Einstein single frequency

To simplify the analysis, the Einstein single frequency (coe) model is used. The Einstein frequency is given by ... 

A more transparent representation of the temperature dependence can be obtained in simple models. Consider for example an Einstein-type model where the phonon spectrum is represented by a single frequency coa. The rate is loosely written in this case in the form... 

Consider now one of these variable and its contribution to the potential energy, z(r) = 27rg 2(7Xz(r)2. This is the potential energy of a three-dimensional isotropic harmonic oscillator. The total potential energy, Eq. (16.82) is essentially a sum over such contributions. This additive form indicates that these oscillators are independent of each other. Furthermore, all oscillators are characterized by the same force constant. We now also assume that all masses associated with these oscillators are the same, namely we postulate the existence of a single frequency Ms., sometimes referred to as the Einstein frequency of the solvent polarization fluctuations, and Ws are related as usual by the force constant... 

Where the integral covers the range from the minimum energy of that type of vibration, (fi )min, to the maximum, (fij)max- Einstein was the first to approximate the vibrational frequencies of a monatomic crystal to a single frequency. We introduce a rather more severe form of this approximation a single weighted mean frequency, cje, is chosen to be representative of all six external vibrations of a molecule in its crystal. 

The Einstein-Planck relation indicates that the energy of a photon of monochromatic (single-frequency) radiation depends only on its wavelength or frequency. A beam of radiation is more or less intense depending on the quantity of photons per unit time and per unit area, but the quantum energy (E) per photon is always the same for a given frequency of the radiation. 

In the above analysis it is assumed that the atoms vibrate with a single frequency Q, as in the Einstein theory, whereas in a real crystal there is a distribution of vibrational frequencies with an appropriate cut-off, as considered in the Debye theory. In any case, it is clear that a measurement of the /-factor by Mossbauer spectroscopy can provide knowledge concerning phonon properties, such as their frequency distribution and density of states. Similar information can also be obtained from an analysis of the second-order Doppler shift. Unfortunately, the restriction imposed by the relative timescales, typically as discussed earlier, normally... 

The Einstein relationship between absorption and fluorescence is strictly valid only for a system that absorbs and fluoresces at a single frequency. This condition clearly does not hold for molecules in solution, which have broad absorption and emission spectra. Expressions corresponding to Eq. (5.12) can be derived for such systems, but before we do this let s look at some of the general features of molecular fluorescence spectra. 

As we noted in Sect. 5.1, the Einstein relationship between absorption and fluorescence (Eq. 5.12) assumes that absorption and emission occur at a single frequency, which is not the case for molecules in solution. However, the overall rate of fluorescence by a molecule with broad absorption and emission bands can be related to the integrated absorption strength by expressions that were developed by Lewis and Kasha [18], Forster [19], Strickler and Berg [20], Birks and Dyson [16] and Ross [21]. 

But by the Einstein relation we know that the energy of a single photon on frequency oi is given by jod, and hence the total energy in tire field is... 

The collective modes of vibration of the crystal introduced in the previous paragraph involve all the atoms, and there is no longer a single vibrational frequency, as was the case in the Einstein model. Different modes of vibration have different frequencies, and in general the number of vibrational modes with frequency between v and v + dv are given by... 

Results of the next section indicate that the three-dimensional lattice acts much like an Einstein solid with a single phonon frequency vE. Viscoelastic experiments (5) on rosin in the glassy state are consistent with the sharp distribution of relaxation times thus predicted. In the dielectric case where each oscillator contains a dipole,... 

Phonons At least two phonon branches are involved in the observed absorption the acoustic phonons and the optical 46-cm "1 branch. Our model includes a single acoustic branch [with cutoff frequency f2max, and isotropic Debye dispersion hfiac q) = hQmaxq/qmax] and an optical dispersionless branch (Einstein s model, with frequency /20p). 

The two main nuclear modes affecting electronic energies of the donor and acceptor are intramolecular vibrations of the molecular skeleton of the donor-acceptor complex and molecular motions of the solvent. If these two nuclear modes are uncoupled, one can arrive at a set of simple relations between the two spectral moments of absorption and/or emission transitions and the activation parameters of ET. The most transparent representation is achieved when the quantum intramolecular vibrations are represented by a single, effective vibrational mode with the frequency Vy (Einstein model). 

The vibronic envelope ECWD (v) in Eq. [129] can be an arbitrary gas-phase spectral profile. In condensed-phase spectral modeling, one often simplifies the analysis by adopting the approximation of a single effective vibrational mode (Einstein model) with the frequency Vv and the vibrational reorganization energy Xy. The vibronic envelope is then a Poisson distribution of... 

Microwave radiation has a wavelength on the order of 1.0 cm. Calculate the frequency and the energy of a single photon of this radiation. Calculate the energy of an Avogadro s number of photons (called an einstein) of this electromagnetic radiation. 

For ionic solids, measurement of the ionic conductivity, <7 , has long provided a method for studying their atomic diffusion [25, 209, 225, 226] (see also Chapter 3). The measurements are usually made with an alternating current (AC) bridge operating at a fixed frequency, f (typically >1 kHz), to avoid polarization effects. The early studies were restricted to measurements on single crystals, and in this case (7i and the tracer diffusion coefficient were seen to be related by the Nernst-Einstein equation [25] ... 

We can guarantee this by following Einstein and asserting that all of the spectral weight is concentrated in a single vibrational frequency, coe. That is, we approximate the full density of states with the replacement... 

The g((o) can be used to provide a simple approximation to the lattice vibrations, the Einstein approximation. Let us begin by agreeing that single characteristic frequencies, could be chosen to individually represent each of the six types (three translational and three rotational) of external mode. The ( 6)y value is the density of states weighted mean value of all of the frequeneies over which that external mode, j, was dispersed. Normalising as discussed above ... 

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