A nucleus with quadrupole moment Q finds itself in a cylindrically symmetric electric field with a gradient (?E_z/?z)₀ along the z axis at the position of the nucleus.

(a) Show that the energy of quadrupole interaction is

(b) If it is known that Q = 2 X 10^?8 m² and that W/h is 10 MHz, where h is Planck's constant, calculate (?E_z/?z)₀ in units of e/4??₀?³₀, where ?₀ = 4??₀h²/me²/me2 = 0.529 X 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 a and semiminor axis b. Calculate the quadrupole moment of such a nucleus, assuming that the total charge is Ze. Given that Eu¹⁵³ (Z = 63) has a quadrupole moment Q = 2.5 X 10^?28 m² and a mean radius

R = (? + b)/2 = 7 X 10^?5 m

determine the fractional difference in radius (a ? b)/R.

, az Jo