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)