Homogeneous and inhomogeneous broadening

The fact that the lineshape (18.49) is Lorentzian is a direct consequence of the fact that our starting point, the Redfield equations (10.174) correspond to the limit were the thermal bath is fast relative to the system dynamics. A similar result was obtained in this limit from the stochastic approach that uses Eq. (10.171) as a starting point for the classical treatment of Section 7.5.4. In the latter case we were also able to consider the opposite limit of slow bath that was shown to yield, in the model considered, a Gaussian lineshape. 

To understand the physical difference between these limits we have to realize that interaction with the environment can affect the spectral behavior of a molecular system in two ways, static and dynamic, both derived from the random character of this interaction  

It is important to understand that the origin of the different frequency shifts experienced by different molecules is the same stochastic frequency modulation 5 y (Z) of Eq. (10.171), only that in the limit considered each molecule encounters a different instantaneous realization of this stochastic variable, which persists on the timescale of the measurement. In this limit the observed lineshape is determined not by the dynamics of m(z) but by the probability P u) ) that at any time the molecule is characterized by the instantaneous transition frequency coT If the normalized absorption profile of an individual molecule is given by a a — co ), where a co) peaks at m = 0 and dcoalco) = 1, the observed lineshape is 

Now consider the opposite limit where the thermal motion in the environment is fast relative to the molecular processes under discussion, in particular relative to the timescale of the molecule-photon interaction that leads to absorption. Now each 

The Redfield equations that lead to Eqs (18.43) or (18.46) were obtained under the assumption that thennal environment is fast relative to the system and therefore correspond to this homogeneous limit. Consequently the absorption spectrum (18.49) obtained from these equations corresponds to a homogeneous lineshape. In contrast, the classical stochastic theory of lineshape, Section 7.5.4, can account for both limits and the transition between them. We will see in the next section that an equivalent theory can be also constructed as an extension of the Bloch equations (18.43). 

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N. Steigerwald, M. L. and Brus, L. E. (1988) Electronic states of semiconductor clusters - homogeneous and inhomogeneous broadening ofthe optical-spectrum./. Chem. Phys., 89, 4001 1011. 

The width of a band in the absorption spectrum of a chromophore located in a particular microenvironment is a result of two effects homogeneous and inhomogeneous broadening. Homogeneous broadening is due to the existence of a continuous set of vibrational sublevels in each electronic state. Inhomogeneous broadening results from the fluctuations of the structure of the solvation shell... 

Each spectral line can arise from a species with a particular concentration and transition dipole moment matrix element and a particular linewidth determined by the extent of homogeneous and inhomogeneous broadening. The magnitude of absorption as a function of frequency is given by Beer s law ... 

As expected, we find that the total response function Xa=i Ri = Xa=4 Ri = J2i=1 Ri = 0 (i.e., for each possible time ordering) vanishes exactly in the harmonic case, defined by A = 0 and /x2i 2 = 2 /r10 2. Furthermore, it can be easily seen that in the case of a strict separation of time scales of homogeneous and inhomogeneous broadening, the line shape function becomes g(t) = t/T2 + a212/2 and the total response function reduces exactly to the result obtained within a Bloch picture (see, for example Refs. 52 and 75), e.g.,... 

Another important parameter giving information on the adlayer interaetions is the bandshape. The experimental phenomena causing a band broadening have been classified in two groups homogeneous and inhomogeneous broadening. 

Fig. 3.4.1 Homogeneously and inhomogeneously broadened lines, (a) Echo train generated by repeated refocussing of the FID (CPMG method, cf. Fig 2.2.10(b)). (b) The Fourier transform of the slowly decaying echo envelope is the homogeneously broadened line, (c) The Fourier transform of the fast decaying echo is the inhomogeneously broadened line.

A. P. Ahvisatos, A. L. Harris, N. J. Levinos, M. L. Steigerwald, L. E. Brus, Electronic States of Semiconductor Clusters Homogeneous and Inhomogeneous Broadening of the Optical Spectrum. The Journal of Chemical Physics 1988,89,4001. 

Figure 1. Determination of T times by extrapolation for the case of non-negligible saturation for homogeneously and inhomogeneously broadened lines (after Fretier, 1979).

Fig. 1. Schematic representation of homogeneous and inhomogeneous broadening. Identical centers, i.e. atoms with the same velocity or ions in identical sites, have similar homogeneous characteristics. All centers contribute to make up the inhomogeneously broadened transition.

AP Alivisatos, TD Harris, NJ Levinos, ML Steigerwald, LE Brus. Electronic states of semiconductor clusters—Homogeneous and inhomogeneous broadening of the optical-spectrum. J Chem Phys 89 4001-4011, 1988. 

Usually, the discussion on the line shape of the broadening of spectroscopic lines is made in terms of homogeneous and inhomogeneous broadening [2, 10]. 

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