Crystallography and Diffraction Techniques Myoglobin
Types of diffraction - X-ray diffraction - Electron diffraction - Neutron diffraction Enhanced visibility of hydrogen atoms by neutron crystallography on fully deuterated myoglobin Myoglobin diffraction pattern 1962 Nobel Prize by Max Perutz and Sir John Cowdery KendrewMax PerutzSir John Cowdery Kendrew
X-ray Diffraction
Water
Light
Electron
Constructive
Destructive
Diffraction from atoms
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1 A About 1 Å
Wave of mater
Wave of electrons The electrons are accelerated in an electric potential U to the desired velocity:
Crystal diffraction
Gas, liquid, powder diffraction
Surface diffraction
Diffraction by diffractometer
Example of spots by diffractometer
X-ray Crystallography
Electron density
Deformation Electron Density
Macromolecule X-ray Crystallography
Generation of X-rays
What is K and K (for Cu) ? K : 2p 1s K : 3p 1s
X-ray tube
An optical grating and diffraction of light
Lattice planes
Lattice planes => reflection
Lattice planes review
Bragg ’ s Law
2dsin(theta)=n lumda
Bragg ’ s Law
Atomic scattering factor
intensity
Phase and intensity
Electron density
Diffraction of one hole
Diffraction of two holes
Diffraction of 5 holes
2D four holes
From real lattice to reciprocal lattice Real holesReflection pattern Crystal lattice is a real lattice, while its reflection pattern is its corresponding reciprocal lattice.
TEM image of Si? or Diamond? Real lattice viewed from (110) direction. Si Diamond
Electron Diffraction
Conversion of Real Lattice to Reciprocal Lattice PPP PPP PPP PPP PPP PPP PPP PPP PPP PPP
Ewald Sphere and Diffraction Pattern 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 Paul Peter Ewald (1888~1985)
Ewald Sphere
A vector of reciprocal lattice represents a set of parallel planes in a crystal lattice 2d sin = n (1/d hkl )/(2/ ) = sin (hkl)
Reciprocal Lattice and Ewald Sphere
Detector, Reciprocal Lattice and Ewald Sphere
3D View of Ewald Sphere and Reciprocal Sphere
Techniques of X-ray diffraction Single Crystal and Powder X-ray Diffractions many many many very small single crystals
Diffractometers for Single Crystal and Powder X-ray Diffractions
Single Crystal and Powder X- ray Diffraction Patterns
The powder XRD method
Formation of a cone of diffracted radiation
XRPD on film electron diffraction of powder sample
Finger Print Identification Finger Print Identification for Known Compounds by comparing experimental XRPD to those in PDF database
Some peaks may not be observed due to preferred orientation For example, layered structure such as graphite.
X-ray powder diffraction patterns of crystalline and amorphous sample
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 >500 nm) added to the sample: B 2 =Bs 2 -Br 2.
2d sin = Some equations to calculate cell parameters (d-spacings)
X-ray powder diffraction patterns for potassium halides
Structure Factor, Intensity and Electron Density R 1 = ||F o | - |F c ||/ |F o | F calc F obs
Electron density maps by X-ray diffraction
Scattering of X-rays by a crystal-systematic absences
Systematic Absences
Systematic absence for C-center: (x,y,z) ≣ (x+1/2, y+1/2, z) F hkl = (1/V) f j exp[2 i(hx j +ky j +lz j )] = (1/V) f j [cos2 (hx j +ky j +lz j )+isin2 (hx j +ky j +lz j )] = (1/V) f j {cos2 (hx j +ky j +lz j )+cos2 [h(x j +1/2) +k(y j +1/2)+lz j )]}+i{sin2 (hx j +ky j +lz j ) +sin2 [h(x j +1/2)+k(y j +1/2)+lz j )]} j=1 N N/2
let 2 (hx j +ky j +lz j )= j cos(A+B)=cosAcosB-sinAsinB sin(A+B)=sinAcosB+cosAsinB (1/V) f j cos2 (hx j +ky j +lz j )+cos2 h(x j +1/2)+k(y j +1/2)+lz j )]} +i sin2 (hx j +ky j +lz j )+sin2 h(x j +1/2)+k(y j +1/2)+lz j )]} =(1/V) f j cos j +cos j + h+k))+i[sin j +sin j + h+k))]} =(1/V) f j cos j +cos j cos h+k)]+i sin j +sin j cos h+k)]} ={[cos h+k) + 1]}/V f j cos j + isin j ] So when cos h+k) = -1 that is when h+k = 2n+1, F hkl = 0 Condition for systematic absences caused by C-center: For all (hkl), when h+k = 2n+1, I hkl = 0
F hkl =(1/V) f j cos2 (hx j +ky j +lz j )+isin2 (hx j +ky j +lz j )] =(1/V) f j { cos2 (hx j +ky j +lz j )+cos2 (-hx j +k(y j +1/2)-lz j )] +i sin2 (hx j +ky j +lz j )+ sin2 (-hx j +k(y j +1/2)-lz j )]} For reflections at (0 k 0) F hkl = (1/V) f j {[cos(2 ky j )+ cos(2 ky j )cos(k )] + i[sin(2 ky j )+ sin(2 ky j )cos(k )]} =[(cos(k )+1)/v] f j [cos(2 ky j )+ i[sin(2 ky j )] Systematic absences for 2 1 //b where (x,y,z) ≣ (-x,y+1/2,-z) So the conditions for 2 1 //b screw axis: For all reflections at (0 k 0), when k = 2n+1, I hkl =0
Conditions of Systematic Absences I-center: for all (hkl), h+k+l = 2n+1, I hkl = 0 F-center: for all (hkl), h+k = 2n+1, h+l = 2n+1 k+l = 2n+1, I hkl = 0 (or h, k, l not all even or all odd) c-glide (b-axis), for all (h0l), l = 2n+1, I hkl = 0 n-glide (b-axis), for all (h0l), h+l = 2n+1, I hkl = 0 d-glide (b-axis), for all (h0l), h+l = 4n+1, 2 or 3, I hkl = //b screw axis, for all (0k0), k = 3n+1, 3n+2, I hkl = 0 其他類推
Setup of Conventional Single Crystal X-ray Diffractometer
Electron diffraction Electron diffraction e - 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 Ni–Mo alloy (18 % Mo) with fcc structure. Weak spots result from superlattice of Mo arrangement.
Secondary diffraction of electron diffraction Extra reflections may appear in the diffraction pattern The intensities of diffracted beam are unreliable
Neutron diffraction
Antiferromagnetic superstructure in MnO, FeO and NiO MnO Fe 3 O 4 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).
Schematic neutron and X-ray diffraction patterns for MnO