Curvature term for the shell energy of deformed nuclei

Nuclear Physics A144 (1970) 545--557; ( ~ North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher

CURVATURE TERM FOR THE SHELL ENERGY OF D E F O R M E D N U C L E I H. SCHULTHEIS, R. SCHULTHEIS and G. S ~ S S M A N H Institut fiir Theoretische Physik der Universitiit Frankfurt, 6 Frankfurt]M, Germany ? Received 26 February 1969

(Revised 8 December 1969) The shell energy of arbitrarily deformed nuclei is approximated for nuclear fission theory as a function depending on the curvature of the nuclear surface. Determining the minimal potential energy which is derived from the liquid-drop model modified by this shell term, intrinsic quadrupole moments Qo, shell, and binding energies of about 50 nuclear ground states are calculated and compared with experimental values.

Abstract:

1. I n t r o d u c t i o n

When developing a theory of nuclear fission, it is important to calculate the shell energy as a function of nuclear shape. Formulations of such a kind have already been proposed in numerous papers 1- s). In recent calculations of barrier energies, Myers and Swiatecki 6-8) successfully made use of a mass folmula with a shell-energy term, which depends on nuclear shape for small deformations and disappears as the nucleus is distorted away from the spherical shape. This method seems to be unsuitable for an investigation of the asymmetric fission o f heavy nuclei, as it would completely neglect shell energies during the formation of nuclear fragments near the scission point, which is the phase most critical for their mass ratio. Therefore, in this paper we propose an equation, which approximates the shell energy of the constituent parts of the nucleus during its increasing constriction. Thereby we hope the influence of magic numbers on asymmetric fission of heavy nuclei will become intelligible. For this purpose, we describe the shell energy of deformed nuclei by a summation extended over all energy components of spherical nuclear sectors. The single summands are calculated according to the Myers-Swiatecki method s) for non-deformed nuclei. We approximate the shell structure of the nucleus locally by inscribed spheres, which are defined corresponding to the nuclear surface curvature (cf. sects. 2 and 3). The dependence of the liquid-drop energy on curvature has been assumed in numerous papers 9-14), and recently Strutinsky has developed a curvature correction 23, 24) of shell effects. A purely hydrodynamic mass formula does not allow any theoretical description of stable ground state deformations, which are definitively measured, Only a mass * Now at the Sektion Physik der Universitat Miinchen, Theoretische Physik, 8 Miinchen, Germany.

545

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H. SCHULTHEISet aL

formula extended by a shell energy favours a deformed shape contrary to a spherical one for m a n y nuclei. Therefore, as a test of the practical application of our shell term to nuclear fission, we calculate in this paper the ground state deformations by determining the minimum of the potential energy and compare masses and intrinsic quadrupole moments of about 50 nuclei with experimental values. 2. Local shell effect We reduce the shell energy of any deformed nucleus to that of spherical nuclei, adapting a suitable sphere to its surface at every point, so that the nucleons of the sphere locally determine the shell effect. Thus a surface density of the shell energy is defined; this is to be integrated (see fig. 1).

Fig. 1. Qualitative picture, which shows a deformed nucleus and one of the spheres inscribed to calculate its shell energy. The sphere fits the nuclear surface thus defining the number (N*, Z*, A*) of nucleons contained in it. In this way we approximate the whole nuclear surface by spheres, and determine the nuclear shell energy by a weighted mean of the Myers-Swiatecki shell function S (N*, Z*) of all inscribed spheres. As a loose approximation, which will be modified in sect. 3, we designate each sphere in such a way that its surface approaches that of the nucleus in the neighbourhood of the point at which the shell energy density is to be determined. Therefore, in an ideal case the curvature of the sphere should coincide with that of the nuclear surface; nevertheless this is attainable at umbilical points only [cf. for example ref. 15)]. Therefore in general, the curvatures of copunctal normal sections [cf. for example ref.15)] of the nuclear surface could only be reproduced on the average by one inscribed sphere, i.e. its radius is the mean value of the curvature radii of all normal sections. Since according to Euler's formula [cf. for example ref. 15)], the curvature of each normal section is uniquely defined by the two principal curvatures 1/R1 and l/R2; the radius R of the sphere, which approximates the nuclear surface, can be expressed by the radii of principal curvature R1 and R2. As numerical computations have

SHELL ENERGY OF DEFORMED NUCLEI

547

demonstrated, the results do not depend critically on the method by which the mean value R(R1, R2) is defined; because of the differential geometric properties of principal curvatures each average curvature of normal sections lies between these. Calculating the ground state deformation we choose in this paper the arithmetic mean R = ½(R1 + R2).

We have expressed the shell energy of the sphere with the radius R approximating the nuclear surface by means of the Myers-Swiatecki s) shell function S(N, Z). They describe the shell energy of spherical nuclei as a function of their neutron and proton numbers N and Z, which satisfy the relation

~F(N) +F(Z) S(N, Z) = C L (½A)g"

cA~rl.

Therein F(m) denotes

F(m) - 3 M~-M~_I ( m - M r _ l ) ---f(m 3 g- - M i _ l ) , 5 Mi-Mi_ 1 with m lying between the magic numbers Mi_ 1 and M~: M~_ ~ < m < Mi. The two parameters C and c are specified according to Myers and Swiatecki s) and also the magic numbers, some of which we have added by extrapolating 16) a harmonicoscillator potential with a spin-orbit coupling term. To calculate the shell energy of the spheres described above, first of all we determine the portion of protons and neutrons contained in the spheres. Assuming the density of the two kinds of nucleons as constant inside the nucleus, we find real numbers Z*, N* and A* associated with each of these spheres as their shares of protons, neutrons and nucleons. As a consequence of constant nucleonic density, the number of nucleons A* is to A as the ratio of the volume of the sphere (-~Tc)R 3 t o the total nuclear volume (~rOr3A. Thus A*, N* and Z* result from the nucleon numbers N, Z and A of the whole nucleus and from the curvature radius R of the sphere by solving the equations

A* = R3/r 3, N* = A*N/A, Z* = A * - N * . For a fixed nucleus N* and Z* can be expressed only as a function of the curvature radius R. Now we are able to state - as mentioned above - the surface density of the nuclear shell energy as a function r/(R) of its surface curvature, so that its total shell energy follows from this density by integration with respect to the nuclear surface E s h e l 1 ----

~ r/(R)dS;

~/(R) is assumed in such a way, that the curvature energy ~ / d S of the above-mention-

548

H. SCHULTHEIS et aL

ed sphere (with the radius R), which approximates the surface, agrees with the value of the shell function S(N*, Z*) of spherical nuclei according to Myers and Swiatecki 6, 7) -

S(N*,

Z*)

4~R 2 In this expression, the originally discrete variable N and Z are replaced by N* and Z*, which now are continuous functions of R and in addition of N, Z. 3. Total shell effect

In calculating the shell energy we have so far associated each single point of the nuclear surface with an energy density in accordance with the curvature at this point. Thus the curvature of the surface in the proximity of the point in question is without significance; i.e. for the shell-energy calculation, shell effects of surface points close to each other are said to be independent. This assumption certainly simplifies the real conditions; for example if one cuts a nucleus in half, then according to the existing formulation of the shell energy, the local shell structure of the curved part of a half-nucleus will be unchanged, though quantum mechanics certainly will yield different solutions in consequence of new boundary conditions. Based on this, a procedure quite contrary to the local shell-energy ansatz used up to now will be more appropriate; instead of associating a shell energy density with each point of the nuclear surface and then calculating its average, the shell energy is deduced from the average curvature of the whole surface. Instead of the ansatz (cf. sect. 2) Esh,ll = ~

S(N*(R),Z*(R)) dS, 4nR 2

the corresponding term then should be put in the form =

where -~ is taken as mean value of all curvature radii R of the nuclear surface. For we set ~=

~dS

dS,

as Strutinsky and Tyapin lo)and Hilf and SfiBmann ~ ) have formulated. This second ansatz simplifies the nuclear structure in a loose way, for the fission of heavy nuclei in approximate magic fragments indicates that there exists a certain independence of shell structure corresponding to parts of the nucleus which are still connected. For this reason a realistic ansatz of the shell energy should allow for either effect, the local and the total one.

SHELL ENERGY OF DEFORMED NUCLEI

549

This will be achieved for instance by the formulation Esho, = ~ S(N*(R), Z*(.q)) dS 4~cR2 with a function R(R1, R2, R) which has to be defined in order that its value lies between R and R. Because there is no information on how to choose this function, we have assumed for simplicity, a weighted mean of the values R and R for R, and have determined the weights by fitting the theoretical quadrupole moments Qo to the measured ones.

4. Binding energy On account of simplicity in papers investigating nuclear fission the nucleus is mostly assumed to be permanently axisymmetric. "/his restriction to solids of revolution is favourable in considering numerical calculations, since the Coulomb energy can be evaluated numerically with a minimum fuss by use of a parametric representation in cylindrical coordinates. Moreover, for solids of revolution, differential geometric relations of particular simplicity are valid. The lines of curvature especially, can instantly be delineated and therefore it is easy to evaluate the principal curvature. Following Lawrence's 17) parametrization Hasse, Ebert, and SiJl3mann is) derived a system of solids of revolution, clearly arranged for that purpose, which describes nuclear shapes of a wide variety by three parameters only: the surface of the nucleus is represented in cylindrical coordinates p, z, q~, the z-axis being the axis of the body. In addition, in describing ground states we confine ourselves to solids which are symmetrical about the origin. In this special case the surface of revolution is bounded by the algebraic curve

p (z)

=

(Z2o- :)(z22 + : ) ,

z

[-

Zo, Zo],

where 0~, Zo, Zz denote parameters. Because of volume conservation during nuclear deformations ~ is specified by z0, z2, and Rc = rcA~: g--

Sc Z 3 / 1 _2

oW~o + z2,)

z o is the distance from the origin to the end of the solid of revolution and Zz describes nuclear constrictions; if z z = 0, the origin is a zero of the function p(z). For z2 ~ oo the curve p(z) tends to an oval shape, and, especially, to a circle if z o equals R c. To specify the equilibrium deformation a nuclear shape has to be determined that is preferable to other shapes of the same volume by having minimal potential energy, i.e., one has to search for the minimum of its shape-dependent portion that is a function of the shape coordinates z o and zz. For that purpose the liquid-drop part of the energy has to be added to the curvature-shell term described in sect. 3. 1he corresponding terms are taken over partially from Myers and Swiatecki 6-s), the

550

H. $CHULTHEISet aL

approximations of the terms depending on the nuclear shape (surface and Coulomb energy) used by them are replaced by the accurate expressions. Then the binding energy has the following form: -

binding energy =

(i) volume energy with a symmetry energy portion + (ii) pairing energy + (iii) surface energy with a symmetry energy portion + (iv) Coulomb energy with a correction for a diffusion of the charge distribution + shell energy (cf. sect. 3).

(i) The volume term derived from the liquid-drop model is assumed to depend on the relative neutron excess according to the papers mentioned above. It has the following form:

where al and x are parameters. (ii) As pairing energy I 6 =

1~ MeV/A ½ for odd nuclei for odd-mass nuclei I - 11 MeV/A ~r for even nuclei

is added 19,20). These two terms are taken unchanged. (iii) The surface term involves a symmetry portion introducing two parameters a2 and x according to Myers and Swiatecki. For deformed nuclei the A~ term

has to be multiplied by the ratio of the nuclear surface area to a spherical one of the same volume. We have squared the nuclear surface area IS[ numerically according to Romberg's integration method 21). Then the surface term of binding energy yields Es = a2

1-re

A~

4nRc2

.

(iv) The electrostatic energy of a homogeneous charged solid of revolution is evaluated after the method used by Lawrence 17). Thereby the Coulomb energy is reduced to a triple integral, by which the numerical computations are essentially shortened, Moreover, Myers and Swiatecki 6, 7) add a correction to the Coulomb energy for a diffusion of the charge distribution near the nuclear surface

~2e2 (d)2Z2 2 rc

A-'

SHELL ENEROY OF O ~ F O ~ D NUCLEI

551

where the parameter d is an electrostatic measure of the unsharpness of the nuclear surface. According to Stanford electron-scattering experiments 0.5461 fm is inserted for d. So the Coulomb energy of homogeneous charged point-symmetrical nuclei together with this surface correction is given by pzo

pzo/z

Ec=4~a2Jo dzp2(z)ZJo dyp2(zY) ×

["

sin 2 • d~

30 141-y) l +(z2(1-y)2 + p2(z )+ p2(zy)_2o(z)o(zy) 2 rc

cos

-A-

where er is the constant charge density. According to Myers and Swiatecki a) we take the following values as parameters of our mass formula: in the liquid-drop part of the mass formula at = 15.4941 MeV for the volume energy; a2 = 17.9439 MeV for the surface energy; tc = 1.7826 for the symmetry term; r c = 1.2249 fm, and in the shell function of spherical nuclei C = 5.8 MeV and c = 0.325. As numerical calculations have shown, the ground state deformations are very insensitive to variations of the values of the shell parameters. Opposed to that the binding energy depends essentially on the value of c, which fixes the mean shell energy. But this is not significant for the determination of ground state deformations. Using the values listed above as fitted by Myers and Swiatecki 6), we obtain corresponding binding energies, the deviations of which from experimental data are even smaller than those they have computed.

S(N,Z)

5. Results

By loosely fitting the quadrupole moments of 24°pu, 232Th, 18"W, 168Er and s 6Gd to experimental values we obtained the weights of local and total shell effects. They amount to 0.3 for R and to 0.7 for R having an error of 0.05: R = 0.3 R+0.7/~. Thus the share of the total shell effect prevails. With these values quadrupole moments, shell, and binding energies of about 50 nuclei are computed. Thereby we confined ourselves to doubly even nuclei, since their experimental intrinsic quadrupole moments Qo can be determined accurately by applying the electric-quadrupole transition propability B(E2) between the 0 + ground state and the first 2 ÷ state. Fig. 2 shows the intrinsic quadrupole moments evaluated by us, and the corresponding experimental ones derived from B(E2) values, Stelson and Grodzins 22) have compiled. As the independent variable we have chosen the neutron number N,

552

H. SCHULTHEIS

et al.

1I

6O

6O

/00

120

I~0'

Neutron number N

Fig. 2. Calculated (dots on the solid line) and experimental (dashed line) intrinsic quadrupole moments Qo versus neutron number N. Data related to isotones are replaced by their mean. in order that the influence of magic numbers (82, 126) becomes evident. To make the plot clearer, values of the same kind are connected by lines, and data related to isotones are replaced by their mean. Experimental and theoretical values are in good agreement except for the neighbourhood of magic numbers (1.31 b mean deviation). There, near N = 82 and N = 126, experimental quadrupole moments involve the effect of single nucleons, by which deviations from (vanishing) quadrupole moments of spherical nuclei arise. In fig. 3 the theoretical shell energy is compared with the experimental one that designates the difference of the total energy and the energy of the liquid-drop model. In this diagram we have assumed experimental deformations to be those of spheroidal nuclei with the measured quadrupole moments. Moreover, in the lower half of the figure the difference of theoretical and experimental binding energies is plotted; these differ from the shell-energy deviation, because the liquid-drop energy corresponds to the theoretical deformation. 2 0

z ~

~

/o

' 4

'

I---,.

'

'

~ ~ -2

Fig. 3. A comparison of the calculated shell energy with the experimental one. The experimental deformations are assumed to be those of spheroidal nuclei. In the lower half of the figure the difference of theoretical and experimental binding energies (theor. --exp.) is plotted.

SHELLENERGYOr DEFO~D m.JCLE~

553

In this graphical representation of results the nuclear shell structure is conspicuous, for at the magic number N = 82 and the nearly magic number N = 124 one obtains (corresponding to the experimental curve), even for the theoretical shell energy, very low values; thus the particular stability of magic nuclei is correctly reproduced. The shell energy is well approximated all over the plotted N interval, valid for binding energy as well. The average error of the binding energies plotted is 0.66 MeV, and the maximum one is 1.94 MeV (22STh).

I0

Sm 152

O-L

-5

-~o-I0

'

-

. . 0. . .

'5

/o

5

z (fro)

Fig. 4. The calculated shape of a nucleus in the ground state.

Fig. 4 shows the shape of 152Sm in the ground state computed by means of our shell term. This nucleus demonstrates in a clear manner that spheroids often represent a rather inexact approximation of nuclear shape; the shape will be reproduced by a somewhat flattened boundary curve more accurately than by an ellipse.

•

~-0 ..~

\

60

,,

80 I00 120 Neutron number N

NO

Fig. 5. Deviations of binding energies (B=,~--Bc=~=) calculated in this paper (solid line) and those calculated by Myers and Swiatecki 6) (dashed line).

554

H. 8CHULTHEI8 e t al.

In order to show to what extent the remaining errors of our shell term are caused by the shell function S(N, Z) - which we have taken over from Myers and Swiatecki s) fig. 5 contains the deviations of the binding energies Bexp-Bea~o of the two papers from measured values, to which the zero line corresponds. The two curves run to some extent parallel except for the region of rare earths (from N ~ 90 to N ~ 120), the energy of which is approximated by the ansatz of Myers and Swiatecki 6) in a less accurate way.

-

.~- e ~-~ i

/o

' k

' ~o',io',~o

Neutron number N

Fig. 6. Deviations of quadrupole moments calculated in this paper (solid line) and those calculated by Myers and Swiatecki e) (dashed line).

In the same manner in fig. 6 the deviations of the intrinsic quadrupole moments Qo are plotted. In that graph too the errors of both papers have similar dispersions. That is why for the most part the remaining error of the shell term is probably implied by Myers' and Swiatecki's shell function of spherical nuclei S(N, Z) and could be diminished above all by a more suitable function. Following sect. 4 the nuclear ground state deformations hardly depend on the two parameters C and c of the shell function S(N, Z). Therefore a new fitting of these parameters will scarcely alter the quadrupole moments; at best a correction of binding energies seems possible. The same concerns (as shown in figs. 7 and 8 for 168Er and 206pb ) the parameters of the liquid-drop model. These contain in addition to the particular terms Coulomb energy Ec, surface energy E s, and shell energy EsheH also the total energy E and the sum of the two shape-dependent parts of the liquid-drop model E c + E s. All these curves are plotted as functions of the ratio of the semiaxes zo/Po of spheroidal nuclei (z2 = ~ ) . In any case the shell energy specifies the position of the minimum of the potential energy and therefore determines the deformations in the ground state, In this plot nuclei are assumed to be spheroidal to illustrate the course of energies; in quantative calculations of ground state deformations such nuclear shapes described by merely one parameter have turned out to be too simplified (cf. fig. 4). Because the ground states of numerous nuclei have been computed in a quite satisfactory way using the parameters of ref. a) a new fitting of these would not appear to justify the expenditure of computer time.

SHELL ENERGY OF DEFORMED NUCLEI

16

blate

8

prolate

- ' ~ "

o

,,

,,

0.8

555

--

I,

Y,

0.9 ,Z) I.I Deformation Zo/9o

-~_ Eshell

....

1.2

Y°

1.3

1.4

Fig. 7. Shape-dependent parts of the binding energy of Z6SEr: Coulomb energy Ec, surface energy Es, shell energy Esh=n, total energy E, and the sum Ec+Es. All plots are against the ratio of the semiaxes of spheroidal nuclei. (To make the plot clearer, suitable constants are added to the energies.)

zo/po

P b 206

Me2~-_-:4 .... -.~ ..... •

I

...

,/-",

~--

--

I ./_.. ,.~..-..~.3. /

X

Eshell

E

/ /

",,,

" ~ f e i',,\~'¢11/ + E s 12

•

8

"f

Ee,

0

0.8

0.9 1,0 I.I Deformation Z o / 90

1.2

1.3

1.4

Fig. 8. Shape dependent parts of the binding energy of 2°6Pb (cf. fig. 6). Recently, Strutinsky 2 3) has published graphs of deformation energies that are more similar to figs. 7 and 8 than are those of Myers and Swiatecki 6), though they are the result of a microscopic treatment: Strutinsky adds the energies of realistic Nilsson level diagrams to the liquid-drop energy, whereas we use the expression which Myers and Swiatecki 6) have determined for an ideal Fermi gas. Yet we replace the rough shell-damping function of Myers and Swiatecki, which ough t to describe the splitting of shells with deformation, by a mean over the surface. The newer version of this damping function 8) is more oscillatory, and also reproduces one secondary minimum of the shell energy, but it disappears monotonically with increasing deformation contrary to the Strutinsky shell correction. As figs. 7 and 8 show, our shell term seems to approximate the results of the microscopic theory more accurately: besides one secondary minimum that results from the characteristic Nilsson level splitting, as Swiatecki s) has pointed out, we obtain further minima in

S(N,Z)

556

H. SCHULTHEIS et aL

agreement with Strutinsky 23). Altogether, the energy dependence on deformation is m o r e oscillatory, similar to ref. 23), and distinct shell structures and relatively strongly b o u n d equilibrium deformations exist even in rather distorted nuclei (see ref. 2 4)). This m a y become i m p o r t a n t in the theory o f nuclear fission (cf. ref. 25)), because our a p p r o a c h would possibly permit one to reduce th~ calculational efforts. In addition, the transitions between spherical and distorted nuclear shapes o f fig. 2 are in g o o d agreement with Strutinsky's 24) results (especially the asymmetry o f the rare-earth quadrupole m o m e n t s due to the n o n - u n i f o r m filling up o f p r o t o n and neutron shells). Those disagreements o f Myers and Swiatecki on the one h a n d and Strutinsky on the other that are caused merely by the shell function S(N, Z), but not by the dependence on the deformation, could not be removed in this paper. A n example are the signs 23) o f the shell correction o f mid-shell nuclei. So far our treatment o f the deformability seems to diminish the most i m p o r t a n t deviations o f the semi-empirical m e t h o d f r o m the microscopic one with respect to the shell energy dependence on deformation. Whether our shell term will prove efficient also for strongly deformed nuclei, to which it can in principle be used, must be the result o f further calculations in the proximity o f the saddle and the scission point. We are grateful to Mr. J. M i n t o n for revising the English manuscript. The w o r k was in part supported by the Bundesministerium fiir Bildung und Wissenschaft.

References 1) N. Zeldes, Nucl. Phys. 7 (1958) 27 2) F. S. Mozer, Phys. Rev. 116 (1959) 970 3) H. Kiimmel, J. H. E. Mattauch, W. Thiele and A. H. Wapstra, Prec. Conf. on nucleidic masses, Vienna 1963, (Springer, Wien, New York, 1964) p. 42 4) W. J. Swiatecki, Prec. Conf. on nucleidic masses (idem.) p. 58 5) N. Zeldes, Prec. Conf. on nucleidic masses (idem.) p. 11 6) W. D. Myers and W. J. Swiatecki, University of California, UCRL 11980 (1965) 7) W. D. Myers and W. J. Swiatecki, Nucl. Phys. 81 (1966) 1 8) W. D. Myers and W. J. Swiatecki, University of California, UCRL 17070 (1966); Prec. Int. Syrup. Why and how should we investigate nucleides far off the stability line, Lysekil, Sweden (August 21-27, 1966) 9) D. L. Hill and J. A. Wheeler, Phys. Rev. 89 (1953) 1102 10) V. M. Strutinsky and A. S. Tyapin, JETP (Sov. Phys.) 18 (1964) 664 11) E. Hilf and G. Siil3mann, Phys. Lett. 21 (1966) 654 12) R. Schade, Universitat Frankfurt a.M., thesis (1968) 13) R. W. Hasse, Nucl. Phys. All8 (1968) 577 14) H. v. Groote and E. Hilt', Nucl. Phys. A129 (1969) 513 15) H. W. Guggenheimer, Differential geometry (McGraw-Hill, New York, San Francisco, Toronto, London, 1963) pp. 210, 211; D. Laugwitz, Differentialgeometrie (Teubner, Stuttgart, 1960) p. 48 16) G. Eder, Kernkrafte (G. Braun, Karlsruhe, 1965) p. 78 17) J. N. P. Lawrence, Phys. Rev. 139 (1965) B 1227 18) R. W. Hasse, R. Ebert and G. Sfi0mann, Nucl. Phys. AI06 (1967) 117 19) A. H. Wapstra, Handbuch der Phys. ed. S. Fliigge (Springer, Berlin 1958, vol. 38/1) p. l

SHELL ENERGY OF DEFORMED NUCLEI

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20) A. E. S. Green and D. F. Edwards, Phys. Rev. 91 (1953) 46 21) P. Henrici, Elements of numerical analysis (Wiley and Sons, Inc., NewYork, London, Sydney, 1965) p. 259 22) P. H. Stelson and L. Grodzins, Nucl. Data 1 No. 1 (1965) 21 23) V. M. Strutinsky, Nucl. Phys. A95 (1967) 420 24) V. M. Strutinsky, Nucl. Phys. A122 (1968) 1 25) S. Bjernholm and V. M. Strutinsky, Nucl. Phys. A136 (1969) 1