On the stationarity of the nuclear-surface energy

Nuclear Physics A422 (1984) l-l 1 @ North-Holland Publishing Company

ON THE STATIONARITY

OF THE NUCLEAR-SURFACE

M. FARINE*

ENERGY+

and J.M. PEARSON

Laboratoire de Physique Nucltfaire, LXpartement de Physique, Unioersite’ de Montrkal, Montrial, QusZbec, Canada H3C 337 Received

17 November

1983

Ahstrati We show that the specific surface energy of semi-infinite nuclear matter is stationary with respect to the limiting central density only in the charge-symmetric case. The failure of the “6 = 0” theorem in the asymmetric case is demonstrated numerically within the context of the-scaling model, and formally within the context of a very general constrained energy-density method. The non-zero value of 6 is shown to be related to the existence of the neutron skin.

1. Introduction The surface properties of finite nuclei are conveniently studied by referring to the system known as semi-infinite nuclear matter [see ref. ‘), for example, and papers quoted therein]. This consists of a medium which in two directions, n and y, say, is of infinite extent and has constant neutron and proton densities, n, and nP, respectively, i.e.

%l,p(r) = n”+(z) 7 but has a well-defined surface perpendicular fall to zero for large negative values of 2:

(1.1)

to the z-axis, so that both densities

lim n,(z) = lim n,(z) = 0. z-r-m

z-b-c.3

(1.2)

.

On the other hand, for large positive values of z the local properties of the medium approach those of infinite nuclear matter with any assigned values of the neutron and proton densities: lim n,(z) = ncn , I-cc

(1.3a)

lim I~J z) = nCP,

z-m ’ Work supported * Present address:

June 1984

by NSERC of Canada. Physics Department, Lebanese

University,

Mansourieh-El-Metn,

Lebanon.

M. Fake,

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energy

These limiting densities deep below the surface define the asymmetry I according to %Il =$n,(l+l),

(1.4a)

ncp=In,(l

(1.4b)

--I) ,

where n, is the limiting value of the total density n(z)=n,(z)+n,(z).

(1.5)

When the semi-infinite system is in equilibrium, with no external forces acting, we have %= n,W > (1.6) where no(I) is the saturation density of infinite nuclear matter for the given asymmetry I. The value of this in terms of the saturation density n,, of symmetric infinite matter is found by expanding the energy per nucleon e&n, I) as follows: e,(n,I)=e,(no,,I=0)+J1*+ir?sK?1*+3L771*+...,

(1.7)

where rl-

--

~---OO

(1.8)

P

no0

and .F, L and K are respectively the volume-symmetry, symmetry-density, and incompressibility coefficients of infinite nuclear matter. Then minimizing e,( iz, I) with respect to n we have (3.9) The surface energy per unit area of the semi-infinite system is then given by 4,

n,) =

* {E( z; II,, I) - e&n,, 04~; I -03

n,, 01 dz,

(1.10)

where %(z; n,, I) and n(z; n,, I) are respectively the local energy and number densities for the given asymmetry I and limiting density la,. The calculation of the energy of a finite nucleus by a macroscopic model such as the droplet model 2*3),and in particular the determination of the saturating value of the central density n,, requires a knowledge of the variation of u with I and n,. This knowledge is also required for a study of the giant isoscalar monopole resonance, the so-called breathing mode. The variation of the specific surface energy with n, and I can be conveniently parametrized by making an expansion similar to that of eq. (1.7): a(l,n,)=ao+u7)+b1*+c~*+dr11*+...,

(1.11)

where W*=W(I=O,n,=noo). Central to the development

(1.12)

of the droplet model is the statement that u=o,

(1.13)

M. Farine, J.M. Pearson / Nuclear-surface

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i.e. in symmetric semi-infinite nuclear matter u is stationary with respect to n, at the equilibrium density, n ,,,,. This is referred to as the “& = 0” theorem and a proof has been offered by Myers and Swiatecki *). Alternative proofs were also given in refs. 4V5). The present paper is concerned with the question as to whether or not this theorem is also valid for the asymmetric case, I # 0. It is to be particularly noted that the proofs of refs. 4,5) were limited to the I = 0 case, while one has the impression that the proof of ref. *) was intended to be more general. From (l.ll), (1.12) and (1.13) we have au

0an,

=$(2c7+&).

(1.14)

I

Thus at the saturation density n,,(l) we have from (1.9)

d-(-$_=( -~n,,ii+~)l’, where we have introduced

(1.15)

the surface incompresibility &r

a*@ 2c 2 I=0 =&’ nE=nw

(>

(1.16)

Now this latter quantity is known to be never zero: from an analysis of the breathing mode it is clear that it is essentially negative [see, for example, ref. 6)], and this is borne out by explicit Hartree-Fock calculations “). It follows that if the c? = 0 theorem holds in the asymmetric case then d cannot vanish, i.e. in the expansion (1.11) there will have to be a term linear in q for the asymmetric case, even though it vanishes for I = 0. On the other hand, if d does vanish then the 6 = 0 theorem will not hold for I # 0 (at least, not at the equilibrium density, no(l)). We shall see in the present note that we have both failure of the & = 0 theorem in the asymmetric case, and non-vanishing of d. This is shown explicitly in sect. 2, where we present the results of scaled Hartree-Fock calculations, generalizations of those of ref. “). Considerable insight into these results is obtained in sect. 3, where we examine the problem within the framework of the constrained energy-density formalism, as developed for semi-infinite nuclear matter in ref. ‘). Some implications of our results are discussed in sect. 4. 2. Scaling calculations To determine ti for any given value of I it is necessary to calculate the semi-infinite system for densities different from the equilibrium value no(l). The simplest way to do this is by scaling.

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M. Farine, J. M. Pearson / Nuclear-surface

energy

TABLE 1 Force S3 (see text for explanations)

I2

no(I) (fm-‘)

0 0.04 0.06

0.1453 (=noo) 0.1447 0.1445

cr(n,, I)

(MeV fin-‘)

kdn

-g a(n,,

1.07 1.17 1.21

1)

-0.01 0.10 0.15

In ref. “) we have performed scaled Hartree-Fock calculations on semi-infinite matter for the symmetric case, I =O. Here we extend these calculations to the asymmetric case, for the same two Skyrme-type forces, S3 [ref. ‘)I and SkM [ref. 8)]. The generalization is quite straightforward and we refer to ref. “) for details. We present the results for the two forces in tables 1 and 2, respectively. In the second column of each table we show the equilibrium values of the density, n&1). These follow eq. (1.9) very closely with the independently calculated values of the macroscopic parameters L and K (respectively 9.9 MeV and 355 MeV for S3, 49.3 MeV and 217 MeV for SkM). As for &, we present it in the form of the dimensionless parameter shown in the last column of each table. Clearly, while the 6 = 0 theorem holds very well for I = 0 it manifestly breaks down in the asymmetric case. The deviation from zero of this parameter lies well outside the range of numerical error, and in fact it follows the linear dependence on 1’ indicated by eq. (1.15). However, using the values of d: given in ref. 4), it is found that only a small part of this dependence comes from the surface compressibility term: the bulk of it comes from the second term, with d = 2.5 MeV * fm-’ for S3 and 3.6 MeV . fm-’ for SkM. 3. Energy-density

method

with constraint

To gain some insight into this non-vanishing of &, we generalize to the asymmetric case the energy-density formalism developed in ref. ‘) (sect. 2) for symmetric semiinfinite nuclear matter under constraint. TABLE 2 Force SkM (see text for explanations)

0 0.04 0.06

ncdn

no(I) Urn-7

a( n,, I) (MeV. fm-‘)

c dno, 0

0.1603 ( = noo) 0.1558 0.1535

1.06 1.21 1.30

0.03 0.19 0.26

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M. Farine, J.M. Pearson / Nuclear-surface energy

Taking as independent

variables the total density n(z), defined in eq. (1.5), and q(z) = n,(z) - n,(z) 7

(3.1)

rather than n,(z) and n&z), the energy density 8 appearing in (1.10) has the form 8( n, q, n’, q’), where primes denote derivatives with respect to z, and all arguments are functions of z, IZ, and I. Then minimizing the total energy, subject to the constraint that neutrons and protons must be separately conserved, we have the variational principle 6

m {gJYn,q, n’,q’)-en-hq}dz=O, I --m

(3.2)

where E and A are Lagrange multipliers. We choose now for the energy density the rather general expression

%(n,4, II’, s’>=f(fi, 4) +g(n’, 4’)

(3.3)

with arbitrary f and g (the principal results that we shall derive hold for a still more general choice). The Euler-Lagrange equations now become (3.4a)

(3.4b) In order to handle the limit of infinite nuclear matter it will be convenient express f in terms of n and i, where i(z) =$J,

to

(3.5)

rather than in terms of n and q. We note that lim i(z)=I, L-m

(3.6)

where I is the asymmetry defined by eqs. (1.4). In place of eqs. (3.4) we now have (3.7a) (3.7b) [It must be realized that we have the same quantify f in both (3.4) and (3.7), so that the functional form of f expressed in terms of n and i will be different than for n and q. Note also that it is not possible to write a variational principle directly in terms of n and i, since unlike q there is no conserved quantity associated with i.]

M. Farine, J.M. Pearson / Nuclear-surface

energy

In the limit z + co, deep below the surface, eqs. (3.7) reduce to

(> 0ii” = af iii af

=&+zA,

(3.8a)

no(Z

(3.8b)

Now the energy per nucleon in infinite nuclear matter is e,(n, I) =g

(3.9)

so that (3.8a) becomes

(2 1

e,(n,Z)+n

=&+zA.

(3.10)

I

Then at the saturation density no(Z) we have e&no, I) = E+ZA,

(3.11a)

=2JZ. rql

(3.11b)

where from (3.8b) and (1.7)

Eqs. (3.11) may be regarded as one form of the theorem of Hugenholtz and Van Hove generalized to the asymmetric case 12). If the semi-infinite system is to have a limiting density II, different from no(Z) a constraint s(n) must be added to the energy density. We have then in place of (3.3) K!(n, 4) = @n, 4) + s(n) 9

(3.12)

whence eqs. (3.7) become (3.13a) (3.13b) where S = ds( n)/dn . In the nuclear-matter

limit the energy per nucleon in the constrained e,(n,Z)=e,(n,Z)+s/n.

Then equilibrium will be given by

(3.14) system is (3.15)

M. Fake,

J.M. Pearson / Nuclear-surface

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It will then be a trivial matter to choose s(n) in such a way as to have saturation at any required density n = n, [see ref. “)I. At the same time eqs. (3.13) reduce in this limit to

(> df

ii

i=,

+s=&+zA

(3.17)

in place of (3.8). Turning now to (1.10) for the surface energy and using (3.3), we find

(3.18) Writing f

=f(n(z, 4,0,

i(z, n,, 0) (3.19)

‘f(470, we have ($)I

=(Z)i(g)I

+(S)“(E), (3.20)

Also, since g=

g(n’, 4’)

7

(3.21)

we have

(3.22) Integrating this term by parts, we see that it can be replaced by

8

M. Furine, J.M. Pearson /

~u~~eur-~ur~~~e en-y

where we have used eqs. (3.13). Eq. (3.18) now becomes

(3.24) If now the constraint s(n) vanishes, so that n,= no(I), (3.24) reduces to

where we have used eq, (3.1 la). Next we recall that for semi-in~nite nuclear matter the neutron-skin thickness is given by (3.26) Using (1.4) this is easily seen to reduce exactly to (3.27) Then using (3.11b) we obtain from (3.25) our final expression:

(3-28) It will be seen that al# explicit reference to the constraint s(n) has vanished, so that the result will be valid for alI conceivable forms of the constraint, including the one that corresponds to scaling. We see from (3.28) that any departure from the ci = 0 theorem must be associated with the existence of a neutron skin. Indeed, b must be non-zero whenever there is a neutron skin, unless its thickness 8(nC, I) is always proportionaI to 8;‘. We now return to the scafing case, noting that (3.28) wiII be applicabie there. Since 8 will vary as E;“~ in any scaling model we have (3.29) whence (3.28) becomes ii=$H(l-1~)6(no,I>. The following general expression for the neutron-skin

(3.30) thickness, vafid for small f,

M. Farine, J. M. Pearson / Nuclear-surface

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is now to be noted:

where r,, is the charge-radius

constant, $lr;=-)

1

(3.32)

no0

and Q is the so-called surface-stiffness coefficient ‘). This result was first derived from the droplet model 2), but it has a much wider validity and can be derived within the framework of the above formalism, defining Q in terms of the coefficient b of eq. (1.11) according to b=

-- 9 J2 16mg Q.

(3.33)

Using (3.31), the scaling-model relation (3.30) becomes, for small I, (3.34) (note particularly that the validity of (3.31) does not depend on the scaling model). Comparing now with (1.15) we see that within the context of the scaling model the coefficient d appearing in eq. (1.11) is not independent of the other macroscopic parameters. However, applying eq. (3.34) to the scaling-model results of tables 1 and 2 enables us to extract a value of Q. With J = 28.2 and 30.7 MeV for S3 and SkM, respectively, we find the following values of Q: S3 : 45 MeV (52) ,

SkM : 28 MeV (32).

The figures in parentheses refer to the values obtained directly from the calculated surface energy [see ref. ‘) and ref. 4), respectively, for these values]. We thus have a very close check on the overall consistency of our entire development. 4. Discussion We have shown that whereas the surface energy of symmetric semi-infinite nuclear matter is stationary with respect to the limiting density at equilibrium, this is not so in the asymmetric case. This we have demonstrated numerically within the probably somewhat limited context of the scaling model, but the formal development of sect. 3 shows that our result has quite general validity. The most obvious implications of this result concern the I-dependence of breathing-mode energies, i.e. the so-called surface-symmetry incompressibility Ksr of Blaizot “). We shall examine this problem elsewhere.

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M. Farine, J.M. Pearson / Nuclear-surface

The droplet-model

description

but only if one goes beyond if one includes

of static properties

the present

the well-known

order

energy

of nuclei will also be modified,

of approximation.

surface-compressibility

term

More

explicitly,

cn2 of eq. (1.11)

[see

refs. “.‘l) for a discussion of this term], then it will also be necessary to include the d$* term discussed in this paper. However, we do not believe that pushing the droplet model beyond its present level of development is worthwhile: it becomes enormously complicated ‘O*“), and quantitative improvement can be more easily obtained through with Skyrme-type Of greater

the Skyrme-ETF forces.

significance

method,

is the intrinsic

i.e. the extended

physical interest

Thomas-Fermi

method

of the result that & vanishes

if, and only if, there is no neutron skin. A simple physical understanding of this observation would be most welcome. As a final conclusion, consider the change in surface energy of a spherical nucleus in the droplet-model description when its radius changes from R to R +dR. With the specific surface

energy

u no longer

a constant

we have

8rlRu+&rR*g

dEs=

dR >

= B?rR(a--~nb) Thus we can write for the change the surface area

in surface

dR. energy

(4.1) resulting

from a change dS in

dE,=GdS,

(4.2)

G= cr(l-$nb_la) .

(4.3)

where

While the surface energy per unit change in area is It is 6, and not a, that determines all measurable matter

in the same

experiments, pressure

per unit area is given by u the given rather by 6. Only when 6 is the true surface tension, in forces. If we could measure the

way that

then it is 6 rather

at the center

chemists

measure

that

change in surface energy = 0 will u and G be equal. the sense that it is 6 that surface tension of nuclear

of liquids,

than v that would be determined.

of our nucleus

e.g. by capillary In any case, the

will be given by P=26/R,

(4.4)

with G replacing the usual u. This distinction between surface tension and specific surface energy is perhaps the most important consequence of the breakdown of the ci = 0 theorem. And from the last columns of tables 1 and 2 we see that the distinction is by no means negligible. [Note, however, that in deriving eq. (4.3) we assume that the change in surface area of the finite nucleus arises from a change in its volume, spherical symmetry being conserved. It is, of course, possible to change the surface area of a nucleus

M. Farine, J.M. Pearson / Nuclear-surface

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without any volume change at all, simply by deforming it; in this case we would have a = 6, assuming no curvature effects.] The Centre de Calcul of the Universitd de Montrtal is thanked for its continuing support. References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12)

M. Farine, J. C&t&and J.M. Pearson, Nucl. Phys. A338 (1980) 86 W.D. Myers and W.J. Swiatecki, Ann. of Phys. 55 (1969) 395 W.D. Myers and W.J. Swiatecki, Ann. of Phys. 84 (1974) 186 J.M. Pearson, M. Farine and J. Cat&, Phys. Rev. C26 (1982) 267 M. Farine, J. C&e, J.M. Pearson and W. Stocker, Z. Phys. A309 (1982) 151 J.P. Blaizot, Phys. Reports 64 (1980) 171 M. Beiner, H. Flocard, Nguyen van Giai and P. Quentin, Nucl. Phys. A238 (1975) 29 H. Krivine, J. Treiner and 0. Bohigas, Nucl. Phys. A336 (1980) 155 M. Farine, J. CBtC and J.M. Pearson, Phys. Rev. C24 (1981) 303 J.M. Pearson, Nucl. Phys. A376 (1982) 501 F. Tondeur, J.M. Pearson and M. Farine, Nucl. Phys. A394 (1983) 462 M. Farine, these doctorale, Universite de Montreal (1982); L. Satpathy and R. Nayak, Phys. Rev. Lett. 51 (1983) 1243