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Phonon Modes for 2D Lattice Vibrations

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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The positions of masses within a two-dimensional periodic array of cells can be described by summing the lattice vector , representing the origin of each of the lattice cells, and a relative vector to the position of each of the masses. With representing the equilibrium position of the mass in cell , the position of that mass is .
Let , with direction , represent the equilibrium separation of the mass in cell from the mass in cell . If the harmonic coupling between these masses has magnitude , then the system of equations describing the vector displacement for the mass in unit cell from the equilibrium position is given by
.
In general, we have one such equation for each pair. A trial solution of the form: can be used to decouple this system, resulting in a single equation for each mass of the form
.
This describes all the steady-state lattice vibrations, the normal modes, that can be modeled by this trial solution. Here is a vector in reciprocal space, effectively parameterizing the angular velocity . The vector is an eigenvector of the equations of motion of the system for this assumed solution, where are the eigenvalues of this system. For an -atom basis, there are such eigenvalues per point, each resulting in a different characteristic motion.
The rank of the resulting eigenvalue problem depends on the number of masses per unit cell, but the complexity of the matrix expression depends on the number of neighboring interactions that are considered. For example, given lattice vectors , diagonals , and a one-atom basis, where each unit cell contains a single mass coupled with harmonic oscillator forces between only nearest neighbors, the normal mode solutions follow from the solution of the eigenvalue problem
.
Controls are provided to display the dynamics associated with each of the characteristic angular frequencies , for given reciprocal vector values .
Three tabs are provided in this Demonstration. The primary tab displays the dynamics of the solution for particular pair of values. In that tab, selecting "Play" for the time control animates the lattice vibrations. A scaling control is provided to alter the initial magnitude of the eigenvectors, tantamount to picking the initial time boundary value constraints. Note that it is possible to select physically unrealistic scaling factors that allow for collisions that are not modeled by this system.
A second tab provides the dispersion relation, the dependence of angular velocity on all points.
Finally, a parameters tab provides controls for the spring constants , the primitive unit cell lattice vectors , and the positions of the masses within each unit cell of the lattice. Additional mass position locators, up to five total, may be added or deleted by Alt-clicking the lattice cell in the desired location. For mass locator removal, it is necessary to adjust one of the other locators so that the removal takes effect. Note that it is possible to select equilibrium mass positions that are too close, leading to physically unrealistic dynamics, such as masses passing through each other.
The total number of interactions, even when those interactions are restricted to just the neighboring lattice cells, increases quickly as additional masses per cell are added. For example, with three masses per unit cell, considering only the neighboring and origin cells, there are 29 interactions possible for each mass. To simplify the physical constants dialog, distinct "spring constant" selection is only available for a subset of the possible interactions. This imposes the following respective constraints along the "horizontal," "vertical," "NE diagonal," and "NW diagonal" directions connecting the masses across the cells
, , , .
These are labeled , , , and in the parameters tab, respectively. For example, the first identity above is the imposition of an equality constraint on the coupling constants in the "left" and "right" directions separating masses in neighboring "horizontal" cells (when the lattice is square). Those are the interactions directed primarily along the lattice vector directions (primarily since these directions are also adjusted for the positions of the masses within the respective cells if different). Similarly, the same coupling constants are used for each of the pairs of directions that are directed primarily along the , ±(), ±() directions separating the lattice cells.
When there is more than one mass per unit cell, distinct coupling constants for the intra-cell interactions between the masses within the origin unit cell may be selected (i.e. ). These are labeled in the parameters tab.
General theory describing oscillations around lattice equilibrium points can be found in [1].
Reference
[1] N. W. Ashcroft and N. D. Mermin, Solid State Physics, New York: Holt, Rinehart and Winston, 1976, Chapters 21 and 22.
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