Medium, isotropic

As discussed above, the nonlinear material response, P f) is the most connnonly encountered nonlinear tenn since vanishes in an isotropic medium. Because of the special importance of P we will discuss it in some detail. We will now focus on a few examples ofP spectroscopy where just one or two of the 48 double-sided Feymnan diagrams are important, and will stress the dynamical interpretation of the signal. A pictorial interpretation of all the different resonant diagrams in temis of wavepacket dynamics is given in [41]. 

Consider an isotropic medium that consists of independent and identical microscopic cln-omophores (molecules) at number density N. At. sth order, each element of the macroscopic susceptibility tensor, given in laboratory Cartesian coordinates A, B, C, D, must carry s + 1 (laboratory) Cartesian indices (X, Y or Z) and... 

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

Figure 4.11. Diagrammatic sketches of atomic lattice rearrangements as a result of dynamic compression, which give rise to (a) elastic shock, (b) deformational shock, and (c) shock-induced phase change. In the case of an elastic shock in an isotropic medium, the lateral stress is a factor v/(l — v) less than the stress in the shock propagation direction. Here v is Poisson s ratio. In cases (b) and (c) stresses are assumed equal in all directions if the shock stress amplitude is much greater than the material strength.

If we now transfer our two interacting particles from the vacuum (whose dielectric constant is unity by definition) to a hypothetical continuous isotropic medium of dielectric constant e > 1, the electrostatic attractive forces will be attenuated because of the medium s capability of separating charge. Quantitative theories of this effect tend to be approximate, in part because the medium is not a structureless continuum and also because the bulk dielectric constant may be an inappropriate measure on the molecular scale. Eurther discussion of the influence of dielectric constant is given in Section 8.3. 

Neal and Nader [260] considered diffusion in homogeneous isotropic medium composed of randomly placed impermeable spherical particles. They solved steady-state diffusion problems in a unit cell consisting of a spherical particle placed in a concentric shell and the exterior of the unit cell modeled as a homogeneous media characterized by one parameter, the porosity. By equating the fluxes in the unit cell and at the exterior and applying the definition of porosity, they obtained... 

This equation is identical to the Maxwell [236,237] solution originally derived for electrical conductivity in a dilute suspension of spheres. Hashin and Shtrikman [149] using variational theory showed that Maxwell s equation is in fact an upper bound for the relative diffusion coefficients in isotropic medium for any concentration of suspended spheres and even for cases where the solid portions of the medium are not spheres. However, they also noted that a reduced upper bound may be obtained if one includes additional statistical descriptions of the medium other than the void fraction. Weissberg [419] demonstrated that this was indeed true when additional geometrical parameters are included in the calculations. Batchelor and O Brien [34] further extended the Maxwell approach. 

Note that for a macroscopically isotropic medium, the tensor given by Eq. (Ill) has equal elements along the diagonal, and therefore Eq. (110) is equivalent to... 

In an isotropic medium, as for normal liquids, the Fourier equation holds ... 

Non-uniform temperature distribution in a reactor assumed model based on the Fourier heat conduction in an isotropic medium equality of temperatures of the medium and the surroundings assumed at the boundary critical values of Frank-Kamenetskii number given. 

Conversely, in a membrane model, acetylcholine showed mean log P values very similar to those exhibited in water. This was due to the compound remaining in the vicinity of the polar phospholipid heads, but the disappearance of extended forms decreased the average log P value somewhat. This suggests that an anisotropic environment can heavily modify the conformational profile of a solute, thus selecting the conformational clusters more suitable for optimal interactions. In other words, isotropic media select the conformers, whereas anisotropic media select the conformational clusters. The difference in conformational behavior in isotropic versus anisotropic environments can be explained considering that the physicochemical effects induced by an isotropic medium are homogeneously uniform around the solute so that all conformers are equally influenced by them. In contrast, the physicochemical effects induced by an anisotropic medium are not homogeneously distributed and only some conformational clusters can adapt to them. 

The rate of a chemical reaction (the chemical flux ), 7ch, in contrast to the above processes, is a scalar quantity and, according to the Curie principle, cannot be coupled with vector fluxes corresponding to transport phenomena, provided that the chemical reaction occurs in an isotropic medium. Otherwise (see Chapter 6, page 450), chemical flux can be treated in the same way as the other fluxes. 

In Figure 1 we show that for isotropic medium the renormalized distribution function,... 

In an isotropic medium, vectors such as stress S and strain X are related by vector equations such as, S = kX, where S and X have the same direction. If the medium is not isotropic the use of vectors to describe the response may be too restrictive and the scalar k may need to be replaced by a more general operator, capable of changing not only the magnitude of the vector X, but also its direction. Such a construct is called a tensor. 

In a homogeneous isotropic medium in which D= tE and B= pH, a complex field vector is defined as... 

However, the components of the yj2) e, e tensor are chiral (i.e., only present in a chiral isotropic medium), whereas the components of the tensors y 2) and y(2) meeare achiral (i.e., present in any isotropic medium, chiral or achiral). Hence, only the electric dipole response of chiral isotropic materials is related to chirality. The experimental work on chiral polymers described in Section 4 showed that large magnetic contributions to the nonlinearity are due to chirality. However, such contributions will therefore not survive in chiral isotropic media. In this respect, the electric dipole contributions associated with chirality may prove more interesting for applications. 

Quantitative models of solute-solvent systems are often divided into two broad classes, depending upon whether the solvent is treated as being composed of discrete molecules or as a continuum. Molecular dynamics and Monte Carlo simulations are examples of the former 8"11 the interaction of a solute molecule with each of hundreds or sometimes even thousands of solvent molecules is explicitly taken into account, over a lengthy series of steps. This clearly puts a considerable demand upon computer resources. The different continuum models,11"16 which have evolved from the work of Bom,17 Bell,18 Kirkwood,19 and Onsager20 in the pre-computer era, view the solvent as a continuous, polarizable isotropic medium in which the solute molecule is contained within a cavity. The division into discrete and continuum models is of course not a rigorous one there are many variants that combine elements of both. For example, the solute molecule might be surrounded by a first solvation shell with the constituents of which it interacts explicitly, while beyond this is the continuum solvent.16... 

For isotropic motions in an isotropic medium, the values of the instantaneous and steady-state emission anisotropies are linked to the rotational diffusion coefficient Dr by the following relations (see Chapter 5) ... 

Such a mechanism is not incompatible with a Haven ratio between 0.3 and 0.6 which is usually found for mineral glasses (Haven and Verkerk, 1965 Terai and Hayami, 1975 Lim and Day, 1978). The Haven ratio, that is the ratio of the tracer diffusion coefficient D determined by radioactive tracer methods to D, the diffusion coefficient obtained from conductivity via the Nernst-Einstein relationship (defined in Chapter 3) can be measured with great accuracy. The simultaneous measurement of D and D by analysis of the diffusion profile obtained under an electrical field (Kant, Kaps and Offermann, 1988) allows the Haven ratio to be determined with an accuracy better than 5%. From random walk theory of ion hopping the conductivity diffusion coefficient D = (e /interstitial mechanism, the corresponding mobility is expressed by... 

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 ... 

If the diffusion medium is isotropic in terms of diffusion, meaning that diffusion coefficient does not depend on direction in the medium, it is called diffusion in an isotropic medium. Otherwise, it is referred to as diffusion in an anisotropic medium. Isotropic diffusion medium includes gas, liquid (such as aqueous solution and silicate melts), glass, and crystalline phases with isometric symmetry (such as spinel and garnet). Anisotropic diffusion medium includes crystalline phases with lower than isometric symmetry. That is, most minerals are diffu-sionally anisotropic. An isotropic medium in terms of diffusion may not be an isotropic medium in terms of other properties. For example, cubic crystals are not isotropic in terms of elastic properties. The diffusion equations that have been presented so far (Equations 3-7 to 3-10) are all for isotropic diffusion medium. 

For binary diffusion in an isotropic medium, one diffusion coefficient describes the diffusion. For binary diffusion in an anisotropic medium, the diffusion coefficient is replaced by a diffusion tensor, denoted as D. The diffusion tensor is a second-rank symmetric tensor representable by a 3 x 3 matrix ... 

In an isotropic medium, D is a scalar, which may be constant or dependent on time, space coordinates, and/or concentration. In anisotropic media (such as crystals other than cubic symmetry, i.e., most minerals), however, diffusivity also depends on the diffusion direction. The diffusivity in an anisotropic medium is a second-rank symmetric tensor D that can be represented by a 3 x 3 matrix (Equation 3-25a). The tensor is called the diffusivity tensor. Diffusivity along any given direction can be calculated from the diffusivity tensor (Equation 3-25b). Each element in the tensor may be constant, or dependent on time, space coordinates and/or concentration. 

The one-dimensional diffusion equation in isotropic medium for a binary system with a constant diffusivity is the most treated diffusion equation. In infinite and semi-infinite media with simple initial and boundary conditions, the diffusion equation is solved using the Boltzmann transformation and the solution is often an error function, such as Equation 3-44. In infinite and semi-infinite media with complicated initial and boundary conditions, the solution may be obtained using the superposition principle by integration, such as Equation 3-48a and solutions in Appendix 3. In a finite medium, the solution is often obtained by the separation of variables using Fourier series. 

DC electric fields. DC generation is known as optical rectification. The actual phenomena that will be observed depend on the experimental conditions and whether or not phase matching has been achieved. Three-wave mixing processes in which two beams interact to generate a third beam require the mixing medium to have a non-zero In an isotropic medium, reversal of the... 

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