Bishop, R. F.

Irvine, J. M.

Owen, J. C.

English

A method of lowest order constrained variation advocated by the present authors and previously applied with excellent results to a model Boltzmann problem is generalized. New constraints are formulated for general Fermi systems, and comparable success is demonstrated for the model Fermi problem

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Volume 66B, number 1 PHYSICS LETTERS 3 January 1977 T H E  E Q U A T I O N  O F  S T A T E  F O R  A M O D E L  F E R M I  F L U I D  J.C. OWEN Department of Theoretical Physics, University of Manchester, Manchester M13 9PL, UK and R.F. BISHOP and J.M. IRVINE Department of Theoretical Physics, University of Manchester, Manchester M13 9PL, UK, and Daresbury Laboratory, Science Research Council, Daresbury, Warrington WA4 4AD, UK Recewed 22 July 1976 Revised version recewed 20 October 1976 A method of lowest order constrained variation advocated by the present authors and previously apphed with ex- cellent results to a model Boltzmann problem is generalized. New constramts are formulated for general Fermi systems, and comparable success is demonstrated for the model Fermi problem. In earlier publications [1,2] we have elaborated a scheme for calculating the binding energy for many- body systems by a method of  lowest order constrained varxation (LOCV), which seeks to redress the objec- tions open to the alternatwe lowest-order approach of Pandharipande [3,4] .  We have demonstrated that our approach yields excellent agreement with the essentially exact results [5] for the Bethe "homework problem" over a wide density range. This problem comprises a hypothetical system of neutrons interacting via the repulsive part of the Reid soft-core 1 SO potential (acting in all partial waves), V(r) = 6484.2 e-7X/x MeV, (1) with x = 0.7 r, and r measured in fro. Ad&tionally, Boltzmann rather than Fermi statistics are assumed. Since excellent agreement has now been reached, it seems appropriate to define a new homework problem where the particles obey Fermi statistics. To this end, and since there is still a demand for a reliable low- order approach for application to such real systems as nuclear matter or finite nuclei with realishc poten- tials, we generalize m this work our previous [1,2] method, and impose new constraints suited to dense Fermi systems. The new method is then tested on the Fermi homework problem, and compared with other recent results [6, 10]. As before we adopt a trial wave function, q, = FI f (o )~ , ,  (2) i<l where q~ is the (uncorrelated) ground-state wavefunc- tion of  N independent fermions of  spin-xsospin dege- neracy v and mass m, m a volume ~2 with number density p = N/~2 = Vk3F/67r 2, and Fermi wavenumber k F. As appropriate, the two-body potential and the correlation functxon, both of  which are considered momentum-independent, can be decomposed into relative partial-wave components Vl(r ) and fl(r). Our previous [1,2] LOCV prescription revolved truncation of  the Jastrow cluster series for the energy at the two-body level, but with constraints imposed on the pair correlation function f(r) which are devised to justify this truncation. These constraints assume the form of  a certain bound, 0 <~fl(r) <~ hl(r), (3) and an integral inequahty of  the form, c o  4zrp ~l fO Gt[ f l ( r ) 'h l (r ) ldr<I '  (4) with prescribed forms of  the functions hl, the functio- nals GI, and the parameter I. In addition, the functionals G l satisfy the condition, Gl[hl(r), hl(r)] - O . (5) The two-body energy in the cluster expansion can be expressed as, 25 Volume 66B, number 1 PHYSICS LETTERS Table 1 Energy per particle versus densny using different constraints. 3 January 1977 p E/N (v = 2) (MeV) (fro-3) LOCVN1 LOCVN2 LOCVB1 LOCVB2 E/N (v = 4) (MeV) LOCVNI LOCVN2 LOCVB1 LOCVB2 0.2 98.35 92.03 86.13 83.36 0.4 227.3 222 9 183.7 213.1 0.6 386.8 381.8 311.8 306.4 0.8 569.7 562.5 455.9 488.6 1.0 772.0 766.0 616.4 608.8 1.2 990.7 982.2 791.6 783.9 1.4 1223 1217 979.5 970.2 1.6 1467 1460 1179 1170 89.06 87.37 78 60 76.45 213.1 210.3 181.4 177.8 368.3 364.5 309.7 305 1 548.6 543.7 458.0 453.8 747.7 741.8 626.5 620.8 962.6 958.3 808.4 801.8 1191 1187 1003 994.5 1437 1427 1209 1204 E2/N= 2rrp ~l ! [tt2m-lfl'2(r)+Vl(r)fl2(r)]a2(r)dr , (6) where the functions ¢2(k F r)= a](r)/r 2 are proport ion- al to the squares of the partial-wave components  of  the antisymmetrlzed uncorrelated fermion pair wave- functions, averaged over the Fermi sea. The total  energy per particle also contains a one-body kinetic energy of  ( ~ ) h 2 k 2 / m ,  which is included in the nu- merical results. In the present case the potential  Vl(r ) ts independent of  l, and we shall present results for the two cases where the correlation function f(r) is either state-independent (1.e. independent o f / ) ,  or at most depends only on the channel parity,  1.e. f(r) =f+(r)P+ + f_(r)e_ , where P_+ are projectors on to the even and odd partial waves respectively. In these cases, the summation over l in eq. (6) can be simplified, using the results, ½ ~l [l+(-l)l]¢p2(x)-a2(r)/r2 (7) = (v ¥ 1)/(2v)[1 -+ 9x-2j2(x)], a2(r)-a2(r) +a2(r), where x - kFr. We have shown [2] that eq. (6) can be minimized subject to the constraints ( 3 ) - ( 5 )  as fol- lows. The solutions to each of  the Euler-Lagrange equations, gt (at h2 gt +2h2a t -  6 f  t -  = 0 ,  (8) where gt(r) =- at(r)ft(r), are first obtained in the range 0 <~ r <. dt, where the correlation distances d t are gwen bY ft(dt) = ht(dt); and for r ~ d  t we set ft(r ) = ht(r ). The index t ranges over the number of  distinct correlation functions used. The constant k in eq. (8), which is the Lagrange multiplier for constraint (4), is either set to zero if this constraint is thus satisfied as an inequality, or is varied iteratively until it is satisfied as an equality. Previously [1,2,  7] we have used the above con- stramts in the form, ht(r ) = 1, l= 1, Gt[ft,ht] = [h2 ( r ) - f2 ( r ) ] a2 ( r ) .  (9) The first of  these constraints has been justified for Bose systems on the basis that it is unreasonable to allow the long-range behaviour of  the potential  to have a large effect on the two-body correlation func- tion. Similarly the other two constraints, which imply that the correlations do not expel more than one par- ticle from within their mutual  range of  strong influence (i.e. from within the correlation distances), seem to be a natural choice for consistency within a two-body approximation.  On the other hand, for Fermi systems, the Pauli exclusion principle induces short-range correlations in the system even in the absence of  interactions. This is reflected in the suppression near the origin of  the factor a2(r) of  eq. (7) (assuming state-independence) in comparison with its Bose counterpart  (1 - 1/N)r 2", and hence constraint (4) becomes a weaker constraint for fermions than for bosons. Accordingly we shall reformulate the constraints (9) for fermions, but  in such a manner that for bosons they remain unchanged. We first note that for bosons, the minimum value of E2/.N from eq. (6) is changed only by O(1 /N)  if the first of  eqs. (9) is changed to read ht(r ) = (1 - l /N) -1/2. This purely formal procedure shows that constraint (4) now reads, 26 Volume 66B, number  1 PHYSICS LETTERS 3 January 1977 I I I I I I I00 so F.No-  ,oov.   ~ ~._OCVN2 0 ~ ~  ÷./~ FPY + -so . %  V Lo. o o  -tsc . .-..._......._. - 20C I I I I I I I 0 02 0"4 0"6 08 I0 12 14 16 jo (f~h Fig. 1 Compar ison of  the results of  several approximat ions  for the  u = 2 Fermi homework  problem, displayed as E / N )  x - E / N ) K T ,  where the label x shown in the figure indicates the approximat ion  used. The KT and LOH results are taken from ref. [6],  the FHNC and FPY results are taken from ref. [10], and the remainder f rom the present  calculations p f {1  - f2(r)¢2} dr ~< I .  (10) Taking the equality sign and I =  1 as before, eq. (10) merely expresses the two-body approximation to the normalization condition, which must be satisfied by any exact radial distribution function [8]. Since it is known that imposition of the normalization condition has the effect of stimulating rapid convergence of the energy cluster series [9], this may indicate why con- straints (9) give such demonstrably good results for bosons. Employing a similar procedure for fermions leads by analogy to the choice, ht(r) =Ha(r), I= 1, Gt [ft, ht] = [h2(r) - / 2  (r)] a2(r), (11) where a(r) is given by eq. (7), and use of eq. (7) easily shows that this choice satisfies the Fermi analogue of the normalization condition (10). A simple rearrange- ment and a trwial integration also shows that one of the convergence parameters [2], k, associated with the cluster series, k=aTrp ~ ? [ l -  f l2(r ) la2(r)dr ,  (12) l 0 i i  0 02 0'4 06 0-8 I'0 12 14 1.6 I I I .I I I I I 0 N 2  + : D . , / / ~ / / ' ~ ' ~  NI "~i/' / / ~  o ~ " '¢~2- i /  o 8 -  I o ! i i ~ o 6 -  7 / o. / / /  O 3  j/ r (fro) Fig. 2. The correlat ion functions f (r)  of  several approximat ions for the p = 2 Fermi homework  problem at a density p = 1.0 fm-3 .  The arrows indicate the correlation distances d t for the present  calculations. Notice the  change of  scale at r = 1.0 fm, applicable only to the KT result taken from ref. [6]. is identically zero when the integral constraint of eqs. (4) and (11) operates. We present in table 1 results obtamed by the method outlined above for fermlons of either degene- racy v = 2 or 4, interacting via the homework poten- tial of eq. (1), and employing constraints (3)-(5)  with both the forms expressed in eq. (9) as well as in eq. (11). We refer to these constraints as boson constraints and normalization constraints respectively, and the corresponding methods are labelled LOCVBn and LOCVNn, where the additional integer n indicates the number of distinct correlation functions employed, i.e. the range of the label t in eq. (8). Krotscheck and Takahashi (KT) [6] perform an unconstrained variational calculation for the same v = 2 Fermi problem, with a single parametrized corre- lation function of assumed analytic form, summing an infinite partial series of cluster contributions which reduce to those included in the hypernetted chain ap- proximation in the limit of Bose statistics. We com- pare our results in fig. 1 with the most ambitious cal- culations of KT which attempt to go beyond the lowest order hypernetted chain (LOH) approximation by in- clusion of the leading basic diagrams, but we also show their LOH results for comparison. 27 Volume 66B, number 1 PHYSICS LETTERS 3 January 1977 In a very recent communication, Fantoni and Ro- sati (FR) [10] also examine the same model Fermi problem using their Fermi hypernetted chain (FHNC) and Fermi Percus-Yevick (FPY) formalisms, wherein the dynamic and statistical correlations are both treated to the same level. In particular the FHNC method takes into account all cluster diagrams con- tributlng to the radial distribution function except those containing elementary subdiagrams. In their cal- culations, FR also use a state-independent Jastrow correlation function of  the same single-parameter ana- lytic form used by both KT [6], and Chakravarty et al. (CMW) [11] for the corresponding model Bose pro- blem, but FR have not optimised this choice by mini- mlsing the energy at each density with respect to va- riations in this parameter. They give FHNC results usmg the "opt imum" correlation functions obtained by both KT [6] for the model Fermi liquid, and by CMW [11] for the corresponding model Bose liquid. The differences in energy so obtained, whilst small (~ 1-2%), indicate that optimization might well de- crease their results still further by a few percent. These results are also displayed in fig. 1. (Note that we have used their FHNC values obtained with the CMW correlation functions, since these lie lower in energy.) Of the methods displayed in fig. 1, the FHNC re- sults are certainly the most ambitious in that they sum the largest class of duster diagrams, and until each has been checked for convergence by including more terms (e.g. by extending FHNC to FHNC/4, etc.), they should probably presently be regarded as the most rehable. On the other hand, the difference between the KT and FHNC results can probably be regarded as an indication of  the remaining uncertainty. The following points should be noted: (i) As ex- pected our LOCVN results are in markedly better agreement with both the KT and FHNC results than our LOCVB results, or indeed than the LOH results. Even at nearly ten tunes nuclear matter density, the LOCVN results agree to within 7% with the KT re- sults and to within 4% with the FHNC results, where- as the LOCVB results are out by 15-20% even with the KT results. In addition the LOCVN results lie above the KT results for most densities, by contrast with the LOCVB and LOH results which always lie below; and although all other results lie below the FHNC results, it should again be remembered that these have not yet been optimised. (ii) By contrast with the LOCVB results, the LOCVN results now bear qualitatively the same relationship to the KT results as did our previous results with Boltzmann statistics to the comparable "exact" Bose results of  CMW [11]; and although the LOCVN results lie lower than the FHNC results, their overall agreement is excellent. (ill) Mlowing state-dependence into the correlation function typically gives a decrease in energy of  about 5 - 8  MeV per particle, and although we clearly do not trust our methods to anything approaching this accu- racy, we believe this difference is an accurate indicator of the state-dependence to be expected in the more ambitious calculations. (iv) Finally we compare in fig. 2 the correlation functions obtained by our me- thod with that obtained in the unconstrained KT cal- culation. The long-range behaviour of  the LOCVN correlation functions, which is dictated by the first constraints (11) is now characterised by an "overshoot",  although this is seen to be small and of  quite short range in comparison with the KT result. In conclusion, these results indicate that LOCV cal- culations with properly chosen constraints can pro- bably be made as successful for Fermi as for Boltzmann systems, at least for repulsive short-range potentials of the type considered here. Comparable agreement with "exact" results is also found for other more "realistic" model potentials including attraction. Fur- ther details of  the present method and applications to such other models will appear elsewhere [12]. References [1] J.C. Owen et al., Phys. Lett. 59B (1975) 1. [2] J.C Owen, R.F. Bishop and J.M. Irvme, Ann. Phys. (NY), to be published. [3] V.R Pandharlpande, Nuel. Phys A174 (1971) 641. [4] V R. Pandharlpande, Nucl. Phys. A181 (1972) 33. [5] L. Shen and C.-W. Woo, Phys. Rev. D10 (1974) 371. [6] E. Krotscheck, K. Takahashl, Phys. Lett. 61B (1976) 393. [7] J.C. Owen et al., Phys. Lett. 61B (1976) 147 [8] M.E. Grypeos and E. Mavrommatls, Lett. Nuovo Cun. 5 (1972) 369. [9] G.P. Mueller, J.W. Clark, Nucl. Phys. A155 (1970) 561; M.L. Ristig et al., C3 (1971) 1504. [10] S Fantom, S. Rosati, Lett. Nuovo Cim. 16 (1976) 531. [11] S. Chakravaxty et al., Nucl. Phys. A220 (1974) 233. [12] J.C. Owen, R.F. Bishop and J.M. Irvine, Nucl. Phys., to be published. 28 

The equation of state for a model Fermi fluid

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The equation of state for a model Fermi fluid

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