Space as we know it — the 3-dimensional environment in which we live and in which crystals exist, an environment called direct space in this context — can be represented in a different way as ‘reciprocal space’. Kohn-Sham solvers make use of it a lot.
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In direct space, the distance between A and B is one. The distance between B and C is one too. Two identical distances.
The coordinates of these three points in reciprocal space are: A=(1/1, 1/1), B=(1/1, 1/2), C=(1/1, 1/3). The distance between A and B in reciprocal space is 0.5, the distance between B and C in reciprocal space is 0.16667.
It is clearly not correct that points that are at identical distances in real space, will be at identical distances in reciprocal space. Points farther away from the origin in real space, will be squeezed closer together in reciprocal space.
Consider 3 points in direct space, with coordinates A=(1,1), B=(1,2) and C=(1,3). Calculate the distance between A and B, and between B and C. Now determine the coordinates of these three points in reciprocal space, and find the same two distances in reciprocal space. Based on your observations, assess the statement that “points that lie at similar distances from each other in real space, will lie at similar distances from each other in reciprocal space.”
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