Trivariant

Equation (Bl.8.6) assumes that all unit cells really are identical and that the atoms are fixed hi their equilibrium positions. In real crystals at finite temperatures, however, atoms oscillate about their mean positions and also may be displaced from their average positions because of, for example, chemical inlioniogeneity. The effect of this is, to a first approximation, to modify the atomic scattering factor by a convolution of p(r) with a trivariate Gaussian density function, resulting in the multiplication ofy ([Pg.1366]

By the variance, or number of degrees of freedom of the system, we mean the number of independent variables which must be arbitrarily fixed before the state of equilibrium is completely determined. According to the number of these, we have avariant, univariant, bivariant, trivariant,. . . systems. Thus, a completely heterogeneous system is univariant, because its equilibrium is completely specified by fixing a single variable— the temperature. But a salt solution requires two variables— temperature and composition—to be fixed before the equilibrium is determined, since the vapour-pressure depends on both. 

A trivariate normal distribution describes the probability distribution for anisotropic harmonic motion in three-dimensional space. In tensor notation (see appendix A for the notation, and appendix B for the treatment of symmetry and symmetry restrictions of tensor elements), with j and k (= 1, 3) indicating the axial directions,... 

System having degrees of freedom three, two, one or zero are known as trivariant, bivariant, univariant (or monovariant) and non-variant systems, respectively. 

Fig. 24. Schematic phase relations of the Fe-Cu-Mo-S system at 700 °C. Note Between the two ternary compounds (X-phase and Y-phase) occurs a complete quaternary solid solution series (indicated simply as a solid bar). All trivariant regions are specified in the text thus the bivariant phase assemblages and the univariant volumes can be deduced...

When the number of independent components exceeds by unity the number of phases into which the sj stem is divided (c= +l), the variance is equal to 3 the system is Trivariant. 

In order to know completely the composition of the phases into which a trivariant system in equilibrium is divided, it does not suflSce to know the temperature and the pressure it is necessary to add a third quantity. ... 

Let C, C, C".. . be the precipitates that may be observed. Under the given pressure ic the states of equilibrium between the liquid mixture of the three independent components and the single solid precipitate C, states in which the system is trivariant, are represented by the various points on a limited surface S that is called the domain of the precipUate C. 

In general a solution formed of four independent components may precipitate various solids of definite composition, as C, C, . .. By a method similar to that used for trivariant systems, we may reach the following conclusions ... 

If the representative point is on the boundary line of the two surfaces /S, /S of the two substances C, C, the three coordinates of this point represent the three concentrations of a solution which may, under atmospheric pressure and at 15 C., remain in equilibrium in contact with a solid precipitate composed of C and C in such a state of equilibrium one system of four independent components is divided into three phases, so that it is no longer quadrivariant but trivariant. 

We have in fact here a system formed of three independent components water 0 and the two salts 1 and 2 this system is divided into two phases, the liquid solution for which we shall continue to indicate by S, S2, the two concentrations, and the mixed crystals C this system is therefore trivariant when the temperature and pressure only are given, the composition of each of the two phases capable of remaining in equilibrium in contact with each other is not completely determined it becomes entirely determined if to the temperature and pressure there is added another ven quantity, for example, one of the concentrations s of the solution. 

Formation in solution of a racemic compound.—The precipitation within a solution of one of the substances we have just studied leads to the study of the equilibrium of a system no longer bivariant, but trivariant this study is, from the experimental point of view, much less advanced than the preceding it has given rise nevertheless to several interesting researches among this number is the analysis of the conditions of formation of the double racemate of sodium and ammonium, anal3rsis for which we are indebted to Van t Hoff and van Deventer. ... 

The classification, borrowed from the phase rule, of systems into numovariantf hivariard, trivariant, etc., is therefore of extreme utility it arranges in admirable order a great number of questions in the discussion of chemical equilibria it is, however, neither the... 

Multivariant systems. Trivariant systems, page 118.— xoz. Theory of double salts, 118.—10a. Surface of solubility of a double salt at a given pressure, 119.—103, Case in which the solution may furnish two distinct salts, 120.—Z04, Conditions in which the two precipitates are simultaneously in equilibrium with the solution,... 

All points on the diagram lying above the curve CED represent conditions under which the solution can exist alone at the given pressure p the one phase system is a trivariant system and all the variables T, p and can be varied arbitrarily. 

Let us consider a solution of three substances A, B and C which do not react together, and suppose the solution is in equilibrium with one of the solid components. Then c = 3, r = 0, = 2 whence — 3 and the system is trivariant. We can thus consider the pressure and composition of the solution (p, Xj, x ) and see how the equilibrium temperature changes with these variables. Let us take the pressure as constant,... 

Example Consider a two-phase system of three non-reacting components. We have r =0, r" — 3, so that r = 3 it will therefore be trivariant w = 3. 

A univariate observation, such as a single laboratory result, may be represented graphically as a point on a line, the axis. The results obtained by two different laboratory tests performed on the same specunen (a bivariate observation) may be displayed as a point in a plane defined by two perpendicular axes. With three results, we have a trivariate observation and a point in a space defined by three perpendicular axes, and so on. We lose the possibility for visualization of a multivariate observation when there are more than three dimensions. StiU, we can consider the multivariate observation as a point in a multidimensional hyperspace with as many mutually perpendicular axes as there are results of different tests. The prefix hyper signifies, in this context, more than three dimensions. Such multivariate observations are also... 

Table 3.7. The optimal moment set used to build a trivariate quadrature approximation (M = 3) for iV = 8...

Since for velocity distributions (in three spatial dimensions) three internal coordinates (M = 3) are needed, we discuss in the following example the construction of a quadrature approximation for a trivariate tensor-product QMOM. 

Exercise 3.7 Consider a trivariate distribution (M = 3) of three internal coordinates i, 2, and fs. Let us construct an eight-point tensor-product QMOM resulting from univariate... 

Figure 3.2. Positions in phase space of the eight nodes of the trivariate tensor-product QMOM (i.e. M = 3) obtained with two-point univariate quadratures N = N2 = Ns = 2).

Note what we have gained because the Hamiltonian operator determines the wave function for the system from the variational principle, it follows that any property, Q, of the ground state of an electronic system may be written as a function of the number of electrons, N, and a functional of a real-valued trivariate function, v(r), which we call the external potential. We denote this functional Q[v(r) IV]. Unfortunately, no expression for Q[v(r) IV] with computational utility comparable to Eq. (1) is known. However, the fact that properties of a system can be expressed as a function of N and a functional of a single trivariate function, v(f), does suggest that there might be a computational useful theory in terms of a trivariate function. 

Soares and Hamielec extended the Stockmayer distribution for copolymers containing LCBs, where the mechanism of LCB formation is terminal branching via macromonomer incorporation [89, 90]. The resulting trivariate distribution is given by the expression... 

Figure 5.11 Trivariate distribution for a model polymer. (See insert for the color representation of the figure. )...

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