Lineshape relaxation

This lineshape analysis also implies tliat electron-transfer rates should be vibrational-state dependent, which has been observed experimentally [44]- Spin-orbit relaxation has also been identified as an important factor in controlling tire identity of botli electron and vibrational-state distributions in radiationless ET reactions. 

Condensed phase vibrational or vibronic lineshapes (vibronic transitions create vibrational excitations of electronic excited states) rarely provide infonnation about VER (see example C3.5.6.4). Experimental measurements of VER need much more than just the vibrational spectmm. The earliest VER measurements in condensed phases were ultrasonic attenuation studies of liquids [15], which provided an overall relaxation time for slowly (>10 ns) relaxing small molecule liquids. 

Two different approaches have been followed to calculate the lineshapes within a relaxation model. According to a phenomenological approach based on the modified Bloch equations [154, 155], the intensity distribution of the theoretical Mossbauer spectrum may be written as [156] ... 

The stochastic theory of lineshape has been developed by Anderson and Weiss [157], by Kubo [158], and by Kubo and Tomita [159] in order to treat the narrowing of spectral lines by exchange or motion, a generalized formulation having been subsequently presented by Blume [31]. We consider below an application of the theory of Blume to the specific problem of relaxation between LS and HS states in Mossbauer spectra of powder materials which is based on the formulation by Blume and Tjon [32, 33], Accordingly, the probability of emission of a photon of wave vector Ik and frequency m is given as [160] ... 

Fig. 23. Mossbauer spectra of Fe(J-mph)NO between 84 and 319 K. Solid lines result from a fit by a two-state relaxation model based on the stochastic theory of lineshapes. According to Ref. [164]...

Fig. 60. Crossover from single-chain to many-chain relaxation at T = 343 K. Lineshape analysis for PDMS/d-benzene at c = 5 and 18% double logarithmic plot of — ln/S(Q,t)/S(Q,0) vs. t/s. (Reprinted with permission from [116]. Copyright 1982 J. Wiley and Sons, Inc., New York)...

As we have found in the above section, the damping of the slow mode when it is nearly alone produces (within the Boulil et al. model) a collapse of the fine structure of the lineshapes. That is in contradiction with the RY semiclassical model which predicts a broadening of the lineshape. Of course, because the quantum model is more fundamental, the semiclassical model must be questioned. However, it is well known that the RY semiclassical model of indirect relaxation has the merit to predict lineshapes that may transform progressively,... 

Now, return to Fig. 14. The right and left bottom damped lineshapes (dealing respectively with quantum direct damping and semiclassical indirect relaxation) are looking similar. That shows that for some reasonable anharmonic coupling parameters and at room temperature, an increase in the damping produces approximately the same broadened features in the RY semiclassical model of indirect relaxation and in the RR quantum model of direct relaxation. Thus, one may ask if the RR quantum model of direct relaxation could lead to the same kind of prediction as the RY semiclassical model of indirect relaxation. 

Figure 15. Rosch-Ratner and Robertson lineshapes at 300 K Rosch and Ratner (direct relaxation) a0 = 1, y0 — 1 Robertson (indirect relaxation) a0 — 1.29, y00 = 0.85.

As a consequence, one may infer that the experimental features of the lineshapes of the vs(X-H) that were explained by means of the RY semiclassical model of indirect relaxation would be as well-explained in terms of the RR quantum model of direct relaxation. 

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