(a) Explain the drawbacks of Einstein's theory of specific heat and how it was overcome by Debye. (20 marks) (b) A neutron and a…

(a) Einstein’s theory of specific heat, its drawbacks and Debye’s correction

Background: classical equipartition gives the Dulong-Petit value C_V = 3R for all solids at all temperatures, but experiments show C_V falling to zero as T → 0.

Einstein (1907):

• Each atom is treated as three independent quantum harmonic oscillators, all with the same frequency ν_E, with energies (n + ½)hν_E.
• The mean energy per oscillator is hν_E/(e^(hν_E/kT) − 1) plus zero-point energy, giving C_V = 3R (Θ_E/T)² e^(Θ_E/T)/(e^(Θ_E/T) − 1)², with Θ_E = hν_E/k.
• Success: C_V → 3R at high T and C_V → 0 as T → 0, the first quantum explanation of the fall.

Drawbacks:

1. At low T it predicts an exponential fall, C_V ∝ (Θ_E/T)² e^(−Θ_E/T), much faster than the observed T³ law.
2. The single-frequency assumption is unrealistic: atoms are coupled, so vibrations are collective waves with a range of frequencies.
3. It ignores low-frequency (long-wavelength) acoustic modes, which are the ones still excited at low T.
4. Θ_E is an adjustable parameter with no link to elastic properties.

Debye (1912):

• The solid is treated as a continuous elastic medium carrying sound waves (phonons) of all frequencies from 0 up to a cut-off ν_D, chosen so that the total number of modes is 3N.
• Density of modes g(ν) = 9Nν²/ν_D³ (quadratic, from counting standing waves in a box).
• C_V = 9R (T/Θ_D)³ ∫₀^(Θ_D/T) x⁴eˣ/(eˣ − 1)² dx, with Θ_D = hν_D/k.
• High T (T ≫ Θ_D): C_V → 3R.
• Low T (T ≪ Θ_D): C_V = (12π⁴/5) R (T/Θ_D)³, the Debye T³ law, in agreement with experiment (Θ_D about 105 K for Pb, 428 K for Al, 2230 K for diamond).
• Θ_D can be computed from the velocity of sound, linking thermal and elastic properties.

Limitations of Debye: the real phonon spectrum is not exactly quadratic (optical branches, van Hove singularities), so Θ_D varies slightly with T, and in metals an electronic contribution γT dominates at very low T.

(b) Radiative capture n + p → d + γ

At rest, the initial momentum is zero, so the photon and deuteron share equal and opposite momenta p = E_γ/c.

Energy available (Q-value): Q = (m_n + m_p − m_d)c² = (1.0086652 + 1.0072766 − 2.013553) u × 931.5 MeV/u = 0.0023888 × 931.5 ≈ 2.225 MeV, the deuteron binding energy.

Energy conservation: Q = E_γ + E_R, with recoil energy E_R = p²/(2m_d) = E_γ²/(2m_d c²).

m_d c² = 2.013553 × 931.5 ≈ 1875.6 MeV, so E_R ≈ (2.225)²/(2 × 1875.6) MeV ≈ 1.32 × 10⁻³ MeV = 1.32 keV.

Hence E_γ = Q − E_R ≈ 2.225 − 0.0013 ≈ 2.224 MeV.

The recoil takes only about 0.06% of the energy, so it is negligible for most purposes. It matters only in precise measurements, for example when the capture γ-ray energy is used to determine the deuteron binding energy or the neutron mass.

(c) Resistance versus temperature, and Cooper pairs

Normal conductor:

• ρ(T) = ρ₀ + ρ_ph(T) (Matthiessen’s rule). Scattering by phonons gives ρ ∝ T at high T and ρ ∝ T⁵ at low T (Bloch-Grüneisen).
• As T → 0, ρ tends to a finite residual value ρ₀ set by impurities and defects.

Superconductor:

• Above T_c it behaves like a normal metal. At T_c the resistance falls abruptly to zero (below about 10⁻²³ Ω m, persistent currents last for years), as Kamerlingh Onnes found for Hg at 4.2 K in 1911.
• The transition is sharp in pure samples and is accompanied by the Meissner effect (flux expulsion), so a superconductor is more than a perfect conductor.

Formation of Cooper pairs (BCS, 1957):

• An electron moving through the lattice attracts nearby positive ions, creating a region of slightly higher positive charge that persists briefly (ions are slow).
• A second electron is attracted to this distortion. This phonon-mediated attraction can exceed the screened Coulomb repulsion for electrons near the Fermi surface.
• Cooper (1956) showed that any net attraction binds two electrons of opposite momenta and spins (k↑, −k↓) into a pair with zero total momentum and spin, with binding below the Fermi level.
• The pairs (size, or coherence length, of about 10²-10³ nm) overlap and condense into a single coherent ground state separated by an energy gap 2Δ (about 3.5 k_B T_c). Scattering cannot break pairs unless it supplies 2Δ, so current flows without resistance.
• Evidence: the isotope effect (T_c ∝ M^(−½)) confirms the role of lattice vibrations.