VOL. 2, NO. 2
CHINESE JOURNAL OF PHYSICS
OCTOBER, 1964
Some Geometrical Properties of Spherical Shells Implying
the Systematics of Nuclear Radii
J E N N- LIN H W A N G (s & @)
Department of Physics, National Taiwan University, Taipei, Taiwan
(Received May 5, 1964)
Assuming that the nucleon is a ball of radius I( = 1 fm) consisting of an incompressible but a deformable matter, we calculate the number of nucleons Nwhich can
be inlaid closely into a spherical shell of the outer radius
R and the inner radius
R-27. (i) If the nucleons suffer d e f o r m a t i o n e x c e p t t h e d i r e c t i o n a l o n g t h e
radial direction of the shell, N/4 equals to the numbers of particles occupying
the shells of a harmonic oscillator well, i.e. to 2, 6, 12, 20, 30......, for R/t=2.5,
3.5, 4.5, 5.5, 6.5....... (ii) If the nucleons suffer no deformation, N/4 equals to the
numbers of particles occupying the V-th, VI-th, VII-th......energetical shells of a
realistic nucleus i.e. to 22, 32, 44...-*., for R/7=6.0, 7.0, go.......
Applications are made to account for the systematics of nuclear radii. It is
shown that a combination of the above properties forms a foundation of the rules
proposed early by us, that is, the closed shell nuclei of proton (neutron) number,
2, 8, 20, 28, 50, 82, 126 . . . . ..have the radii of proton (neutron) density 2.5, 3.5, 4.5,
5.0, 6.0, 7.0, 8.0 ......frn respectively.
Finally, a configuration of nucleons within the
nucleus which corresponds to the classical picture of nucleus is suggested.
INTRODUCTION
T HE traditional formula of nuclear radius
R = r,A’ ‘3
(1)
was derived from a crude approximation that A nucleons of radius r0 can be packed
into a sphere of R as closely as possible. In view of the data of Hofstadter et. a#‘>.,
r0 = 1.2-1.3 fm
and
for most nuclides,
<r2>-(0.80)2 fm2 o r J <m<rZ>=1.00 fm
for protons and neutrons,
it is suggested that the nucleons in a nucleus do not jostle one another so severely as
imagined in Eq. (1). In other words, within the nucleus the whole space is not filled
up with the nuclear matter, and, as easily estimated from the above data, about one
half is empty.
The pattern of packing nucleons into a nucleus assumed in Eq. (1) is most
primitive. In the present paper another pattern of packing is pointed out, which is
_
_ (1) R. Hofstadter, Revs. Modern Phys. 28, 214 (1956).
84
-
,_
’
SOME GEOMETRICAL PROPERTIES OF SPHERICAL SHELLS
8.5
possible to account for the scheme for calculating the nuclear radius proposed previously
by the authoP). The scheme is based on the following rules:
(1) The closed shell nuclei of proton (neutron) number 2, 8, 20, 28, 50, 82, 126 - - . have the radii of proton (neutron) density 2.5, 3.5, 4.5, 5.0, 6.0, 7.0, 8.0. - -. fm respectively.
(2) The radii of proton (neutron) density of other nuclei can be found by an
interpolation between the above set of numbers 2.5, 3.5, 4.5, 5.0, 6.0. * . . ..fm.
This scheme reproduce almost all the data of Rofstadter’s group for spherical nuclei,
and agrees very well with the result of the Elton’s semi-empirical formula”),
R= 1.121 A”3 + 2.426A-‘/ 3-6.614A-‘.
These rules are of course, derivable from the independent particle model of nucleus.
This point was examined with the Green potential(l). However, since the parameters
involved in the potential had to be also chosen from other experimental sources, it was
dfficult to develop an argument in an elegant style. On the other hand, in the present
stage of many body problem treatment it seems premature to be able to verify these
rules.
In the argument developed below, the nucleon is assumed to be an incompressible
but a deformable sphere. Therefore, at a glance, the argument looks somewhat like a
“marbles in a sack” argument. So far as the author knows, there has been no description in the literature that the “marbles in a sack” argument of this kind revealed important features, especially, relations with the shell model of the nucleus and with the
systematics of the nuclear radii. It should be remembered that the thing reported here
is not a “theory” that explains a phenomenon, but is a “description” of a phenomenon
that can be observed by an elementary mathematics instead of instruments. We merely
point out that this phenomenon may be an origin of the systematics of the nuclear
radii and may be a foundation of our proposed rules. This approach is expected to be
a sort of stepping-stone locating in the gap between the experiment and the theory.
PARTICLE CONTENT OF SPHERICAL SHELLS
As already mentioned in the introduction, the nucleon (proton or neutron) is
assumed to be a ball of radius r consisting of an incompressible but a deformable
matter. Now we take a spherical shell of outer radius R and inner radius R-2r, and
calculate the number of balls that can be inlaid closely into this spherical shell, with
a restriction that deformation of the ball along the radial direction of the spherical
shell is not allowed. Two cases are to be considered:
(2) J. L. Hwang, Chin. J. Phys. 1, 24 (1963).
(3) L.R.B. Elton, Nuclear Phys. 5, 173 (1958).
(4) J.L. Hwang and S.H. Yee, Chin. J. Phys. 1, 28 (1963); J.L. Hwang and Y.S. Yang, Chin. J. Phys. 2, 32 (1964).
.,
:
86
JENN-LIN HWANG
(i) the balls suffer deformation and the shell is filled up with these deformed balls, and
(ii) the balls do not suffer any deformation and only contact with each other in the shell.
In the former case, if the spherical shell is viewed from the outside, no gap can be
observed between balls. But for -the latter case, gaps remain around each ball.
The volume of the space occupied by a ball in the spherical shell is shown in
Fig. 1, and is given by
Fig. I. Vo!ume of the space occupied by a ball of radius r in a
spherical shell of the outer radius R and inner radius
R-27. It is given by Eq. (1).
where sin8, = r,‘( R - r).
Since the volume of the whole shell is
V=j;[R3-(R-2r)“3
the number of balls that can closely be inlaid into the shell for the case (i) is equal to
(2)
where c replaces R,h-. As shown in Table 1, when C= 2.5, 3.5, 4.5, 5.5 . . . . ., Nl given
by Eq. (2) is almost exactly equal to 2 x 4, 6 x 4, 12 x 4, 20 x 4 . - - - - *respectively. The
numbers 2, 6, 12, 20.*-** *are just respectively the numbers of protons (neutrons) occupying the I-st (Is-level) II-nd (I+level), III-rd (Id, 2s-levels) IV-th (If, 2p-levels),
V-th (lg, 2d, 3s-levels). . . - - *energetical shells of a hypothetical nucleus with an isotropic harmonic oscillator potential well, when C= 5.0, N,=16 x 4=8 x 8. The number
8 is the number of protons (neutrons) occupying the IV-th (lf,,2 level) shell of a
I.
SOME GEOMETRICAL PROPERTIES OF SPHERICAL SHELLS
realistic nucleus.
c
87
For other values of 5, no interesting evidence can be found.
Nl
____
c
N
2.5
3.5
1.964x4
5.989 x 4
4.5
11.995x4
5.0
15.746 x 4
5.5
6.5
19.997 x 4’
29.998 x 4
6.0
7.0
24.745 x 4
3.5.748x4
7.5
41.999x4
8.0
48.749 x 4
8.5
55.999x4
9.0
65.941 x 4
Table 1. Number of balls that can be inlaid
closely into a spherical shell of the outer radius
R and the inner radius R-27, under a condition
that the balls suffer deformation except the
direction along the radial direction of the
shell. Notice that t/,N equals the numbers
of particles occupying the shells of a harmonic
, oscillator well, for C=2.5, 3.5, 4.5, 5.5....... f is
the radius of a ball, and c=R/r.
For the case (ii), the right handed side of Eq. (2) should be multiplied by a factor
a/s/z This factor represents the ratio of the area of a hexagon to that of its inscribed
circle. The reason is as follows: When circles of the same radius spread all over a
plane without intersecting with another, the net space occupied by one circle is a regular
hexagon circumscribing that circle. Similarly if rigid balls of the same radius r are
closely arranged within a spherical shell of thickness 2r and if its outer radius R i s
considerably larger than r, the net space occupied by a ball is approximately a spherical
regular hexagonal base. Therefore Eqs (1) and (2) can reasonably be approximated*
merely by multiplying the factor 2/T/z, i.e.
N,=vz12 - ~*2(‘- 1)[(c-z)+{c(c-2)}1’2 3.
(3)
Table 2 Shows the relation between c and NL. W h e n it= 5.0, 6.0, 7.0, 8.0. - - - . -, the
number of balls Nz which can be inlaid into the shell is approximately equal to 22 x 4,
32 x 4, 44 x 4. . -. . - respectively. The numbers 22, 32, 44.. - - - - are just the numbers
of protons (neutrons) occupying the V-th (2p,,,, If,,,, 2p,, 2, lg,,,-levels), VI-th ( lgl12,
2dj, 2, 2& 2, 3~ 2, 1 h,,&dS), VII-th ( lhs12, 2f7, *, 2f5, 2, 3p3, 2, 3pllz lij, ,-levels)- . - - - energetical major shells of a realistic nucleus. For other values of C, no significance
can be found.
c
NS
2.5
1.781 x4
3.5
4.5
5.431 x4
10.853 x4
5.0
7.145x4
5
N2
Table 2.
Number of balls that can be inlaid
closely into a spherical shell of the outer radius
R and the inner radius R-27 under a condition
that the balls suffer no any deformation. Notice
that i/,Ns equals the numbers of protons
6.0
22.443 x 4
(neutrons) occupying the V-th, VI-th, VII-th.-.*--
7.0
8.0
32.420 x4
44.210~4
energetical shells of a realistic nucleus, for c =6.0,
9.0
57.814 x 4
7.0, go....... t is the radius of a ball, and
C=R/7.
* The exact expression can be obtained from the spherical geometry, and has an intractable form
Ns=2
The error of Eq. (3) is within a half percent.
88
JENN-LIN HWANG
APPLICATION TO OUR RULES
In order to account for the first of our rules, the first four sets of C-N value are
chosen from Table 1 and all sets except for the first four ones are chosen from Table
2. This combination is listed in Table 3. The integers closest to the calculated values
Nsimplified Shell Number of Particles
per Shell
Aggregate of Particles
(Magic Numbers)
C
N
2.5
1.964 x 4
2x4
I
2
2
3.5
5.989 x 4
6x4
It
6
8
4.5 11.995x4
12x4
m
12
20
nuclear radii.
5.0 15.746 x 4
6.0 22.443 x 4
8x8
22x4
lv
V
8
22
28
50
per four rows are chosen
7.0 32.420 x4
32x4
VI
32
82
remaining from Table 2.
8.0 44.210~4
9.0 57.814 x4
44x4
58x4
w
44
58
126
WI
Table 3. Particle content
of spherical shells implying the systematics of
from Table 1 and the
See also the caption of
184
of N are also listed in the third column.
The up-
Fig. 2
The manner in which they are related with
the magic numbers are illustrated in the remaining columns. The meaning of Table 3
may be seen more clearly from a schematic diagram of Fig. 2 .
Henceforth the value
1 fm is assigned to the equivalent uniform radius r of the proton (neutron), so that the
outer radius of the spherical shell
I
R
m,Is?,
II
12 \f6\
6’,
2 1,
\
is just equal to < fm.
\
&B
\( ’!
l3+\
* -
L 4.5-I
CZ
5.0
I
I_
____1:
m,
PI
.P
22’1,
44 1,
32’1,
5A
&!
82:
I
I
I
I
I
I
I
I
I
6 . 0 ------y
I
I__- zo d
I
8.0
cI
Fig. 2.
I
I
I
I
I
I
I
I
I
Y
I
Schematic diagram illustrating the meaning of Table 3. The I-st shell contains 2 protons and
2 neutrons. The II-nd shell contains 6 protons and 6 neutrons. The III-rd one contains 6 protons and 6 neutrons, and so on. The aggregate number of particles gives the magic number.
The number below the figure means the outer radius of each spherical shell in units of fm.
Regularity changes at the IV-th shell. The IV-shell contains only 8 protons and 8 neutrons
instead of 16 protons and 16 neutrons.
SOME GEO&fETRICAL PROPERTIES OF SPHERIC.4L SHELLS
89
For concreteness, we take a nucleus of 82 protons (neutrons) as an example. Fig 2
shows that the equivalent uniform radius of the proton distribution (neutron distribution)
is just equal to 7 fm. If the protons were taken out one by one from this nucleus, the
radius would diminish gradually. And finally if 32 protons (neutrons) were taken out
the radius would be reduced to 6 fm. The decrease of radius is simply assumed to be
proportional to the number_ of particles taken out, since no reason for the sudden change
can be thought. This is nothing but the content of the second rule. On the contrary
if 44 protons (neutrons) were added, the radius would become 8 fm. Therefore the
particles in the V-th shell would invade into the VI-th shell from the inside and those
in the VII-th shell would invade from the outside. In other words the V-th, VI-th
and VII-th shells would overlap. Since the particle capacity of the VI-th shell is
32 x 4= 128, 32 protons and 32 neutrons can enter this shell without any restriction.
The remaining 32 x 2 = 64 vacancies provide for the room against the invasion from the
both sides shells. The meaning of factor 4 in the third column of Table 3 has thus
been interpreted. However, between the IV-th and the III-rd shells the change of
radius is only 0.5 fm, so that the II-nd, 111-i-d and the V-th shells overlap unusually
on the IV-th shell. For this reason ‘o nly 8 protons and 8 neutrons completely enter
the IV-th shell and the 8 x 6 =48 vacancies remain to allow this unusual overlapping.
We have thus shown that the systematics of nuclear radii and therefore our proposed
rules may be accounted for. Needless to say, the concept of the spherical shell introduced here does not imply that all of the orbitals of nucleons consisting a energetical
major shell degenerate into one orbit. As easily seen from the previous independent
particle model calculation( ‘)*(j), their orbitals may be different from each other and a
statistical procedure has averaged thz.m out to form an orbit having a breadth. Further,
on viewing' Fig. 2, it is fantastic to suggest that the nucleons within a real nucleus might
take actually such a pattern of configuration. Actually, on considering a difference
between the classical and the wave mechanical pictures of particles we may expect
that it is true. This point will be discussed(Q in the subsequent paper.
(5) A.E.S. Green, Phys. Rev. 104, 1617 (1956).
(6) J. L. Hwang and C. M. Wu, Chin. J. Phys 2, 90 (1964)