EPSC 501 Crystal Chemistry
Week 1
How to describe and visualize a
crystalline structure
The vocabulary we need to describe the symmetry of
crystalline structures… Let’s re-learn it by applying it
to different examples.
Operations that
repeat a pattern in
3D space
How to fill a unit cell
Have you selected a structure that you will learn
to describe?
Did anyone try to search for a mineral (or a
crystalline structure)?
(Anyone…?)
Hematite Fe2O3
hexaquairon III
(dissolved)
Below:
as a trimer cluster
(Recommended for minerals described in the geoscientific literature)
Clicking on “Mineral” will let you
search an alphabetical index.
Handy if you aren’t sure about
spelling…
Why 15 determinations
of the structure of
hematite?
How do these two
descriptions differ?
This is a plausible final
exam question!
We will ask it again
when we meet next.
What do you see in the
header (first 5-6 lines) of
each data files?
Let’s focus on the first
data file
In each data file, the header indicates:
• Author(s) of the published study
• Where was it published (journal name, volume,
year, pages)
• Title of the paper
• AMC database code
The rest of the file describes the structure of
the crystalline solid that was investigated…
What is summarized in the box?
(I reprinted the line below, in larger type…)
5.038 5.038 13.772 90 90 120 R-3c
The unit cell is characterized by three
directions along which a small subset of the
pattern is repeated by translations.
These translations are the “unit lengths”
along the crystallographic axes a, b and c
5.038 5.038 13.772 90 90 120 R-3c
Unit cell edges a, b and c
There are six broad crystallographic systems,
characterized by different degree of
symmetry.
In some of these systems, the angles
between the three crystallographic axes are
determined by some of the symmetry
elements of the structure.
These angles (alpha, beta, gamma) are
listed after the unit lengths.
angles , , 
5.038 5.038 13.772 90 90 120 R-3c
Unit cell edges a, b and c (in
angstroms, 1 Å = 10-10 m)
How wikipedia presents the structure…
Where are these unit lengths and angles?
5.038 5.038 13.772 90 90 120 R-3c
Axis “c”
Let’s try to put it right side up!
The c axis, conventionally, is
vertical, the b axis is horizontal
(usually in the plane of the page).
The AMC datafile tells us
a, b = 5.038Å; c = 13.772Å
Color was used to distinguish Fe
(blue) from O atoms (red).
Is the information in the AMC data
file consistent with this drawing?
Let’s look at the positions of the
atoms.
Axis “a”
Axis “b”
The position of each atom is given as
a fractional coordinate (a number in
the range of 0 to less than 1) along
the axes a, b and c.
Can you find Fe at (0, 0, 0.3553)?
Can you find an
oxygen atom at
(0.3059, 0, 1/4)?
All the other atoms in
this pattern are in
positions that are
predicted by the
symmetry of the space
group R-3c.
What does the space
group refer to?
Who would bother memorizing the 230 space groups?
Let’s use Table 11.9 from Dyar, Gunter and Tasa.
With some practice, anyone can find quickly the space
group…
But you need to know what degree of symmetry is typical of
each one of the six main crystallographic systems.
The first letter of a space group refers to one of a few
possible types of lattices.
What is a lattice?
“… a collection of imaginary points (“nodes”) that have
identical environments in a homogeneous pattern…”
The space group is the set of symmetry operations that
relate these nodes. The symmetry operations are listed
AFTER the type of unit cell (“R” in space group R-3c).
The 14 Bravais lattices
The “spheres” are called
nodes.
Each node represents
an equivalent (or
identical) point within a
crystal structure.
A node does not
necessarily need to
represent an ion or atom
in a crystal structure.
A node represents a motif
that is repeated in the
structure.
The simplest lattices
repeat a pattern at the
eight corners of a boxlike volume, called the
unit cell.
Space groups with
primitive cells are
possible in each
crystallographic system.
Examples? P1 (triclinic)
P2 (monoclinic)
P222 (orthorhombic)
P4 (tetragonal)
P3 (hexagonal)
P23 (isometric)
Self-test #1:
Can you assign
each space group
in this list to the
correct crystal
class (point group)
and the correct
crystallographic
system?
Are all the systems
represented?
(Answers are on the
back…)
Hexagonal system,
primitive cell
The space group of
hematite is abbreviated
as “R-3c”.
First letters that are not
“P” refer to a nonprimitive type of unit cell.
In non-primitive cells, a
motif is repeated within
the cell (or on its
“faces”), as well as at
each corner of the cell.
Hexagonal system,
rhombohedral cell
What are the
types of cells?
PRIMITIVE (P)
Rhombohedral
(R)
BODY-centered
(I)
ALL-faces
Centered (F)
ONE-face
centered (C)
The “R” cell explains some
of the repetition of the
atomic pattern within the
unit cell of hematite.
But the full space group is
R-3c…
-3 is a rotoinversion axis
“c” is a glide plane
These symmetry elements
also repeat the pattern
throughout the lattice.
http://img.chem.ucl.ac.uk/sgp/mainmenu.htm
If you needed (or wanted) to
know where these other
symmetry elements are
located and where they
repeat the atomic pattern,
you might consult this site…
rotoinversion
Screw axis
But most of us won’t
need to go there…
What you must master about the 230 space
groups and the 32 point groups for this
course…
1) How to look up which one of the 6 crystallographic
systems they belong to…
A reference such as Table 11.9 will do.
2) What system of axes is used to describe the atomic
positions? (This is specific to each system.)
3) Will the mirrors (or glide planes) and rotation axes
be perpendicular or parallel to these axes? (This
influences morphology during growth and some surface
properties.)
4) What type of unit cell does this space group show?
Let’s take a break…
And if you downloaded a data file from the
AMCD, let’s print it and make some
photocopies that we might share within the
group.
By our next meeting January 17, you should:
- Submit via WebCT data on the crystalline structure (or
mineral) you propose to learn to visualize and describe
- Check if the crystallographic data for its structure is in the
AMC database (or another source)
- read from the data its space group (and determine the
corresponding crystallographic system and point group)
- find from the data the fractional coordinates of its atoms
- try showing atoms, bonds and polyhedra using XtalDraw
Next week, we will
- examine the geometry file produced by XtalDraw to
examine the attributes of structural sites
- examine the types of coordination polyhedra and how
they are linked throughout diverse crystal structures
Cooperite, PtS
(this specimen from the Rustenburg
District, South Africa)
(Also reported from the Merensky Reef
of the Bushveld Complex…)
Why are Pt and the position (0, 0, 0.25)
repeated within the description of the unit
cell?
Note the column after “x”,
“y” and “z”, with the title
“occ”. The values of that
column refer to
occupancy, i.e. the
relative proportions of
whatever occupies this
site throughout the crystal
structure.
Are two elements
occupying the same
position in the unit cell?
How is this possible?
The meaning of values of occupancy…
The proportions 0.988 (for S) and 0.012 (for
Pt) indicate that, for every 1000 atoms
detected at coordinates (0, 0, 0.25) – in other
words, if a large enough number of unit cells
were surveyed – you would find a sulfur atom
at this position in 988 out of 1000 unit cells.
Dispersed among them, 12 unit cells would
show Pt at coordinates (0, 0, 0.25).
But a JMOL-3D model (or XtalDraw) cannot
represent different elements at the same site.
The JMOL-3D model
is (hum…) skimpy.
(Were you able to fill
the entire unit cell?)
Is this atom at
(0, 0.5, 0) or
(0, 0, 0.25) ?
This is the site with
dual occupancy
Occupancy is determined from the relative
number and intensities of peaks observed within
the diffraction pattern of excellent single crystals
of a composition known with precision.
Elements with different masses also have a
different X-ray scattering power. Pt re-emits Xrays at a higher intensity than the lighter S
atoms.
If two elements randomly occupy the “same”
site, the diffraction pattern differs from that of a
structure where the elements are in sites
unrelated by a symmetry operation.
By comparing
several data
files published
for cooperite…
… we see
how
occupancy
varies across
specimens!
Does that
affect the unit
cell?
In 1932, these subtleties went unnoticed
(possibly because the effect on the diffraction
pattern was too subtle to detect and interpret
without the computing power now available).
Does any other part of
the description suggest
that the lack of
computer power limited
the precision of a crystal
structure refinement?
The JMOL 3D model accessible at the AMCD
site is too skimpy for our purpose…
We want to fill the entire unit cell!
Let’s try with XtalDraw…
First, download an AMC file on a computer
drive or memory stick you can access. (Save
it under a memorable name, but do not
change the filename extension:
“cooperite.amc”, for example…)
Let’s start XtalDraw and open the file…
The atoms are assigned different radii (much
larger for S, than for Pt).
The unit cell can now be completely filled.
Can we see the same atomic pattern repeated
at the eight corners? (We should since the unit
cell of P4 2 /m m c is primitive…)
Can you show that each Pt is surrounded by
four sulfur, in a square planar arrangement?
This is unusual. Anions in most crystalline
solids surround the cations in a coordination
typically described as 3D polyhedra.
Don’t be surprised to see
the same structure
described differently in
peer-reviewed journals.
In space group P4 2 /m m c, the axes a and b
can be chosen at 45 degrees from those in the
description archived in the AMCD. The squareplanar groups are diagonal through the cell.
Garnet group
An example where
crystalline solids share:
 a single structural
formula
 the same atomic pattern:
Similar coordination
polyhedra but their
elements differ
 the same symmetry
(described by the same
space group)?
These end-members are listed,
and many are discussed in
Locock (2008)
A useful reference (in general):
Nickel and Grice, 1998. The IMA commission on
new minerals and mineral names: procedures and
guidelines on mineral nomenclature, 1998.
Canadian Mineralogist, vol. 36, pp. 913-926.
(IMA: International Mineralogical Association)
A reference for the chemistry of the garnet group:
Locock, A.J. (2008). An Excel spreadsheet to recast
analyses of garnet into end-member components,
and a synopsis of the crystal chemistry of natural
silicate garnets. Computer & Geosciences, vol. 34,
pp. 1769-1780.
Garnet group
What is the definition of an
“end member”?
End members have
idealized site occupancies,
and are useful to describe
compositional variation.
The names in italics are
hypothetical mineral
species, i.e. they have
never been found in nature,
but might have been
synthesized in laboratory.
They are also referred to as
“end members”.
Naming minerals: mineralogical nomenclature in solidsolution series follows a system called “the 50% rule” (in
binary solid solution series, defined by two end-member
mineral species).
In a ternary solid solution (i.e. 3 end-member species), it is
more correct to call this the 100%/n rule or the dominantconstituent rule, in which the constituents are atoms (cations
or anions), molecular groups, or vacancies.
CN=
6
CN=
4
CN=8
All species of
garnet share the
same structural
formula, based on
three distinct
coordination
polyhedra that
share specific
edges or corners.
Dana’s chemical classification
(51) Nesosilicate
Insular SiO4 Groups Only
A chemical
(51.04) with cations in [6] and >[6] coordination
classification
(51.04.03b)Garnet group (Ugrandite series)
of minerals
51.04.03b.01 Andradite Ca Fe (SiO4) I a3d 4/m -3 2/m
that takes
51.04.03b.02 Grossular Ca Al (SiO4) I a3d 4/m -3 2/m
51.04.03b.03 Uvarovite Ca Cr (SiO4) I a3d 4/m -3 2/m
into account
51.04.03b.04 Goldmanite Ca (V,Al,Fe) (SiO4)3 I a3d 4/m 3 2/m
the
51.04.03b.05 Yamatoite? (Mn,Ca) (V,Al)2(SiO4)3 I a3d 4/m 3 2/m
coordination
(51.04.03a)Garnet group (Pyralspite series)
environment
51.04.03a.01 Pyrope Mg3Al2(SiO4)3 I a3d 4/m -3 2/m
of ions.
51.04.03a.02 Almandine Fe3Al2(SiO4)3 I a3d 4/m -3 2/m
The Garnet group
51.04.03a.03 Spessartine Mn3Al2(SiO4)3 I a3d 4/m -3 2/m
51.04.03a.04 Knorringite Mg3Cr2(SiO4)3 I a3d 4/m -3 2/m
covers a broader
51.04.03a.05
composition than
the Majorite Mg3(Fe,Al,Si)2(SiO4)3 I a3d 4/m -3 2/m
51.04.03a.06 Calderite (Mn,Ca)3(Fe,Al)2(SiO4)3 I a3d 4/m -3 2/m
ugrandite series.
3
2
3
3
2
3
3
2
3
3
2
3
Is the garnet structure found only in silicate minerals?
Consider schaferite: Ca2NaMg2V3O12
What element(s) fill the A site in natural garnet?
...in Schaferite?
What element(s) fill the B site?
What element fills the Z site (tetrahedral)?
Why is it said to have the “garnet structure”?
Mineral names are sometimes used to refer to (and
“personalize”) certain types of crystalline structures.
In the chemical classification of minerals, “garnet” are
silicates from the subclass “51” of orthosilicates (or
nesosilicates) with the general formula A2+3 B3+2
(XO4)3 .
Schaferite is from a different chemical class (Dana
class 38): which includes phosphates, vanadates and
arsenates with the general formula (A+ B2+)5 (XO4)3
“Garnet structure” refers to the similar proportions of
and edge-sharing among the 8-, 6- and 4-fold
coordination polyhedra. For this reason, schaferite is
said to be isostructural with the garnet group.
Eventually, for your 2nd assignment, you will
look for an example of a property that varies
with composition.
The structure of the two crystalline solids you
should compare must be identical (you will be
expected to provide the reference.)
The property must have been observed in two
crystalline solids that share the same degree
of symmetry (described by a space group)
but differ somewhat in composition.
One or several of the sites in the structure will
be occupied by a different element.
From the American Mineralogist crystal structure database
Which elements in this description
of uvarovite are filling the same
site throughout the crystal?
Look for lines with he same x,y, z
fractional coordinates….
The files for your own mineral
might show examples of
compositional variation within
that mineral species.
Next time: which site (8-fold, 6-fold or 4-fold) of the
garnet formula viiiA3viB2ivZ4O12 are Ca and Mn filling?
How can we use XtalDraw to show me this?
How can you tell that this fragment of the garnet structure is
from an incomplete unit cell?
From which
system is the
garnet
structure?
We would
expect to see
the same type
of ion at the
eight corners of
a cube-shaped
unit cell.
Full space group: I41/a bar-3 2/d
This is usually abbreviated to: Ia3d
Enough information was left to identify
this space group as isometric (cubic).
What gives it away?
Table 11.9 gives you point groups and
their corresponding space groups.
Keep it handy… we will use it throughout
the course.
32 point groups (= 32 crystal classes)
This list displays abbreviated notations for the
Herman-Mauguin symbols
= 2/m 2/m 2/m
From “Web Mineral” at
http://webmineral.com/data/Pyrope.shtml
Pyrope
a = 11.459, Z = 8; V = 1,504.67 Den(Calc)= 3.56
H-M Symbol (4/m bar-3 2/m) Space Group: I a3d
(Z = no of formula units per unit cell. An I-cell has a minimum Z
of 2, but in garnets, this is multiplied by the presence of glide
planes and/or screw axes that repeat the pattern elsewhere
within the cell.)
You might find the same information in other sources… or not.
Different texts, different database serve different purposes.
- Nesse “Introduction to Mineralogy”?
- Klein & Hurlbut “Mineral Sciences”?
Not all this information is included in files from the American
Mineralogist Crystal Structure Database.
Did anyone choose a
pyroxene?
Whoever picked diopside should be able to interpret this
description… and relate it to the formula of diopside.
1) What fills the
M2, M1 sites?
2) Here, no column for
“occ” after the x, y, z
coordinates. Why not?
Spinels
Another
example of a
group
important in
mineral and
material
sciences…
Like the
garnet group,
the spinels
are described
by a general
formula.