Lineshape function

Well below saturation s l, and so the lineshape function becomes... 

Here, L(v) is a lineshape function that integrates to unity, v is the frequency,/ is the Lamb-Mossbauer factor, and the desired side bands have an area fraction / that is proportional to which hence determines the relative peak heights in a NIS spectrum. More details are provided in Appendix 2 (Part III, 3 of CD-ROM). An equivalent and often more suggestive display of the NIS spectrum is the PVDOS approach, which describes the NIS signal in terms of the partial vibrational density of states ... 

For all of the cases considered earlier, a C(t) function is subjected to Fourier transformation to obtain a spectral lineshape function 1(G)), which then provides the essential ingredient for computing the net rate of photon absorption. In this Fourier transform process, the variable 0) is assumed to be the frequency of the electromagnetic field experienced by the molecules. The above considerations of Doppler shifting then leads one to realize that the correct functional form to use in converting C(t) to 1(G)) is ... 

The experimental observables are either the lineshape function 7(co), as in the classical experiments, or the normal coordinate time correlation function, (2(O)2(0) as in the time domain experiments of Tominaga and Yoshihara [126]. The normal coordinate time correlation is related to the frequency modulation time correlation function by... 

The lineshape function which describes the absorption and dispersion modes of an unsaturated, steady-state NMR spectrum is proportional to the Fourier transform of the function MxID(t) (24, 25, 99)... 

In the integral in expression (51), the only significant contribution comes from the first of the two components of the sum within braces in equation (50). The contribution from q j(t) is about 2co0 times smaller, and it may safely be neglected. (23) Therefore, the final form of the lineshape function is ... 

The lineshape function, which is equal to the Fourier transform of MXID, (98, 99) is given by ... 

If the matrices which appear in the calculations do not exceed the size of 2 x 2 one can analytically derive the lineshape function, and this tends to make the computation rather trivial. Such cases are represented by all A B exchange processes and some other two-site exchange systems which give first-order spectra, as well as by the mutual AB = BA exchange. The same is possible in calculations on spectral fragments of certain more complicated systems as reported in a previous review in this series by Sutherland. (53)... 

Contrary to the point by point approach the diagonalization method consists of the generation of an entire lineshape function in one step. (13, 14, 57-60) Time-consuming calculations are carried out only once. The resulting set of complex numbers can be used for a simple calculation of the lineshape (absorption and dispersion modes) at any desired point on the frequency axis. Thus, the complex matrix from equation (147) can be diagonalized by a similarity transformation using an co-independent complex matrix W ... 

If all possible long-range couplings are considered most of the molecules, which constitute interesting objects for investigation by dynamic NMR methods, represent quite complicated spin systems. We are usually compelled to make some simplifications in the model of exchange by considering lineshape functions for simpler spin systems. Otherwise the computational effort involved may go beyond reasonable limits. In this section we discuss such approximations in detail. 

In order to compare theoretical and experimental spectra, the theoretical lineshape function /abs (a ,g) used in Section IV.B.2 must be replaced ... 

The lineshape function g(V-Vc) is defined assuming a Voigt profile ( 7) which allows for a combination of Doppler and collision... 

In this equation, the brackets, < >, denote an ensemble average over the orientational distribution of emitting molecules, denoted by Q, and the integral denotes an average over molecular conformations, denoted by y. /qpl(X) is a normalized lineshape function, and the constants h and c have their usual meaning. 

Normalized Gaussian lineshape function as a function of frequency... 

The Lorentzian lineshape is obtained for liquid samples under ideal high-resolution NMR conditions10 and is readily derived from the Bloch equations.31 The normalized Lorentzian lineshape function in the frequency domain is given by... 

For a spin system in a solid where the lineshape is determined by the magnetic dipole-dipole interaction, the lineshape function in the FD can be approximated by a Gaussian function gG convolved with a rectangular envelope r 10... 

For an axially symmetric tensor, the lineshape function or intensity function 1(f) is given by9,51,53... 

Since the intrinsic lineshape has finite width, the experimentally observed lineshape is the convolution of 1(f) with one of the lineshape functions g(f). A powder lineshape for an axially symmetric chemical shielding tensor is shown in Fig. 4 and a typical example of a general powder lineshape is shown in Fig. 5. Many systems yield lineshapes close to that of a powder pattern and the mathematical properties of these lineshapes are discussed in detail by Alexander et al.iA... 

Fitting in the frequency domain is readily visualized graphically and the well-established Levenberg-Marquardt method85 is straightforward to implement. This method is applicable to any lineshape function. For example, Marshall et al 2 have used V ( f), an approximation to the Voigt function, when fitting... 

The DF spectra of wurtzite-structure ZnO within the VIS-to-VUV spectral region contain CP structures, which can be assigned to band-gap-related electronic band-to-band transitions Eq with a = A, B,C and to above-band-gap band-to-band transitions E13 with (3 = 1,..., 7. The F -related structures can be described by lineshape functions of the 3DMo-type (3.9 and 3.10), the CP structures with (3 = 3,4 by lineshape functions of the 2DMo-type (3.11), and the CP structures with (3=1,2,5,6,7 can be described by Lorentzian-damped harmonic oscillator functions (3.13). The CP structures Eq are supplemented by discrete (3.14) and continuum (3.16) excitonic contributions. Tables 3.9 and 3.10 summarize typical parameters of the CPs Eq and E, respectively, of ZnO [15]. 

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