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J
m

      1 eV =

J

      1 K =

eV

atomic distances, only about 0.1% of all atoms are at the grain boundaries.

      There are, however, also solids that are not crystalline. These are called amorphous. The amorphous state is characterized by the absence of any long‐range order. There may exist, however, a degree of short‐range order between the atoms.

      This chapter is divided into three parts. In the first part, we define some basic mathematical concepts needed to describe crystals. We keep things simple and mostly use two‐dimensional examples to illustrate the ideas. In the second part, we discuss common crystal structures. For the moment, we will not ask why the atoms bind together in the way they do – this topic will be discussed in Chapter 2. Finally, we delve into a more detailed discussion of X‐ray diffraction, the experimental technique that can be used to determine the microscopic structure of crystals. X‐ray diffraction is used not only in solid state physics but also for a wide range of problems in nanotechnology and structural biology.

, created from two non‐collinear vectors
and
as

      (1.1)

      where

and
are integers. In three dimensions, the corresponding definition is

      (1.2)

and
are often called the lattice constants.

Schematic illustration of a two-dimensional Bravais lattice. Schematic illustration of illustration of (primitive and nonprimitive) unit cells and of the Wigner-Seitz cell for a rectangular two-dimensional lattice. Schematic illustration of a two-dimensional Bravais lattice with different choices for the basis.

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