The Einstein Coefficients

The rate constant for fluorescence can be related to the dipole strength for absorption by a line of reasoning that Einstein developed in the period 1914-1917 [1]. Consider a set of atoms with ground-state wavefunction Pa and excited state wavefunction Pf,. Suppose that the atoms are enclosed in a box and are exposed only to the black-body radiation from the walls of the box. According to Eq. (4.8c), the rate at which the radiation causes upward transitions from Pa to Pb is 

Equation (5.2b) simply defines a parameter B that is proportional to the dipole strength  

B is called the Einstein coefficient for absorption. Because each transition from to Wb removes an amount of energy hvba from the radiation, the sample must absorb energy at the rate Bp(vba)Nahvba- Equation (5.3) often is presented without the factors f and n so that it refers to a sample in a vacuum. 

Light also stimulates downward transitions from Tb to Ta, and from the symmetry of Eq. (4.7) the coefficient for this must be identical to the Einstein coefficient for upward transitions (B). If the excited atoms had no other way to decay, the rate of downward transitions would be 

At equilibrium the rates of upward and downward transitions must be equal. But from Eqs. (5.2b) and (5.4), this would require that 

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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... 

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... 

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 ... 

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. 

At low pressure, the only interactions of the ion with its surroundings are through the exchange of photons with the surrounding walls. This is described by the three processes of absorption, induced emission, and spontaneous emission (whose rates are related by the Einstein coefficient equations). In the circumstances of interest here, the radiation illuminating the ions is the blackbody spectrum at the temperature of the surrounding walls, whose intensity and spectral distribution are given by the Planck blackbody formula. At ordinary temperatures, this is almost entirely infrared radiation, and near room temperature the most intense radiation is near 1000 cm". ... 

Here I stands for the intensity of the spectral hnes N is the atom number density in cm Z is the partition function E and y are the energies and degeneracy s of the upper levels, respectively and A and A are the Einstein coefficient and wavelength, respectively, for the observed transitions. When changing the concentration Nt relative to that Nm the line intensities ft and 7m will likewise change, and according to (6.1) one should obtain a cahbration curve with constant slope (Davies et al. 1995 Ciucci et al. 1999 Hou and Jones 2000). 

Whereas absorption spectra can be obtained at a given temperature via Monte-Carlo t q)e simulations, the reach of equilibrium in an excited state of an isolated cluster is less obvious, and even less is the definition of a relevant temperature. In any case, the final state may be strongly dependent on the excitation process. Here we will ignore the vibrations of the Na(3p)Arn cluster. We assume a Franck-Condon type approximation and that emission takes place from relaxed equilibrium geometry structures on the Na(3p)Arn excited PES. The Einstein coefficients of the lines of emission towards the ground state at energy AE are given by... 

For spheres which aggregate, immobilization of hquid between the particles increases the apparent volume of the aggregate raising the value of the Einstein coefficient kg, depending on the number of particles present and their mode of packing [9]. 

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... 

Substituting the Einstein coefficient A for spontaneous emission according to... 

The intercept of the curve is clearly the Einstein coefficient, 2.5, as suggested by Equation (43). 

As noted above, we have replaced the Einstein coefficient of 2.5 by the more general intrinsic... 

Solution Equation (4.41) gives the Einstein relationship between [r/] and , the volume fraction occupied by the dispersed spheres. The volume fraction that should be used in this relationship is the value that describes the particles as they actually exist in the dispersion. In this case this includes the volume of the adsorbed layer. For spherical particles of radius R covered by a layer of thickness 8R, the total volume of the particles is (4/3) + 4ttR2 8R. Factoring out the volume of the dry particle gives Vdfy(1 + 38RJRS), which shows by the second term how the volume is increased above the core volume by the adsorbed layer. Since it is the dry volume fraction that is used to describe the concentration of the dispersion and hence to evaluate [77], the Einstein coefficient is increased above 2.5 by the factor (1 + 36/Vfts) by the adsorbed layer. The thickness of adsorbed layers can be extracted from experimental [77] values by this formula. ... 

Now we can introduce the energy density p (v m) to transform this result into the Einstein coefficient of absorption, viz. the probability that the molecule (or atom) will absorb a quantum in unit time under unit radiation density. The probability of absorption in the Einstein expression is given by B m p (v m). Under the influence of the radiation polarized In x-directions, the relationship between the field strength E in x-direction and the radiation density is deduced as follows ... 

By completely analogous treatment in which values of aH (0) = 0 and am (0) = 1, are used, the Einstein coefficient for induced emission Jm is found to be given by the equation ... 

The probability of absorption given by the Einstein coefficient of induced absorption Bm can be expressed in terms of M 2,... 

Excited state lifetimes are related to the Einstein coefficients of spontaneous emission Amn and can be approximately calculated from the expression... 

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