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WOLFRAM|DEMONSTRATIONS PROJECT

Phonon Modes for 2D Lattice Vibrations

​
dynamics
ω​
q
​
parameters
mass:
m
1
10
coupling to neighboring:
m
1
horizontal:
k
1
||
a
0.25
vertical:
k
2
||
b
0.25
diagonal:
k
3
|| (
b
+
a
)
0.25
diagonal:
k
4
|| (
b
-
a
)
0.25
A lattice of atoms can be modeled as harmonic oscillators, with forces proportional to the displacements of the atoms from equilibrium positions. The simplest such model introduces coupling only between nearest-neighbor atoms. In this Demonstration, a lattice cell containing one to five atoms is modeled, with nearest-neighbor harmonic coupling to the masses in each nearby cell. Normal mode solutions to these equations of motion are plotted. Controls are provided to alter the coupling "spring constants" and other free parameters, as well as controls to select from the reciprocal space vectors and angular frequencies associated with the normal mode solutions. A time control is also provided to display changes of the lattice through one period of the lattice vibration. A plot of the dispersion relation, showing the angular velocities associated with each reciprocal vector, is also provided.
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