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 of the form
→
R
→→→→
R = n1 a1+ n2 a2 + n3 a3
r
r
rplane, and n1, n2, and n3 range through all integral values aa1
Where are any three vectors
2and a3 are not all in the same
2A general 2-D Bravais lattice of no particular symmetry
P
r
r
r
r
P = a1 + 2a2 = −a1 + 2a2'
ra2'
Q
ra2
r r
r r
Q = −a1 + a2 = −2a1' + a2'
r
Oa1
r
ra2 a
Primitive vectors: or
r
ra1 a2'
Primitive vectors are not unique
3Vortices of a 2-D honeycomb do not form a Bravais lattice
P and Q are equivalent
P and R are not
Q
R
P
4Simple cubic (sc) structure
Atoms per cubic cell: 1
r
r
ˆa1 = ax a3
ra2
r
ˆa2 = ay
r
ra1
ˆa3 = az
Body centered cubic (bcc)
B
A
A and B are equivalent
It is a Bravais lattice
5

Primitive vectors for bcc structure z
r
ra2 a3
ra1 yxr 1
ˆ ˆ
ˆa1 = a(x + y − z);
2r
1a2 = a(−x + y + z);
ˆ ˆ
ˆ
2r 1
ˆ ˆ
ˆa3 = a(x − y + z)
2
6

An alternative set of primitive vectors for bcc structure z
ra3
ra2
ra1 yx
r
ˆa = ax r1
ˆa2 = ay
Less symmetric compared to previous set r a ˆ ˆ
ˆa3 = (x + y + z)
7
2

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
ra3 a2 y
ra1 xr
1a1 = a(x + y)
ˆ ˆ
2r 1
ˆ
ˆa2 = a(y + z)
2r
1a3 = a(z + x)
ˆ
ˆ
2
9Primitive 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 obviorr = 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
ra3
ra2
ra1
Disadvantage: the primitive cell defined as above does not reflect full symmetry of the Bravais lattice
11 Primitive cell of a bcc Bravais lattice
ra3
ra2
ra1
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
ra2 a3
ra1 yx
Volume of the primitive cell is ¼ of the cube
13 Simple hexagonal (sc) Bravais lattice and its primitive cell zr
3a1 = ( a)x + ( )y a
ˆˆ
22r
3a2 = −( a)x + ( )y a
ˆˆ
22r
ˆa3 = cz
ra3
r
An alternative set: a2 r
3a1 = ( a)x + ( )y ay
ˆˆ
r
22a1 r
ˆa = ay r2
ˆa3 = cz 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 cube is not the cubic cell
14 faces (8 regular hexagons and 6 squares)
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 nonprimitive 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)
Simple cubic unit cell bcc: fcc: a
Basis:
ˆ ˆ
ˆ
(x + y + z)
0
2
Simple cubic unit cell aaa
ˆ ˆ ˆ
ˆ
(x + y) (y + z)
Basis:
0
ˆ
ˆ
(x + z)
22
2
18 Diamond structure
• Not a Bravais lattice
• Two interpenetrating fcc Bravais lattice
Bravais lattice : fcc
1
4basis
0
ˆ ˆ
ˆ
(x + y + z)
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
01/2 0
1/4
3/4
3/4
1/4
1/2
1/2 0
0
0
1/2
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:
0a1 + a2 + a3
332
ra3
ra2
ra1
21
Four neighboring atoms form the vortices of a tetrahedron 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: Cs+ at (0,0,0) and Cl- at (1/2,1/2,1/2)
Basis: Cl- at (0,0,0) and Na+ at (1/2,1/2,1/2)
24 Miller indices to index crystal planes x3
ra3
r
ra2 a1 x2 x1
• find the intercepts x1, x2, and x3
• h: k: l = 1/x1 : 1/ x2: 1/x3
r
r
ra3
Find the intercepts on the axes in terms of lattice constants a1 a2
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
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Viewed as if it’s “face centered”
Viewed as if it’s “body 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
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Simple orthorhombic
Base centered orthorhombic Two ways of stretching the same centered tetragonal lattice viewed along c axis stretch stretch
Body-centered orthorhombic
Symmetry reduced, two structures distinguishable face-centered orthorhombic
31 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
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Simple monoclinic Body-centered orthorhombic face-centered orthorhombic
Distort rectangular shape into parallelogram
No distinction between “face-centered” and “body-centered”
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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 tetragonal hexagonal orthorhombic trigonal monoclinic triclinic
Arrow: direction of symmetry reduction
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