Ge214
Pyroxene homework series: part I
A) Cation polyhedra in orthopyroxenes
It it the purpose of this series of homeworks to analyze in detail the optical spectrum of
Fe2+ in orthopyroxene. To analyze the spectrum of any mineral in general, we must first
determine several things:
a) Determine from chemical analysis what spectroscopically active ions are present.
b) Determine what sites the ions of interest occupy in the mineral's structure.
c) Determine the point group symmetry of the cation site.
d) Determine the oxidation state(s) of the ion.
In the case of orthopyroxene, there have been a number of studies which demonstrate
that the only transition element present in any appreciable concentration, namely iron, is
responsible for all the major spectroscopic features. We will accept this as fact and
proceed. Furthermore, there is much evidence to show that Fe2+ is the ion of interest.
(Some orthopyroxenes have no Fe3+ detectable yet still show the same spectroscopic
features as all other orthopyroxenes.)
We will begin our analysis by considering the M(1) and M(2) sites in which Fe2+ may
reside. First we need to determine the point groups for the M(1) and M(2) sites.
1) Your assignment is to determine the exact point groups for the M(1) and M(2) sites
using the metal-oxygen bond distances and bond angles determined from single-crystal
X-ray refinements.
One way to solve this is to obtain projections of the M(1) and M(2) sites of an
orthopyroxene from published X-ray structural data. The projection then enables you to
discuss the relative distortions of the two sites in connection with spectroscopic data and
site occupancy data. You could use a reference such as Ghose (Zeit. Kristallogr., 122,
81-94, 1965) which refines the structure of an orthopyroxene. Tables 2 and 4 of this
paper provide atomic coordinates and atomic linkage data for the pyroxene. Other
references are Hawthorne and Ito (Canad. Mineral. 15, 321-338, 1977) and Sueno et al.
(Amer. Mineral., 61, 38-53, 1976).
Projections of both an M(1) and M(2) coordination polyhedra on (100), (010), and
(001) would be prepared using accurate graph paper or a computer program. The book
by Helen D. Megaw (1973) Crystal Structures: A Working Approach" gives
directions for preparing the projections. The International Tables for X-Ray
Crystallography", Vol. I (1969), provides additional data about the space group and
atomic coordinates.
An easier alternative is to solve this by referring to published structural data. We can
use the illustrations in Burns (1970) Mineralogical Applications of Crystal Field
Theory, page 90, or page 185 in the 2d edition (1993). In either case, compare the sites
and notice their distortions from regular octahedral symmetry. Which site is more
distorted?
There is a body of data which leads us to believe that Fe2+ in the M(2) site is
responsible for much of the absorption features in the orthopyroxene spectrum. For
example, Evans, Ghose and Hafner (J Geology, 75, 306-322, 1967) used Mössbauer
spectra and found that most of the Fe is in the M(2) site in low-iron orthopyroxenes.
Therefore, we will concentrate on the analysis of the spectroscopy of Fe2+ in the M(2)
site.
The point groups determined above, although correct, are not of much use in the
detailed analysis of the pyroxene spectra. The problem is that we need symmetry to aid
us in the interpretation of all sorts of useful things. Therefore, we need to approximate
the symmetry of the M(2) site with a point group of higher symmetry. The reasons for
doing this will be obvious only after we work through the entire analysis and find that we
can, in fact, do something useful when we are done.
Make the following assumptions: Note that there are three pairs of Fe-O bonds of
approximately the same length. Pretend that the two long bonds (2.4 Å and 2.5 Å) are
close enough to be the same. Do the same for the two shortest bonds and the two
intermediate bonds. Assume further that the four longer bonds are in the same plane and
that the two shorter bonds are exactly normal to this plane.
Now determine the point group of the approximated M(2) site.
B) Character Tables and classification of orbital symmetry.
Chapter 4 in Cotton, Chemical Applications of Group Theory takes you through the
concept of the matrix representation of groups and develops the concepts of a character
table. You will need to understand these concepts to proceed.
We will be concerned with the character table for C2v in our analysis of the M(2) site of
orthopyroxenes.
C2v
E
C2
v(xz) v’(yz)
A1
A2
B1
B2
1
1
1
1
1
1
-l
-l
1
-l
1
-l
1
-l
-l
1
z
x
y
x2y2z2
xy
Rz
xz
Ry
yz
Rx
Remember, the entries in the character table are the characters of the irreducible
representation of the group, the trace of a matrix.
Let’s get some practice using the character table before we get into our ultimate
objective.
Problem: Classify the d-orbitals (dz2, dx2-y2, dxy, dxz, and dyz) under C2v.
I’ll do one for you, the dyz orbital:
Remember, the d-orbitals have positive and negative lobes. We will follow the changes in
these lobes when we apply the relevant symmetry operations to them. In C2v, there are
only 4 symmetry operations left. They are indicated in the character table above.
Under the B operation,
Under the C2 operation,
Under the v operation,
Under the v' operation,
it returns to +1 times itself.
it returns to -1 times itself.
it returns to -1 times itself.
it returns to +1 times itself.
By comparison to the character table, we find that anything with these properties has the
symmetry classification B2 in C2v. Because it is an orbital, we will follow the standard
convention and designate it in lower case letters. Thus, the dxy orbital is a orbital in a C2v
environment. Please note that these orbitals are not oriented in the same way as the dorbitals of the octahedron; I did a coordinate transformation so that the orbital z-axis
corresponds to the 2-fold rotation axis of C2v. This is an important point, so be sure that
you understand what happened here.
You work out in detail the other d orbitals under C2v symmetry.