Hume-Rothery’s rule

Hume-Rothery s rule The statement that the phase of many alloys is determined by the ratio.s of total valency electrons to the number of atoms in the empirical formula. See electron compounds. 

Nitrogen and oxygen do not obey Hume-Rothery s rule, since in both cases there is only one nearest neighbour and not 8 — N. This is evidently due to the fact that the bonds in Og and Ng are more stable than two single O—O bonds and three single N—N bonds respectively. 

The answer to this question was provided by Hume-Rothery, who in 1926 made the empirical observation that the widespread occurrence of the / , y and e phases in chemically dissimilar systems and at widely differing compositions is determined, not by the chemical properties of the elements concerned, or by any arguments based on valency concepts, but solely by the relative number of valency electrons and atoms in the crystal structure. This generalization, usually known as Hume-Rothery s rule, is illustrated by the data of table 13.03, from which it will be seen that the / , y and e phases are characterized by electron atom ratios of 3 2, 21 13 and 7 4, respectively. In each case this ratio alone determines the structure, and the relative number of atoms and the particular atoms by which the electrons are contributed appear to be of... 

All the phases shown in table 13.03 will be seen to obey Hume-Rothery s rule. In cases where the composition of any phase has been determined on structural grounds it has, with few exceptions, been found to be consistent with the rule. In the majority of cases, however, the formula quoted is that deliberately chosen to give the appropriate electron atom ratio. In the copper-tin system, for example, the y phase has a purely statistical distribution of atoms, so that structurally no particular composition can be preferred. The range of homogeneity is in this case very narrow, but once the possibility of a formula as complex as Cu31Sn8 is admitted, several of no greater complexity could doubtless be found this particular formula is chosen because it gives the appropriate electron-.atom ratio of 21 13. In such cases, however, it is... 

The simpler of these two structures is the caesium chloride arrangement, found in the phases LiHg, LiTl, MgTl, CaTl and SrTl. This is, of course, also the structure of the / phase in the silver-cadmium system and in other electron compounds (fig. 13.11), and for this reason the systems just mentioned are sometimes quoted as exceptions to Hume-Rothery s rule. Apart from this geometrical resemblance, however, these systems have little in common with the electron compounds, and it seems preferable to regard the Hume-Rothery rule as applicable only to alloys of the T2-B1 type. 

Exactly similar arguments apply to the y phases. Here the extent of the solid solution has often led to the assignation of incorrect formulae, especially since the composition satisfying Hume-Rothery s rule is relatively complex so that a composition differing but little from it can often be expressed by a simpler formula. Disordered structures are again common, and even when an ordered arrangement obtains the distribution of the atoms in closely related structures is not necessarily... 

Hume-Rothery s Rules of Solid Solubility (metallurgy) (1) Complete miscibility can occur only if the unit cells of the two components are essenhally alike. (2) If the diameters of... 

ZintI Phases. Invoking Lewis octet rule, Hume-Rothery published his 8 —N rule in 1930 to explain the crystal stmctures of the p-block elements (Hume-Rothery, 1930, 1931). In this expression, N stands for the number of valence electrons on the p-block atom. An atom with four or more valence electrons forms 8 - N bonds with its nearest neighbors, thus completing its octet. The Bavarian chemist Eduard Zintl (1898-1941) later extended Hume-Rothery s (8 - N) mle to ionic compounds (Zintl, 1939). In studying the stmcture of NaTl, Zintl noted that the Tl anion has four valence electrons and he, therefore, reasoned that this ion should bond to four neighboring ions. 

The anion connectivity of many Zintl phases can be rationalized in terms of Hume-Rothery s (8 V) mle. For example, in BaSi2 (with Si clusters), the Si anion is isoelectronic with the nitrogen group elements, that is, it has five valence electrons. The (8 N) rule correctly predicts that each silicon atom will be bonded to three other sUicon atoms. Similarly, in Ca2Si, Si is isoelectronic with the noble gas elements. Again, the 8 A mle correctly predicts that silicon will occur as an isolated ion. Indeed, this compound has the anti-PbCl2 stmcffire, in which the sUicon is surrounded by nine calcium ions at the comers of a tricapped trigonal prism. 

Finally, from x = 0.98 until x = 1.0, a solution of copper in the zinc matrix is observed. The regions of homogeneous solutions are examples of Hume-Rothery s third rule. The higher valent atom (zinc) is more soluble in the lower valent solvent (copper) then vice versa. 

From Tsai s pioneering discoveries [25,27], we know that atomic size, electronegativity, and valence electron counts play substantial roles in the formation of QCs. These criteria are expressed by the Hume-Rothery rules [30,31]. However, three additional highlights are also important in the consideration of possible candidate systems, at least from the viewpoint of chemists. 

Just as the saturated solubility of sugar in water is limited, so the solid solubility of element B in met A may also be limited, or may even be so low as to be negligible, as for example with lead in iron or carbon in aluminium. There is extensive interstitial solid solubility only when the solvent metal is a transition element and when the diameter of the solute atoms is < 0 6 of the diameter of the solvent atom. The Hume-Rothery rules state that there is extensive substitutional solid solubility of S in >1 only if ... 

Bla67] Blandin, AE, Theoiy of the Hume-Rothery Rules, in Phase Stabdity in Metals and Alloys, P.S. Rudman, J. Stringer, and R.I. Jaffee, Ed., McGraw-HiU, 1967, p. 115-124... 

---