1
Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
Diamond Lattice
2
Chem 253, UC, Berkeley
SSi(hkl)=Sfcc(hkl)[1+exp(i/2)(h+k+l)].
SSi(hkl) will be zero if the sum (h+k+l) is equal to 2 times an odd integer, such as (200), (222). [h+k+l=2(2n+1)]
SSi(hkl) will be non-zero if(1)(h,k,l) contains only even numbers and (2) the sum (h+k+l) is equal to 4 times an integer.
[h+k+l=4n]Shkl will be non-zero if h,k, l all odd:
Diamond Lattice
)(4 sisi iff
FCC:S=4 when h+k, k+l, h+lall even (h,k, l all even/odd)
S=0, otherwise
)()()(1 klilhikhifcc eeeS
Chem 253, UC, Berkeley
Silicon Diffraction pattern
3
Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
4
Chem 253, UC, Berkeley
(1/8,1/8,1/8), (3/8,3/8,1/8),(1/8,3/8,3/8) and (3/8,1/8,3/8).
Bond charges form a "crystal" with a fcc lattice with 4 "atoms" per unit cell.
Bond charges in covalent solid
origin : (1/8,1/8,1/8).
Sbond-charge(hkl) = Sfcc(hkl)[1+exp(i/2)(h+k)+exp(i/2)(k+l)+exp(i/2)(h+l)].
For h=k=l=2:Sbond-charge(222)=Sfcc(222)[1+3exp(i/2)4]=4Sfcc(222) non-zero!
Bond charge position: (0,0,0), (1/4,1/4,0),(0,1/4,1/4) and (1/4,0,1/4).
n
j
lzkyhxij
n
j
diKjk efefS j
1
)(2
1
Chem 253, UC, Berkeley
Nanocrystal X-ray Diffraction
5
Chem 253, UC, Berkeley
Finite Size Effect
Chem 253, UC, Berkeley
Bragg angle:
in phase, constructive
B
B 1For
Phase lag between two planes:
At j+1 th plane:Phase lag:
2j planes: net diffraction at :0
1234
j-1jj+1
2j-12j
2
j
1
6
Chem 253, UC, Berkeley
Bragg angle:
in phase, constructive
B
For
Phase lag between two planes:
At j+1 th plane:Phase lag:
12345
j-1jj+1
2j-12j
2
j
B 2
2j planes: net diffraction at :02
Chem 253, UC, Berkeley
How particle size influence the peak width of the diffraction beam.
Full width half maximum (FWHM)
7
Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
The width of the diffraction peak is governed by # of crystal planes 2j. i.e. crystal thickness/size
Scherrer Formula:
BBt
cos
9.0
222SM BBB
BM: Measured peak width at half peak intensity (in radians)BS: Corresponding width for standard bulk materials (largegrain size >200 nm)
Readily applied for crystal size of 5-50 nm.
8
Chem 253, UC, Berkeley
• Suppose =1.5 Å, d=1.0 Å, and =49°. Then for a crystal1 mm in diameter, the breath B, due to the small crystaleffect alone, would be about 2x10-7 radian (10-5 degree),or too small to be observable. Such a crystal wouldcontain some 107 parallel lattice planes of the spacingassumed above.
• However, if the crystal were only 500 Å thick, it wouldcontain only 500 planes, and the diffraction curve wouldbe relatively broad, namely about 4x10-3 radian (0.2°),which is easily measurable.
Chem 253, UC, Berkeley
Index planeCalculate crystal densityCalculate d spacing
Reciprocal lattice
Index diffraction peaksFind out lattice constant.
Find out structural factors, predicting X-ray diffraction pattern(knowing their relative intensity).
22
Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
23
Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
24
Chem 253, UC, Berkeley
Theorem:For any family of lattice planes separated by distance d, there are reciprocal lattice vectors perpendicular to the planes, the shortest being 2/d.
Orientation of plane is determined by a normal vectorThe miller indices of a lattice plane are the coordination at the reciprocal lattice vector normal to the plane.
Chem 253, UC, Berkeley
Direct Visualization of Individual Cylindrical and Spherical Supramolecular DendrimersScience 17 October 1997; 278: 449-452
Small Angle X-ray Diffraction
25
Chem 253, UC, Berkeley
Small Angle X-ray Diffraction
Triblock Copolymer Syntheses of Mesoporous Silica with Periodic 50 to 300 Angstrom Pores Science, Vol 279, Issue 5350, 548-552 , 23 January 1998
Chem 253, UC, Berkeley
Triblock Copolymer Syntheses of Mesoporous Silica with Periodic 50 to 300 Angstrom Pores Science, Vol 279, Issue 5350, 548-552 , 23 January 1998
26
Chem 253, UC, Berkeley
Descriptive Crystal ChemistryWest Chapter 7,8
Chem 253, UC, Berkeley
Close packing structures: Cubic vs. Hexagonal
27
Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
52.36%
Unit cell symmetries - cubic
6
)2
(34
%3
3
a
a
28
Chem 253, UC, Berkeley
-Iron is body-centered cubic
24
33)43
(34
2%3
3
a
a
68%
BCC Lattice
Chem 253, UC, Berkeley
ar 24
29
Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
CN=12
%05.74)22(
33.14%
3
3
r
rFor BCC; 68.02%
30
Chem 253, UC, Berkeley
CN=12
(0,0,0)(1/3,2/3,1/2)
Chem 253, UC, Berkeley
ar 34
31
Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
32
Chem 253, UC, Berkeley
Rare Gas: Ne, He, Ar, Kr, Xe (ccp; fcc)
Metal: Cu, Ag, Au, Ni, Pd, Pt (ccp)
Mg, Zn, Cd, Ti (hcp)
Fe, Cr, Mo (bcc)
5 μmJ. Henzie, et al. Nature Mater, 11, 131, 2012.
500 μm
Optical Dark Field Micrograph
Packing of Truncated octahedron
33
Truncated octahedron
Yaghi, Science 2008
34
Chem 253, UC, Berkeley
Densest lattice packing of an octahedron
The density of a densest lattice packing of an octahedron was already calculated by Minkowski in 1904. In 1948 Whitworth generalized Minkowski's result to a family of truncated cubes. The density of a densest lattice packing is equal to 18/19 = 0.9473...,
Hermann Minkowski: Dichteste gitterförmige Lagerung kongruenter Körper, Nachr. K. Ges. Wiss. Göttingen, Math.-Phys. KL (1904) (1904), 311 - 355
2 μm
Octahedra
200 nm
J. Henzie, et al. Nature Mater, 11, 131, 2012.
35
Close Packing Octahedra: Minkowski Lattice