460
8 Nuclear Physics – II
Hence the rest mass energy of the nucleus
M(A, Z )c
2
= Zm p c
2
+ ( A − Z )m n c
2
− B
= 92 × 938.3 + 143 × 939.5 − 1, 786
= 218, 929 MeV
Therefore the mass is
218,929
931.5
= 235.028 amu
The plot is based on the semi-empirical mass formula obtained in the Liquid
Drop Model. In this formula the lowering of binding energy at low mass numbers due to surface tension effects as well as at high Z (and hence large A) due
to coulomb energy, are predicted by this model. The jumps in the curves at low
mass numbers (A = 2–20) are attributed to the shell effects explained by the
shell Model. The asymmetry term occurring in the mass formula is explained
by the Fermi Gas Model. From the plot the binding energy per nucleon for
87 Br
is found to be 8.7 MeV and that for
145 La it is 8.2 MeV. The energy released
in the fission is
Q = [B(Br) + B(La)] − B(U)
= 8.7 × 87 + 8.2 × 145 − 1786
= 160 MeV
8.41 The liquid drop model gives the value of Z for the most stable isobar of mass
number A by
Z 0 =
A
2 + 0.015A 2/3
For A = 127, Z 0 = 53.38, the nearest being Z = 53. Hence
127
53 I is stable.
But
127
54 Xe is unstable against β
+ decay or e
− capture.
8.42
27
12 Mg −
27
13 Al = −0.000841 (12–13) + 0.0007668 × 27
−1/3 (12
2 –13
2 )
+0.09966
12 −
27
2
2
−
13 −
27
2
2
= +0.225331 amu
As the right hand side is positive
27
12 Mg is heavier than
27
13 Al, and therefore it is
unstable against β
− decay.
8.3.9 Optical Model
8.43 Introduce the complex potential V = −(U +iW) in the Schrodinger’s equation
∇
2
ψ +
2m
2 (E + U + i W )ψ = 0
( 1 )
8 Nuclear Physics – II
Hence the rest mass energy of the nucleus
M(A, Z )c
2
= Zm p c
2
+ ( A − Z )m n c
2
− B
= 92 × 938.3 + 143 × 939.5 − 1, 786
= 218, 929 MeV
Therefore the mass is
218,929
931.5
= 235.028 amu
The plot is based on the semi-empirical mass formula obtained in the Liquid
Drop Model. In this formula the lowering of binding energy at low mass numbers due to surface tension effects as well as at high Z (and hence large A) due
to coulomb energy, are predicted by this model. The jumps in the curves at low
mass numbers (A = 2–20) are attributed to the shell effects explained by the
shell Model. The asymmetry term occurring in the mass formula is explained
by the Fermi Gas Model. From the plot the binding energy per nucleon for
87 Br
is found to be 8.7 MeV and that for
145 La it is 8.2 MeV. The energy released
in the fission is
Q = [B(Br) + B(La)] − B(U)
= 8.7 × 87 + 8.2 × 145 − 1786
= 160 MeV
8.41 The liquid drop model gives the value of Z for the most stable isobar of mass
number A by
Z 0 =
A
2 + 0.015A 2/3
For A = 127, Z 0 = 53.38, the nearest being Z = 53. Hence
127
53 I is stable.
But
127
54 Xe is unstable against β
+ decay or e
− capture.
8.42
27
12 Mg −
27
13 Al = −0.000841 (12–13) + 0.0007668 × 27
−1/3 (12
2 –13
2 )
+0.09966
12 −
27
2
2
−
13 −
27
2
2
= +0.225331 amu
As the right hand side is positive
27
12 Mg is heavier than
27
13 Al, and therefore it is
unstable against β
− decay.
8.3.9 Optical Model
8.43 Introduce the complex potential V = −(U +iW) in the Schrodinger’s equation
∇
2
ψ +
2m
2 (E + U + i W )ψ = 0
( 1 )