# Are crystal lattice and crystal structure the same thing?

Rhetorical question. A lattice is an infinite array of geometrical points in the space where each point has identical surroundings to all others. Hence a lattice is an abstract mathematical concept. The crystal structure is a real object, obtained by convoluting the lattice with a basis (an atom or a group of atoms).

So why to use misleading definitions such as lattice vibrations, lattice energy, or simply ‘lattice’ instead of ‘structure’?

In most cases it can be understood if ‘lattice’ actually refers to ‘structure’, but in some particular cases this ambiguity is really confusing. A practical example from literature. The concept of nematicity in crystalline materials was introduced in [Nature 393 (1998) 550]; considering a 2-dimensional square lattice “The nematic phase breaks the four-fold rotation symmetry of the lattice, but leaves both translation and reflection symmetries unbroken”. This is clearly impossible for a lattice (reflection symmetries are generator for the four-fold rotation), but not for a real crystal structure (for example an orthorhombic structure with pseudo-tetragonal metric). So the doubt is: did the author gave this definition having in mind a lattice (and in this case their theory must be rejected) or a crystal structure?

About the confusion between lattice and crystal structure see: https://journals.iucr.org/j/issues/2019/02/00/to5189/index.html

• You are clearly aware that the answer to your title question is "no," so perhaps you might consider rephrasing it. The real question is about the use of imprecise terminology in a specific article. Commented Jan 26, 2022 at 18:04
• Not all structures are lattices. Seems more a question of English language usage than physics. Commented Jan 26, 2022 at 18:04
• Perhaps you need to look at what a nematic phase implies. Commented Jan 26, 2022 at 18:08
• @Jon Custer I partially agree; anyway, the concepts that we use to communicate are fundamental for the reciprocal comprehension. How can we talk about physics if we use inaccurate or even wrong language? So when you write “Not all structures are lattices” I cannot understand what you’re really meaning. Structures are not lattices and viceversa, by definition; they are different things existing in different spaces (real and geometrical spaces). Commented Jan 26, 2022 at 18:14
• Concerning the nematic phase: the same definition given in my post is completely overturned by the same author when applied to Fe-based superconductors, where the nematic transition is defined as the desymmetrization from the tetragonal (2-dimensional square lattice) to the orthorhombic phase (where obviously the translation and reflection symmetries of the pristine tetragonal phase are broken). Commented Jan 26, 2022 at 18:22

There is nothing wrong with that paper. The crystal system of that particular matter undergoes a transition through which its holohedry reduces to a smaller one. The highest order symmetry element was a 4-fold rotation, now it's not. Of course, diagonal mirror planes have to disappear as well.

Notice that Crystal system is a characteristic of the lattice. It does not care about the crystal structure (i.e., what arrangement of atoms each lattice point carries).

• First of all, if you reduce the holohedry, the 2-dimensional lattice cannot be anymore a square (if you reduce to a merohedry obviously the 4-fold rotation is still present). Then, as I wrote above, in the paper it is stated “The nematic phase breaks the four-fold rotation symmetry of the lattice, but leaves both translation and reflection symmetries unbroken". These unbroken reflection symmetries contrasts with your reply when you write " Of course, diagonal mirror planes have to disappear as well". Then if you cut out diagonal mirror the lattice will be rectangular not square Commented Aug 2, 2022 at 8:54
• That's correct. The lattice cannot be anymore a square (its crystal system is changed). This is why the diagonal mirrors should also vanish. In the particular transition stated in this paper, the symmetry group of the order parameter loses its rotational symmetry, hence obviously, as you also mentioned, the transition is not endowed with the reduction to a merohedry. I am aware of what confuses you here. $D_{4h}$ to $D_{2h}$ transition is a perfectly good example for a nematic transition. Although its reflection elements subset is smaller, this does not mean the reflection symmetry is broken. Commented Aug 2, 2022 at 10:14
• Your explanation adjust somehow the statement in the paper that should be corrected in this light as "...but leaves both translation and SOME reflection symmetries unbroken”. Anyway there is still the problem of the translation symmetry. Please, explain how is it possible to have a D4h to D2h lattice transition with unbroken translation symmetries of the lattice. I cannot see how this is possible. In the D4h to D2h transition a square lattice becomes a rectangular one, so the translation symmetry is broken with nematic transition. Commented Aug 2, 2022 at 15:39
• Lattice parameters does not dictate the lattice type. Symmetry does. You could still have a "square"-like lattice (the length remains the same), but the point group associated at each lattice point be those of a "tetragonal" one. Commented Aug 3, 2022 at 6:01
• Also let me remind you that when people talk about the breaking of translation symmetry, they mean a lot more important case: the one of the crystallization where the uniformity of space descends onto a discrete one. I don't think if lattice parameters change, it implies translational symmetry breaking. Commented Aug 3, 2022 at 6:04

I studied the theory of crystalline solids and I was probably guilty of using the word lattice without stopping to consider whether I meant an ideal lattice or was just using it to refer to a structure that approximated an ideal lattice. I suppose the temptation to use the word in the latter case was to remind the reader that the structure I was referring to was lattice-like, which the use of the more general word 'structure' would not convey. Sadly, physics papers and text books are ridden with ambiguous language, which acts as a real barrier to comprehension. Your case of the loose use of the word lattice is a very minor transgression compared to some of the impenetrably vague language regularly encountered in physics papers.

• The stacking of "very minor transgressions" magnifies the final confusion. Anyway, I cited a contest where the correct definition (to my opinion) is diriment. Commented Oct 16, 2023 at 6:55