Transition probability relaxation equations

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. 

In Equation (5), we can first notice (i) the factor 1/r6 which makes the spectral density very sensitive to the interatomic distance, and (ii) the dynamical part which is the Fourier transform of a correlation function involving the Legendre polynomial. We shall denote this Fourier transform by (co) (we shall dub this quantity "normalized spectral density"). For calculating the relevant longitudinal relaxation rate, one has to take into account the transition probabilities in the energy diagram of a two-spin system. In the expression below, the first term corresponds to the double quantum (DQ) transition, the second term to single quantum (IQ) transitions and the third term to the zero quantum (ZQ) transition. 

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. 

Needless to say, the so-called combination transitions are also considered in this subspace. Lineshape equations for special forms of the relaxation matrix can also be written in terms of the Hilbert space. However, the notation becomes quite involved. This is probably the source of some erroneous simplifications which consist of neglecting combination transitions in the equations of lineshape. (50)... 

In the realm of theory also, greater demands will be made. As such studies (37—39) as those of Cu—Ni (Fig. 13) and Ag—Pd (Fig. 14) have shown, the d levels of the two species in transition metal alloys tend to maintain their atomic identities, at least when the levels in the pure components are sufficiently well separated in energy. However, neither calculation nor experiment has been done with refinement sufficient for quantitative testing of a theory, such as the coherent potential approximation, designed to describe the d band behavior. In pure metals and intermetallic compounds, band calculations can be compared directly with experiment if transition probabilities and relaxation effects are understood. With care they can be used also in evaluation of the effective interelectronic terms which enter equations such as (18a). Unfortunately, one cannot, by definition, produce a set of selfconsistent band calculation results for a matrix of specific valence electron snpmdl.. . configurations thus, direct estimates for I of Eq. (18a) or F of Eq. (18b) cannot be made. However, band calculations for a set of systems can indicate whether or not it is reasonable to factor level shifts into volume and electron count terms, in the manner of Eqs. (18a) and (23). When this cannot be done, one must revert to a more general expression for a level shift, such as Eq. (1). 

Equation 3.26 shows that efficient MEG, i.e. the predominant generation of multiexcitons, is possible only if the relaxation rate of the biexciton is much faster than that of the exciton (72 yi), because Pi 2 < 1. For strong coupling, the transition probability approaches unity and the population ratio approaches its maximum value of (72/71). This result has a transparent physical meaning for a strongly coupled superposition state, the populations of A bi and A/ex are controlled by the state decay into the two independent thermalisation channels. 

Consider a closed system characterized by a constant temperature T. The system is prepared in such a way that molecules in energy levels are distributed in departure from their equilibrium distribution. Transitions of molecules among energy levels take place by collisional excitation or deexcitation. The redistribution of molecular population is described by the rate equation or the Pauli master equation. The values for the microscopic transition probability kfj for transition from ith level toyth level are, in principle, calculable from quantum theory of collisions. Let the set of numbers vr be vibrational quantum numbers of the reactant molecule and vp be those of the product molecule. The collisional transitions or intermolecular relaxation processes will be described by ... 

The next piece of evidence we have to consider is the almost universal insensitivity of calculated reaction rates when the transition probabilities in the model are varied this can be seen in diatomic dissociation [75.P1], chemical activation [72.R 77.Q], and in thermal unimolecular reactions [79.T2]. The reason for this is as follows. Since measurements are most often made at times long after the internal relaxation has ceased, the (normalised) steady distribution during the reaction is (SolilVoli, see equation (3.9). Moreover, the perturbed eigenvector To is rather similar to the unperturbed eigenvector Sq, with the dominant terms in the perturbation arising from the decay terms In fact, Tq=(1 -5)So, where... 

So far, we have considered only experiments with continuous-wave lasers under steady state conditions. With time-resolved experiments, on the other hand, energy transfer rates and transition probabilities can be obtained. Such measurements were carried out by mechanically chopping the laser beam directed into an external absorption cell together with the microwave radiation. Later, Levy et at reported time-resolved infrared-microwave experiments with an N2O laser Q-switched with a rotating mirror to produce pulses less than 1 /tsec in duration. They observed a transient nutation of the inversion levels of the molecule following the infrared laser pulse. Based on the Bloch equations, the observed phenomena could be explained quantitatively. From the decay envelope of the oscillations a value for the transverse relaxation time T2 was determined. Similar effects were produced by rapidly switching a Stark field which brings the molecules into resonance with the cw microwave radiation. 

Assuming the upward and downward transition probabilities to be and respectively, it is possible to obtain the type of relaxation to be expected by writing the simple rate equations for the change of the two populations. 

Therefore a simple exponential decay is expected with a relaxation rate X, equal to the sum of the upward and downward transition probabilities. One can also define a relaxation time, in analogy with notation used in conventional magnetic resonance, as T = X . Under experimental conditions it is often necessary to consider situations where the initial polarization is less than 100 %. This may be represented by the following equation. 

The validity of Eq. (15.11) even at the limit Na Nm means in fact that for harmonic oscillators relaxing in a heat bath there exists a closed equation for the mean energy. This is intimately connected with the linear dependence of transition probabilities on the vibrational quantum number (see Eq. (14.1)) and implies that the energy relaxation rate is independent of the initial distribution of harmonic oscillators over vibrational states. Still another peculiarity of this system is known the initial Boltzmann distribution corresponding to the vibrational temperature Tq =t= T relaxes to the equilibrium distribution via the set of the Boltzmann distributions with time-dependent temperatures. If (E ) is explicitly expressed by a time-dependent temperature, this process is again described by Eq. (15.11). 

Equation (21) is called the rate equation, and it differs from the relaxation equation, Eq. (13), in that the transition probability is presumed complete over the region of integration in Eq. (13) while it is not complete over the range in Eq. (21). Put less abstractly, in the process described by Eq. (13) a molecule must make a transition to states in the range of integration, while in the process described by Eq. (21) a reactant molecule may make a transition to states outside i.e., it may form a product molecule and then be removed from the system. Thus, if x is any reactant state, one must now expect c x,ao) to be zero, rather than a non-zero equilibrium population as in Section III-A. 

Now Retails is the transition probability from the spin state j3 to the spin state a and Raa/sis = R/3/3aa- The diagonal part of Eq. (5.25) is a second-rank equation of motion for evolution of the density matrix under the effect of a random perturbation. There are two important second-rank relaxation mechanisms the dipole-dipole and the quadrupole interactions. Chapter 2 showed that these interactions and the anisotropic chemical shift can all be written as a scalar product of two irreducible spherical tensors of rank two, that is. 

If em adsorbed molecule is moving in electronically adiabatic potential well the model of two-dimensional anharmonic oscillator can be used for the description of its vibrational spectrum. This spectrum E(,n) (n = (ni,Ti2)) can be calculated by the numerical or analytical integration of the two-dimensional Schrodinger equation. The calculations of the transition probabilities e E, E) (or e(n, n)) for such oscillator have been performed with the help of the pertiu-bation theory or more sophisticated approaches. Provided these transition probabiUties e(n, n) are known for dense enough energy spectrum (AE ksT ), a diffusion model of energy relaxation may be used... 

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