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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: