the muppet’s guide to: the structure and dynamics of solids material characterisation
DESCRIPTION
∂ Probes Resolution better than the inter-atomic spacings Electromagnetic Radiation Neutrons Electrons a bTRANSCRIPT
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
∂
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
∂
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
∂
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
∂
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