Separation internal/external modes

There are three translational dispersion curves and three rotational dispersion curves. In the absence of rotation-translation coupling these vibrations are also distinctly separate. However, external modes and internal modes of the same character will become mixed, the closer they are in frequency the greater will be the mixing. Atomic displacements are no longer controlled by exclusively internal or external forces but by both. Modes of a mostly internal (or mostly external) character now involve displacements of the centre of mass (or deformations of the molecular frame). This is a general feature of all molecular crystals but it is usual to ignore such complications when the frequency of the lowest... 

In molecular crystals, the separation between internal and external modes is of importance. Except for torsional oscillations in some types of molecules, the internal modes have much higher frequencies than the external modes. According to expressions such as Eqs. (2.51) and (2.58), the latter are then the dominant... 

Raman and IR spectra of polyphosphides clearly show the separation of internal and external modes (Figure 25). The external modes are strongly pressure and temperature dependent, whereas the internal cluster modes are nearly... 

Since the disjjersion in k space of the energy of internal modes (vibrons) is generally very small (on the order of a few wave numbers, but see Section IV C), the requirements of energy and wave-vector conservation make the decay of a vibron into two (or more) other vibrons a highly unlikely event. The most generally observed decay channel is thus observed to be a mixed process, where the initial vibron interacts with another vibron and a low-frequency (disjjersed) lattice phonon. (When the frequency separation of internal and external modes becomes less well defined, decay can also involve two external modes.)... 

Qualitatively, the dynamic properties of different polyacene crystals are similar, since both the intermolecular van der Waals forces and also the molecular masses are to first order proportional to the number of C atoms per molecule. Therefore, for example all the sound velocities are of the same order of magnitude (Table 5.4). The same is true for the maximum frequencies of the optical phonons and of the intramolecular osdUations. A clear and general difference is however to be found between smaller and larger molecules in terms of the lowest frequencies of the intramolecular modes with increasing molecular mass, the frequency of the low-energy intramolecular vibrations shift towards lower values. These molecular excitations are either bending or torsional oscillations. This frequency shift can cause the gap and the strict separation of internal and external modes to become less sharp. 

The success of an ISPR process does not depend only on the chosen separation technique but also on the configuration of the bioreactor/separation units and mode of operation. Previous reviews have shown the various possible modes of operation (continuous, batch) and the use of a separation unit outside of the reactor or separation techniques that act right inside the fermenter [19,22,31]. Freeman and coworkers introduced a classification scheme for ISPR process based on batch/continuous operation and internal (within the reactor)/external (outside the reactor) removal of the product [3]. 

Polymer molecules are modeled as having two distinct sets of modes contributing to the partition function in the cell models. The two modes are internal and external modes. The internal modes are used to represent the internal motions of the molecules, and the intermolecular interactions are accounted for by the external modes. Prigogine and Kondepudi [12] proposed the conceptual separation of the two modes. The PVT properties of the polymer systems will be affected by the external modes. A polymer molecule is divided into r repeat units. Each repeat unit has 3 degrees of... 

The Prigogine square-well cell model is based upon conceptual separation of internal and external modes for polymers separately accounted for in the partition... 

In the crystal, the total number of vibrations is determined by the number of atoms per molecule, N, and the nmnber of molecules per primitive cell, Z, multiplied by the degrees of freedom of each atom 3ZN. In the case of a-Sg (Z =4, N =8) this gives a total of 96 vibrations ( ) which can be separated in (3N-6)—Z = 72 intramolecular or "internal" vibrations and 6Z = 24 intermo-lecular vibrations or lattice phonons ("external" vibrations). The total of the external vibrations consists of 3Z = 12 librational modes due to the molecular rotations, 3Z-3 = 9 translational modes, and 3 acoustic phonons, respectively. 

Common Cause or Common Mode Failure—Failure, which is the result of one or more events, causing coincident failures in multiple systems or on two or more separate channels in a multiple channel system, leading to system failure. The source of the common cause failure may be either internal or external to the systems affected. Common cause failure can involve the initiating event and one or more safeguards, or the interaction of several safeguards. 

In the case of crystaUine sohds, more than one equivalent structural unit may be present in the primitive cell. This results in sphttings of the fundamental vibrational modes of these units. In the case of many crystalline solid materials covalent units (e.g. oxo-anions for oxo-salts) are present, together with other groups bonded by ionic bonds (e.g. the cations in the oxo-salts). According to the above group approximation, the internal vibrations of the covalent units can be considered separately from their external vibrations hindered rotations and translations of the group that finally contribute to the lattice vibrations and to the acoustic modes of the unit cell) and those of the other units. The presence of a number of covalent structural units in the primitive cell, causes their internal modes to spHt... 

Qei and Qvtbrot denote electronic and rovibrational partition fimctions, respectively. In general, the contributions of the internal degrees of freedom of A and B cancel in g and gviiroXA)gv iro((B ), such that only contributions Irom the external rotations of A and B and the relative motion, summarized as "transitional modes", need to be considered. Under low temperature quantum conditions, these can be obtained by statistical adiabatic channel (SACM) calculations [9],[10] while classical trajectory (CT) calculations [11]-[14] are the method of choice for higher temperatures. CT calculations are run in the capture mode, i.e. trajectories are followed Irom large separations of A and B to such small distances that subsequent collisions of AB can stabilize the adduct. 

Sample extracts were analysed by capillary gas chromatography with electron capture detection using either Perkin-Elmer series 8310 or 8510 instruments. One microlitre aliquots were injected via a split/splitless injection system operated in the splitless mode. The components were separated on a 25 m WCOT fused silica capillary column (CP-Sil-5, internal diameter 0.22 pm, Chrompack, UK). The stationary phase was non-polar 100% dimethylpolysiloxane. The temperature program for the separation of chlorobenzenes was based upon the work of Lee et al (1986). Quantification was based on the method of internal or external standard. 

---