Transcript Seeing Sub-Angstrom Spacings: X

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,a,b,g) 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
Miller index, hkl
• Choose origin
• Pick 1st plane away from
origin in +a, +b direction
• Find intercepts in fractional
coordinates (“none = ∞”).
(110)
(100)
• h = 1/(a-intercept)
• k = 1/(b-intercept)
• l = 1/(c-intercept)
(150)
• If no intercept, index = 0
(2-10)
• Plane equation:
ha + kb + lc = 1
•
Same set of planes will be described if all 3
Miller indices are inverted (2-10) ≡ (-210)
•
Minus sign → bar
(210)  (210)
Crystals facets correspond to Miller indices
b
• Haüy, 1784
a
• Crystals (like calcite) are made
of miniscule identical subunits
(-120)
(-100)
(-1-10)
(-110)
(-2-10)
(-120)
• Facets can be described by
low-order Miller indices
(3-40)
(340)
Woolfson
Miller indices
Origin at orange dot
Miller index, hkl
(-110)
(100)
• 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).
2
2
2
1
h
k
l
 2  2  2
2
d
a
b
c
• This d-spacing will determine
where powder diffraction
peaks are observed
(150)
(2-10)
Distance between planes (Miller indices)
Bragg’s law derivation
(angle of incidence = angle of diffraction)
q
q
d
q
q q
d
x
a
•
•
•
x
b
x = d sin q
extra distance = 2d sin q = nl
nl = 2d sin q
l = 2d sin q
(for x-ray diffraction)
x
Powder diffractometer
KI (CsCl-type) x-ray pattern
Crystals, diffraction, and Miller indices
(001)
(0-10)
(010)
(100)
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).
Laue condition – vector description
• Extra distance = a cos m - a cos n
• Extra distance
= -(a·S0) + (a·S)
• Extra distance = l(a·s)
s = (S – S0)/l
S0
v1
cosq = v1·v2
v2
a
m a
a
n
S
• Will have diffraction when: a cos m - a cos n = hl (h = integer)
• Will have diffraction when: a·s = h (h = integer)
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
S0
b
a
S
•
•
•
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
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
-2
2
-2
2
-2
2
-2
1 0.00
2 42.60
3 1.10
4 100.30
5 -0.30
6 822.30
7 -0.40
8 656.40
9 1.00
10 73.40
11 0.00
12 4.70
13 1.00
1 611.40
-1 613.90
2 443.90
-2 443.00
3 59.90
-3 56.90
4 55.20
-4 51.80
0.10
1.40
0.30
2.50
0.50
16.70
0.50
13.90
0.80
3.00
1.40
1.60
1.70
14.40
12.10
8.90
8.70
1.50
1.50
1.60
1.50
h=2
h=1
h=0
h=-1
h=2
l=10
l=9
l=8
l=7
l=6
3D view of reciprocal lattice
(Lines connecting dots are unecessary)
Massa
Relation between direct and reciprocal cells
Stout and Jensen
Direct vs. reciprocal lattice
direct or real
Lattice vectors a, b, c
Vector to a lattice point:
= ua + vb + wc
d
Lattice planes (hkl)
reciprocal
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)
Hexagonal lattice
a
b
direct or real
reciprocal
b*
g changes from 120 to 60 deg
a*
Distances between planes (d*)
direct or real
Vector to a reciprocal lattice point:
= 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*)
reciprocal
=(ha*)2 + (kb*)2 + (lc*)2 + 2(ha*) (kb*) +
2(ha*) (lc*) + 2(kb*) (lc*)
= h2a2* + k2b*2 + l2c*2 + 2klb*c*cosa* +
2lhc*a*cosb* + 2hka*cosg*
d*
Density of CaRuO3
• Density = mass / volume
• Mass of unit cell =
(# Ca)(mCa) + (# Ru)(mRu) + (# O)(mO)
• Volume of unit cell =
abc = (3.7950 Å)3
• mamuNA = mgrams
nM

/V
NA
nM

N AV
• M = molar mass
(g/mol)
• n = # f.u. per cell
• NA = Avagadro’s #
(1/mol)
• V = cell volume