Lattice Structure of Crystals

The crystals formed by crystalline solids are defined by the repetitions of atoms in a specific pattern. The simplest repeating arrangement of particles, such as atoms, molecules, or ions, within an ordered crystal is a unit cell. Lattice points, which represent the particles, define the boundaries of the unit cell. The most basic form of a unit cell is a simple cubic unit cell—a unit cell that forms a six-sided box with a fourth (1/4th) of an atom, ion, or molecule located at each corner. The repeated unit cells and the way they fit together produce an array, known as a lattice, of atoms, ions, or molecular fragments arranged in three-dimensional space. In a simple cubic unit cell, each atom contacts the atoms above, below, left, right, front, and back of itself. The number of other particles that each particle in a crystal lattice touches is its coordination number. For a simple cubic unit cell, the coordination number is six because each atom in a lattice is bonded to six other atoms around it. The lattice points of the simple cubic unit cell represent the centers of the atoms in the crystal. Therefore the distance from one lattice point to another, which is the length of one side of the unit cell, is the equivalent of two ionic radii. Thus the ionic radii of the atoms in a crystal can be calculated if the dimensions of the unit cell are known. For example, consider a simple cubic unit cell with side length of 336 picometers (pm). This is the distance between two ionic nuclei of the cell. To calculate the ionic radius of the atoms in this crystal, divide the length of the side by 2. This ionic radius corresponds to the crystal lattice arrangement and atomic radius of polonium (Po). In atoms with no charge, the atomic radius and the ionic radius are the same. In the case of an atom that has lost an electron, for example, its ionic radius will be smaller than its atomic radius. So the ionic and atomic radii may differ. Although the simple cubic unit cell is the most basic crystal lattice configuration, it is not a very efficient way of arranging atoms. Over half the space in a crystal composed of this type of unit cell is empty. However, the cubic unit cell forms the basis of more complex shapes when other atoms lie between the atoms that make up the lattice points. The body-centered cubic (bcc) unit cell is a cubic cell unit with eight corner atoms (only 1/4th of each atom contributes to the unit cell) and one atom in the center of the unit cell. The face-centered cubic (fcc) unit cell is a cubic crystal system with half an atom on each face of the unit cell. Because these arrangements are very simple yet more efficient than the simple cubic unit cell, many pure elements crystallize in this manner. Substances that form crystals with the same structure are called isomorphous. For example, potassium (K), chromium (Cr), and iron (Fe) all form crystals with bcc unit cells and are thus isomorphous, meaning they have the same crystalline form. Atoms in crystals tend to align in ways that minimize empty space. When one layer of atoms is laid down, the space between the atoms forms holes. Imagine round balls laid into a single hexagonal layer. The spaces between the balls form trigonal holes because of the number of atoms that enclose the space. When a second layer of balls is laid over the first layer, with the balls of the second layer sitting on top of the holes of the first layer, new holes are formed. These holes are tetrahedral holes, which are trigonal holes over the sphere of the first layer. If the holes in the second layer line up with the holes in the first layer, they form octahedral holes, which are trigonal holes inverted toward each other. When two layers of atoms alternate, they are arranged in hexagons. The third layer (the repeat of the first layer) covers all the tetrahedral holes of the first two layers. This structure, which is the structure that packs atoms most tightly together in a hexagonal arrangement, is the hexagonal closest packing (hcp) structure. However, when three layers of atoms alternate, another layer must be added between the second and third layers. This layer is stacked in the depressions (not holes) of the second layer. This forms the structure that packs atoms most tightly in a cubic arrangement, called cubic closest packing (ccp) structure.

The manner in which atoms pack into crystal structures cannot be directly observed because the atoms are too small to see even with very powerful microscopes (although microscopic resolution is increasing, and individual atoms have been observed on flat planes). However, the crystalline structure of a solid can be deduced by using X-ray crystallography, a method of determining the arrangement of atoms in a crystal using X-rays.

X-rays are a short-wavelength form of electromagnetic radiation. X-rays have the proper wavelength and energy to interact with regular repeating units in crystalline solids to give structural information. When electromagnetic waves encounter an object, they can either reflect off it or bend around it. The scattering of waves by an object is diffraction. Because electromagnetic radiation has the properties of a wave, diffracted waves can interfere both constructively (amplifying the resulting wave) or destructively (canceling out the resulting wave).

When X-rays encounter a crystalline solid, all incident waves (waves from the source) are in phase and parallel until they strike the crystal, meaning they do not destructively interfere. Once the light strikes the first plane of the crystal, some rays will diffract, bending around the atoms of the crystal lattice, while others will pass between the atoms, moving on to the second plane. Some rays will diffract, while others will pass through. This continues through the entire crystal. Because the crystal is formed by repeating units, the angles of diffraction will be the same. Bragg’s law can be written in equation form as $n\lambda=2d\sin\theta$ , which can be derived using trigonometry. This law was developed in 1912 by the British physicist Lawrence Bragg after it was discovered that crystalline solids make a pattern of reflected X-rays. Lawrence Bragg, along with his father, the British physicist William Bragg, set out to predict when X-ray diffraction on a crystal would actually take place. For incident light striking planes x and y of a crystal, the path traveled by the top ray (plane x) ABC is shorter than the path traveled by the bottom ray (plane y) DEF by the difference in their paths. Where these two paths have maximum constructive interference, the difference between the paths must be an integer (n) multiple of the wavelength ($\lambda$ ). If point B, the diffraction point for the top ray, lies on plane x directly above point E, the diffraction point of the bottom ray, the distance between them is distance d. The top ray follows path ABC, while the bottom ray follows path DEF, with points G and H lying at the extensions of the rays formed from the top ray. Thus triangles BGE and BHE are formed, and the difference between the top ray's path and the bottom ray's path is $GE+EH$. Thus $n\lambda=GE+EH$. The angle at which rays strike surfaces is $\theta$, and trigonometry shows that $EH=GE=BE\;\sin\theta$. BE has already been established as d, so $EH=GE=d\sin\theta$. Thus $n\lambda=2d\sin\theta$. Bragg's law can be used along with the diffraction pattern of X-rays from X-ray crystallography to solve the structures of complex molecules. This technique was famously used to determine the double-helical structure of the DNA molecule.