Physics 251 Spring 1999
Modern Physics
Homework Assignment # 10, Due (in class) Monday April 17

Reading Assignment: Krane, Chapter 10 section 10.5, especially Eqs. 10.19-10.22; Chapter 11 sections 6 (especially pp. 352-354), 8, 9 Problem Assignment:

1.

In class the formula was derived for the mean level of thermal excitation of a harmonic oscillator,

$$\bar{n}=\frac{1}{e^{\hbar\omega/kT}-1}.$$

The vibrational frequency of the Cl₂ molecule is $\nu$=16.8 THz. Calculate the mean level of thermal excitation of the Cl₂ molecular vibration if T=77K, 300K, and 1000K. (77K is a popular temperature for experiments, because it is the temperature of a tank of boiling liquid nitrogen.)

2.

Krane, Chapter 11, p.370, #23.

3.

Krane, Chapter 11, p.370, #24.

4.

Krane, Chapter 11, p.371, #25.

Note: In class the formula was derived for the probability of occupation by an electron of an orbital of energy $\epsilon$,if the orbital is in thermal equilibrium with a bath of electrons at temperature T and chemical potential $\mu$,

$$\bar{n}=\frac{1}{e^{(\epsilon-\mu)/kT}+1}.$$

In solid state physics, the chemical potential $\mu$ for electrons is often called the ``Fermi energy." It lies between the energy of the last occupied state and the first unoccupied state, if the temperature is not too high. In a metal, we can often assume that the orbitals of ``conduction electrons'' are free particle plane-wave states, with energy $\epsilon=\hbar^2 k^2/2m$ where $k=2\pi/\lambda$ is the wave-vector. Boundary conditions on the walls of the ``box'' (the ends of the metal) force k to be quantized, $\vec{k}=(\pi/L)(l,m,n)$ where (l,m,n) are positive integers and L is the size of the box. If n=N/L<sup>3</sup> is the number of electrons per unit volume, then the ``Fermi energy" is $\epsilon_F=\hbar^2 k_F^2 /2m$ and the ``Fermi wave-vector'' k_F is $(3\pi^2 n)^{1/3}$. This is derived from a method of counting states which is in Krane, Chapter 10. Although the algebra is not complicated, I think it is a more proper topic for a different course. I will not require you to memorize these formulas. If the subject is covered on the final exam, the formulas will be available. You should understand the ideas.