Phase relations

Isothermal section in the Th-Sb-N system at 1273 K, suggested from cited literature. 

2 Ternary Nitrides of Thorium with Other Metals 

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The basic requirements for the Mephisto model was satisfactory accuracy, that means prediction of amplitude, position and phase relation between the various signals, and short computation times, typically a few minutes for the simulation of a whole Cscan, compatible with an intensive use. These a priori contradictory characteristics have been contented by means of appropriate approximations based on physical considerations. 

In the previous section we discussed light and matter at equilibrium in a two-level quantum system. For the remainder of this section we will be interested in light and matter which are not at equilibrium. In particular, laser light is completely different from the thennal radiation described at the end of the previous section. In the first place, only one, or a small number of states of the field are occupied, in contrast with the Planck distribution of occupation numbers in thennal radiation. Second, the field state can have a precise phase-, in thennal radiation this phase is assumed to be random. If multiple field states are occupied in a laser they can have a precise phase relationship, something which is achieved in lasers by a teclmique called mode-locking Multiple frequencies with a precise phase relation give rise to laser pulses in time. Nanosecond experiments... 

Since there is a definite phase relation between the fiindamental pump radiation and the nonlinear source tenn, coherent SH radiation is emitted in well-defined directions. From the quadratic variation of P(2cii) with (m), we expect that the SH intensity 12 will also vary quadratically with the pump intensity 1 ... 

CIDNP involves the observation of diamagnetic products fonned from chemical reactions which have radical intemiediates. We first define the geminate radical pair (RP) as the two molecules which are bom in a radical reaction with a well defined phase relation (singlet or triplet) between their spins. Because the spin physics of the radical pair are a fiindamental part of any description of the origins of CIDNP, it is instmctive to begin with a discussion of the radical-pair spin Hamiltonian. The Hamiltonian can be used in conjunction with an appropriate basis set to obtain the energetics and populations of the RP spin states. A suitable Hamiltonian for a radical pair consisting of radicals 1 and 2 is shown in equation (B1.16.1) below [12]. 

It is beyond the scope of these introductory notes to treat individual problems in fine detail, but it is interesting to close the discussion by considering certain, geometric phase related, symmetry effects associated with systems of identical particles. The following account summarizes results from Mead and Truhlar [10] for three such particles. We know, for example, that the fermion statistics for H atoms require that the vibrational-rotational states on the ground electronic energy surface of NH3 must be antisymmetric with respect to binary exchange... 

Coherent states and diverse semiclassical approximations to molecular wavepackets are essentially dependent on the relative phases between the wave components. Due to the need to keep this chapter to a reasonable size, we can mention here only a sample of original works (e.g., [202-205]) and some summaries [206-208]. In these, the reader will come across the Maslov index [209], which we pause to mention here, since it links up in a natural way to the modulus-phase relations described in Section III and with the phase-fiacing method in Section IV. The Maslov index relates to the phase acquired when the semiclassical wave function haverses a zero (or a singularity, if there be one) and it (and, particularly, its sign) is the consequence of the analytic behavior of the wave function in the complex time plane. 

The choice of separation method to be appHed to a particular system depends largely on the phase relations that can be developed by using various separative agents. Adsorption is usually considered to be a more complex operation than is the use of selective solvents in Hquid—Hquid extraction (see Extraction, liquid-liquid), extractive distillation, or azeotropic distillation (see Distillation, azeotropic and extractive). Consequentiy, adsorption is employed when it achieves higher selectivities than those obtained with solvents. 

A significant advantage of adsorbents over other separative agents Hes in the fact that favorable equiHbrium-phase relations can be developed for particular separations adsorbents can be produced that are much more selective in their affinity for various substances than are any known solvents. This selectivity is particularly tme of the synthetic crystalline zeoHtes containing exchangeable cations. These zeoHtes became available in the early 1960s under the name of molecular sieves (qv) (9). 

The phases present in products can differ from those predicted from equilibrium diagrams. Nonequilibrium metastable phases form at solidification rates experienced in commercial ingots. Because of the low rate of diffusion of iron in alurninum, equilibrium conditions can only be established by long heat treatments and are very slowly approached at temperatures below about 550 °C. Small additions of other elements, particularly manganese, can also modify the phase relations. 

The acidic character of siUca is shown by its reaction with a large number of basic oxides to form siUcates. The phase relations of numerous oxide systems involving siUca have been summarized (23). Reactions of siUca at elevated temperatures with alkaU and alkaline-earth carbonates result in the displacement of the more volatile acid, CO2, and the formation of the corresponding siUcates. Similar reactions occur with a number of nitrates and sulfates. Sihca at high temperature in the presence of sulfides gives thiosiUcates or siUcon disulfide, SiS2. 

The existence of tridymite as a distinct phase of pure crystalline siUca has been questioned (42,58—63). According to this view, the only tme crystalline phases of pure siUca at atmospheric pressure are quart2 and a highly ordered three-layer cristobaUte having a transition temperature variously estimated from 806 250°C to about 1050°C (50,60). Tridymites are considered to be defect stmctures in which two-layer sequences predominate. The stabihty of tridymite as found in natural samples and in fired siUca bricks has been attributed to the presence of foreign ions. This view is, however, disputed by those who cite evidence of the formation of tridymite from very pure siUcon and water and of the conversion of tridymite M, but not tridymite S, to cristobahte below 1470°C (47). It has been suggested that the phase relations of siUca are deterrnined by the purity of the system (42), and that tridymite is not a tme form of pure siUca but rather a soHd solution of minerali2er and siUca (63). However, the assumption of the existence of tridymite phases is well estabUshed in the technical Hterature pertinent to practical work. 

Flexible rotors are designed to operate at speeds above those corresponding to their first natural frequencies of transverse vibrations. The phase relation of the maximum amplitude of vibration experiences a significant shift as the rotor operates above a different critical speed. Hence, the unbalance in a flexible rotor cannot simply be considered in terms of a force and moment when the response of the vibration system is in-line (or in-phase) with the generating force (the unbalance). Consequently, the two-plane dynamic balancing usually applied to a rigid rotor is inadequate to assure the rotor is balanced in its flexible mode. 

This measurement ean be aeeomplished by using a meehanieal system or various types of eleetronie systems. All of these systems are expensive and in many eases require repeated ealibration. The meehanieal system (Figure 19-14) is a three-gear, phase-related system whieh measures the displaeement between two gears and the proportionate shaft twist. A third gear is situated so that any variations other than shaft twist will oeeur in the first two gears. This signal is used to eliminate errors eaused by these variations. 

Frequeney domain analysis is eoneerned with the ealeulation or measurement of the steady-state system output when responding to a eonstant amplitude, variable frequeney sinusoidal input. Steady-state errors, in terms of amplitude and phase relate direetly to the dynamie eharaeteristies, i.e. the transfer funetion, of the system. 

The phase relations, stoichiometry and structural chemistry of the metal borates have been extensively studied because of their geochemical implications and technological importance. Borates are known in which the structural unit is mononuclear (1 B atom), bi-, tri-, tetra- or penta-nuclear, or in which there are polydimensional networks including glasses. The main structural principles underlying the bonding in crystalline metal borates are as follows " ... 

The curious phase relations between phosphorus, sulfur and their binai compounds are worth noting. Because both P4 and Sg are stable molecules the phase diagram, if studied below 100°, shows only solid solutions with a simple eutectic at 10° (75 atom % P). By contrast, when the mixtures are heated above 200° the elements react and an entirely different phase diagram is obtained however, as only the most stable compounds P4S3, P4S7 and P4S10... 

Fuller treatments of the phase relations and structures of polyphosphates, and their uses as glasses, ceramics, refractories, cements, plasters and abrasives, are available. 

When water freezes the crystalline form adopted depends upon the detailed conditions employed. At least nine structurally distinct forms of ice are known and the phase relations between them are summarized in Fig. 14.9. Thus, when liquid or gaseous water crystallizes at atmospheric pressure normal hexagonal ice If, forms, but at very low temperatures (—120° to — 140°) the vapour condenses to the cubic form, ice Ic. The relation between these structures is the same as that between the tridymite and cristobalite forms of SiOa (p. 342), though in both forms of ice the protons are disordered. 

The phase relations in the tellurium-halogen systems have only recently been elucidated... 

A.J. Schwartz and L.E. Taimer, Transformadons and phase relations in the TiPd-TiCr pseudobinary... 

Fig. 2. Equilibrium phase relations about the Cu-Au superlattices, from...

Second, Schneider s article reviews recent work (notably by Rowlinson, Kohn and co-workers) on phase relations in binary liquid systems where one of the components is much more volatile than the other (D1, D2, E3, M8, R9). Such systems may have lower critical solution temperatures for these systems, an increase in temperature (and, indirectly, pressure) causes precipitation of the heavy component, thereby providing a possible separation technique, e.g., for the fractionation of polymers. 

Roe, R.-J. and Rigby, D. Phase Relations and Miscibility in Polymer Blends Containing Copolymers. Vol. 82, pp. 103-141. 

Heterodyne is a very efficient tool for detecting the phase of a "coherent" signal i.e. a signal which has a stable phase relation to the local oscillator. The detector is only limited by the quantum fluctuation of vacuum. This property is common use in coherent lidar. Satellite to satellite optical communications using laser as a local oscillator are under development (Fig. 3). 

Despite the occurrence of binary AIB2 borides (see also Fig. 2), no ternary representatives are known (Mn, Mo)B2 found from isothermal sections is a stabilized high-T phase by conversion to lower T by a statistical ( ) metal-metal substitution. Both MnB2 and M0B2 are high-T compounds stable above 1075°C and 1517°C respectively WB2 is claimed but is either metastable or impurity stabilized. Similar examples are observed with (W, Pd>2B5 (M02B5 type) as well as (Mo, Rh),, (B3 and (W, Ni), B3 (Mo,., 83 type). The phase relations in the B-rich section of the Mo(W)-B binaries, however, are not known precisely. 

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