In crystallography, the Laue equations give three conditions for incident waves to be diffracted by a crystal lattice. They are named after physicist Max von Laue (1879–1960). They reduce to Bragg's law.
Take
k
i
to be the wavevector for the incoming (incident) beam and
k
o
to be the wavevector for the outgoing (diffracted) beam.
k
o
−
k
i
=
Δ
k
is the scattering vector (also called transferred wavevector) and measures the change between the two wavevectors.
Take
a
,
b
,
c
to be the primitive vectors of the crystal lattice. The three Laue conditions for the scattering vector, or the Laue equations, for integer values of a reflection's reciprocal lattice indices (h,k,l) are as follows:
a
⋅
Δ
k
=
2
π
h
b
⋅
Δ
k
=
2
π
k
c
⋅
Δ
k
=
2
π
l
These conditions say that the scattering vector must be oriented in a specific direction in relation to the primitive vectors of the crystal lattice.
If
G
=
h
A
+
k
B
+
l
C
is the reciprocal lattice vector, we know
G
⋅
(
a
+
b
+
c
)
=
2
π
(
h
+
k
+
l
)
. The Laue equations specify
Δ
k
⋅
(
a
+
b
+
c
)
=
2
π
(
h
+
k
+
l
)
. Hence we have
Δ
k
=
G
or
k
o
−
k
i
=
G
.
From this we get the diffraction condition:
k
o
−
k
i
=
G
(
k
i
+
G
)
2
=
k
o
2
k
i
2
+
2
k
i
⋅
G
+
G
2
=
k
o
2
Since
(
k
o
)
2
=
(
k
i
)
2
(considering elastic scattering) and
G
=
−
G
(a negative reciprocal lattice vector is still a reciprocal lattice vector):
2
k
i
⋅
G
=
G
2
.
The diffraction condition
2
k
i
⋅
G
=
G
2
reduces to the Bragg law
2
d
sin
θ
=
n
λ
.
