Heat Capacity of Crystalline Solids, Phonons, and the Speed of Sound Flashcards
(21 cards)
What is the internal energy of one mole of a crystalline solid?
U=3nRT
What is the heat capacity at constant volume for one mole of a crystalline solid?
CV = 3nR
What does the Dulong-Petit Law state?
C mol = 3R ≈25Jmol^−1K ^−1 at temperatures where all 6 d.o.f are accessible
How many degrees of freedom does each atom in a crystal have?
6 (3 kinetic + 3 potential)
Why does heat capacity decrease at low temperatures?
Vibrational energy is quantised; not all states are accessible when
𝑘𝐵𝑇 < ℎ𝑓0
What is the Einstein temperature 𝜃𝐸?
θE = hf0 / kB
What is the internal energy per direction in Einstein theory?
u = 1/2*hf0 + hf0 / exp(hf0 / kBT)−1
What happens to internal energy
𝑢 as 𝑇 → 0?
u → 1/2*hf0 (zero-point energy)
What is a phonon?
A quantised collective wave-like vibration of atoms in a crystal lattice.
Why is the Einstein model limited?
It assumes all atoms vibrate at the same frequency, ignoring phonon interactions.
What does real atomic vibration involve?
A range of frequencies from 0 to 𝑓max
What is Debye’s low-temperature heat capacity formula?
CV = ((12π^4) * R)) / 5 * (T/θD)^3
What is the Debye temperature 𝜃𝐷
𝜃𝐷 = ℎ𝑓max / 𝑘𝐵
Why is Debye theory better than Einstein’s at low T?
It accounts for a continuous spectrum of phonon frequencies.
What is the formula for longitudinal wave speed 𝑐L?
cL = √𝜇𝑌/𝜌
What is the formula for transverse wave speed
𝑐𝑇?
cT = √G/ρ
What is the relation between phonons and sound?
Both arise from lattice vibrations;
𝑐𝑇 approximates phonon propagation speed.
What is the definition of 𝜇 in wave speed formulas?
μ = (1-v) / (1−2ν)(1+ν)
where 𝜈 is Poisson’s ratio
Shear Modulus in terms of Poissons Ratio
G = Y / 2(1+v)
Bulk Modulus in terms of Poissons Ratio
K = Y / 3(1-2v)
