Bragg Planes

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

Bragg planes are always perpendicular to S

s0

sS

-s0

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.

The length of S is 1/d

s0

sS

-s0

The length of S is 2sin times the lengths of s and s0, which is 1/. So |S| = 2sin/ = 1/d

1 7 0

2 1 0

• 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.

Using Miller indeces: S = ha*+kb*+lc*

d from S using Miller indeces

€

d =1

S=

1

(ha*)2 + (kb*)2 + (lc*)2=

1

(h /a)2 + (k /b)2 + (l /c)2

Axes a,b,c are all orthogonal.

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

If indeces hkl are doubled, Bragg distance d is halved.

2 1 0

• 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.

4 2 00 1 0 0 2 0

(2 3 3) Bragg planes (4 6 6) Bragg planes

3D Bragg planes

Phase-zero planes intersect the cell axes at fractional coordinates

(1/h,0,0), (0,1/k,0),(0,0,1/l)

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.

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)

a

b

In class exercise:

Calculating the density

• Given the structure factors F(hkl), find the point(s) of maximum e-density. F(hkl) = |F(hkl)|e i

• 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.

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°)

a

b

In class exercise:

Terms we have learned

• Reflection• Structure factor• Bragg planes• Scattering factor• Ewald sphere• Laue conditions• Reciprocal space• Miller indeces• Fourier transform