452
8 Nuclear Physics – II
Fig. 8.9 Ellipsoid of
revolution
The equation of the ellipsoid is
x
2
b 2
+
y
2
b 2
+
z
2
a 2
= 1
In cylindrical coordinates (z
, s
, ϕ
) the equation is
s
2
b 2
+
z
2
a 2
= 1, where s
2
= x
2
+ y
2
Q =
ρ(3z
2
− r
2 )dτ
ρ =
Ze
4π
3
ab
2
Q =
3Ze
4ab
2
a
0
dz
/
b
√
1−(z /2 /a 2 )
0
(2z
/2
− s
/2 )s
/ ds
/
2π
0
dϕ
/
=
2
5
Ze(a
2
− b
2 )
8.14 The quadrupole Q =
4
5
ηZR
2
η =
a − b
R
Tantalum:
R = 1.5 × (181)
1/3
= 8.48fm
η =
5Q
4ZR
2
=
5 × 6 × 100 fm
2
4 × 73 × (8.48) 2 fm
2
= 0.143
The ratio of major to minor axes is
a
b
∼ = 1 + η = 1.143
Antimony: R = 1.5 × (123)
1/3
= 7.458 fm
η =
5
4
(−1.2 × 100 fm
2 )
51 × (7.458) 2 fm
2
= −0.053
a
b
= 1 − 0.053 = 0.947
8 Nuclear Physics – II
Fig. 8.9 Ellipsoid of
revolution
The equation of the ellipsoid is
x
2
b 2
+
y
2
b 2
+
z
2
a 2
= 1
In cylindrical coordinates (z
, s
, ϕ
) the equation is
s
2
b 2
+
z
2
a 2
= 1, where s
2
= x
2
+ y
2
Q =
ρ(3z
2
− r
2 )dτ
ρ =
Ze
4π
3
ab
2
Q =
3Ze
4ab
2
a
0
dz
/
b
√
1−(z /2 /a 2 )
0
(2z
/2
− s
/2 )s
/ ds
/
2π
0
dϕ
/
=
2
5
Ze(a
2
− b
2 )
8.14 The quadrupole Q =
4
5
ηZR
2
η =
a − b
R
Tantalum:
R = 1.5 × (181)
1/3
= 8.48fm
η =
5Q
4ZR
2
=
5 × 6 × 100 fm
2
4 × 73 × (8.48) 2 fm
2
= 0.143
The ratio of major to minor axes is
a
b
∼ = 1 + η = 1.143
Antimony: R = 1.5 × (123)
1/3
= 7.458 fm
η =
5
4
(−1.2 × 100 fm
2 )
51 × (7.458) 2 fm
2
= −0.053
a
b
= 1 − 0.053 = 0.947