Consider a diatomic linear chain with two masses m1 and m2...

Consider a diatomic linear chain with two masses m1 and m2 connected alternatively by two different spring C1 and C2. The displacement of the two types of masses may be different, but we assume that the distance between a mass and the nearest neighbour of the same type is always a (Fig 1). a) Derive the phonon dispersions and plot them in the first Brillouin zone (specify the frequency at the zone boundaries and the maximum point). b) Describe the dispersion curve at long wavelength limit.

Transcript

00:01 Hello students, the phonon dispersion relation for a dynamic linear chain with two different masses and spring constant.

00:07 We can first define the equation of motion of a single atom in the chain.

00:13 So m1 d square x1 by dt square will be equal to minus c1 times x1 minus x2.

00:23 So here m1 is the mass of the atom, x1 is the displacement and c1 is the spring constant between the atom and its nearest neighbor.

00:32 So now we can use the fact that the atoms are connected by springs to write the coupled set of equations.

00:41 So we can write down that m2 d square x1 d square x2 by d2 for the second mass d2 square must be equal to minus c2 x2 minus x3.

00:56 So here the displacements are similarly done.

01:01 Now we can solve for a coupled set of equations of motion and the solution will be the solution for this equation will be x of n will be equal to a sine k a e raised to i omega t that is a solution for this thing right if you add all these values you get a constant here.

01:24 So n is a notation of index in the atom index n the atom is the amplitude of oscillation k is the wave vector omega is the frequency of the oscillation.

01:34 Now let's substitute the solution for displacement into the equation of motion.

01:41 So you get omega will be equal to square root of c1 by m1 plus c2 by m2 times sine k a right.

01:55 So here m1 and m2 are the masses of atoms and c1 c2 are the spring constants.

02:01 Now the first brilloing zone occurs the first brilloing zone occurs and the region in the cave space where the photon phonon dispersion relation is valid.

02:14 So the boundaries of the first brilloing zone will be k is equal to plus or minus pi by a.

02:21 So here a is the lattice constant.

02:24 So that is when this component will be pi right...

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