Types of Forces
An Introduction to Diffraction andScattering

Brendan J. Kennedy
School of Chemistry
The University of Sydney

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-waves

higher 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,Cheron 2007 Kennedy

1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-rays

X-rays• Hard X-ray wavelengths ~ 0.1 Å to 60 Å
• Soft X-ray wavelengths ~ 60 Å to 120 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.

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

• X-rays scatter off of valence electrons (~ 1% off nuclei).

Interaction of XX-rays and Matter

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-Scattering.
coherent, 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.
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.

Cheron 2007 Kennedy

e-e-hυEhυ = electron kinetic energy + electron binding energy
Valence spectroscopy: information on bondingCore spectroscopy: qualitative and quantitative analysis,“chemical shift”

2

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

Coherent

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 
 (1 − cos θ ) me c 

λscattered − λinitial = 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.

where h is Plancks constant; me the mass of the electron; c the speed oflight; and θ the angle the photon is deflected by.

Incoherent

Thompson Scattering•

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

Since the scattered x-ray photon has less energy, it has a longer
wavelength and less penetrating than the incident photon.

Absorption
When X-rays pass through a material, the intensity isattenuated (decreased) through absorption by the elementsin the material.

2I  e 2  1 + cos 2 ( 2θ )I = 02 
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.

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.

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.

Cheron 2007 Kennedy3Absorption

Fluoresence

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:If an X-ray photon of sufficiently
high energy strikes an innershell electron the electron canbe ejected it from the atom.

µ =τ +σHigher energy electrons cascadeto fill vacancy, giving off
characteristic fluorescent Xrays.

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. Further, Compton scattering is not important
at wavelengths less than 1 Å.

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.

Higher energy electrons cascadeto fill vacancy, giving off
characteristic fluorescent Xrays.

Scattering and Diffraction Techniques
3-Dimensional
2-Dimensiional

X-ray crystallography
Phasing methods
Grazing incidence diffraction

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

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.

Cheron 2007 Kennedy

4Other Synchrotron Techniques
Scattering:- Raman (inelastic, due to vibrations)- visible Rayleigh (elastic, information on particle sizeand dynamics)

InfraInfra-red Spectroscopy
Displacement of atoms during vibration lead to distortion of electricalcharge distribution of the molecule.

r
F

reF

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

ωi =

1
2π

Absorption and related techniques:F(x) = -kx where X is displacement from equilibrium ki is the force
constant and µi is reduce mass of a particular motion.

EXAFS (measure fluorescence in an excitation spectrum)

Molecular vibration lead to oscillation of electric charge governed byvibration frequencies of the system.

UV, vis, IR (electronic, vibrations)- absorption (CD, linear, time-resolved)-fluorescence-phosphorescence (luminescence)•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)

Cheron 2007 KennedyµiOscillating 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•ki

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.

5Intensity

The Structure Factor Fhkl

• The intensity of the diffracted beam can be given by:
2I = SLP Fhkl

• For a reflection from a plane hkl. The structure factor
Fhkl is:• 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 givenby:
2LP =1 + cos 2θ

sin 2 θ ⋅ cos θ

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.

Cheron 2007 KennedyFhkl =unit − cell∑i =1

sin 2 θ hkl fi ⋅ exp − Bi ⋅ exp[2πihkl ⋅ xi ]λ2 

• 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) factor
for atom (in Å2)• fi. Is the scattering factor.

Anomalous X-ray Scatteringfi2 = ( f 0 + ∆f ' ) 2 + (∆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 in
the 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.

6Dynamics

12108
6
4
20f' (e.u.)-52cm /gNon-equilibrium• B factors• Lauecrystallographyf"f'-10-15• Diffuse scatter

150m

Equilibriumm

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

f" (e.u.)

Bi Wavelength Dependence

10050
0101520

Energy (keV)

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).

25• Rapid Mixing andSmall angle
scattering

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

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

Cheron 2007 Kennedy

7Effect of Temperature Ag2O

• Diffuse Scatter is the scatter that is notin the Bragg reflections. Every crystal
has it.
• Diffuse scatter can arise from:

Intensity

Ag2O 0.5 Å X-rays

T = 100 K

T = 290 K
25303540

45505560

Angle (2 θ) Degrees
4.746

4.744

Lattice Parameter (Å)

Diffuse Scatter

The latticeshrinks due tounusual motionof the Ag atoms!

4.742

4.740

– Thermal displacement– Occupational disorder– Displacive Disorder– Short Range Order– Stacking Faults

4.738

4.736
050

100

150

200

250300Temperature (K)Cheron 2007 Kennedy8