Physics 541: Condensed Matter Physics

Transcription

1 Physics 541: Condensed Matter Physics In-class Midterm Exam Wednesday, October 26, 2011 / 14:00 15:20 / CCIS Student s Name: Instructions There are 23 questions. You should attempt all of them. Mark your response on the test paper in the space provided. Please use a pen. If in answering a question you sketch a diagram, please provide meaningful labels. Aids of any kind including class notes, textbooks, cheat sheets, and calculators are not permitted. Good luck! 15 points multiple choice questions short answer mathematical points Useful integrals 0 0 x dx e x 1 = π2 6 x 3 dx e x 1 = π x 2 dx (1 e x 1 = 2ζ(3) = ) x 4 dx = 24ζ(5) = 24 ( e x )

2 Multiple choice questions (15 points) Answer by circling one of (a), (b), (c), etc. directly on the test paper. Be sure that your selection is clear and unambiguous. 1. Which of the following represents the state of highest symmetry? (a) amorphous solid (b) quasicrystal (c) homogenous, isotropic gas with no long-range correlations (d) stiff linear molecules in solution that form a nematic (e) crystal with a discrete set of rotation, reflection, and translation invariants 2. Which of the following exhibit collective excitations in response to shear? (yes / no) gas (yes / no) liquid (yes / no) glass (yes / no) regular solid 3. Consider a crystal composed of a single element with atomic number Z. In the Born- Oppenheimer approximation, we approximate the wavefunction of the system as a product of slow (ionic) and fast (electronic) degrees of freedom: Ψ = ψ slow (R 1, R 2,..., R N ) ψ fast (r 1, r 2,..., r ZN ). Integrating out the fast modes leaves us with an effective Hamiltonian H = i ( ) 2 h2 + V (2) (R i, R j ) + V (3) (R i, R j, R k ) + 2M R i i<j i<j<k Which of the following statements is incorrect? (a) V (2) is a function of the separation R i R j only (b) V (3) has large contributions proportional to (R i R j ) (R i R k ) when the bonding is strongly covalent (c) there can be no three-body terms (d) V (3) (R i, R j, R k ) = V (3) (R j, R k, R i ) = V (3) (R k, R i, R j ) (e) the bonding can t be ionic in character 2

3 4. Acoustic phonons with a dispersion relation ω q vq are occupied according to n q = 1 e β( hvq µ) + s What are the correct values of the chemical potential µ and the sign s? (a) µ = 0, s = 1 (b) µ set implicitly by q n q = N, s = 1 (c) µ = 0, s = +1 (d) µ = ε F, s = What does the +1/2 that appears in n q represent? (a) leading-order relativistic corrections (b) quantum zero-point motion (c) the Madelung constant (2 points) 6. To each description below, assign the most appropriate of the following phonon modes/branches: (a) acoustic (b) optical (c) longitudinal (d) transverse Write the corresponding letter in the space provided. Each letter can be appear more than once or not at all. exponentially suppressed contribution to the low-temperature heat capacity T 4 contribution to the low-temperature energy of a three-dimensional crystal atomic motion is perpendicular to the direction of wave propagation Goldstone modes that provide a pathway to restoring the full translational symmetry can only exist in a crystal with more than one atom per unit cell 7. Which of the following statements about solidification is incorrect? (a) systems with only an excluded volume interaction are athermal (b) cohesion in materials is governed by potential energy (electrostatic) but solidification is largely driven by entropic considerations (c) for classical hard spheres, the hcp and fcc configurations (ABAB and ABCABC layer stacking, respectively) are degenerate; both correspond to close packing (d) for classical hard spheres, the ordered phase is a state of low entropy 3

4 8. Which of the following statements is incorrect? (a) the liquid-solid phase transition is first order (b) the liquid-solid phase transition is continuous (c) a crystal has a latent heat of formation (d) liquid can be supercooled below its solidification temperature 9. What kind of (carbon valence orbital) hybridization takes place in the linear molecule acetylene (H C C H)? (a) sp (b) sp 2 (c) sp Which of the following materials is the least ionic in character? (a) Ge (b) GaAs (c) InSb (d) InAs (e) CdTe (f) ZnSe 11. We could arrange a two-dimensional ionic solid in which of the following patterns? (a) Kagomé (b) triangular (c) honeycomb (d) none of the them 12. Which of the following statements is incorrect? (a) the largest atoms are in the bottom-left portion of the periodic table (b) the atoms in the top-right portion of the periodic table are hardest to ionize (c) the halides generally have a high electron affinity (d) the noble gas elements are chemically inert (e) none of the them 4

5 (2 points) 13. Indicate whether these are true or false statements. (t / f) pure crystals are thermodynamically unstable to contamination by impurities (t / f) near special stoichiometric ratios, alloys of two elements can form superlattices (t / f) binary mixtures will always alloy rather than phase separate (t / f) the dynamics of phase separation are diffusive (t / f) quenching and annealing are synonyms for the same process Short answer questions (15 points) Try to provide answers in concise prose. At most a few sentences are required for each question. 14. Two regions of a crystal that have grown out from far separated nucleation centres are generically incompatible and meet along a surface of mismatches. What do we call this surface? (2 points) 15. Describe a method for obtaining a high-quality single crystal. (2 points) 16. Suppose that each atom in a crystal consists of an odd number of protons, neutrons, and electrons and is thus a fermion. Why are its vibrational modes bosonic? (2 points) 17. In simple materials, the acoustic phonons propagate in one longitudinal mode and two transverse modes that are (most often) degenerate. Why would they be degenerate? 5

6 (2 points) 18. What sets the upper limit for the phonon occupation number in a crystal? (3 points) 19. What are x-rays? What energy or wavelength must they have if they re to be used in diffraction experiments for the purpose of characterizing crystal structure? When x-rays scatter from a material, are they interacting with protons, neutrons, or electrons and why? (2 points) 20. In the context of an x-ray diffraction experiment, what is extinction? 21. The free energy of a two-component system must behave in what way (as a function of relative concentration) for the system to be unstable to phase separation? 6

7 Mathematical problems (20 points) b a (12 points) 22. The figure above shows a two-dimensional crystal made up of two kinds of atoms. As depicted, the crystal is compressed in one direction, with the length b slightly smaller than a. (a) Find lattice vectors and a basis for the crystal. 7

8 (b) Compute the area of the unit cell. 8

9 (c) Determine the corresponding reciprocal lattice vectors. 9

10 (d) Sketch the Brillouin zone that results from the Wigner-Seitz construction. 10

11 (e) Using the reciprocal lattice vectors g 1 and g 2 that you computed in part (c), parameterize arbitrary points in the reciprocal lattice with the linear combination G = G(n 1, n 2 ) = n 1 g 1 + n 2 g 2 for integers n 1 and n 2. Assuming atomic form factors f and f, determine the geometric structure factor as a function of n 1 and n 2. 11

12 (f) What condition(s) must hold for f and f if some of the x-ray scattering peaks are to vanish exactly? 12

13 (8 points) 23. The vector u(r + τ ) describes the harmonic displacement of an atom residing in a threedimensional crystal at position R + τ. Here, R denotes a point in the Bravais lattice and τ an element in the basis. Each spatial component of the displacement u a = u e a (with a = x, y, z) has the following second quantized form: u a (R + τ ) = 1 e iq R ξ a,τ (λ) (q) NMτ q λ h ( ) a q,λ + a q,λ. 2ω q,λ Here, the total number of atoms in the crystal N = N u N b is the product of the number of unit cells N u and the number of basis elements N b. The wavevector q ranges over the Brillouin zone and λ over the discrete set of phonon modes. The polarizations ξ a,τ (λ) which are just the eigenvectors of the dynamical matrix are normalized such that 1 N b τ a ξ (λ) a,τ (q)ξ (λ ) a,τ (q) = δ λ,λ. They can also be used to define an effective mass as follows: 1 N b τ a 1 M τ ξ (λ) a,τ (q)ξ (λ ) a,τ (q) = 1 M δλ,λ. The value of M has a very weak wavevector dependence that you should ignore. (a) Show that the squared displacement averaged over all atoms is u 2 = 1 u a (R + τ ) 2 = N R τ a h NM q λ 1 ω q,λ ( a q,λ a q,λ + 1 ). 2 (The expectation value is taken with respect to Harmonic oscillator states that are states of definite phonon occupation number n q,λ = a q,λ a q,λ). 13

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15 (b) Suppose that the crystal has a two-element basis (N b = 2) and thus three acoustic and three optical modes. For simplicity, assume ω q,1 = v q ω q,2 = ω q,3 = v q ω q,4 = ω q,5 = ω q,6 = ω E (a constant) In thermal equilibrium at temperature T, the phonons are populated according to the Bose-Einstein distribution, n q,λ = 1 e hω q,λ/k BT 1. (You must drop the divergent +1/2 contribution!) Show that at low temperatures the average atomic displacement is u 2 = 1 [( ) (kb T ) 2 2M v 3 12 h + 3 h ] e hω E/k B T. ω E v 3 15

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