Lineshape theory

After a discussion of the fundamental concepts in Section II, we present, in Section III, an approach to the lineshape theory of dynamic NMR spectra which comprises the most general case, namely that of a multi-component system where various intra- and inter-molecular exchange processes take place. We believe that a fully correct NMR theory of such an equilibrium has not been put forward yet. Section IV is concerned with the methods of simulation and analysis of complicated dynamic spectra. In Section V, we present our views on solving the numerous practical problems which usually appear upon the application of the theory to the analysis of dynamic spectra. 

We shall shortly consider such fundamental concepts as density matrices and the superoperator formalism which are convenient to use in a formulation of the lineshape theory of NMR spectra. The general equation of motion for the density matrix of a non-exchanging spin system is formulated in the laboratory (non-rotating) reference frame. The lineshape of a steady-state, unsaturated spectrum is given as the Fourier transform of the free induction decay after a strong radiofrequency pulse. The equations provide a starting point for the formulation of the theory of dynamic NMR spectra presented in Section III. The reader who may be interested in a more detailed consideration of the problems is referred to the fundamental works of Abragam and... 

R. Vallauri, Balucani U, and U. Tognetti. Lineshape theory and molecular dynamics in collision induced light scattering. Phys. Rev. A, 79 177-186 (1979). 

An important consequence of the lineshape theory discussed above concerns the effect of the bath dynamics on the linewidths of spectral lines. We have already seen this in the discussion of Section 7.5.4, where a Gaussian power spectrum has evolved into a Lorentzian when the timescale associated with random frequency modulations became fast. Let us see how this effect appears in the context of our present discussion based on the Bloch-Redfield theory. 

The second reason for the lack of exploitation of electroreflectance, at least as a probe of electric field distribution, has been the rather limited development of the theory. Whilst the basic equations describing the effect have been known for some time [9], they are of considerable complexity and the simplifications that have been made, such as the Aspnes "third derivative modulation spectroscopy [10] and the extended lineshape theories of Raccah et al. [11] have regions of applicability that may not include all commonly found experimental conditions. There are two difficulties with these theories. The first is that the electric field strengths found in practice in the semiconductors commonly used in electrochemical research may be too high for simple lineshape theories to be applicable the essential requirement of such theories is that the lineshape should be independent of applied d.c. potential, a result not always found in practice, as discussed below. The... 

As discussed in Ref. 86 other approaches to the lineshape theory also lead to exponential activation of Tj. 

Binsch G 1969 A unified theory of exchange effects on nuclear magnetic resonance lineshapes J. Am. Chem. Soc. 91 1304-9... 

The Time Dependent Processes Seetion uses time-dependent perturbation theory, eombined with the elassieal eleetrie and magnetie fields that arise due to the interaetion of photons with the nuelei and eleetrons of a moleeule, to derive expressions for the rates of transitions among atomie or moleeular eleetronie, vibrational, and rotational states indueed by photon absorption or emission. Sourees of line broadening and time eorrelation funetion treatments of absorption lineshapes are briefly introdueed. Finally, transitions indueed by eollisions rather than by eleetromagnetie fields are briefly treated to provide an introduetion to the subjeet of theoretieal ehemieal dynamies. 

The Compton scattering cannot be neglected, but it is independent of molecular structure. Then, fitting experimental data to formulas from gas phase theory, the concentration of excited molecules can be determined. Another problem is that the undulator X-ray spectrum is not strictly monochromatic, but has a slightly asymmetric lineshape extending toward lower energies. This problem may be handled in different ways, for example, by approximating its spectral distribution by its first spectral moment [12]. 

In the stochastic theory of lineshape developed by Blume [31], the spectral lines are calculated under the influence of a time-dependent Hamiltonian. The method has been successfully applied to a variety of dynamic effects in Mossbauer spectra. We consider here an adaptation due to Blume and Tjon [32, 33] for a Hamiltonian fluctuating between two states with axially symmetric electric field gradients (efg s), the orientation of which is parallel or perpendicular to each other. The present formulation is applicable for states with the same... 

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. 32. Fe-M6ssbauer spectra of [Fe(6-Mepy)3tren](PF )2 for various temperatures recorded at p = 150 MPa. Full lines are spectra calculated using the stochastic theory of lineshape by Blume with sgn(F J sgn(Fh.). According to Ref. [169]...

A theoretical investigation of the use of NMR lineshape second moments in determining elastomer chain configurations has been undertaken. Monte Carlo chains have been generated by computer using a modified rotational isomeric state (RIS) theory in which parameters have been included which simulate bulk uniaxial deformation. The behavior of the model for a hypothetical poly(methylene) system and for a real poly(p-fluorostyrene) system has been examined. Excluded volume effects are described. Initial experimental approaches are discussed. 

When a large nurWber of spins interact, the numerous lines of the splitting pattern overlap and merge into a continuous lineshape whose functional form cannot be obtained from theory. Yet this shape contains useful geometrical information. One means of representing this dipolar lineshape g(w) is by a moment expansion... 

The pure quantum approach of the strong anharmonic coupling theory performed by Marechal and Witkowski [7] gives the most satisfactorily zeroth-order physical description of weak H-bond IR lineshapes. 

Using the case of S = 5/2 as an illustrative example, he demonstrated that it was possible to derive closed-form analytical expressions for the PRE of the form of the SBM equations times (1 + correction term). For typical parameter values, the effect of the correction term was to increase the prediction of the SBM theory by 5-7%. A similar approach was also applied to the S = 7/2 system, such as Gd(III) (101), where the correction terms could be larger. For that case, the estimations of the electron spin relaxations rates, obtained in the solution for PRE, were also used for simulations of ESR lineshapes. 

Theory shows that the lineshape contribution of these factors is characterized by an angle dependence of the type (3 cos 9 — 1), where 9 is the angle between the principal... 

Various theoretical formalisms have been used to describe chemical exchange lineshapes. The earliest descriptions involved an extension of the Bloch equations to include the effects of exchange [1, 2, 12]. The Bloch equations formalism can be modified to include multi-site cases, and the effects of first-order scalar coupling [3, 13, 24]. As chemical exchange is merely a special case of general relaxation theories, it may be compre-... 

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