Physics 418 | Final Exam | Spring 2002 |

1) [40 points]

Consider a three dimensional classical ideal gas of atoms of mass m, moving in a potential

The infinite potential for x<0 may be viewed as a rigid wall filling the y-z plane at x=0. The atoms, therefore, are attracted to this wall, but they move freely in the y and z directions. Let T be the temperature and n = N/A be the total number of atoms per unit area of the wall.

a) Calculate the local density n(x), the number of atoms per unit volume, at distance x from the wall. Note that: . [10 points]

b) Find the pressure that the atoms exert on the wall. How does it vary with temperature? [10 points]

c) Calculate the energy and specific heat per unit area of the wall. [10 points]

d) Find the chemical potential of the gas. [10 points]

2) [30 points]

A linear molecule of N identical atoms has a vibrational spectrum given by the angular frequencies

, for m = 1,2, ..., N-1

a) Show, by explicit calculation, that the vibrational contribution to the specific heat of this molecule at very high temperature, , is

C =
(N-1)k_{B}
[10 points]

b) Show that for lower temperatures, such that ,

C ~ T [10 points]

c) How does C vary with temperature at very low temperatures, such that ? [10 points]

3) [30 points]

Landau theory of a 1st order phase transition:

a) Consider a system with an order parameter m, and free energy density

where d and u are positive constants, and a varies linearly with
temperature, .
For zero ordering field, h=0, the state of the system will be that value
of m that gives the *global* minimum of f(m).

Show that as T decreases from large values, there is a 1st order phase transition at:

and that the jump in the order parameter at this transition is:

Make a *sketch* of f(m) vs. m for T> T*, T = T*, and T <
T*.
[10 points]

[Note that the 1st order transition above occurs at a higher temperature
than the T_{o} where there would be a 2nd order transition if the
cubic term was absent, i.e. d=0. ]

b) Consider now an ordering field h, which is the thermodynamic conjugate to the order parameter m, i.e. h = df/dm.

Make a sketch of the physical h(m) vs. m for T> T*, T = T*, and T < T*.

The susceptibility is defined to be = dm/dh. How does behave as T passes through T* at h=0? Be specific. [10 points]

c) While at h=0, the transition at T* is 1st order, there is a 2nd order
critical point else where in the h-T plane. Find the values h_{c}
and T_{c} that locate this critical point, and find the value
m_{c}
of the order paramter at the critical point. Express your answers in terms
of the constants a_{o}, T_{o}, d and
u.
[10 points]