Lineshapes

The energy spectrum of the resonance states will be quasi-discrete it consists of a series of broadened levels with Lorentzian lineshapes whose full-width at half-maximum T is related to the lifetime by F = Fn. The resonances are said to be isolated if the widths of their levels are small compared with the distances (spacings) between them, that is... 

Raman gain coefficient, whose maximum occurs at exact resonance, - oig = For a Lorentzian lineshape, the maximum gain coefficient is given by... 

Clip acts in phase (the same Fourier component) with the first action of cii to produce a polarization that is anti-Stokes shifted from oi (see fV (E) and IFj (F) of figure B 1.3.2(b)). For the case of CSRS the third field action has frequency CO2 and acts in phase with the earlier action of CO2 (W (C) and IFj (D) of figure Bl.3.2 (b). Unlike the Class I spectroscopies, no fields in CARS or CSRS (or any homodyne detected Class II spectroscopies) are in quadrature at the polarization level. Since homodyne detected CRS is governed by the modulus square of hs lineshape is not a synmretric lineshape like those in the Class I... 

As a result, the CRS lineshape is asyimnetric and more complicated due to this nonresonant background interference. 

Lee S-Y 1998 Forward and inverse transforms between the absorption lineshape and Raman excitation profiles XVith int. Conf on Raman Spectroscopy ed A M Heyns (New York Wiley) pp 48-51... 

Figure Bl.5.14 Possible lineshapes for an SFG resonance as a fiinction of the infrared frequency cojj. The measured SFG signal is proportional to + A/(cojj - + iF)P. Assuming both and F are real and...

Fortunately, for non-integer quadnipolar nuclei for the central transition = 0 and the dominant perturbation is second order only (equation Bl.12.8) which gives a characteristic lineshape (figure B1.12.1(cB for axial synnnetry) ... 

This angular dependence is different from the first-order perturbations so that the conventional teclmique of removing linebroadening in solids, MAS (see below), caimot completely remove this interaction at the same time as removing the first-order broadening. Flence, the resolution of MAS spectra from quadnipolar nuclei is usually worse than for spin-2 nuclei and often characteristic lineshapes are observed. If this is the case, it is... 

Figure Bl.12.5. Zr static NMR lineshapes from ZrO polymorphs using frequency-stepped spin echoes.

Figure Bl.12.8. MAS NMR lineshapes from the central transition lineshape for non-integer quadnipole lineshapes with various (A = / /+l) - 3/4 ...

Advantages. The experiment can be carried out with a conventional fast-spuming MAS probe so that it is straightforward to implement. For recording the satellite transition lineshapes it offers better signal-to-noise and is less susceptible to deadtime effects than static measurements. As the effects differ for each value, a single satellite transition experiment is effectively the same as carrying out multiple field experiments on the central transition. 

Figure Bl.12.13. MAS NMR spectra from kyanite (a) at 17.55 T along with the complete simulation and the individual components, (b) simulation of centreband lineshapes of kyanite as a fiinction of applied magnetic field, and tire satellite transitions showing (c) the complete spiimmg sideband manifold and (d) an expansion of individual sidebands and their simulation.

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

Figure B2.4.1 shows the lineshape for intennediate chemical exchange between two equally populated sites without scalar coupling. For more complicated spin systems, the lineshapes are more complicated as well, since a spin may retain its coupling infonnation even though its chemical shift changes in the exchange.

Figure B2.4.3 shows an example of this in the aldehyde proton spectnim of N-labelled fonnamide. Some lines in the spectnim remain sharp, while others broaden and coalesce. There is no frmdamental difference between the lineshapes in figures B2.4.1 and figures B2.4.3—only a difference in the size of the matrices involved. First, the uncoupled case will be discussed, then the extension to coupled spin systems.

Binsch [6] provided the standard way of calculating these lineshapes in the frequency domain, and implemented it in the program DNMR3 [7], Fonnally, it is the same as the matrix description given in section (B2.4.2.3). The calculation of the matrices L, R and K is more complex for a coupled spin system, but that should not interfere witii the understanding of how the method works. This work will be discussed later, but first the time-domain approach will be developed. 

Application of tiiis approach to equation (B2.4.37) gives equation (B2.4.40). If = -co = -S/2, the synnnetry of the matrix and one additional transfomiation means that it can be broken into two 2x2 complex matrices, which can be diagonalized analytically. The resulting lineshapes match the published solutions [13]. 

Figure B2.4.5. Simulated lineshapes for an intennolecular exchange reaction in which the bond joining two strongly coupled nuclei breaks and re-fomis at a series of rates, given beside tlie lineshape. In slow exchange, the typical spectrum of an AB spin system is shown. In the limit of fast exchange, the spectrum consists of two lines at tlie two chemical shifts and all the coupling has disappeared.

Exchange in the solid state follows die same basic principles as in liquids. The classic Cope re-arrangement of bullvalene occurs in both the liquid and solid state [25], and the lineshapes in the spectra are similar. 

---