Einstein rate coefficients

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). 

We assume the reader is familiar with this theory and merely restate the relations to define units and notation. If E3 Ek we have  

This last relation is actually a purely microscopic equation, and its derivation via thermodynamics in the elementary texts is somewhat circuitous in reality, quantum electrodynamics is required for a proper treatment (see, e.g., [128]). 

In the absence of any radiation field, /(i/kj) = 0 The population Nk of level k simply decays exponentially with a lifetime A 1 to level j. This provides one way of defining the intensity of a spectral line. 

It is also shown in elementary texts that the spontaneous transition rate for emission of a photon by a single atom in the dipole approximation is given by  

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Fig. 6. Reaction rate coefficients for the combination of f-butyl radicals in Aj n-hexa-decane solvent V, n-dodecane solvent , n-decane solvent X, n-octane solvent 8 n-heptane solvent and of allyl radicals in propane ( ) and melhylallyl radicals in isobutane (O) plotted against the Smoluchowski—Stokes—Einstein rate coefficient, eqn. (30). The broken line is of unit slope. The solid line is a comparison of the steady-state (t-> >) Collins and Kimball rate coefficient [eqn. (26)] with the activation rate coefficient, feact = 1011 dm3 mol-1 s 1 and the diffusion-limited rate coefficient 4irRD replaced by eqn. (30). After Schuh and Fischer [40].

Hasinoff noted that the rate coefficient of formation of encounters pairs, fcD, was smaller than predicted from the Smoluchowski—Stokes—Einstein rate coefficient [eqn. (29)]. In aqueous glycerol, this reduction was by 0.14 times, in aqueous polyethylene glycol by 0.30 times, and in aqueous ethylene glycol by 0.11 times. Hasinoff compared these reductions in rate of diffusive rate of formation of encounter pairs with three theories of anisotropic reactivity due to Weller [262], Schmitz and Schurr [257] and... 

Hasinoff noted that the rate coefficient of formation of encounters pairs, feo, was smaller than predicted from the Smoluchowski—Stokes—Einstein rate coefficient [eqn. (29)]. In aqueous glycerol, this reduction was by... 

We now make two coimections with topics discussed earlier. First, at the begiiming of this section we defined 1/Jj as the rate constant for population decay and 1/J2 as the rate constant for coherence decay. Equation (A1.6.63) shows that for spontaneous emission MT = y, while 1/J2 = y/2 comparing with equation (A1.6.60) we see that for spontaneous emission, 1/J2 = 0- Second, note that y is the rate constant for population transfer due to spontaneous emission it is identical to the Einstein A coefficient which we defined in equation (Al.6.3). 

Einstein derived the relationship between spontaneous emission rate and the absorption intensity or stimulated emission rate in 1917 using a thennodynamic argument [13]. Both absorption intensity and emission rate depend on the transition moment integral of equation (B 1.1.1). so that gives us a way to relate them. The symbol A is often used for the rate constant for emission it is sometimes called the Einstein A coefficient. For emission in the gas phase from a state to a lower state j we can write... 

In summary, the so-called Einstein A and B rate coefficients connecting a lower-energy initial state i and a final state f are related by the following conditions ... 

Routh and Russel [10] proposed a dimensionless Peclet number to gauge the balance between the two dominant processes controlling the uniformity of drying of a colloidal dispersion layer evaporation of solvent from the air interface, which serves to concentrate particles at the surface, and particle diffusion which serves to equilibrate the concentration across the depth of the layer. The Peclet number, Pe is defined for a film of initial thickness H with an evaporation rate E (units of velocity) as HE/D0, where D0 = kBT/6jT ir- the Stokes-Einstein diffusion coefficient for the particles in the colloid. Here, r is the particle radius, p is the viscosity of the continuous phase, T is the absolute temperature and kB is the Boltzmann constant. When Pe 1, evaporation dominates and particles concentrate near the surface and a skin forms, Figure 2.3.5, lower left. Conversely, when Pe l, diffusion dominates and a more uniform distribution of particles is expected, Figure 2.3.5, upper left. 

The Einstein A coefficient for the electronic transition to the first excited electronic state at 65075.8 cm-1 is 1.37 ns. Calculate the Einstein B coefficient for the induced transition rate. 

Einstein coefficients The measure of the rate of a transition, whether spontaneous (Einstein A coefficient) or stimulated (Einstein B coefficient). 

The excited-state wavepacket spontaneously emits photons while undergoing transitions to any of the electronically-ground vibrational wavefunctions t (where we have lumped the final state quantum numbers i>/, jf in a single index f). The rate of emission from a given 4% component of the excited wavepacket to a given ground state is given in terms of the Einstein A-coefficient [9],... 

Einstein A coefficient in its (001-000) transition at 2349 cm -1 than that for the corresponding band at 2223 cm -1 in NzO, appears always to be the minor triatomic product emitting in this range. Although these results are presently preliminary (and their interpretation may need to be revised if, for example, rates of relaxation of the excited products are distorting the partially relaxed vibrational distributions), they seem incompatible with the diode laser observations unless reaction (18) produces C02 substantially in vibrational levels with v3 = 0. Further experiments are in progress. 

Finally, we introduce the Einstein A coefficient, An,t,spontaneous decay rate of the n state to the lower lying n state. Explicitly,1... 

The use of the Stokes-Einstein equation (2) relating the diffusion coefficient (D) of a spherical solute molecule to its radius (r), the viscosity of the medium (tj) and the Boltzmann constant (k) permits the rate coefficient ( en) to be expressed in (3) in terms of the viscosity of the medium. In this derivation, the... 

Diffusionally mediated collisions between two floccules of equal size can be described by a second-order rate coefficient KD = 8irRD, where R is the radius and D is the diffusion coefficient of a floccule. Upon invoking the Stokes-Einstein relation, D = kBT/67ri7R, one derives Eq. 6.2. For an introductory discussion of the second-order rate law for particle collisions, see, for example, Chap. 11 in P. C. Hiemenz, Principles of Colloid and Surface Chemistry, Marcel Dekker, New York, 1986. 

Measurements of the free acid Ti vibrational lifetimes were also monitored as a function of base concentration (e.g., pyrrole with acetonitrile) to determine the effect of collisions and hydrogen-bond formation rates. Stern-Volmer plots of 1 /T1 rates versus base concentration enabled extraction of a bimolecular rate constant (kbm) for pyrrole acetonitrile of 2.5 0.2 x 1010 dm3/mol-s, which is slightly larger than the estimated Stokes-Einstein diffusion coefficient (0.73 x... 

Similarly the parent problem for fluorescence of phosphorescence emission in dilute gases or solution is the influence of coupling to foreign molecules on spontaneous emission [189]. The emission rate for an isolated molecule A is given by the Einstein A coefficient,... 

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