Homework 4

NOTE: acceptable formats for turning in your homework are as follows:-
(1) handwritten clearly in blue or black ink
(2) handwritten clearly in 2B or darker pencil
(3) typewritten.
NO other types of work will be marked. In addition, if any of your working / logic is unclear, points will be docked -- in physics, it's not only getting the right answer, it's knowing how to get it using the correct method that matters.

[10] Q1: If equations 6.12, 6.14, and 6.18 in Kittel and Kroemer, concerning a classical ideal gas, are to be self-consistent, what can you say about the sign and the size of the chemical potential relative to the temperature (in energy units)? what strong inequality must hold among the total number of particles in the gas, its volume, the mass of the particles, and the temperature? To what extent would the inclusion of the spin of the particles (see comment (b) at the bottom of p. 162) affect your results?

[10] Q2: Kittel and Kroemer, problem 7-2.

[10] Q3: Kittel and Kroemer, problem 7-6. In part (b), you can assume that there are as many protons as neutrons in a white dwarf, in order to relate the number of electrons in the star to its mass. You might want to try 7--10 as well, but do not hand it in.

[10] Q4: In our derivation of the ground-state energy, U₀, of a fermion gas, we assumed that there is no interaction between the fermions. Yet, at fixed volume, U₀ is not directly proportional to the number of particles, N, in the gas. In a few lines, account for this seemingly counter-intuitive result.

[10] Q5: The second experiment that produced a Bose-Einstein condensate used sodium atoms with concentration n=10²⁰ atoms/m³. The mass of a sodium-23 atom is M=3.82 x 10^-26 kg. Only one spin state was populated.

• (a) Model the atom trap with a rigid box. At what value of the temperature T would you expect condensation to set in as T is decreased?
• (b) What temperature would be required for 90% of the atoms to be in the single-particle ground state?
• (c) A box of volume 1 cm³ contains 10¹⁴ sodium atoms at a temperature T=1.3 x 10^-6 K. The atoms form a dilute gas, and only one spin state is populated. Referring to figure 7.19 in KK, determine whether the heat capacity C_V decreases or increases with temperature when
•      (i) all atoms are sodium-23 atoms;
•     (ii) half are sodium-23 and half sodium-21 (ignore nuclear decay).

[10] Q6: Kittel and Kroemer, problem 7-12.