Fermi gas descriptions of nuclear level densities

Fermi gas descriptions of nuclear level densities

ANNALS OF PHYSICS 207, 1-37 (1991) Fermi Gas Descriptions of Nuclear Level Densities C. A. ENCELBRECHT AND J. R. ENGELBRECHT* Institute of The...

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ANNALS

OF PHYSICS

207, 1-37

(1991)

Fermi Gas Descriptions

of Nuclear

Level Densities

C. A. ENCELBRECHT AND J. R. ENGELBRECHT* Institute

of Theoretical

Physics, 7600 Stellenbosch, Received

University of Stellenbosch. South Africa

June 25, 1990

In this paper the derivation of nuclear level densities from a Fermi gas treatment of the nucleons is surveyed. In fact, there are three classes of Fermi gas models: the infinite, in which an unlimited number of fermions are available for excitation, the Iinite, in which this number is finite but the single-particle spectrum is unbounded, and the truncated Fermi gas (TFG), where this spectrum consists of a finite number of levels. Exact calculations within the TFG are possible by means of combinatorial methods, while the finite model may be analysed by assuming that the assumptions of statistical mechanics apply to the numbers of nucleons in a nucleus and then using a saddle point approximation. The standard Bethe formulae actually correspond to the infinite model and apply to the other models only in the low-energy limit. Furthermore, at very low energies they do not approximate any of the models with high accuracy and should there be corrected, as indicated in the text. For high-energy or hightemperature applications, it is essential to take into account the effects of truncation. The TFG is constructed here in a way which accommodates the two main features in which the results for real interacting nucleons should differ from the Fermi gas picture. In order to make the TFG results accessible for practical applications without having to perform the cumbersome combinatorial calculations for each case of interest, simple approximations are presented for the nuclear level densities, as well as the closely related canonical partition 0 1991 functions, as obtained by means of calculations using the truncated Fermi gas model. Academic

Press. Inc.

1.

INTRODUCTION

Calculation of the nuclear level density is now just over 50 years old. When neutron capture experiments revealed a jungle of closely-spaced levels at excitation energies of some 8 MeV, Hans Bethe immediately published “an attempt to calculate the number of energy levels of a heavy nucleus” [ 11. Although almost all that was known about the nucleon-nucleon interaction at that time was that it was short-ranged and very strong, Bethe already then realized that, unlike the nuclear wave function, the general behaviour of the level density would on the average not be greatly changed by the interactions and the nucleons could, for this purpose, be pictured as a Fermi gas of nucleons. In the intervening years, various ways have been developed in which the effects * Present

address:

Physics

Dept.,

University

of Illinois,

Urbana

61801,

IL.

1 OOO3-4916/91

$7.50

Copyright Q 1991 by Academic Press, Inc All rights of reproduction in any form reserved.

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of interactions on nuclear structure can be incorporated. To perform a microscopic calculation of the nuclear level spectrum for anything other than its low-energy part is a most forbidding problem, however. There has therefore been a continued need for statistical representations of the nuclear level density. Although some of these have through the years become quite intricate and have in some cases succeeded to incorporate detailed shell effects and related features, they have in a certain sense continued to be built upon the point of departure of the original analysis by Bethe [ 1,2]. Although there are many problems for which this point of departure is fully adequate, it is based on certain assumptions which have escaped the notice of many of its users. In this paper we wish to address this defect and to extend the province of Fermi gas descriptions to domains which are in fact excluded from the beginning by the traditional Bethe route. This does not pretend to be the first venture outside the confines of that tradition. In looking at particular problems, other authors have of course already trodden on some of this terrain. Here, however, we wish to present a comprehensive attempt to stake out the borders of the different versions in which the Fermi gas description may appear. Section 2 begins with a brief consideration of the nuclear energy spectrum in general and Section 3 presents, for comparative purposes, some of the explicit patterns in which single-particle levels may appear. Section 3.5 deals with a particular version of the single-particle spectrum, the truncated Fermi gas (TFG), which is proposed as an especially appropriate model for the derivation of statistical results about real nuclei. Two general methods are available for deriving such results from the singleparticle spectrum: the combinatorial method (treated in Section 4) and the grand canonical approach (treated in Section 5). In both cases exact calculations can be performed in specific situations and particular limiting forms apply in certain domains. Section 5.4 also contains a brief discussion of the various approximations which underlie the grand canonical approach in general. Canonical partition functions, which are just Laplace transforms of the level density, are discussed briefly in Section 6. When the physics underlying these calculations is examined, it becomes clear that three distinct sets of primary assumptions should be clearly recognized. In the first, one assumes that there is an infinite sea of fermions which can be excited to higher levels. This is actually the assumption underlying the unrestricted use of the Bethe formulae, although they also apply in other cases in the limits of low energy or temperature. In the second situation, referred to in this paper as the tinite case, there is only a specific finite number N of fermions but they may be excited to an infinity of available levels. When these levels are also restricted to a finite number H, we obtain the third situation, that of the truncated Fermi gas. Different techniques must be used to obtain quantitative results for these three cases. Exact calculations for the finite case can be performed by means of the grand canonical approach while the truncated case requires combinatorial calculations. The infinite case is obtained from asymptotic approximations to these calculations.

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When used to represent the low-energy limit of the other two cases, the commonly used formulae should, however, be modified according to the guidelines provided in Section 4.4. Since the results of the combinatorial analysis of the truncated Fermi gas do not yield analytical expressions, considerable attention is given to the derivation of algorithms to represent these results without having to repeat the cumbersome calculations anew every time. These recipes are provided in Section 4.3 for the nuclear level densities and in Section 6.1 for the canonical partition functions. The results which are obtained for the three cases are compared in Sections 5.3 and 6.1, respectively. These comparisons provide an indication of the regime of physical variables in which the differences between these three underlying models become important.

2. THE NUCLEAR ENERGY SPECTRUM

In the conventional the Hamiltonian:

theory of nuclear structure the point of departure is usually

H = 1 t,,aia,

+ 4 c v,,,a~a~a,a,.

(2.1)

This form is applicable to a system of nucleons with two-body interactions. The Brueckner G-matrix treatment of the many-nucleon problem introduces a density dependence (equivalent to the inclusion of many-body forces) at the level of effective potentials while most theories of the strong interactions suggest that many-hadron couplings already occur at the microscopic level. We shall, however, not consider such effects in this paper. In fact, the emphasis here is on simpler treatments, where even the two-body interactions do not have to be taken into account directly. If the average effect of the nucleon-nucleon interactions is replaced by a mean field, (2.1) may be approximated by an independent-particle Hamiltonian, Ho =I

via,,,

(2.2)

n

which will be diagonal in an appropriate single-particle basis { Ik)}. The set of ak appropriate to a particular nucleus with proton and neutron numbers Z and N will depend on the values of these numbers. If the approximate form (2.2) is taken seriously, the energy eigenstates of a specific (A-nucleon) nucleus would be Slater determinants in which a particular set of A of the single-particle levels, known as the configuration {n}, are occupied. The eigenvalues of Ho consist of sums of the A single-particle energies, Es”,, = 1

IIE In)

E,,

(2.3)

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corresponding to the particular configuration. This description is often referred to as an extreme single-particle model. The approximation (2.2) is usually constructed to provide a reasonable description of the nuclear ground state and low-lying excited states. The single-particle energies sk are composed of kinetic and potential energy contributions: Ek=

(2.4)

tk+Vk.

The latter, however, represent the effect of the mean field generated by two-body interactions with the other nucleons. This means that the A-nucleon energies should actually be written

to eliminate double counting of the interactions. (2.3) corresponds to

The difference between (2.5) and

E~,)-E~~,=((n}lH-H,I{n)).

(2.6)

In order for an expression such as (2.3) to provide a reasonable measure of the total energy, the Ed to be used should really be the values of + $ v/, occurring in (2.5). On the other hand, if (2.6) does not depend too sensitively on n (note that the precise values of the vk do in fact depend on the configuration {n} of occupied levels), energy dijjferences calculated using (2.3) rather than (2.5) would not be affected much. Since we shall always be considering excitation energies E,, = E - E, relative to the N-body ground state energy, this deficiency of the single-particle spectrum will not be serious. The independent-particle representation (2.2) at best provides a rough approximation to the true nuclear Hamiltonian (2.1). Even after extraction of the mean field, residual two-body interactions of appreciable strength remain. In the shell model (2.1) is replaced by

tk

H= Ho+ V,,,.

(2.7)

In the spherical shell model, the single particle energies E,~ have a (2j + l)-fold degeneracy. The eigenstates of H, are used as a shell-model basis and the corresponding energy spectrum (2.3) is even more highly degenerate. The essence of the shell model consists of the diagonalization of the full (2.7) in a basis of known A-particle eigenstates (In)} of H,,. In practice, this basis must be truncated to a finite model space, and the effective residual interaction V,,, is usually adjusted to compensate for this truncation. The result of this (laborious) procedure is a detailed energy spectrum, as well as a potential set of transition matrix elements between all possible nuclear states. In this paper we shall not be interested in any detailed properties but only in the overall distribution of nuclear states as a function of energy. With this in mind, we may

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5

draw attention to two major differences between the spectrum resulting from the shell-model diagonalization (which, presumably, can be adjusted to correspond reasonably well to the actual nuclear energy spectrum) and the extreme singleparticle spectrum (2.3): (a) First, the high degree of degeneracy is broken by the diagonalization process: levels are distributed more evenly over a wide energy interval. (b) Second, the energy of the ground state (and often also those of a small number of associated states) is usually lowered appreciably. The shell-model diagonalization does not change the overall density of states in a particular energy interval (chosen sufficiently large) appreciably, however. This feature was already realized in the pioneering paper [ 1 ] in which Bethe pointed out that, apart from the ground state suppression, one could expect that the level density would “in the average be not very greatly changed by the interaction between the particles.” Instead of performing the shell-model diagonalization, one can also derive average spectral properties from the statistical spectroscopy method [3]. In this approach one uses, besides the single-particle energies sk, also, average values of the two-body matrix elements of the residual interactions to calculate various moments of the energy spectrum. The average nuclear level density can, however, be described quite well by an even simpler model: that of a non-interacting Fermi gas (NFG). In such a model, the Hamiltonian is approximated by (2.2) and the energy of the A-particle nucleus is given by a form similar to Eq. (2.3), but with single-particle energies sk which may be quite different from those used in the shell model. In particular, a truncated Fermi gas (TFG) version of the NFG will be developed in which the degeneracies of the single-particle levels are deliberately removed in order to produce an A-particle energy spectrum which does not exhibit the high degree of degeneracy of the extreme single-particle picture but approximates the milder degeneracy brought about by the effect of the residual interactions. In this way the simplicity of a formalism of non-interacting particles can be combined with a more realistic many-body energy spectrum than would be obtained using the highly degenerate shell model single-particle levels E,,~. In the remainder of this paper the attention will be confined to the consideration of nuclear level densities and partition functions within the confines of a NFG framework. Since the point of departure is inevitably the single-particle energies, various representations of this level spectrum are discussed in Section 3. To construct the energy spectrum of the whole nucleus, two independent avenues are available. In Section 4 we discuss the combinatorial approach while the statistical mechanical approach, based on the use of the grand canonical partition function, is treated in Section 5.

ENGELBRECHT

AND

ENGELBRECHT

3. SINGLE-PARTICLE

LEVELS

In this section the NFG description of nucleons bound in a nucleus is briefly reviewed. Most of the discussion is formulated for a single species of particle; the extension to two flavours of nucleon is implemented later in specific applications only. In this description there are no interactions between the particles so that the many-body Hamiltonian has the form given previously in (2.2). The single-particle energy spectrum { sk} implies a density of single-particle states:

g(e)= 1 &E- Ek).

(3.1)

k

Sums over single-particle

states may then be replaced by integrals:

With (3.1), this replacement is trivial. It is, however, often convenient to approximate (3.1) by a continuous function of E, in which case the transformation (3.2) introduces true integrals over the single-particle energy. The condition EFd&g(E) = N

s0

(3.3)

determines the relation between the Fermi energy sr and the number of particles N. Similarly, the integral

defines the ground-state energy Et of the iv-particle system. This section is devoted to a brief discussion of a few possible versions of such an NFG description. 3.1. Free Particles in a Rigid Box (FRB) The standard version of the Fermi gas used in the statistical mechanical description of infinite systems often employs the device of considering N particles confined to a spatial volume Y = 2 nR3 to simplify taking the thermodynamic limit in which N-r cc and V” + cc while the particle density N/Y remains fixed. This description may also be used for finite spherical nuclei with the implication of confinement to a rigid box of volume V, which may also be expressed in terms of the rigid radius R or, through the relation (3.5)

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in terms of the Fermi momentum as g(c)de=g,V-

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pF. The corresponding 4rcp2 dp (27di)3

06p<:,



state density is expressed

(3.6)

where g, is the degeneracy factor due to intrinsic (e.g., spin) degrees of freedom. This form embodies the translational degrees of freedom in three dimensions of a free particle with energy, PL E=Ij--&’

(3.7)

which implies that g(E) is a continuous function (FRB3):

9

O
(3.8)

Actually, for a finite nucleus this must be viewed as the continuous approximation to a function of the form (3.1), where the sk correspond to the zeros of the spherical Bessel functions j,(p,k/fi) = 0 and where the (21+ 1 )-fold degree of degeneracy is built in. As a matter of interest, we may also write down the expressions corresponding to the assumption of free particles with translational degrees of freedom in a rigid box of volume V” = ITR’ in two dimensions and of volume Y = 2R in one dimension. In the two-dimensional case, where

(3.9)

one obtains the level density function (FRB2): OdE
while in one dimension,

(3.10)

where (3.11)

the expression becomes (FRBl ) OdE
(3.12)

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3.2. Non-interacting

Particles

AND

ENGELBRECHT

in an Oscillator

Potential (OSC)

The confining action of a spherical oscillator well corresponds to a much “softer” box. The three-dimensional oscillator spectrum produces a discrete single particle density, g(E) = + (n2 + 3n + 2) h(& - [n + $1 ho),

n=O, 1, ... .

(3.13)

A smooth approximation to (3.13), yielding the same values for the integral (3.3), produces the function (OSC3), m=&(3

(,),-

06&<

I),

co,

(3.14)

which clearly implies a much stronger energy dependence than (3.8). Here the Fermi energy is taken as sF = (nmax + 2) fro + &#iw, where nmax is the highest occupied level. Note that the oscillator spectrum is independent of the fermion mass m and that the size of the nucleus is determined by the parameter w. In comparison with the 3D oscillator, a two-dimensional oscillator would produce a single-particle level density, g(E) = (n + 1) 6(s - [n + 11 ho),

with a continuous approximation

n = 0, 1, ...)

(3.15)

(OSC2), (3.16)

In one dimension

one obtains n = 0, 1, ...)

g(8) = h(E - [n + $1 ho),

(3.17

which leads to the constant level density (OSCl), (3.18 3.3. Equidistant

Spacing Model (ESM)

A simple representation of the single particle spectrum which is often used is one with a constant energy difference between successive non-degenerate levels: (3.19)

g(c) = 8(~ - nd - 4 d).

It may be used in three different versions: inlinite( ESM - inf ): linite(ESM

-fin):

truncated( ESM - tfg):

nE (0, *l,

+2, ...}

n E { 1, 2, 3, ...} n E { 1, 2, .... K}.

(3.20)

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The smooth approximation

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to this model has the forms: ginA&) = d-’ (3.21)

ge,(E) = d-w&) g&E) = d&‘fl(&) 8(Kd-&)

for the three versions specified above. In the latter two versions, which correspond to the constant level densities encountered in the one-dimensional oscillator (0X1) as well as the case of free particles in a two-dimensional rigid box (FRB2), the Fermi energy of the N-particle system is EF=Nd

(3.22)

E,N = ; N’d.

(3.23)

while the ground state energy is

Physically, one would expect the infinite version to be applicable to systems with an infinite filled fermion sea (and with a natural choice of sF = 0 for the corresponding Fermi energy.). In fact, however, one of the standard models for nuclear level densities is really equivalent to the ESM-inf single-particle energy spectrum. When the ESM is used in the literature (see, for example, [4, 5, or 6]), it is usually presented as being merely an approximation to a more general spectrum, such as (3.8), where the constant g, = l/d is then seen as an average value of (3.8) in the region of the Fermi energy. 3.4. Mean Field Models

Although the NFG description presupposes a formalism without explicit interactions between particles, the effect on each particle of the mean field of all the other particles can be incorporated without any problem. One possibility is to generalize the nuclear medium description implicit in (3.7) to incorporate a general spherically symmetric local mean field (3.24)

where either a constant potential depth or an effective mass approximation used for U(p):

may be

U(p)=u,+p2(d&J. The level density in the momentum representation (3.8) it. becomes

representation

remains (3.6) but in the energy

(E- uop2,

U,
(3.26)

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If the effect of the mean field is incorporated into the nuclear medium description as in (3.24), the result is a continuous single-particle level density (3.26). In the shell model (2.2) the mean field is active over a finite spatial volume only, and the resulting discrete spectrum may be represented by

g(E) =cnlj (2j + 1)6(& - E,[i)’

(3.27)

where the energy values E,,~ may be calculated microscopically (using, for example, the HF, BHF, or BCS approximations), computed for a phenomenological potential well (such as the oscillator well yielding the spectrum (3.13) or a Woods-Saxon potential), or fitted empirically to the systematics of observed nuclear spectra. A characteristic feature of realistic single-particle spectra are the shell effects which necessitate the introduction of spin-orbit interactions in addition to the central potential well. If these shell effects are removed, the single-particle spectrum may, however, be represented on the average by a continuous g(s) which could be expressed as C, c,?‘. In a square well the FRB3 results would select n = i as the dominant term. In view of the fact that the oscillator OSC3-which is clearly too “soft’‘-leads to n = 2, one might expect that the levels resulting from a proper Thomas-Fermi approach could be represented by something like n = 1 for actual nuclei. In practice, the simple ESM approach with n = 0, normalized to the level spacing d at the Fermi energy .sr, for most purposes provides a satisfactory basis for the calculation of excitation energies, partly because the errors made above and below sr go in opposite directions and therefore largely cancel each other. The set of {Q} appropriate to a particular nucleus with neutron and proton numbers N and 2 will of course depend on the values of these numbers. Two sets of levels which are often used, are that of Nilsson [7] and that of P. A. Seeger, which distinguishes between protons and neutrons and was published in Ref. [S]. Although the attention throughout this paper is confined to spherical nuclei, there are no fundamental difficulties in generalizing any of this to deal with nuclei without rotational symmetry. 3.5. Truncated Fermi-Gas

Model (TFG)

The single-particle spectrum (3.6) obtained by confining the particles to a specific spatial volume -Y- clearly extends to infinity. On the other hand, microscopic calculations of the E,~ values, as well as the spectrum produced by a realistic potential well, contains a natural cutoff at the single-particle separation energy (denoted as S, and S, for neutrons and protons, respectively), which is defined as the threshold excitation energy for the dissociation of the nucleus (N, Z) into (N - 1, Z) plus a neutron or into (N, Z - 1) plus a proton. Beyond the separation energy a nucleon is no longer confined to the nuclear volume but enters the quasicontinuous part of its spectrum where the level density is determined by the overall density of matter. The question whether one should use a single-particle spectrum with an infinite or a finite number of levels will be considered elsewhere. For the

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moment we just make the observation that the unbounded spectra FRB3 (3.6) or ESM-fin (3.19), for example, may be truncated by imposing constraints such as 0 < p < p, and 1
= C cTFGal;a,,,

(3.28)

where the single particle spectrum is discrete and finite (in the sense of accommodating a finite number N of filled levels below the Fermi energy). To incorporate the existence of the single-particle continuum (starting at the separation energy), it must also be truncated: the sum in (3.28) runs from 1 to K only. In order to simulate the effect of residual interactions (over and above the mean field) on the actual N-particle energy spectrum, the sTFG will be, in general, taken to be non-degenerate. The shell effects at low excitation energies may in particular cases be accounted for by building appropriate energy gaps into the TFG energy spectrum (see [14] for examples). However, for many purposes the effects of shell and pairing energies can be adequately accounted for by assuming that the actual nuclear ground state energy is lowered by an amount A as compared to the ground state energy appropriate to the pure Fermi gas description. With this in mind, the number of unfilled levels (for protons, say) would for an ESM (n = 0) be given by H,=(S,-A)/d.

(3.29)

To correct for n # 0 one may multiply this estimate by 1 + k n(S,/a,) which would, for n = f, typically change HP by about 5%. Note that the usual single-particle bases used for the shell model would produce values for N + H, and Z + HP which always correspond to the binding of a complete n/-level. The essence of the nondegenerate TFG introduced here, is that no such restriction be placed on the values

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resulting from (3.29). Apart from thus obtaining a nuclear energy spectrum which exhibits the lower degeneracy which residual interactions would cause, this feature in a certain sense also reflects the ambiguity inherent in the very concept of a singleparticle energy for an interacting system.

4. COMBINATORIAL

METHODS

The classic treatment of the level density problem by Bethe [ 1 ] was based on the statistical-mechanical approach, which had been introduced by Sommerfeld [9] and will be discussed in Section 5. It was almost immediately recognized by Goudsmit [lo] that this method was, in the case of an equidistant single-particle spectrum, equivalent to a conceptually much simpler combinatorial approach, in which one simply has to count the number of different single-particle excitations leading to the same value of the total energy. Van Lier and Uhlenbeck [ 1 l] published the first practical application of this idea, which harked back to the eighteenth century investigation by Euler [12] of the number of ways an integer can be partitioned. In the single-particle ESM-tfg model of (3.19) and (3.20), N fermions must be distributed among K = N+ H levels. If the occupation number (0 or 1) of level number k is denoted by nk, one has the obvious relations, Fn,=N

and

~nkEk=E~+E,,,

(4.1)

k

where the sums run from 1 to K, where E,N = 1 N2d denotes the ground-state energy while E,, = Ld (thus, defining the integer L) is the excitation energy. If W:(L) denotes the number of different configurations {nk} producing the same value of L, it also represents the degree of degeneracy of the N-body state with excitation energy E,, , and the overall level density is given by P(L)

= :

W,HW) 6(&,-W.

(4.2)

L=O

In the literature a distinction is sometimes made between the nuclear “state density” (the density of the total number of quantum states, as in (4.2)) and the nuclear “level density,” in which, for example, a state of angular momentum J is counted as one “level” rather than 25+ 1 “states.” We shall, however, never use this terminology and shall denote by “level density” any quantity such as p(E,,) above. It is an easy exercise to show that W:(L) also denotes the number of different ways in which the integer L can be written as a sum ‘of no more than N positive integers, each of which is not allowed to exceed H. In Appendix A, symmetry and recurrence relations (A.2))(A.5) are presented which can be used to generate the W:(L) for all positive values of N, H, and L. If L < H, the value of W:(L) will not depend on H and will be indicated by W,,,(L). This corresponds to the variation

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ESM-fin of (3.20). If, in addition, L 6 N, the value of W,(L) does not depend on N either and will simply be written as W(L). This corresponds to the bottomless ESM-inf of (3.20) and is referred to, in combinatorial theory, as the unconstrained partition. 4.1. The Infinite ESM ( Unconstrained

Partitions)

Although the recurrence relations were already used by Euler to compute tables of W,(L) for specific N and L, the procedure is slow and tedious. An asymptotic formula was derived for the unconstrained partition by Hardy and Ramanujan [13]: 1

W(L)=-

2

exp r [ $1TL

4dL

(4.3)

As discussed in Appendix A, this formula is not very accurate. In their paper Hardy and Ramanujan showed how (4.3) can be modified to achieve any desired accuracy (see, for example, Eq. (A.9) in Appendix A). The major share of their improvement comes from replacing (4.3) by

wtL)JS-l)ex~S where 4n JzL”2

S= TL

)

(4.4)

This reduces the error in (4.3) significantly (already by three orders of magnitude for L = 100). The apparent singularity at L = 0 is spurious, of course. In fact B’(0) and W(1) are both equal to unity. When translated back from partitions to Fermi gas models, Eq. (4.3) turns into the famous Bethe formula: 1 A(&,)

=~

E,, ,/‘% exp

2E,, [ J-l =

3d

1 =-exp2Eex ,h

(4.5)

for a single species of fermion. In the latter form the level spacing d has been replaced by the “level density parameter” a,= x2/6d. As seen from the combinatorial calculation for the ESM model, the Bethe formula corresponds to the assumption that the excitation energy: E,, = Ld

must be

< Hd and < Nd.

(4.6)

These ideas can easily be generalized to the case of two species of fermions. Suppose that W,(L) represents the statistical weight of a state of N neutrons with total energy (L + f N.2) d, while W,(L) represents the statistical weight of a state of Z protons with total energy (L + 4 Z2) d,,. If the spacings d, and d, are equal, the combined statistical weight is given by W,,(L) = 2 L.=O

Wrl(L - LJ w&J.

(4.7)

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This expression holds irrespective of whether these statistical weights refer to the constrained (for example, when L, > Z and/or L, > HP) or the unconstrained case. If for both neutrons and protons we use the asymptotic form (4.3) (and assume the same level spacing d), the combined statistical weight becomes 3 114 4 (4.8) W,,(L)=~exp = jL L

d-1

By analogy with (4.4) the accuracy can be greatly improved expression by where

by replacing this

S=n

(4.9)

Typical errors of 40% for low L-values are reduced to a percent or less. Once again the apparent singularity at L = 0 is a mathematical artefact, since W(0) = 1 and W(1)=2. The corresponding level density which follows from (4.8) is given by p,(E,,)=~exp[n~]=12~~E~,4exP2~,

(4.10)

ex

where a is the sum of the values (here assumed to be equal) of the a, in (4.5) for protons and neutrons. This is the expression which has served as the point of departure of practically all the variations of the nuclear level density formula proposed over the past 50 years. As it should, the unconstrained partition depends on L (corresponding to the excitation energy) but not on the constants N (corresponding to the number of particles) or H (corresponding to the number of unfilled single-particle levels), whose values begin to play a role in the situations discussed in Sections 4.2 and 4.3, respectively. The results which have been obtained in this section, and therefore also the expressions (4.5) and (4.10) for the level density, should therefore apply only to excitation energies for which L < N and L < H. This limitation on the validity of the Bethe formula has been a serious oversight in many treatments of nuclear level densities. 4.2. The Finite ESM (Partially Constrained Partitions) The partially constrained partition, which provides the nuclear level density for the finite (but un-truncated) ESM, can also be calculated by the repeated application of the recurrence relation (A.5). This leads to a polynomial of degree N- 1 in L. The leading orders in this asymptotic expression were already given by Van Lier and Uhlenbeck [ 111: LN-I wN(L)

=

N! (N-

l+w+l)Nwl)

l)!

4L

1 ~

To this order the formula provides reasonable values only for L > (N/2)3.

(4.11)

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15

Replacing the factorials by the Stirling approximation and setting L = gpEex, it is easy to show that this expression corresponds, for the first two leading powers if E,,, to: 2N

N-l

1

PLN)=$

.

(4.12)

This is therefore the expression which replaces (4.5) (which is valid for g, E,, 6 N) when g, E,, > (N/2)3. Also in this limit we may use (4.7) to obtain the corresponding expressions when both types of nucleon are included. One finds that (4.11) is replaced by LN+ZwN,Z(L)

=

I

N! Z!(N+Z-

l)!

1.

I+(N+Z-l)(N2+N+Z2+Z) 4L

(413)

This leads to a nuclear level density: A&,

N

z)

=

g, N+Z-I

gFE,+;(N2+z2) 1

(4.14)

For high excitation energies the exponential dependence on E,, is therefore replaced by a polynomial dependence, and the densities become very sensitive to the precise values of N and Z. 4.3. The Truncated Fermi Gas (TFG) The fully constrained partitions, corresponding to the truncated version of the ESM, can be calculated by means of the generating function (A.1 1) in Appendix A. Actually, this generating function is much more powerful. In a slightly modified form it can also be used to handle any single-particle spectrum in which the level energies are expressed as integer multiples of some energy spacing d. Thus, the spectrum is specified by a set m, < m, < . . . 6 mK of integers and (A.1 1) must be modified to read g,(x, y)=

fi

(1 +.X-P)=

k=l

i

(4.15)

XNJ’b

N=O

L=O

where now, somewhat more generally, Lo=

5 mk k=l

and

Lm=

2

mk,

(4.16)

k=K-Nil

while M = L, - Lo still holds. (These definitions should be interpreted

as implying

16

ENGELBRECHT

AND

ENGELBRECHT

L, = L, =0 if N= 0.) In this more general framework W:(L) and the Eulerian partitions is lost, of course.

the connection

between

Equation (4.15) is a very general expression of the truncated Fermi gas model. The coefficients Wz- N(L) determine the level density p(&,) through (4.2), the sum over L in (4.15) becomes equal to the canonical partition function Q(N, j?) if one sets y = e-pd, whereas g,(x, y) as a whole is the grand partition function if x = ez as well. By using a non-degenerate single-particle spectrum (i.e., a set mi which are all different), the lower degree of degeneracy (than that produced by an extreme single-particle description with degenerate single-particle levels) of the full manybody system can be simulated. By incorporating appropriate shell-model gaps in the set mi, one can even, at least in some cases [14], represent the lowered energy of the ground state (and, possibly, some low-lying states as well) of the interacting system. This version of the TFG was “put on the market” [ 151 in 1982. Such complications can, however, at best be handled by treating specific nuclei in a very individual way. In order to discuss nuclei in a more global sense, we shall here not consider such individualized treatments. Instead, we shall confine our attention to the equidistant case (where mi = 1, 2, .... K) and incorporate the ground state effect by writing: P(-K,) = 00 wex) + PESM(&x - A, N -a

(4.17)

where o,, = 2Jo + 1 is the multiplicity of the ground state while pESM(Eex, N, Z) is the density obtained by convoluting (using (5.1)) the densities obtained separately for neutrons and for protons using truncated equidistant single-particle spectra. The “backshift” of the ground state is therefore represented by the parameter A. To obtain the pESM(Eex - A, N, Z) in Eq. (4.17), we must therefore investigate the ESM-tfg for the neutrons and the protons separately. Through (4.2), the energy spectrum corresponding to N particles and H unfilled levels is given by the coefficients W:(L). As explained in Appendix A, these coefficients can be obtained by means of a fairly laborious recurrence calculation. In lieu of having to perform this calculation anew for every individual nucleus one wishes to consider, one could seek some simple approximation based on the general properties of the coefficients -W;(L). They become equal to 1 for L = 0, 1, NH, and NH- 1, attain a maximum at L = NH/2 (or (NH 2 1)/2 if NH is odd), and are distributed symmetrically about this maximum. Their sum over all L is given by (N + H)!/(N! H!). When the conditions L< N and L< H are both satisfied, WC(L) becomes equal to the unconstrained partition (4.3). As an alternative to calculating the density for each nucleus separately, one may therefore try to represent it by the continuous function: (4.18) where f(x) = f( 1 -x) vanishes at x = 0 and x = 1 and has unit integral between these limits. (Where it may be important to preserve the fact that the end point

FERMI

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17

DESCRIPTIONS

values of the W:(L) are unity and not zero, one may add a term [0(x) 0( 1 - x) f(x)]/d to the right-hand side of (4.18); but for our purposes this difference is unimportant.) The simplest function having the properties required off(x) is f(x)=

r(2M,+ Cr(M,+

2) 1),2xMd(l

-Xv?

(4.19)

The free parameter M, may now be determined so that the correct value is obtained for the variance ( (x2 ) - (X ) ‘)/ (s ) ‘. The result of extensive calculations in the domain NC 50 and H < 30 is that this simplest ansatz for f(x) gives a reasonable approximation to (4.18) if M, is chosen to be Al,=

lS(N-’

+ HP’)-’

- 1.8

(4.20)

(accurate to about 1% in this domain). If this function has the correct variance, its value at x = l/2 reproduces the peak value of pz(E,,) accurately to within about 2.5%. The simple function defined by (4.18) to (4.20) is a good approximation to the single species nuclear density near its peak at E,, = NHd/2. The function decreases to a fraction r of its peak value where .x = E,./NHd is equal to x,=+

[l gcF]

(4.21)

and (4.18) produces reasonable results for values of r as low as 0.1. To rely on this simple function also in the wings of the distribution would be naive. Indeed, for the case N = 24 and H = 12, it is already too low by a factor lo4 if E,, = 8d. The treatment of this region is discussed in Section 4.4. 4.4. Low-Energy

Behaviour

The approximation given by (4.18) and (4.19) provides a reasonable representation of the TFG level density at energies where the constraints become effective. When E,, is still smaller than both Nd and Hd, the results of the unconstrained partition hold and the density should be given by them. As discussed in Appendix A, the asymptotic Bethe formula (4.5) is actually a relatively poor approximation to the partitions for low L-values: with a typical value of 0.5 MeV for d, it would still be 22% too high at an excitation energy of 3.5 MeV. Within the TFG model the combinatorial result is of course exact. Much better results can be had by using the form corresponding to the improved approximation (4.4):

JaoEex- 1e2J&E P(U = 28Jm;

(4.22)

Although these formulas were obtained for values of E,, corresponding to integer multiples of the level spacing d only, one would like to use them as actual con-

18

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AND

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tinuous functions for any value of E,,. To get rid of the unphysical and incorrect (in terms of the exact combinatorial result) turning point in (4.5), zero in (4.22) and infinity at zero excitation energy in both, one may use the simple form p(E,,)

d= 1 + (&,/2d)*

= 1 + (3a,EC,)2/714

(4.23)

for 4aoE,, -C 37r2, where it joins smoothly onto (4.22). In order to illustrate the quantitative significance of the various low-energy expressions, we display their results on Fig. 1. It is clear that the Bethe formula (4.5) does not represent the results of the combinatorial model at all well at low energies. The modified expression (4.22), corrected near the origin according to (4.23), is very much better. This form will be denoted plow(Eex) in the following discussion. We have here considered modifications to the low-energy level density for a single nucleon species. Such expressions for neutrons and protons separately must be combined with the backshift expression (4.17) to obtain the actual nuclear level density. The corresponding approach for canonical partition functions (see Section 6) certainly agrees well with partition functions calculated directly from experimental nuclear energy spectra. It is reasonable to expect, therefore, that the problems encountered in the past with the Bethe formula at low energies (see, for

0

2

4

6

8

FIG. 1. Low-energy dependence of pd on E,,/d for a single nucleon species. The upper curve represents the Bethe formula (4.5), and the lower one the modified expression (4.22). The intermediate curve at very low energies represents the correction (4.23). Circles denote the exact combinatorial values of W= pd. The corresponding excitation energies may be obtained by multiplying the ordinate values by a typical value d = 0.5 MeV, for which the scale therefore runs to about 4 MeV.

19

FERMI GAS DESCRIPTIONS

example, [20, 21, 221, where particular ad hoc remedies are discussed), can be removed by the approach proposed here. The single species nuclear level density is given very accurately by plow(Eex) for x = E,,/NHd smaller than x,, = l/N, and to a good approximation by pmid(Eex)for x larger than x, of Eq. (4.21) where we may choose r = 0.1. To represent the density over all energies of interest, we may adopt the interpolated form

~(Eex)= Pmid(Eex) + C~,ow(Ecx) - Pmid(Eex)lfint(Xh,~2Xr), where the interpolating

(4.24)

function fint(a, X, 6) is unity for x b and

f,,,(a, x, b) = (b - x)2 (b + 2x - 3a)/(b -a)’

(4.25)

in the intermediate region. The results produced by means of this expression will be exhibited on Fig. 2.

5. GRAND CANONICAL

APPROACH

Although the combinatorial method is conceptually simple and direct, it is mathematically complicated except for limiting cases of the ESM. A highly appealing alternative route to the nuclear level density has, ever since Bethe’s pioneering work [ 11, been to determine nuclear properties in terms of equilibrium statistical properties of a nucleon gas. Although this involves a detour through statistical mechanics, the mathematical tehniques are very powerful. As before, the discussion will deal mainly with the hypothetical situation of one type of nucleon only, although the results for the extension to neutrons and protons will often be provided too. The level density for a nucleus with N neutrons and Z protons can alternatively also be constructed from the single species density by means of the convolution integral:

p,( E,,, N, Z) = j-“’ dE’ P,(&, - E’, N) PJE’, Z) 0

(5.1)

which is the continuous analogue of (4.7). The point of departure is a gas of a single nucleon species in equilibrium with a heat bath at an effective temperature kT= l/j3 and a particle reservoir with chemical potential p = U//I The grand (canonical) partition function is then defined as the weighted sum over an ensemble of many-body systems, each composed of an arbitrary number of nucleons and with arbitrary energy, (5.2)

where p(E,,, N) is the level density for a nucleus containing

N nucleons (of one

20

ENGELBRECHT

AND

ENGELBRECHT

species only), Et is the ground state energy, and E,, denotes the nuclear excitation energy. For a discrete distribution of levels, (5.3 1

p(E,,,N)=C6(N-n)C6(E,,-E,“-E,“),



the grand partition

k

function becomes a sum: n

k

In the statistical description the effective temperature determines the average energy of the gas of nucleons (Et + E,,), while the chemical potential determines the average number of nucleons (N). In addition to its dependence on T and p, the grand partition function also depends on the volume V”, which will not be indicated explicitly. The canonical partition function Q(N, /?), on the other hand, refers to systems with a definite number N of nucleons and contains an average over systems with arbitrary energy only. It follows that: (5.5) provides the connection between the two partition functions. The reason behind the grand canonical detour is that the twofold Laplace transform (threefold when both types of nucleon are included) (5.2) can be inverted (see (5.6) below). If the system of fermions can be treated as a non-interacting Fermi gas, the grand partition function acquires a particularly simple form, which can, furthermore, be expressed in terms of Fermi integrals if the single-fermion level density g(E) can be expressed in terms of powers of E. Then, the nuclear level density is expressed in closed form, albeit in terms of complex integrals which must be evaluated by means of a saddle point approximation. 5.1. Saddle Point Inversion The inversion of the Laplace transform (5.2) is given by

This is purely a mathematical result, which is properly defined for any values of the real numbers a, and p1 which will satisfy p exp(a,N-p,E,,) -+ 0 in the limit E,,, N+ co. Even with the extreme energy dependence embodied in the Bethe formula (5.23) any cr, < 0 and fil > 0 will suffice. To evaluate (5.6), one may use a saddle point approximation leading to (5.7)

FERMI

GAS DESCRIPTIONS

21

where the saddle point chemical potential p0 = u,,& and the saddle point inverse temperature fl,, are the coordinates of the stationary point defined by

(5.8)

and where JDI is the determinant

(5.9)

In performing the saddle point integral, one must choose the arbitrary parameters a, and b1 in (5.6) to coincide with the saddle point values ~1~and PO. At low excitation energies (and therefore low To or high &), c(~ must in fact be approximately equal to PO+, which strongly violates the condition on c1r referred to above. It turns out that the saddle point approximation is nevertheless quite accurate under these circumstances, which must mean that the damping effect of the eepEeXfactor must be sufhcient to keep the integral in shape. The saddle point temperature k, T, = l/b0 is sometimes referred to as the nuclear temperature [5] in terms of a statistical model of the nucleus where one views the nucleus as a gas of nucleons in equilibrium at temperature k,T,. This interpretation is not necessary; the Laplace inversion may be considered simply as a mathematical construction. The choice of Q, and j0 in (5.8) is only required when (5.6) is evaluated through a saddle point approximation. All the quantities in (5.7) are defined in terms of the function 3(~, 8). In the case of a noninteracting Fermi gas the quantities N and Et + E,, in (5.4) can be expressed in terms of the occupation numbers of (0 or 1) of a set of singleparticle levels: N=znf, I

E,E + E,, = c n&. I

(5.10)

It is then easy to see that (5.11) which, if the single-particle spectrum is represented by a continuous function g(e) which vanishes for negative E, becomes an integral: ln ~(cI, p) = jv dE g(E) ln( 1 + e’~ 8”). 0

level density

(5.12)

22

ENGELBRECHT

AND

ENGELBRECHT

If g(s) is furthermore expressed in terms of powers of E, the right-hand side of (5.12) can be expressed in terms of Fermi integrals F,,(a), as defined in Appendix B, In CZ?(cr, 8) = (n+ l;B”+’

Fn+l(a),

where we have for simplicity assumed a single term g(s) = CE”. This choice implies the following connection between N, Er, and the Fermi energy sF,

NC&E n+l

and

F

E+Q,

(5.14)

n+2

where g, = g(+) = c&k is the value of g(s) at the Fermi energy. The saddle point parameters q, and PO are therefore defined in terms of N and -L by N= -& Bo E,N+E,,=-

The argument of the exponential given by

(5.15)

Fn(ao)

,“,, Fn+I(ao). Bo

in expression (5.7) for the nuclear level density is

(5.17) which is precisely the entropy (in units of the Boltzmann constant) at the saddle point. The product of pressure and volume is given by (5.13) divided by /3 and is therefore proportional to (5.16) at the saddle point:

(W,=-$ The fluctuations N*= (N)* and

in the nucleon

(E,N+ E,,).

number

depend

(5.18)

on the difference between

(5.19) w, Bo which also occurs in the expression (5.9) for JDI. In terms of the Fermi functions this determinant is given by da

da

--dF,+1dF,+I da

da

1r(l. PO .

(5.20)

FERMI

GAS DESCRIPTIONS

23

The grand canonical detour circumvents the need for constructing antisymmetrized many-body states at the price of introducing fluctuations in particle number into the formalism. Everything is now expressed in terms of Fermi functions and can be calculated numerically. The inversion of (5.15) and (5.16), in order to determine LX,,and PO, requires particular care, but presents no fundamental difficulties if accurate numerical tehniques are used. In two limiting cases the Fermi functions can be expressed in analytical terms, which leads to closed-form expressions for the nuclear level density. These limiting cases are discussed in Sections 5.2 and 5.3. 5.2. Highly Degenerate Limit

We may first enquire under what conditions the Fermi integrals may be approximated by the leading terms in the asymptotic expansion (B.3). This happens when their argument c(~is positive and large compared to unity. Under these conditions the position of the saddle point is given to leading order by (5.21) The condition that CI~ be large therefore translates into the condition excitation energy be L @N’lg,.

that the (5.22)

This can be achieved either by lowering the excitation energy or by raising the number of particles N. In either case, a relatively small fraction of the fermions occupy levels above the Fermi level so that this limit may be referred to as the highly degenerate limit. If all the Fermi integrals are approximated in this way, one obtains the following expression for the nuclear level density corresponding to a single species of nucleon:

d&x,N)=

(5.23)

This is, of course, none other than our friend the Bethe formula (4.5), with the inverse l/d of the constant level spacing replaced by the value of g(s) at the Fermi energy. Thus the Bethe formula does not depend on the equidistant level approximation, as one might have been led to believe by the combinatorial derivation. The degeneracy condition, which explains the fact that this expression is independent of N, also explains why only the behaviour near the Fermi energy is important, and therefore also explains the absence of any n-dependence. On the other hand, it is also clear that the equidistance assumption by itself is not sufficient for this approximation to hold. It is interesting that the validity condition (5.22)

24

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AND

ENGELBRECHT

obtained with the saddle point method is much weaker than the one (4.6) obtained in the combinatorial method. The treatment of the three-dimensional inverse Laplace transform in the case of two flavours of nucleon follows exactly the same pattern. The level density parameter a is now proportional to the sum of the single particle level densities at the Fermi energy for neutrons and for protons: (g, + g,) n2/6. If these are equal, one recovers (4.10). 5.3. High-Excitation (“Classical”) Limit The other situation where the Fermi integrals acquire a simple form is when e”O becomes small compared to unity, which happens when ai is negative and large in magnitude. Under these circumstances the saddle point conditions become

1 Em -= PO (n+l)N

+ (n+l)N (n+2)8,’

eao=&[(n+I)N:T+‘,

(5.24)

and the Fermi integrals may be replaced by the leading term in the series expansion (B.4). In this limit the excitation energy is of the order (n + 1) Nk,T,,, which must be large compared to N’/g,. Although r, is really a fictitious “temperature” (which would give rise to a thermal energy equal to E,,), this limit does involve many fermions above the Fermi level and may be called the classical limit. If each Fermi integral is replaced by the first term in its series expansion, one can derive the following general expression for the nuclear level density:

P(&,,

NJ =

g,JGi[l.(n+2)]Ne(n+2)N 2n[(n + 1)N]2(“f’)“’

g,Ee,+(n+ n+2

l)* N2 w+‘)~~’ 1

.

(525 .

For the ESM case 12= 0 and this expression reduces to: 2N

P(E,,,N)=$

N-l

1

(5.26 )

This has the same leading order term as (4.12) but the following term is larger by a factor of two, illustrating the fact that the two methods do not produce completely equivalent results. Note that the quantity whithin the last pair of square brackets is simply gF(EeX + Et). In addition to the polynomial dependence on E,, and the dependence on the number of particles N which were noted in Section 4.2, we observe that the level density also becomes strongly dependent on the singleparticle spectral exponent n. This procedure can easily be extended to the real situation of neutrons and protons. For the general case one obtains a result related to (5.25) but, of course, even more intricate because of the two nucleon species. In the equidistant case this

FERMI

GAS DESCRIPTIONS

25

expression reduces to one which has the same leading-order term as in (4.14) but, as in the case of (5.26), a next-order term which is larger by a factor of two. 5.4. Discussion This is a convenient point at which to recapitulate the approximations underlying what we have done. The point of departure of this whole investigation is the representation of the nucleus by a non-interacting gas of fermions (NFG). In view of the strong interactions between actual neutrons and protons, this starting point might be thought to be, in fact, a “non-starter” from the beginning. For many aspects of nuclei this would indeed be the case. We believe, however, that in this respect Bethe was right in his half-century old assumption that for statistical properties this assumption can be fully adequate if properly implemented. From the statistical point of view (as reflected in nuclear level densities and canonical partition functions) the cardinal differences between an NFG and a system with interactions are the reduction of the degree of degeneracy and the downward energy shift of the ground state. Within the Fermi gas picture the first of these is achieved by the version of the NFG in which the single-particle spectrum is taken to be totally non-degenerate from the beginning. The second effect of the interactions can be simulated by means of a properly chosen backshift energy A. A second approximation to the single-particle spectrum which is sometimes mentioned explicitly, is its representation by means of a continuous function in order to implement the grand canonical approach. Whereas mathematicians may worry about analyticity and so forth, from the physical point of view this is immaterial: there are in any case one or two easily recognizable artefacts introduced by mathematical idealization but which can be removed with ease. As long as our statistical conclusions are sufficiently coarse-grained, the difference between a discrete spectrum and its continuous representation is unimportant. A step which does constitute a real physical approximation, is the representation of the nucleus by a grand canonical ensemble (GCE). It is well known that the same results are obtained for microcanonical, canonical, and grand ensembles in the statistical limit where the number of particles becomes infinite. When the number of nucleons of a given species is usually smaller than a hundred, this equivalence is by no means a general rule. The statistics of small systems is certainly not a clear-cut affair and the validity of this GCE assumption deserves further investigation. The next approximation is a purely mathematical one, namely the saddle point approximation (SPA) which is employed to turn the GCE assumption into a practically useful technique. We have already pointed out that in the implementation of this technique for the degenerate limit, the saddle-point in fact occurs in a domain where the convergence of the integral could be open to doubt. Nevertheless, the identity of the result with that obtained in the ESM case with the combinatorial method, which is exact within the NFG framework, removes this worry. In the “classical” limit the combinatorial and grand ensemble approaches yield the same leading term in the ESM case. The factor of two difference in the next term is

26

ENGELBRECHT

AND

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presumably a signal of the imperfection of either the GCE or the SPA. These two limiting cases (degenerate and classical) of course by themselves imply severe constraints on the variables and should be used only in the regimes where they are appropriate. The degenerate limit, yielding the so-called Bethe formulas, does in fact have a respectable domain of applicability. Comparison with exact calculations show, however, that the “classical” limit only begins to resemble the true results at very high excitation energies (or temperatures, in the case of canonical partition functions). For practical results of high precision it may also be important to know the single-particle spectrum well (i.e., which power law exponent IZ to use). In the nontruncated versions this could have a strong effect. In the truncated version, n is not so important but the results depend sensitively on the number H of unfilled levels. Both of these problems are problems of practice and not of principle, and will not be discussed further here. What remains to be looked at, are the very fundamental differences between the three situations depicted in Section 3.3, in the special case of the ESM, as the “infinite,” the “finite” and the “truncated” versions of the single-particle spectrum. In the first, the results are those which would obtain if the nucleus contained an infinite number of nucleons which could be excited to higher levels. Although the results obtained in such a picture may hold rigorously in certain domains (of sufficiently low excitation energy or temperature) independently of this underlying assumption, the assumption itself is, of course, totally unrealistic. In the “finite” version, there are truly only N nucleons of a specific kind available, but there is an infinite number of levels to which they may be raised. In the “truncated” version, this number H of available unfilled levels is also taken to be finite. The question as to which of the “finite” or “truncated” versions is applicable to the consideration of a particular physical effect, will not be considered further in the present work. (This question is addressed explicitly elsewhere [ 141 for the partition functions to be used in stellar collapse calculations.) What we shall do here, briefly, is to discuss the techniques which may be used to obtain quantitative results for these three versions of an NFG. The infinite version is well known, of course, and finds its realization in the “Bethe formulas” for densities and partition functions. These expressions emerge very naturally as limiting forms of both the combinatorial and the statistical calculations. At low energies it is advisable, however, to use the improved versions discussed in Section 4.4 rather than the better-known forms canonized by long usage. The grand canonical approach is ideally suited for the investigation of the finite NFG models. The “classical limit” results have very restricted use and these calculations should therefore be done exactly. That means, the Fermi integrals must be inverted numerically to turn the SPA formulae into quantitative results. This is discussed in Appendix B. For the truncated version, the elegant method provided by the way in which the grand partition function transcended the constraints, is lost. In this case the vehicle

FERMI

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27

for an exact calculation is the combinatorial method. Although it could with tremendous exertion also be applied to more complicated spectra if the system was small enough, its straightforward use is really restricted to the ESM. Even there large values of N and H quickly tax the memory capacity of the computer. Fortunately, the results obtained for these exact calculations appear to be represented quite well by simple recipes, which are provided in Sections 4.3 and 6.1. Since the purpose of the present article is clearly not to present realistic results for real nuclei-Ref. [ 141 does consider such cases-we shall illustrate the nature of the results obtained with some of these methods only for the artificial situation of one nuclear species and only for a single representative case, chosen arbitrarily. The curves on Fig. 2 denote the single species level density obtained for the infinite (Bethe) model (curve A), for the finite models (curves B, for the ESM (n = 0) and B, for FRB3 (n = i)) corresponding to N = 24, and for a truncated model (curve C) where, in addition, H is taken to be 12. To relate the results to familiar physical magnitudes, a typical value of 0.5 MeV may be used for the single-particle level spacing d (corresponding to a, = 3.29 MeV’ or a = 6.58 MeV-‘). This puts the peak of curve C at E,, = 72 MeV. At these low excitations the asymptotic “classical” expression (5.25) is still completely inapplicable. For the ESM it is a factor 2.5 x lo4 too high at E,,/d= 144, while for n = 4 this factor is 8.1 x 104! The exact calculations of the finite model begin to deviate significantly from the Bethe formula already when L zz2N and the differences between different n-values become visible around twice that excitation.

Eex/d FIG. 2. Dependence of log( pd) on E,,/d at intermediate energies. Curve A represents the Bethe formula. Curves B, and B, represent exact calculations for the finite model with N = 24, and with n = 0 and n = f, respectively. Curve C represents the TFG with H = 12.

28

ENGELBRECHTANDENGELBRECHT

The truncated ESM model deviates from the others very soon and at peak the density is already more than three orders of magnitude smaller. Thereafter this difference increases even more rapidly.

6. CANONICAL

The canonical partition density:

PARTITION

FUNCTIONS

function is usually expressed in terms of the nuclear level

Q(N, p) = Jo- dE,, e-P’exp(E,,,

Nj.

(6.1)

Instead, it may also be viewed as the result of inverting the Laplace transform (5.5): (6.2)

In the same way as before, a saddle point approximation this integral,

may be used to compute

(6.3 1

where now there is only one saddle point condition:

As in Section 5, a power law single-particle spectrum allows all quantities to be expressed in terms of Fermi integrals. The canonical partition function is a much simpler quantity than the level density in that for two species of nucleon one must just use the product of the partition functions of each species separately, rather than a convolution. The degenerate limit is again obtained when c(~is large compared to unity. Now this no longer translates into a condition on the excitation energy but on /I or on the temperature: NB%-gg,

or

NdP k,T.

(6.5 1

The result of (6.3), which could also be obtained by substituting is

(5.23) into (6.1)

Q(N, Bi = J7C/12crok,Texp(a,k,T), where we have written

n2g,/6/? = aok,T.

The apparent

(6.6)

singularity

at zero

FERMI

temperature is a mathematical expression by

GAS

29

DESCRIPTIONS

artefact which can be removed by replacing this

Q(NJ)=Jw

(6.7 1

when a,k, T < 1. (That this replacement provides a reasonable approximation is borne out by the exact calculations referred to in Section 6.1.) The canonical partition function for two types of nucleon is just the product of the separate partition functions, resulting in (6.8)

where a = 24, is now the level density parameter for two types of nucleon. The classical limit follows when eao is small compared to unity. This translates into the condition NB
Or

Nd
(6.9)

and leads to the result,

1

In+‘lN

.

(6.10)

For n = 0 this expression can be rewritten as (6.11) It is interesting to compare this dependence on E,E with that found for the level density (5.26). When comparing with the degenerate limit, we see that here, too, the exponential dependence on temperature is replaced by a power law which depends both on N and on n. For two types of nucleon one again obtains the product of the individual partition functions. 6.1. Truncated Fermi Gas The partition function (6.6), derived from the Bethe level density formula, is valid for all temperatures only in the case of an infinite fermion sea. For a finite number N of nucleons, the validity regime (6.5) shows that this exponential dependence is still valid at small temperatures. When the temperature becomes large, however, (6.10) (or (6.11) in the special case of equidistant single-particle levels) shows that this goes over into a power law. For a truncated Fermi gas the powerful saddle point method is once again not at our disposal. For the ESM-tfg we can, however, calculate the canonical partition function directly, using the combinatorial techniques discussed in Appendix A. As was

30

ENGELBRECHT

AND

ENGELBRECHT

already pointed out in Section 4.3, the last sum over L on the right-hand side of (A.1 1) is none other than Q(N, /I) is y is replaced by e-p”, where d is the singlenucleon level spacing. Thus, we may write

Q;((rr)= f’;(~),

(6.12)

where the polynomials P;(y) can be calculated by a recursion formula which is very simple, although the calculation uses substantial computer memory and requires high precision. These calculations are discussed elsewhere [14]. In the limits of very high b (low temperature), the resulting partition function becomes equal to the expression (6.6) (or its modification (6.7)) obtained from the Bethe formula, which will be denoted by QB(x) (with x = a,k,T) in the remainder of this section. At very low B (high temperature), it becomes equal to the constant asymptotic value (N+ H)!/N! H!, which will be denoted QA. From (6.11) it was seen that the partially constrained case (infinite H but finite N) at sufficiently high temperatures exhibited at TN dependence. The symmetry between N and H implies that in the case of finite H and infinite N one should similarly obtain a TH dependence. The result of the aforementioned calculations is that for finite N and H there exists a substantial intermediate temperature domain over which the partition function can be represented quite accurately by (6.13) where the exponent M, is given by 1p4, = 11~ + I/H + 318 JNH.

(6.14)

These calculations also yielded simple recipes for interpolating between the three forms Q,, Qp,and QA. The interpolation between the first two can be represented adequately by

QF= Qi4x)+ [Q&l - QAx)l &:,,,(a, x>61, where &(a, b= 3M,/2.

(6.15)

x, 6) is the function we defined before in (4.25), a= M,/2 and The transition between QF and QA is represented very well by the

prescription:

l/Q = l/QF+ l/QA+ U.5)Mp/(Q~Q:)“4.

(6.16)

In this way the partition functions QN( /?) and Qz(/?) for neutrons and protons separately are obtained. Within the framework represented by (4.17) the canonical partition function for the nucleus as a whole is then given by (6.17)

FERMI

31

GAS DESCRIPTIONS

At low temperatures this expression depends very sensitively on the backshift parameter A. In order to make the present paper self-contained, we give its prescription, as obtained from the tits described in Ref. [14]. It was found to be the sum of two contributions A,, + Adc,y, of which: 8.1 MeV A oe= q , ~ fi

10 MeV

(6.18)

+927

with (q,, q2) given by (+ 1, - 1) for eveneven, by (- 1, + 1) for odd-odd, and by (0, - 1) for odd nuclei, represents the pairing effect. Shell effects are of course already reflected in the level spacing parameter. It was found, however, that A must include another contribution, which consist of a negative background value which becomes large and positive when the proton number Z and the neutron number N simultaneously approach any of the magic numbers M,. In actual fact this relationship could be explored only in the three doubly magic regions (20, 20), (20, 28), and (28, 28). Calculations for 34 nuclei in this region were represented by a 3-parameter lit given by A

-9MeV da -- A 213

cc L-1 ,., 1 + d,ifi,

1 )

(6.19)

where

f, = CM;- .a2+ (Mj- lty2, ciix[(~)(~)~i3, and d;, = 0.215 c,-.

(6.20)

For negative values of the sum A,, + A,., the exponential factor in (6.17) will diverge when the temperature becomes very small. Using (6.6) to evaluate the factor QN( /I) Q,( /I), one should determine at which temperature its product with the exponential factor no longer decreases with decreasing temperature. This cutoff temperature should then be used in the exponential factor instead of the actual temperature if the latter is lower. The comparison between the infinite, finite, and truncated Fermi gas models, which was presented for the nuclear level densities in Fig. 2, is continued for the canonical partition functions in Fig. 3. In this case the finite models, represented by B, and B,, deviate more strongly from the infinite Bethe curve A. Once again the asymptotic expression (6.10) yields results which are still several orders of magnitude too high in this region. The TFG curve C detaches itself from the Bethe curve from 2 x 10” K onward. Here too, the differences are more marked than in the case of the level density.

32

ENGELBRECHT AND ENGELBRECH’I

FIG. 3. Dependence of log Q on T9, the temperature in units of lo9 K. The curves A, B,, B,, and C correspond to the same cases as in Fig. 2. The dotted line D denotes the saturation value which is the asymtotic limit of curve C.

7. CONCLUSION

In summary, we have in this paper first recapitulated the rationale for a Fermi gas picture of the nucleus and restated the expected correspondence between such a picture and statistical nuclear properties such as the level density and partition function. In particular, this correspondence is expected to be reasonably good if we use a truncated Fermi gas model with a non-degenerate single-particle spectrum, combined with an appropriate backshift to simulate the downward displacement of the ground state energy. When nuclear properties are deduced within a Fermi gas basis, three totally different frameworks should be distinguished from one another. In the “infinite” picture, there are no constraints on the number of fermions which may be excited. This leads to level densities and partition functions which increase exponentially with energy or with temperature. In the “finite” picture, only a specified finite number of nucleons N are available, which -reduces the rate at which these quantities increase to power laws. The most dramatic change occurs in the “truncated” case, where the number of single-particle levels H available for excitation is also restricted. In that case the level density rises to a maximum and then declines again to zero above some maximum energy while the increase of the partition function begins to flatten until it saturates at a finite value, which is just the integral over the level density curve.

FERMI GAS DESCRIPTIONS

33

Most standard treatments of these quantities correspond to the infinite framework, and the specific results are usually derived as certain limiting cases of either a grand canonical approach or a combinatorial analysis. Exact calculations of the finite case are possible within the grand canonical method while combinatorial calculations are required for the truncated model. The results of such calculations are compared with the expressions corresponding to the infinite case. In the situations encountered in most nuclear physics investigations, the excitation energies or temperatures are low enough for the constraints on N and on H not to be effective, so that the results produced by the infinite model are fully adequate. Even in this case, the formulas usually employed do not really approximate the model well and could be improved. However, under the conditions encountered in stellar collapse and in ultra-high energy heavy-ion reactions, the standard expressions become totally inadequate and many of these problems should be reinvestigated on the basis of the truncated Fermi gas (TFG) model. In this paper we have not only tried to chart the space of Fermi gas models, but also to provide the minimal tools necessary for investigations based on the TFG model. The involvement of the senior author (CAE) with the truncated Fermi gas model began in 1980 in order to provide a basis for correcting some of the errors committed in FEW (see [16], including the concluding quotation) and has, in the intervening years, progressed in several spurts which were separated in time. The detailed consideration of those errors and their removal is presented elsewhere [14]. That this present survey of the Fermi gas domain contains in several places a critique of the way Bethe’s level density formula is used, should by no means be seen as an attempt to detract from the value of his trail-blazing work. In fact, it was at the time hoped that the present investigation would be ready to be presented as a tribute on Bethe’s eightieth birthday, but there were then too many gaps left. Some of these were removed as part of the research towards a master’s degree by the second author [17] and the remaining holes were filled during a visit by CAE to the Max Planck Institute for Astrophysics in Garching. The director is kindly thanked for the use of the facilities, and all the staff for their warm hospitality and helpfulness. The principal financial support which made this visit possible was provided by the Alexander von Humboldt Foundation, while all the phases of this project thrived on the support provided by the Foundation for Research Development and the University of Stellenbosch. Although the publication of this work is no longer contemporaneous with one of those years when Hans Bethe’s age is zero modulo ten, we would in any case like to dedicate it to the inspiration he has sown over so many fields of physics.

APPENDIX

A: COMBINATORIAL

CALCULATIONS

As indicated in Section 4, the degeneracy W:(L) of a nuclear level with excitation energy Ld when constructed out of excitations of N fermions occupying K = N + H equally-spaced (with spacing d) non-degenerate single-particle levels, is

34

ENGELBRECHT

AND

ENGELBRECHT

equal to the number of ways an integer L can be partitioned into no more than N positive integers, each of which is not allowed to exceed H. It is also the number of different Young tableaux of L blocks arranged in no more than N rows and no more than H columns. Clearly, W,“(l)=

1

if

Ndl

and Hbl.

(A.1 1

In accordance with the values pertaining to the nuclear model, we may also define WC(L) to be zero for negative values of N, H, or L, and otherwise, W,H(O) = 1

and

W,H(L) = W;(L)

= 6,,.

(A.2

It is then easy to derive the symmetry relation (A.3

W,H(L) = w;(L), as well as the recurrence relation w;(L)=

1

64.4)

W,“I;(L-JH),

J=O

which can be summed to give w;(L)

= WC- ‘(L) + Wf-

,(L - H).

(A.51

These relations can be used to generate the W:(L) for all positive values of N, H, and L. By its construction, it is clear that the value of WC(L) will not depend on H if L < H, on N if L < N, or an either if both inequalities hold. The latter case, with N and H both > L, is called the unconstrained partition. Instead of writing it as W,“(L), we shall simply suppress the values of N and H which exceed L and write W(L). Similarly, when only one of the constraints are effective, we shall write W,(L). For nonvanishing N and H, W:(L) becomes unity not only at L = 0 but again at L = NH. It peaks at the midway point and satisfies the sum rule: (A.6) L-0

Euler already published tables of W,(L) for N d 11 and L < 70 in Chapter 16 of his monograph on analysis [12]. The application of the recurrence relations is slow and tedious. An alternative approach is presented by Euler’s generating function,

fN(X)=kfI,& =.f

L=O

W,( L)P.

(A.7)

FERMI

35

GAS DESCRIPTIONS

For the unconstrained partition (N= co, in which case f.,,(x) will be denoted as f(x) only) Hardy and Ramanujan [ 131 derived an asymptotic formula by performing a Cauchy integral of f( x)/x” + ’ along a contour enclosing the origin, W(L) =-

1 4,/5L

Ml;I-

exp 7c

(A.8)

This expression does approach the correct value in terms of percentage error but certainly not in magnitude: at L = 7 the error is still 22% and even for L = 100 the error is still larger than 4%. The reason for this slow convergence is to be sought in the extremely singular behaviour of f(x): every point on the unit circle is an essential singularity and the function does not even exist outside this circle! In their paper Hardy and Ramanujan showed that (A.8) can nevertheless be improved in a systematic way by successively removing the singular behaviour at x = 1, at x = -1, at x = exp( +2ni/3) and so forth. The result produced by the first two steps in this sequence is W(L)=J

2(S

-

1 expS+(-1)L(S-2)exp(S/2) )

8rGy2

(A.9)

where and

(A.lO)

This already reduces the percentage error to about 1% for L > 7 while the error at N = 100 becomes 1 in 5 x 10’. The generating function (A.7) may be used to determine the values of the unconstrained partition W(L) and the partially constrained partition W,(L), which correspond to the ESM-inf and the ESM-fin versions of (3.20), respectively. For the fully constrained partition W:(L), corresponding to the ESM-tfg version, one could revert to the repeated application of the recurrence relation (AS), but this is a very inefficient procedure in which the number of steps increases proportionally to N! H!. However, for this case an alternative generating function (A.ll) k=l

APPENDIX

N=O

L=O

B: FERMI FUNCTIONS

The equilibrium statistical mechanical description of non-interacting fermions within the grand canonical ensemble involves the evaluation of the grand canonical

36

ENGELBRECHT

AND

ENGELBRECHT

partition function (5.12). If g( E) can be expressed as a sum of powers (which need not be integral) of E, the integral proportional to sn-’ can be manipulated by partial integration into a term proportional to the Fermi integral

(B.1) which converges for n > - 1. The Fermi integrals of different order, n, are related under differentiation by FL(x) = nFH;,- i(x) for all n > 0. For 12= 0 the right-hand side of (B. 1) becomes integrable, yielding Fe(x) = ln( 1 + e”),

Fd(x)=(e-*+‘l))‘.

(B.2)

For n #O these integrals cannot be expressed in closed form. However, an asymptotic expansion [9] exists, FJx)

=s

[l+(‘;*)$22-2)~,+(“fi1)$24-2)8,+

--],

(B.3)

where the (if denote the usual binomial coefficients while B, = i, B, = 6, ... are Bernoulli numbers. This expansion is totally unreliable for x-values below 2, but leads to reasonably accurate values for x > 6. (Large values of x correspond to low temperatures.) For x
e-‘-$+$-

1

. ,

(B.4)

which becomes, for x = 0, equal to (1 - l/2”) times the Riemann zeta function an + 1). In the exact saddle point calculations referred to in Sections 5 and 6, (B.3) was used to calculate the Fermi integrals in the region x > 5.8 while the leading term of (B.4) was used for x< -3.8. In the intermediate region, values of F,(x) were obtained by interpolating spline curves through sets of consecutive values of the Fermi integrals tabulated by Blackmore [19].

REFERENCES 1. H. A. BETHE, P~JX Rev. 50 (1936), 332. 2. H. A. BETHE, Rev. Mod. Phys. 9 (1937), 69. 3. J. B. FRENCH AND V. K. B. KOTA, Ann. Rev. Nucl. Part. Sci. 32 (1982), 35. 4. T. D. NEWTON, Canad. J. Phys. 34 (1956), 804; Erratum, ibid. 35 (1956), 1400. 5. T. E. 0. ERICSON, Adv. Phys. 9 (1960), 425. 6. A. BOHR AND B. R. MOTTELSON, “Nuclear Structure, Volume I,” Benjamin, New

York,

1969.

FERMI 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

GAS DESCRIPTIONS

37

S. G. NILSSON, Dan. Videnskabern. Selsk. Mar. Evs. Medd. 29 ( 19551, 16. M. HILLMAN AND J. R. GROVER. P/QJS. Rev. 185 (1969), 1303. A. SOMMERFELD, 2. Phys. 47 (1928), 1. S. A. GOUDSMIT, Phys. Rev. 51 (1937), 64A. C. VAN LIER AND G. E. UHLENBECK, Physica 4 (1937), 531. I. EULER, “Introductio in Analysin Inlinitorum,” Vol. I, 1748. G. H. HARDY AND S. RAMANUJAN. Proc. London Math. Sot. 17 (1918), 75. C. A. ENGELBRECHT, J. R. ENGELBRECHT, AND D. P. JOUBERT, Asfrophys. J., to be submitted. C. A. ENGELBRECHT AND D. P. JOUBERT, in “Proceedings, International Conference on Nuclear Structure, Amsterdam, 1982, Contributed papers,” Vol. 1, p. 42. W. A. FOWLER. C. A. ENGELBRECHT, AND S. E. WCOSLEY, Astrophvs. J. 226 (1978), 984. J. R. ENGELBRECHT. M. SC. thesis, University of Stellenbosch. 1988. D. P. JOUBERT, M. SC. thesis, University of Stellenbosch, 1982. J. S. BLACKMORE, “Semiconductor Statistics,” Pergamon, Elmsford, NY, 1962. T. E. 0. ERICSON. Nucl. Phys. 11 (1959), 481. A. GILBERT, F. S. CHEN, AND A. G. W. CAMERON, Canad. J. Phys. 43 (1965), 1248. A. GILBERT AND A. G. W. CAMERON, Canad. J. Phys. 43 (1965), 1446. A. N. BEHKAMI AND J. R. HUIZENGA. Nucl. Phys. A 217 (1973), 78.