The sublimation of a pure solid at equilibrium is given by the condition µ_{v} = µ_{s}. It is a standard result that the chemical potential of the vapor at low pressure p is
µ_{v} = kT ln (kT/pλ^{3}), where λ = h/(2πmkT)^{1/2} is the thermal de Broglie wavelength.
This can be rearranged to give the equilibrium vapor pressure p_{e} in terms of the chemical potential of the solid, as
p_{e} = (2πm/h^{2})^{3/2} (kT)^{5/2} exp(µ_{s}/kT).
Thus, to calculate the vapor pressure, we need a model of the chemical potential of the solid. A typical µ_{s} at low pressure is the 'quasiharmonic' model, which assumes harmonic vibrations of the solid, at its (given) lattice parameter. This free energy per particle
F/N = µ_{s} = U_{0} + <3hν/2> + 3kT<ln(1exp(hν/kT))>,
where we use angle brackets < > to mean average values. The (positive) sublimation energy at zero T , L_{0} = (U_{0} + <3hν/2>), where the first term is the (negative) energy per particle in the solid relative to vapor, and the second is the (positive) energy due to zeropoint vibrations.
The vapor pressure is significant typically at high temperatures, where the Einstein model of the solid is surprisingly realistic (provided thermal expansion is taken into account in U_{0}). Within this model (all 3N ν's are the same) and in the high T limit, we have
<ln(1exp(hν/kT))> = <ln (hν/kT)>, so that exp (µ_{s}/kT) = (hν/kT)^{3}exp(L_{0}/kT).
This gives p_{e} = (2πmν^{2})^{3/2} (kT)^{1/2} exp (L_{0} /kT), so that p_{e}T^{1/2} follows an Arrhenius law, and the preexponential depends on the lattice vibration frequency as ν^{3}. The absence of Planck's constant h in the answer shows that this is a classical effect, where equipartition of energy applies.
This equation is closely followed in practice over many decades of vapor pressure. The T^{1/2} term is slowly varying, and many tabulations of vapor pressure simply express log10(p_{e}) = A  B/T, and give the constants A and B. Calculations along these lines yield values for L_{0} and ν. Some curves are given in diagrams 10 (book figure 1.9) and 11 (book figure 1.10), and extracted values using the above equations are given in Table 1. Note that the sublimation energies are accurately known: the frequencies are good to maybe ±20%, but depend on the use of (approximate) Einstein model.
Element  Lattice Constant (a_{0}) nm 
Sublimation Energy (L_{0}) eV or K 
Einstein Frequency ν (THz) 
Metals: Ag

0.4086 (fcc) at RT

2.95 ± 0.01 eV

4

Fe

0.2866 (bcc)

4.28 ± 0.02

11

Semiconductors: Si

0.5430 (diam)

4.63 ± 0.04

15

Ge

0.5658

3.83 ± 0.02

6

Van der Waals: Ar

0.5368 (fcc) at 50K

84.5 meV or 981 K

1.02

Kr

0.5692

120 meV or 1394 K

0.84

Xe

0.6166

167 meV or 1937 K

0.73

The thing to understand about the above calculation is that the vapor pressure does not depend on the structure of the surface, which acts simply as an intermediary: i.e., the surface is 'doing its own thing' in equilibrium with both the crystal and the vapor. The surface of a Kossel crystal can be visualized by Monte Carlo (or equivalent) simulations, as indicated in diagram 12 (book diagram 1.11). At low T, the terraces are almost smooth, with few adatoms or vacancies (see diagram 4 (book figure 1.4) for these terms). As T is raised, then the surface becomes rougher, and eventually has a finite interface width. We might look at these studies in more detail later: there are distinct roughening and melting transitions at surfaces, each of them specific to each {hkl} crystal face. The simplest MC calculations in the so called SOS (solid on solid) model show the first but not the second transition.
This picture of a fluctuating surface which doesn't influence the vapor pressure applies to the equilibrium case; what happens if we are not at equilibrium? The classic paper in this field is the second reference quoted, known as BCF, and much quoted in the Crystal Growth literature. We have to consider the presence of kinks and ledges, and also (extrinsic) defects, in particular screw dislocations. This paper, and the developments from it, are quite mathematical, so we will only look at a few simple cases here, in order to introduce some terms and establish some ways of looking at surface processes.
First, the idea of supersaturation S = (p/p_{e}), and thermodynamic driving force, Δµ = kT ln S. Δµ is clearly zero in equilibrium, is positive during condensation, negative during sublimation / evaporation. The deposition rate or flux (R or F are used in the literature) is related, using kinetic theory, to p as R = p/(2πmkT)^{1/2}.
Second, the idea that an atom can adsorb on the surface, becoming an adatom, with a (positive) adsorption energy E_{a}, relative to zero in the vapor. (Sometimes this is called a desorption energy, and the symbols for all these terms vary wildly). The rate at which the adatom desorbs is (very roughly) given by νexp(E_{a}/kT), where we might want to specify the preexponential frequency as ν_{a} to distinguish it from other frequencies; it may vary relatively slowly (not exponentially) with T.
Third, the adatom can diffuse over the surface, with energy E_{d} and corresponding preexponential ν_{d}. We expect E_{d} < E_{a}, maybe much less. The adatom diffusion coefficient is then (jump distance a) approximately
The main points that follow from the above considerations are:
1. Crystal growth (or sublimation) is difficult on a perfect terrace, and substantial supersaturation (undersaturation) is required. When growth does occur, it proceeds through nucleation and growth stages, with monolayer thick islands (pits) having to be nucleated before growth can proceed. Early MC studies of these effects are seen in diagram 13 (book figure 1.12).
2. A ledge, or step on the surface captures arriving atoms within a zone of width x_{s} either side of the step, statistically speaking. If there are only individual steps running across the terrace, then these will eventually grow out, and the resulting terrace will grow much slower (as in point 1). In general, rough surfaces grow faster than smooth surfaces, so that the final 'growth form' consists entirely of slow growing faces.
3. The presence of a screw dislocation in the crystal provides a step (or multiple step), which spirals under the flux of adatoms (diagram 14, or book figure 1.13). This provides a mechanism for continuing growth at modest supersaturation. Detailed study shows that the growth velocity depends quadratically on the supersaturation for this mechanism, and exponentially for mechanism 1 (book figure 1.14).
Consider the (001) face of a f.c.c crystal with 12 nearest neighbor bonds, and (small concentrations of) adatoms and vacancies at this surface. The sublimation energy is 3eV and the frequency factor is 10 Thz. Use the appropriate formulations of section 1.3 to do the following:
Effects of vacancies and/or lattice vibrations on the sublimation pressure
Consider the model of the vapor pressure of a solid described in section 1.3, table 1.1 and diagram 10 (book figure 1.9). This model neglects the effects of vacancies, and the model of lattice vibrations is only a first approximation.