26. Crystals – Guillaume Schweicher

17.
Guillaume Schweicher (BE)

Crystals

Texts by Guillaume Schweicher

As indicated in information level 2, crystallography is the branch of material science that studies the arrangement of atoms/ions/molecules in crystalline solids.

It is essential to understand the crystal structure of a material since it determines the material’s physical properties (fusion temperature, solubility, magnetism, etc.). A given material, however, can simultaneously crystallize into several “stable” structures. This phenomenon is called polymorphism. The type of polymorph largely depends on the crystallisation type (cooling from a solution or a melt, etc.). It is understandable to want to know the types of polymorphs of a given material, since only certain ones will produce required physical properties. Unsurprisingly, the pharmaceutical industry greatly focuses their research and development efforts on polymorphs of materials in order to be certain of the solid physical properties of its components.

So, how can we determine the crystal structure of a material and thus be able to study the interactions within, and the growth that induces a specific arrangement of molecules? Nowadays, crystallography experts use a systematic and rapid process: X-ray diffraction. They grow a single crystal – with no structural flaws – of a given material, and then mount it on an X-ray diffractometer to obtain a diffraction pattern (diffractogram).

To determine the structure, they use algorithms  and mathematical tools (statistics, symmetry operations, equation solving) to converge towards the material’s crystal structure.  The best-known method, called the “direct method”, is based on work by Herbert Aaron Hauptman (mathematician) and Jerome Karle (chemist). They both received the Nobel Prize for Chemistry in 1985 for developing a method to determine crystal structures. A Nobel Prize in Chemistry for mathematical methods proves, once again, the importance of mathematics as a universal scientific language and tool.  Ionic solid NaCl, commonly known as table salt, was the first crystal structure to be determined.

As explained in the previous information level, a crystal’s microscopic structure forms by repetition in three-dimensional space, from a unit cell called primitive lattice. Each network lattice point contains a node; the basis for patterns atoms, ions or molecules are found on each lattice node.  The lattice is defined by six parameters: side lengths (a\), \(b\) and \(c\), and angles \(\alpha\), \(\beta\) and \(\gamma\) (see box, Figure 1).

Depending on their geometrical characteristics, crystals can be ascribed to one of seven crystal systems, or primitive lattices. If we place an additional node at the centre of the unit cell or at the centre of each of the six faces, we obtain fourteen crystal networks, called Bravais lattices (Figure 2 bis), which comprise all possible crystal arrangements. There is more: to properly describe the crystal structure of a material, we have to take into consideration the symmetry elements in these crystal networks. The symmetry operations in a crystal structure will hence determine its symmetric group. There are a total of 230 space groups (3D) and 17 plane groups (2D).

Figure 2 bis. The fourteen Bravais lattices

Axial system \((x, y, z)\), used to define directions and crystal planes, contains the same three translation vectors \(a\), \(b\) and \(c\). Since all nodes are geometrically equivalent, the choice of direction (at the network starting point) is arbitrary. Given six parameters \(a\), \(b\), \(c\), \(\alpha\), \(\beta\) and \(\gamma\), we can define:

  • A distance : measurement of axis \(x\), \(y\) and \(z\) according to vectors \(a\), \(b\) and \(c\) (which serve as measurement standards).
  • A direction : defined by three indices\([u, v, w]\). This direction, called reticular line, passes through the origin and lattice node \((u, v, w)\). Examples of directions are shown in Figure 4.
Figure 4. Different directions within a lattice
  • A plane : designated by three indices \((h\ k\ l)\) called Miller indices. Miller indices are the inverses of the intercepts of plane axis \(x\), \(y\) and \(z\) according to lengths \(a\), \(b\) and \(c\). Examples of plane definitions are shown in Figure 5.
Figure 5. Définition de différents plans au sein d’une maille.

Let’s get back to the different components in Crystals, which have crystallised within a monoclinic network (\(a\neq b \neq c\) and \(\alpha \neq \gamma\) and \(\beta =90^o\)) of space group \(P2_1/c\) (abbreviated). Unsurprisingly, the three materials belong to the same space group. Actually, most organic molecules crystallise in a fairly small range of monoclinic, triclinic and orthorhombic space groups. The full name of space group (P2_1/c\) is \(P\ 1\ 2_1/c\ 1\). It includes all the symmetry elements in the crystal:

  • As a rule, the symmetry elements will be considered axis \(b\). \(P\) indicates that the lattice is primitive and that the crystal does not contain a node in the centre of its unit cells (see simple monoclinic Figure 1).
  • \(1\ 2_1/c\ 1\) 1 shows all symmetry elements for directions \(a\), \(b\) and \(c\) respectively (1 for \(a\), \(2_1/c\) for \(b\) and 1 pour \(c\)).
  • \(1\) corresponds to the identical operation and indicates the absence of symmetry elements for the given direction, in other words, an absence of symmetry for directions \(a\) and \(c\) in this case.
  • There are also 2 symmetry elements for \(b\): a helicoidal axis  \(2_1\) parallel to \(b\) (180° rotation along axis \(b\) + half-length translation of lattice \(b\) parallel to \(b\)) and a glide for \(c\) perpendicular to direction \(b\) (mirror symmetry to plane + half-length translation of lattice \(b\) parallel to the plane).

Starting from the lattice point and applying the different symmetry elements, it is possible de completely reconstruct the material’s crystal structure! A similar methodology is used in the overall crystallography field and clearly shows the need for mathematics to describe the solid state of matter.

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