The Structure FactorThe Structure Factor - University of Babylon

The Structure Factor

Suggested Reading

Pages 303-312 in DeGraef & McHenry Pages 59-61 in Engler and Randle

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Structure Factor (Fhkl)

N

F f e2i(hui kv j lwi )

hkl

i

i1

? Describes how atomic arrangement (uvw) influences the intensity of the scattered beam.

? It tells us which reflections (i.e., peaks, hkl) to expect in a diffraction pattern.

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Structure Factor (Fhkl)

? The amplitude of the resultant wave is given by a ratio of amplitudes:

Fhkl

amplitude of amplitude

the wave scattered by of the wave scattered

all by

atoms of a UC one electron

? The intensity of the diffracted wave is proportional to |Fhkl|2.

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Some Useful Relations

ei = e3i = e5i = ... = -1 e2i = e4i = e6i = ... = +1 eni = (-1)n, where n is any integer eni = e-ni, where n is any integer

eix + e-ix =2 cos x

Needed for structure factor calculations

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Fhkl for Simple Cubic

? Atom coordinate(s) u,v,w:

? 0,0,0

N

F f e2i(hui kvj lwi )

hkl

i

i1

Fhkl fe2i(0h0k0l ) f

No matter what atom coordinates or plane indices you substitute into the structure factor equation for simple cubic crystals, the solution is always

non-zero.

Thus, all reflections are allowed for simple cubic (primitive) structures.

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Fhkl for Body Centered Cubic

? Atom coordinate(s) u,v,w:

? 0,0,0; ? ?, ?, ?.

N

F f e2i(hui kvj lwi )

hkl

i

i1

Fhkl

fe2i(0)

fe2

i

h 2

k 2

l 2

Fhkl f 1 eihkl

When h+k+l is even Fhkl = non-zero reflection.

When h+k+l is odd Fhkl = 0 no reflection.

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Fhkl for Face Centered Cubic

? Atom coordinate(s) u,v,w:

? 0,0,0; ? ?,?,0; ? ?,0,?; ? 0,?,?.

N

F f e2i(hui kvj lwi )

hkl

i

i1

Fhkl

fe2i0

fe 2

i

h 2

k 2

fe 2

i

h 2

l 2

fe2

i

k 2

l 2

Fhkl f 1 eihk eihl eikl

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Fhkl for Face Centered Cubic

Fhkl f 1 eihk eihl eikl

? Substitute in a few values of hkl and you will find the following:

? When h,k,l are unmixed (i.e. all even or all odd), then Fhkl = 4f. [NOTE: zero is considered even]

? Fhkl = 0 for mixed indices (i.e., a combination of odd and even).

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