Now, any periodic potential V ( "'r "') which shares the same periodicity as the direct lattice can be expanded out as a Fourier series whose only non-vanishing components are those associated with the reciprocal lattice vectors.
32.
As shown in Figure 2, " k "-vectors outside the first Brillouin zone are physically equivalent to vectors inside it and can be mathematically transformed into each other by the addition of a reciprocal lattice vector " G ".
33.
Where "'G "'is a reciprocal lattice vector that ensures that "'q "'falls in the first BZ and the angles ? i and ? s are measured with respect to the normal to the surface.
34.
For example, the reciprocal lattice vector ( hk ! ) as suggested above can be written in terms of reciprocal lattice vectors as h \ mathbf { b _ 1 } + k \ mathbf { b _ 2 } + \ ell \ mathbf { b _ 3 }.
35.
For example, the reciprocal lattice vector ( hk ! ) as suggested above can be written in terms of reciprocal lattice vectors as h \ mathbf { b _ 1 } + k \ mathbf { b _ 2 } + \ ell \ mathbf { b _ 3 }.
36.
Every crystal is a periodic structure which can be characterized by a Bravais lattice, and for each Bravais lattice we can determine the reciprocal lattice, which encapsulates the periodicity in a set of three reciprocal lattice vectors ( "'b 1, b 2, b 3 "').
37.
By comparing this equation with the definition of a reciprocal lattice vector, we see that constructive interference occurs if \ scriptstyle \ mathbf { K } ~ = ~ \ mathbf { k } \,-\, \ mathbf { k ^ \ prime } is a vector of the reciprocal lattice.
38.
Considering only ( hk ! ) planes intersecting one or more lattice points ( the " lattice planes " ), the perpendicular distance " d " between adjacent lattice planes is related to the ( shortest ) reciprocal lattice vector orthogonal to the planes by the formula : d = 2 \ pi / | \ mathbf { g } _ { h k \ ell } |.
39.
We notice that \ scriptstyle \ mathbf { k } and \ scriptstyle \ mathbf { k ^ \ prime } have the same magnitude, we can restate the Von Laue formulation as requiring that the tip of incident wave vector, \ scriptstyle \ mathbf { k }, must lie in the plane that is a perpendicular bisector of the reciprocal lattice vector, \ scriptstyle \ mathbf { K }.
40.
They are written ( hk ! ), and denote the family of planes orthogonal to h \ mathbf { b _ 1 } + k \ mathbf { b _ 2 } + \ ell \ mathbf { b _ 3 }, where \ mathbf { b _ i } are the basis of the reciprocal lattice vectors . ( Note that the plane is not always orthogonal to the linear combination of direct lattice vectors h \ mathbf { a _ 1 } + k \ mathbf { a _ 2 } + \ ell \ mathbf { a _ 3 } because the reciprocal lattice vectors need not be mutually orthogonal . ) By convention, negative integers are written with a bar, as in for " 3.
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