Broadening Mechanics

The narrow linewidths of liquids allow one to observe weak spin interactions such as the indirect spin-spin (/) coupling and the isotropic chemical shift (7 ). In the solid state, however, linewidths are broadened considerably by much stronger interactions, involving mainly the dipole-dipole interaction (typically 10-100 kHz), chemical-shift anisotropy ( 1 kHz), the quadrupole interaction ( 250 kHz) and the effect of paramagnetic impurities. For these reasons the / and 5 interactions are generally not observed in solids.2 

The Hamiltonian describing the energy of interaction of a solid in an NMRI experiment is given by  

The dipole dipole interaction is the through-space coupling of one magnetic spin with the local field of its neighbours. The interaction depends on the magnitude and 

The situation with a pair of spins is illustrated in Fig. 1, where the static field B0 is along the z-axis. To first order and neglecting terms in the expansion of Eq. (2) which lead to a change in nuclear quantum numbers of 1 or 2 (i.e. Am — 1 and + 2 transitions), the effect of is to split the Zeeman levels into many closely spaced energy levels, thereby causing a distribution of resonant frequencies and consequently a broad line. Eq. (2) has been simplified by the van Vleck formula 2 

One can see from Eq. (3) that the van Vleck dipolar Hamiltonian is the product of a spatial part and a spin part. In liquids, the rapid isotropic molecular tumbling motion, which occurs at frequencies well above the dipolar linewidth, averages the spatial part (1—3 cos2 6tj) to zero, thus nulling the dipolar broadening. In solids, the spins are constrained to vibrate and rotate about their mean positions, resulting in an effective dipolar Hamiltonian which is generally non-zero, and consequently in 

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Of great interest to physical chemists and chemical physicists are the broadening mechanisms of Raman lines in the condensed phase. Characterization of tliese mechanisms provides infomiation about the microscopic dynamical behaviour of material. The line broadening is due to the interaction between the Raman active chromophore and its environment. 

For condensed species, additional broadening mechanisms from local field inhomogeneities come into play. Short-range intermolecular interactions, including solute-solvent effects in solutions, and matrix, lattice, and phonon effects in soHds, can broaden molecular transitions significantly. 

From a comparison between (2.44) and (2.52) one sees that at T = 0 only the intradoublet broadening mechanism works. At higher temperatures, [Pg.26]

Lorentzian line shapes are expected in magnetic resonance spectra whenever the Bloch phenomenological model is applicable, i.e., when the loss of magnetization phase coherence in the xy-plane is a first-order process. As we have seen, a chemical reaction meets this criterion, but so do several other line broadening mechanisms such as averaging of the g- and hyperfine matrix anisotropies through molecular tumbling (rotational diffusion) in solution. 

The purely reorientational broadening mechanism with a single threefold quasidegenerate subbarrier level is characteristic of the valence vibration spectral line for OH groups on Si02 surface. Equation (4.2.22) describes the observed temperature... 

The hne-shape function gives the profile of the optical absorption (and emission) band and contains important information about the photon-system interaction. Let us briefly discuss the different mechanisms that contribute to this function, or the different line-broadening mechanisms. 

This Lorentzian line-shape function has been sketched in Figure 1.4(b). The natural broadening is a type of homogeneous broadening, in which all the absorbing atoms are assumed to be identical and then to contribute with identical line-shape functions to the spectrum. There are other homogeneous broadening mechanisms, such as that due to the dynamic distortions of the crystalline environment associated with lattice vibrations, which are partially discussed in Chapter 5. 

The line-shape function of a given transition informs us on the particular character of the interaction of the absorbing atom with its environment in the solid. In the most general case, this line shape is due to the combined effect of more than one independent broadening mechanism. In this case, the overall line shape is given by the convolution of the line-shape functions associated with the different broadening mechanisms. 

Another, less straightforward way to determine the vibrational lifetime is by studies of the infrared absorption peak shape. Consider a single adsorbed molecule at 0 K. The width of the peak is then determined by the lifetime broadening and in the first approximation it has a Lorentaan shape with a full width at half maximum (FWHM) A = (2nct), t then being the lifetime. However, as usual we have to consider an ensemble of molecules at finite temperatures and then there exist other peak broadening mechanisms that must be taken into account. 

From the discussion in section 3.3 it should obvious that another peak broadening mechanism, at least at higher temperatures, is the anharmonic coupling to low energy modes. We discussed the origin of the broadening in that context and found in Fig. 7 that for c(4 x 2)CO/Ni(l 11) this interaction... 

In this investigation, only electron inelastic scattering and x-ray line broadening were chosen for substantial correction. There were two principal reasons first, these broadening mechanisms account for the largest part of the distortion, and, second their contributions are easily determined or approximated. 

In this system the bands (zones) broaden because of diffusion effects and nonequilibrium. This broadening mechanism is fairly symmetrical and the resulting elution bands approach the shape of a Gaussian curve. This system best explains liquid or gas partition chromatography. The system may be viewed in two ways ... 

In this application of the BWR theory, Hudson and Lewis assume that the dominant line-broadening mechanism is provided by the modulation of a second rank tensor interaction (i.e., ZFS) higher rank tensor contributions are assumed to be negligible. R is a 7 X 7 matrix for the S = 7/2 system, with matrix elements written in terms of the spectral densities J (co, rv) (see reference [65] for details). The intensity of the i-th transition also can be calculated from the eigenvectors of R. In general, there are four transitions with non-zero intensity at any frequency, raising the prospect of a multi-exponential decay of the transverse magnetization. There is not a one-to-one correspondence between the... 

Broadening effects, other than from g- or hyperfine-anisotropies, generally lead to symmetrical absorption curves that are Gaussian or Lorentzian depending upon the broadening mechanism. 

If there are several AP minima of close energy, then at low temperatures one should take into account two-phonon-assisted transitions between these minima. In Ref. [15] (see also Ref. [14]) it was found that the rate of these transitions depends on temperature as 7 3. However, as it was already mentioned above, in Ref. [9] it was found that the contribution of the two-phonon-assisted transitions between different Jahn-Teller minima of the AP to the ZPL width at low temperatures is described by the T5 law. Note that an increase of the Jahn-Teller interaction leads to a decrease of the rate of these transitions. Therefore, in the strong Jahn-Teller interaction limit this broadening mechanism becomes unimportant. 

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