Einstein’s A and B coefficients

Einstein s A and B coefficients. Quantized systems, such as atoms and molecules, emit and absorb radiation of frequency vy = E — Ej /hc in transitions between states i) and j) of energy , Ej. Einstein assumed that the probability that a system in state i) will absorb a photon of energy hcvij is proportional to the density of radiative energy per frequency interval, u(vij) dv. The probability of absorbing a photon in the time interval dt is given by... 

The transition energies are not sharp but rather have finite widths due to thermal vibrations in the solid. Thus, the wave functions and the density of states can be treated as functions of energy. The wave functions can be normalized with respect to energy and a Dirac delta function used for the density of final states to insure conservation of energy. Then Einstein s A and B coefficients can be used to relate the transition matrix elements to experimentally measurable quantities such as oscillator strengths and luminescence lifetimes. For electric dipole-dipole interaction the energy transfer rate becomes... 

As we have done for the derivation of Equation 3.39, we will set B21 = Bi2> which allows us to derive an expression for relating Einstein s A and B coefficients as... 

Equations 3.77 and 3.80 relate the stimulated emission rate and the spontaneous emission rate to the absorption coefficient, respectively. With the help of the same equations, the stimulated emission rate and the spontaneous emission rate can be related to each other. The implications of these equations are that both emission rates can be determined if the absorption coefficient along with its energy (or wavelength) dependence is known. Fortunately, the absorption coefficient is a measurable quantity. Therefore, once measured, the emission rates can be determined. The absorption rate can also be calculated with numerical techniques. Because the absorption coefficient, spontaneous emission rate, and stimulated emission rate can all be determined with the knowledge of Einstein s B coefficient (recall that the A coefficient can be calculated from the B coefficient), calculation of the B is sufficient to determine the absorption coefficient (the last part of Equation 3.78), spontaneous emission rate (Equation 3.79), and the stimulated emission rate (Equation 3.72 as A and B coefficients are related). A succinct description of the calculations leading to Einstein s B coefficient and/or the two emission rates is given below. The B coefficient represents the interaction of electron in the solid with the electromagnetic wave, which requires a quantum mechanical treatment. For further details, the reader is referred to [29]. 

The self-diffusion coefficients of CF and Na" in molten sodium chloride are, respectively, 33 x 10 exp(-8500// 7) and 8x10 exp(-4000// 7) cm s". (a) Use the Nernst-Einstein equation to calculate the equivalent conductivity of the molten liquid at 935°C. (b) Compare the value obtained with the value actually measured, 40% less. Insofar as the two values are significantly different, explain this by some kind of structural hypothesis. 

A charged particle in suspension with its inner immobile Stern layer and outer diffuse Gouy (or Debye-ffiickel) layer presents a different problem from that arising with a smooth and small nonpolar sphere. In movement such particles experience electroviscous effects which have two sources (a) the resistance of the ion cloud to deformation, and (b) the repulsion between particles in close contact. When particles interact, for example to form pairs in the system, the new particle will have a different shape from the original and will have different flow properties. The coefficient 2.5 in Einstein s equation (7.30)... 

Figure 4.23 Binding of antibody during diffusion in the interstitial space of a tumor. Adapted from [118]. In panel a, the bound fraction is plotted versus the dose of antibody given to the animal. The solid symbols represent specific antibody and the open symbols represent non-specific antibody squares indicate IgG and circles indicate Fab fragments. In panel b, an apparent binding constant for the specific antibody was calculated from the mobile fraction and the previously measured diffusion coefficient for non-specific IgG and Fab. The diffusion coefficient in water was estimated, for IgG Dq was 3.9 x 10 cm /s and for Fab Aissue was 6.6 x 10 cm /s. The tumor diffusion coefficient was lower, for IgG Dq was 1.3 x 10 cm /s and for Fab tissue was 2.7 x 10 cm /s. All diffusion coefficients were corrected to 20 °C using the Stokes-Einstein equation.

The presence of mobile species in a particular medium can give rise to a macroscopic concentration gradient obeying Pick s law . The flux of matter in a given x direction is given by J = —D dC/dx, where C is the concentration of mobile species and D the diffusion coefficient. The conductivity is given by cr = CBe, where B is the mobility and e the charge of the mobile ions. The Nernst-Einstein law which connects diffusion coefScient and conductivity is a = DCe /kT, where k is Boltzmann s constant and T temperature. Furthermore, the thermal activation of C and D must be taken into account, as follows ... 

When the laser is tuned to one of the v=0 -> v=l transition frequencies like one of the Ri(j-1) lines a fraction e of the molecules in the relevant ground state v=0, j-1 is excited to the specific vibrational-rotational state v=l,j. In the absence of saturation effects e is proportional to the spectral power density 5(co) of the radiation, Einstein s B coefficient and the period of time t the molecule senses the radiation field... 

FIGURE 15.27 (a) Stimulated absorption, which defines Einstein s coefficient B. (b) Spontaneous emission, which defines Einsteins coefficient A. (c) Stimulated emission, which defines Einstein s coefficient B. In stimulated emission, the two photons have the same wavelength and phase, as indicated. 

When the probe polymer is larger than the network size, the Stokes-Einstein (S-B) type diffusion as shown in Eq. (7) becomes difficult and the polymer diffuses by reptation. In this case, the diffusion coefficient can be expressed by Eq. (8). Pajevic et al. [3] measured the diffusion coefficient D of polyst) ne (PS), which is included in the gel as a probe molecule, by using dynamic light scattering. The gel used was poly(methyl methacrylate) gel, which is composed of 12.5% (v/v) polymer and is swollen by toluene. The results are shown in Fig. 14 [3] where is the diffusion coefficient of PS at each molecular weight (Mf) in toluene and... 

In this extension of the Tsai-Halpin equation (Brassell and Wischmann, 1974), K is a generalized Einstein (1905, 1906, 1911) coefficient equal to 2.5 for a suspension of rigid spheres in a matrix having a Poisson s ratio equal to 0.5, and given approximately for other values of v in the reference cited. Note that K = A + I [see equation (12.5)]. The constant B is defined as in equation (A-6) and ij/ is given by functions such as... 

Comparing with the Boltzmann-Gibbs distribution, this imposes the constraint A(s) = -B(s)f(s)l2kgT, which is known as the Einstein relation. Written this way, the current J(s, t) has a simple physical interpretation the first term in the current is the drift, arising due to the action of the force and characterized by the mean velocity A(s), and the second term represents the diffusive flux (Pick s law) with the diffusion coefficient D x) = B (jc) / 2. 

Einstein Coefficients. Using the spectroscopic constants for the X v = 0 and A Ilj, v = 0 states derived from the A<-X absorption spectrum [8] and assuming Hund s case (b) to hold for these states, the Einstein A coefficients were calculated for 23 purely rotational transitions between the X v = 0, N = 0 to 5 levels and for 79 purely rotational transitions between the A Ilj, v = 0, N = 1 to 5 (and A-doublets) levels [9]. 

It can be observed that the constant B is basically the same as that for Halpin-Tsai equation. Constant A is related to the graieralized Einstein coefficient kg. The value of kg is not a constant but depends on the state of agglomeration of the particles. It has the value of 2.5 for perfect dispersion of spherical particles and perfect matrix-particle adhesion. The value will decrease if there is dewetting and slippage at the matrix-particle interface but will increase if there is agglomeration [74]. However, kg is applicable only to matrix with Poisson s ratio of 0.5. For other Poisson ratios, a conversion is required and can be done easily with a list of relative Einstein coefficients for various Poisson s ratios presented in the works of Nielsen and co-workers [59, 63, 71]. Finally, i f depends on the maximum or critical packing fraction of the filler in the matrix. 

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