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.

Copyright 1997-2003, James H. Wittke

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