Chemistry 4000
Test #2
Name:_______ANWERS___________________
I.D. #_____________________________
October 3rd, 2007
Time: 60 min.
Answer all questions on the question sheet in the space provided. You may use your lecture notes, lab handouts and the Massa
text book. You may use CaRIne Crystallography and Microsoft Excel to provide answers to this test. You may not use email or
web browsers or otherwise access the internet. Model kit and, drawing tools may also be used.
1.
The mineral anatase is a titanium oxide. It is found to crystallize in the space group I41/amd (No. 141). The unit cell
parameters were measured to be a = 3.7842 and b = 9.5146 Å.
a)
A partially erased entry of the first page of the space group entry is found directly below. What is the crystal system of
this space group?
The crystal system is tetragonal. You can get this from the space group name (i.e. from the point group, see below). Or
you can get it from the form of the unit cell axes, or from finding No. 141 in CaRIne for that matter.
b) Show how the symmetry point group that this space group belongs to is determined?
I41/amd: convert all translational symmetry to non-translational. Lose the lattice. Æ 4/mmm
c) Write down, and explain in detail the elements of the “long” name of this space group.
I
41/a
2/m
2/d
This is the lattice symmetry, standing for body centring.
In tetragonal space, the 4-fold axis must be along c. Here it is a 41 screw with an a glide ⊥ to c
In tetragonal space, any 2 fold ⊥ to the primary axis must be on both a and b. There is a normal mirror plane m
perpendicular to each of these 2-fold axes.
Additional 2-folds in tetragonal space must bisect a and c. There is a diagonal glide plane d ⊥ to these diagonal
two-fold axes.
d) Show on the above diagram (with neat markings, arrows, etc.) on the right hand panel only the location of as many
of the symmetry elements indicated by the space group name (see part c) as possible. Please restrict yourself to only one
example of each symmetry element to keep things legible, i.e. no repeats! See colour annotations above.
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2.
The coordinates (in the “origin choice 1” of this space group) were found to be:
Ti
0 0 0
O
0 0 0.208
With the help of CaRIne, build a model of a unit cell for anatase. You may either sketch out the unit cell on paper
using the CaRIne model as a guide, or save to disk the models that you make for submission (by email, after the test is
over) to your instructor. An entry from the International Tables second page follows to assist you in this task.
a)
Which Wyckoff positions do you think the Ti atoms in anatase occupy?
listed)
Looks like 4a (only 0 0 0
Calculate the positions of all the symmetry related Ti atoms in the unit cell of anatase. Hint: these will be needed to
build your model in CaRIne, as the “space group method” fails for this crystal.
000
0½¼
000 + ½½½ Æ ½½½
0½¼ + ½½½ Æ ½1¾ =½0¾
(either answer is acceptable)
In these coordinate listings, either “0” or “1” may be given, but 0 is the better choice. The CaRIne program and a
human artist both supply the additional locations which are related by a unit cell translation, i.e. (0 0 0) defines all
corner locations, and face locations are on both opposite sides, etc.
You may use either primitive or body-centred tetragonal to build the model in CaRIne.. Since the Ti atoms are at (0 0
0), selecting tI will automatically place the Ti at (½ ½ ½) but the oxygen atoms have to each be placed using the
Cell_Creation/List window.
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b)
Which Wyckoff positions do you think the O atoms in anatase occupy?
Looks like 8e (only 0 0 z)
Calculate the positions of all the symmetry related O atoms in the unit cell of anatase. Hint: these will be needed to
build your model in CaRIne, as the “space group method” fails for this crystal. Answers with “1” in place of “0” o.k.
0 0 0.208
0 ½ ¼+0.208 Æ
0 ½ 0.458
½ 0 ¾–0.208 Æ ½ 0 0.542
½ ½ ½-0.208 Æ ½ ½ 0.292
0 0 0.208 + ½ ½ ½
0 ½ 0.458 + ½ ½ ½
½ 0 0.542 + ½ ½ ½
½ ½ 0.292 + ½ ½ ½
Æ
Æ
Æ
Æ
½ ½ 0.708
½ 0 0.958
0 ½ 0.042
0 0 0.792
The resulting structure is shown in two views, at left a conventional unit cell picture, at right a polyhedron model
emphasizing the interlocking units of edge-sharing TiO6 units.
c
z
z
x
y
a
x
y
b
b
a
c)
What is the stoichiometry of anatase?
From the Wyckoff position multiplicity, it is obvious that there are 4 Ti atoms and 8 oxygen atoms “belonging” to each
unit cell. Hence the stoichiometry (in simplest ratio) is TiO2.
d)
Calculate the volume of the unit cell and the density of anatase.
o
V = 3.78422 × 9.5146 = 136.25 Α3
e)
D=
(47.88 + 2 ×15.9994) × 4 ×1.661
= 3.895 g / cm3
136.25
What is the coordination number (number of nearest neighbours at identical or nearly identical distances) of (i) the Ti
atoms, and (ii) the O atoms in anatase. Caution: beware of “edge effects” in your unit cell model. If there is more than
one kind of coordination of any given element type, specify all that exist.
Ti: CN ~6; 2 O at 1.98, 4 O at 1.93 Å which are essentially the same.
O: CN 3; 2 Ti at 1.93, 1 at 1.98 Å which again are essentially the same.
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3.
a) Indicate graphically at least four titanium atoms that are related by one of the 41 screw axes in the unit cell of
anatase. Hints: I suggest a radii scale of 20-25% for your crystal model for this part of the test. If you want to use
CaRIne to answer this question, make a separate model and use colour coding of the individual atoms to indicate the
atoms that you are talking about. Use the “modify atom” button to make these changes. The answer is shown under (a)
below.
(a)
(c)
c
c
z
z
x
y
x
y
b
b
a
a
b) Write down the fractional coordinates of these titanium atoms that are related by the 41 axis. Then show
mathematically how their coordinates demonstrate the screw relationship. Hint: the “modify atom” button in CaRIne
will also tell you the coordinates of atoms. Can you locate these four positions from the coordinates in the International
Tables? Specify which ones (after suitable lattice translations) your choices are.
The four that I have marked in gold in the model above, related by the 41 that is circled on the diagram on page 1 of this
answer guide, i.e. in the top left quadrant of the unit cell, will have the coordinates:
(0 ½ ¼)
(0 0 0)
(½ ½ ½)
(½ 0 ¾)
Rotate by 90° about (¼ ¼ 0) and step up by ¼ in c.
Note that these are precisely the ones listed in the Tables as Wyckoff position 1a, those in the answer to part 2a above.
You could just as well have chosen 4 + 1 titanium ions in any of the other three quandrants of the unit cell.
c) Explain (with a diagram, or with a separate colour coded CaRIne model) which atoms in your unit cell are related by
the normal mirror plane, i.e. an m element, within the unit cell of anatase. The answer is shown under (c) above; note
that this is just one example of m, i.e. the mirror through the center of the unit cell and ⊥ to b. The other one that is not
at an edge is ⊥ to a and also in the middle of the cell. The blue and bronze atoms are mirrored by a plane containing the
normal coloured atoms.
4)
Chromium sources are sometimes used for X-ray diffraction when “softer” radiation is desired (for example, to reduce
radiation damage to a biocrystal). The wavelength of the Kα line of Cr is 2.293606 Å. If radiation of this wavelength
impinges on a clean surface of sodium metal (in a vacuum; sodium valence electrons are easily ionized), calculate the
energy (in Joules) of the ejected photoelectron.
E=
hc
λ
=
6.6262 ⋅10−34 Js × 2.998 ⋅108 ms −1
= 8.661⋅10−16 J
−10
2.293606 ⋅10 m
Fundamental Constants
Atomic mass unit
Avogadro's number
Electron charge (e)
Electron mass
Ti atomic mass
1.6605 × 10-24 g
6.022 × 1023
1.6022 × 10-19 C
9.1095 × 10-28 g
47.88 amu
Planck's constant
Proton mass
Neutron mass
Speed of light in vacuum
O atomic mass
6.626 × 10-34 J s
1.67252 × 10-24 g
1.6749 × 10-24 g
2.998 x 108 m s-1
15.9994 amu
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