In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector,
K
, at right angles. The Bragg plane is defined as part of the Von Laue condition for diffraction peaks in x-ray diffraction crystallography.
Considering the diagram on the right, the arriving x-ray plane wave is defined by:
e
i
k
⋅
r
=
cos
(
k
⋅
r
)
+
i
sin
(
k
⋅
r
)
Where
k
is the incident wave vector given by:
k
=
2
π
λ
n
^
where
λ
is the wavelength of the incident photon. While the Bragg formulation assumes a unique choice of direct lattice planes and specular reflection of the incident X-rays, the Von Laue formula only assumes monochromatic light and that each scattering center acts as a source of secondary wavelets as described by the Huygens principle. Each scattered wave contributes to a new plane wave given by:
k
′
=
2
π
λ
n
^
′
The condition for constructive interference in the
n
^
′
direction is that the path difference between the photons is an integer multiple (m) of their wavelength. We know then that for constructive interference we have:
|
d
|
cos
θ
+
|
d
|
cos
θ
′
=
d
⋅
(
n
^
−
n
^
′
)
=
m
λ
where
m
∈
Z
. Multiplying the above by
2
π
λ
we formulate the condition in terms of the wave vectors,
k
and
k
′
:
d
⋅
(
k
−
k
′
)
=
2
π
m
Now consider that a crystal is an array of scattering centres, each at a point in the Bravais lattice. We can set one of the scattering centres as the origin of an array. Since the lattice points are displaced by the Bravais lattice vectors,
R
, scattered waves interfere constructively when the above condition holds simultaneously for all values of
R
which are Bravais lattice vectors, the condition then becomes:
R
⋅
(
k
−
k
′
)
=
2
π
m
An equivalent statement (see mathematical description of the reciprocal lattice) is to say that:
e
i
(
k
−
k
′
)
⋅
R
=
1
By comparing this equation with the definition of a reciprocal lattice vector, we see that constructive interference occurs if
K
=
k
−
k
′
is a vector of the reciprocal lattice. We notice that
k
and
k
′
have the same magnitude, we can restate the Von Laue formulation as requiring that the tip of incident wave vector,
k
, must lie in the plane that is a perpendicular bisector of the reciprocal lattice vector,
K
. This reciprocal space plane is the Bragg plane.
