Absorption and emission probability

This probability is determined by the amount to which vector E is contained in vector d. This quantity equals the hermitian product of the E and d [140]. The hermitian product can be written as the scalar product of E and d. As a result the probability Wabs of absorption of light, which is characterized by its E-vector, by a classical dipole is proportional to Ed 2. This probability can be divided into a dynamic part Tp 

The explicit form of the factor Ed in laboratory coordinates x y zy in which the orientation of the angular momentum J of the molecule is characterized by angles 9 and p (see Fig. 1.5(e)), can be found by applying the rule of scalar multiplication of vectors in cyclic coordinates (see (A.9))  

In order to be able to apply Eq. (2.2) in practice, let us consider how we can determine the cyclic components E9 of the vector E for completely polarized light. 

The component E+1 describes the light wave which transfers the angular momentum coinciding with the light beam direction, and E l describes that opposed to the former. 

In writing Eq. (2.6) the symmetry properties of the D-functions have been used, as presented in Appendix A (see (A. 11)). A detailed treatment of the properties of D-functions is considered in the quantum theory of angular momentum [379]. 

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In this section, we will study the absorption and emission probabilities for a single two-level atomic center that is illnminated by a monochromatic electromagnetic wave. 

The absorption and emission probabilities for transitions between the two levels are proportional to the vibrational wavefunction, Viq—qo)-Substituting Eq. (8.8) into (8.9) gives the emission spectrum,... 

The second reason to introduce the derivation (6 -9) is to note that all that is required to evaluate the absorption and emission probability F A (t, r) of (9) are matrix elements of the evolution operator exp(-i//r/h). (These matrix elements are the conventional probability amplitudes When considering a situation in which many different kinds of decay processes are involved, e.g. radiative and nonradiative decay, it is not always convenient to deal directly with the matrix elements of exp(-itfr/h), the af(t). Rather, it is simpler to introduce (imaginary) Laplace transforms 16) in the same manner that electrical engineers use them to solve ac circuit equations 33L Thus, if E is the transform variable conjugate to t, the transforms of af(t) are gf(E). The quantities gf (E) can also be labeled by the initial state k and are denoded by Gjk(E). It is customary in quantum mechanics to collect all these Gjk(E) into a matrix G(E). Since matrix methods in quantum mechanics imply some choice of basis set and all physical observables are independent of the chosen basis set, it is convenient to employ operator formulations. If G (E) is the operator whose matrix elements are Gjk(E), then it is well known that G(E) is the Green s function i6.3o.34) or resolvent operator... 

The amount of fluorescence emitted by a fluorophore is determined by the efficiencies of absorption and emission of photons, processes that are described by the extinction coefficient and the quantum yield. The extinction coefficient (e/M-1 cm-1) is a measure of the probability for a fluorophore to absorb light. It is unique for every molecule under certain environmental conditions, and depends, among other factors, on the molecule cross section. In general, the bigger the 7c-system of the fluorophore, the greater is the probability that the photon hitting the fluorophore is absorbed. Common extinction coefficient values of fluorophores range from 25,000 to 200,000 M 1 cm-1 [4],... 

In the first case the so called 0—0 transition will be the most probable one, both for absorption and emission, and therefore the most intense... 

Figure 12.3 outlines the essential features of the PASADENA/PHIP concept for a two-spin system. If the symmetry of the p-H2 protons is broken, the reaction product exhibits a PHIP spectrum (Fig. 12.3, lower). If the reaction is carried out within the high magnetic field of the NMR spectrometer, the PHIP spectrum of the product consists of an alternating sequence of enhanced absorption and emission lines of equal intensity. This is also true for an AB spin system due to a compensating balance between the individual transition probabilities and the population rates of the corresponding energy levels under PHIP conditions. The NMR spectrum after the product has achieved thermal equilibrium exhibits intensities much lower than that of the intermediate PHIP spectrum. 

Finally we note some other properties of these allowed transitions of the lanthanide ions. From Table 1 it becomes clear that in general the 4f—5d bands have a smaller band width than the c.t. transitions, typical values being 1000 and 2000 cm-i, respectively. In this connection it is interesting to find that at low temperatures the 4f- -5d absorption and emission bands often show a distinct and extended vibrational fine structure [Ce3+ (25), Tb + (25), Eu2+ (14, 26), Yb2+ (27)], whereas c.t. transitions do not. From this it seems probable that in the excited c.t. state the interaction between the lanthanide ion and its surroundings is stronger than in the excited 4f 5d state. This is not imexpected. As far... 

Figure 4. IS Probability of absorption and emission by differently oriented anthracene molecules. The dotted cutves represent vector direction of emitted radiation.

Let neq and /ieq rad be the equilibrium spin population differences between the a and y3 levels in the absence and presence of radiation, respectively. Let pap Ppa be the probabilities per second for stimulated absorption and emission of radiation, (a) Show that... 

Values of the radiative rate constant fcr can be estimated from the transition probability. A suggested relationship14 57 is given in equation (25), where nt is the index of refraction of the medium, emission frequency, and gi/ga is the ratio of the degeneracies in the lower and upper states. It is assumed that the absorption and emission spectra are mirror-image-like and that excited state distortion is small. The basic theory is based on a field wave mechanical model whereby emission is stimulated by the dipole field of the molecule itself. Theory, however, has not so far been of much predictive or diagnostic value. 

I 5.1 Einstein Transition Probabilities, 23 I 5.2 Absorplion Intensity of Atoms, 24 I 5.3 Oscillator Strength, 25 1-6 Resonance Absorption and Emission by Atoms, 27... 

Figure 13.1 Vibrational levels and internuclear distance-probability functions for the ground state and first excited singlet of a diatomic molecule. Absorption and emission according to the Franck-Condon principle are illustrated. Adapted from N. J. Turro, Molecular Photochemistry, Addison-Wesley-—W. A. Benjamin, Reading, Mass., 1967. Reproduced by permission of Addison-Wesley,...

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