Bose occupation number

From equation (A2.2.145). the average occupation number of an ideal Bose gas is... 

This formula describes the exchange of a single phonon of wavevector Q, frequency co(0 ) and polarization e(Q,j). n is the Bose factor for annihilation (—) or creation (+) of a phonon, respectively, i.e. the phonon occupation number. 

At this stage, it is convenient to assume a very low density of excited dipoles. In other words, we assume that the exciting external source is sufficiently weak so that at each instant the probability of finding a given dipole in an excited state is very small compared to 1. In this condition, the system satisfies the linear-response approximation. Since the elementary excitations are very dilute (i.e., the occupation numbers are very small), all statistics are equivalent. For the convenience of further calculations (e.g. interaction with photons), the operators B B are assumed to obey Bose statistics21,22 ... 

The temperature-dependent term represents the difference in the occupation numbers of the phonon states involved in the process. As phonons are bosons, the occupation number for a phonon of pulsation u> at temperature T is given by the Bose-Einstein statistics as... 

First let us remark that the replacement of Pauli operators by Bose operators applied in this chapter is only approximate since the occupation numbers of paulions can be either 0 or 1, whereas the occupation numbers for bosons can take all nonnegative integer values 0,1,2,.., .28 Therefore the replacement of the operators Ps and Pj by Bose operators can provoke uncontrolled errors in all... 

In Equation (8), A represents the anharmonic shift and F the anharmonic broadening. Both A and F are proportional to nj, the Bose phonon occupation numbers - 1)" ], and at temperatures such that kT fi(jjj, rij is propor-... 

Here, the factor of 1/2 in the exponent accounts for the zero-point, quantum fluctuations of the system the Bose statistics imply that any mode can have an occupation number that ranges from 0 to oo. Performing the sum, we find... 

Finally, we should note that all that has been said so far is valid for fermionic annihilation and creation operators only. In the case of bosons these operators need to fulfill commutation relations instead of the anticommutation relations. The fulfillment of anticommutation and commutation relations corresponds to Fermi-Dirac and Bose-Einstein statistics, respectively, valid for the corresponding particles. Accordingly, there exists a well-established cormection between statistics and spin properties of particles. It can be shown [65], for instance, that Dirac spinor fields fulfill anticommutation relations after having been quantized (actually, this result is the basis for the antisymmetrization simply postulated in section 8.5). Hence, in occupation number representation each state can only be occupied by one fermion because attempting to create a second fermion in state i, which has already been occupied, gives zero if anticommutation symmetry holds. 

We conclude, therefore, that it is possible to measure the temperature dependence of the dephasing time of LO phonons in GaP and ZnSe and to account for the observation in terms of the simple Bose-Einstein dependence on phonon occupation numbers. Although this procedure does indeed yield quite good agreement between experiment and theory, no theoretical apparatus exists for the prediction of the magnitude of T2 at T=0K. 

The main difference between the quantum theoretical susceptibilities dss( i,j) according to Eqs. (7.68X (7.77) and the respective classical quantities (5.24), (5.25) is the dependence of the former on temperature via the occupation numbers / . /t The retarded dispersion energy between the particles considered, in accordance with the underlying electron-photon-exchange interaction, depends both on Bose statistics and on Fermi statistics. 

In spite of considering two-photon processes, we still find the energy levels (8.17) to be equidistant with respect to the photon occupation numbers n, -I- i. This suggests that it is permissible to neglect the effect of the electron-photon interaction on statistics and to occupy the electron states and the photon states according to Fermi statistics and to Bose statistics. We shall check this question in Section 8.4 by studying the creation and annihilation operators of the resulting quasi-electrons and quasi-photons. 

We noted that the energy terms (8.8) and (8.12) of order zero and two do not depend on the separation of particles 1 and 2. Accordingly, we calculated the free energy of interaction from the fourth order energy expression (8.16). Considering this term, it is certainly consistent to calculate the average occupation numbers of the excited states from Fermi and Bose statistics. The error is at worst of order six in the interaction parameter. 

The renormalization of the quasi-photon energies entails a renormalization of the quasi-electron energies. The renormalized quasi-electron Hamiltonian results from the total Hamiltonian (8.31) by replacing the photon occupation numbers + i by coth hcoJ2kT). These are just the average occupation numbers given by the Bose distribution (16). 

Figure D.l. Average occupation numbers in the Maxwell-Boltzmann (MB), Fermi-Dirac (FD) and Bose-Einstein (BE) distributions, for = 10 (left) and f = 100 (right), on an energy scale in which fXfo = p-be = 1 for the FD and BE distributions. The factor nl(2nmfl in the Maxwell-Boltzmann distribution of the ideal gas is equal to unity.

The B-E distribution applies to bosons (particles with integral spin) for which no restrictions are placed on the occupancy number. The negative 1 in the denominator allows the occupancy of the groimd state to get very large as T 0 which is known as Bose condensation. The transition of He to superfluid He-11 at 2.17 K (the so-called lambda transition) is an example of Bose condensation. 

Fig.2.10. Bose-Einstein distribution function giving the mean occupation number as a function of temperature T for different values of 0s=H

The total number of particles in an ideal Bose gas at low temperaPires needs to be written such that the ground-state occupancy is separated from the excited-state occupancies ... 

In his derivation, Frieden uses the concept of a number-count set nm, each member representing the number of photons counted in a spectral interval. The total number of photons m= x nm is taken as known to be N. In terms of frequencies vm, the values of the object spectrum are given by om = nmhvm, where h is Planck s constant. The number of normal modes or degrees of freedom available for occupation by photons of frequency vw is labeled zm. The Bose-Einstein degeneracy factor... 

Now from (2) it can be shown that for sufficiently large s, the total number of quanta N=2njc increases proportionally to the total rate of supply s. Equation (4) then shows that a critical rate of supply Sq, exists, such that for s>Sq something analogous to Einstein condensation of a Bose gas must occur. This implies that p, which acts as chemical potential, must approach very closely such that the occupation, n, of becomes very large. Since represents a normal mode, it follows that this excitation is coherent. 

At a temperature below T a macroscopic occupation of the ground state with p = 0 develops. The macroscopic occupation implies that the number N 0) of bosons at the ground state is of the order of the total number N of particles in the system. This is the celebrated Bose-Einstein condensation. ... 

In agreement with Pauli principle, for the particles with integer spin, there is no restriction to the number of particles that can be placed in an available sub-level of an energy level. However, the fundamental difference toward the Boltzmaim distribution, somehow similar in occupancy, is that the quantum one (the Bose-Einstein spin based ne) it can be obtained by arranging particles-imderstate also through the permutation of the walls between sub-states in the same way in which are permuted the particles id est, the particles and the walls that are separating them, can be... 

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... 

From the December 22, 1995 when the Science magazine declared the Bose condensate as the molecule of the year , the Bose-Einstein condensation (BEC, in short), basically viewed as the macroscopic occupation of the same single-particle state in a many-body systems of bosons, had received new impetus both at theoretical and experimental levels in searching and comprehending new states of matter (Anderson et al., 1995 Ketterle, 2002). However, with the ever increasing number of experiments revealing quantum phase transitions at atomic scales (Yukalov, 2004), the need for accurate models for this new state of matter became imperative. Yet, although powerful variational and perturbation methods are available (Kleinert et al., 2004), a basic approach, centered on the key object of BEC - die bosonic gas density /> - it is not yet systematically developed and implemented (Vetter, 1997). 

Particles come in two kinds (1) bosons, with symmetric wave functions, in which case any number of them can occupy the same quantum state (i.e., having the same values of n, ri2, and 3) and (2) fermions, with antisymmetric wave functions such that there is a rigid restriction that (neglecting spin) precludes the occupation of a given state by more than one particle. Examples of bosons include photons, phonons, and entities, such as " He atoms, that are made up from an even number of fundamental particles. Examples of fermions include electrons, protons, neutrons, and entities, such as He atoms, that are made up from an odd number of fundamental particles. Bosons are described by Bose-Einstein statistics, fermions by Fermi-Dirac statistics. 

An interesting phenomenon predicted for boson systems where the number of particles is fixed is that of Bose-Einstein condensation. This implies that, as the temperature is gradually reduced, there is a sudden occupation of the zero momentum state by a macroscopic number of particles. The number of particles per increment range of energy is sketched in Fig. 3 for temperatures above and just below the Bose-Einstein condensation temperature Tb. The distribution retains, at least approximately, the classical Maxwellian form until Tb is reached. Below Tj, there are two classes of particles those in the condensate (represented by the spike at = 0) and those in excited states. The criterion for a high or low temperature in a boson system is simply that of whether T >T or T < 7b, where 7b is given by ... 

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