Einstein frequency expression

With the above-mentioned radial distribution function, expression for the Einstein frequency after performing the angular integration Eq. (114) reduces to... 

In this expression K(0) is the so-called Einstein frequency , which can be calculated exactly from the information of the site-site intermolecular potential as well as of the density pair-correlation functions. K 0) is also an equilibrium quantity but with the three-particle correlation functions, whose approximate expression has been derived in [91]. 

In general, the phonon density of states g(cn), doi is a complicated fimction which can be directly measured from experiments, or can be computed from the results from computer simulations of a crystal. The explicit analytic expression of g(oi) for the Debye model is a consequence of the two assumptions that were made above for the frequency and velocity of the elastic waves. An even simpler assumption about g(oi) leads to the Einstein model, which first showed how quantum effects lead to deviations from the classical equipartition result as seen experimentally. In the Einstein model, one assumes that only one level at frequency oig is appreciably populated by phonons so that g(oi) = 5(oi-cog) and, for each of the Einstein modes. is... 

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by ... 

The simplest model to describe lattice vibrations is the Einstein model, in which all atoms vibrate as harmonic oscillators with one frequency. A more realistic model is the Debye model. Also in this case the atoms vibrate as harmonic oscillators, but now with a distribution of frequencies which is proportional to o and extends to a maximum called the Debye frequency, (Oq. It is customary to express this frequency as a temperature, the Debye temperature, defined by... 

The energy density function p v) is defined so that dE—p v)dv is the amount of available radiation energy per unit volume originating in radiation with frequency in the infinitesimal interval [v,v + dv]. Thus, p v) is expressed in the SI units J/(m Hz) = J s/m, so that Bg and Bg have the SI units m /(J s ). Ag is expressed in s The Einstein coefficients defined in this manner are related to the line strength by... 

Elementary physics classes express the so-called Doppler shift of a wave s frequency induced by movement either of the light source or of the molecule (Einstein tells us these two points of view must give identical results) as follows ... 

For solids the matter is not quite so simple, and the more exacting theories of Einstein, Debye, and others show that the atomic heal should be expected to vary with the temperature. According lo Debye, there is a certain characteristic temperature lor each crystalline solid at which its atomic heal should equal 5.67 calories per degree. Einstein s theory expresses this temperature as hv /k. in which h is Planck s constant, k is Bolizmanns constant, and r, is a frequency characteristic of ihe atom in question vibrating in the crystal lattice. 

Note that the above expression is known as the generalized Einstein equation and that the memory function, ((z), is the frequency-dependent friction. 

To elucidate the effect of temperature, we performed calculations of the rate of multiphonon non-radiative transitions. We considered a case when l and l1 belong to different rows of the same representation. The phonons, contributing to a nondiagonal vibronic interaction are considered in an Einstein-like model with the parabolic distribution function (14) (note that the results are not sensitive to the actual shape of the phonon bands) interaction is arbitrary. In this model the Green function is described by simple expression (16). In the case of a strong linear diagonal vibronic interaction one can expand the gr(f)-function into a series and take into account the terms up to the quadratic terms with respect to t gT(—t) iSjt — Ojt2/2. Here = Oq/wq, cD0 is the mean frequency of totally symmetric... 

As stated above, expression (9) for the rate constant of transition in Einstein s crystal was first calculated analytically by the method of the straight search in the pioneer works of Pekar [1] and Kun and Rhys [2]. Their analytic expression remains till now the unique exact expression for multi-phonon transition probability in the time unit. Then, there appeared different methods that permit to derive the integral expressions for the rate constant in the general case of the phonon frequencies dispersion the operator calculation method [5], the method of generating polynomial [6], and the method of density matrix [7]. The detailed consideration of these methods was made in the Perlin s review [9],... 

The integral in the expression (20) can be calculated exactly only in the absence of the frequency dispersion of the phonons, i.e. for Einstein s model of the crystal cos — a>. Then, the expression for the rate constant of multiphonon transition results from the formula (20) ... 

These coefficients can be evaluated for any biopolymer by taking an arbitrary chain length and determining the allowed values of frequencies for any set of boundary conditions. In our calculations we have assumed that the ends of the chain are fixed and we determine the frequency of the modes that permit an odd number of half wavelengths to be present on the chain. The eigenvectors for these frequencies are determined from the dispersion curves for an infinite chain. To make these calculations more compatible with experiment we have determined the absorption cross section which can be related to the Einstein coefficient by the following expression... 

Einstein s hypothesis, then, led to two definite predictions. In the first place, there should be a photoelectric threshold frequencies less than a certain limit, equal to 4>/h, should be incapable of ejecting photoelectrons from a metal. This prediction proved to be verified experimentally, and with more and more accurate determinations of work function it continues to hold true. It is interesting to see where this threshold comes in the spectrum. For this purpose, it is more convenient to find the wave length X = c/v corresponding to the frequency /h. If we express in electron volts, as is commonly done, (see Eq. (1.1), Chap. IX), we have the relation... 

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