Spherical product operators

Table 2.5 Conversion of spherical product operators, spherical coordinate product operators and the basic operations of these operators [2.31].

Coherence transfer pathways (CT pathway) fall in the domain of spherical product operators instead of CARTESIAN operators. Before proceeding any further it is recommended to a necomer to read section 2.2.2 and for addition information references [2.20 - 2.31]. To illustrate the use of coherence transfer pathways in coherence selection, three pulse sequences will be examined. 

We consider the coherence evolution of uncoupled spins 7=1/2 during the pulsed gradient Hahn spin echo sequence schematically shown in Fig. 2a. A suitable basis for the treatment is the spherical product operator formalism. Explanations, definitions, and rules of the spin operator formahsm needed in this context can be found in Ref [2]. Times just before and immediately after RF (radio frequency) and field gradient pulses will be indicated by minus and plus signs, respectively. 

One more thing we can do with spherical operators We can easily derive the expressions given in Chapter 8 for pure ZQC and DQC. Start with the spherical product I+S+ and... 

If the scalar product is formed from spherical tensor operators which both act on the same inner part of a coupled scheme, it is intuitively obvious that... 

The spherical product is continuously withdrawn from the reactor and separated from the unreacted monomer gas in a bag filter separator operating at intermediate... 

The internal interactions, symbolically designated as X), which provides the wealth of information available from NMR, but which are also responsible for broadening of solid-state spectra of protons in solids may be written as a constant, Ca, times a product of coordinate space and spin space operators. In terms of the coordinate space irreducible spherical-polar tensor operators Rlm(0. < ). and the spin space irreducible spherical tensor operators... 

A detailed analysis of the effect of the cubic part of the acceptor Hamiltonian on the binding energies was studied in [4] in the strong s-o coupling limit. This cubic part can be written in terms of vector products of the spherical tensor operators P(2 > and j(2) already used for the spherical part of the Hamiltonian (the relevant properties of these tensors are given in Appendix E) and is ... 

Taking into account all the relevant criteria spin-1/2 nuclei in the liquid phase can generally be described using CARTESIAN, spherical and shift product operators as shown in Table 2.3. The spherical operators are not shown because they can be easily derived from the shift operators, see Table 2.5. 

Since many of the operators that appear in the exact Hamiltonian or in the effective Hamiltonian involve products of angular momenta, some elementary angular momentum properties are summarized in the next section. Matrix elements of angular momentum products are frequently difficult to calculate. A tremendous simplification is obtained by working with spherical tensor operator components and, in this way, making use of the Wigner-Eckart Theorem (Section 3.4.5). A more elementary but cumbersome treatment, based on Cartesian operator components, is presented in Section 2.3. 

These are indeed spherical tensor operators they are obviously related to step operators by means of = +A/V2. The typical group theoretical route for constructing bilinear forms respecting the SO(3) symmetry, contained in U(4), is to make use of tensor products of boson operators. 

The contents of the large square brackets must thus represent the reduced matrix element in accordance with (52). Since the operator does not contain a spin part, we may use (48) and interpret the interaction as a tensor product with a unit scalar operating in spin space. The real space operator is, however, not yet in the form of a spherical tensor operator, but using (46) we arrive at... 

The scalar product of 2 irreducible spherical tensor operators of rank k is defined by... 

C and 5 kg/cm pressure (see Molecularsieves). Selectivity for toluene and xylenes peaks at 550°C but continues with increasing temperature for hensene. The Cyclar process (Fig. 6) developed joindy by BP and UOP uses a spherical, proprietary seoHte catalyst with a nonnoble metallic promoter to convert C or C paraffins to aromatics. The drawback to the process economics is the production of fuel gas, alow value by-product. BP operated a... 

The only difference is that a(0) is now an operator acting in jm) space of angular momentum eigenfunctions. This space consists of an infinite number of states, unlike those discussed above which had only four. This complication may be partly avoided if one takes into account that the scalar product in Eq. (4.55) does not depend on the projection index m. From spherical isotropy of space, Eq. (4.55) may be expressed via reduced matrix elements (/ a(0 /) as follows... 

Many engineering operations involve the separation of solid particles from fluids, in which the motion of the particles is a result of a gravitational (or other potential) force. To illustrate this, consider a spherical solid particle with diameter d and density ps, surrounded by a fluid of density p and viscosity /z, which is released and begins to fall (in the x = — z direction) under the influence of gravity. A momentum balance on the particle is simply T,FX = max, where the forces include gravity acting on the solid (T g), the buoyant force due to the fluid (Fb), and the drag exerted by the fluid (FD). The inertial term involves the product of the acceleration (ax = dVx/dt) and the mass (m). The mass that is accelerated includes that of the solid (ms) as well as the virtual mass (m() of the fluid that is displaced by the body as it accelerates. It can be shown that the latter is equal to one-half of the total mass of the displaced fluid, i.e., mf = jms(p/ps). Thus the momentum balance becomes... 

The algebra of U(4) can be written in terms of spherical tensors as in Table 2.1. This is called the Racah form. The square brackets in the table denote tensor products, defined in Eq. (1.25). Note that each tensor operator of multipolarity X has 2X+ 1 components, and thus the total number of elements of the algebra is 16, as in the uncoupled form. 

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