Physics 224   Fall 2000

		Discoveries and Inventions of Modern Physics 		due 11:00 am Tuesday Nov. 7

PROBLEM SET 6

Oct. 12 Colloquium: ``Investigating Investigating Galaxy Formation in the Early Universe with Quasar Absorption Line Systems''
Jason Prochaska, Carnegie Observatories
3:30 pm, 101 Rowland Hall November 2 Colloquium: ``Gravitation and materials research with a cryogenic torsion pendulum" Prof. Riley Newman, UC Irvine
3:30 pm, 101 Rowland Hall

1.

(20 pts) In the Weiss (mean-field) theory of ferromagnetism the Gibbs free energy ( G=E(M)-HM-TS, where H is the externally applied magnetic field, M is the magnetization, E is the internal energy, and S is the entropy) has the form

G=G_o(T)+a(T)M²+b(T)M⁴+O(M⁶)-MH (1)

where G_o(T) is independent of M, and where the coefficient b(T) is a slowly varying function of T but a(T) is of the form a_o(T-T_C), T_C being the critical temperature in Weiss theory. Assume that T is close to T_C and that M is small. Using the fact that in thermal equilibrium M will take the value which minimizes G, find (a) the equilibrium value of Mfor H=0, for T>T_C and T<T_C; (b) the form of M at T_C as a function of H; (c) the zero-field differential susceptibility $\chi=(\partial M/\partial H)_{T,H=0}$for T>T_C and T<T_C; and (d) the discontinuity in the specific heat at constant H ( $C_H=-T(\partial^2G/\partial T^2)_H$) at the point T=T_C and H=0. In other words find the difference between the limits of C_H(T,H=0) as $T\rightarrow T_C$ from above and from below.

2.

Meissner Effect In deriving flux quantization in a superconductor, we found that the electric current is given by

$$\vec{j}=q\psi^{*}\vec{v}\psi=\frac{qn_p}{m}\left(\hbar\nabla\theta- \frac{q}{c}\vec{A}\right)$$ (2)

(a)

Use this and the appropriate Maxwell equation to show that

$$\nabla^2\vec{B}=\lambda^{-2}\vec{B}$$ (3)

What is $\lambda$ in terms of the density of Cooper pairs n_p, e, the mass of the electron m, and c? $\lambda$ is called the London penetration depth. (Hint: Use some vector identities to simplify the equations. See inside cover of Jackson's Classical Electrodyamics, for example.)

(b)

Suppose that $\vec{B}$ points along the z axis and only varies in the x direction. Suppose the superconductor fills the half space x>0 and there is vacuum for x<0. Show that B_x dies out exponentially as it penetrates the superconductor in the x direction. (Don't worry about the prefactor of the exponential.) In other words the magnetic field dies out exponentially as you go into the superconductor. This is the Meissner effect.

3.

AC Josephson Effect When a static DC voltage V is applied across a Josephson junction, an AC current results. To see how this comes about, notice that an electron pair experiences a potential energy difference qV on passing across the junction, where q=-2e. We can say that a pair on one side is at potential -eV and a pair on the other side is at +eV. Thus the equations of motion become

$$i\hbar\frac{\partial\psi_1}{\partial t}=\hbar T\psi_2-eV\psi_... ...i\hbar\frac{\partial\psi_2}{\partial t}=\hbar T\psi_1+eV\psi_2$$ (4)

where $\psi_1$is the superconducting order parameter on side 1:

$$\psi_1=\sqrt{n_1}e^{i\theta_1}$$ (5)

n₁ is the density of superconducting pairs on side 1. Similarly

$$\psi_2=\sqrt{n_2}e^{i\theta_2}$$ (6)

Assume that the superconductors are identical. Find the current density J as a function of time and of the phase difference $\delta(0)$. $\delta(0)=\theta_2-\theta_1$ is the phase difference at V=0. What is the angular frequency $\omega$at which the current oscillates when a voltage V is applied?