Vegard rule

For the size of the unit cell of M " substituted Fe " oxides, a linear relationship between the Fe oxide and the appropriate structural analogue has been postulated. This is the so-called Vegard rule (Vegard, 1921). In practice, the change in unit cell parameters often deviates from the Vegard line, with the extent and direction of the deviation being different for different substituents and different unit cell parameters... 

L. Piper 2 found that the reflections belonging to the molecular lengths in a mixture of palmitic and stearic acids appear broadened and lie between the positions due to the pure components, a behavior which is in line with the Vegard rule for inorganic mixed crystals. 

A continuous series of solid solutions forms between fee LaN and ThN at 1700 C [4] and at 1900°C [1] with only a small negative deviation from the Vegard rule in LaN-rich solutions. No ternary compound exists at 1200°C and Ng pressures up to 30 atm, at which pressure the solid solution decomposes into the binary nitrides ThgNa (sic, probably Th3N4) and LaN [4]. A miscibility gap is predicted with the calculated critical point of 1450 K [2]. 

A continuous series of fee solid solutions form between CeN and ThN with positive deviations from the Vegard rule [1, 4]. The 1973-K isotherm proposed by Holleck [3] is illustrated in Fig. 9. 

A continuous series of fee solid solutions form between MN (M = Pr, Nd, Sm, Gd, Dy) and ThN with negative deviations from the Vegard rule (in the case of Pr and Nd only small) after quenching from 1700°C [4]. According to Holleck, Smailos [1] the room temperature lattice parameters of Th-Pr-N and Th-Nd-N solid solutions are reported to follow the Vegard rule after quenching from 1900X. 

A complete series of solid solutions forms between PuN and ThN [1,3, 7]. An alloy with the composition (Tho.i5Puo.85)N and containing 0.1 to 0.3% oxygen has been reported to be single phase with the fee lattice parameter a = 498.0 pm indicating an appreciable positive deviation from the Vegard rule [8]. 

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). 

Vegard s law is an empirical rule which holds that an approximate linear relation exists between the crystal lattice parameter of an alloy and the concentration of its constituent elements (L. Vegard, Z. Kristaflogr. 67, 239 (1928). See also A.R. Denton, N.W. Ashcroft, Phys. Rev. A 43, 3161 (1991)). 

According to Vegard s Law, in order to form a solid solution, the cations and/or anions of two compounds must have ionic radii within 15% of each other. The rule is not so rigid if the solid solution content is kept within a limit of no more than 5% of one in the other. That is In the composition, (1- x) MSi04 x NSi04, if x is kept below 5%, then a solid... 

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). 

Vegard [84,85] established that in ionic salts there is a linear relation between the lattice parameters of a mixed crystal and the concentrations of the components. This rule applies also to mixed crystals of organic compounds. Thus, low-temperature phases of compounds CBr4 nCln (n = 0 — 4) are isostructural and form continuous series of mixed crystals [86]. It has been, moreover, identified that the fractional occupancy of the halogen atoms fully controls the lattice dimensions. Lattice parameters as a function of the fractional occupancy of chlorine atom slightly deviate fl om the Vegard s law (Fig. 6.6). 

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. 

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