Relation Between Einstein Coefficients

Inserting for a = 2(l>Ic)k the expression (2.62a) one obtains in the vicinity of the absorption centre frequency the relation between Einstein coefficient Bik and oscillator strength fik... 

In a celebrated paper, Einstein (1917) analyzed the nature of atomic transitions in a radiation field and pointed out that, in order to satisfy the conditions of thermal equilibrium, one has to have not only a spontaneous transition probability per unit time A2i from an excited state 2 to a lower state 1 and an absorption probability BUJV from 1 to 2 , but also a stimulated emission probability B2iJv from state 2 to 1 . The latter can be more usefully thought of as negative absorption, which becomes dominant in masers and lasers.1 Relations between the coefficients are found by considering detailed balancing in thermal equilibrium... 

Transition strengths can be given in terms of Einstein rate coefficients. For a pair of states j > and k > it is shown in elementary texts that these are related in a simple way. If one assumes that, for any pair of microstates i and j, the rate from i to j is equal to the rate from j to i one has the principle of detailed balance). Then, the relation between the coefficients is consistent with thermodynamics (Planck s black-body radiation law). 

The Einstein relation between diffusion coefficient and mobility is assumed to apply for the motion of orientational defects but may not apply for the quantum tunnelling motion of ion states. (It should be noted in passing that some workers define with a sign opposite to that used here.)... 

The absolute magnitude of the calculated value of Qa depends on the way in which the relation between diffusion coefficient and mobility for ion states differs from the classical Einstein form. If we introduce parameters 0+ and 6 and write... 

The relation between diffusion coefficient and dc conductivity is given (approximately) by the Nemst-Einstein equation. 

The relation between the microscopic friction acting on a molecule during its motion in a solvent enviromnent and macroscopic bulk solvent viscosity is a key problem affecting the rates of many reactions in condensed phase. The sequence of steps leading from friction to diflfiision coefficient to viscosity is based on the general validity of the Stokes-Einstein relation and the concept of describing friction by hydrodynamic as opposed to microscopic models involving local solvent structure. In the hydrodynamic limit the effect of solvent friction on, for example, rotational relaxation times of a solute molecule is [ ]... 

Einstein coefficients provide good empirical relations between the rate of a transition and the density of radiation but quantum mechanics has something to say about... 

Here (A) and crp (A) are the cross-sections for absorption and stimulated fluorescence at A, respectively, and mo is the population of the ground state. The first exponential term gives the attenuation due to reabsorption of the fluorescence by the long-wavelength tail of the absorption band. The attenuation becomes more important, the greater the overlap between the absorption and fluorescence bands. The cross-section for stimulated fluorescence is related to the Einstein coefficient by... 

The Einstein coefficients are related to the most fundamental quantity which describes the transition probability, known as the transition moment. During an electronic transition for instance, an electron jumps from one orbital to another. Its distance from the nucleus changes, so there is a change in the instantaneous dipole moment. The greater this change, the more probable the transition because it is the interaction between this transition dipole and the electric vector of light. 

All upward radiative transitions in Figure 3.23 are absorptions which can promote a molecule from the ground state to an excited state, or from an excited state to a higher excited state. We have seen that the probability of these transitions is related ultimately to the transition moment between the two states and thereby to the Einstein coefficient A. In practice two other related quantities are used to define the intensity5 of an absorption, the oscillator strength f and the molar decadic extinction coefficient e. 

Diffusion of small solute particles (atoms, molecules) in a dense liquid of larger particles is an important but ill-understood problem of condensed matter physics and chemistry. In this case one does not expect the Stokes-Einstein (SE) relation between the diffusion coefficient D of the tagged particle of radius R and the viscosity r/s of the medium to be valid. Indeed, experiments [83, 112-115] have repeatedly shown that in this limit SE relation (with slip boundary condition) significantly underestimates the diffusion coefficient. The conventional SE relation is D = C keT/Rr]s, where k T is the Boltzmann constant times the absolute temperature and C is a numerical constant determined by the hydrodynamic boundary condition. To explain the enhanced diffusion, sometimes an empirical modification of the SE relation of the form... 

This is the Einstein relation, and shows the direct proportionality between diffusion coefficient and mobility. 

The relation between conductivity and diffusion coefficient, the Nernst-Einstein relationy is easily derived from (2.40) and (2.49) ... 

In electrochemistry several equations are used that bear Einsteins name [viii-ix]. The relationship between electric mobility and diffusion coefficient is called Einstein relation. The relation between conductivity and diffusion coefficient is called - Nernst-Einstein equation. The expression concerns the relation between the diffusion coefficient and the viscosity and is known as the - Stokes-Einstein equation. The expression that shows the proportionality of the mean square distance of the random movements of a species to the diffusion coefficient and the duration of time is called - Einstein-Smoluchowski equation. A relationship between the relative viscosity of suspension and the volume fraction occupied by the suspended particles - which was derived by Einstein - is also called Einstein equation [ix]. 

Einstein presented an equation relating the diffusion coefficient of species i (Df to the frictional coefficient in a diffusion process [ii], this can be transformed into a relationship between -> ionic mobility iq and -> diffusion coefficient D ... 

In this section we have distinguished between spontaneous and induced transitions, and we have shown how the probabilities for these processes, the Einstein A and B coefficients, are related to each other. The next section deals with experimental measurements of absorption, and the relation between these measurements and the theoretical quantities is explored. 

The relation between the Einstein coefficients A and By is A — Snhc0v3 By. The Einstein stimulated absorption or emission coefficient B may also be related to the transition moment between the states i and j for an electric dipole transition the relation is... 

Here pt and pj denote the fractional populations of states i and j (p( = exp —Ei/kT /q in thermal equilibrium, where q is the partition function) pm and pn denote the corresponding fractional populations of the energy levels, and dm and dn the degeneracies (pf = pm/dm, etc.). The absorption intensity Gji9 and the Einstein coefficients and Bji9 are fundamental measures of the line strength between the individual states i and j they are related to each other by the general equations... 

For transitions between individual states any of the more fundamental quantities Gjh B ji9 Ajh or Mji may be used the relations are as given above, and are exact. Note, however, that the integrated absorption coefficient A should not be confused with the Einstein coefficient Ajt (nor with absorbance, for which the symbol A is also used). Where such confusion might arise, we recommend writing A for the band intensity expressed as an integrated absorption coefficient over wavenumber. 

Comparing Eqs. (6.108) and (6.110) yields the Einstein relation between the mobility and diffusion coefficient... 

Alternatively, Einstein s equation [6] for the relation between the root mean square displacement X and the diffusion coefficient D,... 

The Einstein Relation between the Absolute Mobility and the Diffusion Coefficient... 

There were several aspeets of the Stokes-Einstein relation that reduced it to being only an approximate relation between the diffusion coefficient of an ionic species and the viscosity of the medium. In addition, there were fundamental questions regarding the extrapolation of a law derived for macroscopie spheres moving in an incompressible medium to asituation involving the movement ofions in an environment of solvent molecules and other ions. In the case of the Nernst-Einstein relation, the factors that limit its validity are more subtle. 

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