1. A dilute system at thermodynamic equilibrium consists of 80 independent, indistinguishable particles. Each particle has four energy levels of energy 0, ", 4", and 10" with degeneracies of 100, 200, 400, and 800, respectively. The system is at a constant temperature T = "/fc_B, where is Boltzmann's constant.

(a) Calculate the partition function for this thermodynamic system. [2 marks]

(b) Find the number of particles in each energy level, at equilibrium. [2 marks]

(c) Using Boltzmann's relation, determine the entropy of this system. [5 marks]

(d) Determine the internal energy for this thermodynamic system. [4 marks]

(e) Calculate the entropy directly from the partition function. Comment on your

result, by comparing to your answer for part (c). [4 marks]

(f) Evaluate the Helmholtz free energy of the system. [4 marks]

(g) Comment on which parts of your answers (a-f) would change if the total number

of particles was increased to 8 million. [2 marks]

(h) Using your answers from parts (a-f), comment on how the distribution of particles

will change with an increase in temperature, and explain with reference to the first and second laws of thermodynamics. [2 marks]

2. (a) Determine the contribution to the Helmholtz free energy (kJ/mol) from the translational energy mode at 500 K and 5 bars for monatomic neon. [5 marks]

(b) Calculate the specific heat at constant volume (J/Kmol) for monatomic neutral phosphorus at 2500 K. Use the following electronic data:

State " (cm^-1)

^S3/2 ⁰

D5/2,3/2 11,368

Pi/2,3/2 18,730

^P1/2,3/2,5/2 56,⁰⁰⁰

[10 marks]

(c) Phosphorous forms tetrahedral molecules composed of four atoms, where each

atom forms bonds to the three others. Discuss in this case the major contributions that you would expect to find in calculating the specific heat, with reference to the degrees of freedom of the molecule. Rank these contributions in order of their probable importance, with physical justification. [5 marks]

(d) At low temperatures and higher pressures, explain why the dilute limit approximation and the equipartition theorem cease to apply. [5 marks]

3. In a classic paper in the Physical Review (159 98 (1967)), Loup Verlet demonstrated the ways in which computer “experiments" may be used to extract macroscopic descriptors of a physical system using the methods of statistical thermodynamics. This paper is attached.

(a) State two approaches for the computational simulation of thermodynamic properties that could be used, given a source of information about the potential energy of particles interacting in either the liquid or solid condensed phases. State which of these has been used in this work, and outline the basis of this approach briefly.

[3 marks]

(b) State an advantage that each of these approaches has over the other. [2 marks]

(c) State the hypothesis that describes how these two different computational approaches are related, and explain this connection. [2 marks]

(d) Discuss the general factors that limit the accuracy of a simulation such as this. Give three specific examples, and describe any dependencies between them.

[4 marks]

(e) In this particular work, melting is understood to occur at the temperature at which

the solid and liquid phases coexist. There is however a fundamental problem with finding this that arises from the computational description of the system. Find the section of the paper in which this problem is described, give the page number/section, and explain in your own words the way in which the author has addressed this. [4 marks]

(f) The relationship between temperature and pressure is discussed in this work,

as changing pressure will change the temperature at which the two phases are in equilibrium. Describe what the findings of this simulation tell us about the relative volumes of the solid and liquid phases. Give an example of a system for which the opposite finding would be expected. [3 marks]

(g) The accuracy of the Lennard-Jones potential for this system is considered, by comparing to experimental data. Find the section of the paper in which this is discussed, give the page number/section, and explain in your own words the way the conclusion the author has come to, and how they have decided this.

[3 marks]

(h) In a recent paper (J. Chem. Phys. 140, 044325 (2014)), clusters of argon atoms

containing 110, 195, and 301 atoms respectively were found to have melting termperatures of 45 K, 53 K, and 58 K respectively. Use these data to estimate the melting temperature of bulk argon, and discuss the accuracy of your prediction. Why can this approach be used for clusters but not for calculations performed using periodic boundary conditions? [4 marks]

4. (a) At atmospheric pressure helium cannot be solidified, although the helium dimer exists — albeit as a very weakly bound molecule. Increasing the pressure on bulk helium leads to solidification. At 2.5 MPa, helium has a melting point of about 1 K.

(i) Describe the bonding in the helium dimer including a sketch of the potential

energy as a function of the internuclear distance. [3 marks]

(ii) Name the contributions to the cohesive energy at T = 0 K and explain why helium does not become solid under atmospheric pressure. [3 marks]

(iii) Suggest two possible crystal structures for helium under high pressure. Explain your choices. [3 marks]

(b) The specific heat capacity of silver as a function of temperature T is shown in Figure 1. The Dirac temperature of silver is 215 K.

(i) State the main assumptions that went into Debye's calculation of the heat

capacity in your own words. [3 marks]

(ii) Describe two different approaches that yield the correct high-temperature

limit for the heat capacity of silver. Calculate the high-temperature result and compare with the given heat capacity curve. [4 marks]

(iii) State the temperature dependence of the heat capacity for low temperatures.

[2 marks]

(iv) Calculate the Debye cut-off frequency. Is your result sensible when compared to the frequencies in the given phonon dispersion spectrum?

[3 marks]

(c) Show the derivation of the two-dimensional density of states, DOS, per unit volume for the free Fermi gas. Explain every step of the derivation. Make a sketch of the DOS as a function of energy and state the main difference to the 3-dimensional DOS. [4 marks]

5. 

(a) Three phonon branches are found in the phonon spectrum of solid argon ³⁶Ar at a temperature of T =10 K as shown in Figure 2. Under atmospheric pressure Argon crystallizes in an fcc lattice. The lattice constant at T = 10 K is a = 5.313 A.

FIG. 2. Phonon-dispersion relations of ³⁶Ar along the three symmetry directions. The solid circles are the resolution-corrected measurements at 10 K. The dashed line is a general-force three- neighbor model fitted to the data and the solid line represents a theoretical calculation at 0 K based on the BB potential by Barker et al. (Ref. 17).

(i) Draw the conventional unit cell of solid argon showing the lattice constant.

Explain why there are three phonon branches and describe the types of the different branches observed. [5 marks]

(ii) The spectrum shows the phonon branches in the first Brillouin zone along three different symmetry directions. The phonon energy is plotted over the

reduced unit < =辭\k| (< 2 [0,1]). Explain where the factor of 嘉 comes from. [2 marks]

(iii) State the reason why there are only two phonon branches visible for the [100]

and [111] directions. [1 mark]

(iv) Show that the density p of solid ³⁶ Argon at T =10 Kis about 1600 kg m^-3.

[2 marks]

(v) Describe how you can find the speeds of sound (travelling along the different

symmetry directions) from the given phonon spectrum. Estimate the speed of sound corresponding to the upper phonon branch (labelled by 'L') in the [100] direction. [5 marks]

(vi) Compare your result with the speed of sound calculated by v_s = • The elastic constant for the considered vibration in this symmetry direction is

c = 4.25 GPa. [2 marks]

(b) The phonon dispersion of PbF₂ is shown below in Fig. 3 along with the conventional unit cell of PbF₂:

(i) Describe the phonon dispersion for PbF₂ and compare with the phonon spectrum of Argon. Include the number and type of phonon modes observed, symmetry directions shown and the behaviour of the different modes for small wave vectors and for wave vectors at the Brillouin zone boundary.

[6 marks]

Figure 3: Conventional unit cell and phonon dispersion spectrum of PbF₂ at T = 10 K (M. H. Dickens and M. T. Hutchings J. Phys. C 11 (1978) 461).

(ii) Explain for which phonon modes an optical excitation is possible.

[2 marks]

6. In the figure below band structures of two unknown semiconductors are shown in the top panels. The bottom panels show the optical absorption spectra of these two compounds and of GaAs as well as the conventional unit cell of the crystal structure of GaAs.

(a) This part relates to the optical absorption spectrum and crystal structure of GaAs shown in the bottom panels of Figure 4.

(i) Describe the crystal structure of GaAs. In particular, find the underlying

Bravais lattice, specify the basis, and find the coordination numbers of Ga and As. [4 marks]

(ii) From the optical absorption spectrum find an estimate for the band gap energy of GaAs. GaAs is represented by the red line. [2 marks]

(b) Consider the band structure displayed in the top right panel of Figure 4.

(i) Describe the information contained in the band structure.

(Hint: Include explanations about what is plotted on the x-axis, the zero of the energy scale, the band gap, and the information about D(E)).

[5 marks]

(ii) What can you say about the ratio between the effective masses of the conduction electrons and the holes in the valence band for this compound?

[2 marks]