Oscillator mode, Einstein

The densities of vibrational states for the two potentials (Fig. 15) are actually very similar, exeept for a somewhat wider gap for RIMl between the two peaks at higher frequencies, corresponding to inner modes of CO3. As for the Kieffer s model, the cut off frequencies of acoustic modes were derived from elastic constants by the Voigt-Reuss-Hill approximation[38], amounting to 51, 66 and 90 cm"i. An optic continuum ranging from 113 to 287 cm was used, and four Einstein oscillators at 708,867, 1042 and 1470 cm with appropriate weights represented the internal optical modes. The... 

More general expressions for the entropy of radiation can be obtained by considerations similar to those for an ensemble of absorbers, except that Bose-Einstein statistics must be used because photons are indistinguishbable and any number of photons can be in the same state (Box 2.6) [141-144], Consider an ensemble of N photons with a narrow band of frequencies representing m different oscillation modes. The number of ways of assigning the photons to various oscillation modes s N - -m— ) / m — 1)W (Fig. 4.31B). Using this expression for the multiplicity Q. in Eq. (4.65) gives the entropy of the photons ... 

Similarly, during their effort to understand the thermal energy of solids, Einstein and Debye quantized the lattice waves and the resulting quantum was named phonon. Consequently, it is possible to consider the lattice waves as a gas of noninteracting quasiparticles named phonons, which carries energy, E=U co, and momentum, p = Uk. That is, each normal mode of oscillation, which is a one-dimensional harmonic oscillator, can be considered as a one-phonon state. 

In these equations and hereafter we use simplified symbols for quantities referring to the P-branch line v, 7—l- c—I, /. Namely, Xvy instead of Xu,y-i etc. The second term in (2) represents the rate of spontaneous emission into the oscillating cavity modes. 1 is approximately the effective solid angle subtended by the mirrors after several reflections. (Alternatively, e is the fraction of stable transverse modes.) After threshold eS j is negligible, eS j- Xvj- i i i " i important only before threshold —as a source of noise photons to trigger-on the lasing process. The spontaneous emission terms in (1) are given by =A j+iN y, A j is the Einstein coefficient. In infrared lasers where typically A ) 10s. S, ... 

Einstein assumed that there are 3N independent distinguishable oscillators. The vibrational partition function (Equation (11.26)) corrected for the zero-point energy (see Equation (13.21)) is q = 1 - e )" . The average energy per vibrational mode is given by the derivative in Equation (10.34) ... 

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