Crystallography and Diffraction Techniques

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Crystallography and Diffraction Techniques
Myoglobin
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Types of diffraction

• X-ray diffraction
• Electron diffraction
• Neutron diffraction

Myoglobin diffraction pattern 1962 Nobel Prize by
Max Perutz and Sir John Cowdery Kendrew
Enhanced visibility of hydrogen atoms by neutron
crystallography on fully deuterated myoglobin
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X-ray Diffraction
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Water
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Light
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Electron
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Constructive
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Destructive
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Diffraction from atoms
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Continue
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1 A

• About 1 Å

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Wave of mater
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Wave of electrons
The electrons are accelerated in an electric
potential U to the desired velocity
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Crystal diffraction
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Gas, liquid, powder diffraction
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Surface diffraction
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Diffraction by diffractometer
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Example of spots by diffractometer
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X-ray Crystallography
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Electron density
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Deformation Electron Density
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Macromolecule X-ray Crystallography
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Generation of X-rays
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What is K? and K? (for Cu) ?K? 2p ?1sK?
3p ?1s
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X-ray tube
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An optical grating and diffraction of light
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Lattice planes
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Lattice planes gt reflection
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Lattice planes review
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Braggs Law
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Braggs Law
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Braggs Law
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2dsin(theta)n lumda
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Braggs Law
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Atomic scattering factor
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Atomic scattering factor
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intensity
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Phase and intensity
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Electron density
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Diffraction of one hole
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Diffraction of two holes
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Diffraction of 5 holes
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2D four holes
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From real lattice to reciprocal lattice
Real holes
Reflection pattern
Crystal lattice is a real lattice, while its
reflection pattern is its corresponding
reciprocal lattice.
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TEM image of Si? or Diamond?
Si
Diamond

• Real lattice viewed from (110) direction.

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Electron Diffraction
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Conversion of Real Lattice to Reciprocal Lattice
P P P
P P P
P P P
P P P
P P P
P P P
P P P
P P P
P P P
P P P
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Ewald Sphere and Diffraction Pattern
Paul Peter Ewald (18881985)

• The Ewald sphere is a geometric construct used in
  X-ray crystallography which neatly demonstrates
  the relationship between
• the wavelength of the incident and diffracted
  x-ray beams,
• the diffraction angle for a given reflection,
• the reciprocal lattice of the crystal

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Ewald Sphere
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A vector of reciprocal lattice represents a set
of parallel planes in a crystal lattice
(1/dhkl)/(2/l) sin q
2d sin q nl
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Reciprocal Lattice and Ewald Sphere
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Detector, Reciprocal Lattice and Ewald Sphere
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3D View of Ewald Sphere and Reciprocal Sphere
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Techniques of X-ray diffraction
Single Crystal and Powder X-ray Diffractions
many many many very small single crystals
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Diffractometers for Single Crystal and Powder
X-ray Diffractions
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Single Crystal and Powder X-ray Diffraction
Patterns
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The powder XRD method
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Formation of a cone of diffracted radiation
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XRPD on film
electron diffraction of powder sample
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Finger Print Identification for Known Compounds
by comparing experimental XRPD to those in PDF
database
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Some peaks may not be observed due to preferred
orientation
For example, layered structure such as graphite.
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X-ray powder diffraction patternsof crystalline
and amorphous sample
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Scherrer Formula

• t thickness of crystal in Å
• B width in radians, at an intensity equal to
  half the maximum intensity

However, this type of peak broadening is
negligible when the crystallite size is larger
than 200 nm.
B is often calculated relative to a reference
solid (with crystallite size gt500 nm) added to
the sample B2Bs2-Br2.
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Some equations to calculate cell parameters
(d-spacings)
2d sinq l
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X-ray powder diffraction patterns for potassium
halides
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Structure Factor, Intensity and Electron Density
Fcalc
Fobs
R1 S Fo - Fc/ S Fo
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Electron density maps by X-ray diffraction
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Scattering of X-rays by a crystal-systematic
absences
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Systematic Absences
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Systematic absence for C-center (x,y,z) ?
(x1/2, y1/2, z)

• Fhkl (1/V) S fjexp2pi(hxjkyjlzj)
• (1/V)Sfjcos2p(hxjkyjlzj)isin2p(hxjkyjlzj)
• (1/V)Sfjcos2p(hxjkyjlzj)cos2ph(xj1/2)
• k(yj1/2)lzj)isin2p(hxjkyjlzj)
• sin2ph(xj1/2)k(yj1/2)lzj)

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let 2p(hxjkyjlzj)aj cos(AB)cosAcosB-sinAsinB
sin(AB)sinAcosBcosAsinB
(1/V)Sfjcos2p(hxjkyjlzj)cos2ph(xj1/2)k(yj1
/2)lzj) isin2p(hxjkyjlzj)sin2ph(xj1/2)k
(yj1/2)lzj) (1/V)Sfjcos ajcos
(ajp(hk))isin ajsin (ajp(hk))
(1/V)Sfjcos ajcos ajcos p(hk)isin ajsin
ajcos p(hk) cos p(hk) 1/V Sfjcos aj
isin aj
So when cos p(hk) -1 that is when hk
2n1, Fhkl 0 Condition for systematic
absences caused by C-center For all (hkl), when
hk 2n1, Ihkl 0
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Systematic absences for 21//b where (x,y,z)
?(-x,y1/2,-z)
Fhkl (1/V)Sfjcos2p(hxjkyjlzj)isin2p(hxjkyjl
zj) (1/V)Sfjcos2p(hxjkyjlzj)cos2p(-hxjk(
yj1/2)-lzj) isin2p(hxjkyjlzj)
sin2p(-hxjk(yj1/2)-lzj)
For reflections at (0 k 0)
Fhkl (1/V)Sfjcos(2pkyj) cos(2pkyj)cos(kp)
isin(2pkyj)
sin(2pkyj)cos(kp) (cos(kp)1)/v
Sfjcos(2pkyj) isin(2pkyj)
So the conditions for 21//b screw axis For all
reflections at (0 k 0), when k 2n1, Ihkl0
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Conditions of Systematic Absences
I-center for all (hkl), hkl 2n1, Ihkl
0 F-center for all (hkl), hk 2n1, hl
2n1 kl
2n1, Ihkl 0 (or h, k, l
not all even or all odd) c-glide (b-axis), for
all (h0l), l 2n1, Ihkl 0 n-glide (b-axis),
for all (h0l), hl 2n1, Ihkl 0 d-glide
(b-axis), for all (h0l), hl 4n1, 2 or 3, Ihkl
0 31//b screw axis, for all (0k0), k 3n1,
3n2, Ihkl 0 ????
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Setup of Conventional Single Crystal X-ray
Diffractometer
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Electron diffractione- ? ? ? 0.04 Å

• Can see crystal structure of very small area
• Associated with TEM
• f much larger than that of X-ray can see
  superlattice

NiMo alloy (18 Mo) with fcc structure. Weak
spots result from superlattice of Mo arrangement.
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Secondary diffraction of electron diffraction

• Extra reflections may appear in the diffraction
  pattern
• The intensities of diffracted beam are unreliable

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Neutron diffraction
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Antiferromagnetic superstructure in MnO, FeO and
NiO
MnO
Fe3O4
The most famous anti-ferromagnetic, manganese
oxide (MnO) helped earn the Nobel prize for C.
Shull, who showed how such magnetic structures
could be obtained by neutron diffraction (but not
with the more common X-ray diffraction).
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Schematic neutron and X-ray diffraction patterns
for MnO