Hume-Rothery ‘rules

Just as the saturated solubility of sugar in water is limited, so the solid solubility of element B in metal 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 B in >1 only if ... 

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. 

The formation of solid solutions of metals is one way to change the properties (generally to increase strength) of the metals. Strengthening metals in this way is known as solid solution strengthening. The ability of two metals to form a solid solution can be predicted by a set of rules known as the Hume-Rothery rules, which can be stated as follows ... 

Solid solutions occur widely in inorganic and mineral systems (Table 4.2). The Hume-Rothery rules apply well in these cases, but it is important to add ... 

These days the variables in the Hume-Rothery rules can be made quantitative by making use of computer simulation or quantum mechanical calculations (Section 2.10). 

In the general case a complex behaviour may be expected for the extension of the terminal solid solutions which, for a pair of metals Mb M2, also depends on the stoichiometry and stability of the M (or, respectively, M2) richest phase. However a certain regularity of the dependence of the mutual solid solubility on the position of the metals involved in the Periodic Table may be observed. This can be related to the so-called Hume-Rothery rules (1931) ... 

Person 2 Do Cu and Ni satisfy the third and fourth Hume-Rothery rules for complete solid solubility ... 

If a Brillouin zone is nearly full or nearly empty, the electron concentration may become an important factor in determining alloy stability. This factor forms the basis of the Hume-Rothery rules (8) and is of great importance in intermetallie semiconductors. 

A simple, but nevertheless very interesting electron counting rule for close-packed high nuclearity clusters has been described by Teo127a. The formula N = (mb + rms)n establishes a connection between the total number of electrons (N), the bulk atoms mb, and the surface atoms ms. The rule is based on an extended Hume-Rothery rule and works quite well for all kinds of close-packed clusters up to the Pt38 cluster of Chini61. ... 

Around 1928, Zintl had begun to investigate binary intermetallic compounds, in which one component is a rather electropositive element, e.g., an alkali- or an alkaline earth metal [1,2]. Zintl discovered that in cases for which the Hume-Rothery rules for metals do not hold, significant volume contractions are observed on compound formation, which can be traced back to contractions of the electropositive atoms [2]. He explained this by an electron transfer from the electropositive to the electronegative atoms. For example, the structure of NaTl [3] can easily be understood using the ionic formulation Na Tl" where the poly- or Zintl anion [TF] forms a diamond-like partial structure - one of the preferred structures, for a four electron species [1,2], Zintl has defined a class of compounds, which, in the beginning, was a somewhat curious link between well-known valence compounds and somehow odd intermetallic phases. 

In general, for a mixture of two or more pure elements, there are two types of solid-solution alloys that may be obtained. Type I alloys are completely miscible with one another in both liquid and solid states. As long as the Hume-Rothery rules are satisfied, a random substitutional alloy will be produced. We will see many examples of these alloys in this section for a variety of metal dopants in stainless steels. By comparison, type II alloys are only miscible in the molten state, and will separate from... 

Certain other alloys also with the body-centred cubic structure, LiHg, MgTl, etc., which have already been mentioned, have sometimes been regarded as exceptions to the Hume-Rothery rules. They fall into line with the other P structures only if we assume that Hg or T1 provides one valence electron. Since, however, the radii of the metal atoms in these alloys, as in those with the NaTl structure, are smaller than the normal values, it is probably preferable not to regard these as 3 electron compounds. 

It is not easy to find other data which show that Eg plays a role in the reactions of solids. There is one case for metals which can be used as a test the formation of alloys, which represents, at least in a sense, the reaction of two solid metals with each other. The stability of alloys of varying composition is influenced by factors such as the relative atom sizes, and the number of valence electrons per atom (the Hume-Rothery rules). 

Recently, Teo has used an extended form of the Hume-Rothery rule (developed for intermetallic alloys125 to account for the structure adopted by pseudo-close packed clusters126. His conclusions are in agreement with the earlier work of Johnson, Messmer et al. who found that the density of states of a bare metal cluster rapidly converges to that of the bulk metal for as few as 13 (f.c.c.) or 15 (b.c.c.) metal atoms127. ... 

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. 

Solid-fluid phase diagrams of binary hard sphere mixtures have been studied quite extensively using MC simulations. Kranendonk and Frenkel [202-205] and Kofke [206] have studied the solid-fluid equilibrium for binary hard sphere mixtures for the case of substitutionally disordered solid solutions. Several interesting features emerge from these studies. Azeotropy and solid-solid immiscibility appear very quickly in the phase diagram as the size ratio is changed from unity. This is primarily a consequence of the nonideality in the solid phase. Another aspect of these results concerns the empirical Hume-Rothery rule, developed in the context of metal alloy phase equilibrium, that mixtures of spherical molecules with diameter ratios below about 0.85 should exhibit only limited solubility in the solid phase [207]. The simulation results for hard sphere tend to be consistent with this rule. However, it should be noted that the Hume-Rothery rule was formulated in terms of the ratio of nearest neighbor distances in the pure metals rather than hard sphere diameters. Thus, this observation should be interpreted as an indication that molecular size effects are important in metal alloy equilibria rather than as a quantitative confirmation of the Hume-Rothery rule. 

DFT studies of binary hard-sphere mixtures predate the simulation studies by several years. The earliest work was that of Haymet and his coworkers [221,222] using the DFT based on the second-order functional Taylor expansion of the Agx[p]- Although this work has to some extent been superceded, it was a significant stimulus to much of the work that followed both with theory and computer simulations. For example, it was Smithline and Haymet [221] who first analyzed the Hume-Rothery rule in the context of hard sphere mixture behavior and who first investigated the stability of substitutionally ordered solid solutions. The most accurate DFT results for hard-sphere mixtures have come from the WDA-based theories. In particular the results of Denton and Ashcroft [223] and those of Zeng and Oxtoby [224] give qualitatively correct behavior for hard spheres forming substitutionally disordered solid solutions. 

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