Mean occupation number

Let us consider a fluorescent probe and a quencher that are soluble only in the micellar pseudo-phase. If the quenching is static, fluorescence is observed only from micelles devoid of quenchers. Assuming a Poissonian distribution of the quencher molecules, the probability that a micelle contains no quencher is given by Eq. (4.22), so that the relationship between the fluorescence intensity and the mean occupancy number < > is... 

Equations (163) allow one to derive equations for the proton mean occupation numbers and thermodynamic functions in the mean-field approximation. Substituting variables 8hmi- where i = 1, 2, 3 into Eqs. (157)-(159), we get... 

Suppose that initially there was a single excited mode labeled with an index n. Because of the linearity of the process, one may assume that the mean number of photons in this mode was v = 1. Then the mean occupation number of the m-th mode at x > 0 equals... 

Figure 6. Steady state fluorescence quenching data fluorescence intensity in the absence (/o) and presence (/) of quenchers, z = mean occupation number of quencher molecules per micelle (proportional to analytical quencher concentration). Curve a k = Icq, Eq. 43 curve b k = Icq, Eq. 41 curve c yi = 1 + z/2 (Stem-Volmer reference line).

Figure 9. Quantum yields of photodimeiization of 9-methylanthiacene in micellar solutions of CTAB (O) and CTAC (V) as a function of mean occupation numbers z. Straight lines drawn according to Eq. 51 using — 0.003 M, kiko — 0.011 and 0.016, k /ko = 730 and 1080 for CTAB and CTAC, respectively.

Fig.2.10. Bose-Einstein distribution function giving the mean occupation number as a function of temperature T for different values of 0s=H

Ramesh and Ramamurthy demonstrated a dependency of the cisoid/transoid ratio on the mean occupancy number (S), which refers to number of ACN molecules per micelle. Under constant concentration (0.05 M) of aqueous surfactant SDS, the cisoid/transoid ratio decreased with decrease of [ACN] that is, the ratio was 4.6, 4.6, 4.0, 3.6, 2.0, and 1.7 under S of 9.0, 7.0, 6.0, 3.0, 2.0, and 1.0, respectively. When cetyltrimethylammonium bromide (CTAB) was employed in place of SDS, the ratio decreased from 4.6 to 1.3 at S of ca. 9. These results can be interpreted as follows. With a decrease of [ACN], the S value decreases. Since the probability for formation of Sj-derived cisoid-l increases at high [ACN], cisoid-l is favored under conditions of high S. With a decrease of S, the probability for the S, state to encounter a ground state ACN prior to ISC to the Tj state decreases, and hence the cisoid/transoid ratio decreases. In CTAB micelles, the heavy atom effect of the bromide ion leads to promoted formation of T,-derived transoid-1, whereas in micelles with atoms hghter than bromine, the S,-derived cisoid-l predominates. 

Finally, and probably most importantly, the relations show that changes (of a nonhivial type) in the phase imply necessarily a change in the occupation number of the state components and vice versa. This means that for time-reversal-invariant situations, there is (at least) one partner state with which the phase-varying state communicates. 

In the case of the LiMg momentum density and occupation number density reconstruction of Stutz et al, who collected 6 x 105 6 counts for Li and 6 x 107 counts for LiMg, this would mean that 6 x 10s—6 x 10 counts per spectrum were required, which hardly can be accomplished in a reasonable amount of time even at modem synchrotron radiation sources. 

One of the first consequences of the above ideas was the development of the Orbital Local Plasma Approximation (OLPA) by Meltzer et al. [37-39]. The main ingredients in the OLPA consist in approximating the orbital weight factors by the orbital occupation numbers and adapting the Lindhard-Scharff Local Plasma Approximation (LPA) [10-12] to an orbital scheme whereby the orbital mean excitation energy was originally defined as [37,38]... 

This parabolic trend can be surmised from Eq. (15), where the occupation numbers n are proportional to Z (the parabola should have a minimum at Cm, i.e. or the half-filling of the 5f shell). The same parabolic trend exists in d-transition metals, and is explained in Friedel s model in a similar way. This fact had seemed to early theorists (see, in Chap. A, the discussion of Zachariasen s model) to suggest that the actinides were 6 d-transition metals. In reality, it means that the light actinides are 5f-transition metals, with the 5 f wavefunctions playing the role of d-wavefunc-tions. 

Another means to reduce the scale of the problem is to shrink the size of the CAS calculation, but to allow a limited number of excitations from/to orbitals outside of the CAS space. This secondary space is called a restricted active space (RAS), and usually the excitation level is limited to one or two electrons. Thus, while all possible configurations of electrons in the CAS space are permitted, only a limited number of RAS configurations is possible. Remaining occupied and virtual orbitals, if any, are restricted to occupation numbers of exactly two and zero, respectively. 

The second model, the so-called gradient-flux law, is considered to be more fundamental, although it is based on a more restrictive physical picture. In contrast to the mass transfer model, in which no assumption is made regarding the spatial separation of subsystems A and B, in the gradient-flux law it is assumed that the subsystems and the distance between them, Axa/b, become infinitely small. For very small subsystems the term occupation number loses its meaning and must be replaced by occupation density or concentration. Obviously, the difference in occupation density tends toward zero, as well. Yet the ratio of the two differences, Aoccupa-tion density Axa/b, is equal to the spatial gradient of the occupation density and usually different from zero ... 

The dependence of the used orbital basis is opposite in first and second quantization. In first quantization, the Slater determinants depend on the orbital basis and the operators are independent of the orbital basis. In the second quantization formalism, the occupation number vectors are basis vectors in a linear vector space and contain no reference to the orbitals basis. The reference to the orbital basis is made in the operators. The fact that the second quantization operators are projections on the orbital basis means that a second quantization operator times an occupation number vector is a new vector in the Fock space. In first quantization an operator times a Slater determinant can normally not be expanded as a sum of Slater determinants. In first quantization we work directly with matrix elements. The second quantization formalism represents operators and wave functions in a symmetric way both are expressed in terms of elementary operators. This... 

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