Moments definitions

Then, curium metal is antiferromagnetic and its paramagnetic effective moment definitively supports the picture of a localized 5 f configuration the same is true for what is known about berkelium and californium metals ... 

In the previous sections the MCD equations were developed starting from the transition moment definition of MCD. As described in the introduction, MCD has also been defined through the complex optical rotation ... 

We see that the two most widely used methods of polarization moment definition, namely (D.3) and (D.4), differ both in their transformation properties at turn of the coordinate system (which is not always fully realized), and in normalization. In addition, the definition of the matrix element of the operator j Tq, as assumed in [133], does not permit us to use (D.13) directly for the description of optical coherences. 

The average size of the population is simply equal to the ratio of the first moment, rtii, to the zeroth moment, mo. These moment definitions are slightly different for the population, 7)q(L), when compared to the size distribution, f(L), discussed in Chapter 2. 

For quantum-mechanical systems, the moment definitions in Eqn. (8) can be made into definitions of moment operators by replacing position coordinates with their corresponding position operators. The molecular Hamiltonian for a molecule experiencing an external electrical potential following the convention of Eqn. (7) is... 

With [D ] [P ] there is little difference in magnitude between vt and ft hut, as described below, the introduction of ft eliminates the moment closure problem created by the LCB mechanism. These moment definitions collapse the infinite set of equations for polymeric species into a manageable subset, and many of the moments have precise physical meanings. The zeroth live moment, [/cq], is the concentration of polymer radicals in the system (denoted by [Ptot] in Section 3.2.1), and the first live moment, [/ri],is the concentration of monomer units contained in all growing radicals. Similarly, [fo] is the concentration of all polymer chains in the system, and [f i ] is the concentration of monomer units bound in all polymer chains. These values are used to calculate MW averages, where Wn, is the molecular weight of the monomeric repeat unit ... 

These species balances are next substituted into the moment definitions (Equations 3.57-3.59). The use of generating functions [54, 57, 58] eliminates the tedium (and possible errors) of performing the required series summations both treatments lead to the following equations describing the rates of change of the moments ... 

Eroglu 4 has transformed Eqns. (1) to (9) into Laplace domain and has solved the theoretical expressions of zero reduced and first absolute moments of the tracer in the liquid and the gas analytically, basing on the following moment definitions applicable in moment analysis technique. The fraction of tracer absorbed in liquid phase, which can also be defined as zero reduced moment of liquid phase is... 

Introducing the moment definition, the PBE can be expressed in terms of moments ... 

Specify the initial moments. They-th moments definition approximation (12.312) might be employed to compute the required lower-order moments fi om a measured or presumed initial size distribution. Alternatively, the necessary moments may be computed in terms of a few abscissas and weights representing a population made up oflV classes of particles with dimensions Ci, Cz, C/v, and number densities w, W2,--- u>n. Normally, Nq is a low number, typically Nq = 3. 

Let s see next the significance of regression equation from the distribution function point of view. Firstly let s introduce the moments" definition ... 

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