Transition probabilities absorption

Since the fluorescence intensity of a fluorophore is proportional to the square of the absorption transition probability, then it should vary with the angular dependence of the fluorophore s dipole with respect to the direction of the excitation beam. If the orientation of the dipoles of the Trp residues is modified as the result of the motion of the Trp residues, then the fluorescence intensity will change with the angle of the rotation of the crystal. In the absence of motions, the fluorescence intensity will remain constant at all positions of the crystal, since we are monitoring a definite orientation of the dipoles. One also should note that we are monitoring the fluorescence of two Trp... 

The absorption spectmm, 0(01), is the ratio of transition probability per unit time/incident photon flux. The incident photon flux is the number of photons per unit area per unit time passing a particular location, and is... 

The transition probability for absorption of two photons can be described in tenns of a two-photon cross section 5 by... 

The time constant r, appearing in the simplest frequency equation for the velocity and absorption of sound, is related to the transition probabilities for vibrational exchanges by 1/r = Pe — Pd, where Pe is the probability of collisional excitation, and Pd is the probability of collisional de-excitation per molecule per second. Dividing Pd by the number of collisions which one molecule undergoes per second gives the transition probability per collision P, given by Equation 4 or 5. The reciprocal of this quantity is the number of collisions Z required to de-excite a quantum of vibrational energy e = hv. This number can be explicitly calculated from Equation 4 since Z = 1/P, and it can be experimentally derived from the measured relaxation times. 

It is much more difficult to observe the Mossbauer effect with the 130 keV transition than with the 99 keV transition because of the relatively high transition energy and the low transition probability of 130 keV transition, and thus the small cross section for resonance absorption. Therefore, most of the Mossbauer work with Pt, published so far, has been performed using the 99 keV transition. Unfortunately, its line width is about five times larger than that of the 130 keV transition, and hyperfine interactions in most cases are poorly resolved. However, isomer shifts in the order of one-tenth of the line width and magnetic dipole interaction, which manifests itself only in line broadening, may be extracted reliably from Pt (99 keV) spectra. 

As stated in Chapter 1, transitions involving a change in multiplicity are spin forbidden. However, for reasons which we will consider later, such transitions do indeed occur although with very low transition probabilities in most cases. The intensity of an absorption corresponding to a transition from the ground state S0 to the lowest triplet state Tx is related to the triplet radiative lifetime t ° by the following equation[Pg.114]

The transition probability P is proportional to the square of the microwave power level. Equation (26) shows that if the product of the microwave power level and the relaxation time are sufficiently small so that 2PT 1, the rate of energy absorption in the sample (signal amplitude) will be proportional to the population difference and to the power level. If 2PTi 5>> 1, saturation occurs and the rate of energy absorption will no longer be proportional to the microwave power level. 

In a celebrated paper, Einstein (1917) analyzed the nature of atomic transitions in a radiation field and pointed out that, in order to satisfy the conditions of thermal equilibrium, one has to have not only a spontaneous transition probability per unit time A2i from an excited state 2 to a lower state 1 and an absorption probability BUJV from 1 to 2 , but also a stimulated emission probability B2iJv from state 2 to 1 . The latter can be more usefully thought of as negative absorption, which becomes dominant in masers and lasers.1 Relations between the coefficients are found by considering detailed balancing in thermal equilibrium... 

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. 

The maximum for the12 + - 1II transition appears to lie well below 2000 A16, but this transition probably contributes to the absorption at 1849A together with lS+ - 3n0+... 

The concept of transition moment is of major importance for all experiments carried out with polarized light (in particular for fluorescence polarization experiments, see Chapter 5). In most cases, the transition moment can be drawn as a vector in the coordinate system defined by the location of the nuclei of the atoms4 therefore, the molecules whose absorption transition moments are parallel to the electric vector of a linearly polarized incident light are preferentially excited. The probability of excitation is proportional to the square of the scalar product of the transition moment and the electric vector. This probability is thus maximum when the two vectors are parallel and zero when they are perpendicular. 

Symmetry-forbidden transitions. A transition can be forbidden for symmetry reasons. Detailed considerations of symmetry using group theory, and its consequences on transition probabilities, are beyond the scope of this book. It is important to note that a symmetry-forbidden transition can nevertheless be observed because the molecular vibrations cause some departure from perfect symmetry (vibronic coupling). The molar absorption coefficients of these transitions are very small and the corresponding absorption bands exhibit well-defined vibronic bands. This is the case with most n —> n transitions in solvents that cannot form hydrogen bonds (e 100-1000 L mol-1 cm-1). 

If the incident light is linearly polarized, the probability of excitation of a chro-mophore is proportional to the square of the scalar product MA.E, i.e. cos2 0A, 8 being the angle between the electric vector E of the incident light and the absorption transition moment MA (Figure 5.2). This probability is maximum when E is parallel to MA of the molecule it is zero when the electric vector is perpendicular. 

EXAMPLE 1.4 Consider a phosphor with a three energy level scheme and the absorption spectrum shown in Figure 1.9(a). Assuming similar transition probabilities among these levels, discuss the nature of the excitation and emission spectra and their relationship to the absorption spectrum. 

In the previous section we have seen how to determine the energy levels of an optically active center. Optical spectra result from transitions among these energy levels. For instance, an optical absorption spectrum is due to different transitions between the ground energy level and the different excited energy levels. The absorption coefficient at each wavelength is proportional to the transition probability of the related transition. 

Let us now establish a way in which to relate / x p, or the transition probability given in Equation (5.14), with experimental measurements, such as the absorption spectrum. 

In the spirit of the adiabatic approximation, the transitions between two vibrational states (belonging to initial and final electronic states) must occur so rapidly that there is no change in the configurational coordinate Q. This is known as the Frank Condon principle and it implies that the transitions between i and / states can be represented by vertical arrows, as shown in Figure 5.12. Let us now assume our system to be at absolute zero temperature (0 K), so that only the phonon level = 0 is populated and all the absorption transitions depart from this phonon ground level to different phonon levels m = 0, 1, 2,... of the excited state. Taking into account Equation (5.25), the absorption probability from the = 0 state to an m state varies as follows ... 

Equation (A3.7) shows the equality between the probabilities of absorption and stimulated emission that we have already established for monochromatic radiation in Equation (5.15). Equation (A3.8) gives the ratio of tlie spontaneous to the induced transition probability. It allows us to calculate the probability A of spontaneous emission once the Einstein B coefficient is known. 

The effective lifetimes of all these excited states are determined by radiative as well as collisional deactivation, and which contribution is the more significant depends on pressure and transition probability. The simultaneous recording of the absorption and fluorescence spectra yields information about the ratio of radiative to collisioninduced nonradiative decays. This ratio is proportional to the quotient of total fluorescence from the excited level to total absorbed laser light. Such experiments have been started by Ronn oif... 

Another explanation for their resonance Raman results could be a change in the zwitterionic nature of the merocyanine isomers in the different solvents which may result in changes in the Raman transition probabilities, or the spectral changes could be due to solvent shifts of the absorption spectrum, resulting in a change in the relative contribution of the different vibrational modes to each resonance Raman spectrum. We note that in the same article, the authors report the transient absorption spectra of the merocyanine forms, which clearly show that the BIPS spectrum in cyclohexane has more discrete vibrational modes than are observed in the polar solvents, which show more spectral broadening. Al-... 

In the absence of the reverse absorption the radiative transition probability fquantum yield of fluorescence qmC) and the decay constant l/r (C)= 2 [Pg.200]

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