L--
CHINESE
VOL. 2,.NO. 2,
JOURNAL OF PHYSICS
OCTOBER, 1954
Nudeon Densities of Atomic Nuclei
J E N N- LIN H WANG ($$_#$j)
AND
C H I E N - MING WV ($&fljj)
Department oJ Physics; National Taiwan Umbersity, Taipri, Tniwall
(Received October 28, 1964)
As reported in the previous paper (Chin. J. Phys. 2, 84 1964), two patterns of
particle packing in the spherical shell can account for the systematics of nuclear
radii, and from them the configuration of n u c l e o n s within the nucleus can be
objectured in a classical manner. The aim of the present paper is twofold: First the
possibility of such patterns of packing is examined quantitatively. Second the wave
mechanical correspondence to the objectured nucleon configuration is seeked for.
For these purposes, the nucleon densities of nucleus are defined and calculated
in a classical manner. A universal function is furnished for determining the nucleon
densities of any nucleus. The calculation shows that the maximum relative density
The
in the nucleus is only 0.79 allowing the suggested patterns of particle packing.
calculated densities for nucleus of N=Z=82 and for that of gold-197 are then compared respectively with the Rotenberg’s self-consistent field calculation and with the
shell model wave function calculation of Ross, Mark and Lawson. A good conformity
and a significant correspondence are obtained.
:
N the forgoing paper’) (referred to as I) it has been, shown that the systematics of
nuclear radii or the rules proposed by us are a natural consequence of some geometrical
properties of spherical shells. Two patterns of particle packing in the spherical shells
yield a relation between the nuclear dimension and the magic numbers. Further, as a
by-product, the configuration of nucleons within the nucleus is objectured in a classical
form. The aim of the present paper is twofold: First the possibility of such patterns
of packing is examined quantitatively. Second the wave mechanical correspondence to
the objectured nucleon configuration is seeked for. For these purposes, the nucleon
densities of the nucleus are calculated in a classical manner and are compared with the
self-consistent field calculation’) and with the shell model wave function calculation.3’
I
DEFINITION OF NUCLEON DENSITIES
The picture of nucleus formed in Fig. 2 of I is as follows: Suppose a closed shell
nucleus of the proton number 2 (neutron number N), say 2=82. The radius of its
proton density is 7 fm according to our rules. If the protons were removed one by one
from this nucleus, the radius OF proton density would become smaller and smaller.
When 32 protons were removed, the radius would reduce to 6 fm. Therefore it may
be assumed that these 32 protons has located originally in the spherical shell of the
outer radius 7 fm and the inner radius 5 fm, that is, in the VI-th shell (see Fig. 2 of I).
1) J. L. Hwang, Chin. J. Phys 2, 84 (1964)
2) M. Rotenberg, Phys. Rev. 100, 439 (1955)
3) A. A. Ross, H. Mark and R. D. Lawson, Phys. Rev. 102, 1613 (1956)
90
._
----
NUCLEON DENSITIES OF NUCLEI
Fig. 1. Definition of nucleon density.
A slice of a nucleon (radius
r) chipped by a thin spherical
shell of radius p and thickness
do has the volume de, given
by Eq.( 1). The nucleon density D(p) is then defined as
dv/4q=dp.
dv=$dp
J
dp
91
If the protons continued to leave the nucleus, and
22 of them were removed, the radius would become
5 fm. The original position of these 22 protons are
thought to be in the spherical shell of the outer
radius 6 fm and the inner radius 4 fm, that is, in
the V-th shell. The same idea is applied continually
to the lighter nuclei, and we can arrive at the conclusion that the next 8, 12 -. - protons locate originally
at the IV-th, III-rd. -e-shells. Of course, the picture
of this sort is possible only if an assumption is made
that the removal of nucleons is done always from the
nuclear surface, or energetically, from the top level.
Let us calculate the nucleon density due to a
nucleon of radius r( = 1 fm) which moves only within
a spherical shell of outer radius R and the inner
radius R - 2r. As is shown in Fig ‘1, a slice of the
nucleon chipped by a thin spherical shell of radius p
and thickness dp would have the volume given by
sin 8 dB=27$ l
c
J
-
p2+ (R-r)2-r2
where Bo=cos-1~~p2+(OR-r)2-r3~/2p(R-r)].
defined to be dv,‘4 zp2dp. Then
-
2p(R-r)
dp
3
’
(1)
T h e density D(p) due to this slice is
D(p)=_~_c1-~~~~~~~)-” ] for R-2r<p<R.
(2)
Introducing a parameter t such that
p=R-r+t
(-r<t<r),
and further the parameters
a = t,‘r
( - r<t<r) and g = (R/r) - 1,
we can rewrite D(p) in a form
(3)
=0
for other p,‘r.
Therefore a nucleon in the VII-th shell (q= 7) gives the density
(l-a2)/4.7.(7+a) w i t h - l<=cr<l for 6<p/r1;8,
and that in the III-rd shell (v= 3.5) the density
(1-(r2)/4.3.5.(3.5+a)
w i t h - l<a<l for 2.5<p/r<4.5.
The porton (neutron) density of a given nucleus can be synthesized by means of the
Fig. 2 of I and Eq.(3); the procedure involves (i) multiplying Eq. (3) by the number
_.
A
J. L. HWANG AND C. M. WU
92
of protons (neutrons) that completely enter the shell, (ii) arranging the necessary terms
with reference to Fig. 2 of I, and finally (iii) summing up the overlapping terms. As
an illustration, the proton density of phosphorous-32 (Z= 15) is written down below:
w i t h - l<cu,<O.
F o r 0.5<p/r<l.5,
D= 2 x ,:.r;;&
1
For 1.5<p/r<2.5,
1 -(Y:2
D=2 x p-152----+8x
4.2.5.(2.5+~)
4.1.5*(1.5+cu,)
with
oLklL1
a n d - l<=a:<O.
1 --Lyj2
_ _ _ _
For 2.5<p/r<3.5, D= 8 x ’ - (yL2
4*2.5.(2.5+a,)-+5x 4*3.5.(3.5+CQ)
with O<a?d 1
a n d - l<cu,<O.
For 3.5<p/r<4.5,
D= 5 x
F o r other p/r,
D=O.
--l
- (yJ2
4*3.5*(3.5+CQ)
with O<a,<l.
Since Eq.(3) is a universal function for any nucleus, it is tabulated in Table 1 for all
practical values of 7.
Table 1. Tabulation of Eq. (3). D(Q a) represents the relative nucleon density formed
by a nucleon in a given spherical shell.
71=1.5
a
-1.0
-0.8
-0.6
-0.4
-0.2
:.:
aa
a)
1
a
Shell II
PIT
0.0
0.2
1.7
KY% :
0.2
2.7
0.03555 6
:::
0.8
1.0
1.9
f.:
2:5
0:05079 4
0.02608 7
0.00000
0.4 0.5
0.7
0.9
2.9 3.0
3.2
3.4
0.02896 0.02500 6 0
0.01593 8
0.00558 8
-0.382
1.118 0.12732
maximum
p=5.0
Shell V
0.0
0”::
0.6
0.8
1.0
-~__
-0.101
PIT
z.
4:4
:I
5.0
:::
5.6
5.8
6.0
Wa
0.00000
0.00428
0.00727
0.00913
0.01000
a)
57
27
04
0
0.01000 0
0.00933 08
0.00777 78
0.00571 43
0.00310 34
0.00000
4.899 0.01010 2
maximum
I
71=3.5
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
1.5
1.7
1.9
2.1
2.3
2.5
0.00000
0.02117
0.03368
0.04000
0.04173
0.04000
6
4
0
9
0
-1.0
-0.8
-0.6
-0.5
-0.3
2.5
2.7
2.9
3.0
3.2
-3.1
0.1
0.3
3.4
0.00000
0.01028 6
0.01693 1
0.01785 7
0.02031 3
0.02079 8
3.6
3.8
0.01964 3
0.01710 5
;:; r
0.9
:.;
4:4
0.01339 0.00867 3 35
0.00308 44
/
Shell m
_-
7=7.0
Shell Jo
a
PI?
Wrl, n)
a
PI7
D(rlr a)
-1.0
-0.8
-0.6
-0.4
-0.2
5.0
5.2
5.4
5.6
5.8
0.00000
0.00288 46
0.00493 83
0.00625 00
0.00689 66
-1.0
-0.8
-9.6
-0.4
-0.2
6.0
6.2
6.4
6.6
6.8
0.00000
0.00207 37
0.00357 14
0.00454 55
0.00504 20
0.0
6.0
0.00694 44
6.2
6.4
6.6
6.8
7.0
0.00645 16
0.00546 88
0.00404 04
0.00230 59
0.00000
0.0
0.2
0.4
0.6
0.8
1.0
7.0
7.2
7.4
7.6
7.8
8.0
0.00510 20
0.2
0.00476 19
0.00405 41
K
0:8
1.0
-0.0839 5.916 0.00699 33
maximum
-0.0627 6.937 0.00392 16
Shell 1V
7,I=4.0
a
PIT
-1.0
-0.8
-0.6
-0.4
-0.2
3.0
3.2
3.4
3.6
3.8
0.0
4.0
i.i
:i
0:6
0.8
1.0
-0.146 3.354 0.02084 3
maximum
-9.209 2.241 0.04174 2
maximum
~=6.0
Shell m
NV, 0)
0:9
1.1
1.3
1.5
-1.0
-0.8
-0.6
-0.4
-0.2
_-_-
PlT
/ ~=2.5
0.00000
0.08571 4
0.11852
0.12727
0.12308
0.11111
a
I
Shell 1
:;
5:o
D(Q a)
0.01428 6
0.01193 2
0.00869 57
;:;I;;; 75
-0.127 3.873 0.01587 7maximum
.-i-
1)=8.0
Shell U
7.0
7.2
7.4
7.6
7.8
0.00000
0.00156 25
0.00272 73
0.00345 39
0.00384 62
0.0
8.0
:::
:::
0.00390 63
0.00365 85
0.00312 50
-1.0
-0.8
-0.6
-0.4
-0.2
_._ .~ ~-.-
93
NUCLEON DENSITIES OF NUCLEI
ACTUAL CALCULATION OF NUCLEON DENSITIES
The sum of the proton and neutron densities for a hypothetically large nucleus and
for a real nucleus of 92 protons and 92 neutrons calculated in the foregoing manner are
illustrated in Fig. 2 respectively with a heavy line and with a broken line. The densities
.45
.40
.35
.3.c
*25
.2c
./5
.to
.05
0
NUCLEAR DENSITY OF AN
I-.-.---- PROTON
INFINITELY LARGE NUCLEUS
NUCLEAR DENSITY FOR A
NUCLEUS N=Z =92
( A F T E R N. ROTENBERG; f=z)
NUCLEAR CENSITY FOR A
NUCLEUS FJ =Zp 92
(PRESENT THEORY 1
-
AND NEUTRON
D E N S I T I E S F O R A &h;;;
I
/:a
NUCLEUS
2.0
I
\
\I
6.0
ZG
I
3.0
4.0
50
R A D I U S I N 10-tJcm.
‘., \
*?
\
8.0
Fig. 2. Nucleon densities of some nuclei.
The numbers on the ordinate represent the relative density, so that unity means that the space is
completely filled up by the nuclear matter.
When the sum of proton density and neutron density
(referred to as nuclear density) is meant in this figure, these numbers should be twiced. The absolute
density may be readily obtained by multiplying these numbers by a factor 4.00 x 10!‘gm/cm3
103s particles/cm.s
or 2.39 X
The factor is determined from an assumption that the nucleon is a uniform sphere
of radius 1 fm and of mass 1.674x10-s’gm.
of the latter nucleus (N=Z=92) determined from the Hartree-Fock self-consistent field
calculation by Rotenberg is also illustrated for comparison. The proton density (2=79)
and the neutron density (N= 118) for gold-197 are shown in Fig. 2 and Fig. 3. In
L-._ _. _
J. L. HWANG AND C. M. WU
94
.45
I.--\
/
.40
1
.35
-
.3c
-
-
.2c
-
.i5 7
I
\
A
I
I
I
I
\ --___m- --.
\ /NEUTRON
f
I
DISTRIBUTION
‘\ FROM r: I\yl’
\( AFTER ROSS. MARK AND
-I
LAWSON)
‘\
\
/
-I
.05 -
I
I
\
DENS IPI
THEORY ).
.25
.I0
/
/
PROTON DENSITY
(PRESENT THEORY)
PROTON AND NEUTRON
DENSITIES
I N 79 Afg7
u If8
/.o
3.0
I
2.0
4.0
I
I
5.0
‘6 .0
R A D I U S , IN lO-‘3 cm.
ZO
Fig. 3. Comparison of present calculation with the shell model wave function calculation. If the
ordinate is expressed in units of 4.00
x
1014gm/cm3 or 2.39
x
1033particles,‘cma,
it readily represents
the absolute density.
Fig. 3 they are compared with the results of Ross, Mark and Lawson which were obtained
from the shell model wave functions with the Wood-Saxon potential. The numbers on
the ordinate of Figs. 2-3 represent the relative density, so that unity means the space
is completely filled up by the nuclear matter.
When the sum of both densities is
meant in these figures, these numbers should be twiced. Obviously, no point with the
relative density 1 or more is found.
The maximum value 0.79 is found in the region
3.0 fmLpd4.0 fm where the nucleons suffer deformation and, in addition, three shells
overlap.
Besides a number of conspicuous hills and valleys which are inherent in our classical
picture of nucleons, the densities determined by us interweaves fairly those of Ross et.al.
and Rotenberg, and shows the common features:
First at the center it has a decided
hole of radius 0.5 fm. Second, as mentioned above there is an elevated hill at p = 3.2 fm.
The density increases till p= 3.2 fm and then decreases till p = 4.8 fm. This part might
be looked as a kind of core within a nucleus.
increases as p increases.
Finally beyond p= 4.8 fm it gradually
Therefore for heavy nuclei there is an additional lower hill
NUCLEON DENSITIES OF NUCLEI.
before the density falls off.
95
This might be contrasted with the Johnson-Teller effect.
Finally it should be mentioned that both of neutron and proton densities have the same
shape and the same magnitude except in the vicinity of surface. In order to explain
the superiority of the neutron density revealed by Ross et.aZ. and other authors, it seems
necesssary to take account of the difference in the charge distribution and the matter
distribution of the proton,
We are not going to step further into this respect at the
present stage.
A similar calculation with the Gaussian form of charge distribution in p r o t o n s
suggested by Hofstadter will be reported later.