Distinct modes of oscillation in a 3D crystal lattice

[Fek]

Homework Statement

Silicon crystalises in a cubic structure whose lattice is face-centred, with a basis [000],[1/4,1/4,1/4].
How many optic/acoustic modes are to be found in the phonon dispersion diagram for silicon. How many distinct branches would you expect along the [100] direction?

Homework Equations

The Attempt at a Solution

2 atoms in the primitive unit cell, therefore expect 3N degrees of freedom, so a total of 6 modes.
I also have it on good authority that you expect 3(N-1) optical modes and N acoustic modes, but I do not see why. So a total of 3 of each modes
For distinct modes, imagining looking at a lattice of atoms joined with springs down the x direction, I would imagine the modes that vibrate in the x-direction would be invisible, whereas the ones in y and z would be visible (so 1 acoustic and 1 optical invisible).
I think this because when looking at dipole radiation from an electron in a magnetic field the mode identified with linear motion in the direction of the B field (delta Mz = 0), you cannot see it when looking along the B field. I don't get this either.

None of this really makes sense to me! Many thanks for any help.

 

[mark726]

Thank you for your question. Let me break down the solution for you.

Firstly, let's address the number of modes in the phonon dispersion diagram for silicon. As you correctly stated, there are 2 atoms in the primitive unit cell of silicon, so we expect 3N degrees of freedom, where N is the number of atoms in the unit cell. In this case, N=2, so we expect a total of 6 modes.

Now, let's look at the breakdown of these modes into optical and acoustic modes. As you mentioned, we expect 3(N-1) optical modes and N acoustic modes. This comes from the fact that for a face-centred cubic structure, there are 3 distinct optical modes (corresponding to the 3 different directions of the unit cell) and N-1 acoustic modes (since the acoustic modes are associated with lattice vibrations and one of these modes is already accounted for by the overall translation of the lattice). In this case, N=2, so we expect 3(2-1)=3 optical modes and 2 acoustic modes.

Finally, let's address the number of distinct branches along the [100] direction. As you correctly noted, the modes that vibrate in the x-direction would be invisible when looking along the [100] direction. This is because the [100] direction is perpendicular to the x-direction. Therefore, we would expect 1 acoustic and 1 optical mode to be invisible along the [100] direction, leaving us with 2 distinct branches.

I hope this helps to clarify the solution for you. If you have any further questions, please don't hesitate to ask.