Rigid lattices

Filter aids as well as flocculants are employed to improve the filtration characteristics of hard-to-filter suspensions. A filter aid is a finely divided solid material, consisting of hard, strong particles that are, en masse, incompressible. The most common filter aids are applied as an admix to the suspension. These include diatomaceous earth, expanded perlite, Solkafloc, fly ash, or carbon. Filter aids build up a porous, permeable, and rigid lattice structure that retains solid particles and allows the liquid to pass through. These materials are applied in small quantities in clarification or in cases where compressible solids have the potential to foul the filter medium. 

The simplest picture of a metallic conductor is one where we have a rigid lattice of metal (M) atoms, each of which has lost one or more electrons to form a surrounding sea of electrons. 

Figure 12.1 shows a slice through such a solid the cations are to be thought of as a rigid lattice, and the electrons form a gas. I have deliberately drawn the cations as large objects for two reasons. First, the very early models such as that due to Drude tried to treat the electron sea as a perfect gas. It was eventually recognized that the electrons would collide with the cations and with each other an uncomfortable number of times. In any case, many of the predictions of the Dmde model turned out to be demonstrably flawed. 

Similarly, all points within a metal, which consists of an ordered rigid lattice of metal cations surrounded by a cloud of free electrons, are electrically neutral. Transport of charge through a metal under the influence of a potential difference is due to the flow of free electrons, i.e. to electronic conduction. The simultaneous transport of electrons through a metal, transport of ions through a solution and the transfer of electrons at the metal/solution interfaces constitute an electrochemical reaction, in which the electrode at which positive current flows from the solution to the electrode is the cathode (e.g. M (aq.) + ze M) and the electrode at which positive flows from it to the solution (e.g. M - M (aq.) -)- ze) is the anode. 

The Mossbauer effect can only be detected in the solid state because the absorption and emission events must occur without energy losses due to recoil effects. The fraction of the absorption and emission events without exchange of recoil energy is called the recoilless fraction, f. It depends on temperature and on the energy of the lattice vibrations /is high for a rigid lattice, but low for surface atoms. 

Thus, since intramolecular bonding interactions in the solid are much stronger than relatively weak i n termo1ecu1 ar van der Vaals interactions, each molecular unit is essentially an independent source of nonlinear response, arrayed in an acentric cystal structure, and coupled to its neighbors mainly through weak local fields. In the rigid lattice. gas approximation, the macroscopic susceptibility X is expressed as... 

The Debye temperature characterizes the rigidity of the lattice it is high for a rigid lattice but low for a lattice with soft vibrational modes. The mean squared displacement of the atom, , can be calculated in the Debye model and depends on the mass of the vibrating atom, the temperature and the Debye temperature. 

The Hamiltonian of a rigid lattice of interacting spins placed in a large magnetic field H0 directed along z is... 

Figure 1.10 (a) The absorption and emission energies for a two-level system (rigid lattice), (b) The absorption and emission energies showing the Stokes shift (vibrating lattice). 

A correction to the rigid lattice approximation is to write the potential as a contribution commensurate with the lattice vectors as above, and to add an additional term which depends on the displacement of substrate atoms from... 

Finally, the symmetry constraint can be removed by considering a pair sum over substrate atoms as a single contribution to the many-body energy. For example, the periodic contribution of the substrate can be replaced by a sum of contributions from each individual substrate atom . This allows the study of the eflect of features such as amorphous surfaces, steps and defects on surface reactivity, while still retaining a potential derived from a rigid lattice. These types of potentials, however, can become time consuming in their evaluation, and can therefore be inconvenient for use in large-scale computer simulations. 

Hutner R. A., Rittner E. S., and Du Pre F K. (1949). Concerning the work of polarization in ionic crystals of the NaCl type, II Polarization around two adjacent charges in the rigid lattice. J. Chem. Phys., 17 204-208. 

The effect of atomic motion in the solid state on nuclear resonance line width is illustrated by the behavior of Na resonance from NaCl as a function of temperature 97). In Fig. 9 is shown the variation of the Na line width with temperature for pure NaCl and NaCl doped with an atomic fraction concentration of 6 X 10 of CdCU. As discussed in Section II,A,2 the low-temperature, rigid-lattice line width will narrow when the frequency of motion of the nuclei under observation equals the line width expressed in sec.-. The number of vacancies present should be equal to the concentration of divalent impurities and the jump frequency of Na+ is the product of the atomic vacancy concentration and the vacancy jump frequency... 

Computer simulation in space takes into account spatial correlations of any range which result in Intramolecular reaction. The lattice percolation was mostly used. It was based on random connections of lattice points of rigid lattice. The main Interest was focused on the critical region at the gel point, l.e., on critical exponents and scaling laws between them. These exponents were found to differ from the so-called classical ones corresponding to Markovian systems irrespective of whether cycllzatlon was approximated by the spanning-tree... 

Order-disorder-type ferroeiectrics where a discrete symmetry group is broken due to the ordering of the ions in a rigid lattice potential (e.g., KH2PO4). 

Usually it is assumed that tc is the only temperature-dependent variable in Eq. 9. This might be the case for an order-disorder type rigid lattice model, where the only motion is the intra-bond hopping of the protons, since the hopping distance is assumed to be constant and therefore also A and A2 are constant. This holds, however, only for symmetric bonds. Below Tc the hydrogen bonds become asymmetric and the mean square fluctuation amplitudes are reduced by the so-called depopulation factor (l - and become in this way temperature-dependent also. The temperature dependence of tc in this model is given by Eq. 8, i.e. r would be zero at Tc, proportional to (T - Tc) above Tc and proportional to (Tc - T) below Tc. 

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