HELP on CRYSTALLINE FORM : named after the shape of a solid occupied by a unit cell, the Bravais lattice, of which there are only fourteen:
CUBIC is a six sided solid where all orthogonal axes are at right angles and all sides equal in length. The atoms occupy the 8 corners. Fully occupied, with 1/8 atom in each corner, there is one atom per unit cell. Packing fraction is 0.524.
BODY CENTRED CUBIC (bcc) is cubic, and with an atom also occupying the middle of the cube. Normally there are two atoms per unit cell, with 1/8 atom in each corner and one in the middle, but any number can fit; in the diamond cubic structure there are eight atoms per unit cell (but this does not mean that diamond is dense, far from it, the unit cell in diamond is large. See diamond below). Packing fraction is 0.68 with 2 atoms/unit cell.
FACE CENTRED CUBIC (fcc) is cubic, and with atoms also occupying the centres of the 6 faces. Fully occupied, with 1/8 atom at each corner, and 1/2 on each face, there are four atoms per unit cell. Packing fraction is 0.74.
DIAMOND structure has a tetrahedral bond arrangement between each atom and four neighbours. Each atom has 4 nearest neighbours and 12 next nearest neighbours, with 8 atoms in a (large) unit cube. It is based on a face centred cubic lattice, so is not an independent member of the 14 Bravais lattices. The diamond lattice is relatively empty, with a packing fraction of 0.34 or just 46% of that occupied by the hexagonal close packed structure.
TETRAGONAL (tetr) is almost cubic, but with one longer side. Fully occupied, it is like the cube, with one atom per unit cell.
BODY CENTRED TETRAGONAL (bc tetr) is tetragonal and with an atom in the centre. Fully occupied, like the body centred cube, it has two atoms per unit cell. This is the equivalent of a face centred tetragonal when planes are drawn instead through the short diagonals.
ORTHORHOMBIC is similar to a cube with three orthogonal axes but with three unequal sides, and fully occupied has one atom per unit cell. Three variants exist, one body centred (two atoms per unit cell), one with two diametrically opposed faces occupied (three atoms per unit cell), and another fully face centred (four atoms per unit cell).
MONOCLINIC is based on a 6-sided solid with three un-equal length sides, and one angle not 90 degrees. A face centred variant exists where two extra atoms occupy diametrically opposed faces. Fully occupied, it has one atom per unit cell.
FACE CENTRED MONOCLINIC is monoclinic, but with an extra atom in the centre of the two smallest diametrically opposite faces. Fully occupied, it has two atoms per unit cell.
RHOMBOHEDRAL or TRIGONAL The atoms occupy the 8 apexes of a deformed cube with three equal length axes at three equal angles which are less than 120 degrees but not 90 degrees. Each face is a rhombus, a diamond shaped quadrilateral with equal length sides. Fully occupied, it has one atom per unit cell.
HEXAGONAL (hex) is columnar where the atoms occupy the 6 corners of a hexagon, with each layer stacked vertically directly over the other, AAA. Fully occupied, with 1/6 atom at each of 12 apexes, there are two atoms per unit cell.
HEXAGONAL CLOSE PACKED (hcp) is the same as hexagonal, but with the layers stacked alternately corner over centre, centre over corner, in the manner ABAB, and is the densest packing arrangement for equally sized spheres. It belongs to the hexagonal class and is not an independent member of the Bravais lattices. Face centred cubic is identical to hexagonal close packed when the layers are stacked alternately upon three different centres: two corners and the centre, ABCABC. Packing fraction is same as for fcc, 0.74. If the atomic binding energy between atoms depended only on the number of nearest neighbouring atoms (12 for both hcp and fcc) then there would be no energy difference between the fcc structure and the hcp structure.
HEXAGONAL CLOSE PACKED ABAC (hcp) is where the layers are stacked alternately upon three different centres: one or other corners of a hexagon, or its centre. It belongs to the hexagonal class and is not an independent member of the Bravais lattices. Fully occupied, there are four atoms per unit cell. Packing fraction is same as for fcc, 0.74. The stacking of layers can also be randomly, such that the substance is crystalline in two directions and non-crystalline or glasslike in the third. Polytypism is where the stacking of layers is repeated but with a very long sequence; zinc sulphide crystals are known with a periodicity of 360 layers; silicon carbide with 594 layers.
TRICLINIC The atoms occupy the 8 apexes of a deformed cube where none of the three axes has the same size or angle. Fully occupied, there is one atom per unit cell.
PENTAGONAL SYMMETRY a 'forbidden' 'symmetry' and not one of the Bravais lattices, it is halfway between a glass and a regular periodic structure. See manganese.
GLASSES occur when the atoms are randomly bonded to each other in haphazard ways without periodicity, they are similar to liquids with extremely high viscosities, and are produced by cooling the substance quickly below its melting point before it can crystallise. They are thus supercooled liquids and are usually transparent.
In addition, there are 24 different ways in which these 14 different lattices can be distorted by twisting, translating and/or shearing called space symmetry operations. By combining these 14 Bravais lattices, and the 24 space symmetry operations, the whole panoply of 230 different space groups or crystal shapes are constructed. These include octahedron, dodecahedron, gyroid, trapezohedron, tetrahedron, hexagonal dipyramid, scalenohedron, and ditetragonal prism to name but a very few.
If transparent, the hexagonal, tetragonal and trigonal/rhombohedral structures would be optically anisotropic with one optic axis (uniaxial) and would be bi-refringent, as is calcite. Orthorhombic, monoclinic and triclinic structures with two optic axes (bi-axial) would be tri-refringent, as is mica, KHöAlò(SOû)ò.
Generally, the lowest energy arrangement of atoms with LIKE or zero charges is that of either a hexagonal close-packed or face centred cubic lattice where each has 12 neighbours, whereas the lowest energy arrangement for two differing atoms with OPPOSITE charges is that where the unlike charges are arranged alternately in a cubic lattice, where each has but 6 neighbours. Many ionic salts including sodium chloride, NaÇClé, have such a structure.
NOTES ON CRYSTALLINE PHASES AND POLYMORPHIC CHANGES
Polymorphism is the phenomenon whereby a substance may exist in more than one crystal structure depending upon temperature and or pressure. The crystal lattice parameters a, b, and c, are the lengths of each of the three axes of the Bravais lattices. The crystal structure of a substance represents the lowest energy arrangement of its atoms, at a particular temperature. As the temperature of a substance is increased, the atoms vibrate with greater vigour increasing the average distance between atoms. Under such circumstances, some substances will change from one crystal structure into another which has a lower energy at that temperature. This is a phase change, and occurs at the phase transition temperature. Some elements, when the temperature is raised, will go through several such phase changes before finally reaching the melting point. Plutonium remarkably goes through six such phases: two monoclinic phases, an orthorhombic, a face centred cubic, a tetragonal and a body centred cubic phase as the temperature is raised from 395 Kelvin to 753 Kelvin. Naturally, each phase has a differing density, usually less. The differing phases naturally have differing properties. As with any phase change, there is usually a difference in energies between the two forms, and hence a latent heat is associated with a phase change. The phase transition does not always occur at a sharply defined temperature, but can occur gradually over a range of temperatures, nor does the transition from one form to another always occur instantly, the phase change can take place gradually over sometimes a month or more. Neither is the transformation always complete. Moreover, the transition temperature can exhibit hysteresis, with the upward transition occurring at a higher temperature than the downward transition, as is the case with cerium, or the change may be irreversible altogether. What is more, some transitions are accompanied by abrupt changes in lattice parameters and atomic volume, (as occurs with plutonium, iron or manganese polymorphs) whereas, in others, the changes that occur in lattice parameters and atomic volume with temperature are gradual (as occurs with orthorhombic terbium where the a-axis parameter increases with temperature whilst that of the b-axes decreases with temperature until they become equal at 223K when it becomes hexagonal close packed). Furthermore, the asymmetrical Bravais lattices are anisotropic, having differing properties in one or more directions. Iodine, which crystallises in the orthorhombic system, has a temperature coefficient of thermal expansion that is more than 10 times greater in one crystallographic direction than in the other two. Some polymorphs like diamond are only stable under the high pressures encountered 25 miles deep within the Earth, and thus when present at normal pressures are in a metastable state. Graphite, not diamond, is the stable form of carbon at NTP. Many rocks and minerals also undergo polymorphic changes with pressure or temperature (see silicon dioxide and quartz, crystoballite and tridymite); some are found co-existent at normal pressures and temperatures (see titanium dioxide and rutile, anatase and brookite; or see calcium carbonate and calcite and aragonite)
The crystalline formation for each element is quoted as the crystalline phase that is stable at room temperature. For elements that are normally liquid or gaseous at room temperature, the most stable phase is quoted, which exists below the lowest phase transition temperature. Under higher pressure, certain elements can adopt high-pressure forms or phases, but these are not detailed in this display.
Some elements have more than one stable phase (not necessarily crystalline, some can be amorphous) at certain temperatures. These are usually referred to as allotropes. Some allotropes are polymers, and are not necessarily solid, but can also be liquid or gaseous, such as the dimer oxygen, Oö and the trimer ozone, Oò.
When similar atoms first group together to form clusters they do not at first form crystalline structures, not even when the number of atoms exceeds 20,000. They instead preferentially form concentric shells of atoms, like layers in an onion. In sodium clusters, the number of preferred atoms in a cluster is 2, 8, 20, 40, 58, 92, 138, 198, 254, 338, 440, 562, 694, 832, etc corresponding to closed shells of atoms. The observed shell structure (in sodium and other metallic elements) is due to the pooling of the valence electrons which are free to roam resulting in quantisation and a preference for a certain number of atoms in the cluster. As the number of atoms in a cluster increases, so the cluster exhibits more metallic properties, where the free electrons scatter off other positive ions giving rise to resistance. Clusters of elements without free electrons seem to prefer a different set of magic number of atoms, for solid xenon they are 13, 55, 147, 309, 561, etc. The shells are not spherical, but rather based on deltahedral shapes. Moreover, the xenon shells exhibit fivefold (pentagonal) symmetry, which is not a crystalline symmetry. When the sodium clusters are cooled, they re-arrange themselves into clusters with the same preferential magic number of atoms as xenon. It is not known how many atoms are required in a cluster before the cluster spontaneously re-arranges itself into its normal crystal structure, which for sodium is body centred cubic, or for solid xenon is face centred cubic (at 5 Kelvin), as they still exhibit the shell structure up to 20,000 atoms.
See CRYSTAL LATTICE PARAMS a,b & c. See also the draw file 'Bravais'. ■
