Dr. Orgel

Linus Pauling

29 December 1953

Proposed research - Brillouin zones in intermetallic compounds

The following investigation should be somewhat simpler than the one that I suggested on the resonating-valence-bond treatment of metals. Moreover, it is more closely related to the molecular-orbital method. Perhaps you would be interested to begin work on this problem.

I propose that a study be made of Brillouin zones and electron numbers in intermetallic compounds in which the distribution of atoms of different kinds is such that there is a larger density of valence electrons in one portion of the unit of structure than in another portion.

I think that the structure to use as the first example is that of Cr_{4}Si_{4}Al_{13}, described by Keith Robinson in Acta Crystallographica 6, 854 (1953). This crystal is based upon a face-centered cubic lattice with a_{0}= 10.917, and with one Cr_{4}Si_{4}Al_{13} per lattice point. If we assume chromium to be sexivalent, there are 79 valence electrons per formula, or 3.76 per atom. Robinson has pointed out that the first important Brillouin polyhedron contains about 1.75 electrons per atom, and the second contains about 7.62 electrons per atom. Accordingly the Brillouin-zone theory seems not to apply to this crystal.

About five years ago Dr. F. J. Ewing and I published a paper in Reviews of Modern Physics in which we pointed out that the gamma alloys and other complex intermetallic compounds have in general enough valence electrons, using the large values (approximately 6) for the valences of the transition elements, just to fill a Brillouin zone. We ask accordingly why Cr_{4}Si_{4}Al_{13} does not show similar agreement with the simple theory.

Dr. Orgel
Linus Pauling
29/12/53 - Page 2

First, there is evidence that the valences 6 for chromium, 4 for
silicon, and 3 for aluminum are correct. If the interatomic distances are interpreted in the usual way (see my paper J.A.C.S. 1947) to obtain the bond numbers, the valences found are 4.0 for Al_{0}, 4.7 for Si, 6.1 for Cr, 3.8 for Al_{1}, and 2.6 for Al_{2}. These numbers indicate some electron transfer, to a value somewhat higher than the total possible number of valence electrons, and it is accordingly unlikely that any of the electrons outside of the next inner noble-gas shell are not valence electrons.

A clue is given by the structure of the crystal. We may describe it by saying that there are Cr_{4}Si_{4} complexes arranged in cubic closest packing, and with the space between these complexes filled with aluminum atoms. The complexes are roughly spherical in shape - there are four chromium atoms at the corners of a tetrahedron, and four silicon atoms at the comers of a somewhat larger negative tetrahedron. As an approximation we can say that each complex is a sphere, with volume 8/21 of the total volume per lattice point, that is, the spheres that are arranged in closest packing have volume 38 percent of the total.

The aluminum atoms have about 3 valence electrons apiece, and the Cr_{4}Si_{4} complex accordingly has 16 electrons more than it would have if each chromium and silicon atom had 3 valence electrons. We conclude that the average electron density within this sphere is 67 percent greater than the average electron density outside of this sphere.

Dr. Orgel Linus Pauling 29/12/53 - Page 3

In the ordinary Brillouin-zone theory, however, no consideration is given to the possibility of any other than a uniform electron distribution. The energy levels are calculated for a free electron moving in a constant potential, and the perturbations that produce the Brillouin zones are then obtained by consideration of the electron wavelengths in relation to Bragg reflection from important crystallographic planes. This method works very well for many crystals. Accordingly we may conclude that a refinement of the theory in such a way as to give a 67 percent increase in electron density around the Cr_{4}Si_{4} complexes might well be satisfactory for the crystal Cr_{4}Si_{4}Al_{13}.

I suggest that you attempt to carry out a treatment of the problem of
the motion of one electron in a three-dimensional sinusoidal field with a
potential minima at the points of a face-centered lattice, and maxima in the intervening regions. Mr. Gary Felsenfeld can, I think, give you references to work that has been done on the solution of the wave equation for such a potential function. The parameter defining the potential function is then to be adjusted until the wave functions for the 79 electrons per lattice point describe an electron distribution that places 16 extra electrons in the sphere around each lattice point - that is, places 40 electrons in this sphere, and 39 electrons in the region outside of the sphere. Next a consideration should be made of the wavelengths of the electrons in relation to the spacings of the planes defining the Brillouin zones, in order to see whether Brillouin-zone stabilization would provide an explanation of the

Dr. Orgel Linus Pauling 29/12/53 - Page 4

stability of the observed structure.

There are a number of other substances to which a similar treatment might
be applied. Perhaps you would prefer to begin with CaB_{4}, which is somewhat
simpler. I have carried out a discussion of Brillouin-zone stabilization in
this crystal by a different method. Here the calcium atom is at the
points of a simple cubic lattice, and these atoms are enclosed in a framework
of boron atoms. There are altogether 20 valence electrons in the unit cube -
that is, per formula CaB_{6}. Of these, only 2 are to be associated with the
calcium atom. The calcium atom has volume about 32 A._{3}, slightly over one
half of the volume of the unit cell, 71 A._{3}. The problem accordingly would be
attacked by setting up a simple sinusoidal potential function with maxima
at the lattice points (occupied by the calcium atoms) and minima in other regions,
which are occupied by the boron framework. The single parameter determining the
potential difference between maxima and minima would be given such a value
that when 20 electrons are introduced into the 10 most stable orbitals there
are 2 electrons in the sphere with radius 1.97 about each lattice point and
18 electrons in the remaining regions. The wavelengths in relation to the
principal interplanar spacings should then be investigated to see whether or
not there is significant Brillouin-zone stabilization. I have found that
the x-ray intensity of a plane cannot be taken as a satisfactory measure of
its significance in producing Brillouin perturbation, in such a case. Since
the wave functions tend to evade the calcium atoms, and the electrons are
concentrated in the neighborhood of the boron atoms, it is the scattering
power of the boron atoms for electrons that determines the magnitude of the
perturbation in this crystal.

cc: Prof. Pauling, Prof. Bergman