Bragg plane explained

In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector,

\scriptstyle K , at right angles.^[1] 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 ^{ik} = \cos {( k ⋅ r )} + i\sin {( k ⋅ r )}

Where

\scriptstyle k is the incident wave vector given by:k = (2 \pi)/(λ) \hat{n}

where

\scriptstyle 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 ^{\prime} = (2 \pi)/(λ) \hat{n} ^{\prime}

The condition for constructive interference in the

\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{\theta} + | d |\cos{\theta ^{\prime}} = d ⋅ \left(\hat{n} - \hat{n} ^{\prime\right)} = m λ

where

\scriptstyle m ~\in~ Z . Multiplying the above by \scriptstyle (2 \pi)/(λ) we formulate the condition in terms of the wave vectors, \scriptstyle k and \scriptstyle k ^{\prime} :d ⋅ \left( k - 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,

\scriptstyle R , scattered waves interfere constructively when the above condition holds simultaneously for all values of \scriptstyle R which are Bravais lattice vectors, the condition then becomes:R ⋅ \left( k - k ^{\prime} \right) = 2 \pi m

An equivalent statement (see mathematical description of the reciprocal lattice) is to say that:

e ^{i\left( k - k ^{\prime} \right) ⋅ R} = 1

By comparing this equation with the definition of a reciprocal lattice vector, we see that constructive interference occurs if

\scriptstyle K ~ = ~ k - k ^{\prime} is a vector of the reciprocal lattice. We notice that \scriptstyle k and \scriptstyle k ^{\prime} have the same magnitude, we can restate the Von Laue formulation as requiring that the tip of incident wave vector, \scriptstyle k , must lie in the plane that is a perpendicular bisector of the reciprocal lattice vector, \scriptstyle K . This reciprocal space plane is the Bragg plane.

See also

• X-ray crystallography
• Reciprocal lattice
• Bravais lattice
• Powder diffraction
• Kikuchi line
• Brillouin zone

Notes and References

1. Book: Ashcroft . Neil W. . Mermin . David . Solid State Physics . Brooks Cole . 1 . January 2, 1976 . 96–100 . 0-03-083993-9 . registration .