What does the wavevector represent in the solution for vibrations in a 1d monoatomic lattice?

Question

In my solid state physics textbook (The Oxford Solid State Basics by Steven H. Simon), we model a 1d monoatomic lattice as a chain of atoms connected by springs with spring constant $\kappa$. We define $x_n$ as the position of the nth atom, $a$ as the equilibrium spacing between atoms, and $\delta x_n=x_n-na$ as the displacement from equilibrium of the nth atom. Then the equation of motion (Newton's 2nd) is: $$m\delta\ddot{x}_n=\kappa(\delta x_{n+1}+\delta x_{n-1}-2\delta x_{n})$$ My textbook gives the solution to this differential equation as: $$\delta x_n=Re\{Ae^{iwt-ikna}\}$$ Where $w$ is the angular frequency and $k$ is the wavevector (technically the wavenumber I guess).

I don't understand what the wavenumber represents physically in this case; I don't get why $\delta x_n$ is a travelling wave in the first place. In my understanding, $\delta x_n$ is a function of $t$, so the wavelength would be the length of one complete displacement cycle along the time axis, which would mean the wavelength and period would be the same thing.

What does the wavelength represent here? Specifically, what does the travelling wave represented by $\delta x_n$ propagate through? What is the physical meaning of $k$?

Answer 1

The first thing to note is that the $k$ (\kappa in the textbook?), the spring constant, in your first equation is not the same as the $k$, the wave vector, in your second equation.

Here is a screen shot from the Phet simulation Waves which shows something propagated along a row of red balls (atoms).
I have shown the whole screen shot so that if you run the simulation you can use the parameters that I chose.

Look at atoms with labels $n=6$ and $n=20$.
As shown in the screenshot both atoms are (approximately - more later) at their equilibrium position.
If you run the simulation you will find that the phase difference between the motion of those two atoms is always zero.
The same is true of (pink) atoms at $n=6+4=10$ and $n=40+4=44$.

We call the (minimum) distance between such pairs of atoms whose motion is in phase a wavelength, $\lambda$ and it is related to the wave vector $k = \dfrac{2\pi}{\lambda}$.
So $k$ is a characteristic parameter of the disturbance moving across the screen.

Furthermore the speed of the disturbance (distance moved by a peak/time taken to move that distance) is $c = \frac \omega k$ ie $\omega$ and $k$ are related to one another.

Coming back to (approximately - more later).
With a discrete set of atoms the idea of wavelength is a little more complex.
If you look at the screen shot the atom which would be exactly in phase with the $n=6$ atom would have a label $n=49.5$ which of of course impossible as $n$ has to be an integer.
However this does not mean we cannot use the idea of a wavevector when describing the motions of the atoms.

$\delta x_{\rm n}$ is a function of both time $t$ and position $na$.
The answer to Wave equation: $y=A \sin(\omega t-kx)$ or $y=A\sin(kx-\omega t)$? might help in terms of how to interpret your second equation?