Phase invariance

Figure 2.13. Building blocks of binary phase diagrams examples of three-phase (invariant) reactions. In the upper part the general appearance, inside a phase diagram, of the two types of invariant equilibria is presented, that is, the so-called 1 st class (or eutectic type) and the 2nd class (or peritectic type) equilibria. In the lower part the various invariant equilibria formed by selected binary alloys for well-defined values of temperature and composition are listed. In the Hf-Ru diagram, for instance, three 1 st class equilibria may be observed, 1 (pHf) — (aHf) + HfRu (eutectoid, three solid phases involved), 2 L — (3Hf + HfRu (eutectic), 3 L —> HfRu + (Ru) (eutectic).

Gauge fields (M) that restore local phase invariance are evidently closely related also to the quantum-potential field. The wave function of a free electron, with temporal and spatial aspects of the phase factor separated, may be written as... 

If such a function now describes the behaviour of a particle through some differential equation, it is important to know that the phase factor will also be modified by differential operators such as V and d/dt. The transformed function therefore cannot satisfy the same differential equation as ip, unless there is some compensating field in which the particle moves, that restores the phase invariance. The necessary appearance of this field shows that local phase invariance cannot be a property of a free particle. It is rather obvious to make the connection with electromagnetic gauge invariance since equation (17) with a = qA is precisely the transformation associated with electromagnetic gauge invariance. 

This form satisfies local phase invariance by demanding that A and V transform as... 

This modified equation is just the Schrodinger equation that describes the interaction of a charged particle with the elctromagnetic field. This appearance of interaction with a field is known as the gauge principle. A vector field such as A, introduced to guarantee local phase invariance, is called a gauge field. The local invariance of Schrodinger s equation ensures that quantum mechanics does not conflict with Maxwell s field. 

Karle, J. Triplet phase invariants from single isomorphous replacement or one-wavelength anomalous dispersion data, given heavy-atom information. Acta Cryst. A42, 246-253 (1986). 

We have underlined that positivity and atomicity of the electron density are basic conditions for the validity of traditional DM. What then is the effect of the possible violation of the positivity criterion on DM procedures It has been assessed that positivity is not an essential ingredient of DM. In particular the triplet phase invariants can again be evaluated via a von Mises distribution, but now the value of N has to be replaced by Aeq, where ... 

Ternary Systems.—Wq pass over the binary system FeClg—HgO, which has already been discussed (p. 187), and the similar system HCl—HgO (see Fig. 132), and turn to the discussion of some of the ternary systems represented by points on the surface of the model between the planes XOT and YOT. As in the case of carnallite, a plane represents the conditions of concentration of solution and temperature under which a ternary solution can be in equilibrium with a single solid phase (bivariant systems), a line represents the conditions for the co-existence of a solution with two solid phases (univariant systems), and a point the conditions for equilibrium with three solid phases (invariant systems). 

Solubility is an oft ill-defined term, used rather indiscriminately to refer to small amounts of a solute of one phase dissolved in a solvent of another phase. Invariably, the solvent is a liquid or dense fluid, though it may contain any number of components, while the solute may be gas, liquid, or solid. Solubility problems are really phase-equilibrium problems and are attacked using the general strategies presented in Chapter 10. In this section we describe the three common solubility problems gas solubility, which refers to supercritical gases dissolved in liquids ( 12.2.1) solid solubility, which refers to solids dissolved in liquids ( 12.2.2) and solubilities in near-critical... 

Rag] reviewed the results of [2000Wan] and slightly modified the isothermal seetions of the phase diagram to meet the accepted binary data. [2002Rag] postulated four-phase invariant reactions in the Cu-Fe-Nb system based on die calculated isothermal sections of [2000Wan]. 

Seven four-phase invariant equilibria were postulated by [2002Rag] in the Cu-Fe-Nb system. Six of them occur with the participation of the liquid phase (points Ui to U4, Ei, E2) and one equilibrium (point E3) occurs in the solid state. The reaction scheme is shown in Fig. 2 according to [2002Rag] with some corrections. The corrections concern... 

TP Triple point (coexistence of three phases, invariant system)... 

SU] using the thermodynamic parameters of C-Fe and Cu-Fe systems and magnimde of the phase distribution coefficients of die elements calculated the temperatures and constimtions of four-phase invariant eutectoid equilibria in stable and metastable C-Cu-Fe systems and constracted some isothermal sections. 

There are two three-phase invariant equilibria, T T + FesC, at 1230°C and T + FesC (Cu) in the metastable Cu-Fe-FesC system [1926Ish, 1991Schl, 1991Sch2]. 

Because of the complex flow behavior ofviscoelastic polymer blends its interpretation becomes easier when compared with the simpler, model systems discussed in Sections 2.1.1 and 2.1.2. For the dominant immiscible blends the emulsions of one liquid dispersed in another provide the best model. While compatibilizers are used for stabilization of blends, in emulsion a diversity of surfactants has been employed. As in blends, a progressive increase of concentration of the emulsion minor phase invariably leads to the co-continuity of phases and phase inversion. However, while emulsion phase inversion is controlled mainly by the emulsifier, in blends the inversion mainly depends on the viscosity ratio, A = th/th (Figure 2.3) [4]. 

P " phase invariably contains MgO or other additives, though not usually in the proportion indicated by the idealised formula. 

Table AT Types of three-phase invariant reactions in binary phase diagrams. (Smith, W.F. (1996) Principles of Materials Science and Engineering, 3rd edition, McGraw-Hill, Inc., New York, p. 464).

Systems of type la (without critical phenomena in solid saturated solutions) (Figure 1.14). This is the simplest type of binary system. Most of the divariant (L-G, L-Sa, L-Sb, G-Sa, G-Sb) and monovariant equilibria (L-G-Sa, L-G-Sb, L = G) in the binary system (A-B) are the monovariant and nonvariant, respectively, with equilibria of one-component subsystems (A and B), spreading into two-component region of composition. Only the phase equilibria with two solid phases (invariant eutectic equihbrium (L-G-Sa-Sb) monovariant equilibria (L-Sa-Sb, G-Sa-Sb) and divariant equi-lihrium (Sa-Sb)) appear in the binary mixture as a result of an interaction of phase equilibria that extend fi om one-component subsystems. 

Figure 13.6 Isothermal section of the ternary system A-B-C the physical variable is pressnre (it is assumed that AB2 is a co-crystal and Pab2 is negligible), (a) shows that for solvent greater than Pi vi e.g. PI), AB2 will uptake solvent C so that it will at first partially and then ultimately completely dissolve. If an infinite reservoir of solvent vapour is assumed, the final point of evolution is T, an undersaturated solution. If C stands for water, compound AB2= <1-2-0> is said to be hygroscopic up to deliquescence, (b) Due to the crystallization of the solvate < 1-2-1 >, P nvi has become a metastable invariant. Pj v2 corresponds to the equilibrium between the , and Cvapor- Pinv3 Corresponds to the three phase invariant between < 1 -2-1 >, its saturated solution and Cvapor- At Psoivent = Pi, the solvated co-crystal should irreversibly decompose (because P2 is said to be efflorescent. Because of the stability domain of the solvate,...

Number of degrees of freedom - the number of externally controllable intensive variables (T, P, concentration, etc.) which can be independently altered without bringing about the disappearance of a phase or the appearance of a new phase. Invariant = zero degrees of freedom Univariant = one degree of freedom Bivariant = two degrees of freedom... 

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