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Fig. 12.1 Planck's radiation law.
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STATISTICAL MECHANICS
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The equation of state may be derived as follows. Consider first a plane wave whose electric and magnetic field vectors are E and B. The average energy density is
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The radiation pressure, which is equal to the average momentum flux, is
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Thus the energy density is numerically equal to the radiation pressure. Now consider an amount of isotropic radiation contained in a cubical box. The radiation field in the box may be considered an incoherent superposition of plane waves propagating in all directions. The relative intensities of the plane waves depend only on the temperature as determined by the walls of the box. The radiation pressure on any wall of the box is one-third of the energy density in the box, because, whereas all the plane waves contribute to the energy density, only one-third of the plane waves contribute to the radiation pressure on any wall of the box. To derive U a: T 4 , recall that the second law of thermodynamics implies the following relation, which holds for all systems:
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Using (12.19) we have
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(12.19)
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From PV = U/3 and the fact that P depends on temperature alone we have = 3P = U == u(T) (12.20)
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Hence
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(12.21) The constant C cannot be obtained through classical considerations. If the photon had a finite rest mass, no matter how small, then it would have three independent polarizations instead of two. * There would be, in addition to transverse photons, longitudinal photons. If this were so, Planck's radiation formula (12.14) would be altered by a factor of t. The fact that (12.14) has been experimentally verified means that either the photon has no rest mass, or if it does the coupling between longitudinal photons and matter is so small that
*If the photon had a finite rest mass, it could be transformed to rest by a Lorentz transformation. We could then make a second Lorentz transformation in an arbitrary direction, so that the spin would lie neither parallel nor antiparallel to the momentum.
BOSE SYSTEMS
thermal equilibrium between longitudinal photons and matter cannot be established during the course of any of our experiments concerned with Planck's radiation law.
12.2 PHONONS IN SOLIDS
Phonons are quanta of sound waves in a macroscopic body. Mathematically they emerge in a similar way that photons arise from the quantization of the electromagnetic field. For low-lying excitations, the Hamiltonian for a solid, which is made up of atoms arranged in a crystal lattice, may be approximated by a sum of terms, each representing a harmonic oscillator, corresponding to a normal mode of lattice oscillation. * Each normal mode is classically a wave of distortion of the lattice planes-a sound wave. In quantum theory these normal modes give rise to quanta called phonons. A quantum state of a crystal lattice near its ground state may be specified by enumerating all the phonons present. Therefore at a very low temperature a solid can be regarded as a volume containing a gas of noninteracting phonons. Since a phonon is a quantum of a certain harmonic oscillator, it has a characteristic frequency Wi and an energy i . The state of the lattice in which one phonon is present corresponds to a sound wave of the form
(12.22) where the propagation vector k has the magnitude
(12.23)
in which c is the velocity of sound. t The polarization vector can have three independent directions, corresponding to one longitudinal mode of compression wave and two transverse modes of shear wave. Since an excited state of a harmonic oscillator may contain any number of quanta, the phonons obey Bose statistics, with no conservation of their total number. If a solid has N atoms, it has 3N normal modes. Therefore there will be 3N different types of phonon with the characteristic frequencies (12.24) The values of these frequencies depend on the nature of the lattice. In the Einstein model of a lattice they are taken to be equal to one another. An improved model is that of Debye, who assumed that for the purpose of finding the frequencies (12.24), one may consider the solid as an elastic continuum of volume V. The frequencies (12.24) are then taken to be the lowest 3N normal frequencies of such a system. Since an elastic continuum has a continuous
'In as much as anharmonic forces between atoms, which at high temperatures allow the lattice to melt, can be neglected. tWe assume an isotropic solid, for which c is independent of the polarization vector E.