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1.1. Translational Symmetry, Lattice Points, and Unit Cells

What is a Crystal?

In the broad sense, a crystal refers to a state (and the material in that state) that possesses a definite translational symmetry. Translational symmetry is the property whereby the state is indistinguishable after moving parallel to a certain direction by a certain distance. The unit of “that distance” is arbitrary1, but readers of this page are most likely interested in crystals as solid matter — substances in which atoms, molecules, or ions are periodically arranged at a small scale and possess a macroscopic size (large enough for physical properties to manifest). Hereafter the term “crystal” is used in this sense.

Lattice Points, Crystal Lattice, and Unit Cell

Let us think about the properties of a crystal. The figure below shows an infinitely extending park with translational symmetry2. Suppose you stand at a position \(o\) in the park and look around; the view you see is exactly the same as the view from another position \(a\), obtained by translating from \(o\) without changing the direction of your gaze. You cannot tell from the scenery which position you are standing at. In other words, positions \(o\) and \(a\) have exactly the same surroundings. Therefore, the position reached from \(a\) by advancing further by the vector \(\vec{oa}\) (call it \(a’\)) must also show the same view. The same scenery is not limited to points on the line \(oa\): position \(b\) also shows the same view, and so does the position \(b’\) reached from \(b\) by advancing \(\vec{oa}\).

As shown, there exist infinitely many positions in this park from which exactly the same scenery is visible. Such positions are called lattice points — an extremely important concept. In the figure, lattice points (= observers) were placed on the lawn, but identifying them with rocks or slides would not affect the discussion at all. Where in a crystal the lattice points are anchored is a matter of human choice3. However, if one lattice point is identified with a rock, then all other lattice points must also be identified with rocks. What matters in a lattice point is the vector (distance and direction) between one lattice point and another — called the translation vector. Translating the entire park by a translation vector leaves it indistinguishable from its original state. If there are \(n\) independent translation vectors, the crystal is \(n\)-dimensional. The park example has two independent translation vectors (e.g., \(\vec{oa}\) and \(\vec{ob}\)), so it is a 2D crystal.

Two further concepts to understand alongside lattice points and translation vectors are the crystal lattice and the unit cell. The crystal lattice is the periodic geometric arrangement of lattice points — it describes how all the lattice points are arranged as a whole, not a single point. Connecting the lattice points with red lines gives the figure below: the red lines in two directions are equally spaced, forming a grid pattern. The smallest parallelogram appearing in this grid is called the unit cell.

A unit cell is a parallelogram in 2D and a parallelepiped in 3D. Unit cells stacked together build up the crystal lattice. As with lattice points, only the shape of the unit cell matters (the spacing between the red lines and the angle at which they cross); it is not essential that the corners of the unit cell coincide with any particular feature of the crystal.

Even when the crystal lattice (arrangement of lattice points) is exactly the same, there is a variety of ways to draw lines connecting them, leading to different unit cell shapes. For example, drawing lines as shown below changes the unit cell shape. (The optical illusion of seeing different viewing angles is just that — only the way the red lines are drawn has changed.)

Which is the “correct” unit cell? The answer is “both are correct.” But does connecting lattice points always give an “appropriate” unit cell? The answer is “it depends”4. The details of when to use which unit cell will be explained from the next page onward, but it is important to keep in mind that even when the crystal lattice is identical, there is freedom in how the unit cell is chosen.

Those who have studied crystallography before will have encountered the term “unit lattice” (単位格子). In practice this term is used rather ambiguously: the “unit” suggests something periodic, making it seem synonymous with “unit cell,” yet since it contains “lattice” it might seem synonymous with “crystal lattice.” In fact, “unit lattice” is very often used to encompass both “crystal lattice” and “unit cell.” There is no need to be overly concerned by this. Since crystal lattice and unit cell are intimately related concepts, combining them under “unit lattice” rarely causes problems or misunderstandings. Nevertheless, this website distinguishes between “crystal lattice” and “unit cell.”

Various terms have appeared, so here is a final summary list.

Footnotes

  1. In the broad sense, for example, the sine function is in a “crystal state” along the angle axis, and frictionless pendulum motion is in a “crystal state” along the time axis. ↩︎
  2. Imagine a park extending to the horizon in all directions. Of course, real crystals have finite size and a boundary with the outside world, so a truly ideal crystal in this sense does not exist in reality. ↩︎
  3. Although everyone choosing freely would cause confusion, there are guidelines to some extent. These are discussed in detail on the “6.3. Choice of Axes and Axis Transformation” page, but beginners need not worry about this for now. ↩︎
  4. The park shown as an example contains neither rotation nor reflection operations, so any grid drawn through lattice points gives an “appropriate” crystal lattice. If the crystal includes rotational or reflection operations, crystallographic convention recommends choosing a lattice that reflects those operations as much as possible. ↩︎
  5. Forcing “単位格子” into English would give “unit lattice,” but this usage is not standard. When writing in English, it is safest to rephrase as either “unit cell” or “crystal lattice” depending on the context. ↩︎
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