Chapter 1 Crystal Structure
Outline
• Definition of Crystal and Bravais lattice • Examples of Bravais lattice and crystal structures • Primitive unit cell • Wigner – Seitz unit cell • Miller Indices • Classification of Braivais lattices
1 • An ideal crystal is constructed by the infinite repetition in space of identical structural unit. • The structure of all crystals is described in terms of a lattice with a group of atoms attached to each lattice point.
Bravais lattice basis
A Bravais lattice is an infinite array of discrete points and appear exactly the same, from whichever of the points the array is viewed.
A (three dimensional) Bravais lattice consists of all points with → positions vectors R of the form → → → → = + + R n1 a1 rn2 ar2 n3 a3 r Where are any three vectors a 1 a 2 and a 3 are not all in the same plane, and n 1, n 2, and n 3 range through all integral values
2 A general 2-D Bravais lattice of no particular symmetry
P r r r r = + = − + ' r ' P a1 2a2 a1 2a2 Q a2 r a r r r r 2 = − + = − ' + ' Q a1 a2 2a1 a2 O r a1
r r Primitive vectors: a1 a2 r r or ' a1 a2 Primitive vectors are not unique
3 Vortices of a 2-D honeycomb do not form a Bravais lattice
P and Q are equivalent P and R are not Q
P R
4 Simple cubic (sc) structure Atoms per cubic cell: 1
r r a = xaˆ a3 r 1 a 2 r r a = yaˆ a 2 1 r = a3 zaˆ
Body centered cubic (bcc)
B
A A and B are equivalent
It is a Bravais lattice 5 Primitive vectors for bcc structure z
r r a3 a2
r a1 x y
r 1 a = a(xˆ + yˆ − zˆ); 1 2 r 1 a = a(−xˆ + yˆ + zˆ); 2 2 r 1 a = a(xˆ − yˆ + zˆ) 3 2 6 An alternative set of primitive vectors for bcc structure
z
r a3 r a r 2 a 1 y
x r a = xaˆ r1 a = yaˆ 2 Less symmetric compared to previous set r a a = (xˆ + yˆ + zˆ) 3 2 7 Face centered cubic (fcc) structure
Each point can be either a corner point or a face-centering point It is a Bravais lattice
8 Primitive vectors for fcc structure
z
r r a3 a2
r y a1
x
r 1 a = a(xˆ + yˆ) 1 2 r 1 a = a(yˆ + zˆ) 2 2 r 1 a = a(zˆ + xˆ) 3 2 9 Primitive unit cell
Primitive unit cell is a volume of space that, when translated through all the vectors in a Bravais lattice, just fills all the space without either overlapping itself or leaving voids.
Two ways of defining primitive cell
• Primitive cell is not unique • A primitive cell must contain exactly one lattice point.
10 The obvious primitive cell: r r r r = + + r x1a1 x2a2 x3a3 Is the set of all points of the above form for all xi ranging continuously between 0 and 1.
It is the parallelipiped spanned by the primitive vectors
r a3 r a2
r a1
Disadvantage: the primitive cell defined as above does not reflect full symmetry of the Bravais lattice
11 Primitive cell of a bcc Bravais lattice
r a3 r a2
r a1
3 Primitive cell is a rhombohedron with edge a 2 Volume of primitive cell is half of the cube
12 Primitive cell of a fcc Bravais lattice
z
r r a2 a3
r a1 y
x
Volume of the primitive cell is ¼ of the cube
13 Simple hexagonal (sc) Bravais lattice and its primitive cell
z r 3 a a = ( a)xˆ + ( )yˆ 1 2 2 r 3 a a = −( a)xˆ + ( )yˆ 2 2 2 r = a3 zcˆ
r a3
r An alternative set: a2 r 3 a = ˆ + ˆ y a1 ( a)x ( )y r 2 2 a r 1 a = yaˆ r2 = a3 zcˆ x
14 Wigner – Seitz primitive cell • a primitive cell with the full symmetry of the Bravais lattice • a W-S cell about a lattice point is the region of space that is closer to that point than to any other lattice point
hexagon
Wigner – Seitz unit cell about a lattice point can be constructed by drawing lines connecting the point to all others in the lattice, bisecting each line with a plane, and taking the smallest polyhedron containing the point bounded by these plane
15 Wigner-Seitz cell of fcc
Wigner-Seitz cell of bcc 12 faces (parallelograms)
Truncated octahedron Note: the surrounding 14 faces (8 regular hexagons and 6 squares) cube is not the cubic cell
16 Crystal structure: lattice with a basis
A crystal structure consists of identical copies of the same physical unit, called the basis, located at all the points of a Bravias lattice (or, equivalently, translated through all the vectors of a Bravais lattice)
Honeycomb net
Basis
Lattice: 2-D triangular lattice
17 Describe a Bravais lattice as a lattice with a basis by choosing a non- primitive cell (a unit cell)
A unit cell is a region that just fills space without any over-lapping when translated through some subset of the vectors of a Bravais lattice. It is usually larger than the primitive cell (by an integer factor)
bcc: Simple cubic unit cell
a Basis: (xˆ + yˆ + zˆ) 0 2
fcc: Simple cubic unit cell
a a ( ˆ + ˆ) ( ˆ + ˆ) a Basis: 0 x y y z (xˆ + zˆ) 2 2 2
18 Diamond structure
• Not a Bravais lattice • Two interpenetrating fcc Bravais lattice
Bravais lattice : fcc basis 1 0 (xˆ + yˆ + zˆ) 4
Coordination number : 4 Four nearest neighbors of each point form the vertices of a regular tetrahedron 19 Atomic positions in the cubic cell of diamond structure projected on (100) surface
0 1/2 0
3/4 1/4 1/2 0 1/2
1/4 3/4
0 1/2 0
Fractions in circles denote height above the base in units of a cube edge.
20 Hexagonal close-packed (hcp) structure Two interpenetrating simple hexagonal Bravais lattice Bravais lattice: simple hexagonal
2 r 1 r 1 r Basis: 0 a + a + a 3 1 3 2 2 3
r a3
r a2
r a1
Four neighboring atoms form the vortices of a tetrahedron 21 Both hcp and fcc can be viewed as close-packed hard spheres
hcp fcc Coordination number: 12 for both fcc and hcp
22 Fcc is close packed structure
Try calculating packing density
The (111) plane equivalent to the triangular close packed hard sphere layer
23 Basis consisting of different atoms
CsCl NaCl Bravais lattice: sc Bravais lattice: fcc Basis: Cl - at (0,0,0) and Na + at (1/2,1/2,1/2) Basis: Cs + at (0,0,0) and Cl - at (1/2,1/2,1/2)
24 Miller indices to index crystal planes
x3
r a3 r r a2 a1
x2 x1
• find the intercepts x 1, x 2, and x 3 • h: k: l = 1/x 1 : 1/ x 2: 1/x 3 r r r Find the intercepts on the axes in terms of lattice constants a1 a2 a3
Take the reciprocals of these numbers and then reduce to the smallest three integers h, k, and l.
The results, enclosed in parentheses (hkl), are known as Miller indices 25 Classification of Bravais lattices and crystal structures
Symmetry operations: all rigid operations that take the lattice into itself Rigid operation: operations that preserve the distance between all lattice points. The set of symmetry operation is known as space group
All symmetry operations of a Bravais lattice contains only operations of the following form:
1. Translations TR through lattice vectors 2. operations that leave a particular point of lattice fixed (point operation) 3. successive operations of 1 and 2
The set of point operation is known as point group , a subset of space group
26 (a) A rotation operation through an axis that contains no lattice points (b) An equivalent compound operation involving a translation and a point operation
27 Seven crystal systems (point groups) and fourteen Bravais lattices (space groups)
cubic monoclinic trigonal
tetragonal
orthorhombic triclinic hexagonal
There are only seven distinct point groups that a Bravais lattice can have 28 Cubic system: 3 Bravais lattices
Simple cubic Body centered cubic Face centered cubic
Tetragonal system: 2 Bravais lattices
Obtained by pulling on two opposite faces of a cube
Simple tetragonal Centered tetragonal No distinction between face centered and body centered tetragonal Two ways of viewing the same lattice along c axis One lattice plane
Next lattice plane c/2 above
29 Viewed as if it’s “body centered” Viewed as if it’s “face centered” Orthorhombic system: 4 Bravais lattice
By stretching tetragonal along one of the a axis
Two ways of stretching the same simple tetragonal lattice viewed along c axis
stretch stretch
30 Simple orthorhombic Base centered orthorhombic Two ways of stretching the same centered tetragonal lattice viewed along c axis
stretch stretch
Body-centered orthorhombic face-centered orthorhombic 31 Symmetry reduced, two structures distinguishable not a right angle Monoclinic system: 2 Bravais lattices
Reduce orthorhombic symmetry by Distorting the rectangular faces perpendicular to c axis
Base centered orthorhombic Simple orthorhombic Distort rectangular shape into parallelogram
32 Simple monoclinic Body-centered orthorhombic face-centered orthorhombic Distort rectangular shape into parallelogram
No distinction between “face-centered” and “body-centered”
33 Centered monoclinic Triclinic system: 1 Bravais lattice
Tilt the c axis of a monoclinic lattice
• no restrictions except that pairs of opposite faces are parallel • a Bravais lattice generated by three primitive vectors without special relationship to one another • the Bravais lattice with the minimum symmetry Think: why there are no face-centered or body-centered triclinic?
Trigonal system: 1 Bravais lattice
Stretch a cube along a body diagonal
Hexagonal system: 1 Bravais lattice cubic
hexagonal tetragonal
trigonal orthorhombic
monoclinic
triclinic Arrow: direction of symmetry reduction 34