Dipole relaxations

The perturbation of one spin by the nearby magnetic dipole of another is the commonest, and in some cases the only significant relaxation mechanism for spin-1/2 nuclei. Quantitative calculation of this contribution to relaxation is easier when the two spins lie at a fixed distance, and so this case is treated first. 

If spin I is relaxed solely by the dipolar interaction of a different type of spin S, tumbling with a correlation time at a fixed distance r, then ITx=Wq + 1Wy- tW2 [essentially equation (10)] and IT2 = l2Ti + ( al - -a)]x1)- -Aa. A full calculation based on the principles of the previous sections gives 

Here a = jxly y hh S S- - )l2 n r and (Oi,cos are in rads . The fVg term represents mutually antiparallel spin transitions, the W2 term mutually parallel ones, and the JVj term represents transitions of spin I parallel to Bq which are induced by motions of spin S normal to Bq. 

When spin S is of the same type as I, then a different formula applies  

In a typical heteronuclear example, such as 1 = and S= H, the term ensures that only directly bound protons, if present, relax the nucleus significantly. It is normal in this case to set r = 0.109 nm, although tlm assumption has been disputed. One should note that the term is really where the average is over 

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In a similar fashion. Thermally Stimulated Current spectrometry (TSC) makes use of an appHed d-c potential that acts as the stress to orient dipoles. The temperature is then lowered to trap these dipoles, and small electrical currents are measured during heating as the dipoles relax. The resulting relaxation maps have been related to G and G" curves obtained by dynamic mechanical analysis (244—246). This technique, long carried out only in laboratory-built instmments, is available as a commercial TSC spectrometer from Thermold Partners L.P., formerly Solomat Instmments (247). 

At = 1 it coincides with the well-known Rocard formula for the spectrum of dipole relaxation [90] ... 

Here, is the magnetization of spin i at thermal equilibrium, p,j is the direct, dipole-dipole relaxation between spins i and j, a-y is the crossrelaxation between spins i and j, and pf is the direct relaxation of spin i due to other relaxation mechanisms, including intermolecular dipolar interactions and paramagnetic relaxation by dissolved oxygen. Under experimental conditions so chosen that dipolar interactions constitute the dominant relaxation-mechanism, and intermolecular interactions have been minimized by sufficient dilution and degassing of the sample, the quantity pf in Eq. 3b becomes much smaller than the direct, intramolecular, dipolar interactions, that is. 

A single-quantum transition involves one spin only, whereas the zero- and doublequantum transitions involve two spins at the same time. The zero- and double-quantum transitions give rise to cross-relaxation pathways, which provide an efficient mechanism for dipole-dipole relaxation. 

InterprotoD Distances (A) for 44, Calculated from the Specific, Interproton Dipole-Dipole Relaxation Contributions, and Assnming that —1-8... 

The rate at which dipole-dipole relaxation occurs depends on several factors (a) the nature of the nucleus, (b) the internuclear distance, r, and (c) the effective correlation time, t of the vector joining the nuclei (which is inversely proportional to the rate at which the relevant segment of the... 

In addition to the dipole-dipole relaxation processes, which depend on the strength and frequency of the fluctuating magnetic fields around the nuclei, there are other factors that affect nOe (a) the intrinsic nature of the nuclei I and S, (b) the internuclear distance (r,s) between them, and (c) the rate of tumbling of the relevant segment of the molecule in which the nuclei 1 and S are present (i.e., the effective molecular correlation time, Tf). 

While the rate of change of dipolar interaction depends on t its magnitude depends only on the internuclear distance and is independent of t,. Thus the dipole-dipole relaxation depends on the molecular correlation time T the internuclear distance r, and the gyromagnetic ratios of the two nuclei, y and js -... 

If the intensity of nucleus I before irradiation of the neighboring nucleus S was lo, and its intensity after irradiation of nucleus S was f, than the fractional increase in intensity ti/(5) is given by (7o — I )/h- If we assume that relaxation of nucleus I occurs only through dipole-dipole relaxation by interaction with nucleus S and that a steady equilibrium state has been reached, then the fractional increase in intensity t]/ of nucleus I is given by the equation... 

Let us first consider the nuclei //a, Hys, and as lying in a straight line and equidistant from one another (Fig. 4.5). The central proton, can relax by interactions with two neighbors, //a and He, while //a and He can relax by interaction with only one neighbor, so //b can relax twice as quickly as //a or Hq. If we assume that relaxation can occur only through dipole-dipole relaxation, and that an equilibrium steady state has been... 

The story is even more complicated than we have suggested, because carbon can relax by more than one mechanism. Protons rely on dipole-dipole relaxation, which also works well for protonated carbons but badly for non-proton-ated carbons. But carbon also for example makes use of spin-rotation relaxation, which is particularly active for methyl groups. And the magnetic field dependence of the various mechanisms also differs. We realize that relaxation is a very difficult subject, and if you want to know more then there are plenty of textbooks available ... 

Figure 1. Schematic representation of dependence of the T, and Tt relaxation times on isotropic correlation time (tc) of motion for a C-H fragment assuming dipole-dipole relaxation and 7 T magnetic field.

Bauer, W. S. Yilmaz, S. Wirges, W. Gerhard Multhaupt, R., Optimized poling of nonlinear optical polymers based on dipole orientation and dipole relaxation studies, J. Appl. Phys. 1994,75,7211 7219... 

To study dipole-dipole relaxation, one must distinguish between homonuclear and heteronuclear (unlike) spin-1 pairs. The latter gives rise to the so-called 3/2 effect.29 For an isolated pair of like spin-i nuclei (/= 1) separated by an intemuclear distance r, the treatment of spin relaxation is identical to that for a spin-1 quadrupole system. The Zeeman spin-lattice relaxation time T1Z and spin-spin relaxation time T2 are given, respectively, by... 

The second step of the evolution towards equilibrium is the Zeeman dipole-dipole relaxation. Hartmann and Anderson estimated this time using the hypothesis that p at any time is of the form (22). As a consequence of the shortness of the dipole-dipole relaxation time we may assume that the dipole-dipole system always remains in equilibrium we are thus led to treat the evolution of the Zeeman system as the Brownian motion of a collective coordinate in the dipole-dipole heat bath. We assume that the diagonal elements of p have the form... 

It follows that the spin-spin relaxation time (exactly the Zeeman, dipole-dipole relaxation time) is not r12 but... 

Another transport property of interfacial water which can be studied by MO techniques is the dipole relaxation time. This property is computed from the dipole moment correlation function, which measures the rate at which dipole moment autocorrelation is lost due to rotational motions in time (63). Larger values for the dipole relaxation time indicate slower rotational motions of the dipole... 

Monte Carlo and Molecular Dynamics simulations of water near hydrophobic surfaces have yielded a wealth of information about the structure, thermodynamics and transport properties of interfacial water. In particular, they have demonstrated the presence of molecular layering and density oscillations which extend many Angstroms away from the surfaces. These oscillations have recently been verified experimentally. Ordered dipolar orientations and reduced dipole relaxation times are observed in most of the simulations, indicating that interfacial water is not a uniform dielectric continuum. Reduced dipole relaxation times near the surfaces indicate that interfacial water experiences hindered rotation. The majority of simulation results indicate that water near hydrophobic surfaces exhibits fewer hydrogen bonds than water near the midplane. 

The high level of deuteration enormously reduces the line broadening due to the 1H-1H dipole-dipole relaxation of the methyl protons of Met, lie and Thr. Furthermore, the selective 13C labeling of these methyl groups can be used to resolve the methyl proton resonance overlap. 

The results obtained show that the dipole-relaxational motions in protein molecules are really very retarded as compared to such motions in the environment of aromatic molecules dissolved in liquid solvents (where they occur on a time scale of tens of picoseconds).(82) Dipole-relaxational motions on the nanosecond time scale have been observed for a variety of proteins. For example, such motions were recorded for apohemoglobin and bovine serum albumin0 04 105) labeled with the fluorescent probe 2,6-TNS. 

The fluorescent probe 2,6-TNS and other similar aminonaphthalene derivatives (1,8-ANS, DNS) were considered to be indicators of the polarity of protein molecules, and they were assumed to be bound only to hydrophobic sites on the protein surface. The detection of considerable spectral shifts with red-edge excitation has shown that the reason for the observed short-wavelength location of the spectra of these probes when complexed to proteins is not the hydrophobicity of their environment (or, at least, not only this) but the absence of dipole-relaxational equilibrium on the nanosecond time scale. Therefore, liquid solvents with different polarities cannot be considered to simulate the environment of fluorescent probes in proteins. 

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