The resip iprocal space and unit itcell Bragg ggs la law - - PowerPoint PPT Presentation

In Introduction to cry rystallography The unit itcell ll The resip iprocal space and unit itcell Bragg ggs la law Structure factor Fhkl and and atomic ic scattering factor fzθ

Introduction to crystallography

We divide materials into two categories:

• Amorphous materials
• The atoms are ”randomly” distributed in space
• Not quite true, there is short range order
• Examples: glass, polystyrene (isopor)
• Crystalline materials
• The atoms are ordered
• Short range and long range order
• Deviations from the perfect order are of

importance for the properties of the materials. The specimen Why do we see facets on the surface

Introduction to crystallography

The specimen

Basic aspects of crystallography

• Crystallography describes and characterise the structure of

crystals

• Basic concept is symmetry
• Translational symmetry: if you are standing at one point in a

crystal, and move a distance (vector) a the crystal will look exactly the same as where you started.

a a a a 1D a a a a 2D b The specimen

The lattice

• described as a set of mathematical points in space
• each of these points represents one or a group of atoms,

basis

a a a a a a Basis + Lattice = crystal structure The specimen

Axial systems

The point lattices can be described by 7 axial systems (coordinate systems) x y z a b c α γ β Axial system Axes Angles Triclinic a≠b≠c α≠β≠γ≠90o Monoclinic a≠b≠c α=γ=90o ≠ β Orthorombic a≠b≠c α= β=γ=90o Tetragonal a=b≠c α= β=γ=90o Cubic a=b=c α= β=γ=90o Hexagonal a1=a2=a3≠c α= β=90o γ=120o Rhombohedral a=b=c α= β=γ ≠ 90o

The specimen

Bravais lattice

The point lattices can be described by 14 different Bravais lattices Hermann and Mauguin symboler: P (primitiv) F (face centred) I (body centred) A, B, C (bace or end centred) R (rhombohedral) The specimen

Unit cell

• The crystal structure is described by specifying a repeating element and its translational periodicity
• The repeating element (usually consisting of many atoms) is replaced by a lattice point and all lattice points have the

same atomic environments.

• The unit cells are the smallest building blocks.
• A primitive unit cell has only one lattice point in the unit cell.

a c b α β γ Lattice point Repeating element, basis The specimen

Exaples of materials with a face centered cubic lattice

Copper The specimen

Exaples of materials with a face centered cubic lattice

Silicon The specimen

Exaples of materials with a face centered cubic lattice

ZnS The specimen

What about other symmetry elements?

• We have discussed translational symmetry, but there are also other

important symmetry operations:

• Mirror planes
• Rotation axes
• Inversion
• Screw axes
• Glide planes
• The combination of these symmetry operations with the Bravais

lattices give the 230 space groups

The specimen

Space groups

• Crystals can be classified

according to 230 space groups.

• Details about crystal description

can be found in International Tables for Crystallography.

• Criteria for filling Bravais

point lattice with atoms.

• A space group can be referred to

by a number or the space group symbol (ex. Fm-3m is nr. 225)

• Structural data for known crystalline

phases are available in “books” like “Pearson’s handbook of crystallographic data….” but also electronically in databases like “Find it”

• r f.ex. “Crystallography open

database”: http://www.crystallography.net/cod/ .

• Pearson symbol like cF4 indicate the

axial system (cubic), centering of the lattice (face) and number of atoms in the unit cell of a phase (like Cu).

The specimen

Lattice planes

• Miller indexing system
• Crystals are described in the axial system
• f their unit cell
• Miller indices (hkl) of a plane is found

from the interception of the plane with the unit cell axis (a/h, b/k, c/l).

• The reciprocal of the interceptions are

rationalized if necessary to avoid fraction numbers of (h k l) and 1/∞ = 0

• Planes are often described by their

normal

• (hkl) one single set of parallel planes
• {hkl} equivalent planes

Z Y X (010) (001) (100) Z Y X (110) (111) Z Y X

y z x

c/l a/h b/k

The specimen

Directions

• The indices of directions (u, v and w) can be found from

the components of the vector in the axial system a, b, c.

• The indices are scaled so that all are integers and as small

as possible

• Notation
• [uvw] one single direction or zone axis
• <uvw> geometrical equivalent directions
• [hkl] is normal to the (hkl) plane in cubic axial systems

ua a b x z c y vb wc [uvw] Zone axis [uvw] (hkl)

uh+vk+wl= 0 The specimen

Recip iprocal l vectors, pla lanar dis istances

• The reciprocal lattice is defined by the vectors :

2 2 2

l k h a dhkl   

–The normal of a plane is given by the vector: –Planar distance between the planes {hkl} is given by: –Planar distance (d-value) between planes {hkl} in a cubic crystal with lattice parameter a:

* * *

c l b k a h g hkl   

hkl hkl

g d / 1 

V b a c V a c b V c b a / ) ( / ) ( / ) (

* * *

     

a*=(bcsinα)/V b*=(casinβ)/V c*=(absinγ)/V

The specimen

Elastic scattering

Interaction with sample

Why do we want a monochromatic wave in diffraction studies?

• X-rays are scattered by the electrons in a material
• Electrons are scattered by both the electron and the nuclei in a material
• The electrons are directly scattered and not by an field to field exchange as in the case for X-rays
• The diffraction theory is the same for electrons, X-rays and neutrons.
• Based on Braggs law

Bragg ggs law

Gives the angle when the scattered wave is in phase. Coherent incoming wave Elastic scattering Interaction with sample

Deduction of Braggs law:

θ θ x x ψi Sum ψ scattered in phase In order for the scattered waves to be in phase (resulting in constructiv interference), the difference in the travelled distance of the waves must be a multiple of λ (i.e 2X = nλ). X= d sinθ and hence 2d sinθ= nλ. d Interaction with sample

Effect of adding a scattering plane with d/2

θ θ x x ψi Imaging that the red planes represents the (001) planes in a cubic structure with the same atom (=electron) density on the (002) plane (like in fcc and bcc structures). The (002) plane will now scatter the incoming wave and travel 1/2nλ shorter/longer than the waves scattered from (001). The amplitude of the sum of the waves in this situation is 0 (x’=d/2 sinθ, 2x’= d sinθ and this is equal to 1/2nλ. In this example “d“ represented the planar distance of the (001) planes and “d/2” the planar distance of the (002) plane. This is commonly written as d001 and d002. For a cubic unitcell d001 and d002 is equivalent to d100 and d200. The latter ones are the ones commonly tabulated in d-value tables if the intensity is not 0. d d/2 (001) (002)

x’ x’

Interaction with sample

Example: Bragg angle for (002) relative to (001)

θ θ x x ψi Imaging that the wave length and the crystal are the same as in the previous examplered planes represents the (001) planes in a cubic structure with the same atom (=electron) density on the (002) plane (like in fcc and bcc structures). The (002) plane will now scatter the incoming wave and travel 1/2nλ shorter/longer than the waves scattered from (001). The amplitude of the sum of the waves in this situation is 0 (x’=d/2 sinθ, 2x’= d sinθ and this is equal to 1/2nλ. In this example “d“ represented the planar distance of the (001) planes and “d/2” the planar distance of the (002) plane. This is commonly written as d001 and d002. For a cubic unitcell d001 and d002 is equivalent to d100 and d200. The latter ones are the ones commonly tabulated in d-value tables if the intensity is not 0. d d/2 (001) (002)

x’ x’

Interaction with sample

Atomic scattering factor (X-ray)

Z

Intensity of the scattered X-ray beam

Structure factors

The structure factors for x-ray, neutron and electron diffraction are similar. For neutrons and electrons we need only to replace by fj

(n) or fj (e) .





 

N j x j hkl g

f F F

1 ) (

2 exp( )) (

j j j

lw kv hu i   

X-ray:

The coordinate of atom j within the crystal unit cell is given rj=uja+vjb+wjc. h, k and l are the miller indices of the Bragg reflection g (and represents the normal to the plane (hkl). N is the number of atoms within the crystal unit cell. fj(n) is the x-ray scattering factor, or x-ray scattering amplitude, for atom j.

rj uja a b x z c y vjb wjc

The intensity of a reflection is proportional to:

 g g F

F

Example: Cu, fcc

• eiφ = cosφ + isinφ
• enπi = (-1)n
• eix + e-ix = 2cosx





 

N j j hkl g

f F F

1

2 exp( )) (

j j j

lw kv hu i   

Atomic positions in the unit cell: [000], [½ ½ 0], [½ 0 ½ ], [0 ½ ½ ] Fhkl= f (1+ eπi(h+k) + eπi(h+l) + eπi(k+l)) If h, k, l are all odd then: Fhkl= f(1+1+1+1)=4f If h, k, l are mixed integers (exs 112) then Fhkl=f(1+1-1-1)=0 (forbidden)

What is the general condition for reflections for fcc? What is the general condition for reflections for bcc?

Example: Cu, fcc

• enπi= (-1)n





 

N j j hkl g

f F F

1

2 exp( )) (

j j j

lw kv hu i    Atomic positions in the unit cell: [000], [½ ½ 0], [½ 0 ½ ], [0 ½ ½ ]

Fhkl= fCu (exp2πi(0+0+0) + exp2πi(1/2h+1/2k+0) + exp2πi(1/2h+0+1/2l) + exp2πi(0+1/2k+1/2l)) Fhkl= fCu (exp0πi + expπi(h+k) + expπi(h+l) + expπi(k+l)) : General expression for fcc F100=fCu(1 + expπi(1+0) + exp πi(1+0) + exp πi(0+0))= fCu(1-1-1+1)= 0 F200=fCu(1 + expπi(2+0) + exp πi(2+0) + exp πi(0+0))= fCu(1+1+1+1)= 4fCu F110=fCu(1 + expπi(1+1) + exp πi(1+0) + exp πi(1+0))= fCu(1+1-1-1)= 0 F111=fCu(1 + expπi(1+1) + exp πi(1+1) + exp πi(1+1))= fCu(1+1+1+1)=4fCu

If h, k, l are all odd (ex. 111) or even then: Fhkl= f(1+1+1+1)=4f (intensity) If h, k, l are mixed integers (exs 110) then: Fhkl=f(1+1-1-1)=0 (no intensity)

The coordinate of atom j within the crystal unit cell is given rj=uja+vjb+wjc. (hkl): a plane

The fcc real space lattice results in a bcc reciprocal lattice

200 111 002 a* c* b*