Consider an infinite chain of one-dimensional atoms, as seen below.

• Again, let us focus only on the possible normal modes for the system. Assuming that the positioning of the particles satisfies a normal mode, the envelope function of the particles will give the initial conditions. Define the envelope function for the $n$th particle as

$$x_{n}=\sin{kna} \; \; $$

where $na$ is the location of the nth atom and $k$ is the wave vector of the envelope function. The position function can be generalized by writing it as

$$x_{n}=Re\left[Ae^{ikna}\right] \, \, at \, \, t=0 \; \; .$$

• Let us add the time dependence for the position of the $n$th particle now as well.

$$when \; t>0,$$

$$x_{n}(t) \, = \, Re\left[Ae^{ikna}e^{i\omega_{k} t}\right] \; \; , $$

where $\omega_{k}$ is the frequency of the normal mode associated with this particular envelope function.

Similarly, the position function for the neighboring particles can be written as

$$x_{n+1}(t) \, = \, Re\left[Ae^{ik(n+1)a}e^{i\omega_{k} t}\right] \; \; $$

and

$$x_{n-1}(t) \, = \, Re\left[Ae^{ik(n-1)a}e^{i\omega_{k} t} \right] \; \; . $$

• Now, all we need to define the motion of the $n$th particle is the frequency of oscillation $\omega_{k}$. In all of our previous cases, information about the frequency has been found in the equations of motion derived from the forces on a particle. Let us see what we can find with this strategy; the equation of motion for the $n$th particle is written as

$$m\ddot{x}_{n} \, = \, -\kappa\left(x_{n}-x_{n-1}\right) - \kappa \left(x_{n}-x_{n+1}\right) \; \; .$$

• Let us plug in the expressions for $x_{n}$, $x_{n+1}$, and $x_{n-1}$ into the equation of motion and simplify what we can.

$$-m\omega_{k}^{2}e^{ikna} \, = \, -\kappa\left(e^{ikna}-e^{ik(n-1)a}\right) - \kappa \left(e^{ikna}-e^{ik(n+1)a}\right) \; \; .$$

Factoring out a term of $e^{ikna}$ will then give

$$m\omega_{k}^{2}=\kappa\left(1-e^{-ika}\right) + \kappa\left(1-e^{ika}\right) \; \; ,$$

$$m\omega_{k}^{2}=2\kappa -\kappa\left(e^{ika}+e^{-ika}\right) \; \; ,$$

$$m\omega_{k}^{2}=2\kappa\left(1- \cos{ka}\right) \; \; , $$

$$m\omega_{k}^{2}=2\kappa\left(2 \: \sin{^{2}\frac{ka}{2}}\right) \; \; , $$

$$\omega_{k}(k)=\left|\sqrt{\frac{4\kappa}{m}}\; \sin{\frac{ka}{2}}\right| \; \; . $$

• We have found the normal mode frequency as a function of the envelope function's wave vector. Let's plot the dispersion relation and see what we can learn about how a 1-d infinite chain behaves.

• Ask the class about any interesting characteristics of the dispersion relation they see.

Some important observations:

• There is a maximum frequency at which atoms in a solid can vibrate (This has profound implications for the heat capacity of solids).
• The velocity of sound waves, which have small wave vectors (long wavelength), can be predicted from

$$\omega \approx v_{sound}\, k $$

for small $k$.

• There are not any new normal modes described by $k > \frac{\pi}{a}$. All information for the oscillating system is contained in the range $k < \frac{\pi}{a}$. This region is called the first Brillouin Zone.