## Special topics, solutions

### Problem 1:  (C)

Orbit of the moon

Which number is reasonable?
The semi-major axis of the moon's orbit is 384399 km.
3.84*108 m/(3*108 m/s) = 1.28 s = time for light to make a one-way trip.

### Problem 2:  (C)

Semi-empirical binding energy formula

B(Z,N) = C1A - C2A2/3 - C3Z(Z-1)/A1/3 - C4(N-Z)2/A.
The liquid drop model of the nucleus takes into account that the forces on the nucleons on the surface are different from those on nucleons in the interior, where they are completely surrounded by other attracting nucleons.  This is similar to considering the surface tension to be a contributor to the energy of a tiny liquid drop.  The first term of the semi-empirical formula is the volume term.  It is proportional to the mass number A.  If we divide by A, this term alone would gives a constant prediction for B(Z,N)/A, which for large nuclei is nearly, but not completely, correct.
The second term in the formula is the surface term, which is proportional to A2/3.  It decreases the binding energy because nucleons near the surface of the nucleus are in a less deep potential well.  The surface term is most important for the binding energy of small nuclei, because the ratio of the second to the first term is largest for small A.
The third term of the semi-empirical formula is the Coulomb term.  Each pair of protons contributes to the Coulomb term equally.  The number of protons is Z, so the number of proton pairs is Z(Z-1)/2.  The electrostatic potential energy is proportional to the inverse distance.  The average inverse distance increases with the radius of the nucleus and therefore is proportional to A1/3.  The total electrostatic potential energy term is proportional to Z(Z-1)/A1/3, and it decreases the binding energy.
The fourth term of the formula expresses the fact that the binding energy decreases if a neutron is transformed into a proton, if the number of protons is larger than that of the neutrons.  The same holds if a proton is transformed into a neutron.

### Problem 3:  (C)

Nuclear Reactions

AX = 4 + 40 - 1,  ZX = 2 + 18 - 1.

### Problem 4:  (E)

Atomic and nuclear dimensions

atomic: ~10-10 m , nuclear: 10-15 - 10-14 m.

### Problem 5:  (E)

Nuclear decay, conservation laws

The free neutron decays with a mean life of 885.7 s ~ 14.7 min.
Why are the other processes not allowed?
(A)  charge conservation; (B)  lepton number conservation; lepton and barion number conservation;  (D) energy conservation

### Problem 6:  (E)

β decay

Examples of β- and β+ decay:
β- decay:  n --> p + e- + νe
β+ decay:  p --> n + e+ + νe(bar)

### Problem 7:  (E)

Momentum conservation

The more massive nucleus has less kinetic energy.

### Problem 8:  (A)

The hyperfine interaction

he hyperfine interaction is a coupling between electronic and nuclear magnetic moments.  The nuclear magnetic moment is proportional to the nuclear spin.

### Problem 9:  (B)

Specific heat in solids

Answers (B) and (E) cannot both be correct.
In thermodynamics and solid state physics, the Debye model is a method for estimating the phonon contribution to the specific heat capacity in a solid.  It treats the vibrations of the atomic lattice as phonons in a box, in contrast to the Einstein model, which treats the solid as many individual, non-interacting quantum harmonic oscillators. Debye's realized that there exist a maximum number of modes of vibration in a solid.  He pictured the vibrations as standing wave modes in the crystal. We call these modes phonons.  There exists a maximum allowed phonon frequency now called the Debye frequency.  The Debye model correctly predicts the low temperature dependence of the heat capacity, which is proportional to T3.

### Problem 10:  (E)

Properties of superconductors

Superconductors are perfect diamagnets.