chem 253, uc, berkeley diamond · pdf filecalculate d spacing reciprocal lattice index...
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Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
Diamond Lattice
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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
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Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
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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
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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
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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)
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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.
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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).
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Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
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Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
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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
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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
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Chem 253, UC, Berkeley
Descriptive Crystal ChemistryWest Chapter 7,8
Chem 253, UC, Berkeley
Close packing structures: Cubic vs. Hexagonal
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Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
52.36%
Unit cell symmetries - cubic
6
)2
(34
%3
3
a
a
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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
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Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
CN=12
%05.74)22(
33.14%
3
3
r
rFor BCC; 68.02%
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Chem 253, UC, Berkeley
CN=12
(0,0,0)(1/3,2/3,1/2)
Chem 253, UC, Berkeley
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Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
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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
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Truncated octahedron
Yaghi, Science 2008
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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.
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Close Packing Octahedra: Minkowski Lattice