An Introduction to Diffraction andScattering

Brendan JJ. 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) Atoms2)) Molecules
3) Crystals
4) Particles
Particles, Solids,
Solids Surfaces
Surfaces, Liquids
Liquids, Glasses
Glasses,Gases

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

UVVisIRμ waves
μ-waveshigher energy  lower energy
For particles:d B
deBroglieli wavelengthlth = Pl
Planck’sk’ const./momentumt/t[λ = h/p]h/ ]For an electron accelerated through 100 Volts, λ = 1.2 Å,Why is wavelength important?
To probe
Tb a sample
l you needd radiation
di i withi h a wavelengthlh off similar,
i il orsmaller, magnitude to the size of the “object” under investigation.

samplesampleVisible light

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

X rays
X-rays• Hard X-ray wavelengths ~ 0.1 Å to 6.0 Å• Soft X
X-ray
ray wavelengths ~ 66.00 Å to 12
12.00Å• Interact weakly,weakly ie.
ie penetrating – therefore,
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
te act o oof X
X--raysays aandd Matter
atteScattering.
gcoherent, incoherent
elastic (Thomson), inelastic (Compton)
Absorption.
atoms: can then be emitted as fluorescence, photoelectrons, Augerelectrons
molecules: can emit fluorescence, phosphorescence, transfer heat,(stimulated emission)
Diffraction. The bending of waves due to obstructions and small apertures, aswith crystals.
g of a wave as it ppasses from one medium to another
Refraction. The bending
Reflection. Radiation bouncing back from one medium to the original medium,
where the wavelength << size of the object.

Photoelectricotoe ect c EffectectIonization occurs when a light has sufficient energy to eject
an electron.
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
analysis,“chemical shift”

X-ray Scattering
Coherent
Chscattering
i occurs whenh theh X
Xray particle collides with an atom anddeviates without a loss in energy.

Coherent

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

IncoherentComptonCo
pto Scattering
Scatte gCompton scattering occurs when the incident x-ray photon is deflectedfrom its original path by an interaction with an electron
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
alonglan alteredlt d path.
th Energy
Eandd momentumt
are conservedd iin thithisprocess. The energy shift depends on the angle of scattering and not on
the nature of the scattering medium.

 h scattered  initial   (1  cos  ) 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 xx-ray
ray photon has less energy
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 thef ll i equation:followingi22I0 e
1  cos (2 )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 isi polarisedl i dbby ththe scattering
tt i process, resulting
lti iin a didirectional
tilvariation in the scattered intensity.

Atomic scattering factors as
fcore and fvalence
CFe2+

AbsorptionpWhen 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
pThese 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:  Whereτ = 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
ignored. Further
Further, Compton scattering is not important
at wavelengths greater than 1 Å.

Fluoresence
If an X-ray photon of sufficiently
high energy strikes an innershell electron the electron canbe ejected it from the atom.
atom
Higher energy electrons cascadeto fill vacancy,
aca cy, ggiving
gooff
characteristic fluorescent Xrays.
Higher energy electrons cascadeto fill vacancy, giving off
characteristic fluorescent Xrays.
ySecondary
Secoda y Fluorescenceuo esce ce
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 belement,butt instead wee see 40% FeFe, 28% NiNi, and 32% Co
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.

Scatteringg and Diffraction Techniquesq3-Dimensional
2-Dimensional

X-ray crystallography
g methods
Phasing
Grazing incidence diffraction

LLow-anglel scatterttDynamicsEquilibrium: B factors, diffuse scatter
Non-equilibrium:qLaue 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
fl-phosphorescence (luminescence)Infra--red Spectroscopy
Infrappy
Displacement of atoms during vibration lead to distortion of electricalcharge distribution of the molecule.

rF

reF

atoms (mass) are connectedwith bonding electrons. re isth equilibrium
theilib i
didistancetandd F:
force to restore equilibrium.

1
i 2

ki

i

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 frequencyh=hEnergy is quantized and vibrations are in the range 1011 to 1013 Hz=> 30 - 3,000 cm-1

CircularC
cu a Dichroismc os•

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

•This difference is measured as a function of wavelength,g , and is always
yvery small (<<1/10000 of total). Therefore CD spectroscopy requires
a high flux source that provides stable linear polarization over andextended wavelength range (VUV-Near(VUV Near IR)X-rayay Scatte
Scatteringg by aan Atomto•

Scattering by an atom is essentially thesum of the scattering of the electron“ l d” around“cloud”d ththe nucleus.
l•Scattering from each electron follows
the Thompson equation
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
dimensions, there
will be path differences between thescattered waves. These differences will
always be less than one wavelength, sothe 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 ionion. At higher scattering angles
angles, thefactor will be less.

Intensity• The intensity of the diffracted beam can be given by:
2I  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:y2LP 1  cos 2

sin 2   cosThe Structure Factor Fhkl
• For a reflection from a plane hkl. The structure factor
Fhkl is:Fhkl uniti  cellll



i 1


sin 2  hkl f i  exp  Bi
 exp[2ihkl  xi ]
2


• xI = (x( iyizi) are theth fractional
f til coordinates
di t forf atomti inithe cell
• hkl are the Miller indices for the reflection• Bi is the Debye-Waller displacement (thermal) factor
for 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’
f’ andd f”f” are theth reall andd anomalousl
didispersion
iterms.
• The first term f0 depends on the diffraction angle 22.

Anomalous X
X-rayray Scattering
22f i  ( f 0  f ' )  (f " )2• Anomalous scatteringg or anomalous dispersion
poccurs
when the incident X-ray energy is sufficient to cause
photoelectric x-ray production in a target atom. The
process is called fluorescence.
fThis phenomenon is
responsible for “absorption edge” observed when certain
elements interacting with particular wavelength xx-rays.
rays In
this process a characteristic X-ray photon is produced in
the target;g ; subsequentqinteraction p
produces coherent X rays which are slightly out of phase with other coherently
scattered X-rays. The net result is a reduction of thescatteredtt d intensity
i t
it fromftheth element.
lt

1210864
20f' (e.u.)

-5

ff"
f'-10

m-15
150

2cm /gJust as the absorption
coefficient shows large
jumps at “edges”
edges thescattering power ofelements shows rapidpchanges near edges.
• Minimum absorption is
obtainedb i d near 13
13.31
31 eV
V(0.93 Å)• Maximum ∆f∆f’ is ca 8electrons by measuring at
13.3 and 13.5 eV.

f" (e.u.)

Bi Wavelength Dependence

100m

50
0101520

Energy (keV)25

DynamicsEquilibrium

NNon-equilibrium
ilib i• B factors• Lauecrystallography• Diffuse scatter• Rapid
ap d Mixing
gaanddSmall anglescatteringgThe temperature (B) factor• AtAtoms are nott locatedlt d att fixed
fi d points
i t - undergodvibrations about their mean positions.
• The amplitude of these vibrations increase as thetemperature increases.
• Due to such motion, the scatteringg 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
reductiond ti iin iintensity.
tit
The greatest
reduction is at highangles (low d-values).

Debye-Waller temperature factor B = 8<u>2

Effect
ect oof Temperaturee pe atu e Agg2O
Intennsity

Ag2O 0.5 Å X-rays

T = 100 K

T = 290 K
25303540

45505560

Angle (2) Degrees
4 7464.746

Lattice Parameteer (Å)

4.744

The latticeshrinks
hi k ddue tto
unusual motionof the Ag atoms!

4 742
4.742

4.740

4.738

4.736
050

100

150200Temperature (K)250300Diffuse Scatter• Diffuse Scatter is the scatter that is notin the Bragg reflections.
reflections Every crystal
has it.
• Diffuse scatter can arise from:– Thermal displacement– Occupational disorder– Displacive Disorder– Short Range Order– Stacking Faults

Diffuse
use Scatte
Scatter