Problem Assignments for Section 1, Spring 2011

The first and second Problem Assignments are due to be completed before Spring break,
but I will set individual deadlines before then by email. I expect *three*
of the following problems to be done by the agreed deadline. There will be a
second problem assignment on section 2 and parts of 3. Please
let me know which subjects interest you as a topic for a talk or mini-project for
later in the semester.

These problems test ideas of bond counting, elementary statistical mechanics, diffusion and surface structure. They have typically not been done ‘cold’, but have been used to open a discussion on topics which are best attempted through problem solving rather than by lecturing. There are further problems of a similar type in Desjonquères and Spanjaard, chapters 2 and 3. Make sure that you see me in time if you need some help- you are not expected to be able to do this assignment without some assistance, and you won't be able to complete these problems if you only think about them the night before.

### Preliminary Exercises: Some questions to get us started

1) Most of us don't have the definitions of the thermodynamic functions in foreground memory. Remind yourself of the definitions of entropy S, and the thermodynamic potentials U, F and G, and differentials dU, dF and dG. How are these functions used to discuss equilibrium phenomena?2) We also don't usually carry around statistical mechanical ideas and distributions in our head. Describe briefly the characteristics of the Micro-Canonical, Canonical and Grand-Canonical distributions, and what fluctuations are allowed in arriving at these distributions. (If you haven't done a course on Statistical Mechanics, this material can be understood as the semester proceeds).

3) What is the pressure difference between the inside and outside of a small soap bubble of radius r? Put in numbers to see the effect of surface tension and size, taking care over the units of pressure. Soap bubbles have radii in the 1-10 cm range. But what if we are interested in nano-particles of metals in the size range 1-1000 nm? Construct a table that illustrates the actual pressure differences as a function of surface tension (or energy, see class notes), units J/m

^{2}, and radius (nm, log scale).4) One reason for positive surface entropy is given in the notes, section 1.1, due to slower atomic vibrations (weaker bonding). Can you think of any other reasons, which might argue in the same, or indeed the opposite, direction?

5) More advanced: Why is γ (gamma) the surface density of (F-G), as Blakely (1973) puts it, on page 5?

### Problem 1.1: Bond counting and surface (internal) energies of a static lattice

Consider the (012) and (015) faces on a Kossel (simple cubic) crystal with 6 nearest neighbor bonds.a) Use the analysis of section 1.2 to consider the surface energy of this crystal in terms of the sublimation energy L and the lattice parameter a. Find the ratio of the surface energy of the (012) and (015) faces to that of the (001) face.

b) Repeat this exercise for the (012) and (015) faces of a f.c.c crystal with 12 nearest neighbor bonds. Compare your result with diagrams 7 and 9 (book figures 1.6 and 1.8), and comment on the relative values.

*Note: this problem can be done most readily by drawing the structure and counting bonds. However, there is a general approach you could consider, described by MacKenzie et al. (1962). In either case, it helps to keep your approach as general as you can for the face (01n), until you plug in particular values, n=2 or n=5. If the stereograms shown in figure 1.6 are not familiar, consult a crystallography book such as Kelly and Groves (1970), or use the web-based book comments for section 1.3 or go directly to a 1998 Sussex student project on the Properties of stereograms. This project needs some new web references that we can work on this year.*### Problem 1.2: Local equilibrium at the surface of a crystal at temperature T

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:a) Express the local equilibrium between adatom evaporation and the rate of arrival, R, of atoms from the vapor, to find the concentration of adatoms in monolayer (ML) units. Find the adatom concentration at 1000K if R = 1 ML/sec.

b) Use the chemical potential formulation to express the local equilibrium between the bulk crystal and the surface adatoms, to obtain their equilibrium concentrations at the same temperature, ignoring arrival from, or sublimation to the vapor. Hence decide whether the case described in a) corresponds to under- or over-saturation, and calculate the thermodynamic driving force in units of kT.

### Problem 1.4: Crystal growth at steps and the condensation coefficient

Consider a surface consisting of terraces of width d, separated by monatomic height steps.a) Set up a 1-dimensional equation describing the diffusion of adatoms to the steps in the presence of both adatom arrival and desorption. Explain what boundary conditions you use at the steps.

b) Show that the steady state profile of adatoms between the steps depends on the ratio cosh (x/X)/ cosh(d/2X), where X is the BCF length (section 1.3). Show that the fraction of atoms which get incorporated into the steps, the condensation coefficient is given by (2X/d)tanh(d/2X). Evaluate the limits (2X/d) when much greater than 1 and much less than 1, and give reasons why these limits are sensible.

### Problem 1.5: Surface reconstructions of particular crystals

Consider a surface structure in which you are interested. In metals this could be W and Mo(100) which have transitions below room temperature to 2x1, and 2x1-like structures, or in semiconductors the difference between 2x1, c4x2 and p2x2 superstructures on Si or GaAs(100).Use your chosen system to explore the relation between the structure, the symmetry and size of the surface unit cell, and the diffraction pattern, most obviously the LEED patterns in the literature.

*Note: if you want to study a more complex structure on a lower symmetry substrate, have a discussion first so that you don’t spend your time unprofitably.*