Crystal Structure Continued!

NOTE!!Much of the discussion & many figures in what follows was\constructed from lectures posted on the web by Prof. BeşireGÖNÜL in Turkey. She has done an excellent job of coveringmany details of crystallography & she illustrates her topics withmany very nice pictures of lattice structures. Her lectures on thisare posted Here:

http://www1.gantep.edu.tr/~bgonul/dersnotlari/.Her homepage is Here: http://www1.gantep.edu.tr/~bgonul/.        

NOTE!!Much of the discussion & many figures in what follows was\constructed from lectures posted on the web by Prof. BeşireGÖNÜL in Turkey. She has done an excellent job of coveringmany details of crystallography & she illustrates her topics withmany very nice pictures of lattice structures. Her lectures on thisare posted Here:

http://www1.gantep.edu.tr/~bgonul/dersnotlari/.Her homepage is Here: http://www1.gantep.edu.tr/~bgonul/.        

Crystal Lattices

Bravais Lattices(BL)

Non-Bravais Lattices(non-BL)

All atoms are the same kind All lattice points are equivalent

Atoms are of different kinds. Somelattice points are not equivalent.

Atoms are of different kinds.Some lattice points aren’t equivalent.

A combination of 2 or more BL

2 d examples2 d examples

In General• Mathematically, a lattice is defined by 3 vectors called

Primitive Lattice Vectorsa1, a2, a3 are 3d vectors which depend on the geometry.

• Once a1, a2, a3 are specified, the

Primitive Lattice Structureis known.

• The infinite lattice is generated by translating through a

Direct Lattice Vector: T = n1a1 + n2a2 + n3a3

n1,n2,n3 are integers. T generates the lattice points. Each lattice point corresponds to a set of integers (n1,n2,n3).

Lattice Translation Vectors

Consider a 2-dimensional lattice (figure). Define the

2 Dimensional Translation VectorRn n1a + n2b

(Sorry for the notation change!!)a & b are 2 d Primitive Lattice Vectors, n1, n2 are integers.

2 Dimensional Lattice Translation Vectors

• Once a & b are specified by the lattice geometry & an origin is chosen, all symmetrically equivalent points in the lattice are determined by the translation vector Rn. That is, the lattice has translational symmetry. Note that the choice of Primitive Lattice vectors is not unique! So, one could equally well take vectors a & b' as primitive lattice vectors.

Point D(n1, n2) = (0,2)

Point F(n1, n2) = (0,-1)

The Basis(or basis set)

The set of atoms which, when placed at eachlattice point, generates the Crystal Structure.

Crystal Structure≡ Primitive Lattice + Basis

Translate the basis through all possible lattice vectors

T = n1a1 + n2a2 + n3a3

to get the Crystal Structure or the

DIRECT LATTICE

• The periodic lattice symmetry is such that the atomic arrangement looks the same from an arbitrary vector position r as when viewed from the point

r' = r + T (1)where T is the translation vector for the lattice:

T = n1a1 + n2a2 + n3a3

• Mathematically, the lattice & the vectors a1,a2,a3 are

Primitiveif any 2 points r & r' always satisfy (1) with a suitable choice of integers n1,n2,n3.

• In 3 dimensions, no 2 of the 3 primitive lattice vectors a1,a2,a3 can be along the same line. But,

Don’t think of a1,a2,a3 as a mutually orthogonal set!

Usually, they are neither mutually perpendicular nor all the same length!

• For examples, see Fig. 3a (2 dimensions):

The Primitive Lattice Vectors a1,a2,a3 aren’t necessarily a mutually orthogonal set!

Usuallythey are neither mutually perpendicular nor

all the same length! • For examples, see Fig. 3b (3 dimensions):

Bravais Lattice An infinite array of discrete points with anarrangement & orientation that appears exactly thesame, from whichever of the points the array is viewed. A Bravais Lattice is invariant under a translation

T = n1a1 + n2a2 + n3a3

Crystal Lattice Types

Nb film

Non-Bravais LatticesIn a Bravais Lattice, not only the atomicarrangement but also the orientations must appearexactly the same from every lattice point.2 Dimensional Honeycomb LatticeThe red dots each have a neighbor to

the immediate left. The blue dot has a neighbor to its right. The red (& blue) sides are equivalent & have the same appearance. But, the red & blue dots are not equivalent. If the blue side is rotated through 180º the lattice is invariant.

The Honeycomb Lattice is NOT a Bravais Lattice!!

Honeycomb Lattice

It can be shown that, in 2 Dimensions, there are

Five (5) & ONLY Five Bravais Lattices!

2-Dimensional Unit CellsUnit Cell The Smallest Component

of the crystal (group of atoms, ions or molecules),which, when stacked together with pure

translational repetition, reproduces the whole crystal.

S

ab

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

S

Unit Cell The Smallest Componentof the crystal (group of atoms, ions or molecules),which, when stacked together with puretranslational repetition, reproduces the whole crystal.

Note that the choice of unit cell is not unique!

2-Dimensional Unit CellsArtificial Example: “NaCl”

Lattice points are points withidentical environments.

2-Dimensional Unit Cells: “NaCl”Note that the choice of origin is arbitrary!

the lattice points need not be atoms, butThe unit cell size must always be the same.

These are also unit cells! It doesn’t matter if the origin is at Na or Cl!

2-Dimensional Unit Cells: “NaCl”

These are also unit cells.The origin does not have to be on an atom!

2-Dimensional Unit Cells: “NaCl”

These are NOT unit cells!Empty space is not allowed!

2-Dimensional Unit Cells: “NaCl”

In 2 dimensions, these are unit cells.In 3 dimensions, they would not be.

2-Dimensional Unit Cells: “NaCl”

Why can't the blue triangle be a unit cell?2-Dimensional Unit Cells

Example: 2 Dimensional, Periodic Art!A Painting by

Dutch Artist Maurits Cornelis Escher (1898-1972)

Escher was famous for his so called “impossible structures”, such as Ascending & Descending, Relativity,..

Can you find the “Unit Cell” in this painting?

3-Dimensional Unit Cells

3-Dimensional Unit Cells3 Common Unit Cells with Cubic Symmetry

Simple Cubic Body Centered Face Centered (SC) Cubic (BCC) Cubic (FCC)

Simple Cubic Body Centered Face Centered (SC) Cubic (BCC) Cubic (FCC)

Conventional & Primitive Unit CellsUnıt Cell Types

Primitive

A single lattice point per cellThe smallest area in 2 dimensions, orThe smallest volume in 3 dimensions

Simple Simple CCubicubic ((SCSC))ConventionalConventional Cell Cell ==

Primitive cell Primitive cell

More than one lattice point per cell Volume (area) = integer multiple of

that for primitive cell

Conventional (Non-primitive)

Body Centered Cubic (BCC)Body Centered Cubic (BCC)ConventionalConventional Cell Cell ≠≠

Primitive cell Primitive cell

Face Centered Cubic (FCC) Structure

Conventional Unit Cells• A Conventional Unit Cell just fills space when

translated through a subset of Bravais lattice vectors.• The conventional unit cell is larger than the primitive

cell, but with the full symmetry of the Bravais lattice.• The size of the conventional cell is given by the lattice

constant a.

The full cube is the Conventional UnitCell for the FCC

Lattice

The full cube is the Conventional UnitCell for the FCC

Lattice

FCC Bravais LatticeFCC Bravais Lattice

Conventional & Primitive Unit CellsFace Centered Cubic Lattice

Primitive LatticeVectors

a1 = (½)a(1,1,0)a2 = (½)a(0,1,1)a3 = (½)a(1,0,1)Note that the ai’s are

NOT MutuallyOrthogonal!

Primitive LatticeVectors

a1 = (½)a(1,1,0)a2 = (½)a(0,1,1)a3 = (½)a(1,0,1)Note that the ai’s are

NOT MutuallyOrthogonal!

Conventional Unit Cell (Full Cube)

Conventional Unit Cell (Full Cube)

Primitive Unit Cell(Shaded)

Primitive Unit Cell(Shaded)

Lattice Constant

Lattice Constant

Elements That Form Solidswith the FCC Structure

Body Centered Cubic (BCC) Structure

Primitive LatticeVectors

a1 = (½)a(1,1,-1)a2 = (½)a(-1,1,1)a3 = (½)a(1,-1,1)

Note that the ai’s are

NOT mutuallyorthogonal!

Primitive LatticeVectors

a1 = (½)a(1,1,-1)a2 = (½)a(-1,1,1)a3 = (½)a(1,-1,1)

Note that the ai’s are

NOT mutuallyorthogonal!

Primitive Unit CellPrimitive Unit Cell

Lattice Constant

Lattice Constant

Conventional Unit Cell (Full Cube)

Conventional Unit Cell (Full Cube)

Conventional & Primitive Unit CellsBody Centered Cubic Lattice

Elements That Form Solidswith the BCC Structure

Conventional & Primitive Unit CellsCubic Lattices

a

b c

a

b c Simple Cubic (SC)Primitive Cell = Conventional Cell

Fractional coordinates of lattice points:000, 100, 010, 001, 110,101, 011, 111

Body Centered Cubic (BCC)Primitive Cell Conventional

CellFractional coordinates of the lattice points in the conventional cell:000,100, 010, 001, 110,101, 011, 111, ½ ½ ½a

bc

Primitive Cell = RombohedronPrimitive Cell = Rombohedron

Conventional & Primitive Unit CellsCubic Lattices

Face Centered Cubic (FCC)Primitive Cell Conventional Cell

a

bc

The fractional coordinates of lattice points in the conventional cell are:

000,100, 010, 001,110,101, 011, 111,

½ ½ 0, ½ 0 ½, 0 ½ ½, ½ 1 ½, 1 ½ ½ , ½ ½ 1

Simple Hexagonal Bravais Lattice

Conventional & Primitive Unit Cells

Hexagonal Bravais Lattice

Primitive Cell = Conventional Cell

Fractional coordinates of lattice points in conventional cell:

100, 010, 110, 101,011, 111, 000, 001

Points of the Primitive Cell

a

b

c

120

o

Hexagonal Close Packed (HCP) Lattice:A Simple Hexagonal Bravais Lattice

with a 2 Atom Basis

The HCP lattice is not a Bravais lattice, because the orientation of the environment of a point varies from layer to layer along the c-axis.

General Unit Cell Discussion• For any lattice, the unit cell &,

thus, the entire lattice, is UNIQUELY determined by 6 constants (figure):

a, b, c, α, β and γwhich depend on lattice geometry.

• As we’ll see, we sometimes want to calculate the number of atoms in a unit cell. To do this, imagine stacking hard spheres centered at each lattice point & just touching each neighboring sphere. Then, for the cubic lattices, only 1/8 of each lattice point in a unit cell assigned to that cell. In the cubic lattice in the figure,

Each unit cell is associated with (8) (1/8) = 1 lattice point.

Primitive Unit Cells & Primitive Lattice Vectors• In general, a Primitive Unit

Cell is determined by the parallelepiped formed by the Primitive Vectors a1 ,a2, & a3 such that there is no cell of smaller volume that can be used as a building block for the crystal structure.

• As we’ve discussed, a Primitive Unit Cell can be repeated to fill space by periodic repetition of it through the translation vectorsT = n1a1 + n2a2 + n3a3.

• The Primitive Unit Cell volume can be found by vector manipulation:

V = a1(a2 a3) • For the cubic unit cell

in the figure, V = a3

Primitive Unit Cells• Note that, by definition, the Primitive Unit Cell must

contain ONLY ONE lattice point.• There can be different choices for the Primitive Lattice

Vectors, but the Primitive Cell volume must be independent of that choice.

A 2 Dimensional Example!

P = Primitive Unit CellNP = Non-Primitive

Unit Cell