Crystal

structure

Same building blocks, butdifferent crystals...

Cleaving a rocksalt crystal

The 5 two-dimensional

Bravais-lattices

Symmetries in 2D

6 – OK!

3 – OK! 4 – OK! 5 – No!

7 – No! 8 – No!

Quasi-crystals!

(Quasi-periodicarrangement

using twodifferent ”tiles”.)

Possible rotationsymmetries have1,2,3,4 or 6-foldrotation.

Exercise: Outline the unit cell in

this structure!

14 Bravais-lattices in 3D

P = PrimitiveI = Body-centeredF = Face-centeredC = Side-centered

Primitive cells

A primitive cell can always be constructed like this:

1. Draw lines from a lattice point to allnearest neighbour-points.

2. Add planes perpendicular to the lines, halfway between the lattice points.

3. The smallest enclosed volume (area)is a Wiegner-Seitz primitive cell.

Primitive cells may havemany different shapes ina given structure, but theyalways contain one (and onlyone) lattice point!

Hexagonal structure – a Bravais

lattice with two points in the base

Primitive unit cellfor the Bravais lattice

The hexagonal structure in itselfis no Bravais lattice, since theenvironment is different as seenfrom Q and R, respectively (it isnot invariant under translation)!

Simple cubic Body centered cubic Face centered cubic

N2 (20K), Po Li, Na, Cs, Ba, Fe, NbMo, Cr……

Cu, Rh, Pd, Pt, Ag, Au, Ir, Ne, Kr……..

Cubic crystal structures Data for cubic structures

”Filling factor:”

bcc – Body-centered cubic

Conventional (cubic) cell,and a primitive cell.

The conventional cell has2 lattice points, while the primitive cell has one.

The primitive cell also has (bynecessity) half the volume ofthe conventional cell.

fcc – face-centered cubic

Conventional (cubic) cell,and a primitive cell.

The conventional cell has4 lattice points, while theprimitive cell has one.

Non-primitive cells often havea more immediate and/ormore evident relation to thecrystal structure!

hcp – hexagonal close-packing

Simplehexagonal

Hexagonalclose packed

NaCl och CsCl-structure

FCC Bravais lattice with Na at 0 and Clat the center of the cubic cell.

SC Bravais lattice with Cs at 0 and Clat the cube center.

Crystal planes in an

fcc-structure

An fcc-structure is alsoclose-packed, but withthe hexagonal planes ina different arrangement!

Hexagonal!

Differences

between

close-packing

in fcc and hcp

A A A

AAA A

B B B

BB

C C C

CCC

B

1. The bottom layer is formed by the green circles with centers in A.2. The second layer (black rings) have centers in B.3. The third layer can be formed in two ways, with centers in…

A, so that the 1st and 3rd layers are aligned over each other, andthe order is ABABAB... – hcp-structure (”eclipsed”).

C, so that the three layers are displaced relative to each other, andthe order is ABCABC... – fcc-structure (”staggered”).

Note that fcc-packing results in a Bravais lattice, while hcp does not, sincethe surroundings vary between layers along the c-axis!

Diamond structureFcc-structure with two atoms in the base(or two fcc-structures displaced ¼ diagonal relative to each other...)

The bonds form tetrahedrons; each atom has 4 nearest neighbours, and 12 next nearest neighbours!

Ex: C, Si, Ge, α-Sn

Crystal structures of the

elements

sc

bcc

hcp

fcc

Wiegner-Seitz

primitive cells

Create a primitive cell in the following way:

1. Draw lines from a lattice point to itsneighbour points.

2. Add planes perpendicular to the lines, halfway between the lattice points.

3. The smallest volume enclosed by the linesis a Wiegner-Seitz primitive cell.

Wiegner-Seitz-cell for a bcc lattice(a truncated octahedron).

Wiegner-Seitz-cell for an fcc lattice(a rhombic dodecahedron).

Calculating Miller indices

a b

c

n1

n2

n3• Find the intersection of the plane with the axes,expressed in the unit vectors: (n1a, n2b, n3c)

• Invert the coefficients and multiply with the lowest common denominator to form integers,which then are the Miller indices:

• Intersection on the negative side of the axis is indicated with a bar:

(h k l)

• A direction in the crystal is denoted [uvw], so that [100] is the directionalong the positive a-axis.

1 2 3

1 1 1( ) , ,hkl lcd

n n n

= ⋅

Notation for lattice planes and

directions• A family of equivalent lattice planes (planes with the same symmetryproperties) is denoted {hkl}.

For example, (100),(010),(001) gives {100}

• Similarly, equivalent directions are writted < n1 n2 n3>.

For example, [100], [010], [001] in a cubic crystal gives <100>

• In cubic crystals, the [hkl]-direction is perpendicular to the plane defined by the Miller indices (hkl).

• If the vectors A and B span a crystal plane, a normal vector to the plane is given by

n = A x B

…from which also the Miller indices for the plane are obtained.

Distances between lattice planes2 2

0

1 12 2 hk

a a a ad

k h k h

⋅ = ⋅ ⋅ +

( )4

2 202 2 2

1 1hk

ad a

k hkh

= ⋅ ⋅ + 2 2

2 20

1

hk

h k

d a

+=

2 2 2

2 2 2 2

1

hkl

h k l

d a b c= + +

Assume a square lattice in 2Dwith Miller index (hk0):

Find a relation between the interplanedistance dhk0 and the lattice parametersa/h and a/k !

Generalized to three dimensionsin an orthorombic lattice:

dhk0

a

a/h

a/k

aHalf the area of the

dashed squareThe area of theshaded triangle

. . . .

. . . .

. . . .

. . . .

(110)-planes

. . . .

. . . .

. . . .

. . . .

(440)-planes

dnh nk nl = dhkl / n d440 = d110/ 4

Lattice plane distances Equivalent planes

With the same incidenceangle, the dashed linesresult in diffraction at someother angle!

2 sinhkln dλ θ=

The Bragg angle (θ) is half the diffraction angle of the indident beam (2θ)!

Bragg’s law Bragg’s experiment

hkl

hkl

hklθ

dhkl

X-ray, λ ~ 2 Å

2θ

Intensity

Detector

2θ

hkl

hklhkl

sin θ = λ2 dhkl

Bragg’s law

Diffractograms

I(2θ )

Both KCl and KBr arefcc structures!

The diffractogram for KCllooks like a monoatomic SCwith lattice constant a/2.

The results are differentbecause Cl and Br have different X-ray contrasts!

2θ Chong, Soft Matter, 7, 4768 (2011)

X-ray diffraction on monoolein stabilizedwith 0.5 % (wt) F127 at 37 C.

The diffractogram is interpreted as aprimitive inverted bicontinuous cubic phase(Im3m) with lattice parameter 134 Å.

Structure

in lipid

systems