Needing explanation of Kittel Intro to Solid States chapter 4.

[lylos]

In chapter 4, Kittel is setting up the differential equations for the forces involved in lattice vibrations. I understand the concept, the only problem I'm running into is why he chose the solution he did.

Here goes:
He states that the differential equation is
$$M\ddot{u_{s}}=C(u_{s+1}+u_{s-1}-2u_{s})$$

I understand this. He then goes to say that all displacements must have some time dependence of
$$exp(-Iwt)$$

Again, I understand why he says this. Then if you put this into the differential equation you get the following
$$-w^2 u_{s} M=C(u_{s+1}+u_{s-1}-2u_{s})$$

He then states "This is a difference equation in the displacements u and has traveling wave solutions of the form:"
$$u_{s\pm1}=uexp(IsKa)exp(\pm IKa)$$

This is where I'm running into the issue. Why does he have uexp(IsKa)exp(IKa) to the solution of u(s+-1)?

 

[marcusl]

The difference equation is a second order differential equation (a wave equation, to be exact) like the one for the time dependence, but written instead in the form of discrete differences. You can verify this starting with the definition of the derivative

$$\frac{du}{dx}=\lim_{\Delta x \rightarrow 0} \frac{u(s)-u(s-1)}{\Delta x}$$

Using the difference form is valid since the spacing a between planes is very small. Now take the second derivative to get the same second-order finite difference equation that you wrote above. The solutions to this discretized wave equation are traveling waves

$$u(s+p)=u(0)e^{i[(s+p)Ka-\omega t]}$$

for any p. (This is close to but not quite what you wrote...)

 

[charng]

I can understand your confusion regarding Kittel's choice of solution for the differential equation. Let me try to explain it in simpler terms.

Firstly, it is important to understand that the differential equation given by Kittel represents the forces acting on the lattice vibrations. The left side of the equation represents the mass of the atoms (M) and their acceleration (\ddot{u_{s}}). The right side represents the restoring force (C) due to the displacement of neighboring atoms (u_{s+1} and u_{s-1}) from their equilibrium position (u_{s}).

Now, when solving this differential equation, we are looking for a solution that satisfies the equation for all values of s (representing different atoms in the lattice) and t (representing time). In other words, we are looking for a solution that describes the motion of all atoms in the lattice at any given time.

Kittel suggests that this solution has a time dependence of exp(-Iwt). This is because the motion of atoms in a lattice can be described as a superposition of many different modes of vibration, each with a specific frequency (w). The exponential term represents the amplitude of the vibration at a particular frequency.

Substituting this time dependence into the differential equation, we get the expression -w^2 u_{s} M=C(u_{s+1}+u_{s-1}-2u_{s}). This is essentially the same equation, just written in terms of the frequency (w) instead of the acceleration (\ddot{u_{s}}).

To find the solution for this equation, Kittel uses a method called the difference equation method. This method involves finding a general solution for the equation and then applying boundary conditions to get a specific solution for the lattice. The solution he suggests is in the form of traveling waves, with a wave vector (K) and a phase factor (a). The wave vector represents the direction of propagation of the wave and the phase factor represents the relative phase of the vibrations at different points in the lattice.

Now, why does he have uexp(IsKa)exp(IKa) as the solution for u_{s\pm1}? This is because the displacement of an atom at position s+1 (u_{s+1}) or s-1 (u_{s-1}) can be described as the displacement of the atom at position s (u_{s}) multiplied by a phase