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The lattice sickness pandemic

Massimo Nespolo
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One of the first and most fundamental concepts every crystallographer learns in their (scientific) infancy is that of lattice, i.e. an infinite set of geometric points (known as lattice nodes) that represent the translational symmetry of an ideal crystal. If one node is taken as origin, all the other nodes are obtained by applying to it all the vectors that represent a translation of the crystal pattern, i.e. a rigid motion that does not modify the orientation of any element of the crystal pattern. The number of possible lattices is always infinite. However, lattices obtained by translation vectors that are one-by-one parallel to each other and differ in their length define a single type of lattice. For example, there are an infinite possible number of face-centred cubic (f.c.c.) lattices, which all differ by the length of the basis vector a; they all correspond to a single type of f.c.c. lattice. In one-, two- and three-dimensional spaces there are one, five and 14 unique lattice types, respectively, which are universally known as Bravais lattice types (they could have been named “Frankenheim lattice types” if Moritz Ludwig Frankenheim hadn’t earlier counted 15 instead of 14 types in three dimensions).

A lattice being infinite, crystallographers use a finite portion to represent it: the unit cell. This unit cell has only two requirements: to have lattice nodes at its corners, and to be able to reconstruct the whole lattice by translations parallel to its edges. These requirements allow a large (in principle infinite) number of choices for the unit cell; the description of the crystal structure and the Hermann–Mauguin symbol of the space group depend on this choice. Some choices are particularly convenient; one unit cell has the same symmetry as the lattice (the conventional cell), another unit cell may be particularly useful to compare the structure of a related compound or of a polymorph; still another unit cell can be preferable for studying a twinned crystal; in the end, the “best” unit cell depends on the choice of the investigator. Some unit cells are so often adopted that they have come to be indicated by a letter, for the sake of brevity (P, A, B, C, I, F, R, H, D; the last two are much less often used, though). In some cases, one may find it advantageous to use a unit cell that does not have a special name (letter): in that case, the centring translation vectors are explicitly indicated when the cell is chosen, and in the rest of the publication the author simply uses a non-reserved letter (e.g. X has been used in the literature).

Whatever be the choice of the unit cell, the latter is like an empty box, a container that waits for its contents: the atoms. Indeed, the unit cell does not have an atomic nature, it is a geometric abstraction made up of zero-dimensional points. It goes without saying that the lattice described by that cell is a geometric abstraction itself. Once atoms take their place in the unit cell, and therefore in the infinite unit cells all obtained by the translations, one gets a crystal structure (often shortened just to “structure”); if one puts something else (drawings of flowers, birds, fishes etc.), the result is a crystal pattern, i.e. an extended network of (real or imaginary) objects that have the same periodicity as a crystal structure.

The reader may wonder why I am wasting so much space emphasising something that should be considered a lapalissade.1 Unfortunately, the scientific literature is polluted by completely nonsensical uses of the term “lattice”, which makes this call for attention unexpectedly necessary. I have recently published an article in the Teaching and Education section of the Journal of Applied Crystallography that deals with the confusion between lattice and structure (as well as between periodicity and dimensionality), where I have discussed some examples taken from the recent literature. The purpose of this text in this IUCr Newsletter is to alert a larger number of readers who are probably unaware of the confusion and may be making the same mistake without even realising it. In the following I repeatedly quote Wikipedia and other online resources, not because I consider them to be a reference, but because the world wide web is today a primary source for so many readers who incorrectly think that if a term is used in many different places, then it cannot be wrong. Unfortunately, copying-and-pasting the same mistake does not make it less wrong. As William James stated, “There's nothing so absurd that if you repeat it often enough, people will believe it”.

Practically every reader has most likely met expressions like “lattice vibrations” (a geometric point does not vibrate; an atom does), “lattice energy” (how can a geometric point have some energy associated with it?), “lattice dynamics” (a geometric point does not evolve with time), “ionic lattice” (points cannot have an electric charge) and so on. What the writer has in mind is obviously the structure, but because of an incorrect shortcut, he calls it lattice. This bad habit has such deep roots that often one no longer recognises how meaningless the term being used is. The most representative example is probably that of lattice energy, defined as “a measure of the energy released when ions are combined to make a compound” (https://en.wikipedia.org/wiki/Lattice_energy), a definition that every chemistry student has to metabolise in their first year. Yet, it is sufficient to think for a few seconds to understand how self-contradictory it is. Indeed, “when ions are combined to make a compound” the result is a structure so that what is measured or computed is a structural energy. Unfortunately, the term “lattice energy” is so profoundly rooted that any effort to correct it would certainly be a Cervantian battle against windmills. Let us declare ourselves satisfied if the readers and the authors realise they are facing a shortcut that, for historical reasons, would be hard to fix.

In much the same way, “lattice vibrations” is used as a synonym for phonon. The latter is defined as “a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, like solids and some liquids” (https://en.wikipedia.org/wiki/Phonon). It is therefore self-evident that what vibrate are atoms, not geometric points. In this case, the much more suitable term phonon does exist and is extensively used as well. There is therefore absolutely no reason to forgive an aberration like “lattice vibration”: this expression must simply be banned.

“Lattice dynamics” is another absurdity that you can frequently meet. It is defined as “the study of the vibrations of the atoms in a crystal”, so that we have one more meaningless term used instead of the correct word “phonons”. The BBC bitesize website falls into the same type of trap, when it gives a definition of ionic compounds such that “[a]n ionic lattice is held together by strong electrostatic forces of attraction between the oppositely charged ions”. The reader may enjoy wandering among other masterpieces like lattice water, for example. There are even people pretending to produce lattice waves, or manipulate lattice strain to influence lattice thermal conductivity (energy transmission by means of pure geometry – not even Walt Disney would have had such an idea)! Even more eyebrow raising is to discover that some may actually want to try to damage a lattice. Time permitting, one may play a little game and harvest meaningless titles in publication databases: not recommended for those with a history of depression!

But what is the origin of this confusion? To track down the original sin that has damned crystallographers apparently without hope for redemption is certainly not an easy task. Without any pretension at divination, one may make an educated guess by considering the simplest yet exceptional cases of crystal structures in which atoms occupy positions that coincide with lattice nodes. The cubic close-packing of spheres corresponds to a crystal structure in a space group of type Fm3m with spheres at the origin (x = y = z = 0) and at the centre of each face (two of the fractional coordinates ½, the third 0). This corresponds to Wyckoff position 4a and also to the fractional coordinates of the lattice nodes when the lattice is described by the conventional (F) unit cell. Therefore, the cubic close-packing (c.c.p.) is also called the face-centred cubic (f.c.c.) structure whose lattice, needless to say, is face-centred cubic as well. Yet, the two concepts are by no means equivalent: the structure is made of atoms, and a number of metals crystallise in this type of structure (Ni, Cu, Ag, Au, Pt, Ir, Rh, Pb, Th, Ac), the lattice of geometric points, and in this specific case the atoms are on the positions of lattice nodes. A similar case is that of b.c.c. for the body-centred cubic structure and lattice. A number of metals crystallise in the b.c.c. structure type (V, W, Ta, Cr, Mo, Nb, Mn, Eu, Ba), which correspond to space groups of type Im3m and atoms with coordinates (0,0,0) and (½,½,½), i.e. Wyckoff position 2a. These are also the coordinates of lattice nodes when the lattice is described by the conventional (I) unit cell. These extremely simple yet very special cases are often used in textbooks to introduce crystal chemistry. Unfortunately, a clear distinction between lattice nodes and atomic positions is often missing and the student is misled into believing that the concepts coincide. A clear example is that of the CsCl structure, which is primitive cubic: one type of atom sits on the origin, the other at the centre of the unit cell, but the two atoms being different there is no translation vector relating the two positions as is instead the case for the b.c.c. structure. CsCl crystallises in a space group of type Pm3m, with one atom in position 1a and the other in position 1b. Yet, a number of textbooks and online resources describe CsCl as “cubic centred”, as if the chemical difference between the atoms could be ignored.

The situation starts getting more confused in the case of the diamond structure, which crystallises in a space group of type Fd3m with carbon atoms in Wyckoff position 8a: for origin choice 1, the coordinates in the asymmetric unit are 0,0,0 (they become ⅛,⅛,⅛ for origin choice 2). Now, the multiplicity of the Wyckoff position is 8, which means that there are eight atoms in the unit cell, whereas the F-centred unit cell contains only four nodes. This difference should be sufficient to avoid any possible confusion. Yet, because there are none so blind as those who will not see, the structure of diamond is often described as “two interpenetrating face centered cubic Bravais lattices”. Now, if one imagines taking an f.c.c. lattice L1, making of copy of it (let it be lattice L2) and moving the latter with respect to the former by ¼ along one of the body diagonals, the translation vectors of the resulting construction, which define the lattice, connect two nodes of L1 or two nodes of L2: the lattice is still L1. A vector connecting a node of L1 and a node of L2 does not extend beyond those two nodes, i.e. it does build up a periodic distribution of nodes, the definition of which is a lattice. If one instead takes an f.c.c. structure S1 (e.g. the structure of Ni: Figure 1a), repeats the process above to obtain the structure S2 and moves S2 by ¼ along one of the body diagonals of S1, the result is a new structure (Figure 1b): the sphalerite type of structure, or the diamond-type of structure if the two types of atoms become one (i.e. if one ignores the colour difference). Needless to say, Ni does not crystallise in this of type of structure; but if we replace Ni with C (the cell parameters are modified accordingly) we do get the structure of diamond. Both metallic Ni and diamond have a lattice of type f.c.c., but they have a different types of space group: Fm3m the former, Fd3m the latter (Figure 1). Interpenetrating two lattices does not make any sense: interpenetrating two structures does.

 
[Figure 1]Figure 1. (a) An f.c.c. lattice, as well as the structure of metallic nickel, in which Ni atoms occupy the positions of lattice nodes. (b) Two copies of the structure of nickel, one in its original position (red), the other shifted by ¼ along one of the body diagonals (yellow). The colour difference is used simply to emphasise the logical process of constructing the new structure.

The result is isostructural to sphalerite, ZnS, or, if the colour difference is ignored, to diamond. Despite the 1:1 correspondence between the atomic positions and the positions of the lattice nodes in the case of metallic nickel, the atomic nature is fundamental in constructing a new, derivative structure: to move lattice nodes does not make any sense.

Even worse is the case of the hexagonal close-packed structure, which corresponds to space groups of type P63/mmc and atoms with coordinates ⅔,⅓,¼ and ⅓,⅔,¾ (Wyckoff position 2d) or ⅓,⅔,¼ and ⅔,⅓,¾ (Wyckoff position 2c) i.e. away from the lattice nodes, yet the meaningless term hcp lattice is frequently used.

Finally, why does all this matter? Is this just pedantry? Well, for a start, terms like lattice and structure have been defined on an international basis by the IUCr, in much the same way as most other physical terms are. For instance, we would not be allowed to use, say, g the acceleration due to gravity, as the same as the force due to gravity. Or, for example, energy when one means entropy. And so on. The reason for the importance of using correct terminology is to enhance communication and ensure that different people do not end up talking at cross purposes.

Paraphrasing Zizi’s Lament,2 so many authors may today sing “I am in love with the lattice sickness ...”.

Notes

1. A truism.

2. From Gregory Corso’s text for Leonard Bernstein’s Songfest: "I am in love with the laughing sickness …".

14 May 2019