Subatomic Physics

18.9. References 575
18.33. Consider a completely asymmetric nucleus, with ω1 >ω2 >ω3. What is
the spectrum of single particle levels in such a nucleus if ω1/ω2/ω3 = α/β/1.
(Hint: Use Cartesian coordinates.)
18.34. Compare molecular and nuclear spectra. Discuss the energies and energy
ratios involved in the three types of excitations. Discuss the corresponding
characteristic times. Sketch the essential aspects of the spectra.
18.35. Show that the term l = 1 in Eq. (18.35) corresponds to a translation of the
nuclear c.m. Draw an example.
18.36. Find a relation between the coefficients αlm and α∗ l,−m
the reality of R and the properties of the Y m
l .
in Eq. (18.35) by using
18.37. Use Eq. (18.35) to draw a deformed nucleus described by α30 �= 0,allother
α =0.
18.38. Verify the solution (18.39).
18.39. Show that for an incompressible irrotational nucleus the semiempirical mass
formula gives for the coefficients B and C in Eq. (18.38)
B = 3
8π AmR2
C = 1
�
2asA
2π
2/3 − 3
5
Z 2 e 2
18.40. Show that vibrational motion implies the existence of excited vibrational
states. (Hint: Consider the nuclear density and show that the density is
always constant if only one state exists. Then consider a small admixture of
an excited state.)
18.41. Discuss a plot of the energy ratio E2/E1 for even-even nuclei. Indicate where
rotational and where vibrational spectra appear. Compare the corresponding
excitation energies E1.
18.42. Why can a state 3 + turn up in the level N = 3, but not N = 2, in Fig. 18.12?
18.43. Consider nonazimuthally symmetric quadrupole deformations,
�
R = R0 1+ �
m
R
α2mY m
2
�
�
.
α20 = β cos γ, α22 = α2,−2
= 1
√ 2 β sin γ.