1000 Solved Problems in Modern Physics

452 8 Nuclear Physics – II

Fig. 8.9Ellipsoid 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 ,wheres′^2 =x′^2 +y′^2

Q=

∫

ρ(3z′^2 −r′^2 )dτ′

ρ=

Ze

( 4 π

3

)

ab^2

Q=

3 Ze

4 ab^2

∫a

0

dz/

∫b√ 1 −(z/ (^2) /a (^2) )
0
(2z/^2 −s/^2 )s/ds/
∫ 2 π
0
dφ/

=

2

5

Ze(a^2 −b^2 )

8.14 The quadrupoleQ=^45 ηZR^2

η=

a−b

R

Tantalum:

R= 1. 5 ×(181)^1 /^3 = 8 .48fm

η=

5 Q

4 ZR^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