Average occupation

Flere the zero point energy is ignored, which is appropriate at reasonably large temperatures when the average occupation number is large. In such a case one can also replace the sum over by an integral. Each of the triplet n can take the values 0, 1, 2,. . ., co. Thus the sum over can be replaced by an... 

From equation (A2.2.145). the average occupation number of an ideal Bose gas is... 

In terms of free-energy surfaces, multiple electron transitions correspond to multiple transitions between various free-energy surfaces of the initial and final states, and the system in fact moves along some effective potential profile. Multiple electron transitions allow one to speak about an average occupation of the... 

Time Average Occupancy Level Time Fraction (HoursAVeek)/168... 

The second property is that each term of the PF is proportional to the probability of occurrence of the particular state it represents when the system is at equilibrium. We shall use mainly the second property of the PF. The next section is devoted to this aspect of the theory. Once we have the probabilities of all possible events we can compute average quantities pertaining to the system at equilibrium. Of these, the average occupation number, or the binding isotherm, will be the central quantity to be examined and analyzed in this book. 

Since site a can be either empty [with probability 1 - P(a)] or occupied [with probability P(a)], 9 is the average occupation number for the site a. Clearly, 0 S 6 1. When forming the sumn in Eq. (2.1.7) or (2.1.8), we sum over all average quantities 0,. and obtain the average occupation number for the entire molecule. Clearly 0[Pg.28]

Clearly, both P a b = 0) and P alb =1) can be interpreted as individual Bis for site a (and similar definitions apply to site b). It should be noted that all three individual Bis defined above can, in principle, be measured experimentally. The conditions of the experiments are different. In P alb = 0) [Eq. (2.1.11)], we follow the average occupation of site a while maintaining site b empty. On the other hand, in P aJb =1) [Eq. (2.1.12)], we follow the average occupation of site a while we secure the occupation of site b. In 6 [Eq. (2.2.9)], we follow the occupation of site a while leaving site b unrestricted to bind ligands under the same conditions as if we were to measure 0, or n, but monitor the binding on a only. 

When 5 1, the effective binding constant will be dependent on C and therefore 0 is not a simple Langmuir isotherm. Clearly, when we focus on site a and follow its average occupation as a function of X (or C), the effective binding constant fc is affected by what happens at site b. 

A structure can be defined as possessing long-range order if at least two sets of positions can be distinguished by a different average occupation. These classes are usually called sub-lattices. The simplest example of an order/disorder transformation occurring in a b.c.c. lattice may be described in terms of two interpenetrating simple cubic arrays. If the occupation probability of each species is the same on both sublattices, then this is equivalent to a fully disordered b.c.c. stmcture, A2 (Fig. 7.1). 

In order to device an effective averaging procedure one still has to specify formulas for the average occupation numbers n . 

The conservation of the number of particles introduces a new Fermi level and helps to define the averaged occupation numbers, e g., in the HF case ... 

The averaged occupation numbers h, are a formal ingredient in Strutinsky s averaging method. Combining Eqs (15) and (17) yields their explicit form ... 

Introducing the averaged occupation numbers nf, n°, nj from Eqn (17) into Eqn (39), one obtains the averaged parts of the above ... 

In order to determine the averaged occupation numbers n,. for the considered atoms (3 < Z 30), we have solved numerically the system of highly non-linear equations composed from the particle number conservation condition (17) and the plateau condition (24). It was found that the condition (25) is fulfilled with desired accuracy (with less than 1% discrepancy for 5iEhfr) for values of M > 35, and the roots Yo and X found at M = 35 allow a correct determination of the n. ... 

The first set of equations govern the Cj amplitudes and are called the CI- secular equations. The second set determine the LCAO-MO coefficients of the spin-orbitals (f>j and are called the Fock equations. The Fock operator F is given in terms of the one- and two-electron operators in H itself as well as the so-called one- and two-electron density matrices yij and Tyj i which are defined below. These density matrices reflect the averaged occupancies of the various spin orbitals in the CSFs of VP. The resultant expression for F is ... 

Next suppose at t = 0 the bath is in thermal equilibrium with the temperature j3 1, while S is in level N0. The average occupation number of the n-th bath oscillator is given by... 

To obtain a 2CK fixed point we assume that Vd is tuned to make the average occupancy of the small dot nd = 1, creating a local spin-/. We further... 

Figure 2 Typical PEPT output for a spouted bed (Left to Right single trajectory time-averaged velocity vectors time-averaged occupancy , showing denser annular region and leaner spout region) (see Plate 8 in Color Plate Section at the end of this book).

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