The Fascination of Crystals and Symmetry

Unit 1.8

by Frank Hoffmann & Michael Sartor The 7 Crystal Systems Unit Cell – Definition

. The unit cell is the unit, which builds up the whole crystal structure by repeated translations along all three spatial directions. z

z

y y

x

x Unit Cell – Attributes

z Metric . it is defined by six cell/lattice parameters . the three cell/lattice constants, the lengths of the edges (a, b, and c) c . and three angles between these edges (α, β, and γ) β α b y a γ Symmetry . it contains all present symmetry elements . defines the minimum size of the unit cell x

Chemical Composition (Stoichiometry) . the chemical content of an unit cell corresponds to the chemical composition of the considered compound (!)

H2O HO HO HO H2O H2O H2O H2O H2O H2O 2H:1O HO HO HO H2O H2O H2O H2O H2O H2O The 7 crystal systems (metric, system of coordinates)

z Classification of unit cells Every imaginable crystal of the world belongs to one of altogether 7 possible crystal systems c β α b y a γ

restrictions for cell constants cell angles x triclinic none none monoclinic none a = g = 90° orthorhombic none a = b = g = 90° tetragonal a = b a = b = g = 90° trigonal a = b a = b = 90°; g = 120° hexagonal a = b a = b = 90°; g = 120° cubic a = b = c a = b = g = 90° symmetry hexagonal crystal family The 7 crystal systems (metric, system of coordinates)

z Classification of unit cells Every imaginable crystal of the world belongs to one of altogether 7 possible crystal systems c β α b y a γ maximum restrictions for cell constants cell angles symmetry x triclinic none none 1 monoclinic none a = g = 90° 2/m orthorhombic none a = b = g = 90° mmm tetragonal a = b a = b = g = 90° 4/mmm trigonal a = b a = b = 90°; g = 120° 3m hexagonal a = b a = b = 90°; g = 120° 6/mmm cubic a = b = c a = b = g = 90° m3m symmetry The 7 crystal systems (metric, system of coordinates)

z Warning! You will find such a table in almost every text book of crystallography. c β α b y However, in most text books this table looks like this – γ a and this is simply wrong!

cell constants angles x triclinic a ≠ b ≠ c a ≠ b ≠ g ≠ 90° monoclinic a ≠ b ≠ c a = g = 90°, b ≠ 90° orthorhombic a ≠ b ≠ c a = b = g = 90° tetragonal a = b ≠ c a = b = g = 90° trigonal a = b ≠ c a = b = 90°; g = 120° hexagonal a = b ≠ c a = b = 90°; g = 120° cubic a = b = c a = b = g = 90° The 7 crystal systems (metric, system of coordinates)

z VIMs (Very Important Messages)

. The cell parameters give only an indication of the underlying symmetry! c β α b y . It is not the metric which determines the symmetry! γ a . It is the other way round: the symmetry determines the metric, although not in a biunique way…. x . Crystals, which belong to the triclinic crystal system (according to their symmetry), may have coincidentally cell parameters like this:

a = b = c a = b = g = 90°

. Vice versa: A crystal of the cubic crystal system has (necessarily and due to the symmetry!) always the following cell parameters:

a = b = c a = b = g = 90° The 7 crystal systems (metric, system of coordinates)

z Exercise An analysis of a crystal gave the following unit cell parameters: a = b = 12 Å, c = 12.1 Å a = b = g = 90° c Which of the different crystal systems could this crystal belong to? β α b y a γ tetragonal orthorhombic monoclinic triclinic ! x