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 • Bragg’s 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 100=010=001. For a primitive cell, we count according to h2+k2+l2 Quick question – why does 100=010=001 in cubic systems?

5. 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.

6. 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:

7.  = 1.54 Å  = 1.22 Å 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.

9. 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 • Let’s 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)?

10. Question  = 12.32o, so 2 = 24.64oBUT….

11. 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: h+k+l =2n allowed F: h,k,l all odd or all even

12. PRIMITIVE BODY FACE h2 + k2 + l2 All possible h+k+l=2n h,k,l all odd/even 1 1 0 0 2 1 1 0 1 1 0 3 1 1 1 1 1 1 4 2 0 0 2 0 0 2 0 0 5 2 1 0 6 2 1 1 2 1 1 8 2 2 0 2 2 0 2 2 0 9 2 2 1, 3 0 0 10 3 1 0 3 1 0 11 3 1 1 3 1 1 12 2 2 2 2 2 2 2 2 2 13 3 2 0 14 3 2 1 3 2 1 16 4 0 0 4 0 0 4 0 0 Reflection Conditions So for each Bravais lattice:

13. 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

14. 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

15. Try another… Highest common factor = So a = Å In real life, the numbers are rarely so “nice”! Lattice type?

16. …and another Highest common factor = So a = Å Watch out! You may have to revise your hcf… Lattice type?

17. 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)

18. 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!