Nuclei number density

There are two important limiting cases of isothermal crystallization, namely the instantaneous nucleation, where the nuclei are there from the beginning, and the sporadic nucleation, where the number of nuclei increases linearly with time. 

For quiescent crystallization under a constant temperature, in the case of instantaneous nucleation with a constant number density No, one obtains Nq t) = NoH t) for the activated quiescent nuclei number density, where H(t) is the Heaviside unit step function, zero for f 0 and unity for t 0. Then the rate of the nuclei number density is Ng = NoS t), with (5(f) being the Dirac delta function concentrated at f = 0. Equations 4.1 and 4.3 lead to the familiar Avrami equation  

According to Koscher and Fulchiron (2002), for quiescent instantaneous nucleation, the value of No has an exponential dependence on the degree of 

The flow-induced nuclei number takes the following form as suggested by Eder and Janeschitz-Kriegl (1997)  

Several expressions for the function/have been proposed by different authors based on different assumptions about the driving force for the enhancement of nucleation. The suggested driving forces include shear rate, recoverable strain, the first normal stress difference, the change in free energy induced by flow, the effect of the combination of shear rate and strain, etc. 

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This means that, to calculate the relative crystallinity, we have to calculate only the fictive volume fraction q>. But to obtain growth rate and the nuclei number density. 

N(t) - JV (/)/iSo being the nucleus number density. At time / -> 00 the number of nuclei tends to its saturation value ... 

Ns = Nl /So being the saturation nucleus number density. Finally, the current /. (t)of progressive two-dimensional nucleation with overlap can be... 

The analysis of steady-state and transient reactor behavior requires the calculation of reaction rates of neutrons with various materials. If the number density of neutrons at a point is n and their characteristic speed is v, a flux effective area of a nucleus as a cross section O, and a target atom number density N, a macroscopic cross section E = Na can be defined, and the reaction rate per unit volume is R = 0S. This relation may be appHed to the processes of neutron scattering, absorption, and fission in balance equations lea ding to predictions of or to the determination of flux distribution. The consumption of nuclear fuels is governed by time-dependent differential equations analogous to those of Bateman for radioactive decay chains. The rate of change in number of atoms N owing to absorption is as follows ... 

As Ix(Nucleus) increases with an increase of the number density of nuclei, this clearly confirmed that the number density of nuclei increases during the induction period. Thus, it is concluded that the nucleation during the induction period is directly confirmed experimentally for the first time. 

The abundance of each element is fixed by its binding energy, which characterises its strength as an entity, and the temperature and density of free neutrons and protons attacking the nucleus (Fig. A3.1). If, as is usually the case, nuclear equilibrium is reached before a significant number of radioactive decays have had the time to occur, an auxiliary constraint can be imposed the total number density of protons and neutrons, both free and bound, must preserve the mean n/p ratio. 

The vector model of a single spin is the vector representation of the complex number in the individual density matrix of a single nucleus. This density matrix consists of only one complex number thus there is only one vector in the model. In the case of more than one nuclei, the density matrix is larger, there are more single quantum coherences and more vectors belong to one spin set in the model. Moreover, in case of a strongly coupled spin system, the density matrix has different numerical form for different basis sets of the vector space of the simulation (the basis can be one of the ) and

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The measured reflectivity, R(Q), depends upon the neutron refractive index profile perpendicular to the interface, defined as the z-direction. The neutron refractive index is a function of the scattering length density, Nb, which is the product of the number density N, in units of nuclei per cm3, and the neutron scattering lengths, b, of the nuclei present. Since the neutron scattering length varies from nucleus to nucleus, chances in the nature and composition of the surface result in changes in reflectivity. 

What appears below is a brief discussion of neutron scattering formalism (for an in depth exposition see, for example, the classic paper by van Hove). We assume the scattering is from point particles, which is reasonable, because thermal neutron scattering involves wavelengths on the order of 1A and the length scale of the interaction between an atomic nucleus and the neutron is around five orders of magnitude smaller. If we consider an individual particle j at position rj at time t, the number density can be expressed as... 

Fig. 6. Results of the model calculations for an initial mixture containing COj and no CO (Compos. 5 of Table 5), Number densities reduced to a solar distance of 1 a. u. are plott against distances from the nucleus, a Inner coma < 10 km from the nucleus and species with densities > 3(X) cm . b Coma between 10 and 10 km. Ordinates show densities of less abundant constituents (ions) between 4 and 250 cm" ...

Figs, b are continuations of Figs, a to lower number densities, starting with a distance of 100 km from the nucleus and containing only ion and electron densities. Note that the abundances of CN and which give the dominant contribution to the visible... 

Neutron wavelength Macroscopic number density Microscopic particle density operator Nucleus cross-section Nucleus coherent cross-section Nucleus incoherent cross-section... 

The current associated with the growth of an isolated disc shaped nucleus is therefore expected to increase linearly with time. In practice, of course, it is extremely difficult to observe the growth of a single nucleus, so that it is necessary to take into account the way in which the growth centres are formed. The limiting cases of Equation (9.21) and (9.22) are useful here. If the nucleation of growth centres is essentially instantaneous, the net current density corresponding to a number density of Nq of isolated centres is... 

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