Mechanism relaxation

A very similar equation [5] applies to the relaxation rate in the rotating frame. 

Other relaxation mechanisms. The above principles may be applied much more widely than to CH and HH pairs. A few examples are relevant to polymer relaxation. 

Intermolecular dipolar relaxation. Nearby protons on other molecules, including solvent, will not contribute significantly to the relaxation of protonated carbons, because of the r dependence deduced above. They do have a detectable influence on unprotonated carbons, however, typically contributing 0.1 Hz to 1/Ti. They can also influence the proton resonances, especially if they are held fairly close to the proton in question by, for example, some polymer entanglement. In this case f(t) will include not only any angular 

Shift anisotropy. If the chemical shift depends upon molecular orientation, as it does in an aromatic ring, then this will also contribute to V t) as a result of motionally induced fluctuations in the electron shielding field. The contribution increases with and typically contributes 0.02 Hz to 1/Ti, in high-field spectrometers. Again, if it arises in partially oriented molecules then it affects lineshape, this time asymmetrically. This is discussed further in chapters 5 and 7. 

Here we enunciate the principal mechanisms of nuclear relaxation, for the mechanisms differ in the effectiveness with which they cause relaxation, such that before useful information can be obtained from a system by studying relaxation phenomena, it is essential to determine the dominant mechanism(s) responsible for relaxation. Use of relaxation data to study motions in macromolecules is considered in Section 4(e)(iii). 

There is no straightforward and completely rigorous procedure for determining the relative combinations of the various relaxation mechanisms, except where one mechanism clearly dominates (e.g., if the maximum possible nuclear Overhauser effect (NOE) for a resonance is obtained, dipolar relaxation must dominate its relaxation or an increase in relaxation rate in proportion to the square of the applied field must be due to chemical shift anisotropy). Hence, the study of molecular motion in proteins from relaxation data is performed most readily on nuclei directly bonded to H, and so principally relaxed via dipole-dipole interactions (see Section 4(e)(iii)). 

The net macroscopic magnetization along the z axis produced by the nuclei A and B in the sample, designated and respectively, is linearly proportional to the difference between the energy level population sizes, i.e. 

Identical equations apply to the equilibrium magnetization values A° and if iV is 

Essentially identical equations apply to transverse relaxation 

Interaction with Randomly Fluctuating Magnetic Fields 36) 

The magnetic field b(t) is, for instance, created by another spin (nuclear spin or the spin of an unpaired electron) in that case, it is proportional to 1/r (r distance between the two spins). Its time dependency arises from the orientation of r and/or from the distance fluctuation. However, in this section, we shall disregard the origin of b t) and we only rest on its general properties (assuming an isotropic medium) which arise from the random nature of molecular motions  

To understand the two latter points, one can think of a quantity whose modulus is non-zero but which can take opposite values with the same probability. 

The coherence of a random field can be evaluated by its correlation function, i.e.. 

Conversely, a rf field is totally correlated because it is represented by a sine (or cosine) function and, as a consequence, its value at any time t can be predicted from its value at time zero. The efficiency of a random field at a given frequency co can be appreciated by the Fourier transform of the above correlation function 

This section deals with some of the main factors which contribute to the width of ESR spectral lines, an understanding of which is important in achieving maximum resolution of spectra and hence the maximum amount of information. 

In general, Heisenberg s uncertainty principle relates line width with the lifetime of the excited state by the equation 

Spin-lattice relaxation predicts a line width of (27tTi) , where Tj is the relaxation time for the transfer of energy from the spin system to the lattice. has been found to be sensitive in particular to  

Theoretical analyses of spin-lattice relaxation have been performed by Kubo and Tomita 420), Redfield (579), and Kivelson (390). This mechanism is applicable to solids and liquids. The line shape for this type of relaxation is Lorentzian. 

Because the local magnetic field produced by an electron at a distance r(cm) is given to a first approximation by 

---

A second type of relaxation mechanism, the spin-spm relaxation, will cause a decay of the phase coherence of the spin motion introduced by the coherent excitation of tire spins by the MW radiation. The mechanism involves slight perturbations of the Lannor frequency by stochastically fluctuating magnetic dipoles, for example those arising from nearby magnetic nuclei. Due to the randomization of spin directions and the concomitant loss of phase coherence, the spin system approaches a state of maximum entropy. The spin-spin relaxation disturbing the phase coherence is characterized by T. ... 

Understanding VER in condensed phases has proven difficult. The experiments are hard. The stmcturally simple systems (diatomic molecules) involve complicated relaxation mechanisms. The stmctures of polyatomic molecules are obviously more complex, but polyatomic systems are tractable because the VER mechanisms are somewhat simpler. 

There is arbitr iriness in describing phenomena as either physical or chemical, but in some sense the nuclear relaxation mechanisms we have discussed to this point are physical mechanisms, based as they are on rotational motions of molecules, magnetic dipole-dipole interactions, quadrupolar interactions, and so on. Now we discuss a nuclear relaxation mechanism that is chemical in origin. 

Consider a nucleus that can partition between two magnetically nonequivalent sites. Examples would be protons or carbon atoms involved in cis-trans isomerization, rotation about the carbon—nitrogen atom in amides, proton exchange between solute and solvent or between two conjugate acid-base pairs, or molecular complex formation. In the NMR context the nucleus is said to undergo chemical exchange between the sites. Chemical exchange is a relaxation mechanism, because it is a means by which the nucleus in one site (state) is enabled to leave that state. 

Nuclear dipole-dipole interaction is a veiy important relaxation mechanism, and this is reflected in the relationship between 7, and the number of protons bonded to a carbon. The motional effect is nicely shown by tbe 7 values for n-decanol, which suggest that the polar end of the molecule is less mobile than the hydrocarbon tail. Comparison of iso-octane with n-decanol shows that the entire iso-octane molecule is subject to more rapid molecular motion than is n-decanol—compare the methyl group T values in these molecules. 

Though the accuracy of description of flow curves of real polymer melts, attained by means of Eq. (10), is not always sufficient, but doubtless the equation of such a structure based on the idea of relaxation mechanism of non-Newtonian polymer flow, correctly reflects the main peculiarities of viscous properties. Therefore while discussing the effect a filler has on the viscosity properties of polymer melts, besides the dependences Y(filler modifies the characteristic time of relaxation. According to [19], a possible form of the X versus

[Pg.86]

The existence of characteristic time D"1 must lead to the appearance of specific relaxation effects. This relaxation mechanism has nothing in common with visco-... 

A possible approach to interpretation of a low-frequency region of the G ( ) dependence of filled polymers is to compare it with a specific relaxation mechanism, which appears due to the presence of a filler in the melt. We have already spoken about two possible mechanisms — the first, associated with adsorption phenomena on a filler s surface and the second, determined by the possibility of rotational diffusion of anisodiametrical particles with characteristic time D 1. But even if these effects are not taken into account, the presence of a filler can be related with the appearance of a new characteristic time, Xf, common for any systems. It is expressed in the following way... 

Bone, F. Transient Relaxation Mechanisms in Elongated Melts and Rubbers Investigated by Small Angle Neutron Scattering. Vol. 82, pp. 47— 103. 

K Relaxation mechanism was also proposed [130-133], The % relaxation originates from cyclic delocalization of % electrons in the double bond through the hyperconjugation with a bonds on the saturated ring atoms under control of the orbital phase property [134, 135],... 

M.L. Williams, R.E. Landel, and J.D. Ferry, The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming Uquids, J. Am. Chem. Soc., 77, 3701-3707, 1955. 

The process of spin-lattice relaxation involves the transfer of magnetization between the magnetic nuclei (spins) and their environment (the lattice). The rate at which this transfer of energy occurs is the spin-lattice relaxation-rate (/ , in s ). The inverse of this quantity is the spin-lattice relaxation-time (Ti, in s), which is the experimentally determinable parameter. In principle, this energy interchange can be mediated by several different mechanisms, including dipole-dipole interactions, chemical-shift anisotropy, and spin-rotation interactions. For protons, as will be seen later, the dominant relaxation-mechanism for energy transfer is usually the intramolecular dipole-dipole interaction. 

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 deviation from the value of l.S indicates that other relaxation mechanisms contribute to the relaxation of spin i. The extent of intramolecular dipole-dipole interactions for spin i is given by ... 

A potential source of systematic error is the quantity pf, which may not be negligible, and indicates that relaxation mechanisms other than dipole-dipole interactions contribute to the relaxation of the spin in question. 

When other relaxation mechanisms are involved, such as chemical-shift anisotropy or spin-rotation interactions, they cannot be separated by application of the foregoing relaxation theory. Then, the full density-matrix formalism should be employed. 

---