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4 "born": Classical dynamics of a Linear Chain of Atoms

"born" is a simple simulation of lattice dynamics (free, undriven motion) on a one dimensional chain. Initial condition options include standing or traveling waves, or traveling or "standing" packets. Other options include: periodic or fixed boundary conditions; an impurity mass to illustrate local impurity modes and scattering; a quartic anharmonic potential to illustrate local anharmonic modes. The student can define the mean wave vector, amplitude and width of the packets (or pulses) that define the initial conditions.

"born" illustrates clearly the distinction between group and phase velocities, the idea of periodic boundary conditions, the absence of group propagation for wave vector at the edge of the Brillouin zone, the equivalence of solutions with wave vectors differing by a reciprocal lattice vector, and the inverse relation between rate of spreading of a packet and its initial width.

The first illustration here shows a packet being scattered by a light mass impurity on the chain. Two still frames are taken from the movie: the first shows the wave packet traveling towards the blue impurity atom; the second is taken after the scattering and shows both the transmitted and reflected packets.

movie (80K). .


Another illustration shows the remarkable abilty of deviations from the simple perfect harmonic chain to alter dramatically the behavior of the system. The first clip is the evolution of the ideal chain with initial conditions corresponding to a localized disturbance. The effect of the non-linear dispersion is evident in the rapid diffusion of the excitation throughout the chain. The still frames show the system after it has evolved for roughly the same amount of time for each case.

movie (92K)

The second centers the initial disturbance on a single light imurity mass. The excitation in this case remains localized: an example of a "light mass impurity mode."

movie (30K)

The third treats all of the atoms as equivalent (full translational symmetry) but includes a quartic term, in addition to the quadratic, in the potential. Again there is a localized mode, despite the translational symmetry of the chain. The simulation offers challenges to students at all levels of sophistication! (Can you invent a SIMPLE handwaving argument for the stability of this "quartic anharmonic mode"?)

movie (49K)

The third treats all of the atoms as equivalent (full translational symmetry) but includes a quartic term, in addition to the quadratic, in the potential. Again there is a localized mode, despite the translational symmetry of the chain. The simulation offers challenges to students at all levels of sophistication! (Can you invent a SIMPLE handwaving argument for the stability of this "quartic anharmonic mode"?)

movie (49K)


``born'' also gives a graphical display of the time dependence of the displacement of 5 adjacent atoms on the chain, useful for detailed analysis of the motion. The figure below shows the graph for the case of the anharmonic localized mode, the blue trace being for the central atom in the disturbance, and the others for the next four atoms down along the chain.
Table of contents for Chapter 4 of Simulations for Solid State Physics
  1. Introduction
  2. Plane waves--infinite medium
    1. Traveling waves
    2. Energy and momentum
    3. Standing waves
  3. Normal modes of finite chain
  4. Wave packets
    1. Traveling versus standing packets
    2. Group velocity
    3. Dispersion
  5. Pulse propagation
  6. Impurities
    1. Scattering
    2. Localized modes
  7. Anharmonicity
  8. Summary
  9. Appendix: "born" -- the program
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