Equivalent operator treatment

We have already seen that the crystal field potential can be rather simply expressed through a set of radial parameters and the irreducible tensor operators Tk m which refer to a certain, symmetry-predetermined combination of spherical harmonic functions (see Table 8.12). Such an expression can be written in several equivalent forms 

Energy levels of free ions in weak octahedral and tetrahedral crystal fields 

Here y m are the spherical harmonic functions Q m = yj47r/(2k + 1) y, m is the Racah tensor operator = rk Ykm is the irreducible tensor operator Pk m (not to be confused with the Legendre polynomials) are unnormalised homogeneous polynomials of Cartesian coordinates proportional to the function rk Ykm + Yk m) Ok are referred to as equivalent operators which are constructed of only the angular momentum operators. 

The passage from the spherical harmonic functions and/or their combinations to the equivalent operators is based on the Wigner-Eckart theorem and consequently on the replacement theorem. The reduction of a matrix element to the reduced matrix element and a Clebsch-Gordan coefficient 

Therefore, instead of the irreducible tensor components, a proper combination of the angular momentum operators (Jz, J+, J ) is applicable. However, the expansion coefficients (the potential constants) need to be redefined to account for the proportionality factor a (k,j). A set of equivalent operators is compiled in Table 8.17. It is quite practical to handle the equivalent operators since they can be easily constructed by matrix multiplications with the help of computers. 

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Winter et al. [41] could demonstrate that the intra-operative treatment of retinal detachments with PFCLs, followed by an exchange to intraocular gas, resulted in total coating of the retinal surface with the fatal effect of complete sealing, creating an interruption of essential transretinal exchange processes. Equivalent scenarios have to be taken into consideration in the situation of a completely filled eye or if the droplets of an endotamponade media show a non-spherical shape. 

Using the operator equivalent technique of Stevens (32), and following Elliott and Stevens (33), one may also express the splittings in terms of the better known A parameters (c.f. also Abragam and Bleaney (34)), which are in turn simply related to the Bk parameters which occur in the tensor operator treatment of crystal field splittings. Using the relationship given by Wyboume (35), the crystal field and Bk parameters are found to be connected by the expressions... 

If the system is at equilibrium then F is at a minimum with respect to such arbitrary redistribution. A comparison of (2.97) and (2.98) shows that 8F=0 is consistent with p(z) = p, a chemical potential which is constant throughout the system. Minimizing F and imposing constancy on p are equivalent operations in quasi-thermodynamic treatments. 

Vacuum Treatment. Milk can be exposed to a vacuum to remove low boiling substances, eg, onions, garlic, and some silage, which may impart off-flavors to the milk, particularly the fat portion. A three-stage vacuum unit, known as a vacreator, produces pressures of 17, 51—68, and 88—95 kPa (127, 381—508, and 660—711 mm Hg). A continuous vacuum unit in the HTST system may consist of one or two chambers and be heated by Hve steam, with an equivalent release of water by evaporation, or flash steam to carry off the volatiles. If Hve steam is used, it must be cuUnary steam which is produced by heating potable water with an indirect heat exchanger. Dry saturated steam is desired for food processing operations. 

A fully relativistic treatment of more than one particle has not yet been developed. For many particle systems it is assumed that each electron can be described by a Dirac operator (ca ir + p mc ) and the many-electron operator is a sum of such terms, in analogy with the kinetic energy in non-relativistic theory. Furthermore, potential energy operators are added to form a total operator equivalent to the Hamilton operator in non-relativistic theory. Since this approach gives results which agree with experiments, the assumptions appear justified. 

Passivation under a wide range of operating conditions and in all boiler designs is a goal of all BW treatment programs. It is achieved through the formation of a stable, continuous, uniform, self-limited, and nonporous magnetite film. (This objective also applies to the maintenance of cuprous oxide, the copper equivalent of a film of passive iron.)... 

Most kinetic experiments are run in batch reactors for the simple reason that they are the easiest reactor to operate on a small, laboratory scale. Piston flow reactors are essentially equivalent and are implicitly included in the present treatment. This treatment is confined to constant-density, isothermal reactions, with nonisothermal and other more complicated cases being treated in Section 7.1.4. The batch equation for component A is... 

The various response tensors are identified as terms in these series and are calculated using numerical derivatives of the energy. This method is easily implemented at any level of theory. Analytic derivative methods have been implemented using self-consistent-field (SCF) methods for a, ft and y, using multiconfiguration SCF (MCSCF) methods for ft and using second-order perturbation theory (MP2) for y". The response properties can also be determined in terms of sum-over-states formulation, which is derived from a perturbation theory treatment of the field operator — [iE, which in the static limit is equivalent to the results obtained by SCF finite field or analytic derivative methods. 

Some aspects of reactor behavior are developed in Chapter 5, particularly concentration-time profiles in a BR in connection with the determination of values of and k2 from experimental data. It is shown (see Figure 5.4) that the concentration of the intermediate, cB, goes through a maximum, whereas cA and cc continuously decrease and increase, respectively. We extend the treatment here to other considerations and other types of ideal reactors. For simplicity, we assume constant density and isothermal operation. The former means that the results for a BR and a PFR are equivalent. For flow reactors, we further assume steady-state operation. 

The microbial activity of wastewater under anoxic conditions is lower compared with aerobic conditions (Abdul-Talib et al., 2001). This is important to consider, because a low nitrate uptake rate (NUR) compared with the oxygen uptake rate (OUR) in units of electron equivalents means a reduced transformation rate of the most biodegradable fractions of the organic matter. As mentioned under the point on injection of air, this may have implications in terms of treatment. Furthermore, a relatively low NUR value also has operational advantages because of a reduced demand for nitrate to suppress sulfide formation. 

The hfs (or quadrupole) tensors of geometrically (chemically) equivalent nuclei can be transformed into each other by symmetry operations of the point group of the paramagnetic metal complex. For an arbitrary orientation of B0 these nuclei may be considered as nonequivalent and the ENDOR spectra are described by the simple expressions in (B 4). If B0 is oriented in such a way that the corresponding symmetry group of the spin Hamiltonian is not the trivial one (Q symmetry), symmetry adapted base functions have to be used in the second order treatment for an accurate description of ENDOR spectra. We discuss the C2v and D4h covering symmetry in more detail. 

As mentioned previously (see p. 237), the 1,2,3-benzotriazinones function as masked diazonium compounds, and the so-called Pologenowe dyes developed in Poland are operated on this principle and have been in use there since 1959. There are three major advantages associated with the use of 1,2,3-benzotriazinones as azo-dye equivalents (1) The equilibrium that theoretically exists between the heterocyclic system and the diazonium compound normally lies completely to the side of the cyclic compound. Opening of the heterocyclic ring can be easily accomplished thermally, and hence dye formation can be induced by heat treatment processes. (2) Many 1,2,3-benzotriazinones are reasonably water soluble and the sodium salts even more so consequently, complete permeation of fabric fibres by the dye precursors immediately prior to generation of the dye can be ensured. (3) The colors produced are fast, and some of them can be made highly water insoluble. ... 

Operationally, this is equivalent to the method of using two simultaneous one-sided tests to test the (composite) null hypothesis that the treatment difference is outside the equivalence margins versus the (composite) alternative hypothesis that the treatment difference is within the margins. ... 

According to the vendor, capital costs of an Ultrasorption system is about 60% of an equivalent GAC system. Operational costs and disposal of the treatment media are approximately half of those associated with GAC units. Manpower requirements are the same in both cases (D147SOM,... 

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