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, Ca_{3}Al_{2}Si_{3}O_{12}. The standards that were used are:

Oxide Weight Fraction 


Al_{2}O_{3} 
SiO_{2} 
CaO 
Corundum 
1.000 


Quartz 

1.000 

Wollastonite 

0.517 
0.483 
The appropriate values of a (Albee and Ray 1970) are:

a 


Al_{2}O_{3} 
SiO_{2} 
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 
bAl = 1.00 x 1.000 = 1.000 
Quartz 
bSi = 1.00 x 1.000 = 1.000 
Wollastonite 
bCa = 1.10 x 0.517 + 1.00 x 0.483 = 1.052 
The measured Xray 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:
So,
KAl_{2}O_{3} 
(29146 / 140126) x (1.000 / 1.000) = 0.2080 
KSiO_{2} 
(19764 / 56471) x (1.000 / 1.000) = 0.3500 
KCaO 
(26971 / 34358) x (0.483 / 1.052) = 0.3604 
These Kfactors 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:
Al_{2}O_{3} 
(1.00 x 0.2080 + 1.05 x 0.3500 + 1.18 x 0.3604) / 0.9184 = 1.090 
SiO_{2} 
(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:
Al_{2}O_{3} 
1.090 x 0.2080 = 0.2267 
SiO_{2} 
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:
Al_{2}O_{3} 
(1.00 x 0.2267 + 1.05 x 0.4025 + 1.18 x 0.3806) / 1.0098 = 1.088 
SiO_{2} 
(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 betafactors to get new concentrations:
Al_{2}O_{3} 
1.088 x 0.2080 = 0.2263 
SiO_{2} 
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 
Al_{2}O_{3} 
0.2263 
SiO_{2} 
0.4002 
CaO 
0.3735 
In BenceAlbee 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. BenceAlbee calculations require very little computing time or memory, but with more powerful online computer this is no longer a significant advantage and the more complicated ZAFtype corrections can be made.
Back: 5.3. BenceAlbee Matrix Corrections  Next: 5.4. ZAF Matrix Corrections  Home: Course Overview
Copyright 19972003, James H. Wittke
Last update: 01/18/2006 01:47 PM.