Crystal susceptibility tensor

The equation for the ellipsoid of the crystal susceptibility tensor referred to an orthogonal system of coordinates. V. y. z may bo written as... 

CS. 11,20, 21 X=Xi -Xi with thenb plane vertical (Z lying along the unique fourfold crystal axis c, Xi in the ab plane) anisotropies and K measured in 10 emu/mole molecular susceptibility tensor is identical to crystal susceptibility tensor (K, =Z, X, =z,). The TIP corrected /Cy and X I obey the Curie-Weiss law with 0p =3K, 0p = lOK... 

Most of the four above-mentioned properties for Raman spectra can be explained by using a simple classical model. When the crystal is subjected to the oscillating electric field = fioc " of the incident electromagnetic radiation, it becomes polarized. In the linear approximation, the induced electric polarization in any specific direction is given by Pj = XjkEk, where Xjk is the susceptibility tensor. As for other physical properties of the crystal, the susceptibility becomes altered because the atoms in the solid are vibrating periodically around equilibrium positions. Thus, for a particular... 

One result of studying nonlinear optical phenomena is, for instance, the determination of this susceptibility tensor, which supplies information about the anharmonicity of the potential between atoms in a crystal lattice. A simple electrodynamic model which relates the anharmonic motion of the bond charge to the higher-order nonlinear susceptibilities has been proposed by Levine The application of his theory to calculations of the nonlinearities in a-quarz yields excellent agreement with experimental data. 

The crystal field does, however, have a dramatic effect on the magnetic anisotropy of lanthanide complexes. For complexes of less than cubic symmetry the three principal values of the susceptibility tensor are unequal. For uniaxial symmetry, Xx — Xy Xz and for biaxial symmetry Xx Xy Xz- Very extensive studies632-640 have been carried out on the single crystal susceptibilities of the D3d lanthanide hexakis(antipyrene) triiodides over the temperature range 80-300 K, and crystal field parameters were obtained. This crystal field-induced anisotropy is responsible for the effectiveness of lanthanide complexes as NMR shift reagents, and single crystal anisotropies of lanthanide complexes have been determined in this connection also.563... 

Fig. 9. Relationship of two symmetry-related molecular susceptibility tensors to monoclinic crystal axes. (b) Definitions of principal crystal susceptibilities in monoclinic crystals.

No effective method for the determination of the complete susceptibility tensors of triclinic crystals was available. Many papers had been published on this subject, yet, surprisingly, no reliable technique had been devised. 

The principal molecular susceptibilities, calculated as the penultimate step in reproducing the experimental crystal susceptibilities, were found to lie far from any M-L bond directions or from any suitable bisectors of these vectors nor did the magnitudes of the principal molecular susceptibilities describe anything like an axially symmetric susceptibility tensor. It is clear why so much difficulty was experienced earlier in the interpretation of the crystal susceptibilities. 

Obviously, such an isotropic model does not apply longer to Ba2CaCoFe2Fi4, where the presence of Co2+ ions in octahedral sites causes the susceptibility tensor to remain anisotropic even above TN, as it may be seen on the single crystal susceptibility measurements given in Fig. 24 [26],... 

The hrst suffix of each tensor component gives the row and the second the column in which the component appears. The xis term, for example, measures the component of the polarization parallel to X2 (usually the y direction in a Cartesian coordinate system) when a field is applied parallel to X3 (the z direction). The susceptibility tensor must conform to any restrictions imposed by crystal symmetry, see Eqs. 6.5-6.9. 

The oriented gas model was first employed by Chemla et al. [4] to extract molecular second-order nonlinear optical (NLO) properties from crystal data and was based on earlier work by Bloembergen [5]. In this model, molecular hyperpolarizabilities are assumed to be additive and the macroscopic crystal susceptibilities are obtained by performing a tensor sum of the microscopic hyperpolarizabilities of the molecules that constitute the unit cell. The effects of the surroundings are approximated by using simple local field factors. The second-order nonlinear response, for example, is given by... 

Both through-bond and pseudocontact contributions can be easily factorized into a series of products of two terms, each term depending either on the nucleus i (topologic and geometric location) or from the lanthanide j (electronic structure and crystal-field effects). For axial complexes, that is, possessing at least a three-fold axis as found in triple-stranded helicates, the molecular magnetic susceptibility tensor written in the principal magnetic axes system is symmetrical xx = mag-... 

In a seminal contribution, Bleaney demonstrated that when the crystal-field splitting of the ground multiplet is smaller or comparable to kT, a situation often met with lanthanide complexes, the anisotropic part of the axial paramagnetic susceptibility tensor originates from second-order effects and can be simply estimated by the product of magnetic constants Cj, characteristics of the electronic configuration of each lanthanide (i.e., Bleaney factor), multiplied by the second-rank crystal-field parameter Bq (Eq. (34), Bleaney, 1972). 

Valencia et have used proton pseudocontact chemical shift data in conjuction with x-ray crystal structures to determine the solution geometries of the lanthanide complexes with the ligand Py2N6Ac4, which contains four acetate pendant arms. The aqueous solution structures of diamagneticLa and Lu complexes were characterized by their COSY NMR spectra. The structures of the paramagnetic complexes were determined by fits of H pseudocontact chemical shift data and spin relaxation data to a model that assumed rhombic magnetic susceptibility tensors. 

If a nematic liquid crystal has negligible conductivity the results of Sections 11.2.1-11.2.5 for the Frederiks transition induced by a magnetic field may be directly applied to the electric field case. To this effect, it suffices to substitute H by E and all components of magnetic susceptibility tensor Xij hy correspondent components of dielectric permittivity tensor s,y. From the practical point of view the electrooptical effects are much more important and further on we discuss the optical response of nematics to the electric field. 

Munn et have attempted to relate the macroscopic refractive indices and susceptibility tensor for the iodoform and sulfur (Sg) crystals and the crystalline complex CHal.SSg to the molecular polarizabilities and hyperpolarizabilities. The attempt is only partially successful and illustrates the difficulties that will be encountered in projects such as that described in ref. 246. Penhuis and Muim have performed calculations of the linear and non-Unear optical properties of layers of Langmuir-Blodgett films and find that the internal field varies little after the first layer. They have investigated Langmuir-Blodgett... 

---