MSE 603 / Diffraction
April 22, 2002
p.8
Diffraction from a crystal:
In a crystal, charge density is periodic and can be expanded into a Fourier series:
1
iH⋅r j
− iH⋅r
ρ (r ) =
, where FH = ∑ f j e
is the so-called structure factor, Vc is
∑ FH e
Vc H
j
the unit-cell volume. Vector H represents a Fourier component, and is called the
reciprocal space vector. The concept of reciprocal space is so central to diffraction
analysis that we probably should spend a few minutes on its basics.
In real space, or direct space, a crystal is made of a periodic lattice (called Bravais lattice)
plus a bunch of atoms located at around each lattice site (or atoms within a unit cell).
A unit cell in a Bravais lattice is defined by three unit
cell vectors: a1, a2, a3, which should be non-coplanar.
a2
a3
a1
Unit cell volume V c = a 1 ⋅ (a 2 × a 3 ) .
Any lattice point in direct space can be represented by a lattice vector Rn,
R n = n1 a1 + n 2 a 2 + n 3 a 3 , where (n1, n2, n3) are integers.
Reciprocal space or Fourier space of a crystal is defined by three reciprocal unit cell
vectors: b 1 = 2π (a 2 × a 3 ) V c , b 2 = 2π (a 3 × a 1 ) V c , b 3 = 2π (a 1 × a 2 ) V c . Any
lattice point in reciprocal space can be represented by a reciprocal space vector:
H = ( hkl ) = hb 1 + kb 2 + lb 3 , with (hkl) being integers. Reciprocal space is orthogonal
to direct space such that a i ⋅ b j = 2πδ ij .
The physical meaning of the reciprocal space vector H = (hkl ) is the following:
(i) The direction of H = (hkl ) is perpendicular to the set of atomic planes that intercept
the real space lattice (a1, a2, a3) at (a1/h, a2/k, a3/l); and (ii) The magnitude of H = (hkl )
is H = 2π / d , where d is the spacing between the lattice planes defined by H = (hkl ) .
For cubic crystals, d = a h 2 + k 2 + l 2 , where a is the lattice parameter of the cubic
unit cell. For other crystal systems, one can find formulas in e.g. Cullity’s book. For
computer programs, it is often easier to use the notion of metric tensor and dot product to
evaluate d: d = 2π H ⋅ H . Or: one can use the formula for a general triclinic lattice.
Note: two equivalent conventions are used in literature.
(i) H = 2π / d , k = 2π / λ ⇔ Plane wave e − ik⋅r , often used by physicists.
(ii) H = 1 / d , k = 1 / λ
Qun Shen (qs11@cornell.edu)
⇔ Plane wave e − i 2πk⋅r , often used by crystallographers.
Cornell High Energy Synchrotron Source (CHESS), Wilson Laboratory
MSE 603 / Diffraction
April 22, 2002
p.9
From what we learned earlier, the scattering amplitude S(Q) is given by the Fourier
transform of the charge density ρ(r) of the scattering object:
S ( Q ) = ∫ ρ ( r ) e iQ⋅r dr .
V
For a crystal, any position r can be decomposed into r = R n + r ' , where Rn is the lattice
vector and r ' indicates positions inside a unit cell. Thus,
S ( Q ) = ∑ ∑ ∑ ∫ ρ ( r ' ) e iQ⋅( R n + r ' ) dr '
n1 n 2 n 3 Vc
= ∑ ∑ ∑ e iQ⋅R n ∫ ρ ( r ' ) e iQ⋅r ' dr '
n1 n 2 n 3
Vc
We now write Q in reciprocal space:
Q = hb 1 + kb 2 + lb 3 , then Q ⋅ R n = 2π ( hn1 + kn 2 + ln 3 ) .
Assuming Ni unit cells exist along ai direction (i = 1, 2, 3), then each of the lattice sums
gives rise to a sin(Nx)/sin(x) function, and the integral over the unit cell volume yields the
structure factor FH. Therefore, apart from a phase factor, we have:
S (Q) = F (Q)
sin(πhN 1 ) sin(πkN 2 ) sin(πlN 3 )
,
sin(πh )
sin(πk )
sin(πl )
and the scattered intensity:
2
2
2
 sin(πhN 1 )   sin(πkN 2 )   sin(πlN 3 ) 
I (Q) = F (Q) 
 
 
 .
 sin(πh )   sin(πk )   sin(πl ) 
2
This interference function shows strong peaks at (hkl) = integers, and almost zero
everywhere else for any real crystals with N1, N2, N3 greater than say 10. Thus we see
that scattered intensity is significant only when the momentum transfer Q is equal to a
reciprocal space vector H:
2
2
2
 sin(πhN 1 )   sin(πkN 2 )   sin(πlN 3 ) 
I ( H ) = F ( hkl ) 
 
 
 .
 sin(πh )   sin(πk )   sin(πl ) 
2
|F(hkl)|2N12N22N32
1/N1
1
Qun Shen (qs11@cornell.edu)
2
3
hÆ
Cornell High Energy Synchrotron Source (CHESS), Wilson Laboratory
MSE 603 / Diffraction
April 22, 2002
p.10
We can also see that the condition Q = H is equivalent to Bragg’s law: since
Q = 4π sin θ / λ and H = 2π / d , we get: 2d sin θ = λ .
Another useful concept in diffraction analysis is
the Ewald sphere construction in reciprocal
space (see Figure at right). Any reciprocal
nodes that fall on the Ewald sphere give rise to
Bragg diffraction peaks.
Ewald sphere:
radius 2π/λ.
kH
H
k0
A diffraction peak can have a finite width, with
three types of contributions: size broadening,
mosaic spread, and lattice strain. In reciprocal
space units, these contributions give rise to:
O
∆Q = 2π / L , both transverse and longitudinal, for crystal with average domain size L,
∆Q = H ⋅ ∆θ , transverse to H, for crystal with mosaic spread ∆θ; and
∆Q = H ⋅ ∆d / d , along H (longitudinal), for crystal with non-uniform strain ∆d/d.
The Q- and the directional dependence of these
contributions can be used to distinguish different possible
causes to an observed diffraction peak width.
In an experiment using x-ray diffractometer, a transverse
scan is realized by a “rocking” scan or q scan, while a
longitudinal scan is done by rocking the specimen by ∆θ
followed by moving the detector by an increment 2∆θ.
Qun Shen (qs11@cornell.edu)
θ−2θ scan:
longitudinal
θ-scan:
transverse
kH
k0
O
Cornell High Energy Synchrotron Source (CHESS), Wilson Laboratory