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Solid State Chemistry, a subdiscipline of Inorganic Chemistry, primarily involves the study of extended solids. •Except for helium*, all substances form a solid if sufficiently cooled at 1 atm. •The vast majority of solids form one or more crystalline phases – where the atoms, molecules, or ions form a regular repeating array (unit cell). •The primary focus will be on the structures of metals, ionic solids, and extended covalent structures, where extended bonding arrangements dominate. •The properties of solids are related to its structure and bonding. •In order to understand or modify the properties of a solid, we need to know the structure of the material. •Crystal structures are usually determined by the technique of X-ray crystallography. •Structures of many inorganic compounds may be initially described in terms of simple packing of spheres.
Solids
Close-Packing
Square array of circles Close-packed array of circles Considering the packing of circles in two dimensions, how efficiently do the circles pack for the square array? in a close packed array?
Layer A Layer B hcp hexagonal close packed ccp cubic close packed
Face centered cubic (fcc)
has cubic symmetry.
Atom is in contact with three atoms above in layer A, six around it in layer C, and three atoms in layer B. A ccp structure has a fcc unit cell.
The coordination number of each atom is 12.
Coordination Number
hcp ccp
Primitive Cubic
atoms per unit cell Spheres are in contact along the face diagonal, thus l = d√2. The fraction of space occupied by spheres is:
- 74 ( 2 ) ( / 2 ) 4 3 4 3 3 ^ d d V V unitcell spheres Packing Efficiency: ratio of space occupied by spheres to that of the unit cell It is the most efficient (tied with hcp) packing scheme. V ( ) numberofformulaunits ( / 6. 022 10 (ing/atom )) ( / ) 3 23 3 cm FW Densityg cm unitcell The density expression is: The majority of the elements crystallize in ccp(fcc), hcp, or bcc. Polonium adopts a simple cubic structure Other sequences include ABAC (La, Pr, Nd, Am), and ABACACBCB (Sm). Actinides are more complex. Occurrence of packing types assumed by elements
Pressure-temperature phase diagram for iron
Molecules contain mirror planes, the symmetry element is called a mirror plane or plane of symmetry.
- σh(horizontal): plane perpendicular to principal axis
- σd(dihedral), σv(vertical): plane colinear with principal axis
- σv: Vertical, parallel to principal axis
- σd: σ parallel to Cn and bisecting two C 2 ' axes
Planes and Reflection (σ)
Hermann-Mauguin: m
Inversion, Center of Inversion (i)
A center of symmetry: A point at the center of the molecule. (x,y,z) → (-x,-y,-z). Tetrahedrons, triangles, and pentagons don't have a center of inversion symmetry. Ru(CO) 6
C 2 H 6
Hermann-Mauguin:
Rotation-reflection, Improper axis (Sn) •This is a compound operation combining a rotation (Cn) with a reflection through a plane perpendicular to the Cn axis σh.(Cn followed by σh) σCn=Sn (Schoenflies) •It may be viewed as a combination of a rotation (1/n of a rotation) and inversion. (Hermann-Mauguin) 𝒏 Hermann-Mauguin: n molecular spectroscopy solid state Schoenflies Hermann- Mauguin S 1 ≡ m (^2) ≡ m S 2 ≡ i (^1) ≡ i S 3 6 S 4 4 S 6 3 Equivalent Symmetry elements in Schoenflies and Hermann-Maguin Systems 1
Hermann-Mauguin inversion axis
4
n
Lattices and Unit Cells A crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern of ‘building blocks’, extending in all three spatial dimensions. -the ‘building block’ is known as the unit cell. Simplest regular array is a line of evenly spaced objects (one-dimensional). a The line of dots is called the lattice, and each lattice point (dot) must have identical surroundings. a The choice of unit cell is arbitrary. Lattice + basis = crystal structure
CRYSTAL STRUCTURE = lattice + basis (atoms or ions)
lattice basis
Lattice points are associated to a basis
Crystal Lattices
Space lattice a pattern of points that describes the arrangement of ions, atoms, or molecules in a crystal lattice. Unit Cell the smallest, convenient microscopic fraction of a space lattice that:
- When moved a distance equal to its own dimensions (in various directions) generates the entire space lattice.
- reflects as closely as possible the geometric shape or symmetry of the macroscopic crystal. Although crystals exist as three dimensional forms, first consider 2D. A crystal contains approximately Avogadro’s number of atoms, ions, or molecules. The points in a lattice extend pseudo infinitely. How does one describe the entire space lattice in terms of a single unit cell? What kinds of shapes should be considered? Square 2d array Select a unit cell based on,
- Symmetry of the unit cell should be identical with symmetry of the lattice
- Unit cell and crystal class (triclinic…) are related
- The smallest possible cell should be chosen
Five Types of Planar 2-D Lattices
Translational Symmetry Elements – Glide Plane and Screw Axis
Glide plane – combination of translation with reflection. a
+ ,^
Glide direction
x y z a/ Comma indicates that some molecules when reflected through a plane of symmetry are enantiomorphic, meaning the molecule is not superimposable on its mirror image. Glide plane
Glide plane
1. Reflection at a mirror plane
2. Translation by t/2 (glide component)
Axial glides a, b, c
a/2, b/2, c/
Diagonal glides n
Vector sum of any two a/2, b/2,
c/2, e.g. a/2 + b/
Diamond glide d
Vector sum of any two a/4, b/4,
c/4, e.g. a/4 + b/
Translational symmetry elements: –combination of translation with rotation. –uses the symbol ni, where n is the rotational order of the axis (twofold, threefold, etc.) and the translation distance is given by the ratio i/n. Example of a 21 screw axis c
21 Screw axis
x y z c/ Screw axis
http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/press.html
Three-Dimensional Unit Cells
The unit cell of a three-dimensional lattice is a parallelepiped
defined by three distances (a, b, c) and three angles (a, b, g).
7 Lattice systems 14 Bravais lattices Different ways to combine 3 non-parallel, non-coplanar axes Compatible with 32 3-D point groups (or crystal classes) Combine 14 Bravais Lattices, 32 translation free 3D point groups, and glide plane and screw axes result in 230 space groups.
Seven Lattice systems
Seven Crystal Systems or Classes
Minimum symmetry
requirement
Four threefold axes at 109°28’ to each other One fourfold axis or one fourfold improper axis. Any combination of three mutually perpendicular twofold axes or planes of symmetry (H) One sixfold axis or one sixfold improper axis; (R) one threefold axis One twofold axis or one symmetry plane None
