Generalization of Ashcroft and Mermin Problems 22.2 and 22.3.
Consider a diatomic chain consisting of alternating masses
M_1 and M_2 > M_1.
The lattice constant (i.e., the equilibrium separation between adjacent
masses of the same type) is a.
The masses interact via nearest-neighbor harmonic potentials, with
spring constants G and K>G alternating along the chain.
- (a)
- Calculate the dispersion relation for lattice vibrations in this system.
Your answer should reduce in the appropriate special cases to Ashcroft
and Mermin Eqs. (22.37) and (22.94).
- (b)
- Determine the eigenvector corresponding to each normal mode at
wavevector q=0, for each of the special cases
(i) M_1 = M_2 and (ii) G=K.
- (c)
- Repeat part (b) for q=pi/a.
- (d)
- Show how the normal modes you have derived reduce to those
of a monatomic chain when (i) M_1 = M_2 and G -> K;
and (ii) G=K and M_1 -> M_2. Consider not only
the dispersion relation, but also the eigenvectors at q=0
and q=pi/a. Is there any inconsistency between (i) and (ii)?