1. 2   y k y  rows ky =ky = Diffraction kyky yy Reciprocal space k x, k y Real space x, y

2. Reciprocal space is seen in diffraction patterns. Reciprocal space is momentum space = k-space. The momentum p = ħ k is a key quantum number of electrons, phonons, … in solids. |k| = 2  / Everything is backwards in reciprocal space: Large distances  x in real space transform into small k-vectors k x = 2  /  x in reciprocal space. Waves are diffracted perpendicular to the diffracting lines or planes. This generates a 90 0 rotation of diffraction patterns. Reciprocal space versus real space

3. Rosalind Franklin’s x-ray diffraction pattern of DNA, which led to the double-helix model (Linus Pauling’s copy)

4. Single helixDouble helix Test patterns for simulating diffraction from DNA

5. Explaining the diffraction pattern of a helix d 2/d2/d Diffraction pattern of a green laser pointer Vertical X Horizontal line grating Vertical dots Horizontal X Side view of a helix (screw): Two tilted gratings

6. X-ray diffraction pattern of DNA b p  Diffraction pattern (negative) The DNA double helix p = period of one turn b = base pair spacing  = slope of the helix  2 p2 p 2  b

7. X-ray diffraction image of the protein myoglobin Protein crystallography has become essential for biochemistry, because the structure of a protein determines its function. This image contains about 3000 diffraction spots. All that information is needed to determine the positions of thousands of atoms in myoglobin.

8. Low Energy Electron Diffraction (LEED) at surfaces 1D chain structure2D planar structure (7x7) D k = 2  /D K = 2  /d d Real spaceReciprocal space

9. Diffraction conditions k 0 k

10. Connect the Bragg condition with momentum conservation G k 0 k k0k0 k = k 0 + G

11. Energy and momentum conservation in diffraction Origin G-vector Energy conservation: Ewald sphere |k| = |k 0 | Momentum conservation: Vector triangle k = k 0 + G G arrow starts at the origin (000)