The atoms having in each case, the mass m and are located at positions with x i i ∈ Z. The rest position of the ith atom is i · a where a is the lattice constant of the crystal.
The interaction between the atoms can be modeled in a simple approximation, as a spring force of the spring constant D between adjacent atoms. The i-th atom causes therefore the (i - 1) th power of the atom has a size Fi→i−1 = D (xi − xi−1 − a).
(1) Set the equation of motion for the position of the ith atom in the crystal lattice and show that the equations of motion of the atoms are solved by standing waves of the form
xi (t) x = sin (a i k) sin (ω t) + a i. Look for the solution ω as a function of D, m, a and k, and determine the maximum value of ω as a function of the parameters of the crystal. Also sketch the history of ω as a function of k.
|k | = 2 π/λ
ω = 2 π f
The Attempt at a Solution
Equation of motion for the position of the i-th atom:
Mx ̈i = C(xi+1 + xi-1 2xi)
I don´t know how i should go on.