Ge 214: Pyroxene homework series: part II
Energy states of Fe2+ in C2v symmetry.
We know from Tanabe-Sugano energy level diagrams in Cotton, Burns, or any other
reference, that Fe2+ has two energy states in an octahedral environment, namely T2g and Eg
. What the energy level diagrams do not tell us is what are the states of the Fe2+ system
when it is in a lower symmetry such as C2v. We are going to have to work this out. Because
both the T and E states are degenerate, we should suspect that these degeneracies might be
lifted in the lower symmetry. This will be a moderately long, but not particularly
complicated process.
1) We would already know the answer to the problem if the Fe2+ M(2) site were still an
octahedron. Our plan is to start with this knowledge and to see how the octahedral states
split when the site is changed to the lower symmetry. First we must determine which
symmetry elements originally present in octahedral symmetry are still preserved in C2v
symmetry. Because the C2v site can be directly derived from the octahedral site by simple
geometric distortion without adding or subtracting atoms, and without breaking bonds
(topologically identical sites), it should be evident that the symmetry elements of the C2v
site are a subset of the symmetry elements of the octahedral site. But how to determine
which ones are preserved?
We can work this out in full detail via the methods explained in chapter 9 of Cotton. I will
ask you to proceed along a less detailed route which makes use of the symmetry properties
of the site and the information about symmetry encoded in the character tables.
Our primary problem is that a coordinate transformation has occurred in going from the Oh
to the C2v site. Note that the oxygen atoms are on the x, y and z axes of the Oh site, but the
z (C2) axis of C2v bisects the Fe-O bond axes. To determine which symmetry elements in
Oh symmetry are preserved in C2v symmetry, pretend that the O-Fe-O bond angles are
90.00 in the M(2) site. Establish (draw) the coordinate system for an octahedral site,
mentally place the M(2) site on top of it, and rotate the M(2) symmetry coordinate system
so that the site is oriented along the octahedral symmetry axes; that is to say, rotate the site
until the Fe-O bonds are placed along the octahedral axes. There are a number of ways you
can do this (the x, y, and z axes in Oh are equivalent); they will all lead to the same answer.
Just so we all do it the same way, put the C2 axis of the C2v site in the horizontal plane of
the octahedral coordinate system.
Now identify and list the symmetry elements that are preserved. The symmetry elements
of Oh are listed on the top of the character table for Oh.
2) The next step is to use our knowledge of the remaining symmetry elements and the
states which exist in octahedral symmetry to derive the states of Fe2+ in the low symmetry
M(2) site. Construct a subset of the octahedral character table with just the following
entries: T2 and Eg (the only two states of interest from octahedral Fe2+) and just the
symmetry elements which are preserved in C2v symmetry. These are the only symmetry
elements and entries in the character table with which we must be concerned. If we can
solve the problem for this subset, we will have solved it for C2v.
The Experimentally Determined Energy states of Fe2+ in C2v symmetry.
We will again examine the experimental spectroscopic data for Fe2+ in orthopyroxene to
get a feel for what it is we still need to do. The prominent features of the data are band
splitting and polarizations. The previous parts of the homework derived the fact that the
splittings should occur. Next we have to deal with the polarizations.
1) Read section 5.41 in Burns, Mineralogical Applications of Crystal Field Theory (1970),
which will give you some of the experimental data. The analysis that follows has been
rendered out of date by later work on the west coast.
2) Attached, are two orthopyroxene spectra. The upper figure shows the alpha (dots), beta
(dash) and gamma (solid line) spectra of a 100 m thick bronzite (Fs14) in the visible and
near-infrared range. The bottom figure shows only the gamma spectrum of two 0.5 mm
thick crystals in the visible through the mid-infrared range. The solid line is for
hypersthene (Fs39.5), and the dotted line is for synthetic enstatite (Fs0.00). Compile a list of
band positions and polarizations of all Fe2+ bands (both in terms of the indicatrix direction
and in terms of the morphological axes). Also determine the molar absorptivity (epsilon
parameter) for each band.
The Origin of Polarizations and Fleochroism.
We will use the observed polarization properties of the Fe2+ spectrum in the pyroxene
M(2) site to establish the order of the C2v states. To do this, we must first establish some
predictions about the polarization dependence of each possible electronic transition. We
will then compare our predictions (models) to the actual data and hope that afterwards,
only a single model will remain viable.
1) Electronic transitions, in general are polarized. They correspond to a redistribution of
electronic charge in three-dimensional space. These redistributions of electrons have
vector properties and must match the electric dipole of the exciting radiation if there is to
be effective coupling of energy between the incident radiation and the electron cloud.
We want to know the polarizations of transitions such as:
Ground State  Excited State
That is to say, we want to know along which C2v molecular axis (or axes) a particular
electronic transition (such as A1  B1) will occur. Once we know this, we need only to
align a linear polarizer in this direction to make the transition occur (this assumes that we
know the orientation of the molecular axes with respect to the morphological axes of the
crystal).
Quantum mechanics tells us the intensity of an electronic transition will be proportional to
an integral of the form:
or
 is an operator that corresponds to the components of an electric dipole having the form:
 = ( k1 •X + k2 •Y + k3 •Z)
where X, Y, and Z are the molecular axes.
We want to evaluate integrals of the form:
or by components:
From purely symmetry considerations, we can tell if the integrals are nonzero. This is
analogous to evaluation of integrals of the odd (sine) and even (cos) functions.
Assume for a moment that the A1 state is the ground state. (Actually, ground states are
often A1 states) We need to evaluate a whole series of integrals of the type:
Where Al, Bl, A2, and B2 are the designations for all the possible states.
Because Al, Bl, A2, and B2 are not only designations for the possible states but are also
the designation for their symmetry, we can forget the details of the (complicated)
mathematical description of the wavefunctions, and can instead concentrate ONLY on the
symmetry properties.
Thus we need to evaluate integrals of the type:
From the character table, we learn that the molecular z axis in C2v has the symmetry Al.
(You can work this out by going through all the symmetry operations with the axis [+ and
- lobes, don’t forget] or you can read it directly from the character table; the letter z in the
Al row indicates that anything which is mathematically described by a function first order
in just z [such as the z axis which is, after all, just +z] has Al symmetry.)
Thus, we need to evaluate integral (1) which has components with the symmetry property:
A2 • Al • Al
Is this integral non-zero?
Here is how we proceed:
We need to work out a multiplication table for C2v wherein we multiply all possible
symmetries by each other. Take the case of
A2 • Al • Al
We can do this with pairs of functions:
(A2 • Al) • Al
This answer is equivalent to A2. In fact, you can show that Al times anything else
is the something else.
Our problem of evaluating (A2•Al) • Al has now been reduced to one of evaluating
(A2)•Al
We already know the answer to this is A2.
In order for the integral to be non-zero, the integrand must be invarient under the operation
of the symmetry elements for the group. This demands that the totally symmetric
representation (Al) must be a component of the integrand. (This comes back to the even,
odd function problem; we now are just worried about it in three dimensions).
Thus, we can now conclude that the transition from an Al state to an A2 state will not
occur if it is stimulated with linearly polarized light vibration in the direction of the
molecular Z axis.
You work out the answer for the X and Y components of this transition. You should now
conclude that this transition is not allowed for any vibration direction of linearly polarized
light. It is forbidden.
Now work out the answer for all three components of a possible transition from a) an Al
state to an A2 state.
b)
an Al state to a Bl state.
c)
an Al state to a B2 state.
As you can now see, only one polarization component is allowed for each of these possible
transitions.
2) Work out the multiplication table for all possible
pairs of symmetry elements. Fill in the rest of the
table
This will allow you to fill in the remaining entries in the following table which states the
polarization directions for all possible transitions in C2v.
Our final step will be interpret the experimental data on the basis of our theoretical
modeling.
Comparison of Models to Real Data
1)
Notice that the highest energy absorption band is polarized in the a direction of the
indicatrix. Use your projections of the crystal structure to establish which molecular axis
of the C2v M(2) site is oriented exactly on (or at least close to) the a direction.
2)
Now use your table of selection rules (polarizations) from part VIII to list all
possible transitions which are polarized along the molecular axis closest to the a indicatrix
direction. These are the candidates for the transition: ground state to the highest excited
state. Can you establish which transition is the actual one in these orthopyroxenes? (Hint,
the answer is yes.) Do it.
3)
Once you have identified the states involved in the highest energy transition, you
have established the ground state of Fe2+ in this site. From the polarization data you should
be able to assign the remaining experimentally observed transitions to particular excited
states. Construct an energy level diagram for Fe2+ in the M(2) site where the energy scale
in linear and to scale, where the states are identified, and the polarization rules indicated
for each state.
4)
We have one state left over and unaccounted for. This remaining state has not yet
been experimentally detected. You can read about where some theoretical calculations
predicted it should be in the article by Goldmann and Rossman (1977) The spectra of iron
in orthopyroxene revisited: the splitting of the ground state. American Mineralogist, 62,
151-157.
Another source for some more information on the spectroscopy of pyroxenes is in the
Reviews of Mineralogy series, Volume 7, CT Prewitt, editor, Pyroxenes, chapter 3,
Pyroxene Spectroscopy