The Muppet’s Guide to: The Structure and Dynamics of Solids Kinematical Diffraction.
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Transcript of The Muppet’s Guide to: The Structure and Dynamics of Solids Kinematical Diffraction.
The Muppet’s Guide to:The Structure and Dynamics of Solids
Kinematical Diffraction
∂
Kinematical Diffraction Theory
Violates Energy
Conservation
Far-Field, Fraunhofer
regime
2
exp q rr r id
dd
V
Assumes weak scattering
Works surprisingly well!
∂
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
∂
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
∂
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
∂
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
∂
Magnetic component
2
02e
em t i S H
m
The electron oscillates creating a dipole moment along :x
Rotation of incident polarisation
∂
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?
∂
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
∂
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
∂
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
∂
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 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:
∂
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, University of Zürich, 2009
X-ray atomic form factor
Finite size of atom leads to sinq/l fall off in intensity with angle
∂
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.
∂
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.
∂
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
∂
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.
∂
Alignment
X-ray BeamG
onio
met
erCritical that the
diffractometer/goniometer rotation axis is well aligned to
the incident x-ray beam.
∂
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.
∂
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
∂
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.
∂
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
∂
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
∂
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
∂
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
∂
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
∂
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
∂
Peak Shape
Peaks are clearly NOT Gaussian! What can we learn from the peak shape?
Nano-catalyst material in a matrix
∂
‘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
∂
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
∂
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
∂
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)
∂
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
∂
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