Absorption Einstein coefficients

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. 

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,... 

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). 

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... 

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". ... 

Einstein coefficient of absorption for the pump wavelength calculated as B 531 In(lO) / where c is the velocity of light and is Avogadro s number. Other details of the modeling may be found in reference 3. 

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... 

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... 

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 ... 

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

Consider a system in which matter and radiation are in equilibrium in a closed cavity at temperature T. (This equilibrium situation does not generally hold in spectroscopy, but the transition probabilities are fundamental properties of the interaction between radiation and matter and cannot be affected by the presence or absence of equilibrium.) As before, let be greater than (0). The probability of absorption from state n to state m is proportional to the number of photons with frequency near vmn the number of photons is proportional to the radiation density u(vmn). Hence the rate of absorption is given by Bn t,mNnu(i mn)t where Nn is the number of molecules in state n and Bn m is a proportionality constant called the Einstein coefficient for absorption. From the discussion following (3.46) and from (3.47), it follows that... 

These equations are similar to those of first- and second-order chemical reactions, I being a photon concentration. This applies only to isotropic radiation. The coefficients A and B are known as the Einstein coefficients for spontaneous emission and for absorption and stimulated emission, respectively. These coefficients play the roles of rate constants in the similar equations of chemical kinetics and they give the transition probabilities. 

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. 

N, Ng...population of excited, ground states in two-level systems Bu, B. ..Einstein coefficient for absorption, stimulated emission A.Einstein coefficient for spontaneous emission... 

The Einstein coefficient of absorption of radiation for the longitudinal (along the z axis) and the transverse (perpendicular to the z axis) electromagnetic fields can be obtained from these matrix elements and are given by... 

These coefficients can be evaluated for any biopolymer by taking an arbitrary chain length and determining the allowed values of frequencies for any set of boundary conditions. In our calculations we have assumed that the ends of the chain are fixed and we determine the frequency of the modes that permit an odd number of half wavelengths to be present on the chain. The eigenvectors for these frequencies are determined from the dispersion curves for an infinite chain. To make these calculations more compatible with experiment we have determined the absorption cross section which can be related to the Einstein coefficient by the following expression... 

We now finally define the Einstein coefficient for induced absorption as ... 

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. 

Special Topic 2.1 Einstein coefficients of absorption and emission... 

We should have in mind two conditions which limit the applicability of the relation (1.16). First, the absorption of a photon by a molecule, crystal, solution, etc. is not necessarily related to an electronic excitation for some spectral region. In such a spectral region the relation between the Einstein coefficients is changed, i.e. the relation (1.14), which makes the relation (1.16) invalid.6... 

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