Transcript Unit cell - National Chung Cheng University

Crystallography and Diffraction
Techniques
Myoglobin
Types of diffraction
- X-ray diffraction
Myoglobin diffraction pattern
1962 Nobel Prize by Max Perutz
and Sir John Cowdery Kendrew
- Electron diffraction
- Neutron diffraction
Enhanced visibility of hydrogen atoms by neutron
crystallography on fully deuterated myoglobin
X-ray Diffraction
Water
Light
Electron
Constructive
Destructive
Diffraction from atoms
Continue
1A
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
Bragg’s Law
Bragg’s Law
2dsin(theta)=n lumda
Bragg’s Law
Atomic scattering factor
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 holes
Reflection pattern
Crystal lattice is a real lattice, while its reflection
pattern is its corresponding reciprocal lattice.
TEM image of Si? or Diamond?
Si
Diamond
Real lattice viewed from (110) direction.
Electron Diffraction
Conversion of Real Lattice to
Reciprocal Lattice
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Ewald Sphere and
Diffraction Pattern
Paul Peter Ewald
(1888~1985)
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
Ewald Sphere
A vector of reciprocal lattice
represents a set of parallel
planes in a crystal lattice
(hkl)
2q
q
(1/dhkl)/(2/l) = sin q
2d sin q = nl
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 Xray Diffraction Patterns
The powder XRD method
Formation of a cone of
diffracted radiation
XRPD on film
electron diffraction
of powder sample
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: B2=Bs2-Br2.
Some equations to calculate cell
parameters (d-spacings)
2d sinq = l
X-ray powder diffraction patterns for
potassium halides
Structure Factor, Intensity
and Electron Density
Fcalc
Fobs
R1 = S ||Fo| - |Fc||/ S |Fo|
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)
N
Fhkl = (1/V) S fjexp[2pi(hxj+kyj+lzj)]
j=1
=(1/V)Sfj[cos2p(hxj+kyj+lzj)+isin2p(hxj+kyj+lzj)]
N/2
=(1/V)Sfj{cos2p(hxj+kyj+lzj)+cos2p[h(xj+1/2)
j=1
+k(yj+1/2)+lzj)]}+i{sin2p(hxj+kyj+lzj)
+sin2p[h(xj+1/2)+k(yj+1/2)+lzj)]}
let 2p(hxj+kyj+lzj)=j
cos(A+B)=cosAcosB-sinAsinB
sin(A+B)=sinAcosB+cosAsinB
(1/V)Sfj{cos2p(hxj+kyj+lzj)+cos2p[h(xj+1/2)+k(yj+1/2)+lzj)]}
+i{sin2p(hxj+kyj+lzj)+sin2p[h(xj+1/2)+k(yj+1/2)+lzj)]}
=(1/V)Sfj{cos j+cos (j+p(h+k))+i[sin j+sin (j+p(h+k))]}
=(1/V)Sfj{[cos j+cos jcos p(h+k)]+i[sin j+sin jcos p(h+k)]}
={[cos p(h+k) + 1]}/V Sfj[cos j+ isin j]
So when cos p(h+k) = -1
that is when h+k = 2n+1,
Fhkl = 0
Condition for systematic absences caused by C-center:
For all (hkl), when h+k = 2n+1, Ihkl = 0
Systematic absences for 21//b where (x,y,z) ≣(-x,y+1/2,-z)
Fhkl =(1/V)Sfj[cos2p(hxj+kyj+lzj)+isin2p(hxj+kyj+lzj)]
=(1/V)Sfj{[cos2p(hxj+kyj+lzj)+cos2p(-hxj+k(yj+1/2)-lzj)]
+i[sin2p(hxj+kyj+lzj)+ sin2p(-hxj+k(yj+1/2)-lzj)]}
For reflections at (0 k 0)
Fhkl = (1/V)Sfj{[cos(2pkyj)+ cos(2pkyj)cos(kp)]
+ i[sin(2pkyj)+ sin(2pkyj)cos(kp)]}
=[(cos(kp)+1)/v] Sfj[cos(2pkyj)+ i[sin(2pkyj)]
So the conditions for 21//b screw axis:
For all reflections at (0 k 0), when k = 2n+1, Ihkl=0
Conditions of Systematic Absences
I-center: for all (hkl), h+k+l = 2n+1, Ihkl = 0
F-center: for all (hkl), h+k = 2n+1, h+l = 2n+1
k+l = 2n+1, Ihkl = 0
(or h, k, l not all even or all odd)
c-glide (b-axis), for all (h0l), l = 2n+1, Ihkl = 0
n-glide (b-axis), for all (h0l), h+l = 2n+1, Ihkl = 0
d-glide (b-axis), for all (h0l), h+l = 4n+1, 2 or 3, Ihkl = 0
31//b screw axis, for all (0k0), k = 3n+1, 3n+2, Ihkl = 0
其他類推
Setup of Conventional Single
Crystal X-ray Diffractometer
Electron diffraction
e-  l  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
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).
Schematic neutron and X-ray
diffraction patterns for MnO