## Classics in materials science: Vegard’s law of linear relationship between lattice parameter and alloy composition

Let us consider a pure metal, say copper; let us consider the case in which we remove some copper atoms and substitute for them with gold atoms; since gold atoms are slightly bigger than copper atoms, it is natural to expect that the lattice parameter in this copper based alloy (compared to pure copper) increases with increasing amounts of gold; this expectation is indeed borne out by experiments; for example, the following data (taken from Table 1 of [1]), clearly indicates such a behaviour in copper-gold alloys:

Copper-0.23 atomic % Au: 3.616 Å
Copper-2.8 atomic % Au: 3.632 Å
Copper-10.0 atomic % Au: 3.669 Å

One can go ahead and ask a more quantitative question, namely, as to what the relationship between the increase in the lattice parameter and alloy composition is. One of the most common answers to that question in materials science is that the lattice parameter increases linearly with the alloy composition; this empirical, widely used, and, mostly approximate relationship is known as Vegard’s law; in this post, I would like to discuss the origins of Vegard’s law and its relevance.

In a paper published in Zeitschrift für Physik in 1921 and titled Die Konstitution der Mischkristalle und die Raumfüllung der Atome (which, google translates for me as The constitution of the mixed crystals and the space filling of the atoms), L Vegard, based on his X-ray crystallographic studies on several ionic salts, concluded that the increase in lattice parameter of the solid solution with increasing amounts of solute to be linear — to quote Vegard’s own words [2]:

Die für das System KBr — KCI gefundene Seitenlänge (a) des Elementargitters erfüllt mit großer Genauigkeit die Gleichung der Additivität:
$a_m = \frac{p-100}{100} a_{\mathrm KBr} + \frac{p}{100} a_{\mathrm KCl}$
wo p die Anzahl Molekularprozente KCI im Mischkristall bezeichnet. Welche Allgemeinheit und Strenge diesem Additivitätsgesetz zukommt, müssen weitere Untersuchungen ergeben. Auch die Molekularvolumina folgen annähernd diesem Additivitätsgesetz. Die angegebenen Zahlen aber deuten darauf bin, daß die linearen Dimensionen die Additivität am besten erfüllen.

Here is the translation of Google of the above passage:

For the system KBr – KCI found side length (a) of the Elementary lattice fulfilled with great accuracy the equation of Additivity:
$a_m = \frac{p-100}{100} a_{\mathrm KBr} + \frac{p}{100} a_{\mathrm KCl}$
where p is the number of molecular percent KCI in the mixed crystal means. What general and rigor Additivitätsgesetz this role, need further investigation. Also, the molecular volumes follow approximately this Additivitätsgesetz. The figures But I suggest that that the linear dimensions of the additivity best meet.

Thus, from the above passage, it is clear that Vegard

(i) claims a linear relationship between lattice parameter and alloy composition;
(ii) wants further investigations as to the origins of this behaviour; and,
(iii) though there is a similar behaviour seen with respect to the molecular volumes, he believes the lattice parameter change to be the best described by such a linear relationship.

And, thus was born the Vegard’s law.

An aside
: By the way, notice Vegard’s use of the word “mixed crystal” in both the title and body of his apper — in spite of Heycock and Neville’s recommendation nearly 17 years earlier to discard the terminology in favour of solid solution.

Unfortunately, I am not fluent enough in German to understand all of Vegard’s arguments and conclusions (for example, did Vegard arrive at his conclusion by just looking at three data points? his Table 2, reproduced below makes me think so — but I could be wrong); what is more, I do not have access to another of Vegard’s paper (published in 1928, I understand — L. Vegard, Z. Kristallogr. 67, 239 (1928).) which is also mentioned, almost invariably, in the same breath as this one, whenever a reference to the law is made. Finally, it is not clear as to how the result from the study of ionic crystals got carried over to metallic systems — a topic, which, by itself, is very interesting and might be a worthy supplement to Cahn’s The Coming of Materials Science, if only somebody looks into it.

Having made the above disclaimers, from the literature, it is clear is that this law has been used widely in spite of known deviations and its empirical nature; here is a quote from a paper of Denton and Ashcroft [3] (which paper also forms the basis for most of wiki’s short entry on Vegard’s law):

In an early application of x-ray diffraction to the analysis of crystal structure, Vegard observed that in many ionic salt alloys a linear relation held, at constant temperature, btween the crystal lattice constant and concentration. This empirical rule has since come to be known as “Vegard’s law,” although in subsequent extensions of the rule to metallic alloys the majority of the systems have been found not to obey it.

I believe, one reason for this popularity of the law is the fact that the lattice parameter change with concentration is one of the important pieces (and, as Denton and Ashcroft note in their paper, rather fundamental piece) of information in understanding many interesting properties of alloys — for example, the wiki entry on the law mentions the determination of semiconductor band gap energies — and, a linear relationship is the minimum that one needs to assume (not to mention that linearity comes with its own attractions!).

While deviations from Vegard’s law and the reasons behind the same continue to be active areas of study (as is clear from a simple Google Scholar-ing of the words Vegard’s law), a related, but slightly different question is also of interest: under what conditions is Vegard’s law strictly obeyed or expected to be obeyed? During my search for materials for this post, I found a couple of answers to this question; one is from the paper of Denton and Ashcroft (mentioned above) based on their density functional theory:

In conclusion, the results of the (parameter-free) density-functional theory presented here demonstrate that, independent of other factors [the relative volume per valence electron in crystals of the pure elements, Brillouin-zone effects and electrochemical differences between the elements], a simple geometric difference in atomic sizes can play a significant role in determining the crystal structure — in particular, the form of the lattice constant-concentration relationship — of binary alloys. They suggest, furthermore, that the linear relationship predicated by Vegard’s law for alloys at constant temperature may also extend — for sufficiently small atomic size disparities — to the fluid-solid coexistence curve, along which the temperature varies.

The second one, which is very interesting is from the work of Thorpe and Garboczi [4]:

We study a triangular network containing two kinds of Hooke springs with different natural lengths. If the two spring constants are the same, we can solve the model exactly and show that Vegard’s law is obeyed, irrespective of whether the bonds are arranged randomly or in a correlated way.

Let me end this post with a reference to the solid state chemistry text of Anthony West which contains a very interesting discussion on the deviations from Vegard’s law [5].

#### References

[1] S G O’Hara and B J Marshall, Elastic constants of copper-rich alloys with gold, Physical Review B, 3, 4002-4006, 1971.

[2] L Vegard, Die Konstitution der Mischkristalle und die Raumfüllung der Atome, Zeitschrift für Physik, 5, 17-26, 1921.

[3] A R Denton and N W Ashcroft, Vegard’s law, Physical Review A, 43, 3161-3164, 1991.

[4] M F Thorpe and E J Garboczi, Elastic properties of central-force networks with bond-length mismatch, Physical Review B, 42, 8405-8417, 1990.

[5] A R West, Solid state chemistry and its applications, John Wiley and Sons, 1991.