Quantitative Analysis: Bence-Albee Corrections

5.3.1. Example Bence-Albee Calculation

Unknown concentrations are determined by inserting K values into the equation for b in the place of concentrations and finding initial concentrations. These new concentrations are reinserted to find new bs and the iteration is repeated until convergence in achieved. Two or three iterations usually result in values very close to the theoretical composition. In this example, the "unknown" will be a synthetic grossularite garnet, Ca₃Al₂Si₃O₁₂. The standards that were used are:

	Oxide Weight Fraction
	Al₂O₃	SiO₂	CaO
Corundum	1.000
Quartz		1.000
Wollastonite		0.517	0.483

The appropriate values of a (Albee and Ray 1970) are:

	a
	Al₂O₃	SiO₂	CaO
Al	1.00	1.05	1.18
Si	1.54	1.00	1.07
Ca	1.08	1.10	1.00

Note that all a >= 1, indicating that absorption dominates.

Calculating the bs for the standards yields:

Corundum	b-Al = 1.00 x 1.000 = 1.000
Quartz	b-Si = 1.00 x 1.000 = 1.000
Wollastonite	b-Ca = 1.10 x 0.517 + 1.00 x 0.483 = 1.052

The measured X-ray counts corrected for background and deadtime were:

	Unknown	Standard
Al	29146	140126
Si	19764	56471
Ca	26971	34358

First, we need to make an initial approximation of the composition of the unknown. To do this we calculate initial K factors for the unknown:

Bence Albee 2

So,

K-Al₂O₃	(29146 / 140126) x (1.000 / 1.000) = 0.2080
K-SiO₂	(19764 / 56471) x (1.000 / 1.000) = 0.3500
K-CaO	(26971 / 34358) x (0.483 / 1.052) = 0.3604

These K-factors total to 0.9184. Next, we make an approximation of the bs using these K values and the as above. For this initial approximation, the K values are used as estimates of the concentrations:

Al₂O₃	(1.00 x 0.2080 + 1.05 x 0.3500 + 1.18 x 0.3604) / 0.9184 = 1.090
SiO₂	(1.54 x 0.2080 + 1.00 x 0.3500 + 1.07 x 0.3604) / 0.9184 = 1.150
CaO	(1.08 x 0.2080) + 1.10 x 0.3500 + 1.00 x 0.3604 / 0.9184 = 1.056

Now we can use these bs and the K values to calculate a better approximation to the actual concentrations:

Al₂O₃	1.090 x 0.2080 = 0.2267
SiO₂	1.150 x 0.3500 = 0.4025
CaO	1.056 x 0.3604 = 0.3806

The new sum is 1.0098. This total can be improved by iteration. We refine the estimates of the bs using the weights just calculated:

Al₂O₃	(1.00 x 0.2267 + 1.05 x 0.4025 + 1.18 x 0.3806) / 1.0098 = 1.088
SiO₂	(1.54 x 0.2267 + 1.00 x 0.4025 + 1.07 x 0.3806) / 1.0098 = 1.148
CaO	(1.08 x 0.2267 + 1.10 x 0.4025 + 1.00 x 0.3806) / 1.0098 = 1.058

And use the refined beta-factors to get new concentrations:

Al₂O₃	1.088 x 0.2080 = 0.2263
SiO₂	1.148 x 0.3500 = 0.4018
CaO	1.058 x 0.3604 = 0.3816

The new total is 1.0094. Continued iteration would improve this number a little. Compare the values for the ideal composition:

	Weight Fraction
Al₂O₃	0.2263
SiO₂	0.4002
CaO	0.3735

In Bence-Albee corrections, standards of compositions similar to the unknowns are often used to minimize Z effects. In addition, because the effect of fluorescence in silicates is very small, only absorption need be considered. Bence-Albee calculations require very little computing time or memory, but with more powerful on-line computer this is no longer a significant advantage and the more complicated ZAF-type corrections can be made.

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Last update: 01/18/2006 01:47 PM.