Using Neural Networks for Quantitative X-Ray Analysis

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Using Neural Networks for

Quantitative X-ray Analysis

John N. Driscoll, PID Analyzers,LLC, Sandwich, MA

&

Walter Johnson, Physics Dept.,Suffolk Univ., Boston, MA

Paper #1360 - 10


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Introduction

The overlap of Pb and As spectra is a well

known phenomenon in XRF

At Pittcon 2002 (paper #1968P ), wepresented a neural network that recognized

overlapping Pb and As peaksin an XRFspectrum. This was done using a singleneuron in the network.

Now, we attempt for the first time to predict

the concentrationsfrom Pb and As X-rayspectra using input, output and hiddenneurons.


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A technique for the analysis ofmaterials that is based onfundamental principles ofspectroscopy

Atomic spectra are well knownSince we are dealing with atoms,

absorption and emission lines areat the same wavelength

Absorption and emissioncoefficients are well known and canbe used for correction of spectra(inter element corrections) via

libraries

What is XRF?


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Model XR1000


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Fits to As & Pb


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Goals:

To design and test a simple neural network which can use data from

selected channels of an X-ray spectrometer to determinequantitatively the amounts of Pb and As present.

Considerations:

1) The major peak in both As and Pb (K-alpha and L-alpharespectively) exactly overlap. The smaller beta peaks are

distinguishable and must be used.

2) The neural net should have as few neurons as possible to allow

expansion to other elements.

3) The training set for the neural net should cover a large number of

relative proportions of As and Pb and ideally should be generated

by computer from normalized shapes which well represent the data.


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What does a neural network do?

A neural network is a mathematical procedure modeled after biological

brains (network of neurons) to produce a desired set of output valuesfrom a set of input values.

Structure: The net has inputs which are numbers usually scaled

between 1 and 1. Each input is fanned out to many different

processing units (neurons) and each of these sends its output to moreneurons. The final set of neurons (the output layer) gives the answer as

numbers each ranging from 1 to 1.

Function: Any given neuron weights each of its inputs by some

different factor (any positive or negative number), adds up all of these

weighted inputs, checks to see if a numerical threshold has beenreached and fires or does nothing. If the neuron fires, then it sends

out its own signal (a value of 1) to other receiving neurons. Not firing

results in an output of 0.


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Training: The net goes through a training process where it is presented

with many different patterns of numbers at its inputs and also the desired

output for each pattern. The software used to produce the net starts witha random set of weights. Then one member of the training set is

randomly selected and slight changes are made in all of the weights so

the net is more likely to produce the desired output than before the

weights were changed.

Back propagation: This training process typically starts with the output

layer of neurons and proceeds backwards to the input layer (back

propagation). It is repeated many times (hundreds of thousands or

millions) until the weights are correct and the desired results are

obtained for most if not all input patterns.

Testing: The final step is to test the behavior of the network on data it

has not seen (i.e. which was not part of its training set). If it does not

work then more neurons may be added to the net to enhance it learning

capabilities. In general the fewer neurons needed to solve a problem, the

better.


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As-10,000 PPM, Pb- 0 PPM

Gaussian fits to peaks

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As spectrum with Mg(CO3)2 backgroundSpectral shape used for generation of neural net training sets


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Pb spectrum with Mg(CO3)2 background

Spectral shape used for generation of neural net training sets

Pb-10^4 PPM Pb, As- 0 PPMGaussian fits to peaks

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Pb & As SampleMatrix

Sample

#

ppm Pb ppm As

B blank Blank

1 1000 9000

2 2500 7500

3 5000 5000

4 7500 2500

5 9000 1000

6 10000 0

7 0 10000


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Generation of neural net training sets

The Gaussian fit for As , the Gaussian fit for Pb, and the linear fit for the

background were each separately normalized to one. Multiplying each by an arbitrary number of counts and adding the three

contributions together produced a spectrum containing some mix of As,Pb, and background. The determination of the meaning of the area of eachcontribution was determined from the samples with only As and only Pbpresent. (Samples 6 and 7).

N1= total counts in the As spectrum (area of the Gaussian fit)

B1= total counts in the background for sample with only As present

(N1/B1)= 1(1) where 1= density of As in the sample in PPM Using the fit for the sample with only 104 PPM As and background

present, we determined the proportionality constant 1. A similarprocedure was used for the Pb spectrum.

Many spectral shapes were generated for training the net by varying thedensity of As and Pb present


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Ratio of transition

probabilities


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Combined As, Pbspectra


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Comparison of Measured & Calculated (NN)Spectra

Fit to PbAs: 18P386A

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Generation of Neural Net Training Sets

The Gaussian fit for As , the Gaussian fit for Pb, and the linear fit for the

background were each separately normalized to one.

Multiplying each by an arbitrary number of counts and adding the threecontributions together produced a spectrum containing some mix of As,Pb, and background. The determination of the meaning of the area ofeach contribution was determined from the samples with only As andonly Pb present. (Samples 6 and 7).

N1= total counts in the As spectrum (area of the Gaussian fit)

B1= total counts in the background for sample with only As present

(N1/B1)= 1(1) where 1= density of As in the sample in PPM

Using the fit for the sample with only 104

PPM As and backgroundpresent, we determined the proportionality constant 1. A similarprocedure was used for the Pb spectrum.

Many spectral shapes were generated for training the net by varying thedensity of As and Pb present


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Pb spectrum fit

The same procedure was used to fit the pure Pb spectrum-

a main Gaussian and a smaller Gaussian shoulder locatedat 0.972 of the energy of the main peak.

The counts in channel x due to Pb L-alpha and L-beta are

given by :a2 (x) + b2 (x)

where the subscript 2 refers to Pb rather than As.

Background

The background is well represented by 3 straight lines oneunder the alpha peak , another between peaks and the lastunder the beta peak.


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Neural Net Determination of Relative Amounts of As and Pb

A neural network has the capability of learning verynon-linear relationships between many input values

and one or more outputs. To determine the relative amounts of As and Pb in a

sample, different portions of the spectrum from thealpha peak, K- beta (As) and L-beta (Pb) are usedalong with background outside the peak areas. Inparticular ratios of the counts in these regions of thespectrum can be used to determine the ratio (g1/g2)where

g1= fraction by weight of As present in the sample

g2= fraction by weight of Pb present in the sample The network must be trained on a very large number

of spectra representing many different values of g1and g2 present in the sample. To produce this trainingset we use analytical functions to generate the

spectra.


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Summary

We demonstrated previously (Pittcon2002) that we could identify Pb and Asoverlapping peaks using a single neuron

We wanted to modify our neural net

training program by running the matrix ofconcentrations in slide # 12 andincreasing the number of spectra in ourtraining set

We found that with 5 input neurons, twooutput neurons and a hidden neuron, wewere able to calculate concentrations thatagreed with the concentrations prepared

and calculated by xrf software

Using Neural Networks for Quantitative X-Ray Analysis

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