8.2 Problems 435
8.2.2 ElectricPotentialandEnergy ........................
8.7 Derive an expression for the electrostatic energy of a spherical nucleus of radius
Rassuming that the chargeq=Ze is uniformly distributed homogeneously in
the nuclear volume.
8.8 A chargeqis uniformly distributed in a sphere of radiusR. Obtain an expres-
sion for the potentialV(r) at a point distantrfrom the centre (r<R).
8.2.3 Nuclear Spin and Magnetic Moment ..............
8.9 Suppose a proton is assumed to be a classical particle rotating with angular
velocity of 2. 6 × 1023 rad/s about its axis. If it posseses a rotational energy of
537.5 MeV, then show that it has angular momentum equal toh.
8.10 (a) A D 5 / 2 term in the optical spectrum of^3919 K has a hyperfine structure with
four components. Find the spin of the nucleus.
(b) In (a) what interval ratios in the hyperfine quadruplet are expected?8.11 Given that the proton has a magnetic moment of 2.79 magnetons and a spin
quantum number of one half, what magnetic field strength would be required
to produce proton resonance at a frequency of 60 MHz in a nuclear magnetic
resonance spectrometer?
8.12 In a nuclear magnetic experiment for the nucleus 25 Mn^55 of dipole moment
3. 46 μN, the magnetic field employed is 0.8 T. Find the resonance frequency.
You may assumeJ= 7 / 2 ,μN= 3. 15 × 10 −^14 MeV T−^1
8.2.4 Electric Quadrupole Moment........................
8.13 Show that for a homogeneous ellipsoid of semi axesa, bthe quadrupole
moment is given byQ=^25 Ze(a^2 −b^2 )
8.14 Estimate the ratios of the major to minor axes of^18173 Ta and^12351 Sb. The
quadrupole moments are+ 6 × 10 −^24 cm^2 for Ta and− 1. 2 × 10 −^24 cm^2 for
Sb (Take R= 1 .5A^1 /^3 fm)
[Saha Institute 1964]
8.15 Show that the electric quadrupole moment of a nucleus vanishes for
(a) Spherically symmetric charge distribution
(b) Nuclear spinI=0orI= 1 / 2
8.16 Show that for a rotational ellipsoid of small eccentricity and uniform charge
density the quadrupole moment is given byQ=^45 ZRΔR. Assuming that the
quadrupole moment of^18171 Ta is 4.2 barns, estimate its size.