Line Lorentzian

The spin-spin relaxation time, T, defined in the Bloch equations, is simply related to the width Av 2 Lorentzian line at the half-height T. Thus, it is in principle possible to detennine by measuring... 

The exponential decay of the A population corresponds to a Lorentzian line shape for the absorption (or emission) cross section, a, as a fiinction of energy E. The lineshape is centred around its maximum at E. The fiill-width at half-maximum (F) is proportional to... 

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. 

In the remainder of this section, we compare EISFs and Lorentzian line widths from our simulation of a fully hydrated liquid crystalline phase DPPC bilayer at 50°C with experiments by Kdnig et al. on oriented bilayers that, in order to achieve high degrees of orientation, were not fully hydrated. We consider two sets of measurements at 60°C on the IN5 time-of-flight spectrometer at the ILL one in which the bilayer preparations contained 23% (w/w) pure D2O and another in which bilayer orientation was preserved at 30% D2O by adding NaCl. The measurements were made on samples with two different orientations with respect to the incident neutron beam to probe motions either in the plane of the bilayers or perpendicular to that plane. 

Analysis of neutron data in terms of models that include lipid center-of-mass diffusion in a cylinder has led to estimates of the amplitudes of the lateral and out-of-plane motion and their corresponding diffusion constants. It is important to keep in mind that these diffusion constants are not derived from a Brownian dynamics model and are therefore not comparable to diffusion constants computed from simulations via the Einstein relation. Our comparison in the previous section of the Lorentzian line widths from simulation and neutron data has provided a direct, model-independent assessment of the integrity of the time scales of the dynamic processes predicted by the simulation. We estimate the amplimdes within the cylindrical diffusion model, i.e., the length (twice the out-of-plane amplitude) L and the radius (in-plane amplitude) R of the cylinder, respectively, as follows ... 

The bracketed term in Eq. (4-60b) describes a Lorentzian line shape for the NMR absorption band. The maximum in the band occurs at the resonance frequency, wq. Expressed in units of X0W0T2/2, the maximum value of x" s 1 at one-half this maximum peak height we find, by substitution, that (wq — w) = IIT. Using w = 2 ttv to convert to frequency (in Hz) gives (vq — v) = 3-7 T 2. However, the peak width is twice this, or... 

The Fourier transform of a pure Lorentzian line shape, such as the function equation (4-60b), is a simple exponential function of time, the rate constant being l/Tj. This is the basis of relaxation time measurements by pulse NMR. There is one more critical piece of information, which is that in the NMR spectrometer only magnetization in the xy plane is detected. Experimental design for both Ti and T2 measurements must accommodate to this requirement. 

Here te, tc are the correlation times of rotational and vibrational frequency shifts. The isotropic scattering spectrum corresponding to Eq. (3.15) is the Lorentzian line of width Acoi/2 = w0 + ydp- Its maximum is shifted from the vibrational transition frequency by the quantity coq due to the collapse of rotational structure and by the quantity A due to the displacement of the vibrational levels in a medium. 

The width of this Lorentzian line is half as large as that found in (3.37). This, however, is not a surprise because the perturbation theory equation (3.23) predicted exactly this difference in the width of the line narrowed by strong and weak collisions. This is the maximal difference expected within the framework of impact theory when the Keilson-Storer kernel is used and 0 < y < 1. 

Fig. 3.3. Lorentzian line shape (solid line) and experimental CARS data (points) of liquid nitrogen (T — 77 K) from [136].

The best resolution of Q-branch rotational structure in a N2-Ar mixture was achieved by means of coherent anti-Stokes/Stokes Raman spectroscopy (CARS/CSRS) at very low pressures and temperatures (Fig. 0.4). A few components of such spectra obtained in [227] are shown in Fig. 5.9. A composition of well-resolved Lorentzian lines was compared in [227] with theoretical description of the spectrum based on the secular simplification. The line widths (5.55) are presented as... 

Inverting (7.44) and substituting the result into (7.39), one can see that the spectrum consists of a pair of Lorentzian lines... 

As a result, (7.71) consists of Lorentzian lines, centred near the eigenvalues of the free rotator. Their widths can be found as follows ... 

Eq. (7.80) describes a Lorentzian line centred near the zero of frequencies, of width and integral intensity Vf/B. Owing to this, a component appears in the complete spectrum, which is forbidden for an unperturbed linear rotator. Its maximum intensity... 

Apparently, the time-domain and frequency-domain signals are interlinked with one another, and the shape of the time-domain decaying exponential will determine the shape of the peaks obtained in the frequency domain after Fourier transformation. A decaying exponential will produce a Lorentzian line at zero frequency after Fourier transformation, while an exponentially decaying cosinusoid will yield a Lorentzian line that is offset from zero by an amount equal to the frequency of oscillation of the cosinusoid (Fig. 1.23). 

Lorentzian line at zero, (b) The FT of an exponentially decaying consinusoid FID gives a Lorentzian line offset from zero frequency. The offset from zero is equal to the frequency of oscillation of the consinusoid. (Reprinted from S. W. Homans, A dictionary of concepts in NMR, copyright 1990, p. 127-129, by permission of Oxford University Press, Walton Street, Oxford 0X2 6DP, U.K.)... 

The simplest definition of sensitivity is the signal-to-noise ratio. One criterion for judging the sensitivity of an NMR spectrometer or an NMR experiment is to measure the height of a peak under standard conditions and to compare it with the noise level in the same spectrum. Resolution is the extent to which the line shape deviates from an ideal Lorentzian line. Resolution is generally determined by measuring the width of a signal at half-height, in hertz. 

Absorption-mode spectrum The spectrum in which the peaks appear with Lorentzian line shapes. NMR spectra are normally displayed in absolute-value mode. 

Dispersion mode A Lorentzian line shape that arises from a phase-sensitive detector (which is 90 out of phase with one that gives a pure-absorption-mode line). Dispersion-mode signals are dipolar in shape and produce long tails. They are not readily integrable, and we need to avoid them in a 2D spectrum. 

Lorentzian line shape The normal line shape of an NMR peak that can be displayed either in absorption or dispersion mode. 

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])...

Thus, the experimental Mossbauer spectmm of a thin single-line absorber is a Lorentzian line, with full-width at half maximum twice the natural Une width of the separate emission and absorption lines Texp = 2E. 

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... 

Cahbration spectra must be measured at defined temperamres (ambient temperature for a-iron) because of the influence of second-order Doppler shift (see Sect. 4.2.1) for the standard absorber. After folding, the experimental spectrum should be simulated with Lorentzian lines to obtain the exact line positions in units of channel numbers which for calibration can be related to the hteramre values of the hyperfine splitting. As shown in Fig. 3.4, the velocity increment per channel, Ostep, is then obtained from the equation Ustep = D,(mm s )/D,(channel numbers). Different... 

For relaxation times t 1 ns, the spectra can be described as three Lorentzian lines with different line width, and for relaxation times around 10 ° s, the spectra appear as asymmetric doublets with line widths that decrease with decreasing relaxation time. In the theoretical spectra in Fig. 6.2, the EFG was assumed uniaxial... 

Fig. 7.39 " Ta (62 keV) Mossbauer spectrum of in W metal versus Ta metal absorber at room temperature. The solid line represents the fit of a dispersion-modified Lorentzian line to the experimental data the dashed line shows the dispersion contribution (from [179, 185])...

Fig. 7.40 Ta (6.2 keV) Mossbauer spectra obtained with a tantalum metal absorber and sources of diffused into various cubic transition metal hosts. The solid lines represent the results of least-squares fits of dispersion-modified Lorentzian lines to the experimental spectra (from [186])...

Fig. 7.72 Pt (99 keV) Mossbauer spectrum of the one-dimensional conductor K2[Pt(CN)4] Bro.3o 3H20 at 4.2 K (source Au in platinum at 4.2 K). The solid line represents a single Lorentzian line fitted to the measured spectrum. The dashed line represents the best fit using a sum of two Lorentzian lines with an intensity ratio of 85 15 and with the isomer shifts of the spectra of K2[Pt(CN)4]-3H20 and K2[(Pt(CN)4Br2] (from [333])...

---