Indexing cubic powder patterns

1
Indexing (cubic) powder patterns

• Learning Outcomes
• By the end of this section you should
• know the reflection conditions for the different
  Bravais lattices
• understand the reason for systematic absences
• be able to index a simple cubic powder pattern
  and identify the lattice type
• be able to outline the limitations of this
  technique!

2
First, some revision

• Key equations/concepts
• Miller indices
• Braggs Law 2dhkl sin? ?
• d-spacing equation for orthogonal crystals

3
Now, go further

• We can rewrite

and for cubic this simplifies
Now put it together with Bragg
Finally
4
How many lines?
Lowest angle means lowest (h2 k2 l2). hkl are
all integers, so lowest value is 1 In a cubic
material, the largest d-spacing that can be
observed is 100010001. For a primitive cell,
we count according to h2k2l2
Quick question why does 100010001 in cubic
systems?
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How many lines?
Note 7 and 15 impossible Note we start with the
largest d-spacing and work down Largest d-spacing
smallest 2? This is for PRIMITIVE only.
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Some consequences

• Note not all lines are present in every case so
  beware
• What are the limiting (h2 k2 l2) values of
  the last reflection?

or
sin2? has a limiting value of 1, so for this
limit
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Wavelength

• This is obviously wavelength dependent
• Hence in principle using a smaller wavelength
  will access higher hkl values

8
Indexing Powder Patterns

• Indexing a powder pattern means correctly
  assigning the Miller index (hkl) to the peak in
  the pattern.
• If we know the unit cell parameters, then it is
  easy to do this, even by hand.

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Indexing Powder Patterns

• The reverse process, i.e. finding the unit cell
  from the powder pattern, is not trivial.
• It could seem straightforward i.e. the first
  peak must be (100), etc., but there are other
  factors to consider
• Lets take an example

The unit cell of copper is 3.613 Å. What is the
Bragg angle for the (100) reflection with Cu K?
radiation (? 1.5418 Å)?
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Question

• ? 12.32o, so 2? 24.64o BUT.

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Systematic Absences

• Due to symmetry, certain reflections cancel each
  other out.
• These are non-random hence systematic
  absences
• For each Bravais lattice, there are thus rules
  for allowed reflections

P no restrictions (all allowed) I hkl
2n allowed F h,k,l all odd or all even
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Reflection Conditions

• So for each Bravais lattice

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General rule

• Characteristic of every cubic pattern is that all
  1/d2 values have a common factor.

The highest common factor is equivalent to 1/d2
when (hkl) (100) and hence 1/a2. The multiple
(m) of the hcf (h2 k2 l2)
We can see how this works with an example
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Indexing example
? 1.5418 Å
3 4 8 11 12 16
1 1 1 2 0 0 2 2 0 3 1 1 2 2 2 4 0 0
Highest common factor 0.02 So 0.02
1/a2 a 7.07Å
Lattice type? (h k l) all odd or all even ?
F-centred
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Try another
Highest common factor So a Å
Lattice type?

• In real life, the numbers are rarely so nice!

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and another
Highest common factor So a Å
Lattice type?

• Watch out! You may have to revise your hcf

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So if the numbers are nasty?

• Remember the expression we derived previously

So a plot of ?(h2 k2 l2) against sin ? has
slope 2a/?
Very quickly (with the aid of a computer!) we can
try the different options. (Example from above)
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Caveat Indexer

• Other symmetry elements can cause additional
  systematic absences in, e.g. (h00), (hk0)
  reflections.
• Thus even for cubic symmetry indexing is not a
  trivial task
• Have to beware of preferred orientation (see
  previous)
• Often a major task requiring trial and error
  computer packages
• Much easier with single crystal data but still
  needs computer power!