An Introduction to Diffraction andScattering

Brendan J. Kennedy
School of Chemistry
The University of Sydney

Types of Forces
1) Strong forces
2) Weak forces
3) Electromagnetic forces
4) Gravity

Types of Matter
1) Atoms
2) Molecules
3) Crystals
4) Particles, Solids, Surfaces, Liquids, Glasses,
Gases

Electromagnetic Radiation at a
Synchrotron
Energy transmitted in the form of waves or particles
γ-rays

X-rays

UVVisIRμ-waveshigher energy  lower energy
For particles:
de Broglie wavelength = Planck’s const./momentum [λ = h/p]
For an electron accelerated through 100 Volts, λ = 1.2 Å,Why is wavelength important?
To probe a sample you need radiation with a wavelength of similar, orsmaller, magnitude to the size of the “object” under investigation.

samplesampleVisible light

X-raysTo investigate atomic/molecular structure, we use X-rays, electrons,
and neutrons, since they can have wavelengths about the sizes ofatoms.

X-rays• Hard X-ray wavelengths ~ 0.1 Å to 6.0 Å• Soft X-ray wavelengths ~ 6.0 Å to 12.0 Å• Interact weakly, ie. penetrating – therefore, can see
inside of a structure, but most of the beam passes
through, unperturbed.
• 10 x more photons are absorbed than scattered.
• X-rays scatter off electrons (~ 1% off nuclei).

Interaction of X-rays and Matter

Scattering.
coherent, incoherent
elastic (Thomson), inelastic (Compton)
Absorption.
atoms: can then be emitted as fluorescence, photoelectrons, Auger
electrons
molecules: can emit fluorescence, phosphorescence, transfer heat,(stimulated emission)
Diffraction. The bending of waves due to obstructions and small apertures, aswith crystals.
Refraction. The bending of a wave as it passes from one medium to another
Reflection. Radiation bouncing back from one medium to the original
medium, where the wavelength << size of the object.

Photoelectric EffectIonization occurs when a light has sufficient energy to eject
an electron. The kinetic energy of the emitted electron isdiagnostic of the orbital from which it was ejected. Allows
Photoelectron spectroscopy.
e-e-e-hυEhυ = electron kinetic energy + electron binding energy
Valence spectroscopy: information on bondingCore spectroscopy: qualitative and quantitative analysis,“chemical shift”

X-ray Scattering
Coherent scattering occurs when the Xray particle collides with an atom anddeviates without a loss in energy.

Coherent

Incoherent scattering is where theincident X-ray loses some of its energy tothe scattering electron. As total momentum
is preserved, the wavelength of thescattered photon is increased.

IncoherentCompton Scattering
Compton scattering occurs when the incident x-ray photon is deflectedfrom its original path by an interaction with an electron. The electron isejected from its orbital position and the x-ray photon loses energy
because of the interaction but continues to travel through the material
along an altered path. Energy and momentum are conserved in thisprocess. The energy shift depends on the angle of scattering and not on
the nature of the scattering medium.

 h 
λscattered − λinitial =  (1 − cos θ ) me c 
where h is Plancks constant; me the mass of the electron; c the speed oflight; and θ the angle the photon is deflected by.
Since the scattered x-ray photon has less energy, it has a longer
wavelength and less penetrating than the incident photon.

Thompson Scattering•An electron will oscillate in phase with an x-ray beam according to thefollowing equation:

22


I0 e
1 + cos (2θ )I= 2 2
2r  me c 2

where I0 is the intensity of the incident beam; e the charge on theelectron; me the mass of the electron; c the speed of light; andr the distance from the scattering electron to the detector.
Clearly (by the second term) the scattered energy from a single electron
is quite low. Third term, involving the cosine function, is called thepolarisation factor because it indicates that the incoming non-polarised xray is polarised by the scattering process, resulting in a directionalvariation in the scattered intensity.

Atomic scattering factors as
fcore and fvalence
CFe2+

Absorption
When X-rays pass through a material, the intensity isattenuated (decreased) through absorption by the elementsin the material.
Attenuation is due to two effects:
Compton scattering (σ) of the beam, which diverts
photons in directions different from the primary beam:
Photoelectric absorption (τ), which produces
fluorescence or Auger electrons.

Absorption
These two effects may be combined into a single bulk mass
absorption coefficient μ). This coefficient is different for eachelement and wavelength and is defined as a the sum ofphotoelectric and Compton scattering:µ =τ +στ = photoelectric absorption coefficient andσ is the Compton Scattering coefficient
In many materials photoelectric absorption accounts forabout 95 percent of the absorption and Compton scattering
can be ignored. Further, Compton scattering is not important
at wavelengths greater than 1 Å.
WhereFluoresence
If an X-ray photon of sufficiently
high energy strikes an innershell electron the electron canbe ejected it from the atom.
Higher energy electrons cascadeto fill vacancy, giving off
characteristic fluorescent Xrays.
Higher energy electrons cascadeto fill vacancy, giving off
characteristic fluorescent Xrays.

Secondary Fluorescence
The absorption of an incident X-ray to produce another X-ray produces aneffect termed secondary fluorescence. Secondary fluorescence produces anexcess intensity. Some elements undergo self-absorption: K-line X-rays may beabsorbed by other atoms of same element to produce L-line X-rays. For a given
element of interest, any X-rays of energy higher than Ec produced from otherelements can cause fluorescence. The energy at which the mass absorption
coefficient increases abruptly is the absorption edge and corresponds to Ec
exactly.
For example, consider a material composed of 1/3 Fe, 1/3 Ni, and 1/3 Co. One
would expect about 33.3% of the total X-rays observed to be produced by eachelement, but instead we see 40% Fe, 28% Ni, and 32% Co. The Fe in thematerial is fluoresced by Ni-Kα producing Fe-Kα X-rays and absorbing Ni-K α.
Plot of the X-ray absorption curves for Fe (black), Co(red), and Ni (blue), showing the Kα absorption edges.
The locations of the Kα lines for these elements is alsoshown. Note than Ni-Kα x-rays have sufficient energy tofluoresce Fe-Kα x-rays.

Gd2-xTbxZr2O7a) Zr L3-edge XANES spectra of Gd2-xTbxZr2O7. b) Zr L3-edge XANES
spectra of the 6, 7 and 8 coordinate standards, labelled a, b and c, respectively.

17

O K-edge – Growth of Tb4+ in Gd2-xTbxZr2O7CBO K-edge
Gd2-xTbxZr2O7Nµ(E)Nµ(E)

A528 529 530 531

530

535

540

x=0x = 0.4x = 0.8x = 1.2x = 1.6x = 2.0545

550

Absorption Energy (eV)O K-edge XANES spectra of Gd2-xTbxZr2O7 (where x = 0, 0.4, 0.8, 1.2, 1.6, 2).
The inset shows the change in the intensity of Peak C across the Gd2xTbxZr2O7 series.
18

O K-edge – Growth of Tb4+ in Gd2-xTbxZr2O7CO K-edge
Gd2-xTbxZr2O7528 529 530 531

530535540

x=0x = 0.4
x = 0.8
x = 1.2x = 1.6
x = 2.0545

550

Absorption Energy (eV)

Nµ(E)

B

Nµ(E)

Nµ(E)

A

Tb1/3TaO3 (Tb3+)
BaTbO3 (Tb4+)

530

535

540

545

550

Absorption Energy (eV)O K-edge XANES spectra of Gd2-xTbxZr2O7 (where x = 0, 0.4, 0.8, 1.2, 1.6, 2).
The inset shows the change in the intensity of Peak C across the Gd23+
xTbxZr2O7 series. b) The O K-edge XANES spectra of Tb1/3TaO3 (Tb ) andBaTbO3 (Tb4+).
19

CO K-edge
Gd2-xTbxZr2O7528 529 530 531

530535540

x=0x = 0.4x = 0.8x = 1.2x = 1.6x = 2.0545

550

Absorption Energy (eV)

Nµ(E)

B

Nµ(E)

Nµ(E)

A

Tb1/3TaO3 (Tb3+)
BaTbO3 (Tb4+)

530

535

540

545

550

Absorption Energy (eV)Normalized Area of Peak CO K-edge – Growth of Tb4+ in Gd2-xTbxZr2O7

1.0
0.8
0.6
0.40.20.0
0.0

0.40.81.2

1.6

2.0

Composition of Tb (x)O K-edge XANES spectra of Gd2-xTbxZr2O7 (where x = 0, 0.4, 0.8, 1.2, 1.6, 2).
The inset shows the change in the intensity of Peak C across the Gd23+
xTbxZr2O7 series. b) The O K-edge XANES spectra of Tb1/3TaO3 (Tb ) andBaTbO3 (Tb4+). c) Plot of the area of feature C, normalized to the peak area ofTb2Zr2O7, with respect to the composition of Tb (x). All spectra were collected
20in fluorescence mode.

Scattering and Diffraction Techniques3-Dimensional
2-Dimensional

X-ray crystallography
Phasing methods
Grazing incidence diffraction

Low-angle scatter
Dynamics
Equilibrium: B factors, diffuse scatter
Non-equilibrium: Laue method

Other Synchrotron Techniques
Scattering:- Raman (inelastic, due to vibrations)- visible Rayleigh (elastic, information on particle sizeand dynamics)
Absorption and related techniques:
EXAFS (measure fluorescence in an excitation spectrum)
UV, vis, IR (electronic, vibrations)- absorption (CD, linear, time-resolved)-fluorescence-phosphorescence (luminescence)Infra-red Spectroscopy
Displacement of atoms during vibration lead to distortion of electricalcharge distribution of the molecule.

rF

reF

atoms (mass) are connectedwith bonding electrons. re is
the equilibrium distance and F:
force to restore equilibrium.

1
ωi =
2π

ki

µi

F(x) = -kx where X is displacement from equilibrium ki is the force
constant and μi is reduce mass of a particular motion.
Molecular vibration lead to oscillation of electric charge governed byvibration frequencies of the system.
Oscillating molecular dipole can interact directly with oscillating electric vector ofelectromagnetic radiation of the same frequencyhν=hωEnergy is quantized and vibrations are in the range 1011 to 1013 Hz=> 30 - 3,000 cm-1

Circular Dichroism•

CD measures the difference between the absorption of left and righthanded circularly-polarized light by a chiral chromophore.

•This difference is measured as a function of wavelength, and is always
very small (<<1/10000 of total). Therefore CD spectroscopy requires
a high flux source that provides stable linear polarization over andextended wavelength range (VUV-Near IR)X-ray Scattering by an Atom•

Scattering by an atom is essentially thesum of the scattering of the electron“cloud” around the nucleus.

•Scattering from each electron follows
the Thompson equation. Because of
the distance between electrons
scattering within the atom and the factthat the x-ray wavelength is of the sameorder as the atomic dimensions, there
will be path differences between thescattered waves. These differences will
always be less than one wavelength, so
the interference will always be partiallydestructive.

•This phenomenon is called the atomic scattering factor, described by the quantity
f0. This function is normalized in units of the amount of scattering occurring from asingle electron in the Thompson equation. At zero degrees, f0 will be equal to thenumber of electrons surrounding the atom or ion. At higher scattering angles, thefactor will be less.

Intensity• The intensity of the diffracted beam can be given by:2
I = SLP Fhkl
• S is a scale factor that is dependent on the intensityof the source and the detector efficiency.
• LP is a combination of the Lorentz and polarisation
factors which for a non-polarised X-ray beam is given
by:
2LP =

1 + cos 2θ

sin 2 θ ⋅ cosθThe Structure Factor Fhkl
• For a reflection from a plane hkl. The structure factor
Fhkl is:Fhkl =unit − cell

∑

i =1


sin 2 θ hkl f i ⋅ exp − Bi⋅ exp[2πihkl ⋅ xi ]
2λ


• xI = (xiyizi) are the fractional coordinates for atom i inthe cell
• hkl are the Miller indices for the reflection• Bi is the Debye-Waller displacement (thermal) factorfor atom (in Å2)• fi. Is the scattering factor.

X-ray Diffraction• In X-ray diffraction fi is given by:

f i2 = ( f 0 + ∆f ' ) 2 + (∆f " ) 2• ∆f’ and ∆f” are the real and anomalous dispersionterms.
• The first term f0 depends on the diffraction angle 2θ.

Anomalous X-ray Scattering
2

2f i = ( f 0 + ∆f ' ) + (∆f " )2• Anomalous scattering or anomalous dispersion occurs
when the incident X-ray energy is sufficient to cause
photoelectric x-ray production in a target atom. The
process is called fluorescence. This phenomenon is
responsible for “absorption edge” observed when certain
elements interacting with particular wavelength x-rays. In
this process a characteristic X-ray photon is produced inthe target; subsequent interaction produces coherent X rays which are slightly out of phase with other coherently
scattered X-rays. The net result is a reduction of thescattered intensity from the element.

1210864
20f' (e.u.)-5f"f'-10-15150

mcm2/gJust as the absorption
coefficient shows large
jumps at “edges” thescattering power ofelements shows rapidchanges near edges.
• Minimum absorption isobtained near 13.31 eV(0.93 Å)• Maximum Δf’ is ca 8electrons by measuring at
13.3 and 13.5 eV.

f" (e.u.)

Bi Wavelength Dependence

m100
500101520

Energy (keV)25

Dynamics
Equilibrium

Non-equilibrium• B factors• Lauecrystallography• Diffuse scatter• Rapid Mixing andSmall angle
scattering

The temperature (B) factor• Atoms are not located at fixed points - undergo
vibrations about their mean positions.
• The amplitude of these vibrations increase as thetemperature increases.
• Due to such motion, the scattering factor falls off
exponentially.The greatest reduction in intensity is athigh angles (low d-values).

Debye-Waller temperature factor B = 8π2<u>2
Where <u>2 is mean-square amplitude of atomic vibration. This isdirectional and can be anisotropic

The temperature (B) factor
Increasing motion of
an atom induces anangular dependent
reduction in intensity.
The greatest
reduction is at highangles (low d-values).

Debye-Waller temperature factor B = 8π<u>2Effect of Temperature Ag2OIntensity

Ag2O 0.5 Å X-rays

T = 100 K

T = 290 K
2530

3540

45505560

Angle (2θ) Degrees
4.746

Lattice Parameter (Å)

4.744

The latticeshrinks due tounusual motionof the Ag atoms!

4.742

4.740

4.738

4.736
050

100

150

200

Temperature (K)250300Diffuse Scatter• Diffuse Scatter is the scatter that is notin the Bragg reflections. Every crystal
has it.
• Diffuse scatter can arise from:– Thermal displacement– Occupational disorder– Displacive Disorder– Short Range Order– Stacking Faults

Diffuse Scatter