Linewidths relation

Prior knowledge allows to include fixed relations between some of the four parameters (amplitude, phase, frequency position, peak width) describing a symmetrical well-shaped resonance. Signal ratios, chemical shift difierences, linewidth relations and zero-order phase relations can be included. The reduction of the number of unknown parameters leads to a reduced calculation time, better convergence behaviour and improved results. However, the assumptions made to include the prior knowledge must be validated for each experiment. Differences between the parameter values set by the prior knowledge and the actual parameters could lead to systematic errors. 

It is important to realize that the relaxation times might depend on some factors that are properties of the atom or molecule itself and on others that are related to its environment. Thus rotational spectra of gases have linewidths (related to the rotational relaxation times) that depend on the mean times between coUisions for the molecules, which in turn depend on the gas pressure. In liquids, the collision lifetimes are much shorter, and so rotational energy is effectively non-quantized. On the other hand, if the probability of collisions is reduced, as in a molecular beam, we can increase the relaxation time, reduce linewidths, and so improve resolution. Of course, the relaxation time only defines a minimum width of spectral lines, which may be broadened by other experimental factors. 

Similarly to the ESR of the lanthanide ions in insulators, in metallic systems ESR contributes to understanding of the spectroscopic state of the ion in the host lattice and the symmetry and magnitude of the crystalline electric field at the lanthanide site. The g-shift of the resonance may be related to the sf exchange interaction and the spin polarization of the conduction electrons, and the temperature and concentration dependence of the g-shift and resonance linewidth relate to the bottleneck effect in the spin relaxation process. These relationships have been outlined in section 3.5. 

As the spins precess in the equatorial plane, they also undergo random relaxation processes that disturb their movement and prevent them from coming together fiilly realigned. The longer the time i between the pulses the more spins lose coherence and consequently the weaker the echo. The decay rate of the two-pulse echo amplitude is described by the phase memory time, which is the time span during which a spin can remember its position in the dephased pattern after the first MW pulse. Tyy is related to the homogeneous linewidth of the individual spin packets and is usually only a few microseconds, even at low temperatures. 

The amplitude of the elastic scattering, Ao(Q), is called the elastic incoherent structure factor (EISF) and is determined experimentally as the ratio of the elastic intensity to the total integrated intensity. The EISF provides information on the geometry of the motions, and the linewidths are related to the time scales (broader lines correspond to shorter times). The Q and ft) dependences of these spectral parameters are commonly fitted to dynamic models for which analytical expressions for Sf (Q, ft)) have been derived, affording diffusion constants, jump lengths, residence times, and so on that characterize the motion described by the models [62]. 

To extract information about xj from NMR data, the transverse relaxation time Tj may be used as well as the longitudinal time T. For gaseous nitrogen it was done first with Ti in [81] and confirmed later [82] when T was measured and used for the same goal. The NMR linewidth of 15N2 is the inverse of T2, and the theory, relating to Ti to x.1, is well known [39, 83]. For the case of diatomic and linear molecules the formula is... 

The 13C NMR sensitivity can sometimes be a problem, but for the kind of samples studied here the effective concentration of monomer units is several molar which does not place excessive demands on present Fourier transform NMR spectrometers. In addition to the sensitivity of the chemical shift to structure (9), the relaxation of protonated carbons is dominated by dipole-dipole interaction with the attached proton (9). The dependence of the relaxation parameters T, or spin-lattice, and Tor spin-spin, on isotropic motional correlation time for a C-H unit is shown schematically in Figure 1. The T1 can be determined by standard pulse techniques (9), while the linewidth at half-height is often related to the T2. Another parameter which is related to the correlation time is the nuclear Overhauser enhancement factor, q. The value of this factor for 13C coupled to protons, varies from about 2 at short correlation times to 0.1 at long correlation... 

This chapter considers the distribution of spin Hamiltonian parameters and their relation to conformational distribution of biomolecular structure. Distribution of a g-value or g-strain leads to an inhomogeneous broadening of the resonance line. Just like the g-value, also the linewidth, W, in general, turns out to be anisotropic, and this has important consequences for powder patterns, that is, for the shape of EPR spectra from randomly oriented molecules. A statistical theory of g-strain is developed, and it is subsequently found that a special case of this theory (the case of full correlation between strain parameters) turns out to properly describe broadening in bioEPR. The possible cause and nature of strain in paramagnetic proteins is discussed. 

Hagen, W.R. 1981. Dislocation strain broadening as a source of anisotropic linewidth and asymmetrical lineshape in the electron paramagnetic resonance spectrum of metal-loproteins and related systems. Journal of Magnetic Resonance 44 447-469. 

The modulation index is related to the transmission spectra of the gas-filled reference cell, TRef( ), the measurement cell, with its unknown gas concentration, Ty Ca,(/.), and the optical filter, F(/.), all of which are shown in Equation 1. If required, the source spectra can also be taken into account, by using additional spectral functions (although this will usually have little spectral variation over the linewidth of the gas band) ... 

The natural linewidth comes from the lifetime, r, of the upper state of a spontaneous transition, which is related to the Einstein A coefficient so that r = A l faster transitions have shorter lifetimes and vice versa, and similarly an allowed transition will have a short lifetime for the upper state whereas forbidden transitions will have a long lifetime. The lifetime consideration is very important in the laboratory where transitions have to occur on the timescale of the experiment, otherwise they are not observed. Hence in the laboratory allowed transitions are observed and in general (but not specifically) forbidden transitions are not seen. For astronomy this does not matter. So what if a forbidden transition has a lifetime of 30 million years - the Universe is 15 billion years old - if you wait long enough it will happen. The rules of spectroscopy need to be understood but in space anything goes ... 

Bell et al. (2002) investigated the relationship between water mobility as measured by oxygen-17 NMR (transverse relaxation rate obtained from linewidth at half-height) and chemical stability in glassy and rubbery polyvinylpyrrolidone (PVP) systems. Reported results suggest that water mobility in PVP model systems was not related to Tg. The study did not find a link between water mobility and reaction kinetics data (half-lives) for degradation of aspartame, loss of thiamin and glycine, and stability of invertase. 

In gases (atomic or ionic) the electronic energy levels of free atoms are narrow, since they are diluted systems and perturbation by the surroundings is very weak. An important fact derived from the discrete nature of the electronic levels in a gas is the high monochromaticity of the laser lines in this type of laser, compared to that of solid-medium based lasers. The high degree of coherence achievable with gas lasers is also a characteristic feature related to the narrow linewidth. 

The reason why one chose to follow the main liquid-crystalline to gel phase transition in DPPC by monitoring the linewidth of the various or natural abundance resonance is evident when we consider the expressions for the spin-lattice relaxation time (Ti) and the spin-spin relaxation time T2). The first one is given by 1/Ti oc [/i(ft>o) + 72(2ft>o)] where Ji coq) is the Fourier transform of the correlation function at the resonance frequency o>o and is a constant related to internuclear separation. The relaxation rate l/Ti thus reflects motions at coq and 2coq. In contrast, the expression for T2 shows that 1/T2 monitors slow motions IjTi oc. B[/o(0) -I- /i(ft>o) + /2(2u>o)], where /o(0) is the Fourier component of the correlation function at zero frequency. Since the linewidth vi/2 (full-width at half-maximum intensity) is proportional to 1 / T2, the changes of linewidth will reflect changes in the mobility of various carbon atoms in the DPPC bilayer. 

The magnetic susceptibility relaxation is usually more important for than for Ti. In fact, this mechanism is often dominant in determining the proton linewidth in paramagnetic proteins at high magnetic fields (3). Gillis and co-workers have recently developed a theory for the related case of proton linewidth in colloidal solutions of so-called superparamagnetic particles (54,55). 

The linewidth-temperature relation of the polyethylene oxide samples are given in Fig. 5. Despite the large differences in molecular weight, these samples have about the same linewidth, 300-350 Hz, in the crystalline state at 25°C. They all also possess a spherulitic type of morphology. The influence on the linewidth of the different types of supermolecular structures... 

Connor and Hartland (39) display virtually the same behavior. These similarities do not appear to be coincidental since the linewidth is directly related to the spin-spin relaxation parameter, T2, which is sensitive to low frequency motions. The Tj p values are also more sensitive to the lower frequency motions and would thus be expected to behave similarly to T2, or line-widths, in the relaxation studies. 

The results discussed above indicate that the further study of the spin relaxation parameters possess the potential to develop our understanding of the structure of the non-crystalline regions of semicrystalline polymers. Significant progress has already been made in relating the spin-lattice relaxation parameters with that of the pure melt. The linewidths, or spin-spin relaxation parameters, of semicrystalline polymers have been... 

The first term in Eq. (1.34a) represents the resonance scattering. It becomes large when photon (= Hu>) % E — 0. In comparison, the second term is small and is usually neglected. The imaginary term in the denominator contains T, the inverse lifetime (related to linewidth) of the intermediate state t/t >. 

Expressions for determining rate constants from exchange contributions to observed linewidth for unequally populated systems in the fast exchange limit have been derived from the formal solutions to the Bloch equations modified for chemical exchange [3, 127-129]. These equations relate each rate constant to the site populations, chemical shift difference between sites, and spin relaxation times T and T2. For example, the forward rate A i 2 is given by [3, 127] ... 

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