### KarkheckAAPTJuly302014 - American Association of Physics

```The Multipole Expansion of the Electric
Potential and Non-Spherical Nuclei
John Karkheck
Marquette University
Summer Meeting
American Association of Physics Teachers
University of Minnesota
July 30, 2014
Electrostatic Potential
Multipole Expansion
• Monopole term
• Dipole term
General expression
Qij = 1/e ∫ r(r) (3 xixj - r2 dij ) dr
Ellipsoid: (x12 + x22)/a2 + x32/c2 = 1
assume r(r) = Ze/(4p/3 a2c) inside; = 0 outside
then Qij = Q dij
with Q = 2/5 Z(c2 - a2)
Ellipsoids
• Prolate: Q > 0
• Oblate: Q < 0
• Qm = I (2I - 1) /((I + 1)(2I + 3)) Q
I = nuclear spin
Closure
• Arithmetic model
• Geometric model
• Nuclear density
4p/3 a2c = 4p/3 R3
R = R0 A1/3
d = (c - a)/R
R = (a + c)/2
Q = 4/5 Z R2 d
• c3 – 2.5 (I + 1)(2I + 3)/(I(2I-1))
(Qm/Z) c - A R03 = 0
Table 1. Semimajor and semiminor axes obtained via arithmetic and geometric
approaches to closure.
Nucl. Spin Qm (fm2)
14N
1
2.02
40K
4 -7.49
151Eu 5/2
90.3
153Eu 5/2
241
167Er
7/2 357
c
3.516
3.986
6.783
7.461
7.672
Arithmetic
a
d
2.269
4.222
5.998
5.375
5.544
0.431
-0.058
0.123
0.325
0.322
a2c/R3
0.748
1.028
0.935
0.815
0.817
Geometric
c
a
3.707
3.947
6.912
7.792
8.010
2.555
4.185
6.144
5.825
6.002