According to Brandow 7, t 6), one way of doing this is to use a potential U k which is continuous at k r. Our prescription has this property. This choice redefines the expansion for the binding energy. As we shall discuss in subsect. 3.3, it probably decreases the value of the three-hole line graphs but increases the two- and four-hole line contributions 11, 12). This seems to indicate that for our choice of Up, the two- to four-hole line terms include a fraction of the m a n y - (five and more) hole line terms of the conventional expansion (Up--0). This may be necessary for the long-range correlations 7, 16). We return to this point in sect. 4. (iii) A third possibility is that relativistic and mesonic effects must necessarily be included, which implies partially giving up at least the phenomenological model based on nucleons interacting via a two-body potential and obeying the non-relativistic Schr6dinger equation. For instance, the point labelled HEA in fig. 3 refers to the value of B 2 calculated in ref. 32) from a one-boson-exchange potential involving a fictitious a-meson.

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J.-P. J E U K E N N E et al.

According to Green, Dahlblom and Kouki 33), the replacement of this a by a 27: exchange shifts the value of B 2 as indicated by the dotted arrow. This arrow points towards lower densities, i.e. in the right direction to solve the binding energy versus saturation density problem; however, it also decreases the value of IB21. In a more recent calculation, limited to the 1So partial wave, Green and Haapakoski 22) show that replacing the a-meson by an inelastic process involving the A(1236) may provide some additional help. In the present paper, we deal with point (ii) and find that a physically meaningful, and at the same time reasonable, choice for Up brings the saturation point outside the narrow band. We now turn to the three-hole line contribution. For large values of the particle momentum p, all three-hole line graphs should be included on the same footing 4' 7). For low values of the particle momentum p however, graph ld dominates a). It is thus appropriate to choose Up in such a way that graphs ld and le cancel each other for p close to k F. This choice is discussed in refs. 2, 3, 10,12). It is also closely related to a proposal by Da Providencia and Shakin 34). In refs. 10,12), a potential Up(z) is introduced for p < k L, where k L is some cut-off value beyond which Up is set to zero (sect. 1). This Up(z) depends upon an off-shell parameter z, determined by the momentum of the second excited particle q and by the hole momenta j and k (see eqs. (2), (3) and (5), refs. 2, a, ,0,12)). For details , we refer to ref. 12). For p and q close to k r, Up(z) becomes identical to our "physical" potential Up [ref. 14)]. The leading term of the hole-line expansion for the self-energy or mass operator M(k, E) reads 2s)

M(k, E) = ~ n <(j)(kj[g(E + e(j))lkj)A.

(6)

J

The complex optical-model potential can be identified with 25, 3s)

Up + i ~ = M(p, e(p)).

(7)

Brandow 7) suggested using the following approximation in the calculation of the off-shell values appearing in Up(z) [dM(k, E)]

Up(z) ~ Up(zo)+ ( z - Zo) L

dE

E=zo'

(8)

where z o is some average value. We shall return to this approximation in sect. 3. The expression

fk = [1- [dM'l

-I-1= [l-M}<] -1,

(9)

gives for k < k F, and in lowest order, the occupation number of momentum state k. The quantity 1 - f k is approximately equal to the "small parameter" x of the BetheBrueckner theory 7). In sect. 3, we shall see that M~ ~ -0.27, for k F -- 1.4 fm -1 and for our choice for Up. Since a typical value of z - e ( p ) is - 6 0 MeV for p close to

BINDING ENERGY

417

kF, w e expect our potential Up to differ from the "average" Up(z) by about 15 MeV

in this region. This will be confirmed in sect. 3. The introduction of some cut-off momentum k L in refs. lo, 12) is necessary to ensure a proper convergence of the hole line expansion, since Up(z) has been shown 3, 5) to increase likep 2 for large values ofp. One drawback is that the saturation curve depends upon k L in a fairly critical way 12). For instance, graph 1a leads to the saturation momentum 1.40 f m - 1 for k L = 2k F (all kF), while the saturation momentum 1.47 fm -1 is obtained for the cut-off k L = 2.8 fm -1 (all kF). In the case k L = 2.8 fm-1, moreover, no saturation is found for k F < 1.6 fm-1 if diagrams la and 1f are added 12). The contribution of the whole set of three-hole line graphs has been investigated by Dahlblom 13, 36) and others 1). It depends upon the choice of Up. In the case Up = 0, Wong and Sawada 26) use the approximation 36) W3 ~ +1.3(kF/1.36) 5

MeV

(10)

for one of Reid's hard core interaction. This value is calculated from an approximation of the Bethe-Faddeev series 1) which is accurate only for the short-range part o f the interaction. We shall return in subsect. 3.3 to the effect on W 3 o f a change

of up. The four-hole line graph 1f can easily be calculated. It is given by W4 = (M')2W2,

(Ii)

where M' is an average value of IM~,) for k < kF, and where Wa is the value of graph la. As mentioned below eq. (9), the quantity M' can be identified with the small parameter x of the hole-line expansion i, 6). The value ff'4 of the sum of all four-hole line graphs has been discussed by Day 37). Day's estimate is summarized by the formula 26) W3 +

~V4 =

~ F z W z - a F 2 +(1 - ] F ) W 3 ,

(12)

where a ~ 37 MeV, while F is an average value of 1 --fk (eq. (9)). The full arrows attached to the open dots in fig. 3 show the effect of W 3 + I~4 : the saturation points are pushed towards larger densities and remain in the narrow band. The dotted arrow attached to the open dot labelled R H C points to the value of B 2 + W, (eqs. (4) and (I 1)). Finally, we make a comment on the connection between expression (5) and the "physical" single-particle energies. The choice (5) for the particle potential energy Uk corresponds to graph 2a only, while the physical single-particle potential energy is given by the sum 2a + 2b + 2c + .... Hence, we use here only an approximate value for the physical potential. One might identify U k with the complete expression for the physical potential energy but some caution must be exerted. We illustrate this point by considering the contribution o f graph 2c. Its inclusion in U k would not cancel the sum l b + If with lc, because of a factcr ½ associated with the symmetric

418

J.-P. J E U K E N N E et al.

graph 1f. This is discussed, for instance, by Brandow is) who shows that the following relation approximately holds 1~2 -~ (1 +2F2)W2,

(13)

where the quantity # 2 is calculated from Uk = 2a + 2c; we recall that W 2 is calculated from graph 2a alone. In subsect. 3.3, we shall see that eq. (13) is indeed fairly accurate. The Pauli rearrangement graph 2b could also be included in the potential spectrum. It is large (,~ 15 MeV) for hole states and small for particle states 38). This inclusion would thus modify W2 by about - 3 MeV. However, the sum l a + l b + l c would remain essentially unchanged 39). Finally, we note that the physical potential energy used here differs (mainly above kF) from the mean removal energy recently discussed by Koltun ,0). 3. Numerical results

In the present section, we present numerical results obtained from Reid's hard core (RHC) interaction 23). We include all partial waves with l < 2, except 3D 3 . The calculational procedure is based on the Brueckner-Gammel method 2) and is described in ref. 35). In subsect. 3.1, we give for several values o f k F the values of Uk computed from eq. (5), and we compare our results with previous prescriptions. The two-hole line contribution B 2 (eq. (4a)) to the binding energy is calculated in subsect. 3.2. Subsect. 3.3 contains estimates of the size of the three- and four-hole line terms. Finally, we describe in subsect. 3.4 the results which one obtains when the complex kernel in eq. (3) is replaced by a principal value integral. 3.1. S I N G L E - P A R T I C L E P O T E N T I A L

The full line in fig. 4 represents the values of Uk obtained from eq. (5), for k F = 1.4 fro-1. The dot-dash curve corresponds to the values obtained for Uh with the "standard" choice Up = 0. Two main features emerge. Firstly, our prescription leads to an increase of B 2 as compared with the standard choice. Secondly, Up appears to approach a constant forp ~ 5 fro- 1. The increased binding energy is a consequence of the suppression of the gap a t k F ; the repulsive potential energy beyond 2.3 k F only partly cancels this effect. The short dashes in fig. 4 show the results of K6%ler 12) in the case of Reid's soft core (RSC) interaction 23). The three curves between k F and 2k F represent Up(z) with, from bottom to top, z = - 7 0 , - 140 and - 2 8 0 MeV, respectively. The z-dependence of Up(z) is in fair agreement with eq. (8), since here 31" ~ 0.27, as we shall see below. We see that our potential spectrum lies at the lower edge of the off-shell range of values. Therefore, we expect that graph 1d approximately cancels le for our choice of Up. We return to this point in subsect. 3.3. Incidentally, Brandow had previously noted (p. 236 of ref. 16)) that the parameters of the momentum dependent optical-model potential used by Perey and Buck 41) "are rather close

BINDING ENERGY '

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419

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i

,

-

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20

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/ f / /

#, / //,/

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/ / ,

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U D (1.36 fro-l) -

I

~ " -8o>~/

------}

1.0

UK(1.4fm -1) I

2.0 k/k F

I

3.0

Fig. 4. A plot of the single-particle potential versus k/k F, for k F = 1.4 f m - 1. The full curve corresponds to our choice of Up # 0 (eq. (5)) and the dash-dot curve to the case Up = 0, respectively, for Reid's hard core interaction. The short dashes represent K6hler's results 12) for Up(z) with, from b o t t o m to top, z = - 7 0 , - 1 4 0 and - 2 8 0 MeV, respectively for Reid's soft core interaction. The long dashes show Dahlblom's estimate 36) for the potential which would cancel the short-range contribution of the R H C interaction to the three-hole line graphs.

to those which would fit the negative portion of the Brueckner theoretic spectrum of Coon and Dabrowski 3) for particles which propagate not far off the energy shell". This remark corresponds to the similarity discussed above and shown in fig. 4; it also confirms the physical meaning of our potential spectrum (see ref. 35)). The long dashes in fig. 4 represent the potential energy which, according to Dahlblom 36) would cancel W3 (eq. (10)) for k v = 1.36 fm-1. We see that Dahlblom's potential energy is fairly close to ours in the momentum range 3-5 fro- 1 which is relevant for that part of the interaction that he considers, namely the short-range part. Fig. 5 shows the momentum and density dependence of U k in the domain 3 < k < 5 fm -1 and 1.0 < k F < 1.4 fm -1. We see that the repulsion increases with kv.This can be understood in the framework of the impulse approximation, which predicts Up to become proportional to k 3 for largep. Fig. 6 shows that the k 3 behaviour (long dashes) is indeed approximately valid at 5.5 fm-1, but not quite so at 3.8 fm-1. At 4.5 f m - t , the dependence appears closer to kSv (short dashes). This result may be compared with the k F dependence of W3 (eq. (10)) and implies that our prescription has the proper density dependence to minimize the three-hole line contribution for values of k v close to 1.4 fm- 1. We return to this important point in subsect. 3.3. At3.8 fm-1, Up becomes proportional to k~ 1, but is then fairly small.

420

J.-P. J E U K E N N E et al. ~

30

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~ 1 . 3

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i

///

30 ,o

_

i

///

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//

/

>

0

1.0 ,

3

~

5

k (frn-11 Fig. 5. High momentum behaviour of Uk (eq. (5)) for several k F.

1.2 kF (fro-1)

1.4

Fig. 6. Dependence of Uk (eq. (5)) upon kF, for three values of k (full lines). The long and the short dashes represent curves proportional to k 3 and kF5 laws, respectively.

Finally, we note that for large p, Up should approach zero like (sin pc/p), where c is the hard core radius. We checked that this occurs beyond 7 f m - t 3.2. B I N D I N G E N E R G Y

The value of

B2

is given by eq. (4), or, equivalently,

B2 = Z n<(j)[j2/Em + ½Uj] •

(14)

J

The full curves in fig. 7 represent the k F dependence of B 2 calculated from a vanishing particle potential Up (upper curve) and from the prescription (5) (lower curve), respectively. The saturation points lie at B 2 = --9.05 MeV, k F = 1.31 fm -1 for Up = 0, and at B 2 = - 12.20 MeV, k F = 1.30 fm- 1 for Up ~ 0. It is remarkable that the saturation density remains essentially unchanged while we gain about 3 MeV .in binding energy. A similar trend was found by K61aler x2) from the choice (6)-(8), with the cut-off value k L = 2k v (for all kF). K61aler finds that B 2 is then increased by 1.5 MeV, while the saturation momentum decreases from 1.44 to 1.40 f m - x, for the RSC interaction. As we mentioned in sect. 2, his findings are, however, sensitive to the choice o f the cut-off m o m e n t u m k L . The value of B 2 obtained from our prescription (eq. (5)) is represented by a triangle in fig. 3. We see that it falls well outside the narrow band discussed in sect. 2. This is due to the conjunction o f two characteristic features of our choice, namely the suppression o f the gap at k F and the increase with k F of the repulsion at large momenta p (fig. 5).

BINDING ENERGY

m

-12

421

-.~

J Up~O

-16 1.0

I

1.1

11.2

I

13

.

11./.

I

1.5

k F (fro-1) Fig. 7. Saturation curves B(kF) for Up = 0 and Up # 0 (eq. (5)). The full lines give the two-hole line contribution B z (eq. (4a)); the long dashes show B 2 + W 4 (eq. (11)) and the short dashes correspond to B 2 + W~+ if'4 (eq. (12)). 3.3. T H R E E - A N D F O U R - H O L E L I N E C O N T R I B U T I O N S

(i) Three-hole line contributions. T h e expression (10) for the three-hole line contribution corresponds to a vanishing particle potential Up. We mentioned that the short-range contributions to W3 vanish if Up is taken according to the long dashes in fig. 4. Since our choice is close to this curve and has a similar density dependence in the relevant momentum range 4-5 fm-1, our value for W 3 would probably be smaller than (10). One might wonder whether the suppression of a gap at k v, which implies a larger value for the small parameter x will not spoil this conclusion. We believe that this will not occur, for two main reasons. Firstly, our spectrum Up in the domain k v < p < 2.5 fm- 1 is close to the one which cancels the three-hole graph ld which is dominant 8) in this momentum range. Secondly, the short-range contributions to the Bethe-Faddeev series is not much affected by the suppression of the gap at k F, as shown by table 2 of ref. 13). We conclude that the three-hole contribution to B is, with our prescription for Up, probably smaller than the estimate (10). (ii) Four-hole line graphs. We mentioned that one of them, graph If, can easily be calculated and is given by eq. (11). In fig. 8, we plot the occupation number fk (eq.. (9)) versus k, for various kF, in the cases of a vanishing particle potential Up (dashes) and of our choice (5) (full curves). We see thatfk is smaller in the latter case, which corresponds to a larger value of the small parameter ~c. We take for M' in eq. (11) the value of IMP,[ at the most probable value k = 0.75 k F. The dependence of M' upon k v is also plotted in fig. 8. The long dashes in fig. 7 represent the dependence upon k F of B 2 + W 4. The saturation points lie at B 2 + W 4 = - 1 0 . 1 0 MeV, k F - 1.35 f m - 1 in the case Up = 0, and at - 14.55 MeV, k F --- 1.33 f m - 1 for our choice Up ~ 0. The dotted arrows in fig. 3 point towards these values. We note that in the ease Up ~ 0, our result comes quite close to the empirical value.

422

J.-P. JEUKENNE et al. 1

I

I

I

0.30 0.85 0.25 0.80 I

0.75 ... k F = 1.25 fm-1

J

J

J

Is" I 1.0

I I I I

0.20 0.15

I 1!4 1.2 kF (fm-1}

0.85

0.85

0.80

o.8o

0.75 _ k F=1.10

k F = 1.40 fm -1

fm -1

I

I

0.2 k

I

I

0.6 (fm -1)

I

I

0.2 k

0.75 I

I

0.6 {fm -1)

Fig. 8. Momentum dependence of the occupation numberJk (eq. (9)). The long dashes and the full curves correspond to Up = 0 and to Up # 0 (eq. (5)), respectively. The dependence of M' (eq. (11)) upon k F is shown in the top-right corner.

We also performed for k v = 1.4 fm -1 a calculation of if'2 in eq. (13). The value of the sum 0 k of graphs 2a and 2c is given by U k = Re ~ n < (j)(1 J

4- M' j)(jkLq(~(j) + e(k))ljk)A.

(15)

We made a self-consistent calculation of 0 k. We obtained the value - 40.0 MeV for the left-hand side of eq. (13), as compared with - 3 9 . 2 MeV for its right-hand side. Brandow's relation (13) is thus rather accurate. In order words, it is practically equivalent to calculate the value W2 corresponding to Uk of fig. 2a, or to compute I5"2 with Uk given by 2a + 2c and then use eq. (13). The former procedure is usually more convenient. The main interest of relation (13) is to indicate that one could take for Uk the full expression of the single-particle potential energy (all graphs of fig. 2), instead of the leading term alone (graph 2a), provided that one corrects for the effects of double counting and of self-consistency, as expressed by eq. (13). We also compute the value of B E 4- W 3 -~- ~/'4 (eqs. (10) and (12)) for our choice of Up. We note, however, that this quantity may not be very meaningful, since we believe that W 3 is, in our case, smaller than the value given by eq. (10). In fig. 3, the full line arrows point to the corresponding saturation values B24- W 3 + 1 ~ 4 - - 1 5 . 3 0 MeV, ki~ = 1.32 fm -1 (Up # 0), to be compared with B 2 + W 3 4 - 1/~/"4 = - 9 . 7 0 MeV, k F = 1.36 fm-1 for the conventional.choice Up = O. The comparison between the values o f W 4 in the cases Up = 0 and Up # O, respectively, reveals two striking features. Firstly, the upwards shift of the saturation density due to W4 is smaller in the case Up ~ 0 (see fig.? 3). This reflects the fact that the density dependence of M ' is smaller in that case (fig. 8). This property is closely

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ENERGY

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related to the fact that Up increases with k F at high momenta (fig. 5). Secondly, the value of I4/, is larger (in absolute magnitude) for Up # 0 than for Up = 0. This reflects the larger value of M' in the former case (fig. 8). A similar effect was previously observed by K6hler 11.12) for his choice of Up, discussed in sect. 2. According to the discussion in section (i) above, we believe that W 3 is smaller than expression (10), for our choice Of Up. Accordingly a better, although admittedly still quite crude estimate of the sum of the three- and four-hole line contributions to the binding energy is obtained in setting W a = 0 in eq. (12). We call B 4 the corresponding value of the binding energy (a = 37 MeV)

B, = B2 +~F2W2 - a F 2.

(16)

We find B 4 = -15.60 MeV and the saturation momentum kF = 1.34 fm -1. The dash-dot curved arrow starting from the open triangle in fig. 3 points towards these values. We note that a fraction of the four-hole line graphs is given by - s F W 3 , i.e. is proportional to W 3 [ref. 37)]. If that part of the four-hole line contribution is retained, with W 3 given by eq. (10), we find that B2+ i f ' 4 - W3 equals - 1 6 . 5 MeV at the corresponding saturation momentum kF = 1.36 f m - 1. These latter numbers, however, do not appear to be very meaningful; we provide them mainly for indicating the amount of uncertainty which is involved in our estimates. In table 1, we compile the main numerical results for the binding energy and saturation momentum k F obtained in subsects. 3.2 and 3.3. TABLE 1 C o m p i l a t i o n of binding energies a n d saturation m o m e n t a a) Binding energy (MeV)

B2 b) B 2 + W4 °) B2 + W3 + i,~4 d) B4 e)

v~=o

v~#o

- 9.05 - 10.10 - 9.70

-

12.20 14.55 15.30 15.60

k F saturation (fm-1)

v~=o

v~¢o

1,31 1.35 1.36

1.30 1.33 1.32 1.34

a) All partial waves with l < 2 are included, except 3D 3 . b) Eq. (14). ¢) Eq. (11). d) Eqs. (10), (12). ') Eq. (16).

3.4. E F F E C T

OF THE ABSORPTIVE

PART OF THE POTENTIAL

The auxiliary potential Up defined in eq. (5) is real, but contains the influence of the imaginary part of the kernel in eq. (3). While this influence is small for p < 3 fm- a (see fig. 1 of ref. 35)), it becomes sizable beyond this value. This reflects the fact that Wp is then as large as Up (eq. (7)). It is thus of interest to study to what extent the inclusion of the imaginary part is important. In fig. 9, we show the values of the po-

424

J.-P. JEUKENNE et al.

100f

-

'

/

//135

,o1:- r

I

//., //Y"

I

I

-10

> -12 J

,-n -14 -16

3

4

k (fm -11

I

1.0

5

Fig. 9. High momentum behaviour, for several kv, of the single-particle potential obtained from eq. (5) when the + i6 is omitted in eq. (3).

I

1.1

I

I

1.2 1.3 kF (fm-1)

I

1.4

Fig. 10. Saturation curves corresponding to the particle potentials of fig. 9. Same conventions as in fig. 7.

tential energy obtained f r o m eq. (5), when omitting the + / 5 and using a principal value integral in eq. (3). W e note that the repulsive potential in fig. 9 is stronger at high m o m e n t a than in fig. 5. This leads to a smaller binding energy and to a smaller saturation density than f o u n d in subsect. 3.2. This is explicitly shown in fig. 10, where we plot the corresponding values o f B 2 , B 2 -k- W 4 and B 2 Jr W 3 Jr- l'~4 . The saturation points lie at B 2 = - 11.5 MeV, k F = 1.25 f m - 1 ; B2 + W4 = _ 13.75 MeV, k v = 1.27 f m - 1 ; B2 + W3 + if'4 = - 14.7 MeV, k v = 1.24 f m - 1. These values are represented by a cross (B2), a dotted a r r o w (B 2 + I414) and a full a r r o w ( B 2 4- W 3 + I'V4) in fig. 3. The d a s h - d o t a r r o w starting f r o m the cross in fig. 3 points to the equilibrium values B 4 = - 14.9 MeV, k F = 1.25 f m - x The results obtained in the present section for the binding energy and saturation density are smaller than those calculated in subsects. 3.2 and 3.3 (see fig. 3). This corresponds to the fact that above 3 f m - 1 the particle potential Up used here (fig. 9) is larger and increases m o r e rapidly with k F than the one shown in fig. 5. This indicates the importance o f the choice 6 f Up above 3 f m - ~. By the same token, it illustrates the fact that our prescription is quite different f r o m the one used in ref. ~2), where Up is set to zero above the cut-off value k L = 2 k v.

4. Summary and discussion In the conventional calculations o f the average binding energy per nucleon (B) in nuclear matter, the potential energy Up o f excited nucleons is set to zero. One

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425

characteristic feature of these calculations is that they yield either too small a binding energy or :too large a saturation density 31). In the present paper, we show that there exists a prescription for Up which modifies this trend. It consists in identifying Up with the physical potential energy of the nucleons, i.e. essentially with the depth of the optical-model potential. This choice is intuitively attractive; more importantly, we show in sects. 2 and 3, and surrlmarize below, that it shares several properties previously required by several authors. We note that this choice involves no adjustable parameter. The main reason for taking Up = 0 in the conventional calculations derives from the requirement that the three-hole line contribution (I4"3) to B be small, and concerns the momenta p > 3 fm-1. It was shown by Dahlblom 36) that positive, but small, values for Up(p > 3 f m - 1), with the density dependence implied by eq. (10), would fulfill this requirement better than Up = 0. In conventional calculations, however, the choice Up = 0 is eventually retained for computational convenience. Figs. 4 and 6 show that our Up presents very approximately the energy and density dependence required in ref. 36), in the relevant range p > 3 f m - 1. For p < 3 f m - 1, the physical potential energy takes negative values and smoothly joins the hole spectrum at the Fermi momentum k v (fig. 4). This is precisely the feature required by Brandow 6, 7, 15, 16) for a better treatment of the long-range correlations. In addition, the overall shape of our Up is, for p < 3 fm- 1, similar to that obtained in refs. 2,3, ~0,12) from the requirement that the leading contribution (graphs ld + le) to W 3 be small for these values ofp. It is moreover shown in ref. 13) that the use of an attractive potential below 3 fm-1 effectively decreases W 3 . However, it corresponds to an increase of the "small parameter" x of the hole-line expansion. The latter feature leads to the expectation 11, 12) that the four-hole line contribution, in particular its dominant term represented by graph lf, is larger in our case than for the conventional choice Up = 0. This increase of ~ appears unfavourable for the convergence of the hole-line expansion, which is designed to handle the short-range part of the nucleon-nucleon interaction. However, a somewhat larger ~cmay be unavoidable for a better treatment of the long-range part 6, 7). We performed numerical calculations in the case of Reid's hard core (RHC) nucleon-nucleon interaction. The leading contribution B 2 (graph 1a) to the binding energy is found to be -12.20 MeV at the saturation momentu m k r = 1.30 fm-1, for our choice of Up ~ 0. These values must be compared with B 2 -- - 9 . 0 5 MeV, k F = 1.31 fm -1 obtained with the conventional choice Up = 0. Hence, our results may help solving the problem of saturation density versus binding energy (see open triangle in fig. 3). Since we expect the three-hole line contribution W 3 to be small for our choice of Up, we believe that the next largest graph contributing to B is lf. The corresponding contribution is W4 (eq. (11)), while the sum of all four-hole line graphs is the quantity I,~"4 in eq. (12). We calculated the occupation numberfk of hole states, as a function of momentum k and of k v , both for Up = 0 and Up ~ 0. The quantity 1 - f k

426

J.-P. J E U K E N N E et al.

is closely related to the small parameter x = M'. Fig. 8 shows that x(,~ 0.27) is larger for Up ¢ 0, but is then almost independent of kF ; for Up = 0, x ~ 0.20 for k F = 1.4 f m - 1 and decreases for smaller k F. In the case Up = 0, we find B2 + W4 = - 10.1 MeV, kF = 1.35 fm- 1. For Up @ 0, these values become - 14.55 MeV and 1.33 f m - 1, respectively. These results show that, for our choice Up # 0, W4 is larger than in the case Up = 0, but that the increase in saturation density is smaller. These features again help solving the binding energy versus saturation density problem. The sum of two-, three- and four-hole line contributions is approximately given by B 4 = - 15.60 MeV at the saturation momentum 1.34 fm-1. Three main questions arise. (i) What is the origin of the sizable gain in binding energy obtained by taking our choice for the single-particle energy Up ? (ii) How would our results be modified in the case of a soft core interaction, for instance the RSC ? (iii) Could one find a deeper justification for our choice of Up ? We can only give conjectural answers to these questions. (a) The suppression of the gap in Up at kF leads to a sizable increase of B2, of K and therefore also of W4. Since the full sum of the expansion should be independent of Up, one must conclude that the difference between the value of B 2 + W 3 + if'4 in the case Up = 0 and that o f B 4 for Up ~ 0, which is about 6 MeV, i.e. about 20 % of the potential energy, lies in the diagrams with five and more holes. This appears surprizing, since long-range correlations are usually not believed to be important in nuclear matter, in contrast to finite nuclei. However, we note that the very definition of the "long-range correlations" depends upon the choice o£ Up. One tentative but conjectural interpretation o f our results is that the long-range correlations contained in the diagrams with five and more holes are negligible for Up ~ 0, but not for Up = 0. In any case, our findings exhibit the sensitivity of the sum of the many-hole line graphs to the choice o f Up. It would be of great interest if it were possible to investigate this problem in the framework of an exactly soluble model, like, for instance, the one studied in ref. 42). (b) The two-hole line contribution B 2 for the RSC interaction is, in the conventional calculations with Up = 0, larger than for the R H C interaction 1). Hence, it may seem undesirable that we obtain about the correct binding energy from the RHC, since the RSC interaction is usually believed to be more realistic. The numbers given in table 1 refer to the sum of the contributions of those partial waves that we included in our calculation (see sect. 3); the partial waves aD 3 and 1 > 2 would probably increase the binding by only 0.02 MeV at k v = 1.36 f m - 1 [ref. 45)] (see sect. 2). The value o f B 2 in.the case of the RSC is about - t 1.1 MeV [refs. 4,3, 46)]. Hence, the difference between the value of B2 corresponding to RHC and RSC, respectively, is about 2.1 MeV. This number is smaller than the larger difference (3.7 MeV) calculated by Kallio and Day 1,44). We note, however, that the latter authors did not compute the difference between binding energies at saturation: they took k v = 1.36 f m - 1, and moreover did not perform a self-consistent calculation. We checked our

BINDING ENERGY

427

program 35) against Siemens 45, 4.6) in the case Up = 0. We found the same results as his, within 0.2 MeV. The value of B4-B2 is proportional to F 2 (eq. (12)), which is twice as small for RSC as for RHC, in the case Up = 0 [ref. 44)-]. Assuming that this feature persists for our choice Up 4: 0, we conclude that the difference B 2 - B 4 would be reduced from 3.4 MeV (RHC) to 1.7 MeV (RSC). Let us in addition make the rough guess that the increase of IB21 due to taking Up # 0, rather than~ Up = 0, is the same (3.15 MeV) for the RSC as for the RHC. Then, we obtain the estimate B 4 (RSC) ~ - 15.95 MeV. The change in saturation momentum k r is difficult to evaluate. In fig. 3, we represented the RSC value k F -- 1.35 f m - 1 quoted in ref. 26) in the case Up = 0, but the saturation momentum 1.43 fm-1 calculated in refs. 43,46) is probably more accurate 1). Because of the smaller value of F 2 in the case of the RSC interaction, we expect that the shift o f k F due tO if'4 is roughly twice as small for RSC as for RHC, i.e. is about 0.02 f m - 1. If the change in k F due to taking Up # 0 instead of Up = 0 is the same for the RSC as for the RHC, namely -0.01 fro-1, the value of k F for the RSC interaction would only be 0.01 larger in the B 4 than in the B 2 approximation, i.e. reach about 1.44 f m - 1 [refs. 43, 46)1, or 1.36 f m - 1 [ref. 26)-]. Since these estimates are quite rough, it would be of great interest to check them by a detailed calculation. Unfortunately, this is not practicable with our numerical program 35), which cannot be easily used for soft core interactions. Finally, we note that the amazing agreement between theoretical and empirical values emerging from our choice foi- Up is somewhat fortuitous, since there exist a number of theoretical corrections due, for instance, to higher-order graphs, to relativistic and quantumfield effects, to three-body forces, etc. 1). We attach more significance to the general trend of the modifications introduced by the use of our auxiliary single-particle potential Up than to the resulting almost perfect agreement with the empirical values. (c) We have argued in sects. 2 and 3 that the overall characteristics of our choice for the potential spectrum meet, at least semi-quantitatively, the requirements set forward by several authors 6-13, 36). However, this choice is not based on a systematic cancellation of graphs in the Bethe-Brueckner expansion. Rather, we observed that the physical particle spectrum is such as to lead to an approximate cancellation of the three-hole line graphs. An alternative, a priori, justification of our prescription would clearly be desirable. We feel doubtful that this could be achieved in the framework of the hole-line expansion proper, but we note the similarity of our prescription with the so-called A l l approximation 47) of the Green function theory, based on the hierarchy of many-particle Green functions 4s, ~9). We gratefully acknowledge stimulating discussions with Professor H. A. Bethe and Dr. J.-P. Siemens, and thank Dr. B. D. Day for a communication ofref. 31).

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References I) 2) 3) 4) 5) 6) 7) 8) 9)

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