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Figure 6.28. Phase diagram for soft spherical particles in suspension showing the fluid state, the body-centered cubic (BCC) and face-centered cubic (FCC) phases. The vertical axis (ordinate) represents the volume that is inaccessible to one particle because the presence of the others. The excluded volume has been scaled to the cube of the Debye screening length, which characterizes the range of the interaction. [Adapted from A. P. Gast and W. B. Russel, Phys. Today (Dec. 1998.)]

where V is the volume of the solid, the momentum p = M, and the wavevector k is related to the wavelength k by the expression k=2n/L In the nearly free-electron model of metals the valence or conduction electrons are treated as noninteracting free electrons moving in a periodic potential arising from the positively charged ion cores. Figure 6.29 shows a plot of the energy versus the wavevector for a one-dimensional lattice of identical ions. The energy is proportional to the square of the wavevector, E = h2k?l%n2m, except near the band edge where k = ±n/a. The important result is that there is an energy gap of width E%, meaning that there are certain wavelengths or wavevectors that will not propagate in the lattice. This is a result of Bragg reflections. Consider a series of parallel planes in a lattice separated by a distance d containing the atoms of the lattice. The path difference between two waves reflected from adjacent planes is 2d sin ©, where © is the angle of incidence of the wavevector to the planes. If the path difference 2d sin © is a hàlf-wavelength, the reflected waves will destructively interfere, and cannot propagate in the lattice, so there is an energy gap. This is a result of the lattice periodicity and the wave nature of the electrons.

In 1987 Yablonovitch and John proposed the idea of building a lattice with separations such that light could undergo Bragg reflections in the lattice. For visible light (his requires a lattice dimension of about 0.5 pm or 500 nm. This is 1000 times larger than the spacing in atomic crystals but still 100 times smaller than the thickness of a human hair. Such crystals have to be artificially fabricated by methods such as electron-beam lithography or X-ray lithography. Essentially a photonic crystal is a periodic array of dielectric particles having separations on the order of 500 nm. The materials are patterned to have symmetry and periodicity in their dielectric constant. The first three-dimensional photonic crystal was fabricated by Yablonovitch for microwave wavelengths. The fabrication consisted of covering a block of a dielectric material with a mask consisting of an ordered array of holes and drilling through these holes in the block on three perpendicular facets. A technique of stacking micromachined wafers of silicon at consistent separations has been used to build thé photonic structures. Another approach is to build the lattice out of isolated dielectric materials that are not in contact. Figure 6.30 depicts a two-dimensional photonic crystal made of dielectric rods arranged in a square lattice.

Forbidden band

First allowed band

Second allowed -.band

Forbidden band

First allowed band

Second allowed -.band

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