1. Bragg Planes How to do a Fourier transform on paper with no calculations at all.

2. Bragg planes are always perpendicular to S s0 -s0 S s   Since s0 and s are the same length and have the same angle to the reflection plane, S = (s-s0)/ is normal to the plane.

3. The length of S is 1/d s0 -s0  S s    The length of S is 2sin times the lengths of s and s0, which is 1/. So |S| = 2sin/ = 1/d

4. Using Miller indeces: S = ha*+kb*+lc* • Max von Laue says: The vectors S that have amplitude > 0 are the ones where the Bragg planes all line up with the unit cell origins. • This must be true for all unit cells in the crystal (ta+ub+vc) to scatter with the same phase. 2 1 0 1 7 0

5. d from S using Miller indeces Axes a,b,c are all orthogonal.

6. 2 1 0 Where the first Bragg plane cuts the axes • The n=1 Bragg plane (normal to S at distance d) cuts the unit cell axes at 1/h 1/k 1/l

7. If indeces hkl are doubled, Bragg distance d is halved. • All unit cell origins have phase zero. But not all phase-zero Bragg planes must go through a unit cell origin. For example, the n=odd Bragg planes for the 0 2 0 reflection does not touch a single unit cell origin. 0 1 0 0 2 0 2 1 0 4 2 0

8. 3D Bragg planes (4 6 6) Bragg planes (2 3 3) Bragg planes Phase-zero planes intersect the cell axes at fractional coordinates(1/h,0,0), (0,1/k,0),(0,0,1/l)

9. Calculating the structure factors • Draw a plane that intersects the unit cell axes at 1/h, 1/k, and 1/l (careful to consider the sign of h,k,l) • Measure the phase of each atom as its distance from the nearest Bragg plane, divided by d and multiplied by 360°. • Draw the scattering factor for that atom, and sum the scattering factors to get the structure factor.

10. In class exercise: Calculate structure factors: F( 1 1 0) F(-1 1 0) F(-2 1 0) For a unit cell with two atoms: carbon (amplitude 6) @ (0.5, 0.2, 0.0) oxygen (amplitude 8) @ (0.3, 0.4, 0.0) b a

11. Calculating the density • Given the structure factors F(hkl), find the point(s) of maximum e-density. F(hkl) = |F(hkl)|ei • Draw Bragg planes with phase =  (Measure phase in the direction (h,k,l)) • Erase Bragg planes with phase = +180° • After drawing and erasing all F’s, the darkest areas are the locations of the atoms.

12. In class exercise: Find the maximum density point giventhe following structure factors: F( 1 1 0) = 1 ei(108°) F(0 1 0) = 1 ei(180°) F(1 1 0) = 1 ei(-60°) F(-1 1 0) = 1 ei(60°) F(-2 1 0) = 1 ei(-10°) b a

13. Terms we have learned • Reflection • Structure factor • Bragg planes • Scattering factor • Ewald sphere • Laue conditions • Reciprocal space • Miller indeces • Fourier transform