Fission saddle point calculations with the inclusion of shell effects

Volume

27B. number 3

PHYSICS

LETTERS

FISSION SADDLE POINT WITH THE INCLUSION OF

24 June 1968

CALCULATIONS SHELL EFFECTS

E. NARDI Israel

Atomic

Energy

and Weizmann

Commission:

Institute

Soreq

of Science,

Research

Y. BONEH Nuclear Research Center, Negev?

Center

Israel

Rehovot;

Israel

and E. CHEIFETZ Weizmann

Institute

of Science,

Received

Rehovot,

Israel

15 May 1968

Fission saddle point calculations which include a quantitative description of shell effects have been performed on the basis of the liquid drop model. The results show that relatively small shell corrections are decisive in causing asymmetric saddle point shapes.

Shell effects are generally believed to play a prominent role in the process of low energy nuclear fission, particularly in connection with the problem of mass asymmetry [1,2]. This communication contains a description of saddle point calculations of nuclei which are based on the liquid drop model and which contain a quantitative inclusion of shell effects. Since the connection between the saddle point shape and the mass distribution in the fission of heavy nuclei is still an open question, it will be difficult to reach any definite conclusion regarding the mass distribution. However, if one were to assume that the asymmetry of the nuclear shape is essentially unchanged during saddle to scission motion, an asymmetric saddle point would indicate asymmetric mass distribution. The configuration used here to describe the nucleus in the vicinity of the saddle point consists of two spherical clusters Al and A2 connected by a cylindrical neck, (fig. 1). The potential energy of such a configuration is in our model given by, E ‘ECoulomb

fEsurface

fEshell(A1)+Eshell(A2)

.

(1) The first two terms in the sum are the conventional liquid drop energies while the remaining two terms describe energies attributed to the shell structure of each of the spheres. The shell 144

d

Fig. 1. Schematic

representation of the model used in the calculations.

energy attributed to each sphere is equal to the liquid drop shell correction term of the liquid drop mass formula for a nucleus consisting of the number of neutrons and protons of the sphere. No shell correction is applied to that part of the nucleus which consists of the neck. A model which describes the nucleus as composed of two roughly spherical clusters connected by coercing nuclei has been previously suggested in Inglis [3] in connection with the problem of fission mass asymmetry. This model also accounts qualitatively for the behaviour of the odd rotational states of even-even nuclei in the region of Bi. Our calculations are based on the mass for-

Volume 27B. number 3

PHYSICS

mula of Myers and Swiatecki [4] in which the shell term is described by means of a function of Z and N which acquires both positive and negative values. The configuration of fig. 1 is described by three parameters. The parameters used here are, d the distance between the centers of the spheres, A1 the volume of the small sphere, and cy the ratio of the larger to smaller sphere (o = = AZ/A 1). The conservation of the volume of the drop defines uniquely the width of the cylinder. The energy of the system in the three parameters is determined by eq. (1). The calculations assumed constant Z/N throughout the nucleus. The electro-static energy was calculated by exact expressions except for the self-energy of the neck which has the form of a cylinder with concave ends. This self-energy was assumed to be equal to that of a cylinder of equal radius and volume as the neck. The self-energy of the cylinder was calculated by performing a double integration of Beringer’s disc-disc interaction values [5]. The coordinate describing asymmetry is (Y. Mathematical saddle point in the two dimensional space of Al and d at constant cy were calculated for different values of cy . These “quasi saddle points” which are thus termed since they do not describe the saddle point of the nuclear system, were determined in the manner reported by Lawrence [6] by using a saddle point search code. The saddle point of the three dimensional system is the point of minimum of the two dimensional quasi-saddle point versus QIcurve. The calculations were carried out with the aid of a C. D. C. 1604A computer. In order to verify whether results obtained in the present formulation agree with the more elaborate liquid drop calculations, saddle points without inclusion of the shell correction terms of eq. (1) were first calculated. The saddle point shapes obtained by this method for X = 0.6 and X = 0.7 agree with those of Cohen and Swiatecki [7] with respect to the length and maximum width of the drop, but fail to reproduce the shape of the neck. Fig. 2a contains a plot of the quasi-saddle points (without shell corrections) in the two dimensional space Al and d, as a function of 01 for various heavy nuclei. All the nuclei treated are observed to possess symmetric saddle points and their stiffness towards asymmetry tends to increase with X. Both these conclusions are in accord with Cohen and Swiatecki [7]. The energies of the saddle points are seen to

24 June 1968

LETTERS

52 48 44 40 , > 36 z 32, _ 28 24 20, IE

0 iI.1

I

1.4

1

I

1.8

I

I

2.2

I

I

1.0

I

1-1

1.4

I

1.8

I

I

I

2.2

Q Fig. 2. Quasi-saddle point energies as a function of the asymmetry 01for various heavy nuclei: a) calculations without shell corrections b) calculations with the inclusion of shell effects. decrease with increasing value of X as expected. They are, however, higher than those calculated by Cohen and Swiatecki for the same value of X, due to the constraints imposed on the shape by the low number of parameters. Although the shell correction terms are based on the Myers and Swiatecki mass formula the X values of the nuclei are those due to Green [8]. The X values based on the Myers and Swiatecki mass formula are higher than those of Green and due to the limitations of our model, saddle points could not be obtained for these higher values of X. In the remaining part of this letter we will deal with the results for the calculations which include the shell effects. The potential energy of the nuclear configuration is given here by the four terms of eq. (1) and the calculations were carried out in the manner described above. In fig. 2b are plotted the quasi-saddle point versus (Y curve for a number of nuclei between 252Cf and ‘OgBi. The saddle points of the nuclei are all in the range of LY= 1.5 to 1.6 hence they correspond to nuclei of asymmetric shape. The minima in fig. 2b are shifted to values of higher o! as the mass of the fissioning nucleus 145

I

Volume 27B. number 3

PHYSICS

24 June 1968

of the shell energy term. The attenuation factor is denoted in fig. 3 by f.The (Y values of the minima of the curves are essentially independent of the magnitude off. The values of A1 at the saddle point change less than 1% as a function of shell attenuation in the range of zero attenuation to 80% attenuation, while d changes by about 12%. It is also observed that asymmetric saddle points are obtained when the shell correction term is as low as 20’$0of its full value. The sum of the two shell correction terms in this case for cy = 1.6 is equal to -1.7 MeV while for (Y = 1 this sum is equal to 1.6 MeV. Hence it should be stressed that, relatively small values of the shell terms are decisive in causing asymmetric saddle point shapes. The main drawback in the calculations is the asymmetric saddle point of 2ogBi in fig. 2b, which may be due to the crudeness of our model. Calculations which employ a five parameter description of the saddle point shale are presently being carried out in order to resolve this problem and in order to obtain a fuller account of the saddle point configuration.

30

25

The authors are grateful to Professor Z. Fraenkel and Dr. J. R. Nix for their helpful criticism.

Fig. 3. Quasi-saddle point energies of 238U as a function of asymmetry N for different values of the shell

attenuation factor f.

decreases. If one were to associate the cy values of the minima in fig. 2b with the most probable mass ratios of the fissioning isotopes then the above shift in the values of a is in qualitative agreement with experimental results. The composition of the spheres is fairly close to the magic number spheres proposed by Faissner and Wildermuth [2]. The large sphere of 238U for example consists here of 79 neutrons and 52 protons. The effect of the attenuation of the shell correction term on the saddle point results was also investigated. This was accomplished by multiplying the shell terms in eq. (1) by attenuation factors in the range of 1.0 to 0. In fig. 3 are plotted the quasi-saddle points versus N curve for 238U for various attenuation *****

146

LETTERS

References 1. ,, .M. G. Mayer, Phys. Rev. ‘74 (1948) 234. L. H. Faissner and K. Wildermuth, Phys. Letters 3. 4.

5.

6. 7. 8.

2 (1962) 212. D. R. Inglis, Ann. Phys. (N.Y.) 5 (1958) 106. W. D. Myers and W. J. Swiatecki, Nucl. Phys. 81 (1966) 81. R.Beringer, Phys. Rev. 131 (1963) 1402. J. N. P. Lawrence, Phys. Rev. 139 (1965) B227. S. Cohen and W. J.Swiatecki, Ann. Phys. (N. Y.) 221 (1963) 406. A. E.S. Green. Rev. Mod. Phys. 30 (1954) 59.