In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, K {\displaystyle \scriptstyle \mathbf {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 adjacent diagram, the arriving x-ray plane wave is defined by:
e i k ⋅ r = cos ( k ⋅ r ) + i sin ( k ⋅ r ) {\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }=\cos {(\mathbf {k} \cdot \mathbf {r} )}+i\sin {(\mathbf {k} \cdot \mathbf {r} )}}
Where k {\displaystyle \scriptstyle \mathbf {k} } is the incident wave vector given by:
k = 2 π λ n ^ {\displaystyle \mathbf {k} ={\frac {2\pi }{\lambda }}{\hat {n}}}
where λ {\displaystyle \scriptstyle \lambda } 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 ^ ′ {\displaystyle \mathbf {k^{\prime }} ={\frac {2\pi }{\lambda }}{\hat {n}}^{\prime }}
The condition for constructive interference in the n ^ ′ {\displaystyle \scriptstyle {\hat {n}}^{\prime }} 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 {\displaystyle |\mathbf {d} |\cos {\theta }+|\mathbf {d} |\cos {\theta ^{\prime }}=\mathbf {d} \cdot \left({\hat {n}}-{\hat {n}}^{\prime }\right)=m\lambda }
where m ∈ Z {\displaystyle \scriptstyle m~\in ~\mathbb {Z} } . Multiplying the above by 2 π λ {\displaystyle \scriptstyle {\frac {2\pi }{\lambda }}} we formulate the condition in terms of the wave vectors, k {\displaystyle \scriptstyle \mathbf {k} } and k ′ {\displaystyle \scriptstyle \mathbf {k^{\prime }} } :
d ⋅ ( k − k ′ ) = 2 π m {\displaystyle \mathbf {d} \cdot \left(\mathbf {k} -\mathbf {k^{\prime }} \right)=2\pi 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 {\displaystyle \scriptstyle \mathbf {R} } , scattered waves interfere constructively when the above condition holds simultaneously for all values of R {\displaystyle \scriptstyle \mathbf {R} } which are Bravais lattice vectors, the condition then becomes:
R ⋅ ( k − k ′ ) = 2 π m {\displaystyle \mathbf {R} \cdot \left(\mathbf {k} -\mathbf {k^{\prime }} \right)=2\pi m}
An equivalent statement (see mathematical description of the reciprocal lattice) is to say that:
e i ( k − k ′ ) ⋅ R = 1 {\displaystyle e^{i\left(\mathbf {k} -\mathbf {k^{\prime }} \right)\cdot \mathbf {R} }=1}
By comparing this equation with the definition of a reciprocal lattice vector, we see that constructive interference occurs if K = k − k ′ {\displaystyle \scriptstyle \mathbf {K} ~=~\mathbf {k} \,-\,\mathbf {k^{\prime }} } is a vector of the reciprocal lattice. We notice that k {\displaystyle \scriptstyle \mathbf {k} } and k ′ {\displaystyle \scriptstyle \mathbf {k^{\prime }} } have the same magnitude, we can restate the Von Laue formulation as requiring that the tip of incident wave vector, k {\displaystyle \scriptstyle \mathbf {k} } , must lie in the plane that is a perpendicular bisector of the reciprocal lattice vector, K {\displaystyle \scriptstyle \mathbf {K} } . This reciprocal space plane is the Bragg plane.
See also X-ray crystallography Reciprocal lattice Bravais lattice Powder diffraction Kikuchi line Brillouin zone
References