Classical Electrodynamics · Problem 4.6

Magnetostatics, Faraday's Law, Quasi-Static Fields — John David Jackson

Physics — E&M · nuclear-quadrupole-interaction

Problem

A nucleus with quadrupole moment QQ finds itself in a cylindrically symmetric electric field with a gradient (∂Ez/∂z)0(\partial E_z/\partial z)_0 along the zz axis at the position of the nucleus. ((a)) Show that the energy of quadrupole interaction is W=−e4Q(∂Ez∂z)0W = -\frac{e}{4} Q \left(\frac{\partial E_z}{\partial z}\right)_0 ((b)) If it is known that Q=2×10−28Q = 2 \times 10^{-28} m2^2 and that W/hW/h is 10 MHz, where hh is Planck's constant, calculate (∂Ez/∂z)0(\partial E_z/\partial z)_0 in units of e/4πϵ0a03e/4\pi\epsilon_0 a_0^3, where a0=4πϵ0ℏ2/me2=0.529×10−10a_0 = 4\pi\epsilon_0 \hbar^2/me^2 = 0.529 \times 10^{-10} m is the Bohr radius in hydrogen. ((c)) Nuclear charge distributions can be approximated by a constant charge density throughout a spheroidal volume of semimajor axis aa and semiminor axis bb. Calculate the quadrupole moment of such a nucleus, assuming that the total charge is ZeZe. Given that Eu153^{153} (Z=63Z = 63) has a quadrupole moment Q=2.5×10−28Q = 2.5 \times 10^{-28} m2^2 and a mean radius R=(a+b)/2=7×10−15 mR = (a + b)/2 = 7 \times 10^{-15} \text{ m} determine the fractional difference in radius (a−b)/R(a - b)/R.

Related theory

Browse the problem library