Vegard’s rule

As a comment to the atomic dimensions concept, two rules , Vegard s andBiltz-Zen s rules, formulated for solid solutions, and to a certain extent for ordered compounds can be mentioned here. These rules, mutually incompatible, are very seldom obeyed they may however be useful either as approximations or for defining reference behaviours. The first one, Vegard s rule (1921), corresponds to an additivity rule for interatomic distances (or lattice parameters or average atomic diameters). For a solid solution (x = atomic fraction) between two components of... 

As a matter of fact, the volumes of NaAlSi308-KAlSi308 mixtures are not ideal and have positive excess terms. Deviations from simple Vegard s rule proportionality are mainly due to edge h Kroll and Ribbe (1983) provide three polynomial expressions relating the volumes of mixtures to the molar fractions of the potassic component Xor) valid, respectively, for monoclinic symmetry (complete disorder), triclinic symmetry of perfect order (low albite and maximum micro-cline), and the intermediate structural state ... 

Comparison of the potassium content of the cetineite crystals to the composition of the synthesis mixture shows a preferred incorporation of potassium into the structure. Although this compound with 50 atoms per unit cell is a highly complex structure, it seems to follow the empirical Vegard s rule similar to homogeneous mixed crystals and for intermetallic phases of the Laves type as well as to microporous tin(IV)thioselenides of type SnSi.xSex.i.[13]... 

For a simple solid solution the lattice parameter usually follow Vegard s rule with good approximation, i.e. a linear relationship exists between lattice parameter and the concentration of the solute. The slope of this straight line is termed Vegard s slope, Kim published an empirical relation between ionic radius of the metal ion of oxides (dopants) dissolved in ceria (and other fluorite structured oxides)... 

As expected, Th02 forms a complete series of mixed crystals with Am02, whose lattice constants follow Vegard s rule (Figure 7). 

Mixed crystals Te2,oBri, U with x < 0.75 are easily prepared by annealing the respective vitreous form which can be recovered from the melt in a reversible process. The reversibility indicates the close relationship between the glassy and the crystalline state. Crystallized samples were investigated by the X-ray powder technique and were shown to consist of single-phase solid solutions The cell volume versiK composition plot is given in Fig. 25 and obeys to Vegard s rule. 

In X-ray method Vegard s law is applied, which assume that the changes of unit cell is directly proportional to the increase of concentration of solid solution. Usually the measurements of the changes of inter-plane distance (t/) in the function of introduced additive are conducted. Irrespectively of numerous departures from the Vegard s rule it has a significant experimental importance and give the possibility to establish maximum of component content in the solid solutions (Figs. 2.19 and 2.61). 

For compositions and crystal structures, see Tables 3.1-122-3.1-124 [1.217,218,223,224]. Primary solid solutions have the fee structure of Ag and the lattice parameters correspond roughly to Vegard s rule with a few exceptions. Alloys with Pt, In, Mg, Cd, and Zn form superlattice phases with tetrahedral and rhombo-... 

The phase diagram of the Sc-Dy binary system has not been investigated. Kotur et al. (1991) reported about a complete solubility of the components at 600°C. The a and c lattice spacing of the hep solid solution appear to follow Vegard s rule within the experimental error. 

The Sc-Dy-Si phase equilibria at 600"C have been investigated by Kotur et al. (1991) (fig. 55). SoUd solutions of substitution of Sc and Dy of different extent on the basis of binary silicides occur in the system. DySi dissolve only 1 at.% Sc while the isotypic silicides ScsSis and Dy5Si3 are completely miscible. The variations of the lattice spacings vs. composition of solid solutions obey Vegard s rule in case of Sci j Dyj Si, or show slight deviations fi-om this rule (Dy5-j SCj Si3, Dy5 jSc cSi4). Two ternary compounds occur in this system (see table 19). One of them also has a variable composition of Sc and Dy. 

The Sc-Nb Si isothermal section at 800"C presented in fig. 61 has been reported by Kotur (1986). Solid solution of substitution Sc5 1.5Nbo-3.5Si3 occurs in the system. The variation of the lattice parameters show negative deviation from Vegard s rule. Other binary scandium and niobium silicides do not dissolve the third component. T vo ternary compounds exist in the system. Probably Sco.4Nb4.6Si3 is a solid solution of the binary... 

The isothermal section at 800"C of the Sc-Cr-Ge system has been studied by Kotur et al. (1988b) (fig. 79). The binary ScsGe3 germanide dissolves 7.5 at.% Cr. A positive deviation fi Om Vegard s rule is observed for the lattice parameters of this solid solution. Four ternary compounds exist in this system, their crystal structure data are shown in table 20. The compoimd ScCrGc2 (3) has been synthesized independently by Kotur et al. (1988b) and Venturini et al. (1985). 

Only minimal data have been reported on the density of liquid alloys containing R metals. Measurements of for the R-R alloy Pr-Nd indicate this to be an ideal substitutional alloy, one obeying Vegard s Rule ... 

Urusov, V.S. (1992) A geometric model of deviation from Vegard s rule. /. Solid State... 

Vegard s rule, i.e., the assumed linear increase of the lattice constants from that of Tm Se to (hypothetical) Tm Se, is often used to determine the Tm valence from the lattice constants, but this linear interpolation must be handled cautiously, see for example, Vettier [2]. An equation is proposed, based on a linear interpolation oetween the bulk moduli, which leads to a nonlinear relationship between the true valence from lattice constants and the Vegard valence . A calculated curve is in fair agreement with the results of other determination methods for a figure, see the paper, Neumann et al. [3]. 

The lattice parameter of a simple solid solution usually follows Vegard s rule (i.e. a linear lattice parameter dependence on the solute concentration) with good approximation, often for quite a few percent of foreign cations, depending on circumstances. Recently, deviations from Vegard s rule have been described in... 

The mutual solid solubility of the III-Vs is a further property of fundamental importance of this class of compounds. Alloying results in ternary or quaternary (and higher) compounds with electronic and other physical properties that roughly obey Vegard s rule. This makes heterojunctions possible with tailored band structure for bipolar opto- and microelectronic devices. In layer stractures the lattice mismatch can be controlled by appropriate alloying to reduce defect generation or to determine the strain in the layers, if required. 

The existence of a continuous series of solid solutions Lai tEUjB6, CaB -type, Pm3m, was confirmed by Aivazov et al. (1979), who used various physico-chemical methods. Samples were synthesized by borothermal reduction of the corresponding oxides (1900-2000°C). On the basis of the electrophysical properties a supposition is made on an increase of the ionic component in the Me-B bond when Eu is substituted for La. Boron and metal defects were determined from density and lattice parameter measurements (small negative deviation from Vegard s rule). Their X-ray results essentially agree with the data by Mercurio et al. (1974). 

A continuous solid solution (Y, La)B6 with CaB -type of structure, Pm3m, was found from specimens b), but a restricted solid solution appears from alloys a), due to the different melting behavior of YBj (incongruent) and LaBg (congruent), which results in a nearly complete fractional crystallization of the two hexaborides. At variance with the results obtained by Bondarenko et al. (1966) on samples prepared by borothermal reduction, lattice parameters of specimens b) closely obey Vegard s rule and are shown in fig. 36 including superconductivity data. 

Tarascon et al. (1980) investigated the valence transition of Sm in the Sm,, Th,B hexaboride solid solution (Pm3m). For sample preparation as well as methods of investigation employed, see Sm, Y Bj. Th + substitution of Sm was found to decrease the average Sm valence more rapidly than La + or Y. The variation of lattice parameters (pronounced positive deviation of Vegard s rule) is presented in fig. 34c. 

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