Shell Structure of Two-Dimensional Nuclei

INTRODUCTION

I

magic numbers have been explained in terms of the particle capacity of spherical shells. The radii of these spherical shells form a simple sequence of integers and half -integers. This property will be examined in the present paper further for a hypothetical two-dimensional (2D) nucleus. The magic numbers for the 2D nuclei will be determined from the usual schemes for the filling of one-particle levels, and then will be compared with those predicted by the particle capacity of circular rings.

LEVEL SEQUENCE CALCULATIONS

The scheme for shell filling in the 2D nucleus can be determined by the same way as the 3D nucleus. It is well known that a central potential which is somewhat intermediate between a square well and a harmonic oscillator can give a reasonable level sequence if a spin-orbit interaction is added. The central scheme shown in Fig. 1 reveals this situation for 2D nuclei. For a test of numerological preciseness two schemes of level sequence calculation are attempted in the subsequent part.

(a) The Nilsson Scheme

Nilsson(2) found that the level sequence given by the expression (1 ) J. L. Hwang. Chin. J. Phys. 2, 34(1964).

(2) S. G. Nilsson. Dan, Mat. Fys. Medd. 29, No. 16 (1955). 109

GJ

15 (2) 1s ______ -,~,*-_o,-(~ )_---.--. lb _-__?S[Z?

Fig. 1. The levels of an harmonic oscillator, a square well, an intermediate potential shape and an intermediate potential shape plus a spin-orbit interaction for a two-dimensional nucleus.

a3(nZj)

=tw{(A+-:-)

--2r;<Z-s>j-DZ(Z+l)}

can reproduce the experimental one very well. For the 2D case this expression is modified to be

a2(12Zj)=tiw{ (n+l) -2iC<Z*S>j--OZ’},

where o is the circular frequency of the oscillator, A=2n + Z-2,

<Z.s>j+j’-s”-z”) = Z / 2 (j=Z+ -+) - Z / 2 (j=Z-+), and D is a number which takes the values

D=o for = -0.011 for =0.0175 f o r = 0.020 for -2~tion represents the mean

A=O, 1 A = 2 if=3 A=4.

value of the spin-orbit interaction 1 dV

particles occupying the degenerate level of energy EA. But the 2D nucleus is hypothetical and the radius R is not obtainable from the measurement. Nevertheless, as will be shown in the next section, by counting the particle capacity of the space, the radius for the closed shell nucleus with an isotropic harmonic oscillator potential well can be determined. The result is shown in Table 1. For nuclei of proton (neutron) number 22, 44 and 74, fiio is determined to be 3.15, 2.11 and 1.59 respectively, and the level sequences are calculated for these nuclei. The shell formation of the energy levels can be seen readily from Fig. 2.

Sengupta wall of finite

(b) Bounded Harmonic Oscillator Potentials

and Ghosh”’ investigated a harmonic oscillator bounded by a potential height;

V= - Vo + +ZO~Y, r<R

v=o,

The level sequence obtained agrees very well with the experimental one. A similar calculation for the 2D case is possible and is done here.

The radial part of the wave function is given by Rnl=e-P12P(z12)+(1!4)1F~( -&, Z+l, p), OM~

where P=----) ~nl&E?L-z+1

h 2ho 2’ o is the circular frequency of the oscillator,

and E,,! is-the energy. $1 is the confluent hypergeometric function. The rigid wall condition leads

n=4 72’6

Fig. 2. The energy levels calculated by the Nilsson formula for the two-dimensional nuclei of neutron (proton) number 22, 44 and 74. The energy is expressed in units of fi.i.w.

R,,I ( pO) =O, where P O= wMR2/h

and R is the nuclear radius. A suitable modification of the Tricomi expression for the zeros of hypergeometric function gives R,r for fixed values of PO and I as

&-

z+1

31

u+uu-1) +

4Po 2 ---@l 12 ’

jit being the n-th zero of the Bessel function of order 1. For not very large value of p. (po<3n+Z+ 3/2), the energy levels are given by

+ 2(1+1) (Z-1) P;

~_ ___

1

-Using the Thomas-Frenkel expression and the same value of I, 1=250, as used in (a) the total spin-orbit energy can be calculated by the perturbation method, and it is given by

n =4

Fig. 3. The energy levels numbers are 30, 56

On glancing Figs. 2 and makes us incline to suggest

72=6 71=8

for bounded harmonic oscillators. The neutron (proton) and 90. The energy is expressed in units of fL2/2MR2.

3, the tendency of shell formation of energy levels that the magic numbers might be 2, 6, 12, 14, 22, 32.... with a straightforward analogy with the 3D case. However, as will be seen in the next section, we shall have to conclude that two particles occupying the 1 f7,2 level cannot form an individual shell. The magic numbers should be

spin-orbit interaction 1. If the

lf7,2

PARTICLE CAPACITY OF CIRCULAR RINGS

If .the two dimensional nucleon (proton or neutron) is assumed to be a rigid circircle of radius r, then the number of circles N which can be inlaid into a circular ring of inner radius S--r and outer radius &+r can be calculated by an elementary geometry, and it is given by

N= z/sin-’ (117) where 77 = Ro/r.

Next we assign r=l fm as did in the 3D case. Then whenever t7 takes a value listed in the first column of Table 1, N becomes an integer. These values of 77 do not make each ring overlap so much, and do not obstruct the circular motion of nucleons within the ring. Actually the number N is the same with the one obtained by arranging circles of the same radius over a plane as closely as possible but without overlapping with each other (Fig. 4).

Fig. 4 The closest arrangement of circles on the plane. nucleons within the two-dimensional nucleus is pattern.

N 3

The arrangement of generated from this

Referring Fig. 1 we find it reasonable to assume that the maximum number of protons and neutrons existing in a ring is N/3 for each kind and the remaining space of capacity N/3 is vaccant. Then the numbers of particles of one kind in the rings becomes 2, 4, 6, 8, 10.. . . as listed in the fourth column of Table 1, which are just equal to those of the particles occupying the major energetical shells of 2D nucleus with a harmonic oscillator potential. The aggregate of them gives the magic numbers 2, 6, 20, 30, 42.. . . . Unlike the 3D nucleus, there is no evidence of discontinuity in the neighborhood of the shell (ring) corresponding to the 1 f7,z level.

1 2.00 3.86 5.76 7.66 9.57 11.47 13.38 15.29 d?l 1.86 1.90 1.90 1.91 1.90 1.91 1.91 N 6 12 18 24 30 36 42 48

Particles in each shell (ring) _I Magic Numbers Isotropic harmonic 2D nucleus

oscillator r - . for 2D nucleus

2 2 2 2 2 4 4 4 6 6 6 (8) 6 14 12 8 8 (10) 22 22 10 10 10 32 32 12 12 12 44 44 14 14 14 58 58 16 16 16 74 74

(2) The radii of proton (neutron) density of other 20 nuclei can be found by an interpolation between the above set of numbers 3.00, 4.86, 6.76, 8.66. . . . fm. These rules are not so beautiful as those of 3D case”). However, without them we had no mean to know the magnitude of the potential well parameters and could not determine the level sequence. We have thus learned that in the case of 2D nucleus the shell structure is also intimately related with a geometrical property of the space.

ACKNOWLEDGMENT

A part of this paper is based on the work done by Drs. Nai-Li Huang, Fei-Shian Chen and Tsu-Ming Wu several years ago in our department. The authors express their sincere thanks to them for using their results.