Crystal dipole sums

Figure 6-4. Qualitative energy level diagram of the 1 Bu excinm band structure of T<, at A =0 derived by the Ewald dipole-dipole sums for excitation light propagating along the a crystal axis.

The selection of the solvent is based on the retention mechanism. The retention of analytes on stationary phase material is based on the physicochemical interactions. The molecular interactions in thin-layer chromatography have been extensively discussed, and are related to the solubility of solutes in the solvent. The solubility is explained as the sum of the London dispersion (van der Waals force for non-polar molecules), repulsion, Coulombic forces (compounds form a complex by ion-ion interaction, e.g. ionic crystals dissolve in solvents with a strong conductivity), dipole-dipole interactions, inductive effects, charge-transfer interactions, covalent bonding, hydrogen bonding, and ion-dipole interactions. The steric effect should be included in the above interactions in liquid chromatographic separation. 

A four-pulse DEER measurement of the distance between two tyrosyl radicals on the monomers that make up the R2 subunit of E. coli ribonucleotide reductase gave a point-dipole distance of 33.1 A, which is in good agreement with the X-ray crystal structure.84 Better agreement between the calculated and observed dipolar frequency could be obtained by summing contributions from distributed... 

From Eq, (1) it is clear that a model of crystal polarization that is adequate for the description of the piezoelectric and pyroelectric properties of the P-phase of PVDF must include an accurate description of both the dipole moment of the repeat unit and the unit cell volume as functions of temperature and applied mechanical stress or strain. The dipole moment of the repeat unit includes contributions from the intrinsic polarity of chemical bonds (primarily carbon-fluorine) owing to differences in electron affinity, induced dipole moments owing to atomic and electronic polarizability, and attenuation owing to the thermal oscillations of the dipole. Previous modeling efforts have emphasized the importance of one more of these effects electronic polarizability based on continuum dielectric theory" or Lorentz field sums of dipole lattices" static, atomic level modeling of the intrinsic bond polarity" atomic level modeling of bond polarity and electronic and atomic polarizability in the absence of thermal motion. " The unit cell volume is responsive to the effects of temperature and stress and therefore requires a model based on an expression of the free energy of the crystal. 

The total pyroelectric response at constant pressure (or stress) is the sum of two contributions. Primary pyroelectricity, which is due to changes in the magnitude of the dipole oscillation with temperature, accounts for only about 9% of the total response of the crystal at 300 K. The remaining overwhelming contribution is due to secondary pyroelectricity—the coupling of the piezoelectric response and thermal expansion. 

Here the d are the analogs of the a in the expression for the induced dipole-dipole interaction and have been discussed by Margenau (89, 90). Ss and Ss are sums of the type l/(ri/ro) over all like and all unlike ions in the crystal lattice. They have been computed by Lennard-Jones and Ingham (8 ) for various lattice types and are given in Table IX. The actual value of the induced dipole-quadrupole term is about one tenth th t of the induced dipole-dipole term. 

Working out dipole lattice sums, for any wave vector k, became a simpler matter soon after, with computers of the IBM 650 generation, in about 1956-7. Stuart Walmsley (now Reader in Chemistry, University College London) came as my first PhD student after I returned to University College from the University of Sydney in 1956. He had a magic touch with the early computers, and made the first dipole-dipole lattice summations on simple aromatic crystals [220]. Dipole-octupole and octupole-octupole interaction sums were first calculated by Thirunamachandran [221] using the Ferranti Mercury Computer and later extended to quadrupole couplings. These sums are not shape dependent, but were of formidable difficulty at the time. 

In 1996, Munn extended the microscopic theory of bulk second-harmonic generation from molecular crystals to encompass magnetic dipole and electric quadrupole effects [96] and included all contributions up to second order in the electric field or bilinear in the electric field and the electric field gradient or the magnetic field. This was accomplished by replacing the usual polarization of Refs. 72 and 84 by an effective polarization as well as by defining an effective quadrupole moment. Consequently, the self-consistently evaluated local electric field and electric field gradient were expressed in terms of various molecular response coefficients and lattice multipole tensor sums (up to octupole). In this... 

In the following section we derive the sum rule for excitonic transitions in a crystal which can be considered as analogous to the sum rule for dipole transitions in atoms. 

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