Lorentzian line broadening

Figure 40 Static and MAS spectra of polycrystalline (Bu4N)iMo207 (a) H- Mo spectrum with 3477 scans, 5-s recycle delay, and Lorentzian line broadening of 500 Hz (b) H-decoupled Bloch decay spectrum with the same line broadening and number of transients as in (a) (c) 4.7 kHz MAS spectrum with H decoupling, 6970 scans, 5-s recycle delay, and 50 Hz of line broadening. (From Ref. 31.)...

Lorentzian line broadening function. The apodization function most commonly used to emphasize the signal-rich initial portion of a digitized FID. 

The 300 MHz spectrum of pergolide mesylate (10 mg/mL) in dimethylsulfoxide-d6 (2.5 ppm) is shown in Figure 3. The spectrum was obtained on a Varian Unity spectrometer using die following instrumental parameters 5 mm jual probe spectral width, 4416 Hz 90 pulse 64K time-domain data points acquisition time, 7.421 seconds 100 scans and probe temperature, 35°C. The spectrum was provided with 0.1 Hz Lorentzian line broadening. 

Figure 2.28 (A) Observed (left) and calculated (right) NMR spectra of Leu5-enkephalin crystallized from water (A), methanol/H20 (B) and DMF/H2O (C). The isotropic signal at 0 Hz is due to natural abundant solvent molecules. Lorentzian line broadening (lb=1000 Hz) was applied prior to Fourier transformation. The 180° flip frequencies for the calculations are 5.0 x 103 (A), 3.0 x 104 (B), 2.4 x 106 (Q. The asymmetry parameters, tj, for the calculations are 0.02 (A), 0.05 (B), 0.05 (C). t=30 ps was used for the calculations. Reprinted from Ref. [166]. Copyright 1999 American Chemical Society.

Figure 2.10 Optically allowed transitions in eV and oscillator strength /e obtained from the MCLR, Cl or RPA calculations for the lowest-energy structures of NanFn- n = 2-6) and NttnFn-i n = 3-6). In the drawn structures full circles denote F atoms. The electronic states to which intense transitions occur are labeled according to the irreducible representations of the point group. Lorentzian line broadening is indicated by the thin full line. Compare Ref. [21] and J. Pittner, Dissertation, Humboldt University, Berlin (1996)...

It was shown in Ref. [15] that for optical transitions in glasses the TLS dynamics results in spectral diffusion, which shows up in the experiment as a time and temperature-dependent Lorentzian line broadening. The width of this Lorentzian line must be calculated by averaging over the distribution of energies and relaxation rates P E, R). It can be written as ... 

To account for the broadening of the experimental spectrum (related not only to the finite lifetime of the electronic state, but also to the rovibrational structure, collisions, and other aspects of the interaction between light and matter), and in particular to investigate whether certain bands may be hidden in the experimental spectrum due to overlapping bands, simple Lorentzian line broadening is often added in the form... 

Figure 1 Comparison of the photodepletion spectra of Na9 (left) and Nau+ (right) recorded at temperatures estimated to be 560, 105, and 35 K (dotted curves), with optically allowed transitions in eV and oscillator strengths(fine lines) for the two lowest-energy isomers of Na9, (a) C2, and (b) C3, and for the lowest-energy structure of Nan obtained from Cl calculations. The geometries were optimized at the SCF level. The electronic states to which intense transitions occur are labeled and Lorentzian line broadening is indicated by full lines (cf. Refs. 15 and 28) (Reproduced by permission of the American institute of Physics, New York)...

In these Lorentzian lines, the parameter x deseribes the kinetie deeay lifetime of the moleeule. One says that the speetral lines have been lifetime or Heisenberg broadened by an amount proportional to 1/x. The latter terminology arises beeause the finite lifetime of the moleeular states ean be viewed as produeing, via the Heisenberg uneertainty relation AEAt >fe, states whose energy is "uneertain" to within an amount AE. 

Equation (2.27) illustrates what is called the natural line broadening. Since each atom or molecule behaves identically in this respect it is an example of homogeneous line broadening, which results in a characteristic lorentzian line shape. 

Matched filter The multiplication of the free induction decay with a sensitivity enhancement function that matches exactly the decay of the raw signal. This results in enhancement of resolution, but broadens the Lorentzian line by a factor of 2 and a Gaussian line by a factor of 2.5. 

Fig. 2.7 Dependence of the experimental line width Cexp on the effective absorber thickness t for Lorentzian lines and inhomogenously broadened lines with quasi-Gaussian shape (from [9])...

Although Lorentzian line shapes should be strictly expected only for Mossbauer spectra of thin absorbers with effective thickness t small compared to unity, Margulies and Ehrman have shown [9] that the approximation holds reasonably well for moderately thick absorbers also, albeit the line widths are increased, depending on the value of t (Fig. 2.7). The line broadening is approximately... 

There is a second relaxation process, called spin-spin (or transverse) relaxation, at a rate controlled by the spin-spin relaxation time T2. It governs the evolution of the xy magnetisation toward its equilibrium value, which is zero. In the fluid state with fast motion and extreme narrowing 7) and T2 are equal in the solid state with slow motion and full line broadening T2 becomes much shorter than 7). The so-called 180° pulse which inverts the spin population present immediately prior to the pulse is important for the accurate determination of T and the true T2 value. The spin-spin relaxation time calculated from the experimental line widths is called T2 the ideal NMR line shape is Lorentzian and its FWHH is controlled by T2. Unlike chemical shifts and spin-spin coupling constants, relaxation times are not directly related to molecular structure, but depend on molecular mobility. 

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. 

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. 

It is well known that, under conditions of very fast or very slow exchange, dynamic NMR spectra consist of almost purely Lorentzian lines which are only slightly broadened with respect to the corresponding static spectra. The observed line-width can therefore be partitioned into the following approximate contributions ... 

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