Outer direct product

Such localized states as under discussion here may arise in a system with local permutational symmetries [Aa] and [AB], If [Aa] + [S] and [Ab] = [5], the outer direct product [Aa] 0 [AB] gives rise to a number of different Pauli-allowed [A], If the A and B subsystems interact only weakly, these different spin-free [A] levels will be closely spaced in energy. The extent of mixing of these closely spaced spin-free states under the full Hamiltonian, H = HSF + f2, may then be large. Thus, systems which admit a description in terms of local permutational symmetries may in some cases readily undergo spin-forbidden processes, such as intersystem crossing. 

The set ( bj) therefore closes. The other necessary group properties are readily proved and so G is a group. Direct product (DP) without further qualification implies the outer direct product. Notice that binary composition is defined for each group (e.g. A and B) individually, but that, in general, a multiplication rule between elements of different groups does not necessarily exist unless it is specifically stated to do so. However, if the elements of A and B obey the same multiplication rule (as would be true, for example, if they were both groups of symmetry operators) then the product at bj is defined. Suppose we try to take (a,-, bj) as a, bj. This imposes some additional restrictions on the DP, namely that... 

The expression for the intermolecular potential must satisfy two symmetry requirements. First, it must be invariant if we rotate the molecular frame of either of the two molecules through specific Euler angles wg that correspond with a symmetry element of the molecule in question. This means that our basis must be invariant under rotations of the outer direct product group Gp GP-, where GP is the symmetry group of molecule P and GP- that of molecule F Acting with the projection operator of the totally symmetric irreducible representation of the group GP (of order GP),... 

To avoid redundancies introduced by the outer DP A 0 A of a group with itself (Exercise 1.6-3), the inner direct product A E3 A is defined by... 

Bark can roughly be divided into living inner bark or phloem and dead outer bark or rhytidome. The tissues of the bark substance are formed either by primary or secondary growth. The primary growth means direct production of embryonal cells at the growing points of the stem apex and their further development to primary tissues. Epidermis, cortex, and primary phloem are primary tissues (Fig. 6-1). The formation of secondary tissues... 

Pour typical weU patterns for contaminant plume containment are described in Ref. 16. The first is a pair of injection-production weUs. The second is a line of downgradient pumping weUs. The third is a pattern of injection-production weUs around the boundary of a plume. The fourth, the double-cell system, uses an inner ceU and outer recirculation ceU, with four ceUs along a line bisecting the plume in the direction of flow. Two other methods of plume containment are bio filters and a fuimel-and-gate system, which are described in the in bioremediation section. 

While it is inherently probable that product formation will be most readily initiated at sites of effective contact between reactants (A IB), it is improbable that this process alone is capable of permitting continued product formation at low temperature for two related reasons. Firstly (as discussed in detail in Sect. 2.1.1) the area available for chemical contact in a mixture of particles is a very small fraction of the total surface (and, indeed, this total surface constitutes only a small proportion of the reactant present). Secondly, bulk diffusion across a barrier layer is usually an activated process, so that interposition of product between the points of initial contact reduces the ease, and therefore the rate, of interaction. On completion of the first step in the reaction, the restricted zones of direct contact have undergone chemical modification and the continuation of reaction necessitates a transport process to maintain the migration of material from one solid to a reactive surface of the other. On increasing the temperature, surface migration usually becomes appreciable at temperatures significantly below those required for the onset of bulk diffusion within a product phase. It is to be expected that components of the less refractory constituent will migrate onto the surfaces of the other solid present. These ions are chemisorbed as the first step in product formation and, in a subsequent process, penetrate the outer layers of the... 

Like positron emission, electron capture is never observed directly. However, after electron capture, the product atom is missing one of its 1 J electrons, as shown schematically in Figure 22-6b. When an electron from an outer orbital occupies this vacancy in the 1 orbital, a photon is emitted whose energy falls in the X-ray region of the... 

This approach is suggestive of the Hartree approximation of atomic and molecular physics. The outer-shell interactions are important, but complicated because of the correlations involved when they are considered directly. The suggested response to this difficulty is to treat these effects as uncorrelated - as a product contribution to the distribution - but with the product factors optimized by (9.53) to be consistent with the basic data. 

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