Problem Set

Problem 1. Calculate the packing fraction of the simple cubic lattice. Compare it with BCC and FCC, And explain why SC is rarely observed in elemental metals.

Problem 2. Aluminium is FCC with $a = 0.405$ nm and $M = 26.98$ g/mol. Calculate the theoretical Density and compare with the experimental value ($2.70\ \mathrm{g}/cm^3$).

Problem 3. A plane intercepts the crystallographic axes at $2a$, $3b$And $\infty c$. Determine the Miller indices. A direction passes through the origin and the point $(1, -1, 0)$ in units of lattice Constants. Write the direction indices.

Problem 4. Magnesium is HCP with $a = 0.321$ nm, $c = 0.521$ nm. Calculate the ideal $c/a$ ratio And the actual ratio. How many atoms are in the conventional cell?

Problem 5. For a BCC lattice with lattice constant $a$Find the reciprocal lattice vectors and show That the reciprocal lattice is FCC with conventional cubic constant $4\pi/a$.

Problem 6. Construct the first three Brillouin zones of a 2D square lattice. Show that all three Zones have the same area.

Problem 7. A powder sample of copper (FCC, $a = 0.3615$ nm) is illuminated with Cu $K_\alpha$ Radiation ($\lambda = 0.15406$ nm). Calculate the $2\theta$ angles of the first five diffraction Peaks and identify their $(hkl)$ indices.

Problem 8. Calculate the structure factor for the CsCl structure (simple cubic with a two-atom Basis: Cs at $(0,0,0)$ and Cl at $(a/2, a/2, a/2)$). Show that there are no systematic absences.

Problem 9. Derive the structure factor for the perovskite structure (e.g., SrTiO$_3$: Sr at corners, Ti at body centre, O at face centres). Identify which reflections depend on which atoms.

Problem 10. Derive the phonon dispersion relation for a 1D monatomic chain with nearest-neighbour Spring constant $K_1$ and next-nearest-neighbour spring constant $K_2$. Show that the maximum Frequency increases relative to the nearest-neighbour-only case.

Problem 11. The Debye temperature of diamond is $\Theta_D = 2230$ K. Calculate the lattice specific Heat at $T = 100$ K and $T = 500$ K. At what temperature does the Debye $T^3$ law give a 1% Accurate approximation?

Problem 12. Compare the Einstein and Debye predictions for $C_V/C_{\mathrm{Dulong}--Petit}$ as a Function of $T/\Theta$ at $T/\Theta = 0.1$, $0.5$And $1.0$.

Problem 13. Sodium has $n = 2.65 \times 10^{28}\ \mathrm{m}^{-3}$ conduction electrons. Calculate The Fermi energy, Fermi wave vector, Fermi velocity, and Fermi temperature.

Problem 14. Using the tight-binding model for a 1D chain with nearest-neighbour hopping $t$: (a) find the effective mass at the band bottom and band top, and (b) calculate the density of states $g(\varepsilon)$ and show it diverges at the band edges.

Problem 15. A silicon sample is doped with $N_D = 10^{22}\ \mathrm{m}^{-3}$ phosphorus atoms. Calculate the electron and hole concentrations at 300 K ($n_i = 1.5 \times 10^{16}\ \mathrm{m}^{-3}$) And the position of the Fermi level relative to the conduction band edge.

Problem 16. A p-n junction is made from Si with $N_A = 10^{23}\ \mathrm{m}^{-3}$ and $N_D = 10^{22}\ \mathrm{m}^{-3}$. Calculate the built-in potential and the depletion width at 300 K. ($\varepsilon_s = 11.7\varepsilon_0$ for Si.)

Problem 17. A classical paramagnetic salt contains $N = 2.69 \times 10^{27}$ spins/m$^3$ with $J = S = 1/2$ and $g = 2$. Calculate the magnetisation in a field of $B = 1$ T at $T = 300$ K and $T = 4$ K.

Problem 18. Using the mean-field theory, derive the Curie—Weiss law $\chi = C/(T - T_C)$ for a Ferromagnet above $T_C$. Express $C$ in terms of $N$, $\mu$And $k_B$.

Hints and Selected Results:

• Problem 1: \mathrm{APF_}{\mathrm{SC} = \pi/6 \approx 0.524}. SC has the lowest packing efficiency of the three cubic structures, which is why it is energetically unfavourable for most metals (polonium being the exception).
• Problem 3: Reciprocals of $(2, 3, \infty)$ are $(1/2, 1/3, 0)$Giving $(3, 2, 0)$. Direction: $[1\bar{1}0]$.
• Problem 4: Ideal $c/a = \sqrt{8/3} \approx 1.633$. Actual $c/a = 1.623$. 6 atoms per conventional cell.
• Problem 5: $\mathbf{b}_1 = (2\pi/a)(\hat{y} + \hat{z} - \hat{x})$Etc. The 8 nearest reciprocal lattice points at $(\pm 2\pi/a)(\pm 1, \pm 1, \pm 1)/2$ form an FCC pattern.
• Problem 7: First five FCC reflections: (111), (200), (220), (311), (222). Use $2d\sin\theta = \lambda$ with $d = a/\sqrt{h^2+k^2+l^2}$.
• Problem 13: $\varepsilon_F = 3.24$ eV, $k_F = 9.22 \times 10^9$ m$^{-1}$, $v_F = 1.07 \times 10^6$ m/s, $T_F = 3.76 \times 10^4$ K.
• Problem 15: $n = N_D = 10^{22}$ m$^{-3}$, $p = n_i^2/N_D = 2.25 \times 10^{10}$ m$^{-3}$ $E_c - E_F = k_B T\ln(N_c/N_D) \approx 0.214$ eV.
• Problem 16: $V_0 = 0.716$ V, $W \approx 0.35$ $\mu$M.
• Problem 17: At 300 K: $M \approx \mu_0 N \mu_B^2 B/(3k_B T) = 0.078$ A/m. At 4 K: the classical approximation breaks down; use the Brillouin function $B_{1/2}(x) = \tanh(x)$ with $x = \mu_B B/(k_B T)$.

Problem 19. A two-dimensional electron gas (2DEG) is confined to a square of side $L$. Derive the density of states $g(\varepsilon)$ per unit area. Show that $g(\varepsilon)$ is constant for each subband. How does the Fermi energy depend on the electron density $n_{2D}$ in the lowest subband?

Problem 20. Using the Kronig-Penney model, derive the condition for allowed energy bands in a 1D periodic potential $V(x) = V_0$ for $0 < x < b$ and $V(x) = 0$ for $-a < x < 0$ with period $a + b$. Show that band gaps open at $k = n\pi/(a+b)$, and find the gap width in the limit of a weak potential ($V_0 \to 0$).

Problem 21. A type-II superconductor has critical fields $H_{c1} = 0.02$ T and $H_{c2} = 15$ T. (a) Estimate the coherence length $\xi$ and penetration depth $\lambda$. (b) Calculate the Ginzburg-Landau parameter $\kappa = \lambda/\xi$. (c) Determine whether the superconductor is type-I or type-II. (d) Sketch the magnetisation curve $M(H)$.

Problem 22. Explain the quantum Hall effect. For a 2DEG in a strong perpendicular magnetic field $B$, the Hall conductivity $\sigma_{xy}$ exhibits plateaus at integer multiples of $e^2/h$. Derive the condition for Landau level filling $\nu = n h/(eB)$ and show that when the Fermi level lies between Landau levels, dissipationless transport occurs.

Problem 23. Calculate the electron mobility $\mu = e\tau/m^*$ in Si at 300 K given the momentum relaxation time $\tau \approx 10^{-13}$ s and the effective mass $m^* \approx 0.26 m_e$. Estimate the drift velocity in an electric field of $10^5$ V/m. Compare with the saturation velocity in Si ($v_{\text{sat}} \approx 10^5$ m/s).