The Muppet’s Guide to:The Structure and Dynamics of Solids

Kinematical Diffraction

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Kinematical Diffraction Theory

Violates Energy

Conservation

Far-Field, Fraunhofer

regime

2

exp q rr r id

dd

V

Assumes weak scattering

Works surprisingly well!

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What Happens when an EM wave meets a free electron?

Electron:Dipole Moment, PSpin, S

Need to consider the force that the electric and magnetic components exert on the electron.....

An EM wave incident on a free electron will induce motion of both the charge and spin

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Electric component

E 0

0

, expˆ

expˆ

r t E i k r t

E i k t

Electric Component:

Force on Electron:

E eF E

F 0 expˆE e E i k t

Force along e with p phase change

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Electric component

The electron follows the oscillating electric field creating a electric dipole moment along :e

02exp

e

ep t E i t

m

No polarisation changes

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Magnetic component

H 0, expˆr t H i k t

Magnetic Component:

Force on Electron:Zeeman Effect – tries to rotate away from H producing a torque:

0ˆ expMAX

H BF ik E i k t

0

0

expˆ

ˆ expˆ

S H S H i k t

ik S H i k t

Force along x with /2p phase change

F S H2H B

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Magnetic component

2

02e

em t i S H

m

The electron oscillates creating a dipole moment along :x

Rotation of incident polarisation

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Re-radiationActa Cryst. A37, 314 (1981)F. de Bergevin and M. Brunel

We have considered to two cases which produce E-dipole radiation, but what are the relative strengths?

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Force EquationElectron motion is elliptical from the sum of the two forces:

F 0 expˆE e E i k t

F 0ˆ expH Bik E i k t

Ratio Amplitude of Forces:

22 1eE

H B

m cF e MeVF k E E

Magnetic force (amplitude) much weaker than charge force - x-rays measure charge

‘Soft’ x-rays, 500ev 2000EH

FF

‘Normal’ x-rays, 10keV 100EH

FF

61 10EH

II

Atomic scattering factor

arg exp[ ]Ch eV

f r iq r dr Z

arg pC ne ih sfffAtomic scattering factor:

Sum the interactions from each charge and magnetic dipole within the atom ensuring that we take relative phases into account:

arg ( ) iq rch e fiV

f k V k V r e

Atomic scattering factor - neutrons:

Vmb j jr r R

2

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X-ray scattering from an AtomTo an x-ray, an atom consist of an electron density, r(r).

( ) exp q r V

f r i dV In coherent scattering (or Rayleigh Scattering)• The electric field of the photon interacts with an electron, raising it’s

energy.• Not sufficient to become excited or ionized• Electron returns to its original energy level and emits a photon with

same energy as the incident photon in a different direction

Resonance – Atomic Environment

In fact the electrons are bound to the nucleus so we need to think of the interaction as a damped oscillator.Coupling increases at resonance – absorption edges.

The Crystalline State Vol 2: The optical principles of the diffraction of X-rays, R.W. James, G. Bell & Sons, (1948)

Real part - dispersion Imaginary part - absorption

0, spinf q f q fi ff

Real and imaginary terms linked via the Kramers-Kronig relations

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Anomalous Dispersion

6 9 12 15-5

0

5

10

15

20

25

30

35

Sca

tterig

Fac

tors

(el

ectr

ons)

Energy (keV)

Z+f' f''

Ni, Z=28

Can change the contrast by changing energy - synchrotrons

0, spinf q f q fi ff

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Scattering from a CrystalAs a crystal is a periodic repetition of atoms in 3D we can formulate the scattering amplitude from a crystal by expanding the scattering

from a single atom in a Fourier series over the entire crystal

(E, ) exp q r V

f r i dV

(q) E,q exp q T rj jT j

A fi

Atomic Structure Factor

Real Lattice Vector: T=ha+kb+lc

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The Structure FactorDescribes the Intensity of the diffracted beams in reciprocal space

exp q r exp u v w 2jj j

i i h k l

hkl are the diffraction planes, uvw are fractional co-ordinates within

the unit cell

If the basis is the same, and has a scattering factor, (f=1), the structure

factors for the hkl reflections can be foundhkl

Weight phase

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The Form FactorDescribes the distribution of the diffracted

beams in reciprocal space

The summation is over the entire crystal which is a parallelepiped of sides:

1

1

32

2 3

1T 1

2 31 1

q exp q T exp q a

exp q b exp q c

N

n

NN

n n

L i n i

n i n i

1 2 3N a N b N c

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The Form FactorMeasures the translational symmetry of the lattice

The Form Factor has low intensity unless q is a

reciprocal lattice vector associated with a reciprocal

lattice point

1,2,3 1,2,3 1,2,3

sin s sin sq exp s

sin s

i

i

Ni i i i

i ijini i i

N NL i n

s

0

0.5x105

1.0x105

1.5x105

2.0x105

2.5x105

-0.02 -0.01 0 0.01 0.02

Deviation parameter, s1 (radians)

[L(s

1)]

2

N=2,500; FWHM-1.3”

N=500

q d s Deviation from reciprocal lattice point located at d*

Redefine q:

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The Form Factor

0

20

40

60

80

100

-0.6 -0.3 0 0.3 0.6

Deviation parameter, s1 (radians)

[L(s

1)]

2

0

0.5x105

1.0x105

1.5x105

2.0x105

2.5x105

-0.02 -0.01 0 0.01 0.02

Deviation parameter, s1 (radians)

[L(s

1)]

2

The square of the Form Factor in one dimension

N=10 N=500

1,2,3

sin sq i i

ji

NL

s

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Scattering in Reciprocal Space

T

q q exp q r exp q Tj jj

A f i i Peak positions and intensity tell us about the structure:

POSITION OF PEAK

PERIODICITY WITHIN SAMPLE

WIDTH OF PEAK

EXTENT OF PERIODICITY

INTENSITY OF PEAK

POSITION OF ATOMS IN

BASIS

Qualitative understanding

•Atomic shape •Sample Extension

C. M. Schleütz, PhD Thesis, University of Zürich, 2009

X-ray atomic form factor

Finite size of atom leads to sinq/l fall off in intensity with angle

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Practical Realisation

A 4-circle diffo such as in this example gives access

to either vertical or horizontal scattering

geometries but not both.

Limited access due to the c circle. Alternative

designs possible (kappa)

Typically use a 4-circle machine with sample manipulator to align the sample and move in reciprocal space.

Ultimate precision depends on calibration of axes against known standards.

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Scattering – Q space

q/2q2q

q

q/2q2q

Scanning the different axes allows reciprocal (q) space to be probed in different directions.

A coupled scan of q and 2q (1:2) moves the scattering vector normal.

Individual q or 2q scans move in arcs. On a symmetric reflection, a rocking curve (q) measures the in-plane component.

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Laboratory vs. SynchrotronSynchrotron:• High flux with polarisation and

energy control• Complex sample environments• Flexible scattering geometries• Optimised control software• Competitive access and time delays

Laboratory• Easy access• Limited by flux, energy, available

geometries, software, resolution and proprietary constraints

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Sphere of ConfusionDiffractometers / goniometers are mechanical systems engineered to rotate about a fixed point in space. All axes must be concentric otherwise the sample will precess about the focus.

This can cause

• Different parts of the sample to be measured

• The sample to move in and out of the beam

• Limits sample environments

• More general systematic errors

Modern laboratory and synchrotron systems have a sphere of confusion of <30 mm, but this can cause problems if focused beams and/or small samples are used.

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Alignment

X-ray BeamG

onio

met

erCritical that the

diffractometer/goniometer rotation axis is well aligned to

the incident x-ray beam.

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Limitations and TraceabilityAny diffractometer must be calibrated against a standard to ensure

traceability and identify systematic errors (type B). Measurements are limited by:• Energy dispersion – set by the monochromator (Si 111 most

common which has DE/E~10-4).• Angular resolution – set by slits, collimators and angular dispersion.• Mechanical and thermal stability• Electronics (noise)• Number of peaks in a refinement• Calibration (consider relative measurements)

Routine measurements can give a precision of between 10-3 and 10-4 Å in bulk materials.

Accuracy much harder to quantify.

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Powder DiffractionIt is impossible to grow some materials in a single crystal form or

we wish to study materials in a dynamic process.

Powder Techniques

Allows a wider range of materials to be studied under different sample conditions

1. Inductance Furnace 290 – 1500K

2. Closed Cycle Cryostat 10 – 290K

3. High Pressure Up-to 5 million Atmospheres

• Phase changes as a function of Temp and Pressure

• Phase identification

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Powder Apparatus

Bragg-Brentano uses a focusing circle to maximise flux.

q/q system with the specimen fixed

Tube fixed with specimen and detector scanned in 1:2 ratio (q/2q)

Parallel Beam method collimates the beam and uses a fixed incident angle.

Detector scanned to measure pattern. Counts lower than B-B but penetration

and hence probe depth constant.

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PowdersPowder diffractometers often only have limited sample manipulation

and sample preparation is key to obtaining reliable data.

Height errors are the main cause of systematic errors in XRD. The surface

is displaced from rotation axis and this subtends an incorrect angle and

an offset in 2q is introduced.

2 cos2 Height

radius

Will result in incorrect values of the lattice

parameter

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Peak WidthsInstrumental resolution• Angular acceptance of detector• Slit widths (hor. & vert.)• Energy dispersion• Collimation

These are often summarised as the UVW parameters:

Additional terms such as the Lorentz factor relate to how the reciprocal lattice point is cut by the scan type (2q or q/2q). Peak width/shape also depends on detector slits.

2tan tanU V W

q/2q2q

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Peak intensities can be affected by a large range of parameters:

Preferential orientation (texture), Beam footprint, surface roughness, sample volume, temperature etc.

For accurate determination of strain one ideally need a large number of well defined peaks and a refinement, checking for offsets

Peak positions determined from the translation symmetry

of the lattice

Peak intensities determined from the symmetry of the basis (i.e. atomic positions)

Image courtesy J. Evans, University of Durham

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Search and MatchPowder Diffraction often used to identify phases

Cheap, rapid, non-destructive and only small quantity of sample

Inte

nsi

ty

2 A ngle

JCPDS Powder Diffraction File lists materials (>50,000) in order of their d-

spacings and 6 strongest reflectionsOK for mixtures of up-to 4

components and 1% accuracy

Monochromatic x-rays

Diffractometer

High Dynamic range detector

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Peak BroadeningDiffraction peaks can also be broadened in qz by:

1. Grain Size 2. Micro-Strains OR Both

The crystal is made up of particulates which all act as perfect but small crystals

, ,

sin sq i i

ii a b c

NL

s

Number of planes sampled is finite

Recall form factor: Scherrer Equation

2

cosSizeBD

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Particle SizeThe crystal is made up of particulates which all act as perfect but

small crystals but with a finite number of planes sampled.NixMn3-xO4+ (400 Peak)

AFM images (1200 x 1200 nm)

R. Schmidt et al. Surface Science (2005) 595[1:3] 239-248

0

0.2

0.4

0.6

0.8

1.0

-1.0 -0.5 0 0.5 1.0

900C850C

800C

750C700C

650C

2

Inte

nsi

ty

D

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Peak Shape

Peaks are clearly NOT Gaussian! What can we learn from the peak shape?

Nano-catalyst material in a matrix

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‘Grain Size’As the scattering profile is the

Fourier transform of the scattering profile that makes up

the ‘Grain’ one can calculate the inverse Fourier Transform based on the fit to get the real space correlation function and

the correct value of k.

Fit to a Pearson VII function,

transform into reciprocal space and inverse FT

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Peak BroadeningDiffraction peaks can also be broadened in qz by:

1. Grain Size 2. Micro-Strains OR Both

The crystal has a distribution of inter-planar spacings dhkl ±Ddhkl.

Diffraction over a range, ,Dq of angles

Differentiate Bragg’s Law: 2 2 tanStrain B

Width in radians

Strain Bragg angle

dd

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Peak BroadeningDiffraction peaks can also be broadened in qz by:

1. Grain Size 2. Micro-Strains OR Both

Total Broadening in 2q is sum of Strain and Size:

2 2 tancosTotal B

BD

2 cos 2 sinhkl hkl hklB B D

Rearrange

Williamson-Hall plot

y mx c

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Other contributions to widthThe total broadening will be the sum of size and strain dispersion. As the two contributions have a different angular dependence they can be separated by plotting:

2 cos 2 sinhkl hkl hklB B D

Williamson-Hall analysis

Notes on W-H analysis

Likely to be noisy

Slope MUST be positive

Need to be careful if looking at non-cubic systems as the strain dispersion will depend on hkl.

Warning! If extracting widths from lab sources – remember there are 2 peaks at each condition (Ka1 and Ka2 incident energies)

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0.005

0.006

0.007

0.008

0.05 0.10 0.15 0.20 0.25

y=((1.541/d))+(2s)xGrain Size=299 ± 19.5a/a = 0.005 ± 0.001

sin(B)

Wid

th *

cos

(B) Grain size = 30±2nm

Strain Dispersion = 0.005±0.001

Powder Diffraction

0

100

200

300

400

30 40 50 60

Detector Angle (°)

Inte

nsity

(a

rb.

units

)

0

0.05

0.10

0.15

0.20

0.25

0 10 20 30

333422

400

222311

220

y=(1.5412/(4*a2))xa=8.348 ± 0.0036

(h2+k2+l2)

sin2

(B)

Lattice Parameter

Grain Size

Strain Dispersion

Calibration

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StrainPeak positions defined by the lattice parameters:

1

1 1, ,

q exp qN

ini a b c

L i n

Strain is an extension or compression of the lattice,

hkl hkld d

Results in a systematic shift of all the peaks