VIBRATIONS IN CRYSTALS

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Many solid materials, including all metals, are composed of atoms arranged in a lattice arrangement called crystals. There are a variety of crystal structures like cubic, hexagonal, cubic with an atom in the center of the cube, called body centered cubic, cubic with an atom in the center of each face of the cube, called face centered cubic, and others. The particular structure depends on the relative sizes of the atoms that are nestled together to form the crystal. The reason that materials take crystal form is that these neat geometrical structures represent the lowest energy configuration of the collection of atoms making up the material. To dislodge an atom from the crystal structure requires the addition of energy.

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Theoretically, at a temperature of absolute zero, the atoms of a crystal lie at their lowest energy position without moving at all. As thermal energy is added to the crystal it is manifest by vibration of the atoms about this equilibrium location. Within the limits of fairly small vibrations the electric forces bonding the atoms together stretch or compress a bit to a higher energy configuration. Each atom acts as though it were connected to its neighbors by little springs. The added energy is stored in the crystal as the kinetic energy of the atoms in motion and the potential energy of the compressed or stretched springs.

Let us consider the oscillation of the atoms in one-dimensional crystal simulated by the chain of the balls with the masses m1 and m2. The force applied to every atom depends upon the relative displacement of the nearby atoms and stiffness c of the virtual springs. Therefore, the displacement u of every atom in one-dimensional crystal is described by the equations:

These equations determine the oscillatory motion of the atoms:

where a/2 is the distance between the nearest atoms; k=2p/l is the wave factor, l is the wavelength of the wave in crystal. The equations mentioned above have two solutions for w :

where g2=4m1m2/(m1+m2)2; w02=2c (m1+m2)/m1m2. These equations determine two branches of the dispersion curve (so called acoustic and optical branches).

Let us consider the types of oscillation in a crystal  for these two branches. In the case of the long-wave approximation (ak << 1) in acoustic branch the atoms move synchronously and deviation of every atom is about the same at any moment of the time (see this case at the bottom of  the animation); in optical branch the atoms move in antiphase (it is shown on top of animation). For the shortest waves ( k = 2π/λ = 2π/a) in acoustic branch the lighter atoms are still and more heavy atoms oscillate; in the optical branch the situation is inverse (these cases are shown in the middle of animation). Oscillation of atoms in optical branch polarizes  the matter electrically and this type of oscillation can be excited by infra-red optical radiation. This is the reason why this branch was called "optical". We can see that the modes differ in the details of how the atoms move, but they both represent energy stored in the crystal, being passed back and forth between the kinetic energy of the atoms and the potential energy of the "springs". The temperature of the crystal is proportional to the average kinetic energy of the atoms.

There are some limitations to this mechanical description of what is going on in crystal vibrations. Because we are dealing with objects as small as individual atoms, quantum mechanical effects may not be neglected. For example, in a metal sample large enough to work with in a laboratory, there may be millions of individual crystals each with millions of individual atoms. On a laboratory size scale, it appears that we can add energy to the sample in any amount, as if the crystal were really composed of weights and springs. In fact, energy may only be added in multiples of some minimum amount. We call that minimum amount of vibrational energy a "phonon" analogous to the photon energy packet familiar in electromagnetic radiation. Energy addition appears continuous on the laboratory scale because the phonon is so small. On the scale of the atoms in the crystals however, the phonon magnitude is significant, and only vibration modes and amplitudes which differ in energy by whole phonon multiples are allowed.

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