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GEOL0026 2024/25: Dummy Examination
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Answer ALL questions. For Questions 1 – 7 enter your answers on the Answer Sheet
provided. For Question 8 use a new Answer Book. You should also use a separate
answer book for your working out. Credit will be given for incomplete answers, so you
must also hand in your working for each question, clearly labelled with the Question
Number.
Questions 1 – 7 carry equal marks and together comprise 70% of the total mark;
Question 8 comprises 30% of the total mark. Where a question consists of more than
one part (a, b, c etc.) all parts carry equal weight unless stated otherwise.
You may refer to your annotated copy of the GEOL0026 Lecture Booklet and you may
use a non-standard electronic calculator.
1. Using the axes as shown for each of the diagrams, determine the Miller indices of
the 5 sets of planes shown in the diagrams below
(a) (b)
(c) (d)
(e)
GEOL0026 2024/25: Dummy Examination
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2. The image on the next page (labelled Figure 1) shows the space group diagram for
space group Cmc21. The diagram is drawn in the usual orientation, with the a-axis
running down the page, the b-axis running across the page and the c-axis coming out
of the plane of the paper. With reference to the information contained in this
diagram, answer the following questions:
(a) To which crystal system does this space group belong
(b) What is the point group corresponding to this space group
(c) How many different types of special positions exist in Cmc21
(d) Does this space group contain a centre of symmetry
Given an atom at x,y,z,
(e) what are the coordinates of the symmetry-equivalent atom produced by the
mirror plane passing through x = 0
(f) what are the coordinates of the symmetry-equivalent atom produced by the c glide passing through y = 0
(g) what are the coordinates of the symmetry-equivalent atom produced by the 2-
fold screw axis (21 axis) passing through x = , y =
(h) what are the coordinates of the symmetry-equivalent atom produced by the n glide perpendicular to the b-axis and passing through y =
(i) Is it possible for a mineral crystallising in space group Cmc21 to contain only 3
atoms of the same kind in the unit cell
(j) In the diagram the length of the b axis is greater than that of the a axis. Can a
mineral crystallising in space group Cmc21 have a unit cell in which all three axes of
the unit cell are of equal length
GEOL0026 2024/25: Dummy Examination
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Figure 1: Space group diagram for Question 2
3. The hypothetical mineral ABthree-ite, contains two kinds of atom: “A” and “B” and
has the chemical formula AB3. It crystallises with the cubic space group Pm3m, with
a = 3.84 . A diagram of this space group (Figure 2) is shown on the next page.
(a) Given that the unit cell contains one formula unit of AB3, calculate the density of
ABthree-ite.
(Relative atomic masses of A and B are 127.6 and 15.999 respectively; Avogadro’s
number = 6.023 x 10 23)
(b) In AB3-ite the A atoms occupy the 1a sites. With the aid of the space group diagram
provided below, list the fractional coordinates of all of the sets of positions in which
the B atoms might be placed in the possible crystal structures of ABthree-ite.
(c) Using the sets of coordinates that you listed in (b) above, draw sketch projections of
the resulting crystal structures, viewed down the c-axis of the cubic unit cell (you do
not need to include these in your answer). By inspection of your diagrams,
determine how many fundamentally different crystal structures of ABthree-ite have
been formed. To how many B atoms is each A atom bonded in each of these
structures
GEOL0026 2024/25: Dummy Examination
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(d) Calculate the length of the shortest A-B bonds in each of the crystal structures that
you found in part (c) above.
(e) The atomic radii of A and B are 0.56 and 1.40 respectively. By comparing the
sum of these radii with the values you obtained in (d) above, determine which of the
possible structures of ABthree-ite that you found in parts (b) and (c) is correct.
Figure 2: Space group diagram for Question 3
GEOL0026 2024/25: Dummy Examination
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4. A cubic halite crystal, with a = 5.64 is set to rotate about its c-axis, i.e. about
[001], in a rotation camera with radius 30 mm. Its diffraction pattern is then
measured using CuKα radiation (λ = 1.542 ). On the resulting X-ray diffraction
photograph
(a) What is the vertical distance (in mm) between the zero layer line and the layer
line with l = 3
(b) Bragg reflections with Miller indices 220 and 420 are both found on the zero
layer line of this photograph. Which of these reflections lies furthest from the centre
of the diffraction pattern
The crystal is remounted so that it is rotating about [110].
(c) What is the vertical distance (in mm) between the zero layer line and the 4th layer
line
(d) With the crystal rotating about [110], on which layer lines do the following
Bragg reflections now lie: 1 0, 111, 220
5. The first six diffraction lines in the powder pattern of a cubic mineral, measured
with Cu Kα radiation (λ = 1.542 ), were found to have the following 2θ values:
24.5o
, 28.3o
, 40.5o
, 47.9o
, 50.2o
, 58.6o
.
(a) Calculate the d-spacings (in ) of these six lines.
(b) By constructing a table of 1/d
2
values and their ratios, index the powder patterns
and determine the lattice type.
(c) Use the d-spacing of the line with the largest Bragg angle to determine the value
of the cubic cell parameter, a.
(d) From your cell parameter, determine whether the mineral is periclase (a = 4.22
), halite (a = 5.64 ), magnetite (a = 8.40 ) or sylvite (a = 6.29 ).
6. The diagram to the right shows the
relationship between a primitive unit cell
and the corresponding all face-centred
(F-centred) cell of a crystal.
1
GEOL0026 2024/25: Dummy Examination
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(a) Write down the 3 equations giving the lattice vectors of the face-centred cell
(a2, b2, and c2) in terms of those of the primitive unit cell (a1, b1, and c1).
Using the multipliers from these equations, transform the following Miller indices
(h1, k1, l1) indexed on the primitive cell to their corresponding values (h2, k2, l2)
indexed on the face-centred cell: (1,0,0), (1,1,0), (1,2,1), (2,1,0).
(b) If the diffraction pattern from this crystal was indexed using the F-centred unit
cell, will each of the following Bragg reflections be present or be systematically
absent: 002, 201, 110, 111, 321
7. At high temperatures the mineral perovskite, CaTiO3 has a cubic structure, with a
primitive unit cell, containing 1 formula unit, and a ≈ 4 . The Ti atom sits at 0, 0, 0
and the Ca atom at , , ; the 3 oxygen atoms are at , 0, 0; 0, , 0; 0, 0, . The
atomic numbers of the atoms are 22, 20, and 8 for Ti, Ca and O respectively.
(a) By means of the structure factor formula, or by inspection of a sketch diagram of
the crystal structure, calculate the structure factors, F(hkl), for the 100 and 200
reflections in terms of the atomic scattering factors fTi, fCa and fO (you can assume
that the structure is centrosymmetric).
(b) For these two reflections, calculate the values of (1/2dhkl) – which from Bragg’s
Law is equal to sin(θ)/λ.
Graphs of the atomic scattering factors of Ti, Ca and O are given below. Using the
information in the graphs, calculate the numerical values of F(hkl), in electrons, for
these two reflections
.
GEOL0026 2024/25: Dummy Examination
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(c) What are the phases (positive or negative) of these two reflections If the Ca
atom in the unit cell were to be replaced by a Ba atom, with atomic number 56,
would the phase of each of the two reflections be the same, or different
(d) What would be the effect on the 100 and 111 reflections in the powder
diffraction pattern of CaTiO3 if the temperature was reduced so that the crystals
became tetragonal with the a axis slightly smaller than the c axis
8. How do the results obtainable from neutron diffraction differ from those from X-ray
diffraction; in particular, why is neutron diffraction more suitable for studying the
icy materials found in the outer solar system, such as the moons of Jupiter and
Saturn.
Imagine that you are an expert in neutron diffraction working with colleagues who
wish to understand the composition and evolution of these moons. What can you
contribute to this research programme Your account should be illustrated with
examples and should include discussion of: (i) the various results that may be
obtained from neutron powder diffraction both at room pressure and temperature
and at high pressure and low temperature, (ii) the methods that may be used both to
generate the neutons and to provide suitable sample environments, and the basic
diffraction geometries that you might employ.
Note that in writing your account credit will be predominantly based on content;
your answer can, therefore, take the form of short sections of text accompanied by
an appropriate set of annotated diagrams.
3
4
5
6
7
8
9
0 0.05 0.1 0.15 0.2 0.25 0.3
sin(θ)/λ
Atomic Scattering factor of O
f (electrons) O