# 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. Jan 26, 2022 at 18:04
• Not all structures are lattices. Seems more a question of English language usage than physics. Jan 26, 2022 at 18:04
• Perhaps you need to look at what a nematic phase implies. 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). 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). Jan 26, 2022 at 18:22

• 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. Aug 2, 2022 at 10:14