Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
Diamond Lattice
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Chem 253, UC, Berkeley
S fcc  1  e  i ( h  k )  e  i ( h  l )  e  i ( l  k )
Diamond Lattice
SSi(hkl)=Sfcc(hkl)[1+exp(i/2)(h+k+l)].
FCC:
S=4 when h+k, k+l, h+l
all even (h,k, l all
even/odd)
S=0, otherwise
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: 4( f  if )
si
si
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
Bond charges in covalent solid
(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.
origin : (1/8,1/8,1/8).
Bond charge position: (0,0,0), (1/4,1/4,0),(0,1/4,1/4) and (1/4,0,1/4).
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!
n
Sk   f j e
j 1
iK d j
n
  f j e  2i ( hx  ky lz )
j 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:
B
in phase, constructive
For
1   B
1
2
3
4
Phase lag between two planes:
j-1
j
j+1
At j+1 th plane:
Phase lag:
2j-1
2j
  

   j    2
2j planes: net diffraction at 1:0
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Chem 253, UC, Berkeley
Bragg angle:
B
in phase, constructive
For
1
2
3
4
5
j-1
j
j+1
2  B
Phase lag between two planes:
  
At j+1 th plane:
Phase lag:

   j    2
2j-1
2j
2j planes: net diffraction at 2 :0
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:
0 .9 
t
B cos  B
B 2  BM2  BS2
BM: Measured peak width at half peak intensity (in radians)
BS: Corresponding width for standard bulk materials (large
grain 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 crystal
1 mm in diameter, the breath B, due to the small crystal
effect alone, would be about 2x10-7 radian (10-5 degree),
or too small to be observable. Such a crystal would
contain some 107 parallel lattice planes of the spacing
assumed above.
• However, if the crystal were only 500 Å thick, it would
contain only 500 planes, and the diffraction curve would
be relatively broad, namely about 4x10-3 radian (0.2°),
which is easily measurable.
Chem 253, UC, Berkeley
Index plane
Calculate crystal density
Calculate d spacing
Reciprocal lattice
Index diffraction peaks
Find 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 vector
The miller indices of a lattice plane are the coordination at the reciprocal lattice
vector normal to the plane.
Chem 253, UC, Berkeley
Small Angle
X-ray Diffraction
Direct Visualization of Individual
Cylindrical and Spherical
Supramolecular Dendrimers
Science 17 October 1997; 278: 449-452
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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 Chemistry
West 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
Unit cell symmetries - cubic
4 a 3
( ) 
3
2 
%
a3
6
52.36%
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Chem 253, UC, Berkeley
BCC Lattice
-Iron is bodycentered cubic
4
3a 3
(
)
3 3
3
4
%  2

3
a
24
68%
Chem 253, UC, Berkeley
4r  2 a
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Chem 253, UC, Berkeley
Chem 253, UC, Berkeley
CN=12
4 1.33r 3
%
 74.05%
(2 2r ) 3
For 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
4r  3a
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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)
Packing of Truncated octahedron
Optical Dark Field Micrograph
500 μm
5 μm
J.
Henzie, et al. Nature Mater, 11, 131, 2012.
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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
Octahedra
200 nm
2 μm
J. Henzie, et al. Nature Mater, 11, 131, 2012.
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Close Packing Octahedra: Minkowski Lattice
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