Electrical susceptibility first order

The third-rank diamagnetic and paramagnetic contributions to electric field dependent magnetic susceptibility and nuclear magnetic shielding, to first order in... 

Many of the different susceptibilities in Equations (2.165)-(2.167) correspond to important experiments in linear and nonlinear optics. x<(>> describes a possible zero-order (permanent) polarization of the medium j(1)(0 0) is the first-order static susceptibility which is related to the permittivity at zero frequency, e(0), while ft> o>) is the linear optical susceptibility related to the refractive index n" at frequency to. Turning to nonlinear effects, the Pockels susceptibility j(2)(- to, 0) and the Kerr susceptibility X(3 —to to, 0,0) describe the change of the refractive index induced by an externally applied static field. The susceptibility j(2)(—2to to, to) describes frequency doubling usually called second harmonic generation (SHG) and j(3)(-2 to, to, 0) describes the influence of an external field on the SHG process which is of great importance for the characterization of second-order NLO properties in solution in electric field second harmonic generation (EFISHG). 

One can measure the dielectric constant e of gases, liquids and solids by placing the sample in a capacitance cell. From measuring e as a function of temperature, one routinely gets the scalar first-order electric dipole susceptibility... 

Here Pind is the induced dipole moment per unit volume, and X X and X are the first-, second-, and third-order electric susceptibilities of the sample, where the order refers to the power of electric field (not of X). Equation (1) represents the macroscopic (bulk) form for the polarization in terms of single molecule properties, the induced dipole moment per molecule is written as... 

Equations (18)-(21) were given for the case of real susceptibilities. However, they have to be treated as complex quantities if the frequency is close to or within the region of an optical transition in the medium. An example in the domain of linear optics was given in (12) where the imaginary part of the first-order susceptibility, ft) w), was related to the absorption coefficient, of the medium. An example from non-linear optics is the technique of electro-optical absorption measurements (EOAM, p. 167) where the UV-visible absorption is studied under the influence of a static electric field. In EOAM, the imaginary part of the third-order susceptibility, w,0,0), is... 

Other important systems are uniaxial isotropic systems, because the widely studied poled polymers belong to this symmetry class (o°w or Co ). of such systems has seven non-vanishing components of which four are independent, = X k, x9k = xSk = X lx and x9k- For the SHG susceptibility X -2u> o),u)) the number of independent components reduces to three because of intrinsic permutation symmetry in the second and third index. If the uniaxial system is created by poling of an isotropic system by an external electric field, e.g. a poled polymer or liquid, then to first order in the applied field, z, the number of independent components of x -2a> w, w) is only two (Kielich, 1968). It is thus equal to the number of independent components of x -2a) a),o),0) because of (28). 

When the macroscopic third-order electric susceptibility x jkl is measured in the laboratory (the subscripts /, J, K, L refering to the axes in the laboratory-fixed coordinate system), it is first related to the microscopic third-order coefficients [19] ... 

The first order light-matter interaction described by the electrical susceptibility, = w,), may be viewed as quadrature in its simplest form. 

A third class of quantities are calculated by second-order perturbation theory via a one-partide perturbation operator e.g. electric polarizabilities, Van-der-Waals constants, diamagnetic susceptibilities, chemical shifts). These are affected by correlation effects to first order and hence depend... 

A number of exploitable effects exist, due to the non-linear response of certain dielectric materials to applied electric and optical fields. An applied field, E, gives rise to a polarization field, P, within any dielectric medium. In a linear material, the relationship between P and E may be characterized by a single (first-order, second-rank) susceptibility tensor... 

A significant number of techniques, magnetization, AC susceptibility, electrical resistivity, thermal expansion, calorimetry, and elasticity measurements, have been used to determine the transition temperatures. In particular, Watson and Ali (1995) used the first four techniques and the values quoted in Table 167 are an average of these determinations. Many of the transitions are first order, and entropies of transition as determined by Pecharsky et al. (1996) are selected except for Tg, and Tp which are due to Astrom and Benediktsson (1989). A question mark indicates that it is unknown as to whether or not the transition is first order or second order. 

The SmS semiconductor to metal transition was later verified by the direct observation of a discontinuous change in the optical reflectivity at 6 kbar (Kirk et al., 1972). This is consistent with a first order magnetic phase transition which was directly verified by magnetic susceptibility measurements under pressure by Maple and Wohlleben (1971). In the collapsed phase the susceptibility of SmS showed no magnetic order down to 0.35 iC and was almost identical to the susceptibility of SmBa (see fig. 20.10 of volume 2). Bader et al. (1973) measured the heat capacity (fig. 11.16) and electrical resistivity (fig. 11.17) of SmS under pressure. They found a large electronic contribution to the heat capacity ( y = 145 mJ/mole-K ) and a resistivity reminiscent of SmB. Mossbauer isomer shift measurements of SmS under pressure by Coey et al. (1976) reveal the transition from a Sm isomer shift at zero pressure to an intermediate value at pressures above 6 kbar (fig. 11.18). The isomer shift of SmS above 6 kbar was found to be about the same as the isomer shifts for chemically collapsed Smo.77Yo.23S and SmBo at zero pressure. 

The coefficients in the various terms in Eqs. (2a) and (2b) are termed the nth-order susceptibilities. The first-order susceptibilities describe the linear optical effects, while the remaining terms describe the nth order nonlinear optical effects. The coefficients are the nth-order electric dipole susceptibilities, the coefficients G " are the nth-order quadrupole susceptibilities, and so on. Similar terminology is used for the various magnetic susceptibilities. For most nonlinear optical interactions, the electric dipole susceptibilities are the dominant terms because the wavelength of the radiation is usually much longer than the scattering centers. These will be the ones considered primarily from now on. 

In accordance with the described mean-field model, for pure LCs a sharp, weakly first-order I-N transition is typically observed, exhibiting a narrow (only a few mK) two-phase region. Large, externally applied electric or magnetic fields are needed to impose a continuous conversion because the electric or magnetic susceptibility of LCs is low [9-13]. Nevertheless, transition smearing can be observed when strong random fields are introduced to the system, such as the mechanical stresses imposed on LCs confined in random porous media [14]. 

In (2.1) apices i, j, k, I etc. indicate (contravariant) components of the three vectors under consideration, i.e. P, and E. They are, respectively, the dipole moment per volume unit of the perturbed material, the (permanent) dipole moment per volume unit of the unperturbed material, and the perturbing external electric field. Of course, apices run fi om 1 to 3 and we can assume that 1 stands for the X component, 2 for the y component and 3 for the z component of each vector with respect to a common reference system, R = [0,(x,y,z). Partial derivatives in (2.1) depends on two or more indices. These derivatives are the components, in P = 0, (x,y,z), of the various susceptibility tensors. In particular, first order derivatives, which depend on two indices, are 3 = 9 in total and are the components of the linear or 1st order susceptibility tensor, second order derivatives... 

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