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## Vibrational Frequency Spectrum in a 2D Numerically Modeled Solid Lattice

**I have deliberately allowed slider variable ranges that can result in a break up of
the solid. If that happens, the learner just needs to refresh the page, which in Windows means hitting the F5 key.**
In this simulation the bonding is done with a Lennard-Jones (LJ) potential. LJ potentials have extremely large repelling forces when the particles get too close and longer range
weaker attractive forces when the particles are farther apart.
The objective of this simulation is to show that the LJ potential results in higher frequencies than are seen in the spring (parabolic potential) binding forces.
The LJ potential is of the form

`V(r)=4V_(LJ)[(sigma/r)^(2b)-(sigma/r)^b]`

where `V_(LJ)` is the potential minimum, `sigma` is a length scaling parameter, `r` is the particle separation,
and `b=6` is an integer.
The solid is initially set up so that all the particles are separated by a distance that causes the nearest neighbor forces to be zero.
The force between a site and its second nearest neighbor is not initially zero and the combination of nearest neighbor and more distant neighbor forces results in many frequencies.
A lattice site position represents the location of the nucleus of an atom.
An obvious result of that set up is that alternate rows of particles have to be staggered by 1/2 the spacing in the horizontal direction and the vertical
spacing of rows is `sqrt(3)/2` times the horizontal spacing of particles in any column.
The initial configuration is that of a perfectly regular crystal of 6 fold symmetry.
All of the discs initially have the same speed but the direction of the velocity is random in the x-y plane.
The object of this animation is to compute the vibrational frequency spectrum of the lattice sites.
This model is something that neither Einstein nor

Debye
had access to when they devised their calculations of the heat capacity of the solid based on simple spring (parabolic) bonding.
LJ bonding is more realistic because it
takes into account the Pauli Exclusion Principle forces that arise when electron wave functions
overlap too much when the nuclei get closer together.
In order to see the difference between the frequency of an external pair of discs,
which have first harmonic f1, and two different pairs of internal discs, all three cases
are computed. Even the external pair have several significant higher harmonics, 2f1, 3f1, 4f1...
but the internal disc pairs have a wide variety of frequencies.