what were the consideration in driving Laue equation
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In crystallography, the Laue equations relate the incoming waves to the outgoing waves in the process of diffraction by a crystal lattice. They are named after physicist Max von Laue (1879–1960). They reduce to Bragg's law.
In crystallography, the Laue equations relate the incoming waves to the outgoing waves in the process of diffraction by a crystal lattice. They are named after physicist Max von Laue (1879–1960). They reduce to Bragg's law.
The Laue equations
Let a , b , c {\displaystyle \mathbf {a} \,,\mathbf {b} \,,\mathbf {c} } {\displaystyle \mathbf {a} \,,\mathbf {b} \,,\mathbf {c} } be the primitive vectors of the crystal lattice L {\displaystyle L} L, whose atoms are located at the points x = p a + q b + r c {\displaystyle \mathbf {x} =p\,\mathbf {a} +q\,\mathbf {b} +r\,\mathbf {c} } {\displaystyle \mathbf {x} =p\,\mathbf {a} +q\,\mathbf {b} +r\,\mathbf {c} } that are integer linear combinations of the primitive vectors.
Let k i n {\displaystyle \mathbf {k} _{\mathrm {in} }} {\displaystyle \mathbf {k} _{\mathrm {in} }} be the wavevector of the incoming (incident) beam, and let k o u t {\displaystyle \mathbf {k} _{\mathrm {out} }} {\displaystyle \mathbf {k} _{\mathrm {out} }} be the wavevector of the outgoing (diffracted) beam. Then the vector k o u t − k i n = Δ k {\displaystyle \mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }=\mathbf {\Delta k} } {\displaystyle \mathbf {k} _{\mathrm {out} }-\mathbf {k} _{\mathrm {in} }=\mathbf {\Delta k} } is called the scattering vector (also called transferred wavevector) and measures the change between the two wavevectors.
The three conditions that the scattering vector Δ k {\displaystyle \mathbf {\Delta k} } {\displaystyle \mathbf {\Delta k} } must satisfy, called the Laue equations, are the following: the numbers h , k , l {\displaystyle h,k,l} {\displaystyle h,k,l} determined by the equations
a ⋅ Δ k = 2 π h {\displaystyle \mathbf {a} \cdot \mathbf {\Delta k} =2\pi h} {\displaystyle \mathbf {a} \cdot \mathbf {\Delta k} =2\pi h}
b ⋅ Δ k = 2 π k {\displaystyle \mathbf {b} \cdot \mathbf {\Delta k} =2\pi k} {\displaystyle \mathbf {b} \cdot \mathbf {\Delta k} =2\pi k}
c ⋅ Δ k = 2 π l {\displaystyle \mathbf {c} \cdot \mathbf {\Delta k} =2\pi l} {\displaystyle \mathbf {c} \cdot \mathbf {\Delta k} =2\pi l}
must be integer numbers. Each choice of the integers ( h , k , l ) {\displaystyle (h,k,l)} {\displaystyle (h,k,l)}, called Miller indices, determines a scattering vector Δ k {\displaystyle \mathbf {\Delta k} } {\displaystyle \mathbf {\Delta k} }. Hence there are infinitely many scattering vectors that satisfy the Laue equations. They form a lattice L ∗ {\displaystyle L^{*}} L^{*}, called the reciprocal lattice of the crystal lattice. This condition allows a single incident beam to be diffracted in infinitely many directions. However, the beams that correspond to high Miller indices are very weak and can't be observed. These equations are enough to find a basis of the reciprocal lattice, from which the crystal lattice can be determined. This is the principle of x-ray crystallography.