Question

On Wikipedia, look up the mineral ‘Albite’ and retrieve its crystal structure unit cell parameters a, b, c, α, β, γ. For the definition of the three angles, see https://en.wikipedia.org/wiki/Lattice_ constant. (a) Calculate the crystal lattice vectors a1 , a2 , a3 in units of à using the following conventions: a1 is along the x direction, a2 is in the xy plane with a positive y component, and the z component of a3 is positive. Some help: The square of the length of a vector is equal to its scalar product with itself, and the scalar product of two different vectors is equal to the product of their lengths times the cosine of the angle between them. Starting with a1 = (a, 0, 0), you can use these scalar products to generate and solve equations for first the two nonzero components of the vector a2 , and then do the same to calculate the three components of a3 . (b) Calculate the lattice vectors b1 , b2 , b3 of the reciprocal lattice and state what units they are in. (c) Calculate the volumes VR and VK of the real and reciprocal unit cells, respectively, with correct units. Use VR = a1 · (a2 × a3 ), VK = b1 · (b2 × b3 ) = (2π) 3 /VR . You can check the result for VR against the volume formula from https://en.wikipedia.org/wiki/Lattice_constant to see if you made a mistake in the calculation of the set a1 , a2 , a3 . (d) For the real and reciprocal lattice vectors of parts (a) and (b), verify the equation bi · aj = 2πδi ,j Here, i and j means one of the 3 vectors, respectively, · is the scalar product, and δ is the Kronecker-delta.