13 Jul 2026
Lattice Dynamics from the Vibrational Modes of a Crystal Model
Aim
To study the normal modes of a one-dimensional coupled-oscillator model and relate them to lattice vibrations in a crystal.
Apparatus and software
Coupled-spring oscillator model or computer simulation, frequency generator, displacement sensor, and plotting software.
Experimental arrangement
Theory
Atoms in a crystal oscillate about equilibrium positions. If neighbouring atoms are coupled by springs, a displacement of one atom affects its neighbours. For displacement $u_n$ of the $n$th mass in a one-dimensional chain,
\[m\frac{d^2u_n}{dt^2}=K(u_{n+1}+u_{n-1}-2u_n).\]Using a wave-like trial solution $u_n=u_0e^{i(nqa-\omega t)}$ gives the dispersion relation
\[\omega(q)=2\sqrt{\frac{K}{m}}\left|\sin\frac{qa}{2}\right|.\]The slope near $q=0$ gives the acoustic-wave velocity in the model. The normal-mode frequency is set by the restoring force and mass, not by the amplitude in the small-oscillation limit.
Observations
| Mode number | Relative frequency | Character of motion |
|---|---|---|
| 1 | 1.00 | all masses nearly in phase |
| 2 | 1.93 | one phase reversal |
| 3 | 2.71 | two phase reversals |
| 4 | 3.25 | shortest wavelength mode |
Calculation
For the model dispersion relation, take $K/m=1$ and the first non-zero wave number as $qa=\pi/3$. Then
\[\omega=2\sqrt{\frac Km}\sin\frac{qa}{2}=2\sin\frac{\pi}{6}=1.\]The second observed mode has relative frequency $1.93$ compared with $1.00$ for the first mode. Thus its frequency is $1.93$ times the first-mode frequency. The successive phase reversals also show that the wavelength is becoming shorter as the mode number increases.
Maxima Code
Download the lattice-dynamics calculation.
Result
The coupled system possesses discrete normal modes; increasing mode number increases the frequency and decreases the wavelength.
Viva Questions
- What is a normal mode? A collective oscillation in which all parts vibrate at one frequency with fixed amplitude ratios.
- What produces the restoring force? The change in interatomic potential energy when atoms are displaced.
- What is a phonon? The quantum of lattice vibrational energy.
Discussion