A powder diffraction pattern is collected on the hydrochloride salt of a well-known organic substance using Cu Ka₁ radiation (l = 1.54056 Å) and a well-aligned Bragg-Brentano diffractometer. The data are shown below:

1. Using the data supplied in the tables and an indexing program (e.g. ), determine a unit cell and crystal system for this material. In your answer you should list the 2q values of the peaks used, giving a brief reason for your choice, and also the values of the cell parameters.

   [5 Marks]

The positions of 40 reflections were provided. Indexing programs require 20 reflections to work effectively and so 20 'good' reflections were selected from the 40. The strategy I adopted was to select strong reflections - irrespective of whether they had low or high angle 'tails'. The rational for this is that 'tails' would have little effect on the central position of a strong peak, although the presence of tails indicates some underlying problem with the data. I ignored low intensity reflections (e.g. 7.50, 11.77 & 24.54 2q) and relatively weak reflections that were close to more intense reflections (e.g. 15.05 2q) - in the latter case the background of the smaller peak is distorted by the larger peak. I also ignored impurity reflections (e.g. 26.76 2q) and reflections that were so close together that they could not be resolved (e.g. 30.27 & 30.39 2q) - i.e. the dip between the reflections did not drop to the general background count.

Number	2q
1	8.07
2	10.25
3	14.82
4	16.44
5	17.15
6	18.07
7	18.81
8	19.36
9	20.63
10	21.64
11	23.10
12	23.45
13	23.77
14	24.27
15	25.98
16	26.57
17	27.72
18	28.27
19	29.81
20	31.11

The above 2q values were typed into the input page of the zone indexing program. Upon submission of the data, the ten most probable solutions were displayed. I looked for a solution in which at least 19 of the 20 lines were indexed and with a Figure of Merit >4. The first solution presented looked good - all 20 lines had been indexed and the Figure of Merit was 7.4.The unit cell Real-Space parameters were as follows.

Parameter	Value
a	23.696 Å
b	6.193 Å
c	22.013 Å
a	90°
b	97.92°
g	90°

a¹ b¹ c and a = g = 90° and b ¹ 90°
Thus, this is the Monoclinic Crystal System.

Comment: Do not be too hasty in rejecting low-angle weak peaks - higher angle ones yes, but the low-angle ones can often be key reflections for indexing purposes. In addition, when the figure of merit is low (e.g. 7.4), you should be more critical in accepting the solution. In particular, the value of c is out by a factor of 2. It is this that causes the low F20 value. You should have checked to see whether the density of reflections looked reasonable.
3 Marks

2. Using an hkl generating program (e.g. ), or otherwise, determine the Bravais lattice type. Give a logical explanation for your choice.

   [5 Marks]

The hkl generating program produced a list of hkl values for the chosen cell parameters. A study of the data showed that for many of the observed peaks, there were several calculated reflections very near to them. The information supplied says that the data was obtained from a well aligned diffractometer. I, therefore, only considered computed reflections that were within 0.06° of the observed reflection.The computed values can be compared with the observed values in the table below.

Number	2q	Computed 1	Computed 2	Computed 3	Computed 4
1	8.07	8.04	8.10
2	10.25	10.28
3	14.82	14.78	14.81	14.86
4	16.44	16.42	16.45
5	17.15	17.13	17.19
6	18.07	18.06	18.12
7	18.81	18.77	18.81	18.85	18.86
8	19.36	19.35
9	20.63	20.63	20.64
10	21.64	21.64	21.64
11	23.10	23.06	23.10	23.15
12	23.45	23.45
13	23.77	23.74	23.74	23.77
14	24.27	24.24	24.27	24.29
15	25.98	26.00
16	26.57	26.56	26.64
17	27.72	27.70	27.73	27.75
18	28.81	29.80	29.82	29.87
19	30.06	30.06	30.07
20	31.11	31.07	31.10

The red values are unambiquous since there is only one computed reflection within 0.06° of the observed peak. The data suggests a 0.03°error for the first two reflections (these suffer from axial divergence) and approx. 0.01°for the rest. Consequently, these error limits were applied to the values between the red values to obtain a better set of computed values (Blue). This was a fairly difficult process since I tried extrapolate a sensible 2q error from the unambiquous values to the other values. I tried to air on the side of caution. If in doubt, I didn't assign.The computed values for some of the reflections still remain ambiguous and as such all the computed lines for those reflections are left uncoloured (black).

Number	2q	Computed	(hkl)
1	8.07	8.10	0 0 2
2	10.25	10.28	-2 0 2
5	17.15	17.13	1 1 2
6	18.12	18.11	4 0 2
8	19.36	19.35	-3 1 2
11	23.10	23.10	-4 1 3
12	23.45	23.45	-3 1 4
13	23.77	23.77	5 1 0
14	24.27	24.27	-5 1 2
15	25.98	26.00	5 1 2

Comment: You generally (with one exception) have l = 2n here suggesting a doubled c axis.

For the monoclinic crystal system there are two lattice types: primitive (P) and C centred (C). There are no reflection conditions for P centring and for C centring the reflection condition is hkl: h + k = 2n
i.e.h + k =even. In addition, reflections where all three hkl indices were even were ingnored since they provide no symmetry information.

Reflection numbers 5, 13 and 15 satisfy the 'C centred' reflection condition. Thus the Bravais lattice type is mC.

Comment: You need to consider other centred cells for monoclinic since C is just the standard setting. See model answer.

Major comment: Reflection 11 in your above list does NOT satisfy the C-centred condition, so you should deduced a primitive lattice based on your indexing as is.
3 Marks

Minor comment: Why jump to a new page at this point? I cannot see any advantage in a page break just here.

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