Bose factor

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. 

This resembles the Bose factor [exp(/zv//cT) — 1] 1 of quantum optics. Of course, the resemblance is no coincidence, both deriving from similar physics. 

By referring to Eqs.(3) and (41, the imaginary part of the Fourier transform of the OHD-OKE response divided by the Fourier transform of the lAF of the laser pulse is directly comparable to the LS spectrum divided by the Bose factor in addition to Because the dynamical behavior and the fluctuation ought to be correlated with each other through the fluctuation-dissipation theorem, this comparison can verify that the information provided by the two experiments are identical. To our knowledge, however, the direct comparison has not been made yet. In this report, we will describe the details of our experiments and thier results on this problem. 

The LS spectrum is divided by the Bose factor in addition to u . The result of the above calculations is shown as the solid line and the dashed line in Fig. 2. 

By plotting the first-order Raman intensity as a function of temperature, one can determine accurately as the temperature where the intensity of the first-order Raman peaks becomes zero, as shown in Fig. 21.4a and b. For the Tc determination from the SLs spectra shown here, the TO2 and TO4 phonon lines (shown by arrows in Fig. 21.3) are the most suitable because they do not overlap with the second-order substrate features. (However, other optical phonon modes can be used in the same manner, provided that they are clearly observed in the spectra of the ferroelectric phase.) The results, with the phonon intensities normahzed by the Bose factor n -f 1 = (1 exp —h(o/kT)) ... 

The probability of this process being strictly proportional to the phonon population by the Bose factor decreases at low temperatures. Exeept at very low temperatures. there is no detectable intensity difference between the Stokes and anti-Stokes eontributions at variance to Raman scattering. 

Fig. 16. Raman scattering intensities of NdB at different temperatures. The peak at 170 cm" corresponds to the phonon density of states and decreases upon cooling due to the Bose factor. By cooling down from 300 K to 7 K the center of the CEF transitions shifts from 95cm to 92 cm". ...

Equation 1 can be fitted to the experimental spectra after (i) multiplying by x n x) + 1), with X = fuo/k T and n(x) the Bose factor in order to account for the detailed balance, and (ii) convolution with the instrumental resolution function / ( >). The experimental intensity may be in fact written as... 

Fig.11. Dynamic scattering law S(Q,co) of cis-1,4-polybutadiene (PB) below and above the glass transition temperature rg=170 K. Dashed lines are the values expected from the Bose factor. (Reprinted with permission from [32]. Copyright 1993 American Institute of Physics, New York)...

Here a is a constant proportional to the square of the electron-ion exchange constant and n((o) is a Bose factor. The function x(q, ) is the wave number and frequency dependent susceptibility of the interacting RE-ions. It is discussed at length in section 5.1.1 (see for example eq. 17.82). The above expression has been used successfully by Hessel Andersen et al. (1976) to analyse their data for Tb.Y,, Sb. 

S Photon Statistics - The Bose Factor The equations above are normally used in the IR business, but be aware that they contain a small error Jones (1953, 1957) pointed out that the statistics for noise is not actually quite as stated here. Photons have a larger fluctuation rate than we have assumed. We can account for this by including the Bose Factor in any formulas that use the spectral photon irradiance Q(X) ... 

The factor in the square brackets above is the Bose factor it is very close to unity except for wavelengths greater than the peak of the Planck curve, and the effect is normally not considered in performance predictions. However, for a detector whose spectral range is limited to long wavelengths, the factor can be significant. 

Beyond 30 pm, very few additional background photons are collected, but the responsivity (per watt) continues to increase proportional to wavelength, so D increases linearly with wavelength. Inclusion of the Bose factor (Equation 4.20)... 

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