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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!

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First, some revision Key equations/concepts: Miller indices Braggs Law2d hkl sin = d-spacing equation for orthogonal crystals

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Now, go further We can rewrite: and for cubic this simplifies: Now put it together with Bragg: Finally

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How many lines? Lowest angle means lowest (h 2 + k 2 + l 2 ). 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 h 2 +k 2 +l 2 Quick question – why does 100=010=001 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 (h 2 + k 2 + l 2 ) values of the last reflection? or sin 2 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 = 1.54 Å = 1.22 Å

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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 = 12.32 o, so 2 = 24.64 o 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: h+k+l =2n allowed F: h,k,l all odd or all even

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Reflection Conditions So for each Bravais lattice: PRIMITIVEBODYFACE h 2 + k 2 + l 2 All possibleh+k+l=2n h,k,l all odd/even 11 0 0 21 1 0 31 1 1 42 0 0 52 1 0 62 1 1 82 2 0 92 2 1, 3 0 0 103 1 0 113 1 1 122 2 2 133 2 0 143 2 1 164 0 0

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General rule Characteristic of every cubic pattern is that all 1/d 2 values have a common factor. The highest common factor is equivalent to 1/d 2 when (hkl) = (100) and hence = 1/a 2. The multiple (m) of the hcf = (h 2 + k 2 + l 2 ) We can see how this works with an example

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Indexing example 2 d (Å)1/d 2 m h k l 21.764.080.06 25.203.530.08 35.882.500.16 42.382.130.22 44.352.040.24 51.571.770.32 = 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 Lattice type? (h k l) all odd or all even F-centred Highest common factor = 0.02 So 0.02 = 1/a 2 a = 7.07Å

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Try another… In real life, the numbers are rarely so nice! d (Å)1/d 2 m h k l 3.892 2.752 2.247 1.946 1.741 1.589 1.376 1.297 Lattice type? Highest common factor = So a = Å

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…and another Watch out! You may have to revise your hcf… d (Å)1/d 2 m h k l 3.953 2.795 2.282 1.976 1.768 1.614 1.494 1.398 Lattice type? Highest common factor = So a = Å

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So if the numbers are nasty? Remember the expression we derived previously: So a plot of (h 2 +k 2 + l 2 ) 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!

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Discrete Math Recurrence Relations 1.

Discrete Math Recurrence Relations 1.

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