Laue diffraction and the
reciprocal lattice
Lecture 4
Outline
Laue Equations
Reciprocal lattice
Equivalence with Bragg’s Law
Ewald sphere construction
Deviation parameter
Laue Equations
a s in
δ1
Incident wave
δ1
a
γ1
Scattered wave
asinγ1
Constructive interference when:
a(sinγ1 - sinδ1) = hλ
b(sinγ2 - sinδ2) = kλ
c(sinγ3 - sinδ3) = lλ
h,k,l = integers
Laue Equations
r·k
i
ki
ki,kd are unit vectors
with modulus 1/λ
1/λ
kd
r
kd
r·kd
r r r
r r
r " (k d # ki ) = r " K = m
r r
K"a = h
r r
Resolve onto lattice
K"b = k
unit vectors
r r
!
K"c =l
ki
K
with m = integer
with h,k,l = integers
Laue Equations
Diffraction occurs when:
r r
K"a = h
r r
K"b = k
r r
K"c =l
with h,k,l = integers
A general solution to these simultaneous equations is:
r
r
r
r
r
K
=
h
a
*
+k
b
*
+l
c
*
=
g
!
Where a*,b* and c* define a new set of lattice vectors,
which are related to a, b and c according to:
! r r
r
r r
a * "a = 1 a * "b = 0 a * "c = 0
r r
r
r r
b * "a = 0 b * "b = 1 b * "c = 0
r r
r
r r
c * "a = 0 c * "b = 0 c * "c = 1
Reciprocal lattice
This new lattice is referred to as the reciprocal lattice.
In real space:
r
r
r
r
rn = n1a + n2b + n3c
In reciprocal space:
r
r
r
r
r * = m1a * +m2b * +m3c *
!
Several properties of the reciprocal lattice include:
r r
r r
r
r
r r r
V = a"b# c
!a * ⊥ b & c
a* = b " c V
r
r
r r
r r
(volume of unit
b * ⊥ a& c
b
*
=
a
"
c
V
cell of real
r
r r
r
r r
lattice)
c * ⊥ a& b
c* = a " b V
!
!
!
!
(recall
have orthogonal lattice vectors)
! not!all real lattices
!
!
!
!
Reciprocal lattice
Consider a reciprocal lattice vector g such that:
r
r
g = ha * +kb * +lc *
where h,k and l are both integers, and are the Miller
Indices of a plane in real space (h k l)
!
This vector g has two important properties (which we
will prove):
r
r
1
g ⊥ hkl and g =
dhkl
( )
!
!
!
Reciprocal lattice
proofs
r
r
z
Prove: g ⊥ (hkl)
c
r
r
l
b
a
C
1.) + AB =
hkl
k
h
r r
b a
! AB
!= "
!
k h
r
! Is g ⊥ AB ?
!
2.)
B
r
g · AB = 0
r r
!
y
$b a '
r
ha * +kb * +lc * ) " & # ) = 0
(
A
!
b
%k h(
r
k
a
3.) Can repeat to show:
h
x
g ⊥ AC
r r r r r r r r r r r r
r
a * "b = a * "rc = b
r * "ra = c
4.) g
r * "rc = b
r * "a = c " b = 0
⊥ (hkl)
a * "a = b * "b = c * "!
c=1
!
Reciprocal lattice
proofs
r
1
Prove: g =
dhkl
z
C
Shortest distance must be
projection of real lattice
vector
on to a unit vector in
!
direction g (define as n)
r
r r
a r a g
dhkl = ON = " n = " r
h
h g
!
r
c
l
hkl
!
O
r
r r
r
a ha * +kb * +lc * 1
dhkl = "
= r
r
h
g
g
A
x
r
1
g=
dhkl
N
r
b
r
a
h
!
!
B
y
k
Reciprocal lattice
Reciprocal lattice points
each correspond to a
plane in real space
Real space
(001)
Z
(101)
(010)
Reciprocal lattice points
are defined by reciprocal
lattice vectors where:
1
r
d
=
g ⊥ hkl and
r
hkl
g
y
x
(100)
( )
001
101
!
1/d101
!
100
1/d100
1/d001
010
1/d010
Reciprocal space
Real => Reciprocal
The relationship between real and reciprocal determined
r r
r
r r
by:
a * "a = 1 a * "b = 0 a * "c = 0
r r
r
r r
b * "a = 0 b * "b = 1 b * "c = 0
r r
r
r r
c * "a = 0 c * "b = 0 c * "c = 1
c
c*
c*
!
b
b*
a
a*
b*
a*
a* only parallel to a if a, b and c are mutually orthogonal
Diffraction
real space vs. reciprocal space
θ
d hkl
(hkl)
hkl)
ki
Transmitted
Real space
Diffracted
θ
kd
K=g
Reciprocal space
Bragg’s Law
Incident plane wave
ki
kd
Scattered plane wave
θ
d
Parallel
“reflecting”
reflecting”
planes
θ θ
k’s - wave vectors
kd
2θ
ki
K
!
2d " sin# = n$
r r r
k d " ki = K
r 2sin# n
K=
=
r $r d
K=g
!
r 1
K"
&
#
r 1
K"
d
Laue Equations & Bragg’s Law
Laue Equations have soln:
r
r
r
r
r
K = ha * +kb * +lc* = g
!
We have shown that:
r 2sin"B
r
1
K=
& g=
#
dhkl
We recover Bragg’s Law:
kd - ko = K
(hkl)
ki
kd
θB
2sin"B
1
=
#
dhkl
2dhkl sin"B = #
K
Bragg’s Law
Incident plane wave
ki
kd
Scattered plane wave
θ
d
Parallel
“reflecting”
reflecting”
planes
θ θ
k’s - wave vectors
kd
2θ
ki
K
!
2d " sin# = n$
r r r
k d " ki = K
r 2sin# n
K=
=
r $r d
K=g
!
r 1
K"
&
#
r 1
K"
d
Laue Equations & Bragg’s Law
Laue Equations have soln:
r
r
r
r
r
K = ha * +kb * +lc* = g
!
We have shown that:
r 2sin"B
r
1
K=
& g=
#
dhkl
We recover Bragg’s Law:
kd - ko = K
(hkl)
ki
kd
θB
2sin"B
1
=
#
dhkl
2dhkl sin"B = #
K
Diffraction
reciprocal space
Ewald Sphere
(hkl)
s
u
i
d
a
R
2θ
ki
/λ
1
=
kd
g
0
ki = kd = 1/λ
G
“A Valt”
Valt” not “E Walled”
Walled”
Diffraction
Diffraction occurs when the Ewald sphere intersects a
reciprocal lattice vector
For 200 kV electrons, 1/λ = 1/0.00273 nm = 366 nm-1
Length of g = 3 nm-1
ki
Reciprocal lattice
Reciprocal lattice points
each correspond to a
plane in real space
Real space
(001)
Z
(101)
(010)
Reciprocal lattice points
are defined by reciprocal
lattice vectors where:
1
r
d
=
g ⊥ hkl and
r
hkl
g
y
x
(100)
( )
001
101
!
1/d101
!
100
1/d100
1/d001
010
1/d010
Reciprocal space
Diffraction
Diffraction occurs when the Ewald sphere intersects a
reciprocal lattice vector
For 200 kV electrons, 1/λ = 1/0.00273 nm = 366 nm-1
Length of g = 3 nm-1
ki
Diffraction
reciprocal space
Ewald Sphere
(hkl)
s
u
i
d
a
R
2θ
ki
/λ
1
=
kd
g
0
ki = kd = 1/λ
G
“A Valt”
Valt” not “E Walled”
Walled”
Diffraction
Diffraction occurs when the Ewald sphere intersects a
reciprocal lattice vector
For 200 kV electrons, 1/λ = 1/0.00273 nm = 366 nm-1
Length of g = 3 nm-1
ki
Resulting diffraction pattern
First Order Laue Zone
Zero Order Laue Zone
Diffraction
Diffraction occurs when the Ewald sphere intersects a
reciprocal lattice vector
For 200 kV electrons, 1/λ = 1/0.00273 nm = 366 nm-1
Length of g = 3 nm-1
ki
Higher order Laue Zones
Reciprocal lattice is not planar -- it is a true 3-D lattice
Zero order Laue Zone (ZOLZ), First Order Laue Zone
(FOLZ), Higher order Laue Zone (HOLZ)
ki
Resulting diffraction pattern
First Order Laue Zone
Zero Order Laue Zone
Reciprocal Lattice Rods
Each point in reciprocal space contains not a
point, but rather a rod - “rel-rod”
– Result of the finite size of our diffracting crystals
– We’ll derive this in a couple of lectures: for now, just
believe me!
Ewald Sphere and Rel-rods
Presence of rel-rods “relaxes” diffraction requirements
New vector - s - called “deviation parameter”
sz
3g
-2G
-G
0
G
2G
3G
4G
5G
Resulting diffraction pattern
First Order Laue Zone
Zero Order Laue Zone