Thread:  Help with Reciprocal Space 		 		View Single Post

There are different ways to think about reciprocal space.  For one, you can think of it as a purely geometrical thing.  If you have a lattice with vectors $$\mathbf{a}_i$$, then the reciprocal lattice is defined by  $$\mathbf{a}_i \cdot \mathbf{g}_j = 2\pi\delta_{ij}$$.  It's a purely geometrical thing, and the above equation defines a lattice with vectors $$\mathbf{g}_i$$.  In mathematics this would probably be called a dual lattice, but in physics we call it a reciprocal lattice because of the units (g has units of 1 / length).  Another way reciprocal space shows up is if you look at the solution to the Schroedinger equation for a periodic potential, Bloch's theorem says that the wavefunctions are a product of two periodic functions, and of the form  $$\psi_{kn}(r) = u_{kn}(r) e^{i k \cdot r}$$  where k is the so-called pseudomomentum vector, which serves as a quantum number.  k is restricted to the first Brillouin zone, where $$k = l_1 g_1 + l_2 g_2 + l_3 g_3$$ where the l's are restricted to the range [-0.5,0.5].  So k is restricted to wavelengths which are longer than a lattice vector.  The other function $$u_{kn}(r)$$ is periodic within the unit cell, so if you expand it in planewaves, all the planewaves would be like $$n_1 g_1 + n_2 g_2 + n_3 g_3$$ where the n's are integers, thus these wavelengths are all the lattice constants divided by integers.
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