chem 253, uc, berkeley diamond · pdf filecalculate d spacing reciprocal lattice index...

$: Chem 253, UC, Berkeley Diamond · PDF fileCalculate d spacing Reciprocal lattice Index diffraction peaks ... Chem 253, UC, Berkeley Densest lattice packing of an octahedron The density$
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Chem 253, UC, Berkeley


Diamond Lattice

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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


Silicon Diffraction pattern

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(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


Nanocrystal X-ray Diffraction

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Finite Size Effect


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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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


How particle size influence the peak width of the diffraction beam.

Full width half maximum (FWHM)

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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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• 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.


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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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.


Direct Visualization of Individual Cylindrical and Spherical Supramolecular DendrimersScience 17 October 1997; 278: 449-452

Small Angle X-ray Diffraction

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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


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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Descriptive Crystal ChemistryWest Chapter 7,8


Close packing structures: Cubic vs. Hexagonal

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52.36%

Unit cell symmetries - cubic

6

)2

(34

%3

3

a

a

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-Iron is body-centered cubic

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33)43

(34

2%3

3

a

a

68%

BCC Lattice


ar 24

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CN=12

%05.74)22(

33.14%

3

3

r

rFor BCC; 68.02%

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CN=12

(0,0,0)(1/3,2/3,1/2)


ar 34

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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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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

chem 253, uc, berkeley diamond · pdf filecalculate d spacing reciprocal lattice index...

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