Question

Lead has a fcc lattice of constant a = 0.494 nm. The Young's modulus of elasticity for lead is E_Y = 1.6 · 10¹⁰ Nm^{− 2}. If lead melts when the average amplitude of atomic vibrations is 15.8% of the distance between atoms (Lindemann criterion), calculate:

(a) The distance of adjacent atoms in the fcc lattice.

(b) Calculate the Debye temperature θ_D for lead if the phase velocity „c“ of phonon propagation for longitudinal and transverse modes in the fcc lattice has equal value

c² = E_y/ρ

where ρ is the mass density of lead. Compare the calculated temperature θ_D with the value read from the tables.

Hint: remember that the Debye wave vector k_D is related to the atomic density n = N/V by the formula: k_D³ = 6π²n.

(c) Lead melting temperature T_t if the degree of freedom of the harmonic oscillator holds

k_BT_t = E = 2mω_D²x₀² / 2 = mω_D²x₀²

where we used the virial theorem for the harmonic oscillator. Here x₀ is the amplitude of the oscillation at which melting occurs.

(d) The exact value of the melting point for lead is 600.6 K. What is the relative error with respect to the result under (c)?

Answer 1