Imperial Journal of Interdisciplinary Research (IJIR)
Vol-3, Issue-1, 2017
ISSN: 2454-1362, http://www.onlinejournal.in
A Study of Atomic and Nuclear Binding Energies for
Highly Stable Atoms and Stable Nuclei.
Basabdatta Bose
Assistant Professor, Department of Physics. Siliguri College. Siliguri. Darjeeling: 734001.
West Bengal. India
Abstract:
The simultaneous studies of the
stabilities of the atoms as well as the nuclei are
very interesting. Atomic stability corresponds to the
ionization potential (energy) whereas and the
nuclear binding energy
indicates nuclear
stability. Inert gas atoms having very high values of
ionization energy are highly stable. On the other
hand, the binding energies of magic number nuclei
are comparatively high resulting in high nuclear
stability but their ionization energies are not too
high. Here, there will be a comparative study
between two kinds of stability, atomic and nuclear,
for inert gas atoms (highest atomic stability) and
for atoms with magic number nuclei (highest
nuclear stability). For both the cases, there will be
a try to estimate the 1st ionization potential
(energy) , nuclear binding energy, binding energy
of the last neutron and binding energy of the last
proton and finally to do a comparative study.
These nuclei are called magic number nuclei.
Higher stability i.e. higher nuclear binding energies
are well explained with the single particle nuclear
shell model. In that model, different shells and subshells are assumed. The nucleons move in different
shells and sub-shells without disturbing the others.
The maximum number of nucleons that can be
accumulated in the sub-shells s, p, d, f, etc are 2, 6,
18, 32 etc. When the numbers of nucleons are
sufficient enough to fill one or more shells
completely then corresponding nuclei show higher
stability with higher binding energy. These nuclei
are called magic number nuclei.
4
16
40
58
88
120
140
208
2He ,8O ,20Ca ,28Ni ,38Sr ,50Sn ,58Ce
82Pb
are the family members of this group.
Introduction:
1st ionization potential (energy) for inert gas atom
is defined as the energy required (eV) to remove 1st
electron from the outermost completely filled
electronic shell. Whereas for magic number nuclei,
the 1st ionization potential (energy) is the energy
necessary to kick out the electron from the unfilled
shell. The numerical values of the energy can be
determined theoretically. The required formula for
the 1st ionization energy for an atom is
EI= E1x (Z2n/n2).
Where E1= the ionization potential (energy) for
hydrogen atom, n = quantum number of the outer
most shell.
Zn e = effective charge that acts on the outer shell
electrons.
It is well known that the atoms with numbers of
electrons 2, 10, 18, 36, 54, 86
show high chemical stability. That
stability can
be explained on the basis of electronic shell
structure. According to that model, electrons are
available in different shells (k, l, m, n.... etc) and
sub-shells (s, p, d, f....etc.). Maximum numbers of
electrons that can be accommodated in sub-shells s,
p, d, f are 2, 6, 10, 14 respectively. For inert gases,
electrons are distributed in completely filled up
sub-shells. The sub-shell are filled according to 1s2s-2p-3s-3p-4s-3d-4p-5s-4d-5p-6s. So when the
shells and sub shells are completely occupied by
the electrons, the atoms show no tendency to
interact with the other atoms. The electrons are
very tightly bound in those atoms and their 1st
ionization energies are much higher compared to
that of the other elements adjacent to them in the
periodic table. That results in the inert gas atoms to
have very high chemical stability. He, Ne, Ar, Kr,
Xe, Rn are the member of this family.
While studying the nuclear binding energies, of the
nuclei, with number of nucleons (either protons or
neutrons or sum of the two) 2,8,20,50,82,126 show
greater stability compared to their neighbours.
Imperial Journal of Interdisciplinary Research (IJIR)
Theory and Calculation:
A. 1st ionization potential (energy):
B. For nuclear binding energy:
It is possible to estimate the numerical value of
nuclear binding energies either by using the mass
defects calculation or by using Weizsacker semiempirical mass formula assuming atomic nucleus
as a liquid drop.
I).Estimation of binding, using the concept of mass
defect:
Page 753
Imperial Journal of Interdisciplinary Research (IJIR)
Vol-3, Issue-1, 2017
ISSN: 2454-1362, http://www.onlinejournal.in
II). Using Weizsacker semi empirical mass
formula:
In this approach, the nucleus is considered as a
spherical drop of liquid. Here different parameters
come to play for the calculation of the total binding
energy as that is for a drop of liquid. Out of these
parameters only volume effect increases the value
of the binding energy of the nucleus and the others
viz. surface effect, coulomb effect and asymmetry
effect are responsible for lowering it. The
dependence of the value of binding energy on these
parameters is not linear, and can be expressed by
an empirically fitted formula like
Binding Energy EB = a1A – a2A – a3Z(Z -1)/A –
a4(A – 2Z)2/A ± a5/A3/4,where different terms on
the R.H.S are referred to the effective parameters
and Z, A are the atomic and mass numbers of the
nucleus respectively. a1,,a2,a3,a4 and a5, are
constants, with the standard values of 15.8, 17.8,
0.71, 23.7 and 34 respectively. For A and Z both
even, the sign in front of a5 is positive and for at
least one of A and Z odd, the sign is negative. By
using the above formula, nuclear binding energies
of the highly stable atoms as well as the nuclei are
estimated out.
2/3
1/3
After calculating total binding energy (B.E),
binding energy per nucleon (B.E/A) is found out.
C. B.E of the last neutron and the last proton:
With the single particle shell model, neutrons and
protons are separately moving in different shells
and sub-shells without interfering each other. The
energy that to be supplied to kick out a neutron
from the last shell and that for a proton from the
last shell are treated separately and are called
binding energy of the last neutron and binding
energy of last proton respectively. The calculation
for these energies can be done either from mass
defects or using empirical mass formula.
I. Calculation from mass defects:
The binding energies of the last neutron and the
last proton can be calculated by the equations
respectively
En= [mn+ M(Z,A-1)- M(Z,A)]x931MeV.
Imperial Journal of Interdisciplinary Research (IJIR)
Ep=[mp+M(Z-1,A-1)-M(Z,A)]x931MeV. Where mn
, mp are the masses of neutron and proton.
M(Z,A),M(Z,A-1), M(Z-1,A-1) are the masses of
the test nucleus, and that of nucleus with one
neutron less and one proton less respectively.
II) Calculation from semi-empirical mass formula:
The above energies can also be calculated
semi-empirical mass formula
using
i.e, En =X-Y: where X is the total B.E of the test
nucleus and Y is the B.E of the nucleus with one
neutron less. Similarly Ep can be found out.
Calculated results:
Name
Atoms
2He
of
4
20
10Ne
40
18Ar
84
36Kr
132
54Xe
1st I.P (EI
) in eV.
B.E (EB ) in
MeV.
EB/A in
MeV
B.E of the
last neutron
(En) in MeV.
24.59
21.57
15.74
13.98
12.12
28.35
162.63
354.22
744.04
1111.60
7.09
8.13
8.86
8.88
8.42
20.23
13.23
9.34
9.54
8.94
B.E of the
last proton
(Ep)
in
MeV.
19.77
12.67
10.03
9.70
8.39
1. Table for the most stable atoms.
2. Table for magic number nuclei.
Name
Magic
Nuclei.
4
2He
16
8O
40
20Ca
58
28Ni
88
38Sr
120
50Sn
140
58Ce
208
82Pb
of
1st [I.P]
EI in eV.
B.E
(EB) in MeV.
EB /A in
MeV
24.59
13.517
6.223
7.439
5.674
7.354
5.540
7.410
28.35
139.62
354.23
518.58
778.12
1039.3
1172.19
1637.66
7.09
8.73
8.85
8.94
8.84
8.66
8.37
7.87
B.E of the last
neutron
(En) in MeV.
20.23
12.82
9.41
8.23
11.17
9.45
8.93
7.56
1. In Fig: 1, variation of the 1st ionization potential
(energy) EI and the average binding energy (EB/A)
are plotted against atomic no. (Z) of the most stable
atom. The blue line represents the variation of EI
in eV .Where as the red line is for the variation of
EB/A in Mev.
25
EI in eV/EB in MeV
EB=931[mpxZ +mn (A-Z) - MA Z] MeV, Where mp
and mn are the masses of proton and neutron
respectively, and Z and A are the atomic number
and mass number of the atom.
20
15
10
5
0
20
ATOMIC NO. Z
40
60
Fig:1
Page 754
B.E of the
last proton
(Ep) in MeV.
19.77
12.14
7.88
7.28
10.24
8.31
8.10
8.05
Imperial Journal of Interdisciplinary Research (IJIR)
Vol-3, Issue-1, 2017
ISSN: 2454-1362, http://www.onlinejournal.in
EI in eV/EB in MeV
25
20
15
25
EI. in eV/En or Ep in MeV
2. In Fig: 2, variation of the 1st ionization potential
(energy) EI and the average binding energy (EB/A)
are plotted against atomic no. (Z) of magic number
nuclei. The blue line represents EI in eV .Where as
the red line is for the variation of EB/A in Mev.
20
15
10
5
0
10
100
Fig:4
5
0
50
ATOMIC NO .Z
100
Fig:2
3. Variation of 1st ionization potential (energy) EI,
B. E of the last neutron (En) and B.E of the last
proton (EP) are plotted against atomic number Z of
the most stable atoms in the Fig:3 . The blue line
represents the variation of EI in eV. Whereas the
red and green lines are for the variation of (En) and
(EP) in MeV.
25
EI in eV/Enor EP in MeV
50
ATOMIC NO. Z
20
15
10
5
0
20
40
60
ATOMIC NO.Z
Fig: 3
4. Variation of 1st ionization potential (energy) EI,
B. E of the last neutron (En) and B.E of the last
proton (EP) are plotted against atomic number Z of
the magic number nuclei in Fig:4. The blue line
represents the variation of EI in eV. Whereas the
red and green lines are for the variation of (En) and
(EP) in MeV.
Discussion: From the data obtained from the
above calculations, different graphs are drawn and
the nature of the graphs obtained for stable atoms
and for magic number nuclei are compared.
From the Fig.1 and Fig.2, it is observed that
ionization potential (energy) for the stable atoms
are higher than that of the magic number nuclei
(2He4 is a member of both family) as predicted
from electronic shell structure. However the
variations of ionization potential with atomic
number for the two cases are not the same. The
variation shows a steady decline first for both the
cases, but for higher atomic numbers, Fig.1 shows
a steady and slow decrease whereas Fig.2 shows a
periodic variation in ionization potential.
Fig.3 and Fig.4 show the simultaneous variation of
1st ionization potential (energy), B.E of the last
neutron (En) and B.E of the last proton (Ep) with
atomic number Z for stable atoms as well as stable
nuclei. The stable atoms have as high binding
energies as the stable nuclei. The plots of En and Ep
against Z completely overlap for stable atoms and
very close for the magic number nuclei. But for the
stable atoms, En and Ep with Z show same pattern
as the 1st ionization potential. For magic number
nuclei, the pattern is different e.g. Sr, Ce having
low values of ionization potential show high value
of B.E. To conclude, stable atoms like He, Ne, Ar, - etc. have high values of 1st ionization potential(
energy) as well as high values of B.E of last
neutron and last proton. On the other hand, magic
number nuclei with high nuclear stability do not
have that much atomic stability.
References:
[1] Moore, C. E. Ionization potentials --- Analysis
of Optical Spectra NSRDS-NBS34.washington,
D.C, 1970.
[2] Daniel S. Koltum. Total Binding ---Experiments.Phys.Rev.Lett.28,182,1972.
Imperial Journal of Interdisciplinary Research (IJIR)
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Imperial Journal of Interdisciplinary Research (IJIR)
Vol-3, Issue-1, 2017
ISSN: 2454-1362, http://www.onlinejournal.in
[3] Maria. G. Mayer: On closed Shells in Nuclei
phy.Rev.74, 235(1950)
[4]M. G. Mayer: phy.Rev.78, 19 (1950)
[5] J.A.Harrey: phy.Rev.81,53 (1951).
[6] D. Chanda: A Review ---of Atom.IJIR, Vol-2
Issue-11, 2016
[7] Nuclear physics: experimental and theoretical.
H.S.Hans.
[8] Ben R. Mottenson and J.G.Volatin. Phys.
Rev.Lett.5, 511, 1960.
[9] Nuclear physics: S.N.Ghosal.
[10] Yakata Utsuno. Phys .Rev. C 60, 054315.
1999.
[11] Study of Nuclear shell Model, M.M.Avalid...
open Science journal of Modern physics 2(3): 1922, 2015.
[12] R.J.Ede.V,J,Eemery. Procedings of the Royal
Society.1958
Imperial Journal of Interdisciplinary Research (IJIR)
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