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

Material Characterisation

∂

CharacterisationOver the course so far we have seen how thermodynamics plays an important role in defining the basic minimum energy structure of a solid.

Small changes in the structure (such as the perovskites) can produce changes in the physical properties of materials

Kinetics and diffusion also play a role and give rise to different meta-stable structures of the same materials – allotropes / polymorphs

Alloys and mixtures undergo multiple phase changes as a function of temperature and composition

BUT how do we characterise samples?

∂

Probes

Resolution better than the inter-atomic spacings

• Electromagnetic Radiation

• Neutrons

• Electrons

ab

∂

Probes

Treat all probes as if they were waves:

;

hp k p mv

Wave-number, k: 2

k k

Momentum, p:

Photons ‘Massive’ objects

∂

Xavier the X-ray

hcE

Ex(keV)=1.2398/(nm)

Speed of Light

Planck’s constant Wavelength

Elastic scattering as Ex>>kBT

∂

X-ray Sources

http://pd.chem.ucl.ac.uk/pdnn/inst1/xrays.htm

∂

Synchrotrons

Electrons at GeV in a storage ring. Magnetic field used to accelerate in horizontal direction giving x-rays (dipole radiation)

∂

Norbert the Neutron

hmv

222

12 2n

n

hE mvm

En(meV)=0.8178/2(nm)

De Broglie equation:

mass velocity

Non-relativistic Kinetic Energy:

Strong inelastic scattering as En~kBT

∂

Fission

Thermal Neutron 235U

2.5 neutrons + heavy elements + 200 MeV heat

ILL: Flux Density of 1.5x1013 neutrons/s/mm2

at thermal power of 62MW

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Spallation

800 MeV Protons excite a heavy nucleus

Protons, muons, pions .... & 25 neutrons

Pulsed Source - 50Hz

ISIS - Rutherford Appleton Lab. (Oxford). ESS being built in Lund.

∂

Eric the Electron

Eric’s rest mass: 9.11 × 10−31 kgEric’s electric charge: −1.602 × 10−19 C

No substructure – point particle

hmvDe Broglie equation:

mass velocity

Ee depends on accelerating voltage :– Range of Energies from 0 to MeV

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Probes• Electrons - Eric• quite surface sensitive

• Electromagnetic Radiation - Xavier• Optical – spectroscopy• X-rays sensitive to electrons:

• VUV and soft (spectroscopic and surfaces)• Hard (bulk like)

• Neutrons – Norbert• Highly penetrating and sensitive to induction• Inelastic

∂

Interactions

1. Absorption 2. Refraction/Reflection3. Scattering Diffraction

EricXavier

Norbert

∂ a*

b*

100 300200 400

010

120

110

0-10

020

030

0-20

130 230

210

330

310

220 320

1-10 4-103-102-10

Diffraction – a simple context

3D periodic arrangement of scatterers with translational symmetry gives rise to a real

space lattice.

The translational symmetry gives rise to a reciprocal lattice of

points whose positions depend on the real space periodicities

http://pd.chem.ucl.ac.uk/pdnn/diff1/recip.htm

Real Space Reciprocal Space

ab

∂

Interference View

Constructive Interference between waves scattering from periodic scattering centres within the material gives rise to strong scattering at specific angles.

2 sind

a*

b*

100 300200 400

010

120

110

0-10

020

030

0-20

130 230

210

330

310

220 320

1-10 4-103-102-10

2rlp

d

∂

Reciprocal Lattice of Si

f rotation about [001]

(010) plane (110) plane

∂

Basic Scattering Theory

The number of scattered particles per

second is defined using the standard

expression

I I dds 0

Unit solid angle Differentialcross-section

Defined using Fermi’s Golden Rule

INTERACTION POTENTIAL IFina nild

dtial

∂Spherical Scattered Wavefield

ScatteringPotential

Incident Wavefield

Different for X-rays, Neutrons and Electrons

2

exp k r r r r

d dd

i V

∂

BORN approximation:• Assumes initial wave is also spherical• Scattering potential gives weak interactions

02

rexp kp rk r rex V id did

2

exp q rr r id d

dV

Scattered intensity is proportional to the Fourier Transform of the scattering potential

q k k0

2

exp k r r r rd i V dd

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 fi

Vf 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).

( ) 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 EnvironmentIn 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

(ele

ctro

ns)

Energy (keV)

Z+f' f''

Ni, Z=28

Can change the contrast by changing energy - synchrotrons

0, spinf q f q fi ff

∂

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

∂

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

∂

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

∂

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 si

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:

∂

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

∂

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, Univerity of Zürich, 2009

X-ray atomic form factor