Crystal orthogonality

White, shiny crystals orthogonal crystal structure density 1.96 g/cm sublimes at 150°C with decomposition vapor pressure 0.1 torr at 20°C insoluble in water soluble in benzene and paraffin od shghtly soluble in ether. 

The purpose of such a device consists in changing the orientation of the polarization plane of a beam by 90°. That means the initial Stokes vector 1,1,0,0 of a horizontally polarized beam becomes 1,-1,0,0 after passing through the retarder. Retarders are most often birefringent crystals of definite thickness. If the fast and slow axes of such a crystal orthogonal to each other are crossed at 45° with respect to the polarization plane, the retarder rotates the latter by 90°. The Stokes-Mueller transformation corresponding to this experiment should be ... 

It appears from the above discussion that the most satisfactory approach to the quantum mechanical calculation of lattice energies is that developed by Yamashita, in which the parameters of the outer wave functions of the ions are adjusted by a variational method to minimize the total energy of the crystal. Orthogonalization of the simple free ion wave functions seems to produce a result rather worse than that achieved by ignoring the correction. No doubt with the availability of electronic computers Yamashita s method will be extended to crystals in addition to LiF, where it may be necessary to adjust the wave functions of both the ions by a variational method, to allow for the effect of the crystal field. This will produce an exceedingly tedious calculation. Yamashita (1S2) has also used the method described above to show that the 0 ion is stable in the MgO crystal, though not in the gas phase. 

A nematic liquid crystal cell, based on Merck Licrilite E202, was used in these experiments. The rod like liquid crystal molecules preferentially aligned themselves with each other and to an alignment surface in the liquid crystal device. Any birefringence. An, was given as the difference between the two orthogonal refractive indices. As a consequence, any resulting... 

An orientational order parameter can be defined in tenns of an ensemble average of a suitable orthogonal polynomial. In liquid crystal phases with a mirror plane of symmetry nonnal to the director, orientational ordering is specified. 

If we compare with figure C2.2.I I, we can see that this defonnation involves bend and splay of the director field. This field-induced transition in director orientation is called a Freedericksz transition [9, 106, 1071. We can also define Freedericksz transitions when the director and field are both parallel to the surface, but mutually orthogonal or when the director is nonnal to the surface and the field is parallel to it. It turns out there is a threshold voltage for attaining orientation in the middle of the liquid crystal cell, i.e. a deviation of the angle of the director [9, 107]. For all tliree possible geometries, the threshold voltage takes the fonn [9, 107]... 

Institute of Technology (MIT) [193]. Molecules were represented as line drawings on a homemade display (an oscilloscope (Figure 2-122). In addition, the system had diverse peripherals with many switches and buttons which allowed the modification of the scene. The heart of the. system was the. so-called Crystal Ball" which could rotate the molecule about all three orthogonal axes. This prototype cost approximately two million US dollars. 

Figure 6-3. Top Structure of the T6 single crystal unit cell. The a, b, and c crystallographic axes are indicated. Molecule 1 is arbitrarily chosen, whilst the numbering of the other molecules follows the application of the factor group symmetry operations as discussed in the text. Bottom direction cosines between the molecular axes L, M, N and the orthogonal crystal coordinate system a, b, c. The a axis is orthogonal to the b monoclinic axis.

The form of the functions may be closely similar to that of the molecular orbitals used in the simple theory of metals. If there are M interatomic positions in the crystal which might be occupied by any one of the N electron-pair bonds, then the M functions linear aggregates that approximate the solutions of the wave equation with inclusion of the interaction terms representing resonance. This combination can be effected with use of Bloch factors ... 

In this section, we will present the crystal structures of chiral mesogenic compounds exhibiting ferroelectric liquid crystalline phases which are listed in Table 18 [152-166]. Moreover, four compounds of the list show antiferroelectric properties and two compounds form only orthogonal smectic phases. The general chemical structures of the investigated chiral compounds are shown in Fig. 27. 

Smith (91) reported an X-ray crystal structure of a zinc porphyrin polymer (77, Fig. 32) where, unusually, the coordination bond is between a nitro group and the zinc center. The tetranitroporphyrin is highly substituted, and the resulting steric hindrance causes the macrocycle to be noticeably distorted. Adjacent porphyrin planes in the polymer are almost orthogonal. However, there is no evidence of polymerization in solution, and the nitro-zinc interaction is probably too weak to maintain this structure outside the solid state. 

It is possible to ensure that the orbitals we extract for one molecule in the crystal are orthogonal to all other orbitals on all other molecules in the crystal. If this is the case, a determinant wave function can be constructed for the entire crystal. To ensure the required orthogonality, a projection operator is used ... 

Fig. 2. Thermal ellipsoid diagrams for X-ray crystal structures of 1. Left, orange unsolvated form right, yellow toluene-containing crystals (1-C7H8). The toluene, not shown, occupies a position between the two mesityl rings which are nearly orthogonal to the Si=Si bond.4151...

Changing of the flexible scissor-like element, as in 7, to an orthogonal and rigid version of this element, as in 22, reduces the activity of inclusion formation to a certain degree. Nevertheless very different guest molecules are readily accommodated in the crystal lattice of 22, they are proton donors (ethanol, 2-propanol) 47, H-bridge acceptors (dimethylformamide, dioxane), or benzene as an unpolar solvent48 ... 

It is important to stress that ATR absorbance is strongly affected by the sample/crystal contact. Quantitative results are thus difficult to obtain even if the contact is maintained during the sample rotation that is required to record all four polarized spectra. A reference band that does not show significant dichroism is thus most often used to normalize the polarized absorbances in order to obtain quantitative data. For instance, the 1,410 cm-1 band of PET has often been chosen for that purpose, not only for ATR studies but also for specular reflectance (see below) and even transmission studies when the sample thickness is not uniform. It was shown that an appropriate normalization is possible even if no such reference band is available, by using a combination of two bands with orthogonal dichroism [34]. When performing ATR experiments, one should also make certain that the applied pressure does not create artifacts by affecting the structure of the sample. 

Among various superconductors, compounds with the A15 (Cr3Si) crystal structure have the highest critical temperatures. This crystal structure has a simple relationship with the Ll2 structure (Ito and Fujiwara, 1994) as illustrated in Figure 8.9. When the unit cells are aggregated, the face-centered pairs of atoms form uniform chains of transition metal atoms along three orthogonal directions. This feature may be related to the relatively stable superconductivity in compounds with this structure. 

Figure 9.3 Cluster of unit cells of the cesium chloride crystal structure. This figure shows that ions of the same sign in this structure line up along the 100 directions. Thus the three rows are orthogonal to one another. Translation of a (100) plane of ions over its nearest (100) neighboring plane keeps ions of opposite sign adjacent to one another. This is also the case on the (110) planes, but the translation vector is V2 larger than for the the (100) planes.

Figure 5.7 Magnetization (a) and susceptibility (b) of the Dy3 complex obtained on oriented single crystals and powdered samples. The Z direction is orthogonal to the Dy3 plane. The inset in the left panel shows the difference between the X and...

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