WORKING WITH ELECTRON MICROPROBE DATA FROM A HIGH PRESSURE

EXPERIMENT CALCULATING MINERAL FORMULAS, UNIT CELL CONTENT,

Brandon E. Schwab
Department of Geology
Humboldt State University
Arcata, CA 95521
schwab@humboldt.edu

Figure 1. Back scattered electron image of peridotite partial melting experimental run
product showing grains of olivine and orthopyroxene (dark gray), clinopyroxene
(medium gray), spinel (white), with interstitial glass (light gray). Field of view ~200m.

INTRODUCTION
Calculating Mineral Formula from weight percent oxide data
This exercise will introduce you to procedures commonly used to manipulate chemical
data. Specifically, we will examine how chemical analyses can be recast into mineral formula
and how chemical data can be combined with x-ray diffraction data to determine the atomic
content of unit cells. We will also use chemical information to calculate equilibration
temperature for a mantle melting experiment.
The electron microprobe (or electron probe microanalyzer, EPMA) is invaluable in
producing rapid, precise analyses of a wide variety of materials in very small areas. With this
instrument, individual minerals in a rock may be studied in detail. The main drawbacks of
EPMA analysis include the facts that it cannot (1) accurately analyze light elements like oxygen,

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nor (2) determine valence states (e.g., Fe2+ vs. Fe3+). For this exercise we will assume that all
iron is ferrous (Fe(II) or Fe 2+). We can, to a certain extent, get around the oxygen problem when
analyzing silicate and oxide minerals, by assuming that each cation is associated with the
stoichiometric amount of oxygen required to form the cations oxide. In other words, we assume
that oxide and silicate minerals are composed of mixtures of oxides, rather than individual
elements. Instead of reporting an analysis as a list of weight percentages of the elements
(including oxygen), we report them as weight percentages of the common oxides. Mineral
formulae are then calculated from analyses assuming a fixed number of oxygen atoms in the
ideal formula. This approximation is valid as long as the elements are each present in only one
valence state and the mineral is truly stoichiometric. Although these are occasionally
questionable assumptions, we will accept them for this exercise. Lets look at an example.

EXAMPLE:
Consider a real analysis of the mineral forsterite (Mg-olivine). The ideal, end-member
formula for forsterite is Mg2SiO4. In nature, we typically find other divalent metal cations
substituting for magnesium. One such analysis is listed below in terms of weight percentages of
the oxides.
Oxide wt.% oxide
SiO2 40.30
FeO 8.85
MgO 49.58
CaO 0.07
MnO 0.13
NiO 0.42
Total 99.35

In order to calculate the formula for this olivine, we perform the following simple steps
(modified from Deer et al., 1992 p. 678). Olivine has the general formula: (Mg,Fe)2SiO4

Step 1: Obtain the analysis (silicates are reported in terms of oxide weight percentages, which
may or may not be normalized to 100%). List in column I.
Step 2: List (in column II) for each oxide the appropriate molecular weight of the oxide.
Step 3: Divide I by II to obtain the atomic proportions. Column III therefore expresses the
molecular proportions of the various oxides.
Step 4: Column IV is derived from column III by multiplying by the number of oxygen atoms
in the oxide concerned. It thus gives a set of numbers of oxygen atoms associated
with each of the cations. At the foot of column IV is its total.
Step 5: If we require the olivine formula to be based on 4 oxygen atoms, we need to recast
(normalize) the oxygen atom proportions so that they total 4. This is done by
multiplying all of the oxygen proportions (col IV) by a normalization factor defined
as 4/total oxygen proportions. The results of this step are given in column V.
Step 6: Column V gives the formula-normalized number of oxygen anions. We must now
calculate the number of cations associated with each oxygen. Column VI is
calculated by multiplying column V by the ratio of the cation to oxygen in the initial
oxide. For example, in SiO2, there is one Si for every 2 oxygens, so column VI(Si) is
multiplied by . For trivalent cations such as Al2O3, there are 2 Al atoms for every 3

2
 oxygens, so column V is multiplied by 2/3. For divalent cations (e.g., Ca, Mg, Fe2+,
etc.), column VI is the same as column V, and if monovalent cations (e.g., K, Na, H)
are present, column VI is twice that of V.

The number of cations in column VI corresponds to the cations per formula unit (per
fixed number of oxygen atoms). A check on the balance of + and charges in the formula
provides a check on the arithmetic. From column VI, the formula can be written by assigning
cations to the ideal formula. For example, because Al3+ can substitute for Si4+ in the
tetrahedral sites, Al atoms (if present) are commonly added to Si in order to achieve ideal
tetrahedral site occupancy. The remaining Al atoms are placed in the octahedral sites.

The steps above are summarized in the following table:

I II III IV V VI
atomic
proportion number of
Weight Molecular of oxygen anions based number of
percent proportion from each on 4 O [col ions in
oxides mol wt of oxides molecule IVx1.4792] formula
SiO2 40.30 60.08 0.6708 1.3415 1.984 Si 0.992
Al2O3 0 101.96 0.0000 0.0000 0.000 Al 0.000
Cr2O3 0.02 151.99 0.0001 0.0004 0.001 Cr 0.000
2+
FeO 8.85 71.85 0.1232 0.1232 0.182 Fe 0.182
MnO 0.13 70.94 0.0018 0.0018 0.003 Mn 0.003
MgO 49.58 40.3 1.2303 1.2303 1.820 Mg 1.820
CaO 0.07 56.08 0.0012 0.0012 0.002 Ca 0.002
Na2O 0 61.98 0.0000 0.0000 0.000 Na 0.000
NiO 0.42 74.7 0.0056 0.0056 0.008 Ni 0.008
Total 99.37 2.7041 4 3.008
4/2.7041: 1.4792

Formula: (Ca0.002Mg1.82,Fe0.182,Ni0.008Mn0.003)2.015Si0.992O4

Some general rules for writing mineral formulas.

Example: CaMgSi2O6
Charge balance is maintained:
Ca2+ + Mg 2+ + 2(Si4+) = 6(O2-)
2 + 2 + 8 = -(12)

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 Cation size (and radius ratio) dictates CN of cations:
Si 4-fold
Mg 6-fold
Ca 8-fold
VIII
CaVIMgIVSi2O6, where roman numerals indicate coordination of cations

If cations interchange with each other in a given site, then group these cations with
parentheses:
VIII
CaVI(Mg,Fe)IVSi2O6
or
Ca(Mg,Fe)Si2O6

Your mineralogy text should include a list or table of common ions and their typical coordination
numbers. Also, use the general formulas as a guide for ordering the cations in the formulas.
Figure 2 is a chart of cation occupancy for pyroxenes.

Figure 2. Flow chart showing ideal site occupancy and order of assigning atoms to T, M1, and M2 sites of
pyroxene structure (modified from Deer et al., 1992 after Morimoto, 1988).

EXERCISE
Problem #1
In this exercise you will be working with electron microprobe data collected from a high-
pressure mantle melting experiment. Figure 1 is a back-scattered electron (BSE) image of a run
product from just such an experiment. In the highly magnified image you can see several
different types of grains in terms of grayscale or whiteness. The brightness (whiteness) in a
BSE image correlates to the mean atomic number of the material under electron bombardment,
thus the phases in the experimental charge can be differentiated based on their grayscale --
darker gray = lower mean atomic number, brighter (whiter) = higher mean atomic number.
Practically speaking, the analyst positions the sample under the electron beam using the BSE
image as a map and then analyzes the mineral (or piece of glass) of interest. This generally
requires many hours sitting in a dark room. The good news is that I have done this for you!

The following microprobe analyses are from a high-pressure peridotite melting experiment
(Schwab & Johnston, 2001) like the one shown in Fig. 1. These four minerals (olivine,
orthopyroxene (opx), clinopyroxene (cpx), and spinel) are in equilibrium with glass (representing
partial melt) at the pressure (1.0 GPa) and temperature of the experiment.

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 Ortho- Clino-
Olivine pyroxene pyroxene Cr-spinel
1
SiO2 41.05 55.68 53.23 mdl
Al2O3 0.04 3.66 4.49 30.48
FeO* 7.92 5.10 3.74 9.08
MnO 0.14 0.11 0.12 mdl
MgO 48.70 32.54 22.18 18.32
CaO 0.34 2.71 14.61 mdl
2
Na2O mdl 0.06 0.21 mdl
Cr2O3 0.26 1.27 1.72 38.49
Total 98.45 101.13 100.30 96.37

1
Weight percent oxides, unnormalized
2
mdl represents "below minimum detection limit"
2+
FeO* = all iron as Fe

Calculate and write out the formula for each of these four analyses. Base your calculations on
the following definitions of mineral species:
Olivine: (Mg,Fe)2SiO4 4 oxygens per formula unit
Opx: (Mg,Fe)2Si2O6 6 oxygens per formula unit [3 if using (Mg,Fe)SiO3]
Cpx: Ca(Mg,Fe)Si2O6 6 oxygens per formula unit
Cr-Spinel: (Mg,Fe)(Cr,Al)2O4 4 oxygens per formula unit

Setting up a spreadsheet for these calculations is very useful!

Problem #2
X-ray diffraction studies of the cubic Cr-spinel (above) yield the following information regarding
the crystallographic axes and unit cell dimensions for this mineral:
d(100) = a = 8.23
In addition, the density of this mineral has been measured to be 4.12 gm/cm3. Calculate
the unit cell content for this mineral. The unit-cell-content is defined as the number of formula
units contained in the unit cell of the mineral. Hint: Once the formula is calculated (Problem
#1), calculate the volume of the unit cell (cubic unit cell a = b = c, all intersecting at 90) and
the molecular weight (gm/mole) of the formula unit. With this information, you can then
calculate how many formula units are present in the crystallographic unit cell by considering the
following equation:
Number of formula units = VNA/formula weight
where V is the cell volume in cm3, is the density in gm/cm3, NA is Avogadros Number (6.02 x
1023 mol-1). Owing to uncertainties in the analysis, cell parameters, density, etc., the final
number of atoms per formula unit and the number of formula units per unit cell should be
rounded to the nearest whole number. **Helpful conversion: 1 = 10-8 cm, so that 1 cubic =
10-24 cm3.

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Problem #3
Mineralogists and petrologist often talk in terms of percent end member composition for
minerals with solid solution. For example, An55 would refer to a plagioclase feldspar called
labradorite with 55% anorthite (CaAl2Si2O8) and 45% albite (NaAlSi3O8).

The Olivine Group is a solid solution series between magnesian forsterite (Mg2SiO4) and ferroan
fayalite (Fe2SiO4). Use the Mg and Fe cation values you calculated in #1 to determine the
forsterite content (Fo%) of the olivine.

Problem #4
After determining the formulas of the two pyroxene analyses above, plot them on the following
quadrilateral. You could take your calculated mineral formulas, write a reaction for each
pyroxene using the three components (Ca2Si2O6, Mg2Si2O6, Fe2Si2O6), normalize to 100%, and
then plot the results. However, the difference
between the three components is the cations: Ca,
Mg, and Fe. You have already calculated the cation
abundances above, so you can simply normalize the
three cations to 100% and plot the percentages!
Plot the two points on the quadrilateral. Compare
your plot to Figure 3. Provide a mineral name for
each pyroxene. Are there any other significant
chemical components in your pyroxenes? If so,
what are they? Would the consideration of other
components affect the naming of the minerals?
Figure 3. Classification of pyroxenes (modified
after Morimoto, 1988 and Deer et al., 1992).

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Problem #5
The mineral data that you used above came from a series of high-pressure (1.0 GPa) experiments
investigating the melting systematics of peridotite. The coexisting pyroxenes (Opx and Cpx) in
these experiments define a miscibility gap in temperature-composition space from 1235C to
1360C (Schwab & Johnston, 2000). As a result, the CaO content of the clinopyroxene in this
sample can be used as a geothermometer. Using the following equation, calculate the
temperature of last equilibration of the peridotite.

Temp (C) = -2.0914x2 + 44.164x + 1131.4, where x = wt. % CaO in cpx

What is the temperature of the experiment?

Give an example and provide a literature or web reference of another geothermometer.

ACKNOWLEDGEMENTS
This exercise is an offshoot of one developed at the University of Oregon. I am indebted to A.
D. Johnston, J. M. Rice, and K.V. Cashman for innumerable things, including the foundation of
this exercise.

REFERENCES
Deer, W.A., Howie, R.A., and Zussman, J. (1992) An introduction to the rock-forming minerals.
2nd edition, 696 p. John Wiley and Sons, New York.
Morimoto, M. (1988) Nomenclature of pyroxenes. Mineralogy Magazine, 52, 535-550.
Schwab, B.E. and Johnston, A.D. (2000) Contrasting pyroxene phase relations during partial melting
of compositionally variable peridotites. EOS Transactions of the American Geophysical Union,
81, 1299.
Schwab, B.E. and Johnston, A.D. (2001) Melting systematics of modally variable, compositionally
intermediate peridotites and the effects of mineral fertility. Journal of Petrology, 42, 1789-1811.