Note: midterm exam is Thursday Oct. 20, 8:20-9:40 am. The coverage will include Kittel up through sec. 17. On this homework set (#6), the first problem is the only one directly relevant to the exam. Also please note that the policy about computing numbers on the exam will be --

Homework #6 Due Tuesday Oct. 25

1. Consider ^{4}He to be a classical ideal
gas.

a. At p=1 atm and T=4.2 K, what is the thermal wavelength l?2. Kittel, p. 85 problem 18.1

b. Using this picture for the vapor of 4He, what is the chemical potential m_{L}=(dG/dN)_{PT}of the liquid phase, given that it coexists at p=1 atm and T=4.2 K with the vapor (and boils when heat is added)?

c. The numerical value of m_{L}is probably not very different at (p,T)=(0,0) from the value at (p,T)=(1 atm, 4.2 K). Explain qualitatively the interpretation or source of this chemical potential at (p,T)=(0,0). It is OK to neglect such things as quantum physics and the difference between a liquid and a solid. Later we will discuss why it is a liquid rather than a solid.

3. Kittel, p. 85 problem 18.2

4. Kittel, p. 86 problem 18.4 As guidance, separate into parts:

a. Show that the moment µ_{z}in the direction of the field (z direction) is gL_{z}where L_{z}is the angular momentum and the gyromagnetic ratio g is Q/M times some constant. You may assume the charge is uniformly distributed on the surface of the spherical dust particle of mass M.

b. The classical version of statistical mechanics of rotating objects has a partition function which is an integral over the three components of angular momentum and the three angles needed to specify the orientation of the object times the exponential of the classical Hamiltonian (L^{2}/2I -gµ_{z}H) divided by k_{B}T. Compute <µ_{z}> and the susceptibility.

c. Explain why the answer appears different from the Debye-Langevin result 18.50, but is actually very much analogous.

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