X-ray diffraction Meet in the LGRT lab Again, will hand in worksheet, not a formal lab report...
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Transcript of X-ray diffraction Meet in the LGRT lab Again, will hand in worksheet, not a formal lab report...
X-ray diffraction
• Meet in the LGRT lab• Again, will hand in worksheet, not a formal lab report
• Revision exercise – hand in by April 17th class.
Diffraction summarized
• The 6 lattice parameters (a,b,c,) of a crystal determine the position of x-ray diffraction peaks.
• The contents of the cell (atom types and positions) determine the relative intensity of the diffraction peaks.
• If a diffraction peak can be identified with a Miller Index, the unit cell on the phase can usually be determined.
Miller indices
• Choose origin• Pick 1st plane away from
origin in +a, +b direction• Find intercepts in fractional
coordinates (“none = ∞”).
• h = 1/(a-intercept)• k = 1/(b-intercept)• l = 1/(c-intercept)
• If no intercept, index = 0
• Plane equation:
ha + kb + lc = 1
Miller index, hkl
(2-10)
(110) (100)
(150)
• Same set of planes will be described if all 3 Miller indices are inverted (2-10) ≡ (-210)
• Minus sign → bar 10)2(0)1(2
Crystals facets correspond to Miller indices
• Haüy, 1784
• Crystals (like calcite) are made of miniscule identical subunits
b
a
(-120)
Woolfson
• Facets can be described by low-order Miller indices
(-120)
(-110)
(-100)(-1-10)
(-2-10)
(3-40)
(340)
Miller indices
• For a lattice of known dimensions, the Miller indices can be used to calculate the d-spacing between hkl planes (a,b,c = lattice parameters).
• This d-spacing will determine where powder diffraction peaks are observed
Miller index, hkl
(2-10)
(-110) (100)
(150)2
2
2
2
2
2
2
1
c
l
b
k
a
h
d
Origin at orange dot
Bragg’s law derivation(angle of incidence = angle of diffraction)
• x = d sin • extra distance = 2d sin = n• n = 2d sin
d
x
d
x x
a b
= 2d sin (for x-ray diffraction)
Coherent scattering from a row of atoms
Will only happen when emission from all atoms is simultaneously stimulated.(Solid lines represent spatial regions where phase = 0 deg).
a
Laue condition – vector description• Extra distance = a cos - a cos
• Extra distance = -(a·S0) + (a·S)
• Extra distance = (a·s)
a
S
S0
a
• Will have diffraction when: a cos - a cos = h (h = integer)• Will have diffraction when: a·s = h (h = integer)
s = (S – S0)/
v1
v2
cos = v1·v2
a
Laue condition – 2D• Must have coherent scattering from ALL ATOMS in the lattice, no just from one row.• If color indicates phase of radiation scattered from each lattice point when observed at a distant site P, we see
that scattering from rows is in phase while columns is out of phase, making net scattering from all 35 points incoherent and therefore NOT observable
S
S0
b
• Will have diffraction when: a·s = h (h = integer)• Will have diffraction when: b·s = k (k = integer)• Will have diffraction when: c·s = l (l = integer)
Every lattice point related by translational symmetry will scattering in phase when conditions are met
Single crystal diffraction
• Used to solve molecular structure– Co(MIMT)2(NO3)2 example
• Data in simple format – hkl labels + intensity + error
0 0 1 0.00 0.10 0 0 2 42.60 1.40 0 0 3 1.10 0.30 0 0 4 100.30 2.50 0 0 5 -0.30 0.50 0 0 6 822.30 16.70 0 0 7 -0.40 0.50 0 0 8 656.40 13.90 0 0 9 1.00 0.80 0 0 10 73.40 3.00 0 0 11 0.00 1.40 0 0 12 4.70 1.60 0 0 13 1.00 1.70 0 2 1 611.40 14.40 0 -2 -1 613.90 12.10 0 2 2 443.90 8.90 0 -2 -2 443.00 8.70 0 2 3 59.90 1.50 0 -2 -3 56.90 1.50 0 2 4 55.20 1.60 0 -2 -4 51.80 1.50
Direct vs. reciprocal lattice
direct or real
reciprocal
Lattice vectors a, b, c
Vector to a lattice point: d = ua + vb + wc
Lattice planes (hkl)
Lattice vectors a*, b*, c*
Vector to a reciprocal lattice point: d* = ha* + kb* + lc*
Each such vector is normal to the real space plane (hkl)
Length of each vector d* = 1/d-spacing (distance between hkl planes)
Distances between planes (d*)
direct or real
reciprocal
Vector to a reciprocal lattice point: d* = ha* + kb* + lc*
Length of each vector d* = 1/d-spacing (distance between hkl planes)
|d*| = (d*·d*)1/2
=(ha* + kb* + lc*) (ha* + kb* + lc*)
=(ha*)2 + (kb*)2 + (lc*)2 + 2(ha*) (kb*) + 2(ha*) (lc*) + 2(kb*) (lc*)
= h2a2* + k2b*2 + l2c*2 + 2klb*c*cos* + 2lhc*a*cos* + 2hka*cos*