The Scattered X-ray
Joseph H. Reibenspies  Ph.D.
Nattamai Bhuvanesh Ph.D.
X-ray Diffraction Laboratory
Texas A & M University
All rights reserved.  Please do not copy, modify or distribute these slideswithout the consent of the authors.
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X-ray Diffraction Laboratory  1.0.1
X-ray Diffraction Laboratory  1.0.1
What wavelength?
H
10-10m
Angstrom(Å)
Visible Light   4x10-7-9x10-7m
  4000-9000 Å
Bosons (photon)
X-rays    0.1 to  14 Å
Fermions
Neutrons
Electrons
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X-ray Diffraction Laboratory  1.0.1
X-rays and Matter
incoherent scattering
C (Compton-Scattering)
coherent scattering
 (Rayleigh -scattering)
Absorption (XANSE etc)
Beer´s law  I = Io*e-µd
Fluorescence (XRF)
o
Photoelectrons (XPS)
wavelength 
intensity Io
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General Scattering Theory
X-ray photon can be described in terms of anelectric and magnetic component.
Normal to each other and to the direction of thephoton
Collision of the photon with a charged particlecauses the component of the particle to oscillatewith the same frequency.
The oscillating particle returns to the resting stateby emitting a photon that travels outward.
The probability that a given incumbent photonwill result in a scattered photon is <10-20
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X-ray Diffraction Laboratory  1.0.1
 Rayleigh* Scattering(Thomson Scattering)
𝐼 2𝜃 ∝ 𝐼 𝑜  ×  𝐶 𝑓   2 × 1  𝑟 2  ×  1+ 𝑐𝑜𝑠 2 𝜑 2
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
𝑟
 𝐶 𝑓 =  𝑒 2  𝑚 𝑐 2        Cross Section Term
detector
𝐼 ~  1  𝑚𝑎𝑠𝑠 2      As the mass of the charged particle increases the scattered intensity decreases.  
 p+ ~ 2000 x  mass of e 
 Photons are scattered from electrons.
 1+ 𝑐𝑜𝑠 2 𝜑 2    Polarization Term
Polarization = 1 at = 0 and =1/2 at  = /2
 Intensity decreases as  increases.
*Lord Rayleigh aka John William Strutt
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X-ray Diffraction Laboratory  1.0.1
Compton Scattering
X-ray Diffraction Laboratory  1.0.1
An X-ray photon possesses momentum as well as energy
When the recoil of the electron is considered then the scattered photon isinelastic (loses energy).
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
 𝜆 ′ −𝜆=∆𝜆= ℎ  𝑚 𝑒 𝑐  1− cos 𝜑   =0.024 1− cos 𝜑
   >>     (0.048Å) 
Coherent scattering decreases with increasing 
The less dense the “electron cloud” the faster the decrease in intensity
Compton scattering increases with increasing 
The intensity is at the max at when  = 180o.  
 𝐼 𝑐𝑜𝑚𝑝 = 𝐼 𝑡 − 𝐼 𝑐𝑜ℎ = 𝐼 𝑡  1− 𝑓 2  
Estimate Compton Scattering effect if the elemental composition is known.
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recoil
Atomic Scattering
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X-ray Diffraction Laboratory  1.0.1
The X-ray scattering is a maximum  = 0
incident with the X-ray beam
I = total number of electrons
As  increases the destructive interferenceincreases between the X-rays scattered bydifferent regions of the electron cloud.which gives rise to a decrease in thescattered X-ray intensity.
The more diffuse the electron cloud, themore rapid the reduction in the scatteringfunction with scattering angle.
 𝑓 0   𝑠𝑖𝑛𝜃 𝜆  =  𝑖=1 4  𝑎 𝑖  𝑒  −𝑏 𝑖    𝑠𝑖𝑛𝜃 𝜆   2  + 𝑐 𝑖
a,b,c are the Cromer−Mann coefficients∗ for a given element
Atomic Scattering Factors
For routine diffraction experiments, atoms areapproximated by discrete spherical scatteringfunctions.
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X-ray Diffraction Laboratory  1.0.1
*Fox, A. G., O'Keefe, M. A. & Tabbernor, M. A. (1989). Acta Cryst. A45, 786–793
sin/
Scattering vector -   s
X-ray Diffraction Laboratory  1.0.1
So
S
1/
𝑠=  𝑆 𝑜 −𝑆 𝜆
𝑠=  (𝑆 𝑜 −𝑆) 𝜆  

𝑠= 1 𝜆  sin 𝜃+ 1 𝜆  sin 𝜃    

𝑠=2  sin 𝜃  𝜆
s
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Scattering From Two Identical Atoms
𝐴 𝑠 = 1 𝑁  𝑓 𝑛  𝑠  𝑒 2𝜋𝑖𝑠 𝑟 𝑛        where    s=  𝑆− 𝑆 𝑜   𝜆 =  2sin 𝜃  𝜆       
𝐼 𝑠 =  𝐴 𝑠   2 =𝐴 𝑠  𝐴 𝑠  ∗ 
𝐼 𝑠 = 𝑖=1 𝑁  𝑗=1 𝑁  𝑓 𝑖  𝑠  𝑓 𝑗   𝑠 𝑒 −2𝜋𝑖𝑠  𝑟 𝑖 − 𝑟 𝑗     
𝑓𝑜𝑟 𝑖=𝑗 𝑡ℎ𝑒𝑛  𝑓 𝑖  𝑠  𝑓 𝑗   𝑠 𝑒 −2𝜋𝑖𝑠  𝑟 𝑖 − 𝑟 𝑗   =  𝑖=1 𝑁   𝑓 𝑖 (𝑠) 2  
𝑓𝑜𝑟 𝑖≠𝑗  𝑡ℎ𝑒𝑛  𝑓 𝑖  𝑠  𝑓 𝑗   𝑠 𝑒 −2𝜋𝑖( 𝑟 𝑖 − 𝑟 𝑗 ) = 𝑓 𝑖  𝑠  𝑓 𝑗 (𝑠)  cos  2𝜋𝑠  𝑟 𝑖 − 𝑟 𝑗    + cos  2𝜋𝑠  𝑟 𝑗 − 𝑟 𝑖     
𝐼 𝑠 =  𝑖=1 𝑁   𝑓 𝑖 (𝑠) 2  +    𝑖≠𝑗 𝑁  𝑓 𝑖   𝑠 𝑓 𝑗  𝑠  cos 2𝜋 𝑠𝑟 𝑖𝑗           
 𝐼 𝑠  = 𝑖=1 𝑁   𝑓 𝑖 (𝑠) 2  +     𝑖≠𝑗 𝑁  𝑓 𝑖   𝑠 𝑓 𝑗  𝑠   cos 2𝜋 𝑠𝑟 𝑖𝑗              average Intensity
Given     cos 2𝜋𝑠𝑟   =  sin 2𝜋𝑠𝑟  2𝜋𝑠𝑟    and  𝑖=1 𝑛   𝑓 𝑖 (𝑠) 2   = constant
𝐼 𝑠 = 𝑖=1 𝑁  𝑗=1 𝑁  𝑓 𝑖   𝑠 𝑓 𝑗  𝑠   sin 2𝜋 𝑠𝑟 𝑖𝑗   2𝜋𝑠 𝑟 𝑖𝑗             Debye’s Equation (for two identical atoms)
For a diatomic molecule of the same element the maxima will be at  𝑟 𝑖𝑗  𝑠 𝑚 =1.23  given s = 2 sin 𝜃  𝜆  

                                                 𝟐 𝐬𝐢𝐧  𝜽 𝒎   𝝀 = 𝟏.𝟐𝟑  𝒓 𝒊𝒋         or    𝑲𝝀=𝟐 𝒓 𝒎  𝒔𝒊𝒏  𝜽 𝒎       K~1.1 to 1.23
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X-ray Diffraction Laboratory  1.0.1
Scattering from clusters ofidentical atoms
𝐾𝜆=2 𝑟 𝑚  sin  𝜃 𝑚  
K = 1.1 to 1.23
For a VERY VERY simple isotropic (spherical) systemwhere The average contact distances are near 1.5,2.5 and 5 Å we can predict the scattered patterngiven below. (=1.54Å)
2
5.0Å
2.5Å
1.5Å
Limitations : On can DEDUCE the order of magnitude of contact distances.  Exactformulations are difficult due to the nature of the systems involved.
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X-ray Diffraction Laboratory  1.0.1
Non-ordered/ordered Systems
Non-ordered systems (non-crystalline)
 order of magnitude of distances
Large ensemble of distances
Inter and intra-molecular interferences make theinterpretation of the scattered pattern difficult.
Ordered systems (crystalline)
“fixed, discrete, repeated” distances
X-ray Diffraction Laboratory  1.0.1
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Scattering from an linear ordered System
𝐹 𝑠 = 𝑛    𝑓 𝑛 (𝑠)𝑒 2𝜋𝑖𝑠𝑟  
For same element (atom) spaced at equal distance(s) of 𝑎  N>2  times then
𝐹(𝑠)=𝑓 𝑒 2𝜋𝑖𝑠𝑟 +𝑓 𝑒 2𝜋𝑖𝑠(𝑟+𝑎) +…𝑓 𝑒 2𝜋𝑖𝑠(𝑟+𝑁𝑎) 

Squaring this equation obtain: 
𝐼(𝑠)~ 𝑓 2     sin 𝑁2𝜋𝑎𝑠   sin 2𝜋𝑎𝑠    2 

Note :     sin 𝑁2𝜋𝑎𝑠  𝑁2𝜋𝑎𝑠   2 ~ 1  𝑁 2      sin 𝑁2𝜋𝑎𝑠   sin 2𝜋𝑎𝑠    2
X-ray Diffraction Laboratory 1.0.1
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𝑎
Scattering for Ordered Arrays
X-ray Diffraction Laboratory 1.0.1
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Peak sharpens as N increases
𝐼=  𝐹  2 = 𝑓 2     sin 𝑁2𝜋𝑎𝑠   sin 2𝜋𝑎𝑠    2
The diffraction pattern is a convolution of1. scattering factor of the repeat unit
2. interference resulting from scattering byall the repeating units.
The area of the peak remains the same
X-ray Diffraction Laboratory  1.0.1
Scattering from Objects in Arrays
d
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FACTS:
The probability that a given photon will result in a scattered photon is <10-20
The scattering for an atom is simply the value for a free electron multiplied by the atomic number
The INTENSITY of the scattered X-ray will fall off as a function of theta.
Bragg’s Law
d
θ
l
l/d = sinθ
l=dsinθ
(since = l+l then)
 = dsinθ + dsinθ
or
=2dsinθ
3. Since the two waves can be inphase for
  to  n
Then
n=2dsinθ
1. Constructive Interference occurswhen wave 1 and wave 2 are in phase
Wave 1
Wave 2
l
d
θ
θ
l
2. When l+l  then thetwo waves are in phase
Wave 1
Wave 2
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X-ray Diffraction Laboratory 1.0.1
X-ray Diffraction Laboratory  1.0.1
Diffraction and Reciprocal Space
Problem  d and  are on the same side of the equation!
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X-ray Diffraction Laboratory  1.0.1
Unit Cell (real space) andDiffraction (reciprocal Space)
Monoclinic System
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X-ray Diffraction Laboratory  1.0.1
G-Tensor- Matrix
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Scattering for Ordered Systems
X-ray Diffraction Laboratory  1.0.1
𝐹 𝑠 = 1 𝑁  𝑓 𝑛  𝑠  𝑒 2𝜋𝑖𝑠𝑟        

For an ordered system  𝑟=𝑥𝑎+𝑦𝑏+𝑧𝑐

𝐹 𝑠 = 1 𝑁  𝑓 𝑛  𝑠  𝑒 2𝜋𝑖𝑠 𝑥𝑎+𝑦𝑏+𝑧𝑐      

Given 𝑠∙𝑎=ℎ, 𝑠∙𝑏=𝑘,𝑠∙𝑐=𝑙

𝐹 𝑠 = 1 𝑁  𝑓 𝑛  𝑠  𝑒 2𝜋𝑖 ℎ𝑥+𝑘𝑦+𝑙𝑧      

where 𝐼 𝑠 ≈  𝐹 𝑠   2 

For ordered systems 𝐼 𝑠  will have maxima at discrete positions  
ℎ𝑥+𝑘𝑦+𝑙𝑧
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SCD/Powder Patterns
X-ray Diffraction Laboratory  1.0.1
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22
21
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Single-Crystal
Poly-Crystal
Powder
2D- Images
X-ray Diffraction Laboratory  1.0.1
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amorphous
polycrystalline
Crystalline-powder
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crystalline
X-ray Scattered from Solids
X-ray Diffraction Laboratory  1.0.1
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SimulatedRadialDistribution
End of Lecture 2.
Lecture 3- The Experiment
Lecture 4- The Pattern
Lecture 5- The Signal Processing
Lecture 6- The Analysis
Lecture 7- The Reitveld Method
Lecture 8- The Results
X-ray Diffraction Laboratory  1.0.1
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