Braggs Law the Reciprocal Lattice and the Ewald

Bragg’s Law, the Reciprocal Lattice and the Ewald Sphere Construction

Bragg’s Law waves still in phase: constructive interference two neighbouring layers of the crystal lattice

Construct a 2 D “real space” lattice using these parameters b 6Å 110° 8Å a origin “real-space” lattice points Scale: 1 cm = 1 Å

Populate the reciprocal lattice with -2 ≤ h ≤ 2 and -2 ≤ k ≤ 2 24 points: -2 -2 -2 -1 -2 0 -2 1 -2 2 -1 -1 -1 0 -1 1 -1 2 0 -1 - 01 02 1 -1 10 11 12 2 -1 20 21 22

Construct a 2 D Reciprocal Lattice hk = 11 d 11 = 3. 91Å

Formula for calculating dhk Use the Excel spreadsheet provided

Draw the R. L. point corresponding to (1 1) hk = 11 11 * d 11 = 3. 91Å in reciprocal space, use: 1 cm = 0. 04 Å-1. . . so plot the R. L. point 6. 39 cm (i. e. 1/d 25 cm) from the origin

Draw the R. L. point corresponding to (-1 -1) hk = -1 -1 11 * d 11 = 3. 91Å -1 -1 *

Populate the R. L. for -2 ≤ 12 h ≤ 22 2 and -2 ≤ k ≤ 2 * 02 * -12 -22 * * * -21 * -11 b * 01 * 10 -20 * -2 -1 * -2 -2 * * -10 * -1 -1 * -1 -2 * 0 -1 * 0 -2 * 1 -1 * 1 -2 * 21 * 20 * a 2 -1 * 2 -2 *

22 own unit cell The R. L. can be defined 12 by its * 02 * -12 -22 * * * -21 * -11 b * 01 * 10 -20 * -2 -1 * -2 -2 * * -10 * -1 -1 * -1 -2 * 0 -1 * 0 -2 * 1 -1 * 1 -2 * 21 * 20 * a 2 -1 * 2 -2 *

The R. L. can be defined by its own unit cell b 01 b** * 0 11 * 10 *a* a

Take-home Messages • The reciprocal lattice is a regular lattice too • hk in opposite direction to –h – k • Each R. L. point hk is drawn on a line to the (hk) real-space line (plane in 3 D), and at a distance of 1/dhk from the origin • There is a distinct geometrical relationship between the R. L. and the real-space lattice • The scale at which the R. L. is drawn is arbitrary

$Diffraction Geometry and the Ewald Sphere$

Diffraction Geometry and the Ewald Sphere

$The Ewald sphere construction is key to understanding diffraction geometry b d e m$

The Ewald sphere construction is key to understanding diffraction geometry b d e m a e t c rf a dif reciprocal lattice incident X-ray beam crystal Paul Ewald, 1888 -1985