Relaxation phenomenon

The effects of electronic relaxation on the magnetic hyperfine interaction have already been briefly alluded to and will now be discussed in more detail. It was seen in Section 3.5 that the hyperfine field is usually generated by the polarising effects of unpaired electron spins. The direction of the field will be related to that of the resultant electronic spin of the atom. This spin direction is not invariant but can alter or flip after a period of time by one of several mechanisms this is the relaxation phenomenon. 

Influence of the nuclear spin I on the time dependence of the atomic spin direction can be neglected. The precession frequency of the atomic spin S is such that the nuclear spin reacts only to the quantum value S. If the value of is maintained for an average period of time tg which is long compared with the nuclear Lamor precession time I/col (i.e. cottg 1) and the latter is long compared with the lifetime of the Mdssbauer event (i.e. 1) then 

If we have a paramagnetic solid containing ions with 5 =, then there are two orientations of S, namely -f and —These form a degenerate Kramers doublet. Both configurations generate a field H at the nucleus but with 

The collapse of a six-line hyperfine spectrum with decreasing relaxation time is illustrated by the calculated spectra in Fig. 3.8 [44] for a Fe nucleus in a fluctuating magnetic field and a fixed electric field gradient for different values of the electronic relaxation time. If the fluctuation rate is very slow compared to the precession frequency of the nucleus in the field H, the full six-line hyperfine pattern is observed. If the fluctuation rate is extremely rapid the nucleus will see only the time-averaged field which is zero and a symmetric quadrupolar pattern will be seen. At intermediate frequencies the spectra reflect the fact that the 2 - transitions which make up the low-velocity component of the quadrupole doublet relax at higher frequencies than do the I I i and -]- - transitions which 

It will be obvious that there are close similarities between the ordered magnetic and disordered paramagnetic hyperfine spectra. The hyperfine fields originate in similar ways, and the angular properties of the transitions 

In chapter 7, several aspects of conductivity and dielectric relaxation were discussed. Various other properties such as shear modulus, viscosity, refractive index, volume, enthalpy etc. also exhibit relaxational behaviour particularly in the glass transition region. In this chapter, few further aspects of relaxation are discussed. Relaxation of generalized stress or perturbation whether electrical, mechanical or any other form is typically non-exponential in nature. The associated property is a function of time. A variety of empirical functions, (/) t), have been used to describe the relaxation. Some of them have already been discussed in chapters 6 and 7. The most widely used function is the Kohlraush-Williams-Watts (KWW) function (Kohlraush, 1847 Williams and Watts, 1970 Williams et al., 1971). It is more commonly referred to as the stretched exponential function. The decay or relaxation of the quantity is given by, 

Relaxation function (p t) in equation (9.01) can be expressed equivalently through the use of relaxation functions in frequency domain - 

KWW expression is known as stretched exponential function and the reason is as follows. The simple exponential or Debye function is = exp[-(//r )] Therefore, 

Cole etc. (Cole and Cole, 1941 Davidson and Cole, 1958) are all functions in frequency domain. These functions can be derived as special forms of a general function which goes after the names, Havriliac and Negami, HN (Havriliac and Negami, 1966), 

With reference to dielectric relaxation, equation (9.02) can be written as, 

KWW expression is known as stretched exponential function and the reason is as follows. The 

When = 0 and /3=, this gives rise to the Debye relaxation function, which gives rise to a semicircle, when s is plotted against [ (o) = co) - i (cL))]. When a alone is zero. 

In all non-equilibrium systems relaxation phenomena can be observed. Relaxation is the time-dependent return to equilibrium (or to a new equilibrium) after a disturbance. 

Relaxation processes are universal. They are found in all branches of physics mechanical relaxation (stress and strain relaxation, creep), ultrasonic relaxation, dielectric relaxation, luminescence depolarisation, electronic relaxation (fluorescence), etc. Also the chemical reaction might be classified under the relaxation phenomena. It will be readily understood that especially in polymer science this time-dependent behaviour is of particular importance. 

The relaxation process is characterised by a driving force and by a rate constant. The driving force is always connected with the surplus of free energy in the non-equilibrium 

The equation shows that relaxation is strong if t x, whereas practically no relaxation takes place if t x. The relaxation time is temperature dependent it is an exponential function of temperature  

The ratio relaxation time/observation time = x/t is called the Deborah number. It is zero for ideal fluids and infinite for ideal solids. 

A DMTA study of polyolefin-clay nanocomposites has shown that alpha, beta and gamma relaxations of the polymer were affected by polymer chain branching and clay exfoliation level [50]. Salmeron Sanchez and co-workers [51] studied the structure of the system obtained after free radical copolymerisation of ethyl acrylate and hydroxyethyl methacrylate comonomers using dynamic-mechanical and calorimetric techniques. Copolymerisation theory states that the free radical copolymerisation reaction of two monomers may give rise to a copolymer with a different chain composition from that of the random mixture corresponding to the original solution. In this system, the dynamic-mechanical spectra suggested there were two main alpha relaxation processes in the copolymers. 

Leyva and co-workers [52] have studied relaxation phenomena in styrene-butadiene block copolymer with doped polyaniline. 

In many simple cases, the decaying of the transverse components of the nuclear magnetization in the rotating Ifame can be described by a phenomenological differential equation 

Similarly to the transverse case, the longitudinal relaxation can also be described by a phenomenological equation of the form  

The relaxation times Ti and T2 are parameters characteristic of each particular system, whose magnitudes depend on factors such as temperature, physical state of the matter (solid or liquid), molecular mobility, magnitude of the external magnetic field, etc. [4,5]. 

It is universally found that T T2, with the equality occurring mostly in liquids. On the other hand, in crystalline solids one has typically quite short values of T2 and long values of T, which leads to broad resonance lines and poor sensitivity. This is one of the reasons why solid-state NMR is not so easy and informative as the liquid-state counterpart [5]. Moreover, it should be emphasized that the simple behavior described by Equations (2.4.2) and (2.4.4) is not actually observed in many practical cases, where there occurs a distribution of relaxation times leading to a multiexponential behavior for the decaying transverse magnetization and/or the recovery of the longitudinal magnetization. 

The density operator p of a collection of identical, independent nuclei (an ensemble) is defined in such way that the macroscopic average of the expectation value of any observable A over the ensemble is given by [11]  

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Chestnoy N, Hull R and Brus L E 1986 Higher excited electronic states In clusters of ZnSe, CdSe, and ZnS spln-orblt, vibronic and relaxation phenomena J. Chem. Phys. 85 2237... 

The dielectric permittivity as a function of frequency may show resonance behavior in the case of gas molecules as studied in microwave spectroscopy (25) or more likely relaxation phenomena in soUds associated with the dissipative processes of polarization of molecules, be they nonpolar, dipolar, etc. There are exceptional circumstances of ferromagnetic resonance, electron magnetic resonance, or nmr. In most microwave treatments, the power dissipation or absorption process is described phenomenologically by equation 5, whatever the detailed molecular processes. 

Important problems ia coUoid scieace remain to be addressed if the poteatial of coUoids is to be fuUy exploited, amoag them, exteasioa of understanding to more coaceatrated suspeasioas, testiag of predictioas usiag model powders, and examination of relaxation phenomena ia ordered coUoids. Much is known about coUoids and their formation and behavior, but considerably more remains unknown. Thus the fuU potential to control coUoids is not presently realized. 

Anisodiametrical particles of filler appearance of anisotropy of properties and relaxation phenomena, determined by the turn of solid particles in a flow... 

Another largely unexplored area is the change of dynamics due to the influence of the surface. The dynamic behavior of a latex suspension as a model system for Brownian particles is determined by photon correlation spectroscopy in evanescent wave geometry [130] and reported to differ strongly from the bulk. Little information is available on surface motion and relaxation phenomena of polymers [10, 131]. The softening at the surface of polymer thin films is measured by a mechanical nano-indentation technique [132], where the applied force and the path during the penetration of a thin needle into the surface is carefully determined. Thus the structure, conformation and dynamics of polymer molecules at the free surface is still very much unexplored and only few specific examples have been reported in the literature. 

Goldflam R., Kouri D. J. On accurate quantum mechanical approximations for molecular relaxation phenomena. Averaged... 

By far the largest group of small to medium elastomer components comprises seals and gaskets. Relaxation phenomena, which would result in loss of sealing ability, can become important. 

The second reason is related to the misconception that proton dipolar relaxation-rates for the average molecule are far too complicated for practical use in stereochemical problems. This belief has been encouraged, perhaps, by the formidable, density-matrix calculations " commonly used by physicists and physical chemists for a rigorous interpretation of relaxation phenomena in multispin systems. However, proton-relaxation experiments reported by Freeman, Hill, Hall, and their coworkers " have demonstrated that pessimism regarding the interpretation of proton relaxation-rates may be unjustified. Valuable information of considerable importance for the carbohydrate chemist may be derived for the average molecule of interest from a simple treatment of relaxation rates. 

Relaxation phenomena can also be studied by nuclear forward scattering of synchrotron radiation [16, 30]. This is discussed in Chap. 9. 

Although relaxation measurements have been widely used in nuclear magnetic resonance studies of solid catalysts and adsorbed molecules, they have not found such favor in similar ESR work. Relaxation phenomena, however, do play a very important role in any magnetic resonance experiment, whether or not this aspect of the problem is studied. In fact, the temperature at which most ESR experiments are conducted is dictated by the relaxation process. Furthermore, even qualitative data on relaxation times can be used as supporting evidence in the identification of a paramagnetic species. 

This does not mean that the TLS theory is wrong, but only some approximations are to be revised. In fact, the TLS theory does not take into account the processes of absorption and emission of a phonon by a TLS which lead to relaxation phenomena in the tunnelling levels. 

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