Consider a one-dimensional monatomic chain of atoms on a ring of length (Chapter 4)
PHYSICS · Subjective · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Lattice dynamics and thermal properties of solids · Lattice dynamics and thermal properties of solids
Condensed Matter Physics, Ch Einstein, Debye and the heat capacity of solids
Problem
Consider a one-dimensional monatomic chain of N atoms on a ring of length L (Chapter 4), with a single acoustic branch and sound speed v. Following the same reasoning that gave the three-dimensional density of modes in Section 6.5, find the one-dimensional density of modes g(ω) and the cut-off that gives the chain its correct total of N modes, and hence show that at low temperature the chain's heat capacity varies as CV∝T, not T3. You may use ∫0∞x/(ex−1)dx=π2/6 without proof, exactly as the chapter quotes its own three-dimensional integral.