In my view, the central question that Condensed Matter Physics (CMP) seeks to answer is:

How do the properties of a distinct phase in a material **emerge** from the interactions between the atoms of which the material is composed?

CMP aims to find a connection between the microscopic properties and macroscopic properties of a material. This requires determining three things: what the microscopic properties are, what the macroscopic properties are, and how the two are related. None of the three is particularly straightforward. Historically, the order of discovery is usually: macroscopic, microscopic, connection. Making the connection between microscopic and macroscopic can take decades, as exemplified in the BCS theory of superconductivity.

Arguably, the central concept to describe the macroscopic properties is

**broken symmetry**, which can be quantified in terms of an

**order parameter**. Connecting this microscopics is not obvious. For example, with superconductivity, the sequence of discovery was experiment, Ginzburg-Landau theory, BCS theory, and then Gorkov connected BCS and Ginzburg-Landau.

When we discuss (and teach about) crystals and their symmetry we tend to start with the microscopic, particularly with the mathematics of translational symmetry, Bravais lattices, crystal point groups, ...

Perhaps this is the best strategy from a pedagogical point of view in a physics course.

However, historically this is not the way our understanding developed.

Perhaps if I want to write a coherent introduction to CMP for a popular audience I should follow the historical trajectory. This can illustrate some of the key ideas and challenges of CMP.

So let's start with macroscopic crystals. One can find beautiful specimens that have very clean faces (facets).

Based on studies of quartz,

Nicolas Steno in 1669 proposed that ``the angles between corresponding faces on crystals are the same for all specimens of the same mineral". This is nicely illustrated in the figure below which looks at different cross-sections of a quartz crystal. The 120-degree angle suggests an underlying six-fold symmetry. This constancy of angles was formulated as a law by RomÃ© de l'Isle in 1772.

Rene Just Hauy then observed that when he smashed crystals of calcite that the fragments always had the same form (types of facets) as the original crystal. This suggested some type of translational symmetry, i.e. that crystals were composed of some type of polyhedral unit. In other words, crystals involve a repeating pattern.

The mathematics of repeating units was then worked out by Bravais, Schoenflies, and others in the second half of the nineteenth century. In particular, they showed that if you combined translational symmetries and point group symmetries (rotations, reflections, inversion) that there were only a discrete number of possible repeat structures.

Given that at the beginning of the twentieth century, the atomic hypothesis was largely accepted, particularly by chemists, it was also considered reasonable that crystals were periodic arrays of atoms and molecules. However, we often forget that there was no definitive evidence for the actual existence of atoms. Some scientists such as Mach considered them a convenient fiction. This changed with Einstein's theory of Brownian motion (1905) and the associated experiments of

Jean Perrin (1908). X-ray crystallography started in 1912 with Laue's experiment. Then there was no doubt that crystals were periodic arrays of atoms or molecules.

Finally, I want to mention two other macroscopic manifestations of crystal symmetry (or broken symmetry):

**chirality** and distinct

**sound modes** (elastic constants).

Louis Pasteur made two important related observations in 1848. All the crystals of sodium ammonium

tartrate that he made could be divided into two classes: one class was the mirror image of the other class. Furthermore, when polarised light traveled through these two classes, the polarisation was rotated in opposite directions. This is

** chirality** (left-handed versus right-handed) and means that reflection symmetry is broken in the crystals. The mirror image of one crystal cannot be superimposed on the original crystal image. The corresponding (trigonal) crystals for quartz are illustrated below.

Aside. Molecular chirality is

very important in the pharmaceutical industry because most drugs are chiral and usually only one of the chiralities (enantiomers) is active.

**Sound modes** (and elasticity theory) for a crystal are also macroscopic manifestations of the breaking of translational and rotational symmetries. In an isotropic fluid, there are two distinct elastic constants and as a result, two distinct sound modes. Longitudinal and transverse sound have different speeds. In a cubic crystal, there are three distinct elastic constants and three distinct sound modes. In a triclinic crystal (which has no point group symmetry) there are 21 distinct elastic constants. Hence, if one measures all of the distinct sound modes in a crystal, one can gain significant information about which of the 32 crystal classes that crystal belongs too. (See Table A.8

here).

Aside: the acoustic modes in a crystal are the

Goldstone bosons that result from the breaking of the symmetry of continuous and rotational translations of the liquid.

This post draws on material from the

first chapter of

Crystallography: A Very Short Introduction, by

A.M. Glazer.