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Indexing (cubic) powder patterns by 816Igq4

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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!
             First, some revision

Key equations/concepts:
 Miller indices
 Bragg’s Law      2dhkl sin = 
 d-spacing equation for orthogonal crystals

                1     h2 k 2 l 2 
                     2  2  2
               d hkl  a
                 2
                         b   c  
                     Now, go further
                                           1
We can rewrite: d hkl 
                                  h2 k 2 l 2 
                                  2  2  2
                                 a
                                     b   c  
                                                                    a
and for cubic this simplifies: d hkl 
                                                    h      2
                                                                 k2  l2   
                                                                a
Now put it together with Bragg: 2                                           sin   
                                                   h   2
                                                             k2  l2   
 Finally    h   2
                     k l
                       2     2
                                    2a
                                      
                                           sin 
                                             h   2
                                                              
                                                       k2  l2 
                                                                    2a
                                                                         sin 
                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?
                  How many lines?

    hkl    h2 + k 2 + l 2      hkl         h2 + k2 + l2
    100         1           2 2 1, 3 0 0        9
    110         2              310             10
    111         3              311             11
    200         4              222             12
    210         5              320             13
    211         6              321             14
    220         8              400             16


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.
                  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?


  h   2
                   
            k2  l2 
                         2a
                             
                                 sin    or                  4a 2
                                              h 2  k 2  l 2  2 sin 2 
                                                                

sin2 has a limiting value of 1, so for this limit:

                                        
                                         4a 2
                         h 2  k 2  l2  2
                                          
                    Wavelength
                             SPINEL: LAMBDA = 1 .5 4




                 4a 2
                             Lamb d a: 1 .5 4 1 78 Mag n if: 1 .0 FW HM : 0. 20 0
                             Space g rp: F d -3 m:2 Direct cell: 8 . 08 0 0 8 .0 80 0     8 .0 8 0 0 9 0 .0 0   9 0. 00      9 0 .0 0



   h 2  k 2  l2  2




                                                                                               113
                                                                                                                       = 1.54 Å




                                                                                                                       004
This is obviously




                                              111




                                                                                                                                                                            115
                                                                                   022
wavelength dependent




                                                                                                                                                                224



                                                                                                                                                                            333
                                                                                                     222




                                                                                                                                        133
Hence in principle using a   10                     20                             30                      40                                       50                            60



smaller wavelength will
                             SPINEL: LAMBDA = 1 .2 2
                             Lamb d a: 1 .2 2 0 00 Mag n if: 1 .0 FW HM : 0. 20 0
                             Space g rp: F d -3 m:2 Direct cell: 8 . 08 0 0 8 .0 80 0     8 .0 8 0 0 9 0 .0 0   9 0. 00      9 0 .0 0



access higher hkl values




                                                                       113
                                                                                                                 = 1.22 Å




                                                                                                                                              044
                                                                                         004




                                                                                                                             115
                                    111




                                                            022




                                                                                                                 224


                                                                                                                             333




                                                                                                                                                                            335
                                                                                                                                                          135



                                                                                                                                                                      026
                                                                             222




                                                                                                     133




                                                                                                                                                         244
                             10                     20                             30                      40                                       50                            60
          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.
                         high QUARTZ
                         Lambda: 1.54178 Magnif: 1.0 FWHM: 0.200
                         Space grp: P 62 2 2 Direct cell: 5.0800 5.0800          5.5807 90.00 90.00 120.00




                                              011
  1     h2 k 2 l 2 
       2  2  2
 d hkl  a
   2
           b   c  
                                   010




                                                                                  0 0 31 1 2


                                                                                                022 3



                                                                                                              121
                                                                           021
                                                         110



                                                                     020




                                                                                                 01
                                                               012




                                                                                                        120
                                                               111




                        10               20         30                40                       50                   60
         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)?
                              Question
          
  sin 
       1
                           = 12.32o, so 2 = 24.64o                            BUT….
         2dhkl 
            Copper, [W. L. Bragg (Philosophical Magazine, Serie 6 (1914) 28, 255-360]
            Lambda: 1.54180 M agnif: 1.0 FWHM: 0.200
            Space grp: F m -3 m Direct cell: 3.6130 3.6130 3.6130 90.00 90.00 90.00




            10      20       30       40       50       60       70       80
             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
                    Reflection Conditions
So for each Bravais lattice:
                       PRIMITIVE       BODY           FACE
     h2 + k2 + l2      All possible   h+k+l=2n   h,k,l all odd/even
          1               100
          2               110          110
          3               111                         111
          4               200          200            200
          5               210
          6               211          211
          8               220          220            220
          9            2 2 1, 3 0 0
         10               310          310
         11               311                         311
         12               222          222            222
         13               320
         14               321          321
         16               400          400            400
                    General rule

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

                   1 h2  k 2  l 2
                     2
                       
                   d      a2
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
                 Indexing example

                   2
2      d (Å)   1/d    m    hkl
                                    = 1.5418 Å

21.76   4.08    0.06   3    111   Highest common
                                  factor = 0.02
25.20   3.53    0.08   4    200

                                  So     0.02 = 1/a2
35.88   2.50    0.16   8    220


                       11   311
                                         a = 7.07Å
42.38   2.13    0.22

                       12   222
44.35   2.04    0.24              Lattice type?
51.57   1.77    0.32   16   400   (h k l) all odd or all even
                                   F-centred
               Try another…
           2     m                 Highest common
d (Å)   1/d              hkl
                                   factor =
3.892

2.752                              So     a=       Å
2.247

1.946

1.741
                                   Lattice type?
1.589

1.376

1.297


 In real life, the numbers are rarely so “nice”!
                …and another
           2   m                 Highest common
d (Å)   1/d            hkl
                                 factor =
3.953

2.795                            So    a=        Å
2.282

1.976

1.768                            Lattice type?
1.614

1.494

1.398

  Watch out! You may have to revise your hcf…
      So if the numbers are “nasty”?

Remember the expression we derived previously:

                  h   2
                           k l
                             2                 2
                                                         2a
                                                           
                                                                sin 

So a plot of (h2 +k2 + l2) against sin  has slope 2a/
                                         0.5
                                                   y = 0.1089x
Very quickly (with the                   0.4
                                                       2
                                                     R =1

aid of a computer!) we
                             sin theta


                                         0.3
can try the different
                                         0.2
options.                                                                            Primitive

                                         0.1                                        Body Centred
(Example from above)                                                                Face-centred
                                          0
                                               0           1         2        3     Linear (Face- 5
                                                                                        4
                                                                                    centred)
                                                                   sqrt(h2+k2+l2)
                  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!                  Ba2 La0.667 V2 O8
                                   Lambda: 1.54178 Magnif: 1.0 FWHM: 0.200
                                   Space grp: RB3M Direct cell: 5.7527 5.7527 21.0473 90.0 0 90.00 120 .00




                                                              012




                                                                                                         202
                                                                                           113
                                                                                     110
                                                003




                                                                                                               009
                                                                         006
                                                                         104
                                                        101




                                                                                                               018
                                                                                                 107
                                                                               015




                                                                                                       021
                                           10                       20               30                              40

								
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