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)kB                 [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 To 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 hc and Tc that locate this critical point, and find the value mc of the order paramter at the critical point. Express your answers in terms of the constants ao, To, d and u.      [10 points]