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