(a) (i) Discuss the Debye model of lattice specific heat. What are the limitations of the Debye model ? (10 marks) (ii) At very…

(a) Debye model and heat required (10 + 5 = 15 marks)

(i) Debye treated a crystal of N atoms as an elastic continuum whose 3N vibrational modes are phonons with a linear dispersion ω = vk and a common sound speed v, cut off at a maximum frequency ω_D so that exactly 3N modes exist: ω_D = v(6π²N/V)^(1/3). The Debye temperature is θ_D = ħω_D/k_B. The mode density is g(ω) = 9Nω²/ω_D³, and the lattice energy is U = 9Nk_BT (T/θ_D)³ ∫₀^(θ_D/T) x³/(eˣ − 1) dx, with x = ħω/k_BT. Hence, per kilomole,

C_V = 9R (T/θ_D)³ ∫₀^(θ_D/T) x⁴eˣ/(eˣ − 1)² dx.

For T ≫ θ_D, C_V → 3R ≈ 24.9 kJ kmol⁻¹ K⁻¹ (Dulong-Petit law). For T ≪ θ_D the integral tends to 4π⁴/15, giving C_V = (12π⁴/5) R (T/θ_D)³ ≈ 1944 (T/θ_D)³ kJ kmol⁻¹ K⁻¹, the T³ law. The model thus explains the fall of specific heat at low temperature, which Einstein's single-frequency model (exponential fall) could not.

Limitations: the solid is treated as an isotropic continuum with a single sound speed and a linear dispersion to the cut-off, whereas real dispersion flattens near the zone boundary and has different longitudinal and transverse speeds; optical branches (crystals with several atoms per cell) are ignored; the phonon spectrum is taken as harmonic, so thermal expansion and the C_P - C_V difference are not explained; the electronic contribution of metals (γT) is excluded; and θ_D is not constant but varies with T when fitted to experiment, the T³ form being accurate only below about θ_D/50.

(ii) At low temperature the heat capacity per kilomole is C = (12π⁴R/5)(T/θ_D)³, and the heat required is

Q = n ∫ C dT = n (12π⁴R/5θ_D³) (T₂⁴ − T₁⁴)/4 = n (3π⁴R/5θ_D³)(T₂⁴ − T₁⁴).

With n = 2 kmol, R = 8314 J kmol⁻¹ K⁻¹, θ_D = 281 K, T₁ = 10 K, T₂ = 50 K: T₂⁴ − T₁⁴ = 6.25×10⁶ − 1×10⁴ = 6.24×10⁶ K⁴ and θ_D³ = 2.219×10⁷ K³. Then 3π⁴/5 = 58.45, so

Q = 2 × 58.45 × 8314 × (6.24×10⁶/2.219×10⁷) = 2 × 1.367×10⁵ ≈ 2.73×10⁵ J.

(The T³ law is taken as given in the question; strictly it is safe only below about 6 K for rock salt, so the true heat would differ somewhat.)

(b) Photovoltaic effect and solar cell (15 marks)

The photovoltaic effect is the generation of a voltage and current in a junction device when it absorbs light, without an external bias. In a p-n junction solar cell, three processes are required for useful output.

First, absorption: a photon of energy hν ≥ E_g excites an electron across the gap, creating an electron-hole pair; photons with hν < E_g are not absorbed and the excess of hν over E_g is lost as heat. Second, separation: pairs generated within about a diffusion length of the depletion region reach it, and the built-in electric field sweeps electrons to the n side and holes to the p side, building up the photovoltage V_oc and a photocurrent I_L. Third, collection: the separated carriers must reach the front and back contacts and flow through the external load before they recombine, delivering power P = IV.

The efficiency is η = V_oc I_sc FF / P_in, where FF is the fill factor. The factors affecting it are:

• Band gap: a small gap wastes photon energy as heat (thermalisation) and a large gap fails to absorb the low-energy photons; the ideal single-junction limit (Shockley-Queisser) is about 33% near 1.3-1.4 eV, with silicon (1.1 eV) at about 29%.
• Reflection and shading: front-surface reflection, cured by anti-reflection coating and textured surfaces, and shadowing by the metal grid.
• Recombination: bulk and surface recombination, governed by minority-carrier lifetime and diffusion length, and by material purity and defects.
• Series and shunt resistance, which lower the fill factor.
• Temperature: V_oc falls by roughly 2 mV per kelvin per cell, so efficiency drops as the cell heats.
• Illumination: intensity (concentration raises V_oc) and spectrum (air mass), and cell design such as multi-junction stacks.

(c) SU(3) symmetry of baryons and mesons (20 marks)

The strong interaction is nearly independent of the flavour of the light quarks u, d and s, because their masses are small compared with the strong scale. This approximate flavour symmetry is the group SU(3), the unitary transformations of the triplet (u, d, s) with unit determinant, with eight generators (the Gell-Mann matrices). Two generators commute, giving the quantum numbers I₃ (third component of isospin) and hypercharge Y = B + S, related to charge by Q = I₃ + Y/2. The quarks form the fundamental triplet 3: u (I₃ = +½, Y = +⅓), d (-½, +⅓), s (0, -⅔); the antiquarks form the conjugate 3̄. Hadrons fall into irreducible representations, plotted on the (I₃, Y) plane.

Mesons are quark-antiquark pairs, 3 ⊗ 3̄ = 8 ⊕ 1: an octet plus a singlet. Baryons are three-quark states, 3 ⊗ 3 ⊗ 3 = 10 ⊕ 8 ⊕ 8 ⊕ 1: the spin-½ baryons form an octet and the spin-3/2 baryons a decuplet (predicting the Ω⁻ before its discovery). The octets are hexagons with two particles at the centre.

The baryon octet (J^P = ½⁺) contains p (uud), n (udd), Σ⁺ (uus), Σ⁰ (uds), Σ⁻ (dds), Λ (uds), Ξ⁰ (uss) and Ξ⁻ (dss).

The pseudoscalar meson octet (J^P = 0⁻) contains π⁺ (ud̄), π⁻ (dū), π⁰ and η (mixtures of uū, dd̄, ss̄), K⁺ (us̄), K⁰ (ds̄), K⁻ (sū) and K̄⁰ (sd̄); the singlet η′ completes the nonet. The antiparticles of the octet members lie at the reflected positions, so the meson octet is self-conjugate. The symmetry is broken because m_s exceeds m_u and m_d, which splits the masses within a multiplet according to the Gell-Mann-Okubo formula, M = M₀ + aY + b[I(I + 1) − Y²/4].