Bose—Einstein distribution

It is possible to calculate the average energy for a single oscillation mode, following the canonical ensemble methodology [6,11] as 

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Equation 1.20 tells us that there are on average np(cop, T) phonons in the P mode, where this mode contributes energy 

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Given the total density from Eq. (4.17), the temperature follows from the equation of state which depends in turn on what particles are present. For any one species i, with temperature T,. we have from the Fermi-Dirac or Bose-Einstein distribution, Eq. (2.41),... 

Since gt states can be arranged in any order, the previous result must be divided by g3. The particles are indistinguishable and hence no other new arrangements are possible. The number of possible symmetric eigenfunctions correspondind to e, are thus given by the Bose-Einstein distribution... 

One can see the physical meaning of the operator Y for the case where the field is in a thermal equilibrium state. Indeed, by taking the ensemble average with a Bose-Einstein distribution of field modes at temperature T,... 

Figure E.10 (a) Bose-Einstein distribution function, (b) Fermi-Dirac distribution function, and (c) filling of levels by fermions at T = 0 and T=T1>0. The dashed line indicates the Fermi energies p.

One of the simplest procedures to get the expression for the Fermi-Dirac (F-D) and the Bose-Einstein (B-E) distributions, is to apply the grand canonical ensemble methodology for a system of noninteracting indistinguishable particles, that is, fermions for the Fermi-Dirac distribution and bosons for the Bose-Einstein distribution. For these systems, the grand canonical partition function can be expressed as follows [12] ... 

In this equation, h is Planck s constant divided by 2tt, V is the crystal volume, T is temperature, fej, is Boltzmann s constant, phonon frequency, is the wave packet, or phonon group velocity, t is the effective relaxation time, n is the Bose-Einstein distribution function, and q and s are the phonon wave vector and polarization index, respectively. 

Accounting for the difference between the Fermi-Dirac and Bose-Einstein distributions, relativistic fermions may also be related to photons... 

At very low energy, many amorphous materials show a Boson peak , so-called because the temperature dependence of its intensity roughly scales with the Bose-Einstein distribution. Amorphous silica is no exception and has a peak at 40 cm. The origin of this has been controversial but in silica it appears to be related to either transverse acoustic modes or torsions of the Si04 tetrahedra with respect to one another [15]. 

If then A is very small compared to 1, the Bose-Einstein distribution formula passes over into the classical one. It is otherwise in the case when A becomes comparable with 1 (the case of A > 1, i.e. a < 0, cannot occur, for then the denominator vanishes for the energy value = —a/j8, and for smaller values of becomes negative, so that the whole theory becomes meaningless) if -4 1, deviations from the classical properties occur we say then that the gas is degenerate. In this case the subsidiary condition leads to a transcendental equation... 

Bose-Einstein distribution - A modification of the Boltzmann distribution which applies to a system of particles that are bosons. The number of particles of energy E is proportional to [e<, where g is a normalization constant, k is the... 

Since the spectroscopy of this phosphor is incorrectly described in the book on lamp phosphors [2], we add here, also as an illustration of the theory, a few comments on the spectroscopy. In view of its electron configuration (d ), the Mn ion will be octahedrally coordinated. The emission lines are tabulated in Table 6.3. There is a zero-phonon transition (Sect. 2.1) which at low temperatures is followed by vibronic lines due to coupling with the asymmetric Mn -0 deformation and stretching modes, 1/4 and 1/3, respectively. These uneven modes relax the parity selection rule. At room temperature there occur also anti-Stokes vibronics (Pigs. 6.21 and 6.22). The vibrational modes in the excited state and ground state are equal within the experimental accuracy as is to be expected for the narrow A2 transition [25,26]. The intensity ratio of the Stokes and anti-Stokes vibronic lines agrees with the Bose-Einstein distribution [26]. 

Finally, the explicit Bose-Einstein distribution is expressed as... 

The Bose-Einstein distribution (1.162) may be considered to recover the Planck law of black body radiation, i.e., the photon radiation modeling, by considering the following peculiarities ... 

If a quasi-monochromatic light wave I (co) with a statistically fluctuating intensity distribution (7.60) falls onto the photocathode, the probability of detecting n photoelectrons within the time interval dr is not described by (7.58) but by the Bose-Einstein distribution... 

The Bose-Einstein distribution law was derived by S.N. Bose in 1924 to describe a photon gas. Einstein extended it to material gasses. Fermi developed the Fermi-Dirac distribution law in 1926 by exploring the Pauli exclusion principle, and Dirac obtained it independently in the same year by considering antisymmetric wavefimctions. 

A simple example of this, which makes clear the mathematical mechanism by which it occurs, is seen in the ideal Bose gas, which provides an exactly soluble model of a critical point. The number of single-pardde states with energy in the range e to e +de is proportional to de in d dimensions, while the occupancy of a state of energy e is exp[(e -ft)/kTl-1 ", with 0, by the Bose-Einstein distribution law so the density of the gas at given p, and T is expressible in terms of the integral... 

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