Quiz 7 XRD 2000 (Take Home)

Calculation of the intensity of a diffracted beam at a position 2q involves consideration of 1) the intensity scattered from a single electron, 2) a collection of such electrons in an atom, and 3) a collection of atoms in a crystal. In each step of this sequence the amplitude is calculated and summed and in the final step the amplitude is squared to obtain the diffracted intensity.

  1. -Does the intensity scattered from a single electron depend on the Bragg angle 2q?
    -How could this change with the state of polarization of the incident x-ray beam?

  2. -Sketch the typical shape of the atomic scattering factor (Amplitude) versus 2q.
    -Why isn't this curve flat (i.e. why isn't it a constant value in 2    q)?
    -Give the value at 2    q = 0

  3. The scattered intensity due to the unit cell involves a calculation of the phase difference between waves scattered from different atomic positions. This phase difference was calculated for the reciprocal lattice as Dd = (d1 + d2) where d1 = (2p/l)(-S0 . AB), d2 = (2p/l)(S . AB), where AB is the vector separating two atoms in the unit cell.
    -Draw a sketch showing S0/    l, S/l, and AB in real space as well as d1 and d2.

  4. -Draw a sketch showing the reciprocal space construction of S0/l, S/l, and the origin of reciprocal space, O, the center of the sphere of reflection, C and the diffraction angle 2q.
    -How are the positions of the two atoms at A and B represented in reciprocal space?

  5. Two waves, of amplitude A0, having a phase difference f, combine to have an amplitude given by A(x, t) = A1(x, t) + A2(x, t) = A0(x, t) {exp(-i2p s.r1) + exp(-i2p s.r2)}, where s = (S - S0)/l = 2 sinq/l = Hhkl, for a specified diffraction peak corresponding to the hkl plane. Hhkl is the reciprocal space lattice point for the hkl planes.

    If the origin of the vectors ri is the atom A, in problem c above, and if the atom A is at the origin of the real space lattice, u = 0; v = 0; w = 0:

    -show that the diffracted amplitude at the Bragg angle for the hkl plane from the two atoms A and B is given by:

    Ahkl(x, t) = A0(x, t) {1 + exp(-i2p (h uB + k vB + l wB)}, where uB, vB, and wB are the real space distances from atom A at [0 0 0] to atom B.

    -How is this amplituded used to calculate the diffracted intensity at 2qhkl from these two atoms?