One-quantum transitions

Fig. 10. Energy diagram of a IS spin system (two spins 1/2). a and P are the usual spin functions. One-quantum transitions (Ji, Sy S2, physically observable) full arrows. ZQ and DQ (dashed arrows) refer to zero-quantum and double-quantum transitions.

Alternatively, the variation of the logarithm of the observed intensity, IniS", with Ing can be used to determine the nature of a given transition. This function is linear and has a slope of about two for one-quantum transitions and about four for two-quantum transitions. This method has been used on data from triple-axis spectrometers. 

The Fourier component of interaction force F(t) on the transition frequency (2-184) characterizes the level of resonance. Matrix elements for harmonic oscillators m y n) are non-zero only for one-quantum transitions, n = m . The W relaxation probability of the one-quantum exchange as a function of translational temperature To can be found by averaging the probability over Maxwelhan distribution ... 

Rewritten through the energy change in an one-quantum transition, AE = hxo Eq.(2) assumes the form... 

For example, let iicOi = 500 cm , a typical value. It seems plausible that the polyatom moiety s potential energy in this coordinate should change at least on the order of 0.1 cm i upon formation of a complex. Then ai(r) = (0.0004)1/2 = 0.02. Substitution of these values into Eq. (II.9) gives a matrix element of 7 cm i for Vi =1. Third, the vibrational predissociation rate constant for a one-quantum transition should vary linearly with the quantum number of the initial state. Since the energy lost to the van der Waals coordinate in a one-quantum change is independent of the initial state quantum number, the vibrational predissociation rate should vary with the square of the coupling function in Eq. (II.9). This equation indicates that the vibrational predissociation rate should then be proportional to the initial state vibrational quantum number Vi. 

The vibrational relaxation of molecules is an important process in non-equilibrium chemical kinetics. Out of the many different relaxation phenomena the simplest one involves diatomic molecules for which there is no complicated intramolecular energy transfer. If the degree of vibrational excitation is not too high, the most important processes correspond to one-quantum transitions. This results in considerable simplification of relaxation kinetics which simplifies even further under the condition of a constant translational temperature T. 

The harmonic oscillator approximation is the basis for the treatment of the molecular and lattice dynamics of organic systems [19-22]. The approximation of restoring forces linear with the displacements is accepted since the vibrational amplitudes are small for fundamental one quantum transitions. Anharmonicity plays an important and non negligeable role for higher vibrational quantum levels. 

This is not the case below 1 eV, at least in part due to the poor statistics for the histogram results, and the moment results are probably more reliable. The moment results are generally found to be reliable only for one-quantum transitions once the histogram probabilities become large. This is not surprising since they are obtained from only the first two moments (as described in section VI). 

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