X-ray diffraction – the experiment

Learning Outcomes

By the end of this section you should:• understand some of the factors influencing X-ray

diffraction output• be aware of some X-ray diffraction experiments and

the information they provide• know the difference between single crystal and

powder methods

Methods and Instruments

All are based on:

X-ray Source Sample Detector

Sample can be:

• Single crystal

• Powder - (what is a powder?!)

X-rays - interactions

First assumption: X-rays elastically scattered by electrons.

Second assumption: Spherical, discrete atoms

J. J. Thomson’s classical theory of X-ray scattering.

X-ray output is defined through the scattering cross-section.

Very weak interaction. Thus need lots of electrons, and thus many atoms.

J. J. Thomson, “Conduction of Electricity through Gases”

where r0 is the classical electron radius.

20

2

20

2

3

8

43

8r

cm

e

e

Scattering factor

• More electrons means more scattering ( Z)• Scattering per electron adds together, so helium

scatters twice as strongly as H

We define an atomic (X-ray) scattering factor, fj, which depends on:

• the number of electrons in the atom (Z)• the angle of scattering

Function of deflection angle

f varies as a function of angle , usually quoted as a function of (sin )/

http://www.ruppweb.org/xray/comp/scatfac.htm

The more diffuse the electron cloud, the more rapid the reduction in the scattering function with scattering angle.

Deflection angle / atomic number

Different elements show the same trend: note the starting value

http://www.ruppweb.org/xray/comp/scatfac.htm

0

5

10

15

20

25

30

35

0 0.2 0.4 0.6

sin 0 / (A -1)

f in

ele

ctr

on

s

copper

oxygen

(sin ) /

f Z (ish)

For = 0, f is equal to the total number of electrons in the atom, so

f=0 = Z

Ca2+ and Cl- both have 18 electrons.

So at =0 fCa = 18 = fCl

But as increases, Cl- has smaller f as it has a more diffuse electron cloud

What is important?

• Lots of scattering centres • Large enough crystals (lots of planes) • Long range order (otherwise??)

Glass crystallising with temperature

Broad, featureless pattern. Some information can be retrieved (e.g. average atomic distances) but no structure.

Bragg (again!!)

• Look at Bragg set-up with different emphasis

hkl1000’s of planes (1000Å = 1m)

Scattering:angle and Z

Thus the scattering from this plane will reflect which atoms are in the plane. Turn the crystal….

Bragg (again!!)

d expands

Changes d-spacing and atoms within the planes

So we need to either (a) rotate the crystal or (b) have lots of crystals at different orientations simultaneously

hkl

Scattering:angle and Z

Laue Method

White X-ray

sourceCollimator

Fixed single crystal

Detector photographic film or area

detector

http://www.matter.org.uk/diffraction/x-ray/laue_method.htm

Max Von Laue 1879-1960

Nobel Prize 1914

Laue Method

http://www-xray.fzu.cz/xraygroup/www/laue.html

Laue Method

Each spot corresponds to a different crystal plane

USES:

• alignment of single crystal

• info on unit cell

• info on imperfections, defects in crystal

Not so common these days…

4-circle Method

Monochromatic X-rays

Movingdetector

Movingsingle crystal

Crystal can be oriented so that intensities for any (hkl) value can be measured

Actual instrument

http://www.lks.physik.uni-erlangen.de/equipmen/equipmen.html

Now more common to use area detector which removes one circle.

Bruker SMART

Area detector

Output

List of hkl (each spot represents a plane) and intensity

1000’s of data points needed

0 0 -1 0.00 0.11 1 0 0 -2 21.52 0.61 1 0 0 -3 0.15 0.27 1 0 0 -4 245.17 3.01 1 0 0 -5 0.04 0.36 1 0 0 -6 16.09 0.92 1 0 0 -7 0.46 0.40 1 0 0 -8 0.25 0.45 1 0 0 -9 -0.20 0.38 1 0 0 -10 3.44 0.64 1 0 0 -11 -0.31 0.37 1 0 0 -12 -0.04 0.39 1 0 0 -13 -0.15 0.42 1 0 -1 0 0.30 0.15 1 0 -1 -1 237.47 2.80 1 0 1 -1 264.24 2.70 1 0 1 -2 26.53 0.65 1 0 -1 -3 1.63 0.39 1 0 1 -3 2.11 0.32 1 0 -1 -4 2.58 0.46 1 0 1 -4 2.46 0.39 1 0 -1 -5 96.60 2.13 1 0 1 -5 88.31 1.69 1 0 -1 -6 0.65 0.36 1 0 1 -6 0.39 0.38 1 0 -1 -7 2.01 0.58 1 0 1 -7 2.01 0.47 1 0 -1 -8 0.27 0.45 1 0 1 -8 0.32 0.42 1 0 -1 -9 6.73 0.87 1 0 1 -9 6.35 0.68 1 0 -1 -10 -0.46 0.42 1 0 1 -10 0.07 0.38 1 0 -1 -11 0.67 0.51 1 0 1 -11 0.54 0.41 1 0 -1 -12 -0.32 0.47 1 0 1 -12 0.18 0.39 1 0 -1 -13 -0.09 0.47 1 0 1 -13 -0.09 0.45 1 0 -2 0 348.85 3.47 1 0 -2 -1 133.28 2.48 1 0 2 -1 123.01 1.27 1 0 2 -2 148.04 1.54 1 0 -2 -3 60.01 1.75 1 0 2 -3 61.91 1.08 1 0 -2 -4 0.72 0.40 1 0 2 -4 1.20 0.31 1 0 -2 -5 2.93 0.58 1

Uses

• Unit cell determination• Crystal structure determination (primary method)

We will come to the theory later on…

We’ve also used ours to get information on vertebral disks!!

Powder DiffractionBy “powder”, we mean polycrystalline, so equally we can use a piece of metal, bone, etc.

We assume that the crystals are randomly oriented so that there are always some crystals oriented to satisfy the Bragg condition for any set of planes

Monochr.X-rays

Detector -

• Film

• Counter

Film - Debye Scherrer Camera

Camera radius = R

360

4

R2

S

Debye-Scherrer Camera

Now obsolete!

Peter Debye, 1884-1966

Nobel Prize 1936

Counter - Diffractometer

• Bruker D8 Advance

X-ray tube

detector

sample

More detail

Not all are the same…

Stoe Stadi/P

Detector SampleX-ray tube

Furnace Detector

Output

Plot of intensity of diffracted beam vs. scattering angle (2)

The Powder Pattern

The whole pattern is a representation of the crystal structure• Not like some other techniques like spectroscopy• Next section we will examine the uses in more detail,

then the details behind the pattern

Summary

Diffraction experiments consist of a source, a sample and a detector

Samples can be single crystal or “powder” (polycrystalline)

Single crystal is a primary technique for structure determination

Powder diffraction relies on a random orientation of (small) crystallites