Average nuclear properties

ANNALS OF PHYSICS: 55, 395-505 (1969)

Average Nuclear Properties

WILLIAM D. MYERS AND W. J. SWIATECKI

Lawrence Radiation Laboratory, University of CaliJbrnia, Berkeley, Califbrnia 94720

A generalized treatment of average nuclear properties is presented. The theory is developed on two levels: First a refinement of the Liquid Drop Model, called the Droplet Model, is described. The degrees of freedom in this model, in addition to the usual shape variables, are variables specifying deviations from uniformity of the neutron and proton densities. The form of the Hamiltonian defining the Droplet Model, of which only the potential energy part is considered in this paper, is derived by expanding the volume, surface, and Coulomb energies in Taylor series around the standard Liquid Drop Mode1 values. Such an expansion, designed to retain all terms in the total energy up to order Ai’“, PAZ/“, and PA, where I = (N - ZU.4, turns out to contain eleven parameters, two of which may be eliminated. Four of the resulting nine parameters are the standard adjustable parameters of the Liquid Drop Model and five are new coefficients specifying various properties of nuclear systems. (Nuclear compressibility and the curvature correction to the surface tension are two examples.) Minimizing the Droplet Model potential energy with respect to density variations leads to equations, in closed form, specifying the separate neutron and proton radii and the density nonuniformities. The minimized energy expression leads to a refined Droplet Model Mass Formula with nine parameters.

The second level at which average nuclear properties are treated is based on assuming a concrete model of a two-component saturating system, consisting of neutrons and protons interacting by velocity-dependent Yukawa forces (and Coulomb forces). When this model is treated in the Thomas-Fermi approximation a pair of coupled integral equations results, which can be used as the basis of a self-contained model of all average static nuclear properties. The solutions of these equations are discussed in the idealized situations of nuclear matter and semi-infinite nuclear matter, and for finite nuclei both with and without Coulomb energy. One result of these studies is the determination of the values of the five new Droplet Model parameters. Other results have to do with the nuclear density distributions, and the binding energies.

The applications of the Droplet Model and Thomas-Fermi Mode1 discussed in this paper include predictions concerning neutron and proton radii (in particular the presence of a neutron skin), the isotope effect in proton radii, the compression of the nucleus by the surface tension and the dilatation by the Coulomb energy, and the central depression in the densities caused by the Coulomb repulsion. Calculations are made for the surface curvature correction, for the surface symmetry energy, and for a modification to the volume symmetry energy at a large neutron excess. A revised estimate is made for the value of the symmetry energy of nuclear matter. Also treated is the question of whether or not neutron matter is bound, and some discussion is given of the spatial distribution, the energy dependence, and the composition dependence to be expected for nuclear optical model potentials on the basis of the statistical methods used in this paper.

395 1969 by Academic Press, Inc.

595/55/3-I

396 MYERS AND SWIATECKI

CONTENTS

Abstract . . . . . . . . . . I. Introduction . . . , . . .

Il. Droplet Model . . . . . . . A. Qualitative Considerations B. Energy Expression . . . . C. Minimization . . . . . .

III. Theorems . . . . . . . . A. Proofs . . . . . . . . . B. Comments . . . . . . .

IV. Thomas-Fermi Model . . . A. Qualitative Considerations B. One-Component Systems

1. Nuclear Matter . . . . 2. Finite Nuclei . . . . 3. Semi-Infinite System. .

C. Two-Component Systems . I. Nuclear Matter . . . 2. Finite Nuclei . . . . 3. Semi-Infinite System .

D. Coulomb Energy . . . .

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V. Comparison ......... A. One-Component Systems . . . B. Two-Component Systems . . . . . C. Coulomb Energy. ..... . . .

VI. Discussion .................... A. Droplet Model Formulae ............ B. Mass Formula ................ C. Refitting Constants. .............. D. Density Distributions and Nuclear Sizes ..... E. Remark on Comparisons with Experiment .... F. Thomas-Fermi Model ............. G. Optical Potentials ...............

VII. Summing Up .................. Acknowledgments ................... References ......................

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395 396 398 400 405 410 417 417 426 427 427 432 433 439 443 449 449 456 460 467 471 472 474 479 480 480 484 487 488 499 500 501 503 504 504

I. INTRODUCTION

The development of nuclear theory in the past 35 years has been marked by two trends: the microscopic approach and the macroscopic approach. This dual approach is associated with the fact that the number of nucleons in most atomic nuclei is neither very large nor very small. The microscopic approach centers around the single-particle degrees of freedom and is typified by the Shell Model. An example of the macroscopic approach, in which the concern is with the collective degrees of freedom, is the Liquid Drop Model.

AVERAGE NUCLEAR PROPERTIES 397

The relation between a microscopic and a macroscopic approach is that between a more general and a less general theory. The microscopic theory is the more general one and contains the macroscopic one as an average. In practice each approach is useful both in its own right and as a necessary step in a synthesis of the two methods. (For further discussion of this point see Refs. (f-5).)

This paper is concerned with the macroscopic treatment of the average properties of a saturating system, such as a nucleus, consisting of two components (neutrons and protons). Our objective was a systematic refinement of the existing Liquid Drop Model of average properties. The refinements we are concerned with are associated with effects arising from the smallness of nuclei. More precisely the refinements may be regarded as additional terms in an expansion of nuclear properties in terms of the fundamental dimensionless ratio characterizing nuclei, namely the ratio of the interparticle spacing to the nuclear radius. This ratio is given approximately by A-1:3. In introducing these refinements to the Liquid Drop Model we are taking a step in the direction of the microscopic theories in the following sense: One may regard effects of order A as belonging to the domain of nuclear matter studies, effects of order A2j3 as characterizing the Liquid Drop Model. and effects of order 4” as belonging to a microscopic theory. M here indivi- dual particles (consisting of one ‘4th part of the system) dominate the approach. This leaves a gap associated with effects of order A1j3. and the studies described here are concerned with filling this gap.

The reasons for studying such refinements to the macroscopic approach are partly a desire for completeness and a better understanding of the whole problem, and partly of a practical nature. Despite the fact that the energies associated with terms of order A1j3 are several times as large as typical A” terms, such as shell effects. the refinements in the A1/3 category are not associated with prominent experimental features because of their smooth dependence on the neutron and proton numbers. (Contrast this with shell effects.) In fact it is only in the past 5 or 10 years that experimental evidence for the refinements that we shall discuss has been emerging. (The experimental results to which we refer are of the most varied nature. They include. among others, finer details of the nuclear charge distribution, the isotope effect in nuclear radii, curvature corrections to the nuclear surface tension, and the presence of a neutron skin.)

Many of the effects we will consider have been studied before (6, 7). They have often been treated one at a time, with the result that the connection between intrin- sically related phenomena would not be apparent. Our objective was to develop a refined model of average properties that would be capable of describing all these effects systematically from a unified point of view.

In this paper we discuss the problem on two levels, described in Sections II and IV, respectively. The first one is concerned with the introduction of what we have called the Droplet Model. (We use the diminutive of Drop to indicate that the


refined theory would apply to small systems where-in contrast to the Liquid Drop Model assumption-the range of the interparticle forces is no longer negligible compared with the dimensions of the system.) Thus Section II concerns itself with the formulation of a refined model based on an expansion of the volume, surface, and Coulomb energies about the values they have in the Liquid Drop Model. The coefficients in the expansions (of which there turn out to be eleven) are left undetermined.

Section III is devoted to the detailed derivation and discussion of three theorems related to the nuclear surface tension, and completes the development of the Droplet Model.

In Sections IV and V the problem is taken up in an independent way by studying concrete examples of saturating systems, systems consisting of neutrons and protons interacting via a two-body velocity-dependent interaction. The properties of such systems are determined in a statistical approximation analogous to the Thomas-Fermi treatment of electrons in atoms. The study is arranged so that it exhibits the above-mentioned hierarchy of approximations in powers of A-l13. Thus, after studying nuclear matter, we go on to exhibit the Liquid Drop approximation, and finally the analysis is carried to the next stage and contact is made with the predictions of the Droplet Model. The formal structure of the Droplet Model is confirmed and the coefficients it requires are calculated in terms of the assumed interaction between the particles.

The Thomas-Fermi Model (a-20), although we use it as a complement to the Droplet Model, is an independent theory with wider applications than those contemplated by the Droplet Model. The Thomas-Fermi treatment is, in fact, a self-contained theory which, within its limitations, may be asked to answer practi- cally any question about average properties. Some of these more general applications, not related to the Droplet Model, are considered in Section IV.

The principal purpose of this work was the formulation of a refined theory of average properties of saturating two-component systems. The applications of this theory, both in its Droplet Model version and its Thomas-Fermi version, are exceedingly numerous. The actual carrying through of these applications is a considerable undertaking which we plan to pursue in the future. In this paper the applications are limited; they are discussed in Section VI.

Section VII sums up the paper. The principal conclusion is that, to a large extent, it appears possible to fill in the gap that exists between the Liquid Drop Model and the microscopic approach.

11. DROPLET MODEL

The Droplet Model is a generalization and a refinement of the Incompressible Liquid Drop Model of nuclei. Although this generalization eventually leads to a


host of corrections, the two fundamental refinements which characterize the Droplet Model are associated with the finiteness of the thickness of the nuclear surface and the finiteness of the nuclear compressibility. Thus the density distributions of neutrons and protons are no longer considered to be constant inside a sharp boundary and zero outside, as they may be in the Liquid Drop Model. In the Droplet Model the densities are only approximately constant in the bulk of the system, and instead of a sharp boundary there is a diffuse surface region in which the densities decrease smoothly to zero. The density nonuniformities in the bulk and the thickness of the surface region are, however, treated as small quantities.

As usual the first two steps in defining a new model are

(a) The specification of the degrees of freedom.

(b) The specification of the Hamiltonian of the system (i.e., its potential and kinetic energies) in terms of the degrees of freedom.

In the Liquid Drop Model the degrees of freedom are the variables (in general infinite in number) necessary to specify the shape of the sharp boundary of the system. The potential energy consists of a volume energy, a surface energy, and a Coulomb energy, and is a functional of the shape. (This paper does not go beyond discussion of the potential energy-study of the kinetic energy is a separate problem.)

In the Droplet Model the degrees of freedom consist of two sets. The first is. as before, associated with specifying the shape of the (now diffuse) boundary, the second with specifying the (small) density nonuniformities in the bulk. The potential energy consists of a volume energy, a surface energy, and a Coulomb energy, and it is now a functional of the shape and of the neutron and proton density distributions.

In the Liquid Drop Model two classes of equilibrium shapes are of special importance: Spherical equilibrium shapes, whose energy corresponds to the Liquid Drop Model formula for nuclear masses, and the nonspherical equilibrium shapes with degree of instability one (the saddle-point shapes) whose energy provides a description of fission barriers (21,22). In the Droplet Model the same two classes of equilibrium shapes are expected to be present, but the problem of determining the equilibrium configurations is considerably more difficult because of the additional degrees of freedom associated with the nonuniformity of the densities. Thus even the determination of the energy of the spherical equilibrium shapes- a trivial problem in the Liquid Drop Model-is a matter of some intricacy. In this paper we devote most space to the discussion of the spherical configurations of equilibrium in the Droplet Model and to the resulting nuclear mass formula. We also comment on but do not solve the problem of saddle-point shapes.


A. QUALITATIVE CONSIDERATIONS

We derive the basic equations of the Droplet Model from the following general considerations.

The energy of a large class of physical systems may be considered to be given approximately as an integral over all space of an energy density, which is a functional of the particle density (or particle densities, if the system is a mixture of several components). An important subclass of such systems is that of saturating systems. They are characterized by the fact that in the limit of a very large assembly of particles the particle densities tend to uniform values over the bulk of the system, and the total energy has a stable minimum when considered as a function of the uniform bulk densities. A condition for a system of interacting particles to show saturation is that the interparticle attractions have certain properties; in particular, they should be of limited range and they should not be too attractive at short distances or high densities.

If instead of the idealized limit of an infinite saturating system we consider a finite system, deviations from the uniform density will have to be considered, in particular in the surface region. Furthermore, if nonsaturating forces are present- such as the Coulomb repulsions between protons-then density nonuniformities will also be induced in the bulk. The Droplet Model concerns itself with a discussion of these effects considered as perturbations of an idealized saturating system. The expansion parameters in this treatment may be considered to be the ratios of the surface and Coulomb energies to the bulk (or volume) energy. The smallness of the former parameter means a small surface-to-volume ratio. This implies a large system in the sense that characteristic spatial dimensions of the system should be large compared with the thickness of the diffuse surface region (or with the range of the interparticle forces or with the interparticle spacings, since these are all related quantities). In the nuclear case the ratio of the interparticle spacing to the nuclear radius is of order A-1/3, and we may consider this dimensionless ratio as our expansion parameter. We note that the standard Liquid Drop Model retains terms of order A and N3; the Droplet Model extends the expansion to include terms of order A1/3. (See (4) for further remarks on this hierarchy of effects and a discussion of where shell effects enter.)

In addition to keeping terms of higher order in A-li3, the Droplet Model that we formulate also gives a more accurate description of the composition dependence of nuclear properties by including a treatment of higher-order terms in the neutron excess, or nuclear asymmetry Z, defined by (N - Z)/(N + 2). For example, besides the usual symmetry energy proportional to Z2A we include the next term in Z4A. More generally, our development of the Droplet Model has been guided by the following pattern of terms in an expansion of the total energy in powers of the two small quantities A-l13 and 15


1 Order in A-lj3 -

Order A ,43/3 A1 3 in I2 PA ~2~313 (2.1)

l I PA

Retention of only the term in A constitutes the standard nuclear matter approximation. Inclusion of the surface energy proportional to A213 and the symmetry energy proportional to Z2A corresponds to the Liquid Drop Model. The Droplet Model (with some qualifications that are explained later) is characterized by working to the next order of terms along the diagonal, which includes, besides terms proportional to AlI3 and Z4A, terms proportional to Z2A2j3. (One such term is the surface symmetry energy which arises from the dependence of the surface tension on nuclear composition.)

Finally, as regards the electrostatic energy, the Droplet Model includes three refinements beyond the approximation of a uniform sharp distribution of charge. The first two, a correction for surface diffusness and a correction for the charge redistribution associated with finite compressibility and polarizability, are inherent in the Droplet Model. There are no compelling reasons to include the third correction (the exchange correction). We thought it might eventually be well to have this straightforward effect accounted for when discussing the relation of the other refinements of the Droplet Model to the experimental data.

The formulation of the Droplet Model (specialized for the purpose of this discussion to spherical shapes) proceeds according to the following outline. The density distributions pn and pZ of the N neutrons and 2 protons will be characterized by two diffuse surfaces whose effective radii are R, and R, . The definition of such radii is possible in virtue of the assumption (rooted in the saturating character of the nuclear interactions) that the system may be divided into a bulk region and a thin surface region. The neutron radius is defined as that radius which would contain all the N neutrons if the bulk neutron density were extrapolated outward. The effective proton radius R, is defined in an analogous way. In addition we can define an effective nuclear radius R by extrapolating the bulk nuclear density p (equal to pn + p,) outward until it contains A particles.

The densities pn and pZ , within the effective radii R, and R, ) are the degrees of freedom of the (spherical) Droplet Model. The total energy of the system consists of volume, surface, and electrostatic energies.

The volume energy, an integral over an energy density, depends on the bulk density distributions. The surface energy depends on the surface conditions and includes a dependence on the local curvature of the effective surface. The electro-


static energy is a functional of pZ . A series expansion is assumed for the volume energy density and also for the dependence of the surface tension on surface conditions. These series are truncated at suitable low-order terms (higher-order terms being kept in the volume energy than in the surface energy), and the total energy, Coulomb energy included, is minimized with respect to small particle- conserving changes in the density distributions. This leads to a standard variational problem, which in our case results in four equations: two for the average neutron and proton densities pn and pZ (or, equivalently, for the effective radii R, and R,) and two for the functions specifying the deviations of the bulk density distributions pn and pZ from their average values.

The solution of these equations, which is possible in closed form, gives rise to the Droplet Model of nuclei. This description includes the equilibrium energy of the system, which may be used as the basis for a refined nuclear mass formula.

Before formulating the Droplet Model we have to take care of some preliminaries. Consider a nucleus described by the schematic density distributions in Fig. 1.

The bulk functions pn and pZ are shown extrapolated to the effective sharp boundaries R, and R, even though-and it is important to stress this point-the

fit+. 1. Schematic Droplet Model nuclear densities as a function of the radial distance. The solid lines show the bulk functions pn , p z, and p extrapolated out to the effective sharp radii R,, , R, , and R; the Liquid Drop Model radius R, is also shown. T’he dashed lines serve to indicate that the surfaces are really diffuse. The lower part of the figure shows the functions --E and 6.


Droplet Model takes into account the surface diffuseness, as shown schematically by the dashed lines. Figure 1 shows also the total density p extrapolated to its sharp surface R as well as graphs of two dimensionless quantities, E and 6. The former, defined as

(2.la)

specifies the deviation of the density p from its nuclear matter value p. (The ----5 appears in connection with the conventional definition of the compressibility coefficient, and is introduced here for later notational convenience.) The bottom part of Fig. 1 is a plot of the local asymmetry 6, defined by

It specifies the nuclear composition at each point and is analogous to the total asymmetry 1.

The definitions of E and 6 given above apply only in the bulk region. In the surface E and 6 are defined as the smooth outward extrapolation of their bulk behavior.

The density distributions in Fig. 1 are to be considered as resulting from a small perturbation of the Liquid Drop density distribution shown schematically in Fig. 2. The Liquid Drop Model radius R, corresponds to a uniform central density p,, . the same as that of nuclear matter. They are related by the expressions

R, = r,A113 ,

p. = ($7r,3)-1,

I I

RO

FIG. 2. Schematic Liquid Drop Model nuclear densities pn, pz, and p, where p = p,, , as a function of radial distance. There is no spatial variation of the bulk densities in this model and the location of all the surfaces is at R, .


where r,, is the nuclear radius constant. The change between Fig. 2 and Fig. 1 may be thought of as consisting of two parts, whose order is immaterial:

(1) Separate compressions or dilatations of the uniform neutron and proton distributions to the new radii.

(2) Redistribution of the densities from their average values j&, &, and jj to their final values of fn , pz , and p. Since both changes are considered to be small, their effects may be calculated independently. For example, the redistribution could be considered to take place first, for the system in which R, = R, = R, , and the scale changes could be applied subsequently. In any case the degrees of freedom may be considered to be the two radii R, and Rz (or equivalently the average densities fin and &) and the two functions

P”n = pn - in and A = pz - /% *

The radii and average densities are related by

+rRn3p, = N,

The changes in the radii (or in the average densities), and the functions f3, and A are taken to be small quantities.

We note that the pair of dimensionless functions E and 6 are related to the densities by

PIP0 = 1 - 3%

PIlIP = H1 + a

f-%/p = &Cl - 8).

To first order in small quantities we have also

pn = &dl - 3E + S),

pz = &(l - 3E - S),

and the radii can be written in terms of the average values c and 8 as

R = r,,A1i3(1 + C),

R, = r,(2N)‘i3 (1 + C - is),

R, = ro(2Z)1’3 (1 + E + $8).

(2.2a)

(2.2b)

(2.3a)

(2.3b)

(2.3~)


The quantity t, defined by

t = R, - Rz ,

is the thickness of the effective neutron skin, and is an important variable in discussing surface properties. Its relation to 8 is most simply derived by equating two different ways of expressing the uohrme of the effective neutron skin:

$n(Rn3 - RZ3) w 4rrR2t,

where the expression on the left is exact, and the term on the right is an approximation valid for t/R << 1. This equation gives

3tlR ‘: (Rn3/R3) - ( R,“/R3)

N/A ZIA idp y$’

Evaluating the averages to first order in small quantities, we find

31/R = z l--I

1ts --,

Hence 2 I--S __- tlR = 5 , _~ s2 7

or. for 63 & I. t/R = $(Z - 8). approximately. (2.4)

We shall denote by 7 the dimensionless skin thickness. Using the definitions

7 = t/r, and 9 = R/r,, ,

we can write (2.4) as 7 = -;(I - sp.

or approximately T z ;(I .-- 8) ,41/3. (2.5)

B. ENERGY EXPRESSION

With these preliminaries taken care of. we write the total energy of the system as

= ISS volume ep + .cs (3 -:- Ecoulomt, .

surface (2.6)


The volume energy is an integral over the volume +rrR3 of the energy density ep, which consists of the energy per particle e times the density p. The surface energy is an integral over the surface R of a surface tension coefficient U.

The bulk energy per particle e depends on the neutron and proton densities at each point. We write the Taylor expansion

e = -a, + JP + +(KG - 2L& + MS*) + ... . (2.7)

In virtue of the symmetry between neutrons and protons this energy must be a symmetric function of 6, and we have therefore expanded in powers of E and S2 (rather than 6). Equation (2.7) is an expansion to second order in the small quantities E and P; it comprises all the terms that are necessary in extending the nuclear mass formula to include the new terms shown in Eq. (2.1). There is no term linear in E in Eq. (2.7) since the energy per particle e is stationary with respect to deviations of the density from its standard nuclear matter value p. . The coefficient a, is the binding energy per particle of nuclear matter and J is the symmetry energy coefficient. The quantity K is the nuclear compressibility coefficient. The term in L gives the density dependence of the symmetry energy (its sign is chosen so as to make L eventually positive), and M is the coefficient which specifies the deviation of the symmetry energy from a quadratic dependence on 6.

The surface tension coefficient c is a function of the local conditions at the surface -in particular of the neutron and proton densities at the surface-of the neutron skin thickness, and of the surface curvature K (equal to 2/R for a sphere). We write the following Taylor expansion for the surface tension coefficient multiplied by 4~-r,,~ (where this factor is included in order to establish contact with the standard notation for the nuclear surface energy coefficient):

4n-ro20 = a, + Hr2 + 2Pr6, - Gas2 + a&%-l. cw

Here a2 is the usual surface energy coefficient with the dimensions of an energy. The terms in H, P, and G specify the dependence of the surface energy coefficient on T, the neutron skin thickness, and on 6, , the value of the bulk asymmetry 6 extrapolated to the mean surface at r = R. The last term in Eq. (2.8) is a first-order correction to the surface tension coefficient for the curvature of the surface, the curvature being proportional to .CPl.

The absence of terms linear in 7 and 6, is a consequence of the symmetry between neutrons and protons. The Taylor expansion (2.8) is terminated at the quadratic terms in 7 and &-this includes a cross term in T& . (The sign of the term proportional to Ss2 was chosen so the coefficient G would be positive.)

As will be shown in Section III, there is, in the expansion (2.8), no linear term in E* . (The value of E extrapolated to R.) The coefficient of a term in l s2 does not vanish, but such a term leads to higher-order corrections in the sense of the pattern (2.1). For the same reason fewer terms are kept in the expansion of the surface


energy coefficient than in the expansion of the volume energy per particle, since the surface energy is already smaller than the volume energy by a factor A-*j3.

The Coulomb energy in Eq. (2.6) may be written in the usual way as

E coulomb = Be” s.TI’Iss Pz(l) P&J

2 RZ r12

or, if the exchange correction is included, as

Here P,, is the relative probability of approach of two protons, normalized so that without correlations PI2 = 1. Because of the exclusion principle the motion of protons in a Fermi gas is not uncorrelated, and PI2 is not a constant (23). The resulting exchange correction may be written as an integral over an energy density. thus

E. h exe ange

This expression. if worked out for a uniform distribution of Z protons in a volume $rRZ3, gives

E Zi/”

exchange = -c4 ~ 23 ’ z

where 3 3 2/3

c4 = 4 i-1 e2

- and .3z z 2rr 45. (2.9) r. r.

(The proton charge e should not be confused with the energy per particle function e.) Higher-order corrections to the exchange energy arising from nonuniformities in pI are not considered.

The diffuseness correction to the Coulomb energy has been discussed on p. 57 in Ref. (I). It is given by

. % Ediffuseness = --4xZe

! dl II sp.

-7

Here n is the outward normal to the surface and 6p is the difference between the diffuse and the sharp charge densities, 6p = Pdrffuse(M) - Psharp(!?).

When this correction is applied to a distribution of Z protons in a volume $7 Rz3, one finds

Ediffuseness = -C3 $ . (2.lOa) 1.

where

(2.10b)


where pzc is the proton density in the bulk, equal to Z&rRz3). Once again we note that higher-order corrections in this term are not retained.

The basic energy expression characterizing our Droplet Model (specialized to spherical shapes) is now given by

E = JjI, [-al + Ji? + $(Kc2 - 2Ld2 + MS4)] p

+ (a2 + Hr2 + 2Pr6, - G&s2 + a,&l) a2

+ 4e2 jj j j j j, pz(1;lr(2) - c3 g - c4 g . (2.11) z

In the surface energy [the second line in (2.1 l)] we have carried out the integration over the sphere R resulting in an area 4rrR2 = 47~~~9~. The volume integral in the first line extends up to the radius R, and in the last line up to R, .

The energy in Eq. (2.1 I) is a functional of the density variables l and 6 and a function of the radii W and .9& . It is convenient to divide the functions E and S into two parts

E=E+C, s=B+S, (2.12)

where the first part (a number) is the average value of l or 6, and the second part (a function of position) is the deviation from the average. Substitution of (2.12) into (2.11) gives the energy as a function of c and 8 and as a functional of < and 8. We find that after some transformations (explained below), the energy may be written as

E(E, 8; <, 8) = [-al + d2 + &Kc2 - LES2 + @48]A

+ jjjso [Js2 f $Kc2 -t *..I p. + a,~~‘~(1 + 2;)

+ [HT~ + 2P7(8 + 8,) - G(8 + &)2] A2/3 + u3A1j3

+!e” z2 -(

s I 5 l-0 A1'3

l-‘-5+3 i

+1ep.jji,~[~-~(~)'3(-3i-~)-c,~-C1~.

(2.13)

In the first part of the volume energy the integration has been carried out to give A. The second part of the volume energy is a correction term and so R and p have been replaced by their lowest-order values R, and p. . In simplifying (2.13) we have made liberal use of the relations

us c=o and SIS 8 = 0,


which follow from the definitions of 2 and 8. (To the order to which we are working it is immaterial whether the limit of integration in these expressions is R, , R, . or R.) We have also omitted several higher order terms, in particular the influence of density nonuniformities on the corrections associated with L, M, and the surface energy. In the leading part of the surface energy we have written w -= A2j3( 1 -F 2E). but in the corrections associated with H, P, G, a3 , cg , and cq it was only necessary to use the lowest-order result W = 9?Fz = A1j3. The Coulomb energy in the last line of (2.13) was rewritten by use of the transformations

Noting that the first term is the Coulomb energy of a uniformly charged sphere, and that

efh us _ RZ r12

is the potential of such a sphere. given by

$ [g - i(k)‘] for I’ ,C R,.

we find that the Coulomb energy can be written as

If we replace R, in the first term by its expansion

and if we replace R, in the correction terms with its lowest-order approximation R, = R. , we find the result given in (2.13). Note the significance of the combination $(I - 8) in (2.14); according to Eq. (2.4) this is equal to half the ratio of the neutron skin thickness t to the radius R.


Equation (2.13) is the basic equation of our Droplet Model specialized to spherical shapes. It contains two number variables <, 8, two function variables C, 8, two constitution parameters A and Z (or N and Z), and eleven physical coefficients r0 , a, , a2 , a3 , G, H, J, K, L, M, and P. It will turn out that to the order to which we are working all consequences of the theory involve the three surface energy coefficients G, H, and Pin a single combination, so that the actual number of physical coefficients that need be considered is only nine.

Later in this work we develop a statistical theory of nuclei which may be used to relate the nine physical coefficients to four basic adjustable parameters. When used in conjunction with this statistical theory the Droplet Model has four adjustable parameters, the same as the standard Liquid Drop Model, in which r, , a, , a2 , and J are considered as adjustable. [The quantities cQ and cq are not independent coefficients but are related to r, by Eqs. (2.9) and (2.10).]

C. MINIMIZATION

With the Droplet Model energy specified by (2.13) we determine the value of E, 6, EI and 8 that make the energy stationary by demanding that the energy change SE vanish for all small particle-preserving variations. Taking the variation of (2.13), we find

6E = (KAE‘ - LAS2 + 2a2A213 - cl

+ 12JA8 - 2LAE8 + 2MAs3 + A213 [2H7 2 + 2P(8 + 8,) $

+ 2Pr - 2G(S + s,,] - ;c&/ S(S)

+ j& po(KE - $eU k + jjjRop&2J8 - teV> S@). (2.15)

We have used the abbreviations

3 e2 "l=s,, and ~=e[i-i($,“l.

In the last line of Eq. (2.15) we have kept only the leading order terms and have disregarded terms proportional to A 2 3 Using Eq. (2.5), we find dT/d8 = -gA113, / . and hence the coefficient of S(8) may be rewritten as

+ [2P7 - 2G(8 + &)I A2/3 - 2LA8 + 2MAx3. (2.15a)


The coefficients of SC and S(8) may be set equal to zero directly, but particle conservation must be ensured for the variations associated with SZ and S(S) (24,25). Thus, instead of setting the coefficients of SO and S(8) equal to zero, we write

KF - $eV = constant,

258 - $eV = constant,

where the constants are Lagrange multipliers which can be determined by taking the average of each equation. Since the averages of EI and S are zero, the constants are found to be equal to

-#eV and -$eF

respectively. Hence we have the very compact result that (26)

where

3 - Z==eV and

p=v-j7

(2.16)

is the deviation of the internal electric potential of a uniformly charged sphere from its average. We note that

pl 1 Ze

Cl Z = --__ 125 A1/3

The last form (of which we make use presently) is approximately correct when Z w $A, i.e., when Z is small.

The vanishing of the coefficients of SE and S(S) in Eq. (2.15) gives

K< + 2a,A-l/3 - Li32 - cl -?- = 0 ,44/3 '

As is usual in taking variations around a stationary value of the energy, the equations determining the optimum values of the variables need be of lower order than the energy equation itself. Consequently in the second equation we have neglected the higher-order terms which constitute the second line in Eq. (2.15a). (The first

595/55l3-2


term is smaller by A-li3, the second by E, and the last by @, than corresponding terms in the first line.) Hence we can write the final solution for 8 as

3 Cl 22

S= Ifg7T-p + ;; A-1/3&

J 3P * 1 + - A-1/3 _ - _ A-l/3 H 2H

If we use the approximate expression (2.16a) for 8, , and if we introduce the definition

Q = H/(1 -if,, (2.16b)

we find that the expression for 8 can be rewritten as

or

if

The equation for 2 is

1 <=- K ( -2a,A-l/3 + Ls2 + cl $).

(2.17a)

(2.17b)

(2.18)

We note that, to the order to which we are working, the equilibrium values of all the variables (i.e., E, 8, Z, s”) do not depend on G. Moreover the other two surface energy coefficients H and P do not appear separately, but only together with J in the combination Q defined in Eq. (2.16b).

From Eqs. (2.5) and (2.17a) we obtain, for the neutron skin thickness t,

or

if

(2.18a)

(2.19)

9J m A-lJa g la

AVERAGE NUCLEAR PROPERTlES 413

These equations show how the formation of a neutron skin results from two opposing effects: the neutron excess (proportional to Z) tends to produce a neutron skin, the electric repulsion (proportional to Z2/A4/3) tends to reduce its thickness. The driving force in the former is proportional to the symmetry energy coefficient 1, and in the latter it is proportional to the Coulomb energy coefficient c, . The restoring force limiting the extent of a neutron skin is provided by the coefficient Q. In this sense Q may be thought of as an effective stiffness against the pulling apart of the neutron and proton surfaces.

Note that in the absence of Coulomb energy the neutron skin thickness is given by

3 J yoGI

9J ,’

l + 372 A-‘:3

(2.20)

and if

we may write

91A-l/3 .& ]

4Q k

t = zr Jr 12“&1,3 ._~ . . . 2 "Q i 4Q 1

. (2.2&d)

The physical content of Eq. (2.20) may be appreciated by considering two limiting cases: the case of large A and the case of vanishing effective surface stiffness Q. For sufficiently large A the thickness t becomes independent of A, tending to the semi-infinite limit

3 J

t = 5 r” P z. (2.21)

On the other hand, in the case of a vanishing effective stiffness Q we would expect the bulk asymmetry to be zero and all the excess neutrons to move into the surface (since there is no resistance against forming a neutron skin). From Eq. (2.20) we find that as Q + 0 the thickness t becomes

t = $roA1/31

= ;RZ.

Comparing this with Eq. (2.4), we find that this value of t corresponds indeed to a vanishing bulk asymmetry 8. Note that if we had used the expanded version of the formula for t, Eq. (2.20a), the limit Q -+ 0 would have given a meaningless result.

As to the actual value of A at which the asymptotic formula (2.20a) becomes


applicable, an important and unexpected result of this work was the discovery that a nucleus must be very large before the skin thickness attains its limiting form. This is related to the magnitude of the factor $(J/Q)A-I/“, which must be small to justify (2.17b), (2.19), (2.20a) and (2.21). It turns out that in practice this number is not small for typical values of A. Thus we shall find that $(J/Q) is about 4, and the expansion of the denominator in (2.17a) is not useful unless A > 64. The result is that leaving (2.17a) unexpanded provides a much better approximation than the expanded form and, in fact, it is essential to use the unexpanded form in order to have a sensible and useful Droplet Model treatment of actual nuclei (27). Later we discuss the expanded and unexpanded forms and compare their consequences with a Thomas-Fermi treatment (which makes no assumption about the size of A--1/3). The superiority of the unexpanded form will be exhibited.

The poor convergence of Eq. (2.20a) was an unexpected result which calls for some discussion. The size of the coefficient s(J/Q), which is responsible for the poor convergence, depends on the relative sizes of the two physical properties of nuclei described by J and Q. The symmetry energy coefficient J is made up from three contributions: one due to the kinetic energy of the neutrons and protons, one due to the interaction energy between like particles, and one due to the interaction energy between unlike particles. On the other hand the coefficient H, to which Q is proportional, depends only on the unlike interaction. (If there were no interactions between neutrons and protons there would be no restoring force against bodily moving the neutrons with respect to the protons and thus no resistance to the formation of a neutron skin.) Hence, although there is no general reason why H should be small compared with J, it is also not surprising that H is somewhat smaller than J. To summarize: The essence of the relative largeness of $(1/Q) is that it turns out to be more difficult to change the relative neutron and proton densities than to move the neutrons and protons apart, because in the former case one has to overcome the resistance of the kinetic energy as well as that of the interaction energy between like particles.

The total energy of the nucleus according to the Droplet Model is obtained by inserting the values of E, 8, C, and 8 into Eq. (2.13). The two spatial integrals, which together give the effect of the redistribution of neutrons and protons, can be carried out elementarily, with the result

&distribution = -cJ’A~‘~, where

1 c2 = 8;? Cl2 i

912 l/4 K + J 1 .

The redistribution energy is readily shown to consist of a decrease of -2czZ2A113 in the electrostatic energy and an increase of c,Z~A~/~ in the specifically nuclear bulk energy.


The total energy can then be written in closed form:

E = (-al + Js2 + &Kc2 - L& + &ME;3 A

+ u2( 1 + 2s) A2j3 + \fW + 2P7(8 + &) - G(8 + &)2] A’/3 + a,Al ‘3

i Cl $ (1 _ 6 + $TA-li:s) - c2Z3A113 - c3 9 - c4 g .

In this equation E, 8, and & are to be obtained from Eqs. (2.18), (2.17a), and (2.16a). The value of T may be found from Eq. (2.5), but we may also note that, if Eq. (2.18a) and Eq. (2.16a) are used, 7 may be written as

7 = ; 6 (8 + &)

3J8 zzz-- 3.

2Q

Use of this result in eliminating the quantity (8 + &) from the bracket containing H, G, and P in the energy equation gives

1 4 PQ H+5J-gT . 4 GQ' T2

I

Using our definition of Q, Eq. (2.16b), we may rewrite this as

[ 4 PQ” 3 J G

Q+gJ2 ze-P ( il 7-2. The reason for writing the coefficient of 72 in this form is that, as is shown in Section III, a theorem exists which relates the four coefficients H, P, G, and J in the following way:

Hence the coefficient of 72 is simply Q, and the final form of the energy equation at equilibrium is

E = (-al + Js2 + *Kc2 - Ld2 + &Mg4) A

+ a,(1 + 24 A2j3 + Q72A2/3 + a3A1j3

t cl g (1 - c + *Tk-l/3) - C2~3~1~3 - c3 T - c4 g. (2.22)

Equation (2.22) displays explicitly the various physical effects associated with the refinements introduced by the Droplet Model. The first two lines correspond to volume and surface energies, the last line is associated with the Coulomb energy.


The leading terms (with 8 replaced by Z) are

--a,A + JZ2A + a,A213 + 22

c ~ 1 Al/3 '

and correspond to the Liquid Drop approximation. The remaining terms in the first line are changes in the volume energy associated with the adjustments of the neutron and proton radii. The second line contains various corrections to the surface energy, the last term in the second line being the curvature correction. In the last line the term in the brackets corrects the Coulomb energy for a changed proton radius, the second term corrects for charge redistribution (and is composed of a decrease in Coulomb energy and an increase in volume energy), the third is the Coulomb diffuseness correction, and the fourth the Coulomb exchange correction. The magnitudes of these effects are discussed later.

A slight simplification of Eq. (2.22) may be achieved by making use of Eq. (2.18). The result is

E = (-al + J8” - $Ke2 + &Ma”) A

+ (a2 + QT~ + a,A+) A2i3

+ cl g (1 + &TA-‘/~) - c,Z2A113 - cg $ - c, $. (2.22a)

It should be emphasized that both this equation and Eq. (2.22) hold onZy for the equilibrium values of E, 6, and 7.

We note that if, in the absence of Coulomb energy, < and the expanded form of 8 from Eq. (2.17b) were substituted into (2.22) [(2.17b) being justified only for enormously heavy nuclei], we could arrange the energy in the form

E = -a,A + u2A213 + 2a22 a3 - -& A1/3

+ JZ3A - (; $ _ y ~3~313

L2 - ~- g)Z4A. 2K

(2.23)

This is the scheme anticipated in (2. l), and our selection of terms in carrying through the various expansions leading to (2.22) was based on the desire to keep all terms that would contribute to coefficients in this scheme. Note that important lessons are contained in the fact that each of the coefficients of the three new terms (in A1j3, Z2Azi3, and Z4A) is composed of several parts. Thus, contrary to most, if not all, existing discussions of such effects, we see that the coefficient of AlI3 is not just the curvature correction to the surface energy, the coefficient of Z2A213 is not just a surface symmetry energy, and the coefficient of Z4A is not just due to the deviatiou


of the nuclear matter symmetry energy from a quadratic form. We discuss the make up of these terms in more detail after the numerical values of the coefficients have been determined.

III. THEOREMS

In developing the Droplet Model in Section II we made use of two properties of the Taylor expansion of the surface tension coefficient u: the vanishing of the first derivative of u with respect to bulk density changes, and the existence of a relation between the expansion coefficients H, P, G, and J. In this section we discuss three theorems from which these properties can be deduced.

A. PROOFS

We describe two proofs of the relevant theorems, the first following directly from the vanishing of the variation of the total energy, Eq. (2.13).

In Section II we wrote down the variation of E to the lowest relevant order; in particular the last line of Eq. (2.15) included only the bulk terms. In fact, however, the full equations for the vanishing of the variations associated with SE and S(8) should read

(3.1)

and

ISI Ropo(2J8 - $eV) S(8) + AZ/3 [-$ ,(4 1 7rYo2U) S(&) = 0. (3.2)

We proceed to prove the separate vanishing of the higher-order terms in A213 (in addition to the vanishing of the leading bulk terms), and this establishes two of the three theorems with which this section is concerned.

The argument goes as follows. Since Eq. (3.1) must be satisfied for all particle- preserving variations 8Z, let us apply it first to a variation which vanishes at the surface (&, = 0), and then to one which does not. In the first case the second term in Eq. (3.1) is absent and we conclude that KC - jeV’ must be a constant. (One might argue that it need be constant only through the bulk, but since S& is zero at the surface, the quantity KC - $eV is undetermined at r = R, and could even be infinite there. This would indeed invalidate the proof that is to follow, but such singular behavior is excluded by our definition of E” s , which is that Es is the value of z smoothly extrapolated to the surface.)

With the constancy of KC - #eV established, we now consider a particle-preserving variation SZ with a nonvanishing value of &, . The first part of Eq. (3.1)


is now zero, and it follows that the second part must also be zero. Since this is to be true for all values of SE, we conclude that

AW $ (4?7r,%) = 0 s

or au -= acs

0.

Exactly the same argument [involving one variation for which S(&) = 0 and one for which S(8,) f 0] leads to the conclusion that

aa

as,=O* (3.4)

Equation (3.3) proves the absence of a linear term in the Taylor expansion of the surface tension in powers of l s . Equation (3.4) states that the partial derivative of the surface energy with respect to 6, (keeping 8 and hence T ftxed) is zero. From this result follow several interesting consequences.

Applying Eq. (3.4) to the Taylor expansion of the surface tension, Eq. (2.8), we find

2PT - 2G6, = 0, or

G 7 = -s6,.

P (3.5)

This is a relation between the equilibrium values of 7 and 6, in terms of the two coefficients G and P. But in Section II we derived a similar relation between T and 6, by making stationary the leading terms in the energy, and there (Eq. 2.21a) the factor of proportionality was

3J 3J(1-;+j

-- 2 Q ’ ie’ 2 H -

Equating this to G/P, we find the following remarkable relation among the coefficients J, H, P, and G:

35 G -- - 2Q P’

or 3J G

- (1 +j. --_- 2H P (3.7)

Because of the central importance of the theorems contained in Eqs. (3.3) and (3.4) [and the resulting relation (3.7)] we describe an alternative proof. This proof differs in two respects. It is less general in the sense that the Coulomb energy is not


included. It is more general in the sense that it is based directly on the properties of saturating systems rather than on the structure of the Droplet Model energy equation (2.13).

Equation (3.3) states that the surface tension coefficient u has no linear dependence on the bulk density extrapolated to the surface. An alternative way of stating this is that the surface tension would not change (to first order) if the equilibrium density distribution p(r) of a saturating, uncharged system were scaled up to (1 + h-4 PC ), h r w ere 8~ < 1. Thus an equivalent way of stating the result contained in Eq. (3.3) is

du -= dP

0. (3.8)

To prove from first principles that the surface tension is stationary with respect to a scaling up of the density is not trivial. It is easy to show that the surface tension is stationary with respect to density variations localized to the vicinity of the surface, but the discussion of density variations that extend into the bulk, as in the present case, requires some care, since simultaneous bulk and surface energy changes are involved.

We construct a proof of Eq. (3.8) by starting with the equilibrium density distribution of a large nucleus and considering in turn three distinct, infinitesimal density changes which involve surface and bulk variations in different combinations. By manipulating the three equations for the associated energy changes we eliminate all terms not associated with the surface, and show that (du/dp) is zero.

(bi

FIG. 3. The equilibrium density distribution of a large nucleus as a function of the radius. Three different variations about equilibrium are shown. These variations are used to determine the dependence of the surface tension on scale changes.


The three density variations of a large but finite system, of radius R, are illustrated in Fig. 3. In Fig. 3(a) the density change corresponds to a uniform increase from p to (1 + 6p)p. In Fig. 3(b) the density variation is the same as in (a) from R, outward and is zero for r < RI . In Fig. 3, R is the mean radius and R - R, is chosen to be much smaller than R but much larger than the range of the interparticle forces (and thus larger than the diffuseness of the surface). In Fig. 3(c) the density variation consists of a uniform increase by the factor (1 + 6~) in the region between R, and R, , where we again choose R, - R, to be much smaller than R but much larger than the range of the interparticle forces.

Since all the variations are taken about an equilibrium distribution, we know that in each case the following relationship holds:

S(E - LN) = 0. (3.9)

Here E is the total energy, N is the number of particles, and L is the Lagrange multiplier which ensures the validity of Eq. (3.9) for density variations (such as those considered here) which do not preserve the number of particles.

For the density variation in Fig. 3(a), Eq. (3.9) becomes

4 3 rrR3 dw pc S/L + 47rR% 6~ - ; n-R3pcL &L = 0.

dPC

Here w(pC) is the energy density in the bulk, and the first term represents the change in the volume energy. The second term represents the change in the surface energy (6 stands for dcr/dp), and the last term is L times the change in the number of particles.

The variation in Fig. 3(b) leads to the equation

4 dw 3

,.@3 _ 43) _ pc + + 4TR2& 6p - 4 n(R3 - R13) PcL ‘%‘ + 4rrR12k ‘1*. = ‘* 6 3

(3.11)

The meaning of the first three terms is the same as before. The last term allows for a possible contribution to the energy from the presence of the density change at R, . Because of the finite range of the interparticle forces this contribution is localized to the neighborhood of R, and can be written, approximately, as the area 4’rrR12 times a constant k times 6~.

In the same way as above, Eq. (3.9) may be applied to the variation in Fig. 3(c), with the result

(3.12)


The first three terms are the same as before. The last two terms, which come from the density changes at R, and R, , have the same sign, since the energy changes at R, and R, cannot depend (for large R, and R,) on whether the density increases with increasing or decreasing distance from the origin.

Simplifying the three equations and making use of the assumptions that R -- R, < R and R, - R, < R, we may write

; Rpc ($ - L) + ci = 0,

(R - R,) pc [e - L) f 12 + k = 0,

(R, - R,) pc t$ - L) i 2k 17 0.

Eliminating ((dw/c/pc) - L) and k, we find

In the limit R ---f a the bracket tends to unity and consequently I+ must tend to zero. Q.E.D.

Our general proof that 6 x 0 shows that the estimate that C? v 0.30 contained in Ref. (26) was not accurate.

Note that an erroneous proof of the ir = 0 theorem (of which we were initially guilty) could be based on Eq. (3.13) alone. Since we knew that in the limit of an infinite system the Lagrange multiplier L becomes equal to (dw/dp,), we thought this established the vanishing of L?. In fact, however, although the first term in Eq. (3.13) contains [(dw/dpc) - L], which tends to zero, it is multiplied by R, which tends to infinity, so that the product is not necessarily zero.

On the other hand, note that the result 6 = 0 could be obtained by applying a single suitable density variation to a semi-infinite equilibrium density distribution. This variation would be essentially a linear combination of the three variations depicted in Fig. 3(a, b, c), and is shown in Fig. 4. This variation is chosen to con- serve the number of particles, and the density changes at or , x2 , xg are such that any associated energy changes that are linear in 6~ cancel out. It then follows immediately that the only remaining term in the energy change is proportional to ti. and must be zero, since the energy is stationary. (The advantage of proving the C+ = 0 theorem, at least initially, by way of variations for a finite nucleus, followed by the limit R -+ co, rather than by a single variation of a semi-infinite system, is that the limiting process is exhibited and possible pitfalls associated with this process are easily avoided.)


I I \ I X3 X.2 XI X0

FIG. 4. The surface region of an equilibrium semi-infinite distribution of nucleons. The dashed lines represent variations about equilibrium used to determine the dependence of the surface tension on scale changes.

The generalization of the above proof that ac+ks is zero to the proof that a~/a(&) is zero is straightforward. All we have to do is to imagine that Figs. 3(a, b, c) and Fig. 4 refer to one of the two densities of a two-component system. Thus one of the densities [say the neutron density pn(r)] is supposed to undergo the variations depicted in the figures, while the other remains unchanged at its equilibrium distribution. It then follows, without any change in the arguments, that

aa - = 0, aPn

i.e., that the surface tension is stationary with respect to a scaling up of the neutron density alone.

Similarly au -= aPz

0

for a scaling up of the proton density alone. Since the surface tension is thus shown to be stationary with respect to both neutron and proton density changes, it is stationary also with respect to the combinations implied by changes 6~~ and 6(&J [essentially a sum and a difference of the neutron and proton changes]. Hence

au -= 86, 0,

aff as,=

0. (3.13a)

This, in addition to proving once again Theorem (3.3) also proves Theorem (3.4). We now apply the type of reasoning used above, independent of the structure

of the Droplet Model Eq. (2.13), to prove a third general theorem. This theorem leads to the relation between T and 6, contained in Eq. (2.21a), or rather that equation specialized to an uncharged system, for which we may write


This, when combined with the immediately preceding proof that au/G, is zero, will enable us to establish the fundamental relation (3.6) in a way which does not concern itself with the Coulomb energy and does not depend on the Droplet Model energy Eq. (2.13).

FIG. 5. The surface region of an equilibrium two-component semi-infinite distribution of nucleon% The dashed lines represent variations used to establish the equilibrium relationship between 7 and 8.

Let Fig. 5 represent the equilibrium density distribution of a two-component semi-infinite system, with a certain equilibrium value of 8 and 7. The bulk neutron and proton densities are &, and pZ and the mean neutron and proton surfaces are at x~’ and xg’, so that the skin thickness t is xi;“’ - xl”‘. Consider a density variation depicted by the dashed lines. This variation consists of small equal and opposite neutron and proton density changes of magnitude 01 in the regions between x1 and x2 and between x8 and xq . This results in an outward shift of the mean neutron surface by a, given by

a = Ict/pn , (3.15)

where

I = (x1 - x2) - (x3 - x4).

The mean proton surface moves inward by a(&/&), so that the change in the neutron skin thickness is given by

22 = 4 + pnlpz) (3.16)

The neutron and proton density variations are chosen so as to leave the total bulk density unchanged.

The change in the asymmetry 8 in the region between x1 and x2 is

(3.17)

In the region between xS and xp the value of S(8) is the negative of the above.


Hence the change in the bulk energy (per unit area of the semi-infinite system) is

or, if Eq. (3.17) is used, ae

-2culay:.

Here e is the bulk energy per particle. The change in the surface energy (per unit area) is

Since the original system was in equilibrium the total energy change must vanish:

ae _ au --2d s + a(1 + Pnlpz) x = 0.

Using Eq. (3.19, we find

ae 1 1 s=i (,+$)$.

Multiplying both sides by $nro3 (equal to pi’), using the definition 7 = t/r,, , and making the approximation j& m pz m $p,, , we find

$4 3 ae

xro%) = - - . 2 a8 (3.18)

This equation, which is our third fundamental theorem, establishes a relation between the partial derivative of the surface tension with respect to the neutron skin thickness and the derivative of the bulk energy per particle with respect to the relative neutron excess 8.

Note that, in analogy with the existence of two ways of proving the theorems that ag/acs = 0 and a+& = 0, there are two ways of proving Eq. (3.18). Thus the Droplet Model energy Eq. (2.13) (without the Coulomb energy) could be written in the form

E(S) = e(S)A + 45~r,~A~/~o(~, 8) t ... ,

where we have displayed only terms which depend on 8. The vanishing of variations with respect to 8 gives


Since a7 2

= - f Al/3 (see Eq. 2.5).

and since we have already proved that

we find once again that

au

Note that by using the Taylor expansions of 4nro20 and e we may write

&(4 iv,%) = 2Hr + 2P8 ,

Substitution in Eq. (3.18) then gives

7= 3J- 2P 8

2H

This equation, when combined with the relation 7 = (G/P)8 following from Eq. (3.13a), leads as before to the relation $(J/Q) = G/P, proved this time without reference to the Droplet Model energy equation (2.13).

We may summarize this section as follows. We have proved three theorems, each in two different ways, about the first derivatives of the surface tension coefft- cient, evaluated for an equilibrium configuration of a saturating two component system:

THEOREM 1.

THEOREM II.

THEOREM III.

f (47rro2u) = i $ .


The first way of proving the theorems uses the Droplet Model energy equation (2.13). The second way goes back to the equilibrium density distributions of large or semi-infinite saturating systems and, by applying suitable infinitesimal density variations, deduces the required relations from the vanishing of the first-order energy changes.

B. COMMENTS

In this section we bring out the significance of the theorems we have proved and point out the remarkable simplifications which they produce in the treatment of two-component systems, with or without Coulomb energy. We also comment on some remaining puzzles, especially in the treatment of the Coulomb effects.

We may review our understanding of the problem of a two-component system, obtained so far, in the following way.

For an uncharged semi-infinite or finite but large system in equilibrium there exists a relation between the skin thickness T and the bulk asymmetry 8, given by

For such equilibrium systems one can prove a theorem (Theorem II) that the derivative of the surface tension with respect to the bulk asymmetry is zero. This leads to a second relation between 7 and 8, given by

Elimination of T and 8 gives a relation between coejicients in the Taylor expansions of the volume and surface energies:

35 G =-(I-;$).

2H P

This is an unexpected result: One might have thought that the coefficients H, P, G, and J were quite unrelated, able to assume different sets of values according to the nature of the elementary interactions between the particles constituting the system under consideration. It would in fact be an interesting problem to exhibit the microscopic reason for the relation between the physical properties represented by the above four coefficients.

The practical consequence of the existence of the relation between H, P, G, and J is that the eleven coefficients in the Droplet Model may be reduced to ten. Here, however, we have found another unexpected simplification, namely that the results


do not require the separate knowledge of all the surface coefficients H, P, and G but only of a certain combination of coefficients, which we have called Q. This reduces the number of parameters required by the Droplet Model (without Cou- lomb energy) to nine.

The introduction of the Coulomb energy might well have been expected to modify the relation between the neutron skin thickness T and the asymmetry at the surface, and to have removed the degeneracy in the coefficients which allowed us to reduce the number of relevant Droplet Model parameters from ten to nine. Thus the effect of the electrostatic repulsion is to reduce the neutron skin thickness T, to increase the average bulk asymmetry 8, and to induce a charge redistribution 8. However, when all these effects are calculated, we find that the relation between 7 and Ss , the value of 6 at the surface, is given by the same factor of proportionality as before:

3J,

Moreover, we find that all results regarding the equilibrium configuration of the system and its energy again require only the combination Q (rather than H, P, and G separately). We remain puzzled by this result and do not have a clear understanding of the underlying reason for its validity.

An important implication of Eq. (3.5) is that, within the limitation of our average treatment of nuclei, the surface regions of all large nuclei (with different numbers of particles, different neutron excesses and different electric charges) are “similar” in the sense that a single number, i(J/Q), specifies the ratio of the neutron skin thickness to the surface asymmetry 6, .

IV. THOMAS-FERMI MODEL

A natural choice when undertaking a study of average nuclear properties is a statistical method of calculation like the Thomas-Fermi method for atomic electrons. This method which has been so broadly applied in atomic physics is also applicable to nuclei (a-20). A variational principle is used to obtain an equation for the density distribution of the particles, and the solution of this equation provides the basis for the calculation of other nuclear properties.

A. QUALITATIVE CONSIDERATIONS

The basic assumption of the Thomas-Fermi method is the neglect of the effect of density gradients on the kinetic energy of a system of completely degenerate fermions. The kinetic energy at each point in the system is taken to be that of a

595/55/3-3


Fermi gas whose particle density is the same as the density at the point in question. This means that the momenta of the particles at a given location are uniformly distributed within a sphere in momentum space whose radius (the Fermi momentum P) is related to the particle density p by the expression

p = +n-P3(t/h3), (4.1)

where t is the degeneracy of the particle states due to internal degrees of freedom, h is Planck’s constant, and t/h3 is the density of particles in phase space.

Generally speaking there is a correction to the above relation arising from the presence of density gradients. The validity of the Thomas-Fermi method, which does not include this correction, depends upon how slowly the density varies over a typical particle de Broglie wavelength. On the basis of an earlier analysis of this question in Ref. (28), a criterion for the applicability of the Thomas-Fermi approximation may be established.

Consider a potential well containing a degenerate Fermi gas. Let the potential be given by V(r), as measured with reference to the energy of the most energetic particle in the gas. Thus, at each point, the momentum P of the most energetic particle is related to V by P2/2M = V. Let us denote the magnitude of the gradient of the potential at r by [ V’ [ = / grad V I. From these quantities we may form two characteristic lengths:

X = $ = de Broglie wavelength (divided by 27r) of the fastest particle

L= IV1 - = the distance in which the potential (if linear) would change I V’I

by 100%.

The ratio A/L, which we denote by x, is a dimensionless number in terms of which one may discuss the applicability of the Thomas-Fermi method.

In Ref. (28) a related quantity 5 was introduced, which, using Eq. (7) of that reference, may be written as

.$ = (g g)1’3 * (for E > 0).

By using the relations V = P2/2M and X = ii/P we find that

x = g(-312.

In Ref. (28) it was found that density distributions in a linear potential, when calculated according to the Thomas-Fermi method, would be accurate to better


than 10 % if t exceeded 0.4. This imposes a certain condition on x (X < 2.6) and hence on 1 V’ i/l I/ 1, i.e. on the rate of variation of V. A strikingly simple version of this criterion is obtained by considering instead of the rate of variation of V the rate of variation of the Thomas-Fermi density p associated with V. Since this density is proportional to I V j3j2 we have

1 I V’l -zzz ZIP’1 -=-- L IV1 3p'

and consequently x, defined as X/L, is given by

2 I P’ I x xc---. 3 P

From Eq. (4.1) the relation between p and X is

X = (t/67P)li3 p-li3,

and consequently

$ = (6+/t)1/3 t-313,

P

Putting 5 = 0.4 we find that the Thomas-Fermi method for nuclear problems (where f = 4) is valid to within 10% when

For atomic problems, with I = 2, we have

(4.2a)

This criterion (4.2) or (4.2a) is equivalent to the criterion in Ref. (28), but we do not recall having seen it stated in the above compact dimensionless form free from fundamental constants.

It is noted in Ref. (28) that the above criterion is satisfied over most of the charge distribution of an atom, which explains why the atomic Thomas-Fermi method works so well. In fact the introduction of gradient corrections into atomic calculations has been shown to have little influence on the results.

For nuclei, if we apply the criterion (4.2) to a nuclear density distribution of the Thomas-Fermi type (derived later in this section), with a 10-90 diffuseness of 2.4 fm (the experimental value) we find that the condition for the validity of the


Thomas-Fermi treatment is satisfied in the bulk region of the nucleus and through the surface, out to a point where the density has dropped to one-sixth of its central value. (A similar conclusion is reached by Bethe in Ref. (19)). Consequently the neglect of gradient corrections is expected to have little influence on the outcome of the calculations except for the outer fringe of the density distribution.

Since, according to Eq. (4. l), the particles at each point are uniformly distributed within a sphere of radius P in momentum space, the maximum kinetic energy of particles of mass M is P2/2M and the average kinetic energy is (3/5)(P”/2M). Elimination of P between this relationship and Eq. (4.1) shows that the local average kinetic energy per particle is proportional to the two-thirds power of the local density. This is the characteristic relation of the Thomas-Fermi method.

Not only must the kinetic energy be specified; the potential energy of the system of interacting particles must also be given. The characteristic assumption of the Thomas-Fermi method as regards the potential energy is that the energy should be evaluated disregarding any correlation in the motions of the particles. In this work, dealing with saturating systems, we follow Seyler and Blanchard (16) in choosing a phenomenological interaction which ensures saturation when used in conjunction with the Thomas-Fermi assumptions and is simple enough to allow calculations without further approximations. This interaction consists of a Yukawa force whose strength decreases with increasing relative momentum of the particles, and is of different magnitude between “like” and “unlike” particles. The “like” strength applies to the neutron-neutron and proton-proton interactions, while the “unlike” strength applies to the neutron-proton interaction. This interaction can be written

V(r,p) = - Clike (or unlike)

s [l - (p/b)2],

where

I/ = the potential energy of two particles, C = the strength of the interaction (different for “like” and “unlike” pairs), r = the distance between the particles, a = the range of the Yukawa force, p = the magnitude of the relative momentum of the particles, b = the critical value of the relative momentum at which the attractive force

(whose strength decreases with increasing relative momentum) vanishes and beyond which the force becomes repulsive.

The four quantities a, b, CI , and CU will be treated as adjustable parameters of our Thomas-Fermi treatment of nuclei. The four pieces of data that determine them are the binding energy per particle of nuclear matter, the density of nuclear matter, the nuclear symmetry energy, and the nuclear surface energy. The details


of how the parameters are actually determined are discussed later, but their values are listed in Table I along with the experimental input data.

Of course neither the input quantities nor the resulting parameters are known with the accuracy implied by keeping five significant figures. However, our insist- ence on starting with precisely defined parameters and on working out all numerical results to a high degree of accuracy was often essential in the numerical isolation of higher-order corrections and their comparison with the Droplet Model.

With the kinetic and interaction energies specified one may ask, as in the atomic problem, for the spatial distribution of particles that makes the total energy stationary. This leads to a standard variational problem which in the atomic case results in the nonlinear Thomas-Fermi differential equation. For nuclei this variation results in a pair of coupled integral equations for the neutron and proton distributions. The interaction given in Eq. (4.3) leads to equations which were first discussed by Seyler and Blanchard (16), and we sometimes refer to these nuclear Thomas-Fermi equations as the Seyler-Blanchard equations. The systematic working out of the consequences of these equations leads to a compact theory of the average properties of nuclei analogous to the well-known Thomas- Fermi theory of atoms.

TABLE I

INPUT QUANTITIES ANO INTERACTION PARAMETERS

Input quantities from Ref. (I)

Value Property

15.677 MeV a, , the volume energy coefficient 28.062 MeV J, the symmetry energy coefficient 18.560 MeV as, the surface energy coefficient

1.2049 fm rO, the nuclear radius constant Resulting values of the adjustable parameters

Value

367.56 MeV 289.66 MeV 82.030 MeV

0.62567 fm

Cl, the “like” interaction strength C, , the “unlike” interaction strength (b*/2M), the energy of a particle

with the critical momentum b a, the range of the interaction

With respect to the analogy between the Seyler-Blanchard nuclear theory and the Thomas-Fermi atomic theory, one important difference should be pointed out.


This has to do with exchange corrections to the nuclear interaction energy. In the standard atomic case exchange corrections are simply neglected, since their influence is known to be small. In nuclei exchange effects are not small but their proper treatment is difficult. The approach used here is a compromise in which exchange effects are not neglected, but are treated in a phenomenological way through the velocity dependence of the interaction. Although this artifice has some theoretical basis and does, indeed, lead to a saturating system with apparently reasonable properties, it is not clear how reliable it is in representing detailed aspects of the system. A further limitation of this approach is that the phenomenological two-body interaction we use represents an effective force whose relation to actual free nucleon-nucleon forces is not straightforward.

As is often the case in studies of this kind, notational simplicity may be enhanced and the generality of the results increased by carrying out the calculations in dimensionless form. In the remainder of this section most of the discussion is in terms of dimensionless variables where the basic units are

Distance, a = Yukawa range (0.62567 fm), Momentum, b = critical momentum of the interaction (392.48 MeV/c), Energy, T = critical energy (b2/2M = 82.030 MeV), (4.4)

where M = 938.903 MeV/c2 is the average nucleon mass.

B. ONE-COMPONENT SYSTEMS

Symmetric nuclear systems (neutron and proton densities equal) form an interesting subcategory of the general nuclear problem. Because of the symmetry such systems may be thought of as consisting of only one component, each possible phase-space state being four-fold degenerate to account for the spin and i-spin degrees of freedom. The consideration of the properties of such a one-component system without Coulomb effects forms the subject of this section.

In the Thomas-Fermi method the average kinetic energy at any point is given by

@ST , (4.5)

where T is the unit of energy and

52 = P/b

is the dimensionless Fermi momentum. The potential energy of a nucleon with momentum p1 located at r1 is given by

volume sphere


where C = Ci + CU ,

4/h3 = particle density in phase space, r = lb-r21, P” lP1-PPPI.

The spatial integration is over the nuclear volume and the momentum integration is over the Fermi sphere at point r2 . In dimensionless form this expression can be rewritten as

v(% 7 ~1) = - G ($r jn v dE2 j, s dw, 5 (1 - w2), (4.6) . . . .

where

Since the momentum distribution of the particles is considered to be uniform and isotropic at each point in space, it is possible to perform the momentum integration at once. If the integration is made over the particles at &. and the average taken over the particles at point ?& an expression can be obtained for the average potential felt by particles at g1 . This average potential can be written (after the notation changes g1 + < and &Y2 + g are made) in the following way:

The parameter y in Eq. (4.7) is defined by

it provides a dimensionless measure of the relative sizes of the interaction and kinetic energies, and its value determines the degree of cohesion of the system.

1. Nuclear Matter

For infinite nuclear matter the Fermi momentum 52 is independent of position, so the spatial integration in Eq. (4.7) can be performed. The average potential felt by a particle can be calculated, with the result

v = -@3(1 - $G=),


and the average potential energy per particle is 4~. Combining this with (4.5) gives an expression for 7, the total energy per particle in units of T, which can be written

r] = +y22 - $y@(l - $22). (4.8)

This elementary equation describes the dependence of the energy per particle on the Fermi momentum (and hence on the density) for Fermi particles in a system of infinite spatial extent, where the particles interact via the momentum-dependent interaction of Eq. (4.3). Because of the momentum dependence of the interaction these systems of particles will be found to exhibit the saturation property for a range of values of the parameter y. Even though nuclei are characterized by a specific value of y, it is interesting to examine the algebraic structure of Eq. (4.8) as a function of y in order to gain insight into the behavior of saturating systems of this type. A better understanding of the nature of nuclear matter can then be obtained by observing its position among the continuum of possibilities associated with different values of y.

In Figure 6 the energy per particle 7 is plotted against the Fermi momentum L? for various values of y, and the dot-dashed line in this figure is the locus of the equilibrium points. The position of these equilibrium points can be obtained by solving for the roots of the equation which arises when the derivative of 77 with respect to Q is set equal to zero:

@I - gysz2 + 39 = 0. (4.8a)

One root is Q = 0 and the other roots can be obtained by first putting the remaining cubic equation into standard form,

The negative root of this equation is not of interest in the present discussion. The other two roots are (29)

real and unequal real and equal conjugate complex 1

if b2 is 1

positive zero negative

Here b2 is defined by the equation

a2 + b2 = c2,

where c2 = (~13)~ and a2 = (q/2)2. As used here p represents the negative of the coefficient of the linear term in Eq. (4.9) and q represents the constant term.


FIG. 6. The dimensionless energy per particle 7 as a function of the Fermi momentum D for various values of the strength parameter y. The dot-dashed line is the locus of equilibrium values of ?.

Consequently, c = (l/6)“/“,

a = (l/57).

In Figure 6 it can be seen that the stable and unstable equilibrium points coalesce for some critical value of y. The relations above show that this value yc corresponds to b2 = 0. Its value can be determined by setting a = c with the result that

6312 yc=5.

For values of y < yc no equilibrium exists for nonnegative values of !2, except the trivial one at Sz = 0. For values of y > yc two equilibrium points exist which are located at

.Q = C113 COS ,g, & d? sin Q], c 0


where 8 = cos-l(a/c) and the triangle relationship shown below holds.

Substitution of the equilibrium value of Q into Eq. (4.8) gives, for the equilibrium energy, the expression

The equilibrium value of Q is itself a function of the strength parameter y, given by the relation

4

y = z&(1 - 2P) *

Figure 6 shows that for yc < y < y0 the equilibrium energy 7 is positive. A finite system corresponding to such a value of y would at best be metastable, since its energy could be decreased by dispersing it, particle by particle, into a system of particles at infinite separation and zero binding energy. In fact binding of finite systems becomes possible only when r] < 0, which implies that y > y,, , where

27 2 112 Yo=JT . 0

To find the value of y that corresponds to nuclear matter it is convenient to first solve Eq. (4.10) for Q in terms of (q/L?), which is the dimensionless ratio of the equilibrium energy per particle and the Fermi energy. This results in the relation

) ljz

’ (4.12)

For nuclear matter (see Table I) the equilibrium energy per particle is - 15.677 MeV and the Fermi energy is 33.138 MeV. (This corresponds to r, = 1.2049 fm.) So for nuclear matter

( i - = -0.47308. is

Substitution of this value into (4.12) determines Q, , whose subsequent substitution into (4.10) and (4.11) gives the values of qn and yn , where the subscript IZ identifies these quantities as referring to equilibrium nuclear matter. Their numerical values are listed in Table II along with some other numbers of special interest.


TABLE II /

EQUILIBRIUM VALUES FOR THE DIMENSIONLESS FERMI MOMENTUM AND ENERGYPERPARTICLEFOR SELECTEDVALUESOF~

Strength parameter Equilibrium Q Equilibrium I) ~--.- ~~~~ -~

63:s 1 yc = T

e-1,” 5ii

27 f2 Ijd ro=j s 0

Zero

yn = 6.5536 0.63559 -0.19111

I ym = infinity z--1,2 Minus infinity

FIG. 7. Equilibrium values of TJ as a function of the strength parameter y. The point marked yn corresponds to nuclear matter, the one marked y0 corresponds to 7 = 0, and the point marked yc is where the stable and unstable equilibrium curves meet and below which no equilibrium exists.


Slrength parameter Y

FIG. 8. Equilibrium values of 9 as a function of the strength parameter y. The dashed line marked 52, indicates the asymptotic value of B toward which the stable equilibrium values are tending. The unstable equilibrium values tend to zero as y --t 0~). The point marked y* corresponds to nuclear matter, the point y,, to 7 = 0, and the point marked yC is where the equilibrium family goes around a bend, and below which no equilibrium exists.

Figures 7 and 8 are alternative ways of displaying the locus of equilibrium values. We note that systems corresponding to y = yc are at the critical point where the stable and unstable equilibrium points have come together. They have no stable configuration except zero density. When y = y,, there is a well-established stable equilibrium for infinite systems, but no finite system is yet possible. As y increases bound finite systems become possible, and eventually we reach the point where y = ‘yn , corresponding to nuclei. As y increases beyond y,, the system becomes more and more tightly bound and its properties approach those of a system where y = cc. Figure 8 is particularly interesting since it shows that standard nuclei (i.e., those with N = Z and no Coulomb energy), for which y = yn , are in a sense closer to well-bound systems with y = co than they are to critical systems with y = yc .

Since we know the numerical value of yn we can now fix the value of b, the critical momentum in the interaction. To do this we note that the average kinetic


energy of nuclear matter corresponding to r,, = 1.2049 fm is 19.883 MeV, and this is related to T by Eq. (4.5). By substituting into this relation we find

and from (4.4) we find

T = 82.030 MeV,

b = 392.48 MeV/c.

Even at this early stage of the discussion of the Thomas-Fermi method it is possible to determine the compressibility coefficient of nuclei, K. A standard definition of this quantity may be written as

. equflibrium

where E(p) is the binding energy per particle of nuclear matter as a function of the density p. Using our dimensionless variables and Eq. (4.8) we find

K = T(-$2,” + 6y&,5).

This gives K = 294.80 MeV for nuclear matter,

2. Finite Nuclei

Symmetric saturating systems of the type we have been discussing can form into stable finite aggregates of particles when y > y0 . As was pointed out earlier, the density distribution of such systems (or equivalently the distribution of the Fermi momentum 12) can be determined by the use of a standard variational procedure. In this procedure that density (or Fermi momentum) distribution is sought which will minimize the total energy of the system, subject to the constraint of particle conservation. To undertake such a calculation we need first to express the total energy in terms of 52. Thus

where the first factor under the integral is proportional to the particle density and the second factor 7 is the energy per particle (in units of T), given by

77(5) = WY5) + 341).

In anticipation of the fact that the solutions for the distribution of Q will be spheri- cally symmetric we can specialize to that case. With this restriction and substitution


of (4.7) into the equation above for $4) it is apparent that the angular spatial integrals can be’performed, and the result is

Hence x 11 - W2(5> + QW)l. (4.13)

- sj-m[2dt ev[-I 5 - 5 II - expC-Cl: + 01 523co 0 2E

Similarly the total number of particles is given by

In terms of these quantities the variational condition becomes

S[E - LN] = 0, (4.14)

where L is a Lagrange multiplier. Since

we see that the physical significance of L is that of the system’s separation energy. If we define a dimensionless Lagrange multiplier,

substitute it into (4.14), and perform the indicated variation, we arrive at

x $- [3Q2(5) Q3(0 wo + 3WO Q2(0 SJ%T) - 3WO QYO SQ(c-)

- W5(0 -Q2(13 SQ(O - W2(0 Q”(R wo - 3Q3(5) Q4(0 ~w-)l

- fl3522(5) Ssz(()l = 0. (4.15)


Since the bracket multiplying y/2 is to be integrated over 5 and &? with a spatial weighting factor symmetric in 1 and 5, these variables may be interchanged at will in any term within the bracket. Doing this for those terms in the bracket that have SQ(.!J) as a factor, we find the following bracket:

[6Q2G’) Q”(&l - Q2(5) - %‘W) SQn(Ol.

If this is substituted back into (4.15) and the expression simplified we have

x Q”(E)[l - Q2(<) - j&-P(<)] - A/ 652(C) = 0. (4.16)

Since we require that the integral in (4.16) be zero for arbitrary variations SQ(<) it is necessary either that Q(5) b e zero or that the bracket multiplying SQ(iJ be zero. The function Q(c) which satisfies this latter criterion is the one we seek. Setting the bracket equal to zero and solving for Q2(<) results in the equation

This integral equation for Sz (where Q3 is proportional to the density) is the nuclear analog of the atomic Thomas-Fermi differential equation. The radial integrations in (4.17) extend to infinity. It turns out, however, that as in the case of the Thomas- Fermi treatment of ionized atoms, a nuclear Thomas-Fermi density distribution is confined within a finite sphere. We denote its radius by R. [The density approaches zero at r = R in a way characteristic of the Thomas-Fermi method, namely as (R - r)3/2.] Consequently the integrals over 5 in (4.17) extend only to a finite upper limit, which we designate by 8. We note that the Lagrange multiplier A and the radius B of the system are related by the equation

This relationship is obtained by applying (4.17) at the point 5 = 9, where Sz is zero. The integral on the right is the potential felt by a particle at the edge of the density distribution (5 = E), and the physical meaning of Eq. (4.18) is that the Lagrange multiplier A, which we saw was the separation energy of a particle, is also equal to the total energy of a particle at the edge.


In practice the integral Eq. (4.17) was solved by computer iteration. A value of the radius S was chosen and a first guess was made for the function S(t). This guess was in the form

Q(t) = [l - e-(s-f)]lP Q, , for 5 < E,

which is approximately constant and equal to 1;2, in the central region, and falls to zero at the surface over a distance of a few times the Yukawa range. This function also has the property that the guessed density (proportional to Q3) goes to zero as (S - [)3/2, as we know it must in the final answer. With this guess the Lagrange multiplier /1 could be determined from (4.18), and then a new 52 could be calculated from substituting into the right-hand side of (4.17). The new 52 was then used to determine a new (1, and the process was repeated until the solution Q (whose bulk value is approximately the same as Q, = 0.63559) had converged to the fifth decimal place. Most calculations took about thirteen iterations to converge when starting from the first guess given above. When a number of calculations were being done in a sequence the previous solution could be scaled to give a first guess for the next problem, with the result that convergence was sometimes obtained in as few as eight iterations. The numerical procedures were carried out with ten grid points per unit of Yukawa range. Finer grid-point spacing and more accurate convergence criteria were also investigated to assess their influence on the results.

Three examples of density distributions resulting from the solution of (4.17) are shown in Fig. 9. In this figure the ordinate is the normalized density where

p= gctJ3 ( ) PO Qrl

The abscissa is the radial distance in fermis, where r = at. These distributions have the expected characteristics of a uniform central region of constant density and a relatively thin surface region in which the density falls smoothly to zero. Some other interesting properties can also be recognized by examining such a sequence of calculations. First, although the central density is approximately that of nuclear matter, this is strictly true only of an infinitely large nucleus. For large but finite nuclei the surface tension squeezes the central region, with the result that the central density is increased above that of nuclear matter by an amount which depends on the surface-to-volume ratio (proportional to A-l13). This deviation increases as the nuclei being examined become smaller and smaller until it reaches a maximum at about A = 48. Below this value the surface region extends sufficiently far into the center to make the central density decrease with decreasing mass number, so that for light nuclei the central density is determined mainly by the proximity of the surface.

A second feature of these density distributions is the change in the nature of the


Radius in fermis

FIG. 9. Density distributions of three symmetric (N = 2) nuclei without Coulomb energy as a function of the radial distance. The total number of particles in each of these nuclei is indicated.

surface as the binding energy per particle decreases. Since lighter nuclei are less strongly bound the density begins to go to zero less abruptly, and for very weakly bound systems we would expect to find density distributions with a long tail, an indication that the system is on the verge of “dripping off” particles.

3. Semi-In$nite System

In order to be able to examine the intrinsic nature of the density distribution in the nuclear surface, and to calculate the surface energy coefficient, it is necessary to go to the limit of an infinitely large nucleus so as to avoid extraneous effects introduced by finiteness. Consideration of the surface region in this limit leads to a study of the semi-infinite system, which consists of a plane transition region with nuclear matter on one side and nothing on the other.

In order to obtain the Thomas-Fermi equation which applies in this case it is necessary to make the substitutions

and then take the limit as B -+ co. This procedure results in the equation

Y l,” dx B expC- I ?- - x IlP(x) - ~Q”~>l + A @(T) =

r J,” dx B exp[-I ~-xllQYx)+1 ’

595l55/3-4

444 MYERS AND, SWIATECIU

0

Distance from the mean location of the surface in wits of o

FIG. 10. Semi-i&rite density distributions as a function of the distance, in units of a, from the mean location of the surface. The curve for y = to corresponds to a strongly bound system, the one for yn represents the nuclear case, and the curve for y = 3.6 is for a system which is about to become unbound.

--T----z I

-I

\

0 I 2 3 4 5 Dis!ance in fermis

FIG. 11. Curve A is the semi-infinite nuclear density distribution as a function of the distance from the mean location of the surface. Curve B shows-in relative units-how the source of the surface energy is distributed across the surface. Curve C is the constant part of the velocity- dependent potential corresponding to the density distribution. It is the potential that would be felt at each point by a zero-velocity particle.


where

A = -Y jm dx WTQYx) - WYx)l. 0

Just as in the finite case this equation can be solved by computer iteration. Figure 10 shows three examples of density distributions corresponding to different values of y. The density distributions are plotted relative to their asymptotic value (which is equal to the equilibrium value for an infinite system). The abscissa in this plot is the distance (in units of a) from the mean location of the surface, where “the mean location of the surface” is defined as the position at which a sharp surface would have to be located in order to contain the same number of particles. These three density distributions illustrate once,again the development of a long tail for weakly bound systems, which was noted earlier in connection with finite nuclei. Thus for y = cc we have a strongly bound system with a relatively thin surface and a short tail. The middle curve represents the nuclear case, and the curve for y = 3.6 represents a case in which the system is about to become unbound and a long tail is developing.

We also find that in each case the separation energy n is equal to the equilibrium energy per particle yn which is as it must be for a saturating system of infinite extent.

The curve labeled A in Fig. 11 shows the surface density distribution for the nuclear case (y = m), and this time the abscissa is in fermis. The curve labeled B in this figure shows-in relative units-how the source of the surface energy is distributed across the surface. We discuss this further below. The curve labeled C is the constant part of the potential generated by this density distribution; it is the potential which would be felt at each point by a zero-velocity particle. It is given by

Vo(T) = --Y PO dx iI exp[-I 7 - x II[@(x) - 3Q5(~)l. (4.18a) 0

The solution of the nuclear Thomas-Fermi equation in the semi-infinite case enables us to calculate the surface energy coefficient and the curvature correction to the surface energy coefficient. To show the origin of these terms and to derive a method for calculating them, let us consider the total energy E of a large nucleus,

E = 47r I m r2 dr pe, 0

where the particle density p and the energy per particle e are functions of the radius r. Let ec denote the energy per particle in the central region of the system. Then since

A = 4n- ,)?dr p,


we can write E as

E = ecA + 47r 1: r2 dr p(e - ec),

without any approximation or loss of generality. This equation divides the total energy into a volume term ecA and an integral whose contributions come only from a region localized in the surface. This is easily seen, since p tends to zero outside the surface and (e - e,J tends to zero inside the surface. Because of this localization it is convenient to make the substitution

r=R+n,

where R is the mean radius of the system and n is the outward normal distance from the mean surface. The mean radius R is defined as the radius of a sharp spherical distribution containing all the particles and having the same uniform bulk density. With this change in notation we can write

E = e,A + 47rR%,

where the surface energy coefficient u is given by

(4.19)

CT= I m W + WR) + (n/R)“1 F(n). (4.20) --m

We have used the localization of the integrand in the surface region to replace the lower limit of the integral by - co, and we have defined a new function

F(n) = p(e - e,).

If we use the subscript s to specify quantities evaluated in the semi-infinite case (corresponding to R = cc) we see that the surface energy coefficient in the limit of R+ co is given by

m lim u = crs = R-VW s dn F&z),

--m

which is an integral over a function available from our semi-infinite Thomas-Fermi calculation. The limit of e, , occurring in F s , is eao , the energy density of nuclear matter.

Since we wish to include a curvature correction to the surface energy coefficient in our discussion of the Droplet Model, we must calculate the next term in a Taylor series expansion of u about the semi-infinite case. Expanding in terms of the curvature K where, for spherical systems, K = 2/R, we have

(4.21)


When evaluating the derivative in (4.21) it is useful to recognize that although u depends in general both upon the surface curvature and upon changes in the falloff of the density induced by the curvature, this latter effect is of higher order, since 0 is stationary with respect to density changes. Differentiating (4.20) yields

where

(4.22)

the derivative in (4.22) being taken with p(n) considered as independent of K. If (4.19) is combined with (4.21) the energy may be written in the form

E = --a,A + a,A213 + asAll + *a* ,

where

a, = -em

ar,,2ba 64~~ a2 = 7 3 - T SW dv.fsW, --m

a2rob3 128 1~~ a3 = 7 3 - T Irn dv[vf,(v) + fs’(v)l. --oD

(4.23)

(4.24)

The integrals in (4.23) and (4.24) are to be taken over the functions f and f’ in the semi-inlinite case, where v is the normal distance from the mean surface in units of a. The functionf(v), the dimensionless form of the surface energy function F(n), is given by

f(v) = Q3Wh(v~ - %I, (4.25)

where LP is proportional to p, and 7 is the energy per particle in units of T. The function&‘(v), the dimensionless analog of&‘(n), can be written as

(4.26)

where 8 is the mean radius in units of a, and (2/E) is the dimensionless form for the curvature. The derivative in (4.26) can be written

m= a(2/5) - $ & Q3w?W - $01, (4.27)

where we have made use of Eq. (4.25). If we substitute for r) from Eq. (4.13) and


then make use of the fact that the ,difIerentiation in (4.27) can be performed with the density (or Q) held fixed, we arrive at the expression

?f(d - = $ P(v) & 1% jr 6” df exPb-l5 - t II - exp[-Cl; + 01 a(2/8) 255

Making the substitutions

differentiating with respect to 8, and taking the limit in (4.26) as 8 -+ CO, results in the expression

dx(v - x) B exp[-l v - x I] @&)[l - @Z(v) + QQ)],

With these definitions all the integrals in (4.23) and (4.24) can be performed numeri- cally .over the function deduced’from the solution to the Thomas-Fermi equation in the semi-infinite case. The result is

(. , ,,

I m’ dvfs(v) = 0.037292, -al

J i dv v&(v) = -0.03758, -co

s m ‘-co

dvfs’(v) = 0.05565.

The surface energy coefficient a2 defined by (4.23) is to be fixed at 18.560 MeV according to Table I. Substituting this’value into (4.23) and solving for the Yukawa range a, which is the only unknown parameter, yields the result

a = 0.62567 fm. ~’

With a fixed we can calculate a3 , with the result

a3 = 516.18 jrn dv[vf, +fs’] MeV --m

= -19.42 + 28.76 MeV

’ = 9.34,MeV;


With the above choice for a the surface thickness (defined as the distance in which the density falls from 90 to 10 ‘A of its bulk value) for semi-infinite nuclear matter is about 2 fm. This is to be compared with the experimental value for finite nuclei of about 2.4 fm (32).

C. TWO-COMPONENT SYSTEMS

In the previous sections we considered only symmetric (N = 2) systems. Nuclear matter was assumed to consist of Fermi particles with each phase-space state degenerate because of the spin and i-spin internal degrees of freedom. The dimensionless interaction strength y that was used is equal to the sum of the like and unlike interaction strengths a! and /3, where the following relations hold:

and

32~~ ab 3 Cl 01=--7i- r 3 ( 1

p+C(p.

In the previous work the density distributions were considered in terms of the dimensionless Fermi momentum Sz, and the two-component system will be treated in exactly the same way but in terms of the separate Fermi momenta @ and Y for the neutrons and protons.

1. Nuclear Matter

The generalized form of Eq. (4.8) for a homogeneous two-component system of infinite extent is readily shown to be

’ @@” + Y5) - f [@( 1 - $D2) + ?P(l - $P)] I - j?@?P[l - 3(@2 + Y2)] I

rl= CD3 + Y3 . (4.28)

For a symmetric system, @=Y=Q,

and (4.28) becomes identical with (4.8). It is worthwhile to make a thorough investigation of (4.28) in the same way as was done for the one-component system earlier. However, the problem is a bit more involved for a two-component system, since there are two interaction strengths and two Fermi momenta. To reduce the com- plexity somewhat we deal only with those values of the interaction strengths 01 and p which represent nuclear matter; and we choose a more illustrative representation for the densities than the one in terms of CD and Y.


Before examining (4.28) we make the substitutions

r* = &(@,” + P),

s = CD* - Y* (4.29)

aJ* + ?P* *

The quantity r in this symmetric representation of the densities is called the “density parameter,” since r3 is proportional to the total density in the same way as @ and ?P3 are proportional to the neutron and proton densities. These proportionality relationships are

pn = T (%j” @3,

pz+ ;3y* 0 , and for the total density,

167~ b 3r3 -- P=3h * 0

The other new quantity 6 is the asymmetry of the system, equal to

6 - Pn - Pz . P

Before actually embarking on the discussion of (4.28) in terms of r and 6 it is necessary to fix the interaction strengths 01 and p to correspond to nuclear matter. We already have one relationship between them, which is

a+p=yll = 6.5536. (4.30)

Another relation between 01 and /3 can be obtained from the symmetry energy coefficient of nuclear matter. (The value we use for this quantity is given in Table I.) In order to find this relation we make a Taylor-series expansion of (4.28) about its equilibrium value. We thus obtain an algebraic expression for the symmetry energy coefficient as well as a number of other interesting quantities. (For a treatment which is similar in some aspects to this one see Ref. (3Z)).

The first step in this process is the substitution of (4.29) into (4.28) and expansion in powers of S, with the result

7) = [$P - 4(a + /3) P(1 - y-2)] + [p - Q(a - rs> l-3 + #(2a - j?) P] 62

+ IA 2 P-&a - 5/l) rq 64 + **.. (4.31)


When S = 0 only the first bracket remains, and, as we expect, it is identical with (4.8). The other terms give the dependence of the energy per particle on successively higher powers of the asymmetry 6. This equation is now in a form that can easily be expanded in a Taylor series in the square of the asymmetry 6, and in E, the deviation of the density parameter from its equilibrium value, where

The result of this expansion is the equation

7) = 7)n + (6Jof + w1S2) + +(W& - 2W,ES2 + w4S4) + *-’ , (4.32)

where

w. = ;r2 - #(a + 6) r3(1 - @), (4.33a)

w1 = p- &X - pj r3+ g2a - pj r5, (4.33b)

w2 = grz- 3(a + p) ryi - 4r2), (4.33c)

~~=fr~-f(~-p)r3+$2,+3)r~, (4.33d)

~,==&yj+5p)r5. (4.33e)

These equations are all to be evaluated at the stable equilibrium corresponding to nuclear matter, which is characterized by S = 0 and o0 = 0. The latter condition is the same as Eq. (4.8a) in the section describing the one-component system. From this earlier discussion we already know that at the stable equilibrium corresponding to nuclear matter

yn = 6.5536, 7jn = -0.1911, r, = 0.63559.

Reintroducing dimensions into Eq. (4.32), we find that the energy per particle may be written as

e = -6 + JS2 + &(KE2 - 2L& + jj4s4)+ . . . .

where --a, = d’,

J= w,T, K= w2T, L = w3T,

M = w4T,

(4.34a) (4.34b) (4.34c) (4.34d) (4.34e)


Equation (4.34b) for the symmetry energy coefficient J, when combined with (4.33b) and the fact that J = 28.062 MeV, leads to the equilibrium relation

$rz - $(a - lg)r3 + Q(2a - /3)r5 = 0.34209,

which is a second relationship between CII and /I in addition to (4.30). Solving this pair of equations results in

a = 3.6652 and

/3 = 2.8884 and

Cl = 367.56 MeV,

Cu = 289.66 MeV. (4.35)

Substitution of these values back into (4.33) and (4.34) gives the results shown in Table III. (Our value of K is almost half again as large as that obtained in Ref. (32), but the value of L agrees almost exactly.)

The values of Cl , C,, , a, and b deduced in this section define an effective, velocity-dependent Yukawa interaction between nucleons, which reproduces the principal average nuclear properties when used in conjunction with a Thomas- Fermi treatment.

TABLE 111

COEFFICIENTS IN THE TAYLOR-SERIES EXPANSION OF THE ENERGY PER PARTICLE ABOUT STANDARD NUCLEAR MATTER

Dimensionless coefficient Corresponding coefficient in MeV

r)n = -0.19111 a, = 15.677 WI = 0.34209 J = 28.062 w, = 3.5938 K = 294.80 WQ = 1.5060 L = 123.53 w* = 0.03259 M = 2.673

An idea of the influence of asymmetry on the energy per particle of an infinite two-component system can be obtained by examining Figs. 12-14. These figures are somewhat similar to Figs. 6-8. In both cases the effective strength of the binding is being varied, although in Figs. 6-8 this is done by varying the strength parameter y whereas in Figs. 12-14 it is done by varying the composition, the strength parameters Q: and /3 being held fixed at their nuclear-matter values.

Figure 12 is much like Figure 6, since they are plotted to the same scale and the curve labeled 6 = 0 in Fig. 12 is identical to the one labeled y = yn in Fig. 6. The other solid curves in Fig. 12 represent the energy per particle of infinite nuclear matter for various values of the asymmetry. The dot-dashed line traces out the


TABLE IV

EQIJILILNUMVALUESOF THE ENERGY PER PARTICLEANDTHE DENSITY PARAM? TER FOR SELECTH) VALUES OF 6

A B C

6 r

0.0 0.63559 1.0 0.43554 1.3155 0.29678

rl Energy per particle (MeV)

-0.19111 -15.677 -0.01273 -1.044

0.02064 1.693

FIG. 12. The dimensionless energy per particle 7 as a function of the density parameter r for various values of the composition 8. The dot-dashed line is the locus of equilibrium values of 7, and the dashed line is a Taylor series approximation to this locus. Point A corresponds to nuclear matter, Point B to neutron matter (which is slightly bound for our choice of parameters), and Point C is where the stable and unstable equilibrium points coalesce and beyond which no equilibrium exists.


locus of the equilibrium points of these curves to show how the equilibrium value of r depends on asymmetry. The three labeled points have the following significance: A is the point corresponding to the equilibrium of symmetric nuclear matter, B is the point corresponding to the equilibrium of completely asymmetric nuclear matter (neutron matter), and C is the point at which the stable and unstable equilibrium points on the energy per particle curve coalesce for some value of 6 which is greater than unity. The coordinates of these points are given in Table IV.

Figure 12 and Table IV indicate that for our present choice of parameters the theory predicts that neutron matter is bound but that the binding is small. A small change in the nature of the phenomenological interaction might make neutron matter unbound. In fact, we show later that the choice of parameters in (4.35) is

FIG. 13. The equilibrium energy per particle q as a function of the asymmetry 6. The light line corresponds to the Liquid Drop Model assumption that the curve is purely quadratic, and the dashed line is an improved approximation resulting from a Taylor expansion about the minimum. Point A corresponds to nuclear matter, Point B to neutron matter (which is slightly bound for our choice of parameters), and Point C is where the stable and unstable equilibrium curves meet and beyond which no equilibrium exists.


0.0 0.5 Ip 1.5 Asymmetry 3

FIG. 14. The equilibrium density parameter r as a function of the asymmetry 6. The dashed line is a Taylor series approximation to the upper part of the curve. Point A represents nuclear matter, Point B represents neutron matter, and Point C indicates where the family of equilibrium configurations goes around a bend and beyond which no equilibrium exists.

probably not the best one. We have reasons to believe that the improved choice will probably lead to the prediction that neutron matter is not bound.

Before continuing our consideration of Figs. 12-14 let us pause for a moment to discuss how these figures can help us assess the range of validity of the Taylor-series expansion of Eq. (4.32). One way of doing this is to compare the equilibrium values of r and 7 predicted by (4.32) with the true values corresponding to (4.28). Setting the derivative of q with respect to E equal to zero in (4.28) results in the following relationships for the equilibrium values of E and 7:

E= 2 62,

?1 = qn + w,s2 + 9 (

WQ2 64 wq - - 02 1 ,

- qn + WP2

- E. *3

(4.36)

(4.37)

(4.38)


In Figure 12 Eq. (4.38) has been plotted with a dashed line for comparison with the dot-dashed line discussed earlier.

In Figure 13, which is a bit like Fig. 7, the equilibrium energy per particle 77 is plotted against the asymmetry 6. Both the stable and unstable equilibria are shown and the points A, B, and C are labeled. The thin parabola represents the curve

which corresponds to the usual assumption of the Liquid Drop Model that the equilibrium energy per particle is quadratic in the asymmetry. The more accurate approximation provided by (4.37) is plotted as a dashed line.

Figure 14, the analog of Fig. 8, gives the value of r for both the stable and unstable equilibrium points as a function of the asymmetry 6. The points A, B, and C are labeled, and the Taylor-series prediction given by (4.36) is plotted as a dashed line.

This completes the discussion of the infinite two-component system, and we now turn to finite systems.

2. Finite Nuclei

As with the one-component system considered earlier, it is possible for two- component systems to form stable finite aggregates of particles. The density distributions of such finite systems can be determined by the use of a variational procedure. In this procedure we determine what distribution of neutrons and protons minimizes the total energy of the system subject to the separate constraints of neutron and proton conservation. As with the one-component system we actually solve for the spatial distribution of the dimensionless Fermi momenta @ and Y, and so the first step is to write the total energy in terms of these variables. Thus

+L(f)“s nuclear ~S5P3mwt3 + Q%G)l + ~3mw2(c) + +vy(c)]}, volume (4.39)

where

+ BY3(CN - w”(r) + WS))I),

and v&) = the same as above with @ and Y interchanged.


As before, we can specialize to the case of spherical symmetry and then perform the angular part of the spatial integrals with the result

- s ; t” d6 expE-I 1s - t II - exp[-(i + 01 ZIE

In the same notation the number of neutrons is given by

and the number of protons is given by

In terms of these quantities the variational conditions are

6[E - L,N] = 0 (4.40a)

and 6[E - LZ] = 0, (4.40b)

where L, and L, are Lagrange multipliers which represent the neutron and proton separation energies. In dimensionless form

and &a = LIT

A, = L,IT.

Taking the variations indicated in (4.40) results in two coupled integral equations for the neutron and proton density distributions. To clarify the procedure we carry out the neutron variation in some detail.


Substituting into (4.4Oa) and taking the indicated variation results in the expression

x [ 5 (3@(5) @3<n S@,(5) + 3@3(5) @2(o S@(S) - 3Q4(5) Q3(g) S@,(c-)

- W"(5) Qb2(5) wo - W"(5) @"(if) wo - 3Q3(5) @"(O S@(tl>

B + 3 (3Q2(5) Y3@ wo - 3Q4(5) Y3(5) S@5(0 - W2(0 Y5(0 wcl

+ 3Y3(5) @'"(5> SW) - W5(0 @"(5) wn - 3Y3($ @p"(g) wm]

- A, jr 5" d[ 3@"(iJ S@(<) = 0. (4.41)

Just as in Eq. (4.15), we recognize that since the square bracket in (4.41) is to be integrated over both 5 and 5 with a spatial weighting factor symmetric in 5 and 5, it follows that these variables can be interchanged at will in any term inside the bracket. Doing this for those terms in the bracket that have SQ(g) as a factor results in the equation

+ pwxl - @2(5) - w2tn>1 - A\ =a) = 0. (4.42)

Since we require that the integral of (4.42) be zero for arbitrary S@(g, the coefficient of S@(l) must equal zero. The values of @(f;) which satisfy this criterion are solutions to (4.40a) and are the equilibrium distributions we seek. One obvious solution is Q(c) = 0, and the other solution can be obtained by setting the bracket in (4.42) equal to zero and solving for G”(c). This results in the equation

s ;Pds exp[-I 5 - I II - w[-(f; + 01 xx

@q-) = I I s ; P & exp[-I 5 - 6 II - ev[-(ll + 01

I

. (4.43)

255 x (~@"(R + pwo) + 1


This equation and the one which can be obtained from it by exchanging Qb and Y are the coupled equations of the two-component Thomas-Fermi method (the Seyler-Blanchard equations.)

To specify a particular finite nucleus we fix the radii at which the neutron and proton density distributions go to zero. To do this we set

A, = - jm (2 (g exp[-1 G - t II - exp[-6 + 01 0 2&5

x bP(5) - W5(0) + W3(0 - w5@m, (4.44)

where & is the zero density radius for neutrons. A similar relationship involving R UYY, the zero-density radius of the protons, can be obtained for & by exchanging

1 N =22.43

0.5 z = 20.50

A = 42.93

Radius in fermis

IO

FIG. 15. Neutron and proton densities (without Coulomb energy) as a function of the radial distance for three cases representative of stable nuclei. The number of particles these distributions correspond to is shown.

595/.55/3-5


@ and Y in (4.44). The coupled integral equations for a([) and Y(5) can then be solved by computer iteration just as in the one-component case.

Figure 15 shows three examples of solutions of the finite two-component Thomas-Fermi equations. These cases were chosen as representative of stable nuclei (except for the absence of the Coulomb energy).

3. Semi-Infinite System

For one-component systems we considered the solution of the Thomas-Fermi equation in the limiting case of R + cc. This solution, that of semi-infinite nuclear matter, allowed us to examine the intrinsic nature of the surface region and to calculate the surface energy coefficient free from extraneous effects due to finiteness. The same is true for two-component systems. If we make the replacement

and then take the limit as B -+ cc in Eq. (4.43), we obtain

mdx i exp[-I 7 - x llbP(~) - 3@5(xN 0

+ 1B(Y3W - 3WxNl - &

s * dx 3 ew[-I 7 - x ll[~@Yx> + BY3WI + 1 . (4.45)

0

This equation and its complement, which is formed by interchanging Cp and Y, are the coupled equations for the two-component semi-infinite density distribution. There is a new facet to the two-component problem which was not present in the one-component case. This is the possibility that the neutron and proton density distributions can go to zero at different points. We can specify the neutron zero- density point by X, , where

&= j;dx 4 exp[- I xQ - x Ilb(@3(~) - %Q5(xN + IB(Y3(x) - %Y5(xN1.

This equation and its complement, formed by interchanging @ and Y, when substituted into (4.45) and its complement, completely define the Thomas-Fermi problem where the neutron and proton density distributions go to zero at 1, and x P.

Three examples of two-component semi-infinite density distributions are shown in Fig. 16. Each of the examples consists of three curves; the lower two curves


Distance from the mean location of the surface in fermis

FIG. 16. The neutron, proton, and total densities as a function of the distance from the mean location of the surface, for various values of the bulk asymmetry 6. The vertical bar indicates the mean location of the surface of the total density, and the small vertical bars on either side of it indicate the locations of the mean neutron and proton surfaces. The dot-dashed lines are the loci of the mean neutron and proton surfaces. The loci of the asymptotic values of the densities are shown by dashed lines (note the decrease in the total density as 6 increases).

represent the neutron and proton densities, while the upper curve is their sum. The densities are in units of the density of symmetric infinite nuclear matter, and as the bulk asymmetry 6 increases the asymptotic density falls below unity, as we would expect from our discussion of Fig. 12. The densities are plotted relative to the mean location of the total-density surface. This mean location, which is the zero of distance for each plot, is indicated by a long vertical line. The vertical tick


marks-one on each side of the mean surface-specify the mean locations of the surfaces of the neutron and proton density distributions. These tick marks are connected by dot-dashed lines to illustrate the increase in the neutron skin thickness with increasing 6. The plot itself is an isometric representation with the bulk asymmetry coordinate extending outward toward the reader. The density distributions, which extend to infinity, have been terminated in the figure at a point far enough from the surface to insure that they have attained their bulk values.

Since we now have solutions available for the two-component semi-infinite Thomas-Fermi problem, we can proceed to calculate a number of new Droplet Model coefficients. There are three ways to calculate the effective surface stiffness Q, and we can also calculate the surface energy coefficients H, P, and G.

In Sections II and III we showed that for two-component systems with an equilibrium relationship between T and 8, the surface energy coefficient could be written as (az + QT~), where Q is called the efictive surface stiffness. We also showed that the equilibrium relationship between T and 8 in the absence of Coulomb energy is

T=3Ja 2Q *

(4.46)

In Fig. 17 we have plotted 6/7 against TV. (In the semi-infinite case there is no need to distinguish between 6 and 8.) The value of this ratio in the limit as T -+ 0 is 0.3813, which may be used with Eq. (4.46) and the fact that J = 28.062 MeV to deduce

Q = 16.05 MeV. (4.46a)

A second way of calculating Q is to actually evaluate the surface tension of a

FIG. 17. The points represent values of the ratio of the bulk asymmetry 6 to the skin thickness 7 plotted as a function of 3. The intercept at P = 0 of the curve drawn through these points is used to determine the coefficient Q.


sequence of equilibrium systems as a function of T, using Eq. (4.23). In the two- component case the functionf, in Eq. (4.23) is given by

This expression is the two-component analog of the one-component relation, Eq. (4.25). The first factor (W + Y3) represents the total density and the second factor is the difference between the energy per particle at each point and the energy per particle in the bulk.

Since we expect the surface energy coefficient to be given by

Es = a, + Qr2, (4.46b)

we can determine Q by taking the limit of d&/T2 as 7 --+ 0, where LIE, = EB - a2 . Figure 18 is a plot of A&/T~ as a function of TV. The extrapolation to T = 0 of the sequence of calculations shown in this figure gives the result

Q = 16.0 MeV. (4.46~)

The numerical agreement of the values of Q determined in these two distinct ways provides a partial confirmation of the Droplet Model theory which led to Eqs. (4.46) and (4.46b).

In general the surface energy of a two-component system is a function of both T

and 6, , where 6, is the value of the bulk asymmetry extrapolated to the surface.

FIG. 18. The points represent values of the ratio of the change in the surface energy A& to the square of the surface thickness T plotted as a function of 9. T’he intercept at ~2 = 0 can be used to infer the effective surface stiffness coefficient Q.


In Eq. (2.8) we wrote the first few terms in the Taylor expansion of the surface energy coefficient (E, = 437r,,%). This equation can be written

Es = a2 + Hra + 2Pr6 - GA2, (4.46d)

where we have replaced 6, by 6 and have omitted the curvature term as, since we are discussing unchanged, semi-infinite distributions. Of course, this equation simplifies to Eq. (4.46b) for systems with an equilibrium relationship between T and 6.

In order to determine the separate coefficients H, P, and G it is necessary to consider nonequilibrium semi-infinite systems. To do this the semi-infinite Thomas-Fermi problem was first solved for the case in which 7 = 0 and 6 = 0. The resulting neutron and proton density distributions were then scaled to produce a certain asymmetry 6, or moved relative to each other to produce a desired value of the surface thickness 7.

Figure 19 illustrates how H was determined from a series of calculations of the surface energy coefficient for nonequilibrium two-component density distributions where T was varied and 6 held fixed at zero. The quantity (AE,/T~) was plotted as a function of TV, where

AE, = Es - a2.

Extrapolation of these results to T = 0 yields

H = 9.42 MeV.

9.5 I

>

ii!

.= 9.3-

: s

G 9.2 -

9.1 0.0 0. I 0.2 0.3

Surtoce thickness squored,r”

(4.46e)

FIG. 19. The ratio of the change in the surface energy coefficient AEe to the square of the surface thickness 7, plotted as a function of P. The intercept at 7 = 0 gives a value of 9.42 MeV for the coefficient H.


0.005 0.010

8ulk asymmetry squared, 8’

FIG. 20. The ratio of the change in the surface energy coefficient AEs to the square of the bulk asymmetry 6, plotted as a function of P. The intercept at P = 0 gives a value of 45.4 MeV for the coefficient G.

Figure 20 shows a simiiar plot, which was used to determine the value of G. Here a series of calculations was made of the surface energy coefficient for nonequilibrium two-component systems for which 6 was varied and T was held fixed at zero. The quantity (AE,,@) was plotted as a function of 8 and extrapolation of these results to 6 = 0 yields

G = 45.4 MeV.

The determination of P requires knowledge not only of a2 but also of H and G. Using the values of these coefficients that we have just calculated, we can now cal-

FE. 21. The quantity &(A& - HP + GP)/rS plotted as a function of 9. The intercept at I* = 0 gives a value of 17.55 MeV for the coefficient P.

466 MYERS AND SWLWECKI

culate P on the basis of a series of calculations of the surface energy coefficient of a nonequilibrium two-component semi-infinite system in which both T and 6 are varied. Figure 21 shows the result of such a calculation in which the quantity

?&lEs - HT~ + GS2)/d

is plotted against 72 and extrapolation of these results to 7 = 0 yields

P = 17.55 MeV.

Since we now know all the separate coefficients H, G, and P we can calculate Q in yet another way. In Section II, Q was originally defined by the equation

Qc H l-2P’

35

Using this relationship and the values of H and P we have just calculated, we have

Q = 16.15 MeV,

which agrees with the previous results to within the accuracy expected from the numerical procedures.

Another important result which we can now confirm is the relationship among Droplet Model coefficients derived in Section III, which is

G 3J -= P

--. 2Q

Using the results above, we find

G - = 2.59 P

and

3J -- = 2Q

2.61,

which agrees, once again, to within the accuracy of the numerical procedures. Finally we can make use of the procedure we have established for calculating

the surface energy coefficient for nonequilibrium systems to show that the surface energy does not depend linearly on E, where E represents a scale change in the overall density distribution. By simply scaling the semi-infinite density distribution up and down from its equilibrium value and calculating the surface energy in each


FIG. 22. The ratio of the change in the surface energy coefficient AEs to the scale parameter l , plotted as a function of E. The fact that the curve passes through the origin shows that the surface energy coefficient does not depend linearly on l .

case we can obtain the results shown in Fig. 22. In this figure we have plotted AE& as a function of E, where

E = _ 1 Pbulk - PO 3 PO ’

The smooth line drawn through the points passes through the origin, showing that the surface energy coefficient has no linear dependence on E. The slope of this curve at E = 0 yields the result that the coefficient of the quadratic term in a Taylor expansion of the surface energy coefficient in terms of E is equal to -470 MeV.

D. COULOMB ENERGY

To generalize the two-component Thomas-Fermi method to include the Coulomb energy it is necessary to go back to the total energy equation given by (4.39). To this earlier equation we add the Coulomb energy and its exchange correction and then carry out the variational procedure as before. Thus the direct part of the Coulomb energy is

0


where e is the proton charge, and p is the proton density. For spherical symmetry the angular part of the spatial integral can be performed, with the result

where substitution has been made in terms of dimensionless quantities and the Coulomb strength parameter p has been included with the definition that

= 0.022949.

The energy associated with correcting the Coulomb energy for exchange effects is (Ref. (23))

Ex = - $1 d3rk4,

where k is the Fermi wave number given by k = P/h. In terms of dimensionless quantities we have

where we set

and we defined the coefficient h by

x - 3e2b hT

= 0.016670.

If both the direct and exchange parts of the Coulomb energy are combined their contribution to the total energy can be written as

y (f,” Tf 5” dZ: 1% ,; 5” d( (’ + ‘);; E; - E ’ y3(5) y3@ - hy4(5)j.


If we take the variation of this energy contribution with respect to changes in Y we find that the corresponding energy change is

When this is combined with the equation for Y that is the analog of (4.42) the resulting equation is

jr 5” 4 3Y2(5) j’y2K) - jr E” 4 / pp[-I 5 - 5 II - exp[-t5 + 01 x4

-(5iO-li-51 255

py1yq - #Y(j) - A, I

SY(tg = 0. (4.47)

In the usual way, we recognize that for (4.47) to be true the coefficient of SY([) must vanish. The resulting integral equation can be written

Y2(0 = (4.48)

where we require that

A, = j; 5" 4 I""'[-' 9, - f II - exp[-(G + 01 [a(y3(s) _ sys(6j) 2EyE 5

+ /g(cp(f) - 34j5(~)] - G + $- 8, (4.49)

The two equations (4.43) and (4.48) are the coupled integral equations for the density distribution when the Coulomb energy and its exchange correction are included. They can again be solved by computer iteration.

Figure 23 shows three examples of density distributions representative of stable nuclei, as calculated from the equations derived above. A new feature in these


FIG. 23. Neutron and proton densities (with Coulomb and Coulomb-exchange energies included) as a function of the radial distance for three cases representative of stable nuclei. The number of particles these distributions correspond to is shown.

distributions is the central depression in the densities, caused by the Coulomb repulsion.

Equations (4.43) and (4.48) along with the auxiliary conditions of Eqs. (4.44) and (4.49) are the most general statement of the Seyler-Blanchard form of the nuclear Thomas-Fermi Model. Their simultaneous solution (by computer iteration) yields the neutron and proton density distributions of the nucleus being studied. On the basis of these density distributions and the known interaction almost any question can be answered regarding the average properties of the system.

To make the meaning of the integral equations clearer consider Eq. (4.48). The numerator consists of two terms. The first, the integral, is the potential that would be felt by a zero-momentum particle at 5. This potential comes in two parts; the first from nuclear forces, the second from Coulomb forces. The nuclear part of the potential consists of an integral over a spatial weighting function (a Yukawa with the angular integrals performed because of the spherical symmetry) times a bracket containing the neutron and proton densities (0” and Y3) along with part


of the velocity dependence of the force (@” and Y5) and the like and unlike strength parameters (11 and /3. The Coulomb part of the potential is an integral over the Coulomb spatial function and the proton density (!P) multiplied by the Coulomb strength CL. The second term in the numerator of Eq. (4.48) is the Lagrange multiplier (1,. The multiplier is equal to the proton separation energy and Eq. (4.49) shows that it is also equal to the potential felt by a zero-momentum proton located at the boundary of the proton density distribution 8, . If the denominator is moved to the left-hand side of Eq. (4.48) the terms are easier to identify. The first such term would be the contribution to the velocity dependence of the nuclear potential due to the momentum of the particles at 5. The second term [u1”([)] represents the kinetic energy of the particles, and the last term is the exchange correction to the Coulomb energy.

V. COMPARISON

The program proposed in the introduction has now been completed. In an earlier section the formal structure of the Droplet Model was developed. In the last section not only was the Thomas-Fermi Model developed for its own sake but it was also used to estimate the numerical values of the coefficients appearing in the Droplet Model. Even though these coefficients are only approximate (to get the best set of coefficients we will have to refit the experimental masses, using a Droplet Model mass formula) the Droplet Model predictions can be compared with experiment. As we pointed out in the Introduction, such comparisons are often difficult because the smooth dependence of Droplet Model effects on N and 2 makes it difficult to isolate them experimentally. In Section VI we treat some of the more interesting experimental results, but we defer an exhaustive comparison until some time in the future when the Droplet Model coefficients will have been redetermined.

There is another type of comparison possible. We used only infinite and semi- infinite nuclear matter Thomas-Fermi calculations to determine Droplet Model coefficients, and so one way to test the applicability of the model to systems similar to nuclei is to use it to predict the properties ofjinite Thomas-Fermi systems. In this way we can evaluate the Droplet Model both as regards the basic structure of the approximation, which is based on Taylor expansions of the energy, and as regards the range of validity in A-l13 and Z2 of the predicted numerical results. Such comparisons are attractive because of their broad scope. Thomas-Fermi calculations can be performed for symmetric or asymmetric nuclei, with or without Coulomb energy, and for nuclei with hundreds or even thousands of particles.

First we compare Droplet Model predictions with finite Thomas-Fermi systems which are symmetric (Z = 0) and have no Coulomb energy, then we consider


asymmetric systems (Z # 0) without Coulomb energy, and finally we make a comparison that involves Coulomb energy.

In all these comparisons the Droplet Model formulae from Sections II and III are used in combination with the specific values found for the coefficients in Section IV. A convenient listing of these results is given in Section V1.A.

A. ONE-COMPONENT SYSTEMS

For symmetric systems without Coulomb energy the Droplet Model energy Eq. (2.23) becomes

i? = E/A = -15.677 + 18.56 A--li3 + 7.00 A-2/3 + ... . (5.1)

The new term predicted by the Droplet Model is proportional to A-2/3. Its coefficient is composed of the difference between a 9.34-MeV curvature correction to the surface energy, and a 2.34-MeV correction due to a decrease in surface area caused by the surface compressing the nuclear interior. To see whether this prediction is borne out we can compare Eq. (5.1) with the plot in Fig. 24 of the energy per particle

FIG. 24. The energy per particle of symmetric finite nuclei without Coulomb effects plotted against A-118. The straight line (- 15.677 + 18.56 A-llS) corresponds to keeping only the volume and surface energy terms, the two lowest-order terms in the expansion of this function in powers of A-lp.


of a series of finite, symmetric nuclei calculated according to the Thomas-Fermi Model without Coulomb energy. In the figure the dashed line corresponds to the Liquid Drop approximation

E, = -15.677 + 18.56 A-*i3. (5.2)

The fact that the Liquid Drop Model prediction in Fig. 24 gives good agreement with the actual energy all the way down to A = 4 may explain why it has been difficult to obtain evidence for higher-order terms in the mass formula from the experimental masses themselves. This agreement stems from the fact that the next two terms in the energy expression tend to cancel over the region of interest. To determine the next two terms in the Taylor expansion of the Thomas-Fermi results beyond those in (5.2) we can make a plot like that in Fig. 25. In this figure the ordinate is (E - 15.677 + 18.56 A-1/3)/A-2/3 and the abscissa is A-lj3, so the intercept is the coefficient of A-2/3 in the Taylor expansion, and the slope is the coefficient of the A-’ term. Hence, from Fig. 25 it is possible to deduce that the Thomas-Fermi calctilations are represented by an energy equation

.!?,, = -15.677 + 18.56 A-lj3 + 6.98 A-2’3 - 12.3 A-l +- ... , (5.3)

which confirms the Droplet Model prediction for the coefficient of Ae2i3 in Eq. (5.1) to within the accuracy expected in the numerical procedures. We have also determined the coefficient of an additional term, one proportional to A-l, in Eq. (5.3) beyond those predicted by the Droplet Model.

Besides extending the Liquid Drop Model to include a new term in the energy the

FIG. 25. The remainder obtained when the volume and surface-energy terms (-15.677 + 18.56 A-l/*) are subtracted from the energy per particle +?? of symmetric finite nuclei is divided by A-z’s and the result plotted against A-‘la. The intercept of this curve at API5 = 0 is 6.98 MeV, the coefficient of APIS in the Taylor expansion of the energy per particle.


FTG. 26. The mean radius of symmetric finite nuclei without Coulomb effects is plotted against A113 (solid line). The dot-dashed line is the Liquid Drop Model prediction R = 1.2049 #la fm, and the dashed line is the Droplet Model prediction R = 1.2049 A1IS - 0.1517 fm.

Droplet Model also makes an interesting prediction concerning the nuclear radius. Combining Eqs. (2.3a) and (2.18) for the case of no Coulomb energy and 6 = 0 leads to the equation

R = 1.2049 A1lS - 0.1517 fm.

Figure 26 shows this prediction as a dashed line. The usual assumption that R = r&l3 is represented by a dot-dashed line, and the results for a series of Thomas- Fermi calculations are shown by a solid line. This figure confirms the Droplet Model prediction, but it also shows that it is not a bad assumption to take R proportional to A113.

B. TWO-COMPONENT SYSTEMS

When two-component systems are considered comparisons become more involved, since there are now t’wo parameters I and A. To check the validity of the symmetry energy terms in the Droplet Model energy equation let us substitute the numerical values of the coefficients and rewrite Eq. (2.23) as

AE,,, = E + 15.677 - 18.56 A-li3 - 7.00 A-2/s = (28.062 - 95.0 A-li3 - 24.54Z2)12.


lim (dE88ym/12) P-i0

If we now plot

against A-li3 for a sequence of Thomas-Fermi calculations, the resulting curve should tend to

(28.062 - 95.0 AI-‘/~), (5.4)

for large A. The dashed line in Fig. 27 is this prediction, and the solid line is drawn

00 3-

\

I-

o- 0.0

n 1000 125

I I r-- 7

-I

‘\ \

\

0.2

FIG. 27. The small crosses represent calculations of the effective value of the coefficient of ZaA in the nuclear energy equation, plotted against A-lls. The solid line connects these points, and the arrows set off a region corresponding to observed nuclei. The dashed line (28.062 - 95.0 A-lj3) is one form of the Droplet Model prediction. The dot-dashed line (28.062 + 126.0 A-1/3)/(1 + 3.936 A-1/8)8 is the preferred form of the Droplet Model prediction.

595/55/3-6


through the Thomas-Fermi results which are represented by crosses. The fact that the Thomas-Fermi results tend to the expected curve for large A confirms the Droplet Model expression (5.4). The figure also shows that even for very large values of A the Thomas-Fermi results begin to deviate substantially from (5.4). This feature of the Thomas-Fermi systems provides the basis for an important new insight into the dependence of the energy on asymmetry.

As was pointed out earlier during the derivation of the Droplet Model, the relation between the central asymmetry and the total asymmetry is given by Eq. (2.17a), which becomes (without Coulomb energy)

S= I’ 1 + 3.936 A-llS ’ (5.5)

Because of the size of the coefficient which multiplies A-II3 it is! a very poor approximation to expand this function (27). Just how superior Eq. (5.5) is to the Taylor expansion of 8 can be seen in Fig. 28. The upper set of four; curves in this figure

FIG. 28. Asymmetry as a function of A-‘/$. The two sets of heavy lines represent the values of the bulk asymmetry 8 of a series of nuclei without Coulomb energy. The light lines connect the values of the total asymmetry Z for these same nuclei, the do&dashed lines represent one Droplet Model prediction for 8, and the dashed lines represent the preferred form of the Droplet Model prediction for 8.


shows the dependence of the asymmetry on size, for a family of nuclei with a fixed skin thickness. (In the actual calculations the distance between the neutron and proton zero-density radii was held fixed.) The heavy line passes through four points representing the value of the bulk asymmetry 8 for Thomas-Fermi nuclei. The point at A-1/3 = 0 is for the semi-infinite case and the other three points are for finite nuclei. The light line passes through the value of the total asymmetry Z for these same nuclei, and it is obvious from the figure that Z is a very poor approximation to 8 in the region of actual nuclei. The prediction of Eq. (5.5) for the value of 8 for these nuclei is shown by a thin dashed line that is close to the true values. When a Taylor expansion of Eq. (5.5) is used to calculate 8 the results lie on the dot-dashed line, which would be quite useless in actual applications.

The lower set of four curves in Fig. 28 corresponds to a different choice of the skin thickness.

If Eq. (5.5) is substituted back into the Droplet Model energy equation the result is

E = - 15.677 + 18.56 AH3 + 7.00 A--2/3

+ (28.062 + 126.0 A-““) (1 + 3.936 A-l,3)2

- 24-54 (1 + 3&4fj A--1/3)4

In the same way as before we can compare this with the Thomas-Fermi calculations. The new prediction is that the curve in Fig. 27 should tend to

(28.062 + 126.0 A-‘/3)/(1 + 3.936 A-1/3)2

t I I / I 0.0 0.1 0.2

6”

FIG. 29. Effective coefficient of Z’A in the nuclear energy equation plotted against A-l13. The region of ordinary nuclei is indicated by arrows on this line, and the intercept 24.5 MeV at A-Ii3 = 0 serves to con&m the Droplet Model prediction.


as A-l13 goes to zero. This prediction is represented by the dot-dashed line in Fig. 27. Its correspondence with the Thomas-Fermi results is much better than that provided by the dashed line.

To confirm the value of the coefficient of Z4 in the Droplet Model Eq. (5.6) we can use a plot similar to Fig. 27. By subtracting all the other terms in Eq. (5.6) from the Thomas-Fermi result for E, dividing by Z4, and then taking the limit as Z-t 0 for a series of nuclei, we can obtain the results shown in Fig. 29. In this figure the line is a plot of the effective value of the coefficient of Z4 for the Thomas-Fermi calculations as a function of A-lj3. The asymptotic value of this coefficient as A + co confirms the Droplet Model prediction of 24.54 MeV.

In addition to considering Droplet Model predictions for the energy of finite Thomas-Fermi systems we can also make a comparison of the predicted and actual density distributions. For example, the Droplet Model predicts that the deviation dp, of the central density of a large nucleus (without Coulomb energy) from the nuclear matter value p0 is given by

4c _ PC - PO --- PO PO

= -3< = 0.3777A-113 - 1.257 s2. (5.7)

In this expression the first term represents the increase in density due to squeezing of the nuclear interior by the surface and the second term accounts for the fact that the equilibrium density of asymmetric nuclear matter (8 f 0) is less than that of

FIG. 30. The fractional deviation dpc/p, of the central density of finite nuclei from the nuclear matter value p0 as a function of A-‘IS for various values of the bulk asymmetry 8. The dashed lines are the prediction (0.3777 A-l’* - 1.257 8*).


symmetric nuclear matter (6 = 0). Figure 30 shows a comparison between this prediction and the results obtained for a series of Thomas-Fermi calculations. The smooth curve representing the series of Thomas-Fermi calculations with 8 = 0 agrees with the predicted behavior-given by the dashed line-for large A. Eventually the falloff of the density in the surface begins to cause a decrease in the central density. As one goes toward smaller and smaller nuclei this effect begins to dominate, until at about A = 48 the central density begins to decrease as the nuclei become smaller.

The other two solid curves correspond to asymmetric solutions to the Thomas- Fermi equations. They are shown in Fig. 30 to confirm the dependence on 8 of the Droplet Model expression (5.7). As can be seen in the figure, the dashed lines from Eq. (5.7) agree with the Thomas-Fermi results in the limit of large A.

C. COULOMB ENERGY

The Droplet Model introduces a number of corrections to the Coulomb energy. To see if the Droplet Model accurately represents the Coulomb dilation and redistribution effects we can compare a series of Thomas-Fermi calculations with the Droplet Model predictions for the central density. Unfortunately it is difficult to make a comparison analogous to the one in Fig. 30. For large A, where the curves in Fig. 30 agree with theory, the Coulomb energy is becoming so large that its higher-order effects interfere with a comparison. For small A we don’t expect very close agreement, as can be seen in Fig. 30. Nevertheless there is a nice way to test the Droplet Model. In Fig. 31 we have plotted the fractional deviation of the central density of a very large nucleus (A w 1700) from the nuclear matter value. This deviation is plotted against the relative Coulomb strength, where we have slowly increased the strength of the Coulomb energy from zero in steps of 0.1 times its final value. By doing this we are able to check the Droplet Model expression, which is

dp, = 0.3777 A--l13 - 1.257 sz - (0.007296 -$ + 0.005472 +j F, PO

where

8 = (I + 0.01676 --$Fj (1 + 3.936 A+)-l,

and F is the relative Coulomb strength. The nucleus being considered in Fig. 31 is large enough so that no difficulty is encountered from finiteness effects, and by making the comparison for a reduced strength of the Coulomb force we can avoid higher-order effects. The Droplet Model prediction is the dashed line, and it agrees quite well with the Thomas-Fermi calculations. When the disagreement begins to


FIG. 31. The fractional deviation Ap,/p, of the central density of a huge nucleus (A M 1700) from the nuclear matter value p. as a function of the relative Coulomb strength. The dashed line is the Droplet Model prediction.

get large at F = 0.4 the central density has been reduced to 75 % of its nuclear matter value and the proton density is only 60 % of its nuclear matter value. These deviations are more extreme than one expects in ordinary nuclei.

VI. DISCUSSION

In the preceding section we investigated the applicability of the Droplet Model to saturating, two-component systems by comparing it with the Thomas-Fermi Model. It is now appropriate to make a few comparisons between these models and the experimental results and to discuss interesting aspects of the models that were omitted earlier in order to enhance the continuity of the development.

A. DROPLET MODEL FORMULAE

In this section we collect the most useful of the Droplet Model relations. They are expressed in terms of the equilibrium values of the two number variables E and 8, the two constitution parameters A and I (or N and Z), and the physical coefficients listed below.


The basic coefficients from Ref. (I) are

a, = 15,677 MeV, the volume energy coefficient, a2 = 18.560 MeV, the surface energy coefficient, J = 28.062 MeV, the symmetry energy coefficient, r0 = 1.2049 fm, the nuclear radius constant.

Once r, is known the Coulomb coefficients can be calculated, with the results

c1 = 0.7170 MeV, the Coulomb energy coefficient, c2 = 0.0001479 MeV, the Coulomb redistribution coefficient, ca = 0.840 MeV, the Coulomb diffuseness coefficient, cq = 0.5475 MeV, the Coulomb exchange coefficient.

In calculating cg by means of Eq. (2. lob) we used the semi-infinite Thomas-Fermi density distribution calculated in Section IV.B.3.

The Thomas-Fermi calculations were ‘used to determine the remaining coefficients, whose values and suggested names based, on their physical significance (discussed in Section II) are

u, = 9.34 MeV, the curvature correction coefficient, G = 45.4 MeV, the surface symmetry coefficient, H = 9.42 MeV, the surface skin coefficient, K = 294.80 MeV, the compressibility coefficient, L = 123.53 MeV, the density-symmetry coefficient,

M = 2.673 MeV, the symmetry anharmonicity coefficient, P = 17.55 MeV, the skin-symmetry coefficient, Q = 16.04 MeV, the effective surface stiffness coefficient.

The two variables z and 8 are defined in terms of the average bulk densities & , & , and ,j in the following way:

where p0 is the density of standard nuclear matter which is related to r,, by p0 = (@Os)-l. The equilibrium values of these variables for a particular nucleus are given by

L(z+;$g )/( 1 + $5 A-‘/3),

1 c=.-- K ( -2a,A-f/3 + L82 + Cl -&). (6.2)


In terms of these variables the nuclear radii are given by

R = r&l”(l + <),

Rn = r,,(2l19’/~ (1 + B - QS),

Rz = r,,(2Z)1/3 (1 + < + -@),

and the neutron skin thickness t is given by

(6.3)

(6.4)

The total energy is given by

E = [--al + J@ + Q(KC2 - 2Ld2 + A4831 A + c,Z2A113

+ [a,(1 + 23 + QT~] A2j3 + a3A1i3

+ c1 w$ (1 - Q + $7A-‘/a) - ~c,Z~A~/~ - c3 5 - c4 g . (6.5)

A somewhat more compact form of this equation, in which certain terms are combined, is provided by Eq. (2.22a).

Finally, if we consider only the nuclear part of the energy and if we substitute from Eqs. (6.1), (6.2), and (6.4) for 8, ?, and T-making a Taylor expansion of 8 before substituting-we get

E = --alA + u2A2j3 + (a3 - T) A1/3

+ J12A - ($; - k?&) 1ZA-W

L2 --;M)14A. 2K

This last expression is applicable only for very large A because we expanded the denominator in the expression for 8. The main purpose Eq. (6.6) is to exhibit the intrinsic structure of the Droplet Model approximation in terms of a hierarchy in A-lJ3 and P.

After once again cautioning the reader that the approximate Droplet Model coefficients employed here are not expected to give the best possible results when applied to experimental nuclear properties, we would like to give a sample calculation using the above formulae. Consider, for example, a nucleus where N = 126, Z = 82, and A = 208. The total asymmetry I of this nucleus is given by

Z = 0.2115,


and the bulk asymmetry 8 is given by (6.1) which becomes

8 = (0.2115 + 0.01543)/(1 + 0.6643)

= 0.1363.

The first term in the numerator of this expression is Z, the second is the contribution to the bulk asymmetry from the Coulomb energy which is forcing protons from the central region of the nucleus. The second term in the denominator is the one mentioned earlier when we discussed the possibility of making a Taylor expansion of (6.1). Clearly, since this term is not small compared with the leading term in the denominator, such an expansion is not accurate, Note, too, that the commonly made assumption that the bulk asymmetry 8 is equal to the total asymmetry Z is quite poor, since (in this case) 8 is only 64 % of I.

The amount of dilatation is proportional to <, where for the nucleus considered here the expression (6.2) becomes

< = -0.02125 + 0.00779 + 0.01327

= -0.00019.

The first term is the compression due to squeezing by the surface. The second term is the dilatation due to the fact that asymmetric nuclear matter has a smaller equilibrium density than standard nuclear matter, and the last term is the Coulomb dilatation. These terms almost cancel, leaving only a small compression.

Substituting c and 8 into Eqs. (6.3) results in the predicted radii,

Thus

R = 7.138 fm, R, = 7.264 fm, R, = 6.894 fm.

R, -R, = 0.370 fm.

This value of the neutron skin thickness agrees approximately with Eq. (6.4) which gives

t = 0.358 fm.

Finally consider the Droplet Model energy given by Eq. (6.5), which becomes

E = [-3260.8 + 108.4 + 0.0 + 0.1 + 0.1 + 5.91 + [651.6 - 0.2 + 49.6 + 55.31 + [834.2 - 11.8 - 27.2 - 32.91 MeV,

= -3146.3 + 756.3 + 762.3 MeV, = - 1627.7 MeV.

484 MYERS AND SWIA'IECKI

The first bracket contains the volume terms; its first member is the volume energy, the second term is the symmetry energy, the third the compressibility correction (small because < is small), the fourth is a small contribution from the coupling of the asymmetry and the equilibrium density, the fifth term is a higher-order symmetry energy term and the last is the specifically nuclear part of the redistribution energy. The second bracket contains the surface energy terms; the first is the basic surface energy, the second is a correction for the decrease in surface area as a result of compression of the bulk, the third is a correction for the presence of the neutron skin, and the last is the curvature correction. The last bracket contains terms associated with the Coulomb energy; the first is the basic Coulomb energy; the second is a correction for the redistribution of particles caused by the Coulomb forces, the third is the surface diffuseness correction, and the last is the Coulomb exchange correction, The sum of the terms in each bracket is shown in the second equation above, and the last equation gives the total energy as -1627.7 MeV. This is a binding energy of 7.83 MeV per particle. We must once again caution the reader against comparing these numbers with an actual nucleus. The coefficients in these calculations have not been readjusted to fit the experimental data.

B. M&s FORMULA

Equation (6.5) (or 2.22a) is the Droplet Model mass formula. A discussion of the refinements included in the formula can best be given on the basis of Eq. (6.6), which is itself not suitable for numerical applications because of the poor convergence of one of the expansions used in deriving it. The nuclear part of the energy equation which is given in Eq. (6.6) has been arranged so that the new terms arising from the Droplet Model can readily be identified. These are the terms proportional to A1i3, Z2A2/3, and Z4A. Note that each of the coefficients of these new terms consists of two contributions. .1 (

The term in Al/s has sometimes been referred to as the curvature correction to the surface energy. The coefficient of Ali3 consists of two parts, the first being u3, which is, in fact, the contribution from the curvature correction. The second ljart of this coefficient is -2a22/K, which represents the decrease in energy thut-results when the surface tension (proportional to a2) causes the surface area to decrease at the expense of compressing the nuclear interior (whose compressibility is K).

Even the coefficient a3 itself has its origin in two physical effects, a fact ‘that has sometimes been overlooked. When discussing how this coefficient is determined from the Thomas-Fermi Model we found that two separate calculations were required. The first was an integral over a function F’ which gave the change in the surface energy coefficient to be expected because of the increased exposure of the surface particles when the surface is curved. This contribution amounted to 28.76 MeV. The second contribution to a3 was from an integral over the function nF


which takes into account the fact that surface curvature actually reduces the number of particles in the surface and hence reduces the effective surface energy coefficient. This term contributed - 19.42 MeV to a3 , with the final result that a3 = 9.34 MeV. By substituting the values of the coefficients into the expression for the coefficient of Ali3 we have (9.34-2.34)A1i3 MeV, or 7.00 Alla MeV.

An informative example which shows how the two contributions to a3 may cancel exactly is provided by Preston in Ref. (33). In a calculation of the energy of a spherical nucleus with a sharp surface, using a square-well interaction between nucleons, Preston found a volume energy, a surface energy and a term proportional to A”, but the curvature correction (a term in A1j3) was absent. An analysis of Preston’s calculation shows that this is due to an exact cancellation of the two effects described above. The important lesson of this is that in discussions of the nuclear curvature terms no significance should be attached to calculations in which one of the above two physical effects is disregarded.

The coefficient of 12A2j3 consists of two contributions. The first term, -(9/4)(J”/Q), represents the decrease in energy that takes place when the bulk region of the nucleus (whose symmetry energy coefficient is J) decreases its asymmetry 8 by increasing the neutron skin thickness t against the resistance provided by the effective surface stiffness (whose coefficient is Q). The second term, (2a,L)/(K), describes the change in the volume symmetry energy produced by a change in the bulk density. (The compression of the bulk by the surface tension is proportional to a,/K, and the rate of change of the symmetry energy with bulk density is proportional to L.) By substituting the values of the coefficients into the expression for the coefficient of Z2A2/3 we have -(110.5-15.5) 12A2j3 MeV, or - (95.0) 12A2j3 MeV.

The coefficient of 14A consists of two terms. The first term, -(L2)/(2K), represents the fact that the symmetry energy of nuclear matter depends on the equilibrium density (through L). The equilibrium bulk density itself depends on the asymmetry 6 through L (making L2), and the change in the equilibrium density is resisted by the compressibility K. The second term, (1/2)M, is the coefficient of s4 in an expansion of the bulk symmetry energy about standard nuclear matter. Substitution of the numerical values into the coefficient of 14A yields -(25.88 - 1.34) Z4A MeV, or -(24.54) Z4A MeV.

In earlier discussions we have pointed out that the Droplet Model and the Thomas-Fermi Model both indicate that a substantial reassessment is required of conventional ideas regarding the nuclear symmetry energy. The basic defect of the conventional approach is the implicit assumption of the Liquid Drop Model that the average bulk asymmetry 8 is equal to the total asymmetry I. This assumption causes a misinterpretation of the sign of the coefficient of 12A2/3 in the mass formula, and causes the symmetry energy of nuclear matter inferred from the mass formula to be incorrect.


If a Liquid Drop Model mass formula, modified to include a term in Z2A2j3, is fitted to experimental masses, results like these are obtained for the symmetry energy:

(28.062 - 33.222 A-1/3)Z2A MeV, Myers and Swiatecki, Ref. (I)

(30.586 - 53.767 A-1/3)Z2A MeV, Seeger, Ref. (34). (6.7)

[Myers and Swiatecki constrained the coefficients in expression (6.7) to have the same ratio as their volume and surface energy coefficients; Seeger determined both independently.] Since these authors made the implicit assumption that 8 = Z and made no reference to a neutron skin, their result that the coefficient of Z2A2/3 is negative gave the impression that the dependence of the surface tension on composition is negative. This interpretation is quite incorrect. We have shown that the dependence of the surface tension on composition is contained in the coefficient Q, and Q is positive. The surface energy does not decrease with asymmetry, it increases. (This is the same result as Wilets found in Ref. (35)) The coefficient of Z2A2/3, however, is negative, as we showed in an earlier paragraph, and this is in accord

FIG. 32. Effective symmetry energy coefficient as a function of A-1/3. The dashed lines represent empirical results obtained by fitting nuclear masses, and the shaded region represents ordinary nuclei. The solid curve passes through the shaded region and shows approximately what the Droplet Model would predict. The predicted symmetry energy coefficient of infinite nuclear matter is 35.08 MeV.


with the experimental results. The reason behind this apparent disagreement is easily explained: The bulk of the nucleus can decrease its energy by pushing neutrons into the surface and hence becoming less asymmetric. This increases the surface energy and decreases the bulk energy but the net effect is still a decrease in the total energy; hence the negative sign for the coefficient of Z2A2/3.

The Droplet Model (in combination with the Thomas-Fermi calculations) also shows that symmetry energy formulas like (6.7) give a poor determination of the symmetry energy coefficient of nuclear matter. Straight lines corresponding to the expressions (6.7) are plotted in Fig. 32 to illustrate this point. The shaded region on the right of the figure lies between these lines and gives an indication of the region occupied by ordinary nuclei. The Droplet Model predicts that the symmetry energy should have a form like the one in Eq. (5.6), and if we simply scale up this expression so that it will pass through the shaded region in Fig. 32 we find that the Droplet Model expression analogous to expressions (6.7) is

[(35.08 + 157.5 A-‘i3)/(1 + 3.936 A-1/3)2] Z2A MeV.

This result, which we feel more accurately represents real nuclei, is shown as a solid line in Fig. 32. The value it gives for the symmetry energy of nuclear matter is 35.08 MeV. This prediction is significantly larger than the 23.69 MeV found by Green, (36) using a conventional Liquid Drop Model mass formula with only an Z2A symmetry energy term. It is also larger than the 30.586 MeV found by Seeger (34). It is interesting to note that the larger value predicted here is in agreement with the value found in a recent work by Nestor et al. in Ref. (37), but it is not so large as the 43 MeV found by Falk and Wilets in Ref. (32).

C. REFITTING CONSTANTS

We have mentioned repeatedly that the set of nine physical coefficients used in this work is only a first approximation to the values that would best represent nuclei in terms of the Droplet Model. In the future we plan to redetermine these coefficients by fitting experimental masses with a Droplet Model mass formula.

In actual practice we do not expect that the data on nuclear masses are sufficient to permit a meaningful determination of all the coefficients. Consequently, it is our intention to use the Thomas-Fermi Model to extrapolate from the well determined coefficients to the poorly determined ones.

All the Droplet Model coefficients can be calculated from the Thomas-Fermi Model once the four parameters of the phenomenological interaction are specified. We plan to adjust these four parameters until we obtain a set of coefficients which best reproduces nuclear masses when used in the Droplet Model mass formula.

In the past we have emphasized the desirability of including fission barriers when


adjusting mass formula coefficients to fit the experimental data (I). The determination of these barriers involves calculation of the energy of nonspherical equilibrium shapes, a problem which is difficult even in the Liquid Drop Model (21,22). However, there appear to be no difficulties in principle involved in formulating the calculation either at the Thomas-Fermi or Droplet Model level of analysis. In the former a pair of integral equations similar to but more complicated than the Seyler- Blanchard equations would have to be solved for nonspherical (but still axially symmetric) shapes. In the latter case the calculation would be quite similar to that in the Liquid Drop Model, where the sum of the Coulomb and surface energies is made stationary with respect to shape variations of the boundary. In the Droplet Model the relevant energy would include corrections for curvature, compressibility, redistribution, and the presence of a neutron skin. (To leading order the diffuseness and exchange energies are independent of shape.) At the moment the unavailability of calculated Droplet Model fission barriers is the single serious obstacle to a refitting of the nuclear masses with a Droplet Model Mass Formula.

D. DENSITY DISTRIBUTIONS AND NUCLEAR SIZES

The stable equilibrium shapes of nuclei in the Liquid Drop Model are spherical, and their radii are given by R = r,,N3. In the Droplet Model the equilibrium shapes are also spherical, but the radii no longer correspond to the assumption made in the Liquid Drop Model that the central density of all nuclei is the same as p0 (the density of standard nuclear matter). Since nuclear compressibility is included in the Droplet Model, the radius of a particular nucleus is determined from the competition between the surface tension, which tends to decrease the radius, and the Coulomb repulsion, which tends to increase it.

Besides including nuclear compressibility the Droplet Model also recognizes the possibility of the creation of a neutron skin when the number of neutrons is greater than the number of protons. The existence of a neutron skin implies that the mean radius of the neutrons is greater than that of the protons, and that both these radii are different from the mean radius of the nucleus. The expressions (6.3) are the Droplet Model equations for these three radii, and equation (6.4) is the formula for the neutron skin thickness.

In Ref. (38) Johnson and Teller estimated that the neutron skin thickness would be l/3 to l/2 of the total surface thickness. However, we find in our calculations that the skin thickness of a typical heavy nucleus is about 0.36 fm, which is about 15 % of the total surface thickness. In Ref. (35) Wilets found (using a Thomas-Fermi method which differs in a significant way from the one used in this work) that the skin thickness was small, but he did not give a numerical value. Experimental evidence for the neutron skin is becoming available (39-41), and comparisons with the Droplet Model predictions will soon be possible.


Not only does the Droplet Model provide a means of calculating the neutron skin thickness, but it also permits calculation of the “isotope effect” in the nuclear charge radius. It is observed experimentally that the proton radius of a nucleus increases more slowly when neutrons are added than one would expect from the Liquid Drop Model relation R = r,A1fs, (30). The Droplet Model shows that this slow increase is to be expected, and permits one to make a quantitative estimate of the size of the effect. In a future work we plan to make a numerical comparison between the Droplet Model predictions and the experimental results (the experimental results show not only the smooth effects that are predicted here but also quite a bit of scatter because of shell effects).

The Droplet Model gives expressions for calculating the depth and the radial dependence of the central depression in the proton density caused by the Coulomb redistribution Eq. (2.16). Such a depression has always been expected and more experimental evidence for it is beginning to accumulate from electron scattering (30). More data are needed before a decisive comparison can be made with theory.

In the remainder of this section we discuss in detail the predictions that the Drop- let Model makes about the sizes of the charge distributions of finite nuclei. We illustrate these predictions by listing a calculation of sizes for two groups of nuclei. The first consists of the seven nuclei between &a40 and 83Bi20s whose charge distribution sizes were measured in 1956 by electron scattering experiments (47). The second group consists of the three isotopes of calcium, Ca4’J,44.48, for which size measurements were reported more recently (48).

We would like to confine our discussion of sizes to a single size parameter, which should give a sensible measure of the overall spatial extension of the charge distributions in question. Among the many choices of such a parameter (half-density radius, mean-square radius, effective sharp radius) we find it in many ways most satisfactory to choose a number proportional to the Coulomb energy integral extended over the charge distribution, in other words a number proportional to the double average of the reciprocal of the distance between pairs of points in the distribution:

7 = s.f drd h2) rii’ d3rl d3r2 12

.f.f &I) p(r2) d3rl d3r2 '

The ratio of such an average to its value for a sharp uniform spherical distribution of radius r,gP is, of course, just the ratio of the nominal Coulomb energy of the charge distribution to the Coulomb energy of the standard sharp sphere. It provides an integral measure of the size of the distribution in question. (By “nominal Coulomb energy,” which we shall denote by 8, we mean the Coulomb energy without the exchange correction.)

The above measure of size is sensitive not only to the radius of a nucleus but also to the diffuseness of the surface and to bulk non-uniformities. Thus of two distribuT


tions with the same radius the one with a more diffuse surface has a lower Coulomb energy and is larger according to our measure. This is as it should be since diffusing the surface moves particles from smaller to bigger radii and thus increases the extension of the distribution. Similarly a central depression in a charge distribution moves particles from the center towards the surface and should be reflected in an increase of a measure of size. This is true of our measure, since the Coulomb energy is indeed lowered by a central depression.

The use of the nominal Coulomb energy d as a measure of size is also convenient in our case since an explicit formula for d is available in terms of A and 2. Thus, using Eq. (6.5), we have

d = I.21 m$ (1 - g + &A-V) - 2c,Z2A1P - ca q

= gL (1 _ < + &A-1/3 - 2 2 A2/3 - 2 A-zj3),

where gr,, equal to c&Z~/A~/~, is the nominal Coulomb energy of the standard sharp spherical drop. (The suffix L stands for “Liquid Drop.“)

Substituting numerical values for the coefficients we find the following explicit formulae for calculating the measure of size d for a nucleus

d = gL(l - c + &7A-l13 - 0.0004126 A2i3 - 1.1715 A-2/3),

where

& = 0.717 Z2/A113,

c = Cl + E, + e3 ,

Cl = -0.1259 A-1/3

2, = 0.419082

e3 = 0.002432 Z2/A413

8 = (I + 0.01676 Z2/A5/3)/(1 + 3.9364 A-‘j3) +A-113 = +J-113 + &72A-1/3

+T~A-~/~ = 1.3122 SA--1/3

j$~~A-l/~ = -0.005588 Z2/A513.

From Eq. (6.8) we see that there are four principal effects which modify the size of the charge distribution of a nucleus. The term c describes an overall dilatation or compression. The term &A-l/3 may be considered as describing a differential compression of the protons (differential with respect to the neutrons) by half the neutron skin thickness. The term (2c2/cl)A213 gives the fractional increase in our


TABLE V

ANALYSIS OF NUCLEAR SIZES

1 2 3 4 5 6 7 8 9 - -

80 - bs c t1o-w3 c/All3 gs &L Diffuseness Io __--

Is

fm fm fm MeV MeV MeV MeV 03 -- ~~-

&a40 3.64 2.5 1.0643 78 83.86 -12.11 71.75 -8.01 23Vb1 3.98 2.2 1.0732 100 102.28 - 12.56 89.72 - 10.28 ZKO 58 4.09 2.5 1.0506 130 134.26 -14.97 119.29 -8.24 &115 5.24 2.3 1.0775 360 354.01 -25.29 328.72 -8.70 j1Sb’22 5.32 2.5 1.0726 380 376.02 -25.82 350.20 -7.84 ,9Au’8” 6.38 2.32 1.0965 790 769.05 -38.37 730.68 -7.51 s3Bi208 6.47 2.7 1.0902 840 832.32 -39.92 792.40 -5.67

10 11 12

Overall compression -

- i, -E2 -c

(%) (%I (%)

&a4” 3.681 -0.002 -0.711 2.968 0.0211 -0.0394 -0.0183 23Vb1 3.395 -0.121 -0.680 2.594 0.1697 -0.0377 0.1321 &069 3.234 -0.100 -0.772 2.362 0.1547 -0.0427 0.1119 JrP 2.589 -0.338 -1.044 1.207 0.2842 -0.0578 0.2264 51Sb122 2.539 -0.414 -1.045 1.080 0.3146 -0.0579 0.2566 ,QAlF 2.164 -0.680 -1.324 0.160 0.4029 -0.0733 0.3295 s3Bi208 2.122 -0.743 -1.351 0.028 0.4209 -0.0748 0.3462

13

-6

(%I

14

t1

fm

15 16

Neutron skin -~___~ -~

tz t fm fm

17 18 19 20 21 22 23 - _

Differential compression Redistribution Diffuseness

J,71,‘-W &&l/3 &A-1/3 - ~c,Z~A’/~ -cc,Z2/A

(%I (%) Cd) MeV (%I MeV (%I -- __- __--.--

0.256 -0.478 -0.222 -0.40 -0.48 -8.40 - 10.02 1.899 - 0.422 1.478 -0.58 -0.57 -8.71 -8.52 1.649 -0.456 1.193 -0.84 -0.63 - 10.39 -7.74 2.425 -0.493 1.932 -3.45 -0.97 - 17.53 -4.95 2.632 -0.484 2.147 -3.82 -1.02 - 17.90 -4.76 2.874 -0.523 2.350 -10.74 - 1.40 -26.61 -3.47 2.943 -0.523 2.421 -12.09 -1.45 -27.69 -3.33

59sl55/3-7


TABLE V (continued)

24

Correction to

&L

(%I

25

&D

MeV

26 27 28

Total Compression plus

&D - IS compression redistribution

8s

(%) 0 0

d.3 40 -7.75 77.36 -0.82 2.746 2.266 29V51 -5.02 97.15 -2.86 4.072 3.502 &05Q -4.82 127.79 -1.69 3.555 2.925 48 I n 116 -2.78 344.17 -4.41 3.139 2.169 S1Sb’2a -2.55 366.43 -3.57 3.227 2.207 ,9AU’9’ -2.36 750.90 -4.94 2.510 1.110 t&l ‘209 -2.33 812.89 -3.23 2.449 0.999

29 30 31 32 33 34 35 36 37

8s-B; Is-cfs’ 8, - 8; cfD - CFP; R R/All3 “T’ ds’ ___ - -__

&g’ Z2e2r3/R4 8-g ___

8’$ s;

fm fm MeV MeV (%) MeV fm MeV (%I (%)

&a40 3.9323 1.1498 87.88 72.75 5.97 17.02 77.71 0.37 -0.45 2*V61 4.1870 1.1291 109.15 96.32 3.37 17.02 99.80 0.20 -2.66 Q,C058 4.3501 1.1174 144.78 124.42 3.85 14.86 130.46 -0.35 -2.05 asIn116 5.4119 1.1128 383.29 353.76 1.63 15.58 360.22 -0.06 -4.46 61Sb’28 5.5200 1.1129 407.09 371.53 2.08 16.40 379.84 0.04 -3.53 ,gA~19’ 6.5236 1.1211 826.51 781.99 0.97 15.84 790.14 -0.02 -4.97 ,,3Biao9 6.6632 1.1226 893.35 827.08 1.45 15.97 840.13 -0.02 -3.24

measure of size caused by redistribution of change. The last term gives the increase in size due to diffuseness.

The overall compression term has been further split up into three contributions, denoted by <I , c2 and c3 . These are (in order) the surface tension compression, the asymmetry dilatation, and the Coulomb dilatation.

The differential compression $TA-~/~ has been split up into two parts, the differential compression due to asymmetry (&J-l/“) and the differential dilatation due

to the Coulomb repulsion (+7ZA-1/3).


Table V summarizes the result of applying the above formulae to seven nuclei listed in Column 1. Columns 2-9 give some background information on these nuclei. Thus Column 2 gives the experimental radius parameter c in a Woods- Saxon form factor and Column 3 gives the 10-90 fall-off distance, as determined in Ref. 47. Column 4 gives c/All3 and Column 5 gives the nominal Coulomb energy 8s , as deduced in Ref. (47) from the experimental charge distributions. (S stands for Stanford.) Column 6 is the nominal Coulomb energy 4 of a sharp Liquid Drop with radius r&i3, where r,, = 1.2049 fm. Column 7 lists a nominal diffuseness correction based on a standard 10-90 fall-off distance of 2.4 fm. The sum of Columns 6 and 7 is the Coulomb energy used in our old mass formula (I). It is denoted by 8o , the sufhx 0 standing for “old.” Column 9 shows the percentage deviation of go from the experimental value 8s. This is the “Coulomb radius discrepancy” noted in Ref. (I).

Columns 10-12 give the three contributions to the overall compression --i, which is listed in Column 13. (A positive number means a compression, a negative number a dilatation.) Columns 14 and 15 give the two contributions (in fermis) to the neutron skin thickness. Their sum is listed in Column 16. Columns 17-19 convert these numbers into differential compressions of the charge distributions (expressed as percentages). Column 20 gives the change in the Coulomb energy due to charge redistribution, and Column 21 converts it into a percentage (of &) and thus gives the increase in our measure of size due to redistribution. Column 22 is the diffuseness correction in MeV and Column 23 converts it into a percentage increase of size due to diffuseness. Column 24 adds up all the percentage changes in our measure of size. When this percentage correction is applied to 8L we find 8,, , given in Column 25, which is the nominal Coulomb energy corresponding to our Droplet Model (with radius parameter r,, = 1.2049 fm). Column 26 gives the percentage difference between J$ and the Stanford value 8s.

Note the following features of the results. The Droplet energy& is always smaller than 8L . This is seen to be due to the dominant role of the diffuseness correction. Thus, taken us a whole, finite nuclei are always more tenuous than nuclear matter. (Their central regions might be denser than nuclear matter, but this is more than offset by the rarefied surface region.) The effect of diffuseness on size is most pronounced for light nuclei; for heavy nuclei the diffuseness effect is more nearly comparable with the opposing compressive effects.

The overall compression is biggest for light nuclei. The differential compression is biggest for neutron-rich, and thus (usually) heavy nuclei. The two effects together add up to a very roughly constant compression of between 3 and 4% to 24% between Ca and Bi (Column 27). The redistribution effect is of opposite sign and varies from --$ to l+ ‘4. The sum of compression and redistribution effects generally decreases from 2-3 for the lighter nuclei to 1 % around Bi (Column 28). This counteracts to a certain extent the effect of diffuseness which varies from


-10 to about -3 % (for the diffuseness parameter used in our model, which is smaller than the experimental diffuseness).

As regards the light which our analysis throws on the question of the Coulomb radius discrepancy we must be careful not to jump to the conclusion that the smaller deviations of 8n from 6s) as compared with the deviations of go from 8s) are evidence that the Droplet Model removes, to a large extent, this discrepancy. As stated before, the Droplet Model coefficients have not been readjusted to nuclear masses and fission barriers. It is only after such a re-adjustment has been made that it will be possible to formulate the question of the presence or absence of a Coulomb radius discrepancy at the Droplet Model level of analysis. Calculations of nuclear sizes made with the Droplet Model which use the unadjusted value r, = 1.2049 fm have a significance as regards relative but not absolute values.

When making comparisons of the Droplet Model Coulomb energies with the Stanford values &” in Column 5 we became concerned about the accuracy of the numbers for 8s given in Ref. (47). These numbers, as quoted, suggest an accuracy of two significant figures, not adequate for our purposes. We consequently subjected the values of 8s to a test of smoothness based on a comparison with an analytical formula for the energy of a slightly diffuse Woods-Saxon distribution:

p(r) cc (1 + e(7-c)lZ)-1.

Using Ref. (I), p. 57, we may write this energy as

&I = 2 Z2e2 n2 Z2e2 z 2 S

--- - ( 1

- 5R 2RR

to second order in z/R, or

bo = 2 Z2e2 n2 Z2e2 z 2 S

__--- - ( i 5R 2R R

(6.10)

to third order in z/R. In 8.. we have included the cubic term in an expansion of the energy in powers

of z/R. This term is proportional to (z/R)~ times the leading term, but the numerical value of k is unknown at the present time.

In the above equation, R is the effective sharp radius of the charge distribution. Its relation to the half density radius c is

R = c (1 + $ (f)2 + terms of order (fj4).

This may be shown by demanding that the integrals

s R

4m2 dr 0


and

s m 47rr2 dr(1 + e+c)/*)-l 0

be equal. (One treats R - c as a small quantity.) Column 29 lists the values of R and Column 30 gives R/A1/3. (Note that R/A1i3 is more nearly constant than c/A’/“.)

Column 31 gives the leading term in the Coulomb energy:

&p = 3 Z2e2 5-F’

Column 32 gives 8s’ and column 33 lists

We note that the second order formula 8s’ appears to be inaccurate by several percent. In order to see if the deviations have the form of the cubic term in Eq. (6.10) we have listed in Column 34 the value of gs - &’ divided by

Z2e2 z 3 RR’ i 1

If Eq. (6.10) is a good approximation this quotient should be constant. This is seen to be remarkably well satisfied and we thus find an average (empirical) value of 16.1 i 1 for k.

Column 35 shows 8; obtained with this value of k, and column 36 gives the percentage difference (8s - Si)/Si which is in most cases less than 4 %. A large part of this difference is presumably due to the rounding error in 8s . We note that if &i is rounded to the nearest integer the Stanford values 8s are reproduced exactly. Thus we conclude that the values of 8s quoted in Ref. (47) have the appearance of two figure accuracy only in virtue of the one-in-a-million coincidence that all the three-digit entries happen to end with zero.

It would appear also that our formula for 8: is very accurate for elements around and above calcium. For discussing relative values it is even preferable to the Stanford values 8s (which have been rounded to the nearest integer).

In order to extend further the convenience provided by an accurate analytical formula for the Coulomb energies associated with experimental charge distributions of the Stanford type, we have generalized Eq. (6.10) to include distributions with a central depression. This is specified in the Stanford analysis by a parameter w as follows:

p(r ) cc (1 + 4r/c>“> .

1 + e(T-c)/z


We found the following approximate formula for the Coulomb energy of this distribution:

&,, = 3 Z2e2 1 + (8/7)w + (l/3)w2 r2 Z2e2 -7 2 ’ + w

s --- -

5 R 11 + (3/W12 2 R ;;, 1 + (3/5)w ( 1 + 16.1 7 (f)“.

The first term is the exact energy, obtained by elementary integration, of a sharp distribution proportional to 1 + w(~/R)~ for r < R. The second term is the diffuseness correction generalized to non-uniform distributions: as might have been expected the modification of the standard diffuseness correction is by the factor (1 + w)/(l + &v), which is the ratio of the new (sharp) density at the surface to the old. The last factor is the semi-empirical third order correction, whose dependence on w is disregarded.

For numerical applications we may write this semi-empirical formula (the coefficient 16.1 is empirical) as

&‘“(MeV) = 0 86394 2” ’ + (8/7)w + (l13Jw2 _ 7 1056 2222 . Lt!%- + 16.1 2223. s R [l + (3/5)w12 * R3 I+ (3/5)w R4

(6.11)

We have used e2 = 1.43990 MeV fm; R and z are in fermis. When using this formula it is necessary to recalculate the relation between R and c, z in order to include w. Demanding that the integrals

s

a dr r2 1 + +4r/c)2

0 1 + e(r-c)/z

and

J

R

dr r2[1 + w(r/c)2] 0

be equal, we find the following relation correct to third order in z/c:

1 + 2w rr2 z 2 R=c[lf 1+ w 3 (--) + terms or order (z/c)41

= c [ 1 + 3.2899 1+2w z2

~ 1+w

(11 - c

(6.12)

In Table VI we applied the Droplet Model to three isotopes of Ca. Columns 2-8 give the experimental properties of these nuclei. Column 9 is the Liquid Drop Coulomb energy and Column 10 shows the percentage changes in the Coulomb energy in going from Ca 4o to Ca44 and Ca48. The decreases of 3.12 and 5.89 % reflect the increase of the radius according to A113. In contrast, the experimental


TABLE VI

SIZES OF CALCIUM ISOTOPES

1 2 3 4 5 6 7 8

c ff,

W c/A113 t10-so R R/A”” fm fm fm fm

aoCa40 3.602 0.576 0 1.0532 2.531 3.9049 1.1418 2. c a a4 3.6805 0.5664 0 1.0425 2.489 3.9672 1.1237 .&a@ 3.6719 0.5281 0.079 1.0104 2.321 3.9401 1.0841

TAVLE VI (continued)

9 10 11 12

MeV (%) MeV (%I ~ .___._~ ~- -.

83.86 0 77.95 0 81.24 -3.12 77.23 -0.92 78.92 -5.89 77.92 -0.04

TABLE VI (continued)

13 14 15 16 17 18 19 20 21 22

Diff. Comp. Skin Comp. Redistribution Diffuseness Corr. to

i t &A-‘/3 -2~~i?/A’/~ -cc,Z2/A &L A&D

8, --

&&40) -. -

03 fm (%I MeV 02) MeV (%) (PO MeV c4 - -_ --- .~-~-~.---_ ~~_

&@ 2.968 -0.0183 -0.222 -0.40 -0.48 -8.40 -10.02 -7.75 77.36 0 &aP4 2.842 0.1195 1.405 -0.42 -0.52 -7.64 -9.40 -5.67 76.63 -0.94 &a4” 2.604 0.2381 2.719 -0.43 -0.54 -7.00 -8.87 -4.09 75.69 -2.16


TAVLE VI (continued)

23 24

Size anomaly

Experimental Droplet (%I (%I

- - 2.20 2.18 5.85 3.13

percentage changes, deduced using Eqs. (6.11), (6.12) and shown in Column 12, indicate a much smaller increase in size for Ca4” and a negligible (0.04 %) increase in size for Ca48. The remaining columns in Table VI show the Droplet Model results analogous to, though less detailed than, those in Table V. Column 20 shows the total Droplet Model correction and Column 21 gives &D . The Column 22 gives the Droplet Model prediction for the percentage changes in the size parameter (or Coulomb energy) &D . For Ca44 the prediction is of an increase in size of 0.94 % (compared with a Liquid Drop increase of 3.12 % and an experimental increase of 0.92 %). For Ca48 the prediction is of an increase of 2.16 % compared with an increase of 5.89 % according to the Liquid Drop Model and an experimental increase of 0.04 ‘A. We may say that with respect to the Liquid Drop Model there is a size anomaly of (3.12-0.92) = 2.20% for the pair Ca44-Ca40, and that the Droplet Model appears to account for this effect. In the case of the pair Ca48-Ca40 the size anomaly is (5.89-0.04) = 5.85 % and the Droplet Model accounts for between one-half and two-thirds of this figure. This is shown in Columns 23 and 24.

Finally we checked the size anomaly for the Ca4*-Ca40 pair by using the most recent experimental measurements reported in Ref. (49). These are as follows:

for Ca40. for Ca481

c = 3.6685, z = 0.5839, w = -0.1017 c = 3.7369, z = 0.5245, w = -0.0300.

Using Eq. (6.11) and (6.12) we find &;(40) = 77.73 MeV, &:(48) = 78.42 MeV and &‘&(40) = 0.89 %. Hence the size anomaly for this pair is (5.89 + 0.89) = 6.78 %, close to the old value of 5.85 %.

The above discussion illustrates the importance of the neutron skin and the associated differential compression of the charge in comparisons involving the isotopes of an element. The big difference in the Liquid Drop and Droplet Model predictions as regards isotopic sizes is due to the growth of a neutron skin as neutrons are added. Column 14 shows this increase and Column 15 gives the result-


ing percentage effect of the differential compression of the protons. In the case of Ca48 the total of the compression and redistribution effects reaches the relatively high total of 2.60 + 2.72 - 0.54 = 4.78 %. This reduces considerably the 8.87 % increase in size due to diffuseness (Column 19).

We conclude that the Droplet Model which includes compression, redistribution and neutron skin effects accounts for part of the anomalous behavior of the sizes of the calcium isotopes but it does not give a quantitative description of the experimental findings for the Caa-Ca40 pair. We discuss the question of a proper comparison of Droplet Model predictions with experiment in the next section.

E. REMARK ON COMPARISONS WITH EXPERIMENT

Since shell effects may introduce fluctuations in quantities of interest such as radii, skin thicknesses, central depressions, isotope effects, etc., it would be meaningless to attempt an evaluation of the Droplet Model by a comparison of the predictions with measurements on a few nuclei, such as Ca, Au, or Pb. A few isolated measurements will not be sufficient because the average effects one wants to isolate have to contend with considerable “noise” from shell effects.

On the other hand, even when oscillations in the value of a quantity of interest are larger than the average statistical effects one wishes to study, the problem is by no means insoluble if enough data are available over a large region of the periodic table. Thus even if the shell fluctuations in nuclear masses or radii should turn out to be comparable to or greater than, say, the average charge dilatation or redistribution effects predicted by the Droplet Model, this would be no reason to give up the attempt to confront the theory with experiment. What is needed is a sufficiently extensive body of experimental data.

An analogy that serves to bring out this relation between average nuclear properties and shell-effect fluctuations is provided by the topography of the earth, in which the average shape (a somewhat oblate spheroid) is modified by sharp local fluctuations (mountains). On the one hand the determination of the earth’s radius and oblateness from a few local triangulations, especially in mountainous country, would be wellnigh impossible. On the other hand, given methods that perform suitable averages over the sharp fluctuations of the earth’s surface (such as experiments with artificial satellites), one is able to determine the shape of the earth with great accuracy, including higher harmonics. Thus the pear-shaped component of the earth’s figure may be measured quite accurately even though its amplitude is much less than the amplitude of the fluctuations associated with local mountain ranges.

In summary, then, a comparison of the Droplet Model with experiment will be possible only when sufficient data have been accumulated on the particular nuclear property of interest, but, once this is achieved, the comparison should be meaningful and informative.


F. THOMAS-FERMI MODEL

In Section IV we discussed the Thomas-Fermi treatment of a two-component saturating system. The fundamental equations of this treatment, specialized to spherical shapes, are Eqs. (4.43) and (4.48), which are slightly generalized forms of the Seyler-Blanchard equations (16). We have discussed some of the solutions of these equations in Sections III and IV. Here we would like to point out that the Thomas-Fermi treatment of neutrons and protons interacting by means of short-range velocity-dependent forces (and additional electrostatic forces between protons) defines a concrete nuclear model that can be studied exhaustively under all conceivable conditions, including situations beyond the realm of nuclei found in nature. All that is required to achieve this is a (numerical) solution of the well-defined and apparently well-behaved equations (4.43) and (4.48). We shall mention some examples of situations we have not investigated, but which might be worth studying.

It is possible to remove the specialization of the Seyler-Blanchard equations to spherical symmetry. The solution of the generalized equations would be much more difficult but would enable one to study deformed configurations, in particular saddle-point shapes for fission from which the associated fission barriers could be calculated. Such studies might also throw some light on reactions between heavy ions and nuclei in which the geometry of the nuclear system at the moment of contact is not spherical.

Another problem that might be of interest and which the Thomas-Fermi method would also help to clarify is the nature of extremely neutron-rich systems. Even if, as we have reason to believe, neutron matter is unbound, what are the most neutron- rich systems that would still be bound and what do their charge and matter distributions look like? Would the proton and neutron distributions have comparable radii, or would the protons tend to form a much smaller sphere within the neutron sphere ?

It is also known that for a sufficiently highly charged nucleus (idealized as a charged drop) a bubble would form at the center as a result of the electrostatic repulsion (24,25,42,43). This problem could be investigated by solving the Seyler-Blanchard equations for neutrons and protons contained between concentric spherical surfaces. An interesting question about such nuclear cavitation, which cannot be answered within the liquid drop treatment but which the Thomas-Fermi Model could answer, is whether a proton bubble would perhaps appear before a neutron bubble, or, more generally, how the charge density would differ from the mass density.

The investigation of the possible existence and stability of extremely large hollow nuclear systems-nuclear “soap bubbles” with atomic dimensions-would call for a generalization of the Seyler-Blanchard equations to include the effects of the


electronic cloud that would in practice surround the hollow nucleus. (The effect of the electrons on the energy and stability of the system would become important for such large nuclei (43).) In that case the study of a three-component system (neutrons, protons, and electrons) would be necessary, using a coupled system of the Seyler-Blanchard and atomic Thomas-Fermi equations. The question whether such systems could possibly be of practical interest would be clarified in this way.

We mention these speculations because, even though they appear to have only a small chance of being of practical interest, they are not quite without interest and their study seems to be a rather straightforward undertaking, given the well- behaved nature of the Seyler-Blanchard equations.

Coming back to more concrete problems that have not been discussed in this paper, but which could be dealt with by the Thomas-Fermi method, we should mention the distribution of the angular momentum of the neutrons and protons in a nucleus. Just as in the case of the atomic electrons, the Thomas-Fermi method is capable of describing the average relation between the particle number and the angular momentum of the quantum orbits that are being filled (44,45). Such predictions might be of use in the extrapolation of nuclear properties to superheavy nuclei and to regions away from the valley of stability.

G. OPTICAL POTENTIALS

Any derivation of the nuclear optical model potential from two-body forces is a difficult many-body problem, and so no special claims are made for the simple potential we obtain in the Thomas-Fermi Model. This is especially true since the two-body interaction which is used in the Thomas-Fermi Model is a phenomenological velocity-dependent force empirically determined so as to reproduce average nuclear ground-state properties. Its velocity dependence is used to simulate the short-range repulsion of the real interaction and to provide an approximate treatment of exchange effects. Still, even with these deficiencies, it is interesting to examine the space and energy dependence of the nuclear potential predicted by the Thomas-Fermi treatment.

This potential V(r, p) is given by Eq. (4.5a), or, in dimensionless form, by Eq. (4.7). It is linear in the kinetic energy p2/2M and may be written as

L’ = V, + (K.E.) >: /L (6.13)

where V, and p are functions of position. The function V, is given by (b2/2M)v, , where

%(C) = - & I nuclear d3.$ exp[--l 5 - 4 11 J-p(g){1 - “Qyf)2

IC-51 5 I’

VOlllUl~


The function p is given by

In the semi-infinite limit v0 reduces to the expression given by Eq. (4.18a) and the function p reduces to

CL(~) = Y 1: dx Q expl-I 7 - x II Q”(x).

In addition to Eq. (6.13) we have the energy conservation condition

K.E. + V = T, (6.14)

where T is the kinetic energy of an (incident) particle outside the nucleus (where the potential is zero). Eliminating K.E. between Eqs. (6.13) and (6.14) we find

Figure 33 illustrates the behavior of this potential for various incident energies of the particle (neutron or proton). The depth of 38 MeV for an incident energy T = 0 seems about right, but the potential changes sign and becomes repulsive at a lower energy (about 62 MeV) than expected (46).

I 1 I ’ I / I I 1 I

20 - T=SOMeV -

T-62.16MeV .-

FIG. 33. The effective potential of symmetric semi-infinite nuclear matter as a function of the distance from the mean location of the surface of the density distribution responsible for the potential. The potential is shown for various values of the incident energy T.


As regards the spatial distribution of the potential, it should be noted that the potentials in Fig. 33 are plotted relative to the mean location of the surface of the density distribution. It is easy to see from this figure that the mean location of the surface of the potential lies substantially outside that of the density. This is also an observed property of experimental optical-model potentials.

The Thomas-Fermi Model potential for neutrons or protons has an easily calculated dependence on the bulk asymmetry of the target. This could be used as a guide in formulating optical-model potentials which include this effect. Probably the most significant point that can be made regarding the determination of the composition dependence of optical-model potentials is one related to an effect we have mentioned a number of times before. This is the fact that 6, the central asymmetry of a nucleus, is very poorly represented by the total asymmetry I. Attempts to introduce a dependence on Z into the optical potential may lead to confusing results because of the poor correspondence. This difficulty can easily be removed by the use of Eq. (6.1).

VII. SUMMING UP

We have presented a refined theory of average nuclear properties. In doing this we have attempted to be comprehensive in treating all effects entering to the same order in small quantities. Such a comprehensive theory is provided by the Droplet Model, which gathers together many effects whose interrelations were not originally evident. Together with the Thomas-Fermi statistical theory, which we investigated in some detail, the Droplet Model provides a compact framework within which to discuss phenomena intermediate in magnitude between Liquid Drop Model and microscopic effects.

In some respects the theoretical developments outlined here are ahead of the experimental results. In the Droplet Model we have isolated five new coefficients describing nuclear properties beyond the four standard coefficients of the Liquid Drop Model. Present experimental information on these coefficients is very limited and uncertain. Perhaps one of the most important future tasks, both for experiment and for theoretical developments like this one, is the improved understanding of these coefficients.

Two reasons for investing some effort in this direction are the intrinsic information about nuclear systems that these coefficients contain and the possible importance of such refinements for extrapolating to unfamiliar situations, such as very heavy or very neutron-rich nuclei. However, the consequence of the Droplet Model which may be the most important one in the near future is the improvement in the values of the four basic Liquid Drop Model coefficients that result when Droplet Model refinements are included in the mass formula. These four quantities (which


are volume, surface and symmetry energies and the nuclear radius constant) are of central importance in the comparison of theories of infinite or semi-infinite nuclear matter with experiment. We have already pointed out that the Droplet Model indicates that a significant change is required in the currently accepted value of the symmetry energy of nuclear matter. Another question of considerable interest is whether refitting of nuclear masses (and fission barriers) by use of the Droplet Model will remove the serious discrepancy between the nuclear radius constant obtained from electron scattering experiments and the one obtained from nuclear masses (I).

Finally, we hope that by extending the treatment of average nuclear properties to order N3 we will have contributed to the synthesis of the macroscopic and microscopic approaches to nuclear structure.

ACKNOWLEDGMENTS

We wish to thank James R. Nix and Chin-Fu Tsang for many helpful discussions. The assistance of Stanley G. Thompson and the members of his Fission Group is gratefully acknowledged.

This work was done under the auspices of the U. S. Atomic Energy Commission.

RECEIVED: March 27, 1969

REFERENCES

1. WILLIAM D. MYERS AND WLADYSLAW J. SWIATECKI, Nucl. Phys. 81, 1 (1966). 2. V. M. STRUTINSKY, Shell Effects in Nuclear Masses and Deformation Energies, Nucl. Phys.

A95,420 (1967). 3. Nuclides Far Off the Stability Line, Proceedings of the Lysekil Symposium, 1966, Arkiv Fysik

36 (1967), see Sessions VII and XI. 4. WILLIAM D. MYERS, Toward a Thomas-Fermi Mass Formula, in “Proceedings of the Third

International Conference on Atomic Masses, Winnipeg, 1967.” 5. JAMES RAYFORD NIX AND WLADYSLAW J. SWIATECKI, Nucl. Phys. 71,1(1965); see Appendix A. 6. JAMES WING, A Comparison of Nuclidic Mass Equations With Experimental Data, Argonne

National Laboratory Report ANL-6814, Jan. 1964. 7. JAMFS WING, Systematic Comparison of Semiempirical Nuclidic Mass Equations, in “Proceed-

ings of the Third International Conference on Atomic Masses, Winnipeg, 1967.” 8. P. GOMBAS, Acta Phys. Hung. 1, 329 (1952); 2, 223 (1952); 4, 267 (1954); 5, 511 (1956). 9. N. V. FINDLER, Nucl. Phys. 8, 338 (1958).

10. LAWRENCE WILETS, Rev. Mod. Phys. 30, 542 (1958). Il. A. R. BODMER, Nucl. Phys. 17, 338-420 (1960). 12. YASUO HARA, Progr. Theor. Phys. (Kyoto) 24, 1179 (1960). 13. W. MACKE, P. RENNERT, AND W. SCHILLER, Nucl. Phys. 86, 545 (1966). 14. M. A. NAQVI, Nucl. Phys. 10, 256 (1959). 15. KAILASH KUMAR, K. J. LECOUTEUR, AND M. K. ROY, Nucl. Phys. 42, 529 (1963).


16. R. G. SEYLER AND C. H. BLANCHARD, Phys. Rev. 124,227 (1961); Phys. Rev. 131, 355 (1963). 17. V. M. STRUTINSKII AND A. S. TYAPIN, Zh. Eksperim. i Teor. Fiz. (U.S.S.R.) 45, 960 (1963)

[Engl. Transl.: Soviet Phys-JETP 18, 664 (1964)]. 18. A. S. TYAPIN, Yadern. Fiz. 1, 581 (1965) [Engl. Transl.: Soviet J. Nucl. Phys. 1, 416 (1965)]. 19. H. A. BETHE, Phys. Rev. 167,879 (1968). 20. K. A. BRUECKNER, J. R. BUCHLER, S. JORNA AND R. J. LOMBARD, Phys. Rev. 171,1188 (1968). 21. STANLEY COHEN AND WLADYSLAW J. SWIATECKI, Ann. Phys. 22,406 (1963). 22. V. M. S~UTINSK~I, N. YA. LYASHCHENKO, AND N. A. POPOV, Zh. Eksperim. i Teor. Fiz.

(U.S.S.R.) 43, 584 (1962) fJZng1. Transl.: Soviet Phys.-JETP 16, 418 (1963)]. 23. H. A. BETHE AND R. F. BACHER, Rev. Mod. Phys. 8, 82 (1936). 24. W. J. SWIATECKI, Ph.D. thesis, University of Birmingham, England, Oct. 1949. 25. W. J. SWIATECKI, Proc. Phys. Sot. (London) A63, 1208 (1950). 26. W. J. SwIAlXCKt, Phys. Rev. 83, 178 (1951). 27. ROBERT MATHEWS (Theoretical Group, Lawrence Radiation Laboratory, Berkeley), personal

conversations. 28. W. J. S\NIATECKI, Proc. Phys. Sot. (London) A68, 285 (1955). 29. GRANINO A. KORN AND THERESA M. KORN, “Mathematical Handbook for Scientists and

Engineers” McGraw-Hill, New York (1961). 30. H. R. COLLARD, L. R. B. ELTON: AND R. HOFSTALITER, “Nuclear Radii,” Vol. 2 New Series,

Numerical Data and Functional Relationships in Science and Technology Springer-Verlag, New York (1967).

31. R. A. WEISS AND A. G. W. CAMERON, Abstract CD2, Bull. Am. Phys. Sot. 12, 41(1967). 32. DAVID S. FALK AND LAWRENCE WILETS, Phys. Rev. 124, 1887 (1961). 33. M. A. PRESTON, “Physics of the Nucleus” p. 124, Addison-Wesley, Reading, Massachusetts

(1962). 34. PHILIP A. SEEGER, A Model-Based Mass Law, in “Proceedings of the Third International

Conference on Atomic Masses, Winnipeg, 1967.” 35. LA~XENCE WILETS, Phys. Rev. 101, 1805-9 (1956). 36. A. E. S. GREEN, “Nuclear Physics” McGraw-Hill, New York (1955), especially Chapt. 9. 37. C. W. NESTOR, JR., K. T. R. DAVIES, S. J. KRIEGER, AND M. BARANGER, Nut. Phys. A113,

14 (1968). 38. M. H. JOHNSON AND E. TELLER, Phys. Rev. 93, 357 (1954). 39. DENYS H. WILKINSON, Comments on Nuclear and Particle Physics 1, 80 (1967). 40. DENYS H. WILKMSON, Comments on Nuclear and Particle Physics 1, 112 (1967). 41. CLYDE E. WIEGAND AND DICK A. MACK, Phys. Rev Letters 18, 685 (1967). 42. P. J. SIEMENS AND H. A. BETHE, Phys. Rev. Letters 18, 704 (1967). 43. W. J. SWIATECKI, Super-Heavy Nuclei, unpublished talk given at the Research Progress

Meeting, Feb. 13, 1968, Lawrence Radiation Laboratory, Berkeley. 44. E. U. CONDON AND G. H. SHORTLEY, “The Theory of Atomic Spectra” Cambridge University

Press, England (1935). 45. P. GOM~AS, Die statistiche Theorie des Atoms und ihre Anwendung (Springer, Wien, 1949). 46. F. BJORKLUND, Phenomenological Analysis with Spherical Potentials, in “Proceedings of the

International Conference on the Nuclear Optical Model, Tallahassee, 1959.” 47. B. HAHN et al., Phys. Rev. 101, 1131 (1956). 48. K. J. VAN OOSTRUM et al., Pkys. Rev. Letters 16, 528 (1966). 49. J. B. BELLICARD et al., Proc. Znternational Conference on Nuclear Structure, Tokyo 1967, p. 539.

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