46 PROBLEMS
becomes animperfectcrystalwith defects (of the Frenkeltype). The
number of interstitial sites into which an atom can enter is of the same
order as N. Let be the energy necessary to remove an atom from a
latticesite to an interstitial site. Show that, in the equilibrium state at
temperature such that the following relation is valid:
4.100 Two-Dimensional Debye Solid (Columbia,
Boston)
An atom confined to a surface may be thought of as an object “living” in a
two-dimensional world. There are a variety of ways to look at such an atom.
Suppose that the atomsadsorbed on the surface are not independent but
undergocollective oscillations as do the atoms in a Debye crystal. Unlike
the atoms in a Debye crystal, however, there are only two dimensions in
which these collective vibrations can occur.
a)b)Derive an expression for the number of normal modes between
and and, by thinking carefullyabout the total number of
vibrationalfrequencies for N atoms confined to a surface, rewrite it
in terms ofNand the maximumvibrationfrequencyallowed due
to the discreteness of the atoms.
Obtain an integral expression for the energy E for the two-dimensional
Debye crystal. Use this to determine the limiting form of the tempera-
ture dependence of the heat capacity (analogous to the Debye law)
as for the two-dimensional Debye crystal up to dimensionless
integrals.4.101 Einstein Specific Heat (Maryland, Boston)
Derive an expression for the average energy at a temperature of a
quantum harmonic oscillator having natural frequency
Assuming unrealistically (as Einstein did) that the normal-mode vi-
brations of a solid all have the same natural frequency (call it
find an expression for the heat capacity of an insulatingsolid.
Find the high-temperature limit for the heat capacity as calculated
in (b) and use it to obtain a numerical estimate for the heat capacity
of a piece of aninsulatingsolidhaving anumberdensity
of Would you expectthis to be a poor or a
good estimate for the high-temperature heat capacity of the material?
Please give reasons.a)b)c)