- Solve As Soon As Possible All Parts Of The Question 1 (88.82 KiB) Viewed 59 times
solve as soon as possible all parts of the question
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solve as soon as possible all parts of the question
solve as soon as possible all parts of the question
4.11 A linear chain consists of N+1 atoms of type A, whose mass is m and N + 1 atoms of type B, whose mass is M. The two types of atom alternate along the chain. The type-A atom at one end and the type-B atom at the other end are fixed a distance L = (2N+ 1)a apart, while the remaining atoms oscillate in the direction along the chain. Harmonic forces characterized by a 'spring constant' K act between neighbouring atoms. (a) Find the dispersion relation for the normal modes of vibration of this chain. You should find that it has a low-frequency branch (called the acoustic branch) and a high-frequency branch (called the optical branch). (b) Investigate the behaviour of the dispersion relation when m becomes equal to M. (c) Identify circumstances under which the frequency of the optical branch is almost independent of wavelength. Under the conditions of (c), we construct a simplified version of the excitation spectrum by assuming that all the optical modes have the same frequency wo, while the acoustic modes can be treated in the Debye approximation. Assuming that wo> wp. the density of states is g(w) = L ω < wo ле N8(w wo) w > wr) - where is the speed of sound. (d) Find the Debye frequency wp for this model. (e) Find the dependence of the specific heat on temperature for kT << hop and KT »hwo.
4.11 A linear chain consists of N+1 atoms of type A, whose mass is m and N + 1 atoms of type B, whose mass is M. The two types of atom alternate along the chain. The type-A atom at one end and the type-B atom at the other end are fixed a distance L = (2N+ 1)a apart, while the remaining atoms oscillate in the direction along the chain. Harmonic forces characterized by a 'spring constant' K act between neighbouring atoms. (a) Find the dispersion relation for the normal modes of vibration of this chain. You should find that it has a low-frequency branch (called the acoustic branch) and a high-frequency branch (called the optical branch). (b) Investigate the behaviour of the dispersion relation when m becomes equal to M. (c) Identify circumstances under which the frequency of the optical branch is almost independent of wavelength. Under the conditions of (c), we construct a simplified version of the excitation spectrum by assuming that all the optical modes have the same frequency wo, while the acoustic modes can be treated in the Debye approximation. Assuming that wo> wp. the density of states is g(w) = L ω < wo ле N8(w wo) w > wr) - where is the speed of sound. (d) Find the Debye frequency wp for this model. (e) Find the dependence of the specific heat on temperature for kT << hop and KT »hwo.