Question

1.(a) What is meant by the "Harmonic Approximation", as used in the context of describing the interatomic potential energy as a function of separation in solid state materials? [2] (b) Discuss the extent to which the "Harmonic Approximation" can be used to accurately predict the thermal vibration characteristics of crystalline systems and their ability to store thermal energy. Explicitly consider the temperature range over which these predictions seem to map sensibly onto real measurements. [8] (c) Consider the "Harmonic Approximation" in the context of thermal expansion and compare to predictions made using anharmonic descriptions, comment on the temperature range over which it seems to reasonably reflect measured behaviour and compare and contrast this to your answer for part (b). 151
a) b) and c)

Answer 1

Se! potential energy an by atom interatomic distance in a Crystal as experienced function of figure. a ro as shown in Below U 1 k (r-roje 1 re energy is minionum no 91 and as r increases the r=ro In Harmonic potential energy 2 (r-ro) Here - K Force Potential at increases decreases than er and is equilibrium distance. r - cupproximation is quadratic in interatomic Sepauertion ie U L Spring between neighbouring atons. Harmonic approximation potential energy is symmetric about Equilibrium point. (b) have ocas U20) + :3 (&_80) + at kas] By Eies kein approximation Chcroft osullators oscillating with frequency specific Heat q (T) to In 2 we 3123 Яo Wow we have all are Same 3험문 2 i e KAT Ł WE tw 2 é Tot 1) these OF Harmonic approximation ka upto valid. temperatures them is Less T <sook potential energy 0 (9) when take we other terms ie third order terms in the then it is not Symmetric about the Equilibrium position. Amplitude of oscillations very High. can be All Transport properties explanned using Anharmonic Elastic Constants are Results of Anharmonic approsin rero a = - Overalp - T Here (i > 200k are 12 approximation