Zeroth moment

The electroneutrality condition can be expressed in temis of the integral of the charge density by recognizing the obvious fact that the total charge around an ion is equal in magnitude and opposite in sign to the charge on the central ion. This leads to the zeroth moment condition... 

Another important characteristic of the late stages of phase separation kinetics, for asynnnetric mixtures, is the cluster size distribution fimction of the minority phase clusters n(R,z)dR is the number of clusters of minority phase per unit volume with radii between R and + cW. Its zeroth moment gives the mean number of clusters at time r and the first moment is proportional to die mean cluster size. 

It is an ordinary differential equation (time dependenee only) where the zeroth moment (/io) is the total erystal number (per unit volume of suspension), the first (/ii) the total erystal length (lined up end to end), the seeond (7 2) is related to total surfaee area, the third (7 3) total volume (and lienee mass) and so on. 

Although the experimental and simulation time scales differ, the CFD simulation (Figure 8.29(a),(c),(e)) for the zeroth moment (Mq) indicates that once the particles reach the observable size, they will appear approximately in the experimentally observed regions (Figure 8.29 (b),(d),(f)). Predicted velocity vectors are superimposed on supersaturation profiles in Figure 8.30. 

Also, the zeroth moment of the differential molecular weight distribution, DMWD, may be obtained by integration of the simplified equation ... 

Moment Analysis. The zeroth moment is the molar concentration of polymer and is expressed by Equation 5. The first moment is proportioned to the mass of polymer formed and is related to monomer concentration Equation 3, The second moment WA(t) is expressed by... 

The zeroth moment, mQ, was mentioned above. It is nothing more than the total area of the gradient [see Figure 1.12(a)]. If the area vanishes, the signal is independent of spin position. This is the case, for instance, for a bipolar gradient with positive and negative lobes of equal area [see Figure 1.12(b)]. 

A more sophisticated means of quantitating peak asymmetry is through the theory of peak moments.1314 Briefly, the zeroth moment is the peak area, the first moment is the average retention time... 

Relationship 23 provides a method for evaluating the parameter "a" that is defined by Equation 2A. The cumulative molar concentration of polymeric species PT0T was numerically evaluated via integration of population density distributions. The contribution of network molecules to the zeroth moment of the distribution is negligible. Results are presented by Figure A and show that... 

The zeroth moment (k = 0) is simply the area under the distribution curve ... 

Zeroth Moments. The zeroth moment for each of the models is unity. From Eq. (65) the zeroth moment is defined as ... 

Let us now consider the action of the two equal and opposite pulsed gradients (referred to as a bipolar pair) of amplitude +g and length (5, separated by time A, as shown in Fig. 7, and in the absence of relaxation. When there is no motion the first pulse, +g, will cause a phase shift which is proportional to the zeroth moment (Eq (14)) ... 

The second, equal and opposite, gradient pulse will have a zeroth moment given by... 

More interesting are the results of taking integrals over the three-dimensional momentum space c Q, of f(t, x, p), as well as of products of powers of momentum components with f(t, x, p), leading to various moments of these quantities. The zeroth moment is the number density ... 

The summation over final states /> has been carried out by the completeness relation, or equivalently, by matrix multiplication. Thus, the zeroth moment is just an equilibrium average of the square of the perturbation amplitude. The first moment can likewise be expressed as an equilibrium property. 

The zeroth moment (n=0) gives the total intensity and is related to theory by familiar sum formulae (Chapter 5). For nearly classical systems (i.e., massive pairs at high temperature and not too high frequencies), the first moment (n=l) is very small and actually drops to zero in the classical limit as we will see in Chapter 5. The ratio of second and zeroth moment defines some average frequency squared and may be considered a mean spectral width squared. A complete set of moments (n = 0, 1,... 00) may be considered equivalent to the knowledge of the spectral line shape,... 

If one now chooses the appropriate induced dipole model, Eqs. 4.1 through 4.3, or a suitable combination of these, with N parameters po, >7, R0,. .., and one has at least N theoretical moment expressions available, an empirical dipole moment may be obtained which satisfies the conditions exactly, or in a least-mean-squares fashion [317, 38]. We note that a formula was given elsewhere that permits the determination of the range parameter, 1/a, directly from a ratio of first and zeroth moments it was used to determine a number of range parameters from a wide selection of measured moments [189]. In early work, an empirical relationship between the range parameter and the root, a, of the potential is assumed, like 1/a 0.11 a. That relationship is, however, generally not consistent with recent data believed to-be reliable. 

Moments are averages of the spectral profile, a(v) or the spectral density, J(v). The zeroth moment (n = 0) in essence gives the total intensity of the profile. The first moment (n = 1) may be understood as the product of mean width and total intensity the second moment is the product of the mean squared frequency times total intensity, etc. 

We note that in the definition of moments, we have used frequency v in units of cm-1. Other units of frequency are sometimes chosen instead of the experimentalist s standard use of frequency in wavenumber units (cm-1), angular frequencies co = 2ncv are the most likely choice of theorists, which leads to different dimensions of the moments, and to the appearence of factors like powers of 2nc. We note, moreover, that in the early days of collisional induction studies the zeroth moment yo was defined without the hyperbolic cotangens function, Eqs. 5.6. For the vibrational bands (hco 3> kT), the old and new definitions are practically identical. However, for the ro to translational band substantial differences exist. The old definition of yo was never intended to be used for the far infrared [314] only Eq. 5.6 gives total intensity in that case. 

M s is the Hamiltonian of the system and Tr designates the trace. For the computation of the zeroth moment, we start from Eq. 5.1 and write for the difference of Boltzmann factors... 

We note that in a classical formula Planck s constant does not appear. Indeed, the zeroth moment Mo of the spectral density, J (o), does not depend on h, as the combination of Eqs. 5.35 and 5.38 shows. On the other hand, the classical moment y of the absorption profile, a(cu), is proportional to /h because the absorption coefficient a depends on Planck s constant see the discussions of the classical line shape below, p. 246. In a discussion of classical moments it is best to focus on the moments Mn of the spectral density, J co), instead of the moments, yn, of the spectral profile. 

Table 6.3. Various computed zeroth and first moments M , with and without lowest-order Wigner-Kirkwood quantum corrections, for a hydrogen-argon mixture at 195 K. Units are 10-34 J amagat-N for the zeroth moment, and 10-21 W amagat N for the first moments, with N = 2 and 3 for binary and ternary moments, respectively. An asterisk means that Wigner-Kirkwood corrections were not made for the entries of that line. The superscripts 12 and 122 stand for H2-Ar and H2-Ar-Ar, respectively [296].

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