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Introduction to Indexing2015 DENVER X-RAY CONFERENCE
BASIC TO ADVANCED XRD MATERIAL ANALYSIS WORKSHOP
Tom Blanton, ICDD
Co-Presenters
Slides from this presentation include material from Jim Kaduk, Polycrystallography and IIT, Suri Kabekkodu, ICDD,
and Earle Ryba PSU, presented at the ICDD XRD Clinics
Unit cell
a
b
c
ϒ
αβ
Smallest repeat unit in a crystalDefined by lattice constants
lengths a,b,cangles α,β,ϒ
Crystal systems7 CRYSTAL SYSTEMS, 14 BRAVAIS LATTICES
◦ Cubic – primitive, body, face centered◦ a=b=c α=β=ϒ=90°
◦ Tetragonal – primitive, body centered◦ a=b≠c α=β=ϒ=90°
◦ Orthorhombic – primitive, body, face, base centered◦ a ≠ b≠c α=β=ϒ=90°
◦ Hexagonal – primitive◦ a=b≠c α=β=90° ϒ=120°
◦ Rhombohedral (Trigonal) – Primitive◦ a=b=c α=β=ϒ≠ 90°
◦ Monoclinic – primitive, base centered◦ a ≠ b≠c α=ϒ=90° β≠ 90°
◦ Triclinic (Anorthic) - primitive◦ a ≠ b≠c α ≠ β ≠ ϒ≠ 90°
230 SPACE GROUPS
◦ Cubic – 36
◦ Tetragonal – 68
◦ Orthorhombic – 59
◦ Hexagonal – 27
◦ Rhombohedral (Trigonal) – 25
◦ Monoclinic – 13
◦ Triclinic (Anorthic) – 2
Note: sometimes a rhombohedral unit cell is transformed into a hexagonal cellar = 1/3(3ah
2 + ch2)1/2
sin(αr/2) = 3/[2(3 + (ch/ah)2)1/2]
Miller indicesMiller indices form a notation system in crystallography for planes in
unit cells
Represented as (hkl)
Procedure for naming lattice planes
1. Designate origin and define a, b, c axes◦ (plane cannot intersect origin)
2. Find intercepts on each of a, b, c axes
3. Take reciprocals of intercepts
4. The reciprocals are the h, k, l
Put inside parentheses w/ no commas
(121)
(010)
( 111)
Indexing a powder pattern• Assign Miller indices to the peaks of a powder diffraction pattern
• Determine the lattice constants of a unit cell
• One way to do this is to compare the measured interplanar spacings with those given in a particular entry in the Powder Diffraction File
• If, however, no data exist in the PDF for the material for which the pattern was measured, then a different, perhaps more prolonged procedure must be followed in order to index the pattern and subsequently determine its lattice parameters and symmetry.
Why index a pattern?• Strong evidence for phase purity
• A necessary prelude to determining the symmetry and crystal structure (a “gateway technology”)
• Permits phase identification by lattice matching techniques
• Relates peaks to each other and to the crystal lattice (even in the absence of a structure) –permits the analyst to derive more information from the pattern
hn
Indexed powder XRD pattern of NaCl
(111)
(200)
(220)
(311)(222)
(400) (331)(420)
NaCl unit cell with the (111) plane highlighted
The Structure Factor F(hkl)This is a summation (over all the atoms in the unit cell) of the product of . . .
(1) their atomic scattering factor(2) their phase relative to the distance between the Miller planes(3) the effect of their thermal motion
nnnn
lzkyhxief 2
Sum over all atoms in unit cell
Atomic scattering factor, (= z at 2θ of 0°), decreases with increasing 2θ
Phase angle of scattering of each atom relative to the hkl Miller plane that passes through the unit cell origin
Atomic displacement factor describes intensity loss at higher 2θ values due to motion of the atom – Bn is temperature dependent
22 /sin22
1
nnnn BlzkyhxiN
n
nhkl eefF
The Structure Factor F(hkl) - BCC metal
Phase angle of scattering of each atom relative to the hkl Miller plane that passes through the unit cell origin
In a BCC metal, there are two atoms that define the unit cell, all other atoms are related by symmetry (0,0,0)
(½, ½, ½)
Plugging into x,y,z you get:
When h+k+l = an odd number
lkhii efef nn 02
0 nn ffWhen h+k+l = an even number
nnn fff 2
nnnn
lzkyhxief 2
𝑒𝜋𝑖= 𝑒3𝜋𝑖= 𝑒5𝜋𝑖 … = cos𝜋 + 𝑖sin𝜋 = −1 𝑒2𝜋𝑖= 𝑒4𝜋𝑖= 𝑒6𝜋𝑖 … = cos2𝜋 + 𝑖sin2𝜋 = 1
Destructive interference
Constructive interference
BCC Tungsten, PDF
NaCl Sodium Chloride, PDF
CsCl – cubic, centering?
Cs on corners, Cl body center
Primitive Cubic!
Unit Cell and atomic coordinates
Knowing the lattice constants, the possible diffraction pattern d-spacings (2 position) can be calculated
Knowing the atomic coordinates, the intensity of the possible diffraction peaks can be calculated
Calculating d-spacings from unit cells12
2 2 2
2
2 2 2
d
h k l
ah k l
hkl
( )A
12
2 2
2
2
2
2 2 2
d
h k
a
l
ch k l
hkl
( )A C
1 4
32
2 2
2 2
2
2 2 2
d
h hk l
a
l
ch hk k l
hkl
( )
( )A C
12
2
2
2
2
2
2
2 2 2
d
h
a
k
b
l
ch k l
hkl
A B C
1 22
2
2 2
2
2 2 2
2
2
2 2 2
d
h
a
l
c
hl
ac
k
bh k l hl
hkl
sin sin
cos
sin
A B C E
cubic
tetragonal
orthorhombic
hexagonal
rhombohedral
monoclinic
1 1 2
1 2 32 2
2 2 2 2
3 2d a
h k l hk hl kl
hkl
( ) ( )(cos cos )
cos cos
Calculating d-spacings from unit cells
2
cos cos 1 cos 1 cos
1 cos cos cos cos 1
cos 1 cos 1 cos cos1
1 cos cos
cos 1 cos
cos cos 1
hkl
h h ha a a
h k lk k kb b ba b c
l l lc c c
d
h k l hk hl kl2 2 2A B C D E F
triclinic
Most indexing programs work internally in terms of Q
12
2 2 2
dh k l hk hl kl
hkl
A B C D E F
is the basis for most indexing strategies.
From Bragg’s law:
1 4102
2
2
4
dQ
hkl
hkl
hkl sin
( )
Indexing a cubic patternFirst remember that a cubic cell can be:
primitive (no restrictions on h,k,l for reflection to occur)
body centered (h+k+l must be an even number for reflection to be observed)
face centered (h,k,l must be all even or all odd integers for reflection to be observed)
When h2 + k2 + l2 is small d is large
12
2 2 2
2
2 2 2
d
h k l
ah k l
hkl
( )A
Indexing a cubic patternh2 + k2 + l2 Primitive I centering (h+k+l=2n) F centering (hkl all odd or all even)
1 (100) - -
2 (110) (110) -
3 (111) - (111)
4 (200) (200) (200)
5 (210) - -
6 (211) (211) -
7 - - -
8 (220) (220) (220)
9 (300),(221) - -
Indexing a cubic pattern
Observed d-spacings (Å)2.0341.7621.2461.0621.017
Indexing a cubic pattern
d(Å) (hkl) a(Å)
2.034 100 2.034
1.762 110 2.492
1.246 111 2.158
1.062 200 2.124
1.017 210 2.274
d(Å) (hkl) a(Å)
2.034 110 2.877
1.762 200 3.524
1.246 211 3.052
1.062 220 3.004
1.017 310 3.216
d(Å) (hkl) a(Å)
2.034 111 3.523
1.762 200 3.524
1.246 220 3.524
1.062 311 3.522
1.017 222 3.523
Primitive I centered F centered
1/d2 = (h2 + k2 + l2)/a2 or a = d (h2 + k2 + l2)1/2
Indexing a cubic pattern
(111)
(200)
(220) (311)
(222)
Indexing a non-cubic pattern
1. Test a known result2. Determine peak positions3. Run indexing program
Indexing programsTraditional:ITOTreor (N-Treor)DICVOL
Non-traditional:Global optimizationTopographicMonte Carlo searchGenetic algorithm
Along with some proprietary programs
62 peaks found
Try indexing on the first ~20 peaks
Look for unit cells that account for all peaks
Indexing result
Indexing resultThe test data were from PDF entry 00-011-0646, CuSO4 . 5H2O
Comparison of unit cells:
00-011-0646 (reduced cell) Indexinga 5.955Å 5.9621Åb 6.117Å 6.1130Åc 10.710Å 10.7168Å 77.44 77.31 82.37 82.31 72.69 72.64
Suggestions
• Test known materials• Carefully prepare the sample
• Internal standard• Carefully align the diffractometer• Examine peak results
• Missing or extra peaks• Consider profile fitting
• Synchrotron = peak resolution
• Good FOM• Match known compound with chemistry• Density / Z• Systematic absences correspond to a common
space group• Account for all diffraction peaks
Suggested references• Fundamentals of Powder Diffraction and Structural Characterization of
Materials, A. Pecharsky and P. Zavalij
• ICDD XRD Clinics: XRD 2 Advanced, Rietveld Refinement and Indexing• http://www.icdd.com/education/xrd.htm
Suggested referenceA. Altomare, C. Cuocci, A. Moliterni, and R. Rizzi, “Data Processing – Indexing”, Chapter 3.4 in International Tables for Crystallography Volume H: Powder Diffraction (2014)