Transverse relaxation rate equation

Table I reports the observed NMR linewidths for the H/3 protons of the coordinating cysteines in a series of iron-sulfur proteins with increasing nuclearity of the cluster, and in different oxidation states. We have attempted to rationalize the linewidths on the basis of the equations describing the Solomon and Curie contributions to the nuclear transverse relaxation rate [Eqs. (1) and (2)]. When dealing with polymetallic systems, the S value of the ground state has been used in the equations. When the ground state had S = 0, reference was made to the S of the first excited state and the results were scaled for the partial population of the state. In addition, in polymetallic systems it is also important to account for the fact that the orbitals of each iron atom contribute differently to the populated levels. For each level, the enhancement of nuclear relaxation induced by each iron is proportional to the square of the contribution of its orbitals (54). In practice, one has to calculate the following coefficient for each iron atom ...

The linear relationship between H NMR transverse relaxation rate and (1 av) is shown in Figure 30 for pregelled potato starch (Hills et al., 1999). The change in slope at about 0.90 c/w corresponds to the bulk water break (i.e., the removal of bulk water) in a corresponding adsorption isotherm. Equation... 

The measured spin relaxation parameters (longitudinal and transverse relaxation rates, Ri and P2> and heteronuclear steady-state NOE) are directly related to power spectral densities (SD). These spectral densities, J(w), are related via Fourier transformation with the corresponding correlation functions of reorientional motion. In the case of the backbone amide 15N nucleus, where the major sources of relaxation are dipolar interaction with directly bonded H and 15N CSA, the standard equations read [21] ... 

Fig. 6. Temperature variations of observed 0 transverse relaxation rates (a), reduced transverse relaxation rates (b) and reduced chemical shifts (c) calculated from Swift and Connick equations for Bq = 14.1 T (largest effect), 4.7 T, and 1.4 T (smallest effect).

Table XVII lists the e.f.g. tensor invariant calculated on the basis of equations (38) and (39) and of the interatomic distances a (= riMb ci) and b ( = rNb-Br) and the Cl and Br charges, e, and 2- Since the transverse relaxation rates are proportional to both and the correlation time and since this latter is proportional to the molecular volume, numerical values are given for the product g r. From Table XVII it is seen that excellent agreement is obtained between the experimentally observed relaxation rate R and g -r (both normalized).

The observed relaxation time is time-averaged value of s e and Sj. The fraction of Na ion under the slow-motion condition is expected small, but the transverse relaxation rate constants (s ) are expected larger than Sfr j by 10 to 100 times. Therefore, we applied this equation to the observed relaxation rate constants to estimate the 2 compartments. One problem is accurate knowledge of the value of s for 23Na+ in the agar gel. 

Eqs. 3 and 4 show the electron-nuclear relaxation-rates for the spin-lattice (longitudinal) and spin-spin (transverse) relaxation-rates. In both of these equations, the first terms reflect the dipolar term, and the second part reflects the scalar interaction. The dipolar term contains a distinct r distance term between the electron of the metal atom and the respective carbon atom in contrast, the scalar term has none. 

Equation 3 neglects effects of anisotropic motion on both longitudinal and transverse relaxation rates 2). Recent experiments using deuterium NMR on samples similar to those studied here show significant nuclear electric quadrupole splittings that imply an anisotropic component in the water molecule motion ( ). Such motional anisotropy will depress and elevate T]. 

Similar equations arise for the transverse relaxation rate l/7 2 . The problem now is to extract the pore distribution, W , from the observed RJ,t). From a mathematical point of view, this could be done by a Laplace inversion of Eqn (28.3). Examples of this method can be found in the review paper of W. P. Halperin et al. Other methods have used a prerequisite distribution that must be verified a posteriori. This has been carried out recently for the nuclear relaxation of and of methanol and nitromethane adsorbed on an organic polymeric resin crosslinked by paramagnetic divalent metal ions (Fig. 28.2). The results have been interpreted with a fractal distribution of categories of quasi-disconnected spherical pores, each being composed of N" spherical pores of radius R = RJ 1), with = log(/ o/ n.in)/log Introducing iht fractal dimension Df through the relation N a , leads to... 

The n-site Bloch-McConnell equations describe the evolution of nuclear spin magnetization in the laboratory or rotating frames of reference for molecules subject to chemical or conformational interconversions between n species with distinct NMR chemical shifts. Trott and Palmer used perturbation theory to approximate the largest eigenvalue of the Bloch-McConnell equations and obtain analytical expressions for the rotating-frame relaxation rate constant and for the laboratory frame resonance frequency and transverse relaxation rate constant. The perturbation treatment is valid whenever the population of one site is dominant. The new results are generally applicable to investigations of kinetic processes by NMR spectroscopy. 

Longitudinal and transverse nuclear relaxation profiles differ in the high field part. In fact, the equation for the transverse nuclear relaxation rates contains a non-dispersive term, depending only on Xd. Therefore the transverse relaxation does not go to zero at high fields, as longitudinal relaxation does, but increases because Tie increases (until it increases to the point where it becomes longer than x or Xm)-... 

The physical basis of current MRI methods has its origin in the fact that, in a strong magnetic field, the nuclear spins of water protons in different tissues relax back to equilibrium at different rates, when subject to perturbation from the resting Boltzmann distribution by the application of a short radio frequency (rf) pulse. For the most common type of spin-echo imaging, return to equilibrium takes place in accord with equation 1 and is governed by two time constants T and T2, the longitudinal and transverse relaxation times, respectively. 

In these equations, AI is the absorption intensity, T2A and T2P are the transverse relaxation times for the two sites, vd is the resonance frequency of the diamagnetic site, and v is the sweeping frequency. For a mixed second-order rate law,... 

The underlying strategy for extracting dynamical information from NMR relaxation data is based on the equations for either longitudinal (Tfl) or transverse (T2 ) relaxation rates. If relaxation is dominated by the magnetic dipole-dipole interaction between like-spin nuclei, then... 

In general, care must be taken to recognize both intra- and intermolecular contributions to relaxation however, intermole-cular relaxation is often neglected in discussions of liquids at surfaces. With this assumption there is direct access to characterization of liquid dynamics at a surface (, 3,4). A study of the relaxation rate at different temperatures or frequencies provides a measure of the correlation time for water at the surface of a protein if the interproton distance, r, in the water molecule is known. The temperature dependence for longitudinal and transverse relaxation times predicted by equations 3 and 4 is shown schematically as the solid lines in Figure 1. At the position of the minimum in the longitudinal relaxation time, wTp is about 0.616 and Ti/To is about 1.6. 

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