An earlier demonstration (Section 7.2) shows that adding a quartic term to the well…
PHYSICS · Multi-part · 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 Thermal expansion and lattice thermal conductivity
Problem
An earlier demonstration (Section 7.2) shows that adding a quartic term −ϵx4 to the well U(x)=21Kx2−βx3 shifts ⟨x⟩ by nothing at first order. Consider instead adding a general extra term −γxn, with n a positive integer, n≥3.
((a)) Show that this term shifts ⟨x⟩ at first order if and only if n is odd.
((b)) Hence say which of the next two Taylor terms beyond the cubic one, the quartic (x4) or the quintic (x5), could in principle add a further first-order contribution to the thermal expansion.