8.2 Problems
435
8.2.2 Electric Potential and Energy
8.7 Derive an expression for the electrostatic energy of a spherical nucleus of radius
R assuming that the charge q = Ze is uniformly distributed homogeneously in
the nuclear volume.
8.8 A charge q is uniformly distributed in a sphere of radius R. Obtain an expression for the potential V (r ) at a point distant r from 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 × 10
23 rad/s about its axis. If it posseses a rotational energy of
537.5 MeV, then show that it has angular momentum equal to h.
8.10 (a) A D 5/2 term in the optical spectrum of
39
19 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 μ N , the magnetic field employed is 0.8 T. Find the resonance frequency.
You may assume J = 7/2, μ N = 3.15 × 10
−14 MeV T
−1
8.2.4 Electric Quadrupole Moment
8.13 Show that for a homogeneous ellipsoid of semi axes a, b the quadrupole
moment is given by Q =
2
5
Ze(a
2
− b
2 )
8.14 Estimate the ratios of the major to minor axes of
181
73 Ta and
123
51 Sb. The
quadrupole moments are +6 × 10
−24 cm
2 for Ta and −1.2 × 10
−24 cm
2 for
Sb (Take R = 1.5 A
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 spin I = 0 or I = 1/2
8.16 Show that for a rotational ellipsoid of small eccentricity and uniform charge
density the quadrupole moment is given by Q =
4
5
ZRΔR. Assuming that the
quadrupole moment of
181
71 Ta is 4.2 barns, estimate its size.
435
8.2.2 Electric Potential and Energy
8.7 Derive an expression for the electrostatic energy of a spherical nucleus of radius
R assuming that the charge q = Ze is uniformly distributed homogeneously in
the nuclear volume.
8.8 A charge q is uniformly distributed in a sphere of radius R. Obtain an expression for the potential V (r ) at a point distant r from 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 × 10
23 rad/s about its axis. If it posseses a rotational energy of
537.5 MeV, then show that it has angular momentum equal to h.
8.10 (a) A D 5/2 term in the optical spectrum of
39
19 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 μ N , the magnetic field employed is 0.8 T. Find the resonance frequency.
You may assume J = 7/2, μ N = 3.15 × 10
−14 MeV T
−1
8.2.4 Electric Quadrupole Moment
8.13 Show that for a homogeneous ellipsoid of semi axes a, b the quadrupole
moment is given by Q =
2
5
Ze(a
2
− b
2 )
8.14 Estimate the ratios of the major to minor axes of
181
73 Ta and
123
51 Sb. The
quadrupole moments are +6 × 10
−24 cm
2 for Ta and −1.2 × 10
−24 cm
2 for
Sb (Take R = 1.5 A
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 spin I = 0 or I = 1/2
8.16 Show that for a rotational ellipsoid of small eccentricity and uniform charge
density the quadrupole moment is given by Q =
4
5
ZRΔR. Assuming that the
quadrupole moment of
181
71 Ta is 4.2 barns, estimate its size.