The classical theory

Ho is the unperturbed Hamiltonian, describing the Zeeman interaction and Hx(t) is the dipole-dipole Hamiltonian, time-dependent through variation in the orientation of the r/s vector. The DD Hamiltonian can conveniently be expressed using scalar contraction of irreducible tensors (16). 

Using a simple kinetic model, Solomon demonstrated that the spin-lattice relaxation of the I and S spins was described by a system of coupled differential equations, with bi-exponential functions as general solutions. A single exponential relaxation for the I spin, corresponding to a well-defined Tu, could only be obtained in certain limiting situations, e.g., if the other spin, S, was different from I and had an independent and highly efficient relaxation pathway. This limit is normally fulfilled if S represents an electron spin. The spin-lattice relaxation rate, for the nuclear spin, I, is in such a situation given by  

Hamiltonian. The bar denotes ensemble average. Between the first and the second lines of Eg. (8), we assume that the perturbation is stationary (the product + is independent of t). In the third line, we 

The last assumption is very fundamental. It results in time-independent transition probabilities and makes a clean theory possible. It requires that the product of the time scale of the decay time for the tcf (called the correlation time and denoted x ) and the strength of the perturbation (in angular frequency units) has to be much smaller than unity (17-20). This range is sometimes denoted as the Redfield limit or the perturbation regime. 

A common assumption in the relaxation theory is that the time-correlation function decays exponentially, with the above-mentioned correlation time as the time constant (this assumption can be rigorously derived for certain limiting situations (18)). The spectral density function is then Lorentzian and the nuclear spin relaxation rate of Eq. (7) becomes  

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The classic theory due to van der Waals provides an important phenomenological link between the structure of an interface and its interfacial tension [50-52]. The expression... 

A homogeneous metastable phase is always stable with respect to the fonnation of infinitesimal droplets, provided the surface tension a is positive. Between this extreme and the other thennodynamic equilibrium state, which is inhomogeneous and consists of two coexisting phases, a critical size droplet state exists, which is in unstable equilibrium. In the classical theory, one makes the capillarity approxunation the critical droplet is assumed homogeneous up to the boundary separating it from the metastable background and is assumed to be the same as the new phase in the bulk. Then the work of fonnation W R) of such a droplet of arbitrary radius R is the sum of the... 

At very low densities It Is quite easy Co give a theoretical description of thermal transpiration, alnce the classical theory of Knudsen screaming 9] can be extended to account for Che Influence of temperature gradients. For Isothermal flow through a straight capillary of circular cross-section, a well known calculation [9] gives the molar flux per unit cross-sectional area, N, In the form... 

This critical field called coercivity or switching field is also equal to If a field is appHed in between 0 and 90° the coercivity varies from maximum to zero. In the case of this special example the appHed field = H/. Based on the classical theory, Stoner-Wohlfarth (33)... 

I. N. Sneddon and M. Lowengmb, Crack Problems in the Classical Theory of Elasticity,]ohn Wiley Sons, Inc., New York, 1969. 

A cornerstone of the analysis of vaporization processes in a vacuum is the classical theory of gases. In this theory, a gas is assumed to consist of noninteracting molecules which undergo elastic collisions with one another and... 

It is usual in the classical theory to assume that the stress rate is independent of the hardening parameters, since the elastic behavior is expected to be unaffected by plastic deformation. Consequently, the stress rate relation (5.23) reduces to... 

In the classical theory of plasticity, constitutive equations for the evolution of the isotropic and kinematic hardening parameters are usually expressed as... 

The plasticity equations presented so far are still more general than the equations usually considered in the classical theory of plasticity. Linearity and symmetry assumptions, inherent in most classical treatments, are yet to be made. Particularly simple assumptions are made here to serve as an example. 

The classical theory of contact mechanics, due to Hertz, treats the bodies in contact with a hard wall repulsive interaction, i.e. there is no attractive interaction whatsoever, and a steep repulsion comes into play when the surfaces of the bodies are in contact. The Hertzian theory assumes that only normal stresses exist, i.e. the shear stress in the contact region is zero. Under these conditions, the contact radius a), central displacement (3) and the distribution of normal stress (a) are given by the following expressions ... 

The classical theory of methylation with diazomethane was developed by Arndt from a different basis. It depends on the postulate (which can be traced back to von Pechmann " 0 of direct methylation mobile hydrogen in an acid compound is directly replaced by the methyl group, i.e., the methyl group appears in the place which the hydrogen previously occupied. For the reaction of tautomeric substances with diazomethane, the following equation is applicable ... 

We consider first the polarizability of a molecule consisting of two or more polarizable parts which may be atoms, bonds, or other units. When the molecule is placed in an electric field the effective field which induces dipole moments in various parts is not just the external field but rather the local field which is influenced by the induced dipoles of the other parts. The classical theory of this interaction of polarizable units was presented by Silberstein36 and others and is summarized by Stuart in his monograph.40 The writer has examined the problem in quantum theory and finds that the same results are obtained to the order of approximation being considered. 

Both quantum mechanical and classical theories of Raman scattering have been developed. The quantum mechanical treatment of Kramers and Heisenberg 5) preceded the classical theory of Cabannes and Rochard 6). 

The classical theory of scattering provides us with a relatively simple selection rule for Raman activity which can be compared with that for infrared activity. 

Firstly, the classical theories on radical reactivity and polymerization mechanism do not adequately explain the rate and specificity of simple radical reactions. As a consequence, they can not be used to predict the manner in which polymerization rate parameters and details of polymer microstructurc depend on reaction conditions, conversion and molecular weight distribution. 

Nonanalytic Nonlinearities.—A somewhat different kind of nonlinearity has been recognized in recent years, as the result of observations on the behavior of control systems. It was observed long ago that control systems that appear to be reasonably linear, if considered from the point of view of their differential equations, often exhibit self-excited oscillations, a fact that is at variance with the classical theory asserting that in linear systems self-excited oscillations are impossible. Thus, for instance, in the van der Pol equation... 

Substituting Eq. (12) into Eq. (11) permits us to derive the Hookean spring force law, well-known in the classical theory of rubber elasticity ... 

It is also possible to prepare them from amino acids by the self-condensation reaction (3.12). The PAs (AABB) can be prepared from diamines and diacids by hydrolytic polymerization [see (3.12)]. The polyamides can also be prepared from other starting materials, such as esters, acid chlorides, isocyanates, silylated amines, and nitrils. The reactive acid chlorides are employed in the synthesis of wholly aromatic polyamides, such as poly(p-phenyleneterephthalamide) in (3.4). The molecular weight distribution (Mw/Mn) of these polymers follows the classical theory of molecular weight distribution and is nearly always in the region of 2. In some cases, such as PA-6,6, chain branching can take place and then the Mw/Mn ratio is higher. 

On the basis of the classical theory it was shown by Debye2 that cos 9 is given by the Langevin function... 

The experimental points, which surprisingly enough lie very close to a curve of the type required by the classical theory, show deviations as large as 2% from the best quantum theory curve. Nevertheless it is not easy to reject the straight-forward and well-grounded quantum theory calculations given in this paper possibly the discrepancies can be attributed to errors in the experimental measurements. Further measurements testing this point are needed. 

According to the classical theory, the effect of a magnetic field on a system composed of electrons in motion about a fixed nucleus is equivalent to the first order of approximation to the imposition on the system of a uniform rotation... 

The quantum mechanics treatment of diamagnetism has not been published. It seems probable, however, that Larmor s theorem will be retained essentially, in view of the marked similarity between the results of the quantum mechanics and those of the classical theory in related problems, such as the polarisation due to permanent electric dipoles and the paramagnetic susceptibility. f Thus we are led to use equation (30), introducing for rK2 the quantum mechanics value... 

The introduction of the quantum mechanics does not require this picture to be changed essentially. The allowed states of the system can approximate either of two extremes, oscillation and rotation, or can lie between these extremes, approximating neither more closely than the other. For with the quantum mechanics, in contradistinction to the classical theory, the transition from one extreme to the other is unbroken. 

Basically, when analysing the band structures, the equivalent observations apply to typical solid state compounds like thallium halides and lead chalcogenides. In studies on the origin of distortion in a-PbO, it was found that the classical theory of hybridization of the lead 6s and 6p orbitals is incorrect and that the lone pair is the result of the lead-oxygen interaction [44]. It was also noted... 

Overview of the Classical Theory of the Structural Glass Transition The Intrinsic Excitations of Amorphous Solids... 

II. OVERVIEW OF THE CLASSICAL THEORY OF THE STRUCTURAL GLASS TRANSITION... 

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