Coefficient Einstein

The Einstein coefficients prove to be useful for understanding the relationships among the probabilities for spontaneous emission, stimulated emission, and absorption. They are thus valuable for understanding the criteria for achieving laser action, where the competition between spontaneous and stimulated emission in the laser medium is crucial. The Einstein coefficients also lead to important insights into the relationships between the absorption and fluorescence properties of molecules, relationships that are often taken for granted in the chemical physics literature. 

We begin our discussion with an ensemble of identical two-level systems in which the upper and lower state populations are N2 and Ni, respectively. The energy levels are spaced by AE = hv, and the systems are at thermal equilibrium with a radiation energy distribution over light frequencies v given by p(v). It is assumed that only three mechanisms exist for transferring systems between levels 1 and 2 one-photon absorption, spontaneous emission (radiation of a single photon), and stimulated emission (Fig. 8.3). In the latter process, a photon 

However, p(v) also be given by the Planck blackbody distribution at thermal equilibrium [4] 

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Hilborn H 1982 Einstein coefficients, cross sections, f values, dipole moments and all that Am. J. Phys. 50 982-6... 

Einstein coefficient at ai2- When the polarization field at frequency = ... 

Piper L G and Cowles L M 1986 Einstein coefficients and transition moment variation for the NC(A S -X n) transition J. Chem. Phys. 85 2419-22... 

The Einstein coefficients are related to the wave functions j/ and of the combining states through the transition moment R , a vector quantity given by... 

The value of the first coefficient b, for the dispersion of spherical particles is well known and generally accepted. This is Einstein coefficient b, = 2.5, taking into account the viscosity variation of the dispersion medium upon introducing noninteracting solid particles of spherical form into it. Thus, for tp [Pg.83]

As the structure of Eq. (8) shows, the first coefficient for cp -> 0 in the power expansion of this equation is equal to Einstein coefficient (according to formula (9) it is somewhat higher, but not much). 

Einstein coefficient b, in (5) for viscosity 2.5 by a value dependent on the ratio between the lengths of the axes of ellipsoids. However, for the flows of different geometry (for example, uniaxial extension) the situation is rather complicated. Due to different orientation of ellipsoids upon shear and other geometrical schemes of flow, the correspondence between the viscosity changed at shear and behavior of dispersions at stressed states of other types is completely lost. Indeed, due to anisotropy of dispersion properties of anisodiametrical particles, the viscosity ceases to be a scalar property of the material and must be treated as a tensor quantity. 

In the above rather simplified analysis of the interaction of light and matter, it was assumed that the process involved was the absorption of light due to a transition m - n. However, the same result is obtained for the case of light emission stimulated by the electromagnetic radiation, which is the result of a transition m -> n. Then the Einstein coefficients for absorption and stimulated emission are identical, viz. fiOT< n = m rt. 

Fig. 9.7 Plots of G nanocomposite/C matrix VS. Vol./a of MMT for various nanocomposites. The Einstein coefficient kE is shown by the number in the box. The lines show the calculated results from Halpin and Tai s theory with various kE.

Certain features of light emission processes have been alluded to in Sect. 4.4.1. Fluorescence is light emission between states of the same multiplicity, whereas phosphorescence refers to emission between states of different multiplicities. The Franck-Condon principle governs the emission processes, as it does the absorption process. Vibrational overlap determines the relative intensities of different subbands. In the upper electronic state, one expects a quick relaxation and, therefore, a thermal population distribution, in the liquid phase and in gases at not too low a pressure. Because of the combination of the Franck-Condon principle and fast vibrational relaxation, the emission spectrum is always red-shifted. Therefore, oscillator strengths obtained from absorption are not too useful in determining the emission intensity. The theoretical radiative lifetime in terms of the Einstein coefficient, r = A-1, or (EA,)-1 if several lower states are involved,... 

Effect of PVA Molecular Weight on Adsorbed Layer Thickness. Figure 4 shows the variation of reduced viscosity with volume fraction for the bare and PVA-covered 190nm-size PS latex particles. For the bare particles, nre(j/ is independent of and the value of the Einstein coefficient is ca. 3.0. For the covered particles, rired/ t increases linearly with tp. Table IV gives the adsorbed layer thicknesses calculated from the differences in the intercepts for the bare and covered particles and determined by photon correlation spectroscopy, as well as the root-mean-square radii of gyration of the free polymer coil in solution. The agreement of the adsorbed layer thicknesses determined by two independent methods is remarkable. The increase in adsorbed layer thickness follows the same dependence on molecular weight as the adsorption density, i.e., for the fully hydrolyzed PVA s and... 

Effect of PS Latex Particle Size on Adsorbed Layer Thickness. Figure 6 shows the variation of reduced viscosity with volume fraction for 190, 400, and HOOnm-size bare and PVA-covered PS latex particles. The viscosity variation of the different-size bare particles was the same, with an Einstein coefficient of ca. 3.0. The... 

Einstein had an important role to play in the description of absorption with the development of the theory associated with the Einstein coefficients. Consider a transition from a low energy state to a higher energy state with a transition rate w given by ... 

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... 

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

Box 2.2 Einstein coefficients. Transition moment. Oscillator strength... 

Let us consider a molecule and two of its energy levels E) and f 2- The Einstein coefficients are defined as follows (Scheme B2.2) Bn is the induced absorption coefficient, B2i is the induced emission coefficient and A21 is the spontaneous emission coefficient. 

It is interesting to note that for a resonant transition (i.e. coinciding absorption and emission frequencies), the reciprocal of the radiative lifetime is equal to the Einstein coefficient Ai for spontaneous emission (see Box 3.2). 

A being the radiative rate (labeled in such a way because it coincides with the Einstein coefficient of spontaneous emission) and Anr being the nonradiative rate, that is, the rate for nonradiative processes. The solution of the differential equation (1.16) gives the density of excited centers at any time r ... 

This expression can be compared with Planck s formula - Equations (A3.2) or (2.2) - to obtain the following two relations among the Einstein coefficients ... 

The molecule-intrinsic factor in the intensities of emission spectra can be obtained from the well-known Einstein coefficients (see, for example. Refs. [20, 21]). For the two states i and f considered above, whose energies are Ei and Ef, respectively, with Ei < Ef, we define as the Einstein coefficient for absorption, Bfl as the Einstein coefficient for stimulated emission, and Afl as the Einstein coefficient for spontaneous emission. We denote by Ni and Nf the number of molecules with energies Ei and Ef, respectively, and the Einstein coefficients are defined such that, for example, the change in Nf caused by electric dipole transitions to and from i is given 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... 

On the basis of these formulae one can convert measurements of area, which equals the integral in the latter formula, under spectral lines into values of coefficients in a selected radial function for electric dipolar moment for a polar diatomic molecular species. Just such an exercise resulted in the formula for that radial function [129] of HCl in formula 82, combining in this case other data for expectation values (0,7 p(v) 0,7) from measurements of the Stark effect as mentioned above. For applications involving these vibration-rotational matrix elements in emission spectra, the Einstein coefficients for spontaneous emission conform to this relation. 

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