Pure Case

we consider pure BaTiOs. For a ternary system, there are two composition variables, for example, the mole fractions of Ti and O. When there are three sublattices, however, it is more convenient to choose the molecular ratio of the component oxides, BaO and Ti02 and the equivalent ratio of the nonmetalhc component to the metallic components [3, 4] or 

One may start by conjecturing the possible defects of the oxide from its structural and energetic considerations. For the system of perovskite structure, the interstitial defects may be ruled out and hence, the structure elements, the concentrations of whichneedtobeknown,maybetheirregularstructureelements VB3, V , Vo, e, h in addition to the regular structure elements Bal, Ti j, and Oq in terms of the Kroger-Vink notation. 

Letting [S denote the concentration (in number cm ) of the structure element S ([e ] = n, [h ] = p), it is possible to formulate all of the constraints, assuming an ideal dilute solution behavior of defects as  

the mass-action law constants are denoted as Kj j=Re, T, S, t), which may be represented as 

TaUe 10.1 Matrix of majority disorder types in the systems of BaTi03- The top left rectangle demarcated by thick solid lines is for the pure case this rectangle plus the rightmost column for the acceptor-doped case and the rectangle plus the bottom-most row for the donor-doped case. 

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Among the usual advantages of such expressions as Eq. (7-80) and (7-81), one is salient they show forth the invariance of p and w with respect to the choice of the basis functions, u, in terms of which p, a, and P are expressed. The trace, as will be recalled, is invariant against unitary transformations, and the passage from one basis to another is performed by such transformations. The trace is also indifferent to an exchange of the two matrix factors, which is convenient in calculations. Finally, the statistical matrix lends itself to a certain generalization of states from pure cases to mixtures, required in quantum statistics and the theory of measurements we turn to this question in Section 7.9. 

Suppose that a given physical system can be in any one, or in any linear combination of states , < > Each of them, or any linear combination of them, represents a pure case. To each of these j>t we now assign a probability pi, and, furthermore, a statistical matrix p. For instance, if 1 = alaua, then, p m = ajaj. To the mixture, which affirms the presence of with probability p, of a with probability p2, etc., we then allot the statistical matrix. 

Our theorem permits the following inference. The statistical matrix of every pure case in quantum mechanics is equivalent to an elementary matrix and can be transformed into it by a similarity transformation. Because p is hermitian, the transforming matrix is unitary. A mixture can, therefore, always be written in the diagonal form Eq. (7-92). 

This form, however, cannot satisfy Eqs. (7-93) because pfm = (Pl)z im> which equals plm only if pl is 1 or 0. If a statistical matrix were given in an unintelligible, scrambled form and one wanted to know whether it represented a pure case, one would need only to square it and determine if it reproduces itself. 

The structure of the reduced density matrix follows from the symmetry properties of the Hamiltonian. However, for this case the concurrence C iJ) depends on ij and the location of the impurity and not only on the difference i—j as for the pure case. Using the operator expansion for the density matrix and the symmetries of the Hamiltonian leads to the general form... 

Figure 5. Lx)cal entanglement given by the von Neumann entropy, Ey, versus V jt in the pure case.

The procedure was repeated for the density matrix matrix of unpolarized excited states. In Fig. 3 we display = O p (f) / ,0 = 0) 2 for j = 134, derived from the data of Fig. 1, at different times. As in the pure case of Figs. 2, our procedure is able to prefectly reconstruct the true density-matrix at all times considered. 

State-specific memory can be readily constructed for hypnosis that is, state-specific memory may not occur naturally for hypnosis, but it can be made to occur, if you tell a hypnotized subject he will remember everything that happened in hypnosis when he comes back to his ordinary state such will be his experience. On the other hand, if you tell a deeply hypnotized subject he will remember nothing of what went on during hypnosis or that he will remember certain aspects of the experience but not others, this will also be the case when he returns to his ordinary state, in any event he will recall the experiences the next time he is hypnotized. This is not a pure case of state-specific memory, however, because amnesia for hypnotic experiences in the waking state can be eliminated by a prearranged cue as well as by reinducing the hypnosis. 

It is interesting to note that even the more realistic model adhering to the Case II radial and axial drug release from a cylinder, (4.10), can be described by the power-law equation. In this case, pure Case II drug transport and release is approximated (Table 4.1) by the following equation ... 

If the above analysis is repeated for the orbital term in (10.151) we find an almost identical result, except that gs is replaced by 2gi. (because S = 1/2 and L = 1). Consequently for pure case (c) states we obtain the following simple results for the energies Ez,... 

Many of the observed levels have measured g-factors which are closer to the pure case (c) values than to any alternative pure coupling case. However there is extensive rotational electronic coupling which, in many instances, mixes the case (c) states case (e) is then a better limiting basis, as we shall see in due course. First we investigate the electric dipole transition probabilities for the Zeeman components, so that we can understand the pattern of lines illustrated in figure 10.73. 

Pure case (e) effective g-factors are readily calculated from these expressions, and effective g-factors for the final wave functions, expressed as linear combinations of case (e) functions, are also easy to calculate. 

Table 10.26. Observed and calculated g-factors for a selection of near-dissociation levels of He84Kr+ in the v = 4 level of the X state. The calculated values are obtained from the detailed theory described above, but pure case (e) values are also listed...

The appearance of the conjugate sorption data presented so far qualitatively indicates a shift from a more to a less Fickian character as the initial surface moisture content increases. That is, the appearance of the conjugate sorption isotherms obtained by totally immersing the samples violate two of the criteria by which Fickian behavior is defined. The same cannot be said for those samples exposed to less than 100% R.H., particularly at 25°C. This qualitative trend for PMC is further demonstrated by Figures 16 and 17. Here MjVWq is presented as a function of time. For both the thick and the thin sample, as either temperature or relative humidity is increased, the character of the curves progresses towards pure Case II description. That is, the moisture uptake becomes linear with time up to the point where a plateau is achieved in the behavior. 

The singlet distribution function for the species A is defined in complete analogy with the definition in the pure case (section 2.1),... 

In the case of quasi-classical scattering the correlator Gfx.T ) does not change. Hence, as was in the pure case (2), we obtain the following relation between the transition temperature and parameter W ... 

As in the pure case, we suggest the behaviour of G(x,T ) at large distances to be determined by slow fluctuations of the phase ... 

The polarization factor id varies from zero (completely mixed case) to one (pure case). 

The potential energy curve for a pure case (d) /-complex (with a 1 + ion-core) could, in principle (Field, unpublished), be computed ab initio as the degeneracy-weighted average of the Z+l different case (a) A-component potential curves,... 

These energy-level formulas may be expressed in a form similar to that in pure case (a) (i.e., without 2n1/2 3Il3/2 mixing) by introducing a Be// constant... 

There are two classes of transitions between pure case (a) basis functions. Parallel and perpendicular transitions are readily distinguished by their characteristic rotational branch intensity patterns. AA = 0 transitions are called parallel bands. They have R and P branches of comparable intensity and a weak or absent Q branch. A A = 1 transitions are perpendicular bands and have a strong Q branch and R and P branches of approximately half the intensity of the Q branch. 

We have dealt above with a pure case of doublet reaction and other reactions. It is only natural that there can be some quite reasonable deviations. The bonds requiring activation must come into contact with the catalyst. If, for instance, one part of the index of the doublet reaction does not require activation, it can react without coming into contact with the catalyst. Such a half-doublet scheme for esterification in solution was given by the author 37). Hence, the transition from heterogeneous to homogeneous catalytic reactions occurs, and in the extreme case in which both the bonds of the doublet group do not require activation for the reaction, to noncatalytic reactions. 

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