BioPHYSICAL chemistry

where nis a positive integer; this equation is known as Bragg’s law. At

this angle, each of the waves from the other lattice planes will also add

together constructively since those relative pathlengths will be a multiple

of the wavelength. Destructive interference occurs when the pathlength

equals nλ/2.

As a detector is rotated through the angle Θ, it will alternatively measure

strong X-rays followed by weak ones. Notice that this analysis makes a

prediction that if a lattice plane is inserted midway between each of the

existing planes then the additional waves will have a pathlength equal

to a multiple of half a wavelength, resulting in destructive interference.

As discussed below, this corresponds to the difference between primitive

lattices and centered lattices and shows that centered lattices will have

reflections systematically missing compared to primitive lattices.

To measure all of the diffracted waves, the crystal is rotated relative to

the X-ray beam. As the crystal rotates, the distance between the lattice

planes will change depending upon the specific packing of the molecular

protein in the crystal. Therefore, for most angles, it is necessary to consider

all three dimensions of the crystal, which can be grouped into certain

classes, known as Bravais lattices, as described below.

Bravais lattices

A crystal is composed of molecules that are packed such that there is an

exact separation distance between the molecules in all three directions

x, y, and z. Thus, a crystal can be considered to be built from regularly

repeating structural motifs which may be atoms, molecules, or proteins.

Each molecule or protein is packed in a crystal such that

it is possible to move a certain length from an atom of

the protein and come back to the same atom. The pack-

ing of the molecules is represented by a lattice that

specifies the location of a structural motif (Figure 15.4).

The unit cell is the smallest box that contains one of these

repeating patterns, and is commonly formed by joining

neighboring lattice points (Figure 15.5).

In 1850, it was shown by Auguste Bravais that the

different repeating units, termed unit cells, can be

classified into seven crystal systems that reflect the rota-

tional symmetry of the cell (Table 15.1). For example,

a cubic unit cell has four 3-fold axes, denoted by C 3 ,

in a tetrahedral array while a monoclinic unit cell has

one 2-fold axis and a triclinic cell has no rotational

symmetry.

Bravais was able to show that there are only 14 dis-

tinct crystal lattices in three dimensions (Figure 15.6). The

320 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY

Lattice point

Structural motif

Figure 15.4Every crystal can be
considered as consisting of a structural
motif that repeats in space, forming a
lattice.