LASER INDUCED BREAKDOWN SPECTROSCOPY: INVESTIGATION OF LINE

PROFILES, SLURRIES AND ARTIFICAL NEURAL NETWORK PREDICTION

By

Seong Yong Oh

A Dissertation Submitted to the Faculty of Mississippi State University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Engineering with an Emphasis in Applied Physics in the Department of Physics and Astronomy

Mississippi State, Mississippi

December 2007

Copyright by

Seong Yong Oh

2007

LASER INDUCED BREAKDOWN SPECTROSCOPY: INVESTIGATION OF LINE

PROFILES, SLURRIES AND ARTIFICAL NEURAL NETWORK PREDICTION

By

Seong Yong Oh

Approved:

______Jagdish P. Singh David L. Monts Adjunct Professor of Physics and Professor and Graduate Coordinator of Research Professor of Institute for Clean Physics Energy Technology (Co-Major Professor) (Director of Dissertation and Major Professor)

______Fang Yu Yueh Chun Fu Su Research Scientist II of Institute for Clean Professor of Physics Energy Technology (Committee Member) (Committee Member)

______Thomas Philip Chuji Wang Professor of Computer Science and Assistant Professor of Physics Engineering (Committee Member) (Committee Member)

______W. Glenn Steele, Jr. Interim Dean of the Bagley College of Engineering

Name: Seong Yong Oh

Date of Degree: December 14, 2007

Institution: Mississippi State University

Major Field: Applied Physics

Major Professor: Dr. Jagdish P. Singh

Title of Study: LASER INDUCED BREAKDOWN SPECTROSCOPY: INVESTIGATION OF LINE PROFILES, SLURRIES AND ARTIFICAL NEURAL NETWORK PREDICTION

Pages in Study: 129

Candidate for Degree of Doctor of Philosophy

Laser induced breakdown spectroscopy (LIBS) was tested to examine its

applicability to remote and in suit analysis in inaccessible situation. Two types of liquid

sample (slurry) prepared for simulating vitrification of liquid hazardous wastes was

tested. In situ analysis ability makes the LIBS technique practical for analysis of the

slurry samples during vitrification, which is in inaccessible situation. For the first slurry

sample, two slurry circulation systems were devised to overcome major technical

problems associated with LIBS measurements of slurry samples - namely sedimentation

and change in the lens-to-sample distance (L.T.S.D) during measurement. The second

slurry sample contained less water and is able to be managed in a small glass container

during test. We applied direct analysis of slurry sample filled in glass container.

Spectroscopic analysis was performed using two different detection systems:

Czerny-Turner and Echelle spectrometer systems. In particular, spectroscopic analysis of

data from an echelle spectrometer shows the high efficiency for simultaneously

determining physical quantities of all elements of interest.

We also evaluate LIBS technique to tin alloy samples for the purpose of

quantitative analysis by using Echelle spectrometer system. Unknown samples without information of elemental composition were tested to estimate several sample compositions simultaneously. An artificial neural network, calibration method, and

chemical analysis were applied to estimate the elemental concentrations of impurities in

tin (Sn) alloy.

Key words: Slurry, Laser induced breakdown spectroscopy, Line profile

ACKNOWLEDGMENTS

I would like to thank Dr. Kuntz, Dr. Monts and my advisor professor Dr. Singh.

Theoretical part of my thesis is based on a paper written by Dr. Kuntz. When I had trouble in finding the error of computer coding about the Voigt line profile, Dr. Kuntz sent me his FORTRAN coding. With the help of his coding, I found the error and completed the coding. Dr. Monts gave me good correction and desirable direction of my writing. In particular, Dr. Monts’s direction is like lamp in dark for my writing. I really appreciate my advisor, Dr. Singh for hiring me as a graduate student. I would like to thanks Institute for Clean Energy Technology (ICET) for providing the financial assistance.

ii TABLE OF CONTENTS

ACKNOWLEDGMENTS ...... ii

LIST OF TABLES...... vi

LIST OF FIGURES ...... viii

CHAPTER

I. INTRODUCTION ...... 1 1.1 LIBS Advantages and Disadvantages...... 4 1.2 Analysis of Liquid Sample...... 5 1.3 The Objective of This Dissertation...... 6

II. PHYSICS OF LIBS LINE SHAPE & ARTIFICIAL NEURAL NETWORKS...... 7 2.1 Plasma Formation by Laser Pulse...... 7 2.2 Spectral Line Intensity ...... 9 2.3 Local Thermodynamic Equilibrium...... 10 2.4 Voigt Profile...... 11 2.5 Relation between Lorentz and Gaussian half width ...... 16 2.6 Stark Profile for Isolated Neutral Atom Lines...... 21 2.7 Line Broadening Width...... 26 2.7.1 Doppler Broadening Width...... 26 2.7.2 Stark Broadening Width ...... 27 2.8 Self-Absorption...... 28 2.9 Curve of Growth ...... 30 2.10 Spectral Line Intensity in Optical Thin Plasma ...... 35 2.11 Partition Function and Plasma Temperature...... 36 2.12 Artificial Neural Networks ...... 40 2.12.1 Back-Propagation Learning Model...... 40 2.12.2 Cascade-Correlation Learning Model...... 43 2.12.3 Software and Data Analysis...... 44

III. EXPERIMENTAL...... 45 3.1 Spectrometers...... 47 3.1.1 Czerny-Turner Spectrometer System...... 47 iii 3.1.2 Echelle Spectrometer System ...... 53

IV. LIBS APPLICATION TO IN SITU ANALYSIS OF SLURRY SAMPLES PREPARED FOR THE PURPOSE OF SIMULATING THE VITRIFICATION PROCESS OF LIQUID RADIOACTIVE WASTES...... 64 4.1 Experimental Description ...... 64 4.2 Results and Discussion ...... 65 4.2.1 Sample Preparation ...... 65 4.2.2 Spectroscopic Comparison...... 66 4.3 Result Recorded by Czerny-Turner Spectrometer System ...... 67 4.3.1 Reproducibility of Line Emission Intensity...... 67 4.3.2 Laser Power Effect and Calibration Curve of Normalized Intensity...... 69 4.4 Result Recorded by Echelle Spectrometer System...... 72 4.4.1 Electron Density...... 72 4.4.2 Plasma Temperature...... 73 4.4.3 Calibration Curve...... 76 4.5 Conclusion ...... 79

V. COMPARATIVE STUDY OF LASER INDUCED BREAKDOWN SPECTROSCOPY MEASUREMENT USING TWO SLURRY CIRCULATION SYSTEMS...... 80 5.1 Experimental Description ...... 80 5.2 Slurry Handling System...... 82 5.3 Results and Discussions...... 84 5.3.1 Effect of Laser Pulse Rate...... 84 5.3.2 Slurry Circulation Systems ...... 87 5.3.3 Calibration...... 91 5.4 Conclusion ...... 93

VI. TIN ALLOY ANALYSIS COMBINED WITH ARTIFICIAL NEURAL NETWORK PREDICTION ...... 94 6.1 Experimental Description ...... 94 6.2 Data Analysis...... 95 6.3 Results...... 97 6.3.1 Effects of Crater Size...... 97 6.3.2 Voigt Profile of Sn I 333.062 nm...... 99 6.3.3 Reproducibility ...... 101

iv 6.4 Calibration Method and Artificial Neural Network...... 103 6.5 Conclusion ...... 107

VII. CONCLUSION...... 108

APPENDIX

A. CALCULATION OF THE NEEDED SLIT WIDTH TO ACHIEVE CONSTANT BANDWIDTH ...... 110

B. VOIGT PROFILE ...... 114

REFERENCES ...... 119

v

LIST OF TABLES

2.1 The values of Voigt function as a function of a-parameter values46 ...... 14

2.2 The estimated spectroscopic information obtained by Voigt fitting of Al I 309.271 nm by using Origin software. R is correlation coefficient ...... 16

2.3 Values of Lorentzian fraction, Gaussian fraction and empirical function as a function of a-parameter values ...... 19

2.4 List of Fe I spectral information used for Boltzmann plot ...... 39

3.1 Instrument information for three different LIBS systems ...... 46

3.2 Spectral parameters based on HR 460 spectrometer. d: grating spacing, f: focal length of focusing mirror, φ: Ebert angle, SW: slit width (μm), RLP: reciprocal linear dispersion, BW: bandwidth to pass through the instrument91 ...... 50

3.3 Look- up table to achieve the constant bandwidth (Δλ = 0.176 nm) based on HR 460 spectrometer. 1/d: 1200mm-1, ƒ: 460 mm, φ: 16.5°, m: 1 ... 51

3.4 Spectral parameters based on HR 460 spectrometer system ...... 52

3.5 Spectral parameters based on ESA 3000 Analyzer system ...... 57

4.1 List of the Fe I spectral information used for Boltzmann plot ...... 74

5.1 Optimal experiment parameters employed for LIBS measurements ...... 89

6.1 The composition of tin alloy (Sn) employed for LIBS measurement. All values are in wt (%) ...... 95

6.2 The estimated spectroscopic information obtained by Voigt fitting of Sn I 333.062 nm by using Origin software. R is the correlation coefficient ...... 101

6.3 Selected analyte lines and RSD values from four samples (Sn_0 ~Sn_3) .... 101

vi

6.4 Statistical scores in neural training ...... 103

6.5 Comparison of neural predictions and calibration results of Ag, Cu, and Pb from the chemical analysis ...... 105

6.6 Comparison of neural predictions for raw and optimized inputs ...... 106

A.1 Look- up table to achieve the constant bandwidth (Δλ = 1nm) of Czerny- Turner spectrometer 1/d: 1200 mm-1, ƒ: 500mm, φ: 5°, m: 1 ...... 113

vii

LIST OF FIGURES

2.1 Voigt profile as a function of a-parameter calculated using polynomial approximation proposed by Kuntz46 ...... 13

2.2 Theoretical Voigt profile fitted to the experiment data point (Al I 309.271 nm) using Levenberg-Marquardt algorithm installed in Origin software ...... 15

2.3 Plots of empirical functions as a function of a-parameter proposed by Posener,48 Kastner49 and Whiting50 ...... 17

2.4 Plots of Pseudo-Voigt, Voigt line profile. Solid line and Open circles represent Voigt and Pseudo-Voigt profile with 1.2 of a-parameter value ...... 20

2.5 De-convoluted line profile with 1.2 of a-parameter value. Solid line (—), dashed line (– – –) and (---) represent Pseudo-Voigt, Lorentzian and Gaussian contribution ...... 20

2.6 Holtzmark micro-electric field distribution ( R = 0.0 ) and micro-electric field distribution ( HR(β) ) at R = 0.4, 0.6 ...... 23

2.7 Asymmetric line profiles by the electron and ion Stark effect as a function of A-values ...... 24

2.8 Asymmetric line profiles due to electron and ion Stark effect as a function of electron impact parameter we at A-value (0.2) and de (0.0) ...... 25

2.9 Plot of ∆λ/∆λ0 according to absorption coefficient (SA). ∆λ and ∆λ0 correspond to the observed and non-self-absorbed broadening widths, respectively. The equation of this plot was explicitly followed from reference 74 ...... 29

2.10 Series of theoretical COG curves according to a-parameter value for a homogenous plasma ...... 32

2.11 Series of theoretical Duplication curves as a function of a-parameter value for a homogenous plasma ...... 33

viii

2.12 Curves of I/I0 according to absorption coefficient (SA). Solid line 0.46 represents I/I0 = (SA) and triangle represent the numerical result of Equation 2- 46 ...... 34

2.13 Partition function of Fe I based on Irwin’s and Galan’s polynomial approximation ...... 37

2.14 Boltzmann plots of Fe I obtained from tin (Sn) alloy (gate delay 1000 ns and gate width 4 μs) ...... 38

2.15 Neural network architecture with three layers ...... 41

2.16 An artificial neuron ...... 42

2.17 Cascade-correlation architecture after the addition of one hidden units.24 Square boxed connections (□) are frozen. × connections are trained ...... 43

3.1 Schematic diagram of LIBS measurement apparatus ...... 45

3.2 Czerny-Turner optical configuration. θi is the incidence angle, θr is the diffraction angle, m is the diffraction order, f is the focal length of mirror, φ is the Ebert angle, and ρ is the grating rotation angle ...... 48

3.3 The calibration curve of the needed slit width to achieve constant bandwidth based on HR460 spectrometer. Slit-width for constant bandwidth (∆λ = 0.176 nm). 1/d = 1200 mm-1, ƒ = 460 mm, φ = 16.5°, m = 1 ...... 51

3.4 CCD image and LIBS spectrum of HR 460 spectrometer system ...... 52

3.5 Configuration of echelle grating ...... 53

3.6 A part of the Hg spectrum obtained from echelle (a) and HR 460 (b) spectrometer systems, respectively ...... 55

3.7 Experimental setup of ESA 3000 Analyzer (This figure is copied from the manual produced by LLA Instrument GmbH) ...... 57

3.8 CCD image and LIBS spectrum for echelle spectrometer ...... 59

3.9 (a), (b) and (c) indicate CCD image, digitalized spectrum, and averaged spectrum of Mg I 285.213 nm transition ...... 60

3.10 Blooming due to saturation of the Sn I 380.100 nm transition ...... 62

ix

3.11 Two dimensional echelle CCD image of a deuterium (a) and spectrum response of a deuterium continuum source from diffraction orders 85 to 71 (b) ...... 63

4.1 Schematic diagram of LIBS measurement apparatus ...... 65

4.2 LIBS spectra of raw Sludge Receipt and Adjustment Tank (SRAT) slurry around 386 nm. The spectra of dashed line and solid line are recorded by SPEX 500M and echelle spectrometers, respectively ...... 66

4.3 Reproducibility trend of the absolute and normalized intensity of Fe I 382.043 nm line for 7 days. Solid line and dashed line indicate mean value of absolute intensity and normalized intensity for Fe I 382.043 nm ...... 68

4.4 The absolute and normalized intensity variation of Fe I 382.043 nm line as a function of laser energy. The dashed line indicates mean value of normalized intensity for Fe I 387.857 nm ...... 70

4.5 Calibration curve for Na I 330.237 nm line recorded by Czerny-Turner spectrometer system (SPEX 500M) ...... 71

4.6 Electron density of Fe I 381.584 nm and Al I 396.271 nm as a function of gate delay time (gate width 2 μs) ...... 72

4.7 Boltzmann plots of Fe I obtained from the SRAT slurry (gate delay 800 ns and gate width 2 μs) ...... 73

4.8 Excitation and ionization temperatures of SRAT slurry (gate width 2 μs) ... 75

4.9 Calibration curve for Li I 670.790 nm line using glass frit (70 mJ/pulse, gate delay 1 μs and gate width 4 μs) ...... 76

4.10 Calibration curve for Si I 288.158 nm line using glass frit (70 mJ/pulse, gate delay 1 μs and gate width 4 μs) ...... 77

4.11 Calibration curve of Si I 288.158 nm by using silicon dioxide (SiO2) (80 mJ/pulse, gate delay 300 ns and gate width 4 μs) ...... 77

4.12 Saturation of the Ca II 393.365 nm transition on spectrum (80 mJ/pulse, gate delay 300 ns and gate width 4 μs) ...... 78

5.1 Schematic diagram of LIBS and circulation systems for slurry sample ...... 81

5.2 Reproducibility trend of the intensity of Fe I 382.043 nm line obtained with circulation system I ...... 83 x

5.3 Experimental arrangement of direct slurry sampling for studying the effect of laser pulse frequency ...... 85

5.4 Comparison of intensity (a.u.) and RSD value (%) obtained from LIBS measurements with different laser pulse frequencies ...... 86

5.5 (a) LIBS spectrum with circulation system I and (b) LIBS spectrum with circulation system II ...... 88

5.6 The intensity variation of Fe I 385.991 nm line with changes in focal location above and inside the surface for (a) circulation system I and (b) circulation system II. The “0”, + value, and - value position indicate a focal location on, above and inside sample ...... 90

5.7 The intensity variation of Fe I 382.043 nm line during slurry sampling with circulation system I ...... 91

5.8 Calibration curve for Fe I 387.857 nm line with (a) circulation system I and (b) circulation system II ...... 92

6.1 Computed output by neural network training of calibration set as a function of silver (Ag) concentration ...... 96

6.2 L, M and S correspond to large, medium and small circles of ablation traces ...... 97

6.3 The intensity variation of Sn I 333.062 nm (a) and Cu I 327.396 nm (b) line with time during the measurement. L, M and S correspond to large, medium and small circles of ablation traces ...... 98

6.4 The intensity ratio of Cu I 327.396 nm to the reference line Sn 333.062 nm during sampling. L, M and S correspond to large, medium and small circles of ablation traces in Fig. 6.2 ...... 99

6.5 Sn I 333.062 nm line profile. Theoretical Voigt profile fitted to the experiment data point using Levenberg-Marquardt algorithm installed in Origin software ...... 100

6.6 Calibration curves of Bi, Ag, Pb, Cu in reference to Sn I 333.062 nm line ...... 102

6.7 Calibration curve and validation data of Ag I 338.289 nm, Cu I 327.396 nm and Pb I 405.782 nm lines as a function of elemental concentration. Validation data extracting elemental concentration (wt %) are obtained by calibration method, neural prediction and chemical analysis ...... 104 xi

CHAPTER I

INTRODUCTION

Breakdown phenomena that emit bright light can be induced if focused light

energy is incident onto target material placed near the focal point. Based on this principle,

Brech and Cross showed that a laser beam can be used as an excitation source of sample in 1962.1 They observed a micro-size plasma plume and were able to record a spectrum

when a laser was focused onto a metal surface. Laser induced breakdown spectroscopy

(LIBS) is the spectroscopic method which analyzes optical emission spectra from laser-

produced plasma plumes.

The strong electric field of a laser pulse induces ablation, atomization, ionization, and excitation of the sample. The spectral information which reflects the existence of characteristic atoms in the sample can be obtained from these optical emission spectra.

The optical emission spectra decay at their own rates because the laser-produced plasma plume optical radiation source evolves in time after the laser pulse strikes the sample. An intense continuum appears in the early stages of plasma formation due to bremsstrahlung radiation and a strong stark effect. 2 ,3 Time-gated measurement of optical emission spectra is necessary in LIBS experiments. In 1969, Scott presented time-resolved atomic spectra and spatial photographs of laser-produced plasma plumes of an aluminum alloy

1 using a gating device and high speed framing camera.4 The LIBS technique allows in situ

analysis for detecting and monitoring hazardous material. In 1982 and 1987, Radziemski

and Creamers applied the LIBS technique to detect toxic beryllium in air and uranium in

solution.5 ,6 Moreover, the applicability of an optical fiber which delivers the laser pulse to create laser-induced spark plasma on the sample surface promotes use of LIBS as a detection tool in hazardous environments. In 1996, Marquardt et al use a fiber-optic probe to detect the lead paint.7

Individual line emission intensity reflecting the relative population of neutral or ionic excited species in the plasma can provide a determination of multi-elemental concentrations in the sample, assuming on that all elements of the sample are homogeneously distributed.

In 1999, Gornushkin et al. applied the theoretical curve of growth (COG) method

to laser-produced plasma spectroscopy; this can explain the behavior of non-linear

growth in optically thick conditions.8 However, the matching between theoretical and

experimental COGs is difficult because the physical units of the two curves are different.

Ciucci et al developed a calibration-free method that can predict all

concentrations of the sample under plasma conditions in the optical thin regime and

assuming local thermal equilibrium. 9 This calibration-free method requires two

assumptions; (1) plasma is in local thermodynamic equilibrium (LTE) and (2) the

radiation source is the optically thin. Bulajic et al. improved the calibration-free method by considering the effect of self-absorption appearing in the optically thick regime.10

2

The calibration method uses a suitable calibration curve based on absolute intensity and on an intensity ratio. The LIBS signal of a sample without information of its composition is fitted to the calibration curve of a given element and its concentration is predicted. However, this method requires a calibration curve with a similar matrix to reduce the matrix effect. The strength (intensity) of an elemental signal depends on the surrounding sample composition; the phenomenon is called the matrix effect.11 ,12 For example, the spectral responses for silicon (Si) are different from metal alloy, soil and liquid samples even if the three samples have same silicon (Si) concentration.13

Artificial neural networks (ANNs) are computer models which have an ability to

improve the efficiency of the analysis process. ANNs are based on pattern recognition, assigning multivariate data into predefined categorizes.14 Pattern recognition in the ANN technique makes it possible to invert observational data into physical quantities.15 ANNs

have been applied in the research fields of clinic therapeutics,16 bio-medicine,17 atomic- level classification, 18 x-ray spectroscopy, 19 plasma spectroscopy, 20 , 21 Raman spectroscopy22 etc. In 1986, Rumelhart et al. presented a multi-layer neural network

trained with back propagation (BP). 23 Unfortunately, the BP algorithm is very time

consuming for a big neural network. Fahlman et al. developed an algorithm for a multi-

layer neural network designed to overcome the slow-pace learning performance of back-

propagation.24 It is known as the cascade-correlation (CC) learning model. In the case of

LIBS analysis, Philip et al. used artificial neural networks on LIBS spectra to identify

elements present in exhaust gases.25 Sirven et al. employed an ANN to predict chromium

3

(Cr) concentration in soil using LIBS spectra.26 Recently, the LIBS technique has been

popular because of its intrinsic advantages and significant developments of instrument etc

(broadband spectrometer with high resolution and intensified charge coupled device).

Since 1980, the number of publications in the field of LIBS analysis has been growing

exponentially.27

1.1 LIBS Advantages and Disadvantages

Use of an optical fiber, minimal sample preparation and quick on-line elemental

analysis are distinguishable marks of a LIBS probe, which make it practicable to apply in

inaccessible places.28 ,29 The typical laboratory analytical tools are generally unsuitable

for field application. For example, inductively coupled plasma (ICP) requires dissolution

of the sample to spray it into a plasma source in the form of an aerosol. On the other

hand, LIBS technique has the ability for direct detection that can reduce if not eliminates

time involved in sample pre-treatment processes. Moreover, fast direct analysis and

availability of optical fibers can avoid the risk of hazardous reagents and contamination

of the sample. The LIBS technique can analyze conducting as well as non-conducting

materials because of the use of a laser beam.

However, LIBS technique is not as accurate as typical laboratory analytical tools

(e.g., ICP) due to poor signal reproducibility. 30 Poor signal reproducibility mainly

attributed to shot-to-shot plasma fluctuation is the major drawback of LIBS technique.31

The LIBS signal is easily affected by several experimental parameters and environmental conditions, such as the fluctuation of the laser energy, surface features of the sample,

4

nature of the sample materials (matrix elements) and location of focus spot, etc. Due to

the sensitivity of LIBS signal by experimental parameters and environmental conditions,

it results in poor measurement reproducibility. The LIBS technique can be most popular

analytical tool if the reliability of quantitative analysis significantly improves.

1.2 Analysis of Liquid Sample

The LIBS technique fundamentally allows detecting the gases,32 ,33 ,34 solids,35 ,36

,37 and liquids samples.38 , 39 Application of the LIBS technique to liquid sample analysis

is challenging owing to sedimentation, turbulence, and the short lifetime of the plasma on

the slurry’s surface. Since a micro-plasma is created on the surface, sample preparation

with contents homogenously distributed consequently can be an essential parameter to

get analytically useful signals for quantitative purposes in LIBS measurements.

Sedimentation during data acquisition might be a serious obstacle for getting good quality data. Splashing of the slurry sample surface by the laser-induced shockwave also raises fluctuations in the lens-to-sample distance (L.S.T.D.). The geometric configuration of the liquid sample container in which the laser pulse strikes the slurry also influences the

LIBS signals. The ablated slurry samples create droplets and aerosols above the slurry surface, cause fluctuations in the breakdown spatial location, and deplete both the laser energy and the LIBS signal in the detection region. In other words, the spark that should be generated on the surface of the slurry is in the air because the vapor keeps the laser pulse from reaching the slurry surface. It may also disrupt the laser light and emission line’s path.

5

1.3 The Objective of This Dissertation

The first objective of this dissertation is to examine the applicability of LIBS as a remote and the real-time analytical tool that can detect liquid radioactive wastes.

Vitrification of liquid radioactive wastes is an important issue, facilitating the handling and safely store of radioactive waste for a long time.35 Quantitative analytic tools are

necessary to analyze glass melts during the vitrification process of liquid radioactive

wastes. In situ analysis ability makes LIBS technique practicable for analysis of slurry

samples during the vitrification process, which is an inaccessible process. Two kinds of

slurry samples were evaluated in this study. Two slurry circulation systems and direct

analysis of slurry samples filled in a shielded cell were applied to LIBS analysis.

Secondly, we evaluate the LIBS technique for quantitative analysis of tin alloy samples by using an echelle spectrometer (ESA 3000, LLA Instruments, GmbH). Tin alloys contain small quantities of various kinds of impurities. The ESA 3000 permits broadband (200~780nm) detection with high resolution; this makes it possible to simultaneously detect multiple element. Unknown samples without information of their elemental compositions were tested to estimate the sample compositions. Artificial neural network, calibration method and chemical analysis were applied to estimate the elemental concentration of unknown samples.

6

CHAPTER II

PHYSICS OF LIBS LINE SHAPE & ARTIFICIAL NEURAL NETWORKS

The LIBS technique is based on analysis of optical emission spectra from laser- produced plasma plumes on the sample’s surface. The electromagnetic energy of the laser pulse creates a plasma above the sample surface. The plasma is electronically neutral vapors or gas as a whole which consists of atoms, molecules, ions and electrons.2

Therefore, plasmas should be treated statistically to obtain the desired physical properties.

2.1 Plasma Formation by Laser Pulse

After the creation of a plasma on the target surface, the optical emission from different species or different energy states decays at different rates because the laser- produced plasma plume evolves in time. The evolution of laser-produced plasma in time strongly depends on the experimental conditions, such as incident laser power, laser pulse duration, and ambient pressure. This evolution is mainly characterized by three separate regimes:40 target material evaporation of the surface layer, isothermal plasma formation and expansion until the termination of the laser pulse, and adiabatic expansion of the plasma.

During the early stage, the interaction between the high-powered laser beam and the material evaporates the elements on the sample after the initiation of the laser pulse.

7

The evaporated material also interacts with the laser beam due to the finite laser pulse

width (i.e., ~10 ns). The ionization mechanisms of the evaporated species from target include multi-photon absorption processes and collision-induced ionization. The photon energy of a visible laser source is insufficient to ionize atoms and molecules. However, simultaneous absorption of a number of photons above the ionization potential increases the probability of ionization of atoms and molecules. This is given by the following relationship.

W ~ I K (2- 1) where W is the probability of ionization, I is the laser beam intensity and K is the integer number of photons absorbed.41 The collision of atoms with free electrons which have

sufficient energy to ionize the atoms induces ionization as a result of energy transfer.

At this initial stage, an intense continuum spectrum (free-free transitions of

electrons and electron-ion recombination) and heavily broadened ionic lines (Stark

broadening) due to high electron density will be dominant.42 Free electrons which are not bound inside atoms, ions, or molecules experience deflection by the surrounding charged particles (ions or electrons) during collisions. The resulting accelerated electrons emit light according to classical electrodynamic theory. This is known as free-free transition.

Electron-ion recombination radiation appears when a free electron in a continuum state transitions into a bound state through capture by an ion.2 This emission is known as

Bremsstrahlung radiation. An adiabatic expansion of the plasma plume occurs after

termination of the laser pulse; this is a conversion of thermal energy into kinetic energy

8 of elements in the plasma plume. On the other hand, the temperature decreases rapidly with plasma expansion. It should be noted that the plasma temperature slowly decreases at later times because the recombination of ions and electrons releases energy into the plasma.43 This plasma cooling decreases the continuum spectrum (background noise), which reflects the importance of time-resolved detection in LIBS measurement. As the plasma subsequently cools down, molecules are formed by recombination of atoms in the plasma.

2.2 Spectral Line Intensity

According to classical theory, the integrated emission line intensity between atomic (or ionic) upper level j and lower level i is given by2, 44

n + ∞ l 8πhc j g i ⎡ ⎤ I = α 1 − exp( − k (ν)ds) dν (2- 2) 3 ∫−∞ ⎢ ∫0 ⎥ λ n i g j ⎣ ⎦

Here α is a constant factor that depends on the measuring instrument, h is the Planck’s constant (in Js), c is the speed of light (in m/s), λ is the transition wavelength (in m), l is the absorption path length, and ds is the infinitesimal absorption path length.

The absorption coefficient k(ν) (at frequency ν) per unit path length is defined as the fractional ratio of energy lost by passing through the absorbing medium to the energy incident on a layer of thickness with the unit length. The frequency dependence of the absorption coefficient k(ν) is represented by

−y2 a +∞ e k(ν) = k0 dy (2- 3) π ∫−∞ (x − y) 2 + a 2

9

2 3 / 2 e n i f k 0 = 2π (2- 4) m ec b

πΔν b = D (2- 5) ln 2

Δν a = L ln 2 (2- 6) Δν D

(ν −ν ) x = 0 2 ln2 (2- 7) ΔνD

Here, e in equation 2-6 is the electron charge (in C), me is the electron mass (in kg), f is the transition oscillator strength (dimensionless) and ν0 is the frequency of the center of the line (in Hz). ∆νD and ∆νL are the Doppler and Lorentzian line halfwidths, respectively.

The shape of an isolated atomic emission line in an infinitesimal slab of a vapor plume is determined by the frequency dependence of the absorption coefficient.

l The integration equation ( k(ν)ds ) is defined as optical depth at a frequency ν. ∫0

The criterion of optically thin (or thick) plasma condition is determined by the magnitude of optical depth. Optically thin plasma is defined as being when the optical depth is small

l (i.e., k(ν)ds <<1). ∫0

2.3 Local Thermodynamic Equilibrium

The temperature of laser-induced plasmas may vary from one place to another because the plasma is not optically thick at all frequencies. Electrons in the plasma plume excite and de-excite atoms through collisions, which are the dominant part of transition

10 mechanisms in the early stage of plasma formation.2 This condition is known as local thermodynamic equilibrium (LTE).

The LTE condition means that the plasma can be described by the principle of detailed balance for collision transitions. This principle assumes the validity of the

Boltzmann relation under the condition of high electron density. Considering a two energy level system, it can be described by

n j Cij g j = = exp(−(E j− E i ) / kT) (2- 8) ni C ji g i where Cji (Cij) are the collision transition rate from upper (lower) to lower (upper) atomic level; nj and ni are the number densities of the j-th and i-th energy levels; and gj and gi are the degeneracies of the upper and the lower atomic levels, respectively. Here, number density is the number of atom per volume. Therefore, the validity of this collision equilibrium requires that the electron number density exceeds a lower limit. The inequality of the lower limit is derived from the ratio of the spontaneous emission rate to the collision rate for absorption and is given by

18 1/ 2 3 n e >> 1.4 ×10 T ΔE (2- 9)

-3 2 ,45 where T is in K, ∆E is in eV and ne is the electron density in m . Here, ∆E is the largest energy gap for the identified element.

2.4 Voigt Profile

This emission line profile can be described by convolution of Gaussian and

Lorentzian functions. The convolution of a Lorentzian function L(x) and a Gaussian

11 function G(x) is formalized by

+∞ I(x) = G(x) * L(x) = G(y)L(x − y)dy (2- 10) ∫−∞

a G(x) = exp[−(ax) 2 ] (2- 11) π

1 L(x) = (2- 12) 1+ x 2

The Gaussian-Lorentzian convolution is called a Voigt function V(a,x) which is described by the absorption coefficient k(ν):

−y2 k(ν) a +∞ e V(a, x) = = dy (2- 13) ∫−∞ 2 2 k 0 π (x − y) + a

The well-known Voigt function can only be numerically calculated because of the non-existence of an analytical solution.2 The damping constant a-parameter specifies the shape of the Voigt profile, yielding information on the ratio of Lorentzian broadening width to Gaussian broadening width. When the value of the a-parameter approaches zero

(infinity), Voigt function represents a Gaussian (Lorentzian) function. The line profile was calculated using the polynomial approximations of the Voigt profile function proposed by Kuntz.46 The calculation of the Voigt profile was coded and performed in visual C++ computer language (see Appendix B). The selected values and graph of the

Voigt function were compared with the results given by reference 4747 (see Fig. 2.1 and

Table 2.1). For the calculation, the selection of a-parameter values is based on reference

47.

12

1 a=0.0 0.01 0.9 0.1 0.8 0.2 0.7 0.3

0.6 0.5

0.5 0.7

V (a, x) x) V (a, 0.9 0.4 1.0

0.3 2.0 0.2 5.0 0.1 15.0 20.0 0 -4.5 -3 -1.5 0 1.5 3 4.5 x

Figure 2.1

Voigt profile as a function of a-parameter calculated using polynomial approximation proposed by Kuntz46

13

Table 2.1

The values of Voigt function as a function of a-parameter values46

a-parameter x 1.00E-12 0.5 20 1.00E-09 9.999999999988710E-01 6.156890890560770E-01 2.817425718476120E-02 5.00E-08 9.999999999988690E-01 6.156890890560760E-01 2.817425718476120E-02 1.00E-07 9.999999999988610E-01 6.156890890560730E-01 2.817425718476120E-02 5.00E-06 9.999999999739110E-01 6.156890890470980E-01 2.817425718475950E-02 1.00E-05 9.999999998990320E-01 6.156890890201610E-01 2.817425718475430E-02 5.00E-04 9.999997503991200E-01 6.156889992656410E-01 2.817425716726220E-02 1.00E-03 9.999990016002440E-01 6.156887298944250E-01 2.817425711476480E-02 5.00E-02 9.975071341616950E-01 6.147919590265510E-01 2.817408219482030E-02 1.00E-01 9.900659852477800E-01 6.121098388208730E-01 2.817355723797380E-02 9.00E+00 7.098457462159570E-15 3.537807750014820E-03 2.345104896079830E-02 1.00E+01 5.728721455419470E-15 2.856955658307810E-03 2.256304936171660E-02 1.10E+01 4.721759865953560E-15 2.355865797633450E-03 2.165645175874820E-02 1.20E+01 3.959524913256080E-15 1.976245195351090E-03 2.074336400091490E-02 1.30E+01 3.368480932333340E-15 1.681699712675370E-03 1.983417321247650E-02 1.40E+01 2.900836132138260E-15 1.448536619133310E-03 1.893752743688630E-02 1.50E+01 2.524287722779150E-15 1.260721221119780E-03 1.806040463382510E-02 1.60E+01 2.216820658621870E-15 1.107314154587940E-03 1.720824157670420E-02 1.70E+01 1.962374684091800E-15 9.803290183975960E-04 1.638509853953690E-02 1.80E+01 1.749408090127060E-15 8.740223486359540E-04 1.559384028585480E-02 1.90E+01 1.569360664510880E-15 7.841320324052220E-04 1.483631876669710E-02 2.00E+01 1.415774065516810E-15 7.074410084601550E-04 1.411354744939550E-02 2.10E+01 1.283701006878730E-15 6.414839609842040E-04 1.342586099419540E-02 2.20E+01 1.169299661529540E-15 5.843458143229870E-04 1.277305695972690E-02 2.30E+01 1.069549725441690E-15 5.345205808416420E-04 1.215451838353200E-02 2.40E+01 9.820501246602320E-16 4.908107412831180E-04 1.156931755950300E-02 2.50E+01 9.048725796970640E-16 4.522543742370910E-04 1.101630225811540E-02 2.60E+01 8.364540602377490E-16 4.180716167340770E-04 1.049416614467250E-02 2.70E+01 7.755167518744860E-16 3.876248069695160E-04 1.000150536573750E-02 2.80E+01 7.210077712423070E-16 3.603884524715680E-04 9.536863292155800E-03 2.90E+01 6.720532476738120E-16 3.359263488042250E-04 9.098765303128440E-03

14

Considering the symmetry of the observed emission line centered at the peak value, a fitting of the neutral Al I line at 309.271 nm to the Voigt function was chosen and carried out by using Origin software (OriginLab Co., USA) (see Fig. 2.2). A non- linear least-squares fitting was performed by utilizing a Levenberg-Marquardt algorithm installed in Origin software (OriginLab Co., USA). The values of Gaussian and

Lorentizian width were estimated (see Table 2.2). Experimental data points of Al I

309.271nm was measured from tin (Sn) alloy sample by using echelle detection system.

16000

14000

12000 Voigt Profile

10000 Experiment 8000

6000 Intensity (a.u.) (a.u.) Intensity

4000

2000

0 309.027 309.077 309.127 309.177 309.227 309.277 309.327 309.377 309.427 309.477 Wavelength (nm)

Figure 2.2

Theoretical Voigt profile fitted to the experiment data point (Al I 309.271 nm) using Levenberg-Marquardt algorithm installed in Origin software

15

Table 2.2

The estimated spectroscopic information obtained by Voigt fitting of Al I 309.271 nm by using Origin software. R is correlation coefficient

Gaussian Lorentzian Transition λ(nm) R2 Width (pm) Width (pm) 3s23p-3s23d 309.271 18.06 ± 1.74 26.05 ± 1.32 0.99612

2.5 Relation between Lorentz and Gaussian half width

For a given single a-parameter value, the determination of relative Lorentzian and

Gaussian fractions in the Voigt profile is important in LIBS applications. The empirical function defined by Posener is introduced to calculate the fraction of relative Lorentz and

Gaussian component as a function of a-parameter values.48, 49 The empirical function is given by

Δν w(a) = ln 2 t (2- 14) Δν G

Here, w(a) is the empirical function and ∆νt is the total FWHM of the Voigt profile.

Considering other broadening mechanisms causing the Gaussian line width such as instrumental broadening, ∆νD in equation 2- 5, 2- 6 and 2- 7 was replaced by ∆νG. The estimation of w(a) in terms of the a-parameter value can be only calculated by the approximation due to unknown value of ∆νG. For example, an approximation of w(a) proposed by Posener48 is given by w(a) = 1 + a 2 (2- 15)

16

Kastner and Whiting suggested improved approximation of the empirical function. 49, 50

2 3 w(a) = A + Ba + Ca + Da (2- 16)

1 w(a) = (c a + c a 2 + 4 ln 2) (2- 17) 2 1 2 where A = 0.83255, B = 0.531515, C = 0.141917, D = −0.0175063, c1 = 1.0692, and c2 =

0.86639. Here, Equation (2-16) and (2-17) proposed by Kastner and Whiting, respectively provide a fairly accurate improvement.

2

1.8 Posener's Approximation

Kastner's Approximation 1.6

Whiting's Approximation 1.4 W(a)

1.2

1

0.8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 a-parameter

Figure 2.3

Plots of empirical functions as a function of a-parameter proposed by Posener,48 Kastner49 and Whiting50

Note that the fraction of relative Lorentz and Gaussian components as a function of a-parameter values can be derived by

17

a Δν LF = = L (2- 18) w(a) Δν t

ln2 Δν GF = LF = G (2- 19) a Δνt

Here the equations of LF and GF proposed by Tudor Davies and Vaughan51 are the fraction of Lorentz and Gaussian components, respectively. ∆νL is the Lorentzian line halfwidth. Table 2.3 shows a tabulation of the empirical function value divided by the a- parameter value, Gaussian and Lorentz component as a function of a-parameter values.

This result has a good agreement with reference 51.51

Notably, Whiting assumed50 that the Voigt profile can be expressed approximately by the weighting sum of separate Gaussian and Lorentzian function to calculate an empirical function. This equation well-known as Pseudo-Voigt profile is given by52

ΔνL ΔνL V(a,x) = V(a,0){[1− ]g(x) + f(x)} (2- 20) Δνt Δνt where g(x) = exp[− ln 2(x / w(a)) 2 ] (2- 21)

1 and f (x) = (2- 22) 1 + (x / w(a)) 2

An example of a Pseudo Voigt profile with a-parameter value of 1.2 is presented in

Figure 2.4 and 2.5. Lorentzian contribution to the spectrum is dominant in the wing of

Voigt profile.

18

Table 2.3

Values of Lorentzian fraction, Gaussian fraction and empirical function as a function of a-parameter values

a-parameter Lorentzian Gaussian w(a)/a Fraction Fraction 0.1 0.1127 0.938286 8.87314 0.2 0.211716 0.881326 4.72331 0.3 0.298638 0.828775 3.34854 0.4 0.374899 0.78031 2.66738 0.5 0.441789 0.735626 2.26353 0.6 0.500461 0.694435 1.99816 0.7 0.551944 0.656462 1.81178 0.8 0.597151 0.621451 1.67462 0.9 0.636888 0.58916 1.57014 1 0.67186 0.55936 1.48841 2 0.862807 0.359167 1.15901 3 0.92897 0.257806 1.07646 4 0.957467 0.199286 1.04442 5 0.97192 0.161835 1.02889 6 0.980156 0.136006 1.02025 7 0.985263 0.117184 1.01496 8 0.988637 0.102887 1.01149 9 0.990977 0.0916714 1.0091 10 0.992666 0.0826448 1.00739 20 0.998145 0.0415505 1.00186 30 0.999174 0.0277289 1.00083

19

0.4

0.35

0.3 Voigt Profile

0.25 Pseudo Voigt Profile

0.2 V (a ,x) 0.15

0.1

0.05

0 -30 -20 -10 0 10 20 30 x

Figure 2.4

Plots of Pseudo-Voigt, Voigt line profile. Solid line and Open circles represent Voigt and Pseudo-Voigt profile with 1.2 of a-parameter value

0.4

0.35

0.3 Gaussian Contribution

0.25 Lorentzian Contribution

0.2 Pseudo Voigt Profile V (a,x) 0.15

0.1

0.05

0 -30 -20 -10 0 10 20 30 x

Figure 2.5

De-convoluted line profile with 1.2 of a-parameter value. Solid line (—), dashed line (– – –) and (---) represent Pseudo-Voigt, Lorentzian and Gaussian contribution

20

2.6 Stark Profile for Isolated Neutral Atom Lines

Emitting atoms in plasma plume are perturbed by a microscopic electric field generated from ions and electrons surrounding the atom. This Coulomb interaction known as the Stark effect causes an energy shift of an atomic level. The Stark broadening mechanism for non H-like emission lines was introduced by Griem, who applying the electron impact approximation and corrected for quasi-static ion broadening.53

For a non-hydrogenic radiator, the isolated spectral line profile produced by the electron and ion quadratic Stark effect is given by54,55

∞ 1 H R (β) j A,R (x) = dβ (2- 23) ∫0 4 / 3 2 2 π [1+ (x − A β ) ]

-2 -1 Here, jA,R(x) is the relative intensity of the profile per wavelength unit (Wm nm ). The variable x is the reduced wavelength and given by

x = ±(λ − λ 0− d e ) / w e (2- 24) where λ is the wavelength (in nm), λ0 is the center wavelength of the unperturbed line (i n nm), de is the electron shift (in nm) and we is the electron impact parameter (in nm). A is the ion broadening parameter (dimensionless). HR(β) is the micro-field strength distribution at the neutral atoms which is defined as the probability density56 of finding electric field strength (β) in the range from β and β+ dβ. The variable β is the ratio of the normalized Debye screened field strength F to the Holtsmark field strength F0. (i.e.,

2 β=F/F0). The Holtsmark field strength F0 is given by e/r0 , where r0 is the mean inter- distance of singly charged perturbers. F0 can be expressed by the unit of the singly

21

57 charged ion density (ni):

2/3 F0 = 2.61e × ni (2- 25)

The Debye-screened electric field F is the derivative of the Debye and Hückel (DH) potential Φ:58

e Φ = exp(−r / ρ ) (2- 26) r D

−2 F = er [1 + (r / ρ D )]exp(−r / ρ D ) (2- 27)

kT 1/ 2 where ρD = ( 2 ) (2- 28) 4πn ee

Here ne is the electron number density. Considering the situation of a stationary point charge embedded in electrically neutral plasma, Debye et al derived the DH potenial by using the assumption that the charged particles obey the Boltzmann distribution because the temperature is high: eΦ << kT (2- 29)

n e ∝ exp(eΦ / kT) (2- 30)

ni ∝ exp(−eΦ / kT) (2- 31)

The HR(β) function also depends on the Debye shielding parameter (R), which is defined as the ratio of the mean distance between ions and the Debye radius

(dimensionless) (i.e., R = r0/ρD). The Debye shielding parameter (R) value is given by

8.98×10 −3 (n ) 1/ 6 R = e (2- 32) T

22

Bengoechea selected a reasonable R-value (R = 0.6) in a laser-induced plasma by comparing the temperature and the electron density.59 The R values 0.57 and 0.50 were estimated at plasma temperature and electron densities of 11000K and 10+17 electrons per

3 +15 3 cm and of 6000K, 7×10 electrons per cm , respectively. The curves of HR(β) and

Holtsmark based on tabulated value by references 53 and 60 at R = 0.6, 0.4 and 0.0 are shown in Figure 2.6.60 ,61 For very large values of β, the neutral distributions should tend toward the Holtsmark distribution. 62

0.6 Holtsmark's field distribution 0.5 H(β) at R = 0.4

0.4 H(β) at R = 0.6 )

β 0.3 H(

0.2

0.1

0 0 0.51 1.5 2 2.5 3 3.5 4 β

Figure 2.6

Holtzmark micro-electric field distribution ( R = 0.0 ) and micro-electric field distribution ( HR(β) ) at R = 0.4, 0.6

The spectral line profile (Eq. 2-23) parameterized by the A-value was numerically calculated and compared with reference 54 (see Fig. 2.7). The asymmetric trend and shift

23

of the theoretical profile increase as the A-value increases. For a fixed value of

the electron parameter (we), the total FMHW (wt) and shift (dt) of the profile depend on

the ion broadening parameter (A), which increases the asymmetry. The electron

broadening parameter also influences the FMHW and shift of the profile. Figure 2.8 show

that the effect of increasing the value of the electron impact parameter increases the

asymmetry of the line profile.

1.2

1 A = 0.0 A = 0.2 0.8 A = 0.4 A = 0.6 0.6

Intensity (a.u.) 0.4

0.2

0 -15 -10 -5 0 5 10 15 X

Figure 2.7

Asymmetric line profiles by the electron and ion Stark effect as a function of A-values

By inspecting the theoretical profile curves, Greim derived an approximate

formula for the total width and shift as a function of electron impact parameter. The

following formulas are given by63

24

w t ≈ [1+ 1.75A(1− 0.75R)]w e (2- 33)

⎡ d e ⎤ d t ≈ ⎢ ± 2.0A(1 − 0.75R )⎥w e (2- 34) ⎣ w e ⎦

Equation (2- 33) and (2- 34) are reasonably accurate for A ≤ 0.5 and R ≤ 0.8 .

1.0

0.8 we = 2

we = 10 0.6 we = 15

0.4

Intensity (a.u.) (a.u.) Intensity 0.2

0.0

-100 -50 0 50 100

( λ - λ )(a.u.) 0

Figure 2.8

Asymmetric line profiles due to electron and ion Stark effect as a function of electron impact parameter we at A-value (0.2) and de (0.0)

In the far wing, Equation (2-14) yields a useful asymptotic formula for |x| >> 1, given by64

1 3A j (x) = + x > 0 (2- 35) A,R πx 2 4x 7 / 4

25

1 j (x) = x < 0 (2- 36) A,R πx 2

Asymptotic wing formula maintains the asymmetry due to ion broadening parameter (A).

2.7 Line Broadening Width

The observed atomic emission line in the typical LIBS measurements has a finite width caused by instrumental broadening, Stark broadening, Doppler broadening, self- absorption, etc. Instrumental and Doppler broadening approximately correspond to a

Gaussian profile.65,66 Stark broadening which is dominant under typical LIBS conditions can be well described by a Lorentzian profile.

2.7.1 Doppler Broadening Width

Doppler broadening is a result of the well-known Doppler effect, which is the apparent shift in wavelength of a signal from a source moving toward or away from the observer. Doppler effect can be derived from the Lorentz transformation of the

μ propagation four-vector k (k, ω/c), assuming an atom emitting the wavelength λ0 (λ0 =

67 2πc/ω0) at a rest frame has the relative velocity vz to the observer,

v ω = ω γ(1− z ) (2- 37) 0 c

1 where γ = (2- 38) v 1 − ( z ) 2 c

The value of vz is positive for an atom emitting and observer approaching each other and negative for source and observer receding from each other. The Doppler frequency shift

26

along the z-direction can be approximately written as ω = ω0(1− vz/c) = 2πν0(1− vz/c) in the classical limit (vz << c). Substituting vz in terms of frequency into the Maxwell-

Boltzmann distribution, the Doppler shift of the line profile in terms of full-width-half- maximum (FWHM) is given by

ν 8kT Δν = 0 (ln 2) 1/ 2 (2- 39) D c m

Here, ΔνD is the Doppler line halfwidth (in Hz), k is Boltzmann’s constant (in J/K), m is the mass (in kg) of the emitting atom and T is the absolute temperature (K).2

2.7.2 Stark Broadening Width

The broadening width for non-hydrogenic-like emission lines was computed in the electron impact approximation and corrected for quasi-static ion broadening by

Griem.53 Assuming the contribution from quasi-static ion broadening is negligible, the ratio formulation for non-hydrogen-like emission line is given by

Δλ 2w e ≈ ( 16 ) (2- 40) ne 10

Here, ∆λ is the half-maximum line width (in Ångstrom units), ne is the electron density

-3 (in cm ) and we is the reference electron impact half width (in Ångstrom units). The

16 -3 constant (10 ) (in cm ) is the reference electron density. Determination of ne by this method is independent of any assumption regarding LTE.13 Notably assuming an instrumental broadening profile corresponding to a Gaussian profile,68, 69, 70 ,71 Lorentzian

FWHM can be measured from Voigt fitting.72, 73

27

2.8 Self-Absorption

A difference in the line shape between the observed and the truly emitted signal may exist in the case of an optically thick plasmas.2 In optically thick plasmas, the radiation of an atom can be easily reabsorbed by other atoms of same element in lower energy level states. This interaction is known as self-absorption.

The degree of self-absorption can be explained by the self-absorption coefficient

(SA).74 SA is the ratio of the measured peak value of emission intensity in real plasma to the peak value of emission intensity from optically thin plasma. SA is given by

I(λ ) 1− exp(−k(λ )l) 0 = 0 ≡ SA (2- 41) I(0λ 0 ) k(λ 0 )l

Here, I(λ0) and I0(λ0) are the emission intensities of a real plasma and an optically thin plasma at λ0. Emission intensity has a peak value at λ0. Notably, the integration equation

l ( k(ν)ds ) can be replaced by k(ν)l if the absorbing medium is homogenous. The unit ∫0 value of SA (i.e. SA = 1) represents a plasma that is optically thin. On the other hand, SA approaches zero as the plasma becomes thick (see Fig. 2.9). This has been demonstrated in reference 74. The FWHM of ∆λ in real plasma can be parameterized in terms of SA

74 and ∆λ0 as:

α Δλ = Δλ 0 (SA) (2- 42) where α = -0.54.

From the Equation (2.40), the Stark broadening width due to self-absorption is given by

28

α ∗ Δλ = 2w e n e (SA) = 2w e n e (2- 43)

* where ne represents the true electron density of the real plasma. The real electron density

(ne) in optically thin plasma is necessary to obtain the SA-value. To evaluate the real plasma electron density (ne), El Sherbini et al. used the Hα line at 656.27nm emitted by

75 the natural humidity of air. According to mention of El Sherbini et al, the Hα line is not affected by self-absorption because of the small concentration of humid air around the solid target surface. In LIBS measurements, it is desirable to reduce the self-absorption along the optical path between the emission volume and the detector.

30

25

20

Δλ 15 Δλ 0

10

5

0 0.001 0.01 0.1 1 SA

Figure 2.9

Plot of ∆λ/∆λ0 according to absorption coefficient (SA). ∆λ and ∆λ0 correspond to the observed and non-self-absorbed broadening widths, respectively. The equation of this plot was explicitly followed from reference 74

29

2.9 Curve of Growth

Self-absorption under high optical depth conditions may cause a non-linear growth of the integrated intensity as the elemental concentration in the sample increases.

This non-linear growth means saturation of the curve at high elemental concentrations in the sample. Gornushkin applied the theoretical curve-of-growth method to laser-produced plasma spectroscopy to describe saturation of the line intensity due to self-absorption.8

In the case of the resonance transition, total absorption is another form of integrated spectral line intensity modification that facilitates a calculation of the theoretical curve of growth. The resonance transition is the atomic transition when lower state is in the ground state. Assuming the homogenous plasma, the total absorption is given by76

+∞ A t / b = 2 [1 − exp(−k(x)l)]dx (2- 44) ∫0

8, The total absorption (At) keeps linear relation with the integrated spectral line intensity.

76 The theoretical curve of growth (COG) is defined by the double-logarithmic plot of

At/2b vs. n0fl/b. Notably, f is the transition oscillator strength (dimensionless), n0 is the number density of ground state atoms, and l is the absorption path length. b is the

++ π∆νD/ ln 2 . The theoretical COG curve was numerically obtained using Visual C computer language. The trapezoidal rule was applied to this numerical integration. The trapezoidal numerical method with dimensionless integration interval (dx) of 0.25 was applied and compared with the results in reference 76. Figure 2.10 shows a series of theoretical COGs as a function of a-parameter values. A linear relation between the

30

number density of ground state atoms (n0) in the plasma and the analyte concentration in the sample is assumed in order to fit the theoretical and experimental COG. By this hypothesis, the lower limit of n0fl/b corresponds to an analyte concentration of the sample, assuming optically thin plasma. Plots of the lower limit of n0fl/b are almost straight lines, independent of a-parameter values due to a small contribution of the Voigt profile to integration. The curve becomes flat for the largest values of n0fl/b. The wing of the Voigt profile becomes the dominant contribution to integration in this upper limit. Figure 2.10 reflects that a pure Lorentzian (a ~ ∞) has more prominent wings than a pure Gaussian (a

~ 0). At a zero value of the a-parameter (a = 0), saturation of the theoretical COG curve occurs because the Voigt profile is a pure Gaussian.2 Despite the hypothesis, matching between theoretical and experiment COGs is difficult. The simplest way, suggested by

Alkemade, is to move the theoretical COG to the experiment one by adjusting x and y axis until obtaining the best fit.77 This fitting method is generally not accurate to obtain a- parameter values for experimental COGs. This method is only reasonable when the a- parameter value is less than unity (i.e. a

31

1000

a=100.0

10.0 100 5.0

0.9

At/2b 0.2 10 0.1

0.0

1

0.1 1 10 100 1000 10000

n0fl/b

Figure 2.10

Series of theoretical COG curves according to a-parameter value for a homogenous plasma

Duplication is another way to obtain a-parameter values of experimental COGs through fitting. The duplication factor (D) is given by76

A (2n fl) − A (n fl) D ≡ t 0 t 0 (2- 45) A t (n 0fl)

Figure 2.11 shows theoretical duplication curves as a function of a-parameter values. The experimental duplication curve is obtained by doubling the analyte concentration in the sample.8,76 Fitting of the two curves also provides the a-parameter value of the

32 experimental COG. Notably, the D curve has two asymptotic values: 1 and 0.414. A D- value of 1 (0.414) is obtained when n0fl/b approaches 0 (∞).

10

1 a=5.0 a=1.0 Duplication Factor D a=0.5

a=0.1

0.1 1 10 100 1000 10000

n0fl

Figure 2.11

Series of theoretical Duplication curves as a function of a-parameter value for a homogenous plasma

Self-and non-self-absorbed integrated emission line intensities can also be numerically parameterized by using the SA value. The equation is given by

+∞ [1 − exp( −k(x)l )]dx I A t ∫0 β = = = (SA ) (2- 46) I 0 + ∞ 0 A t k(x)ldx ∫0

33

Here, I and I0 are the integrated emission intensity of the real plasma and optically thin

74 0 plasma. El Sherbini et al. calculated the β-value (β=0.46). A t is the total absorption of a non-absorbed emission line.

Figure 2.12 shows the ratio of the self- and non-self- absorbed integrated emission intensity as a function of SA values. At an a-parameter of 5.0, the numerical calculation was executed using Visual C++ computer language. It shows good agreement with the results of reference 74.

1.1

0.9

0.7

I/I0 0.5

0.3

0.1

-0.1 0.0001 0.001 0.01 0.1 1

SA

Figure 2.12

Curves of I/I0 according to absorption coefficient (SA). Solid line represents I/I0 = (SA)0.46 and triangle represent the numerical result of Equation 2- 46

34

2.10 Spectral Line Intensity in Optical Thin Plasma

Under the condition of optically thin homogenous plasma ( k(ν)l <<1), the integrated emission line intensity in Equation (2-4) can approximately be described by2,44

8πhc n j g +∞ l I = α i ⎡1 − exp(− k(ν)ds)⎤dν (2- 47) 3 ∫−∞ ⎢ ∫0 ⎥ λ n i g j ⎣ ⎦

8πhc n j g +∞ ≈ α i k(ν)ldν (2- 48) 3 ∫-∞ λ n i g j

n +∞ 8πhc j g i lΔν D k 0 k(ν) 2 ln 2dν = α (2- 49) 3 ∫-∞ λ n i g j 2 ln 2 k 0 Δν D

+∞ 8πhc n j gi Δν Dk 0 = α l V(a, x)dx (2- 50) 3 ∫-∞ λ n i g j 2 ln2

2 8πhc n j g i πe = α 3 l n i f (2- 51) λ n i g j m ec

hc = αl n jA ji (2- 52) λ

hc = B n jA ji (2- 53) λ

hc exp(−E j / kTe ) = B A ji n x g j (2- 54) λ U(T)

3 1 g j m e c exp(−E j / kT) where f = 2 2 2 A ji , n j = n x g j , B = α l (2- 55) 8π g i e ν U(T)

The area of Voigt profile was calculated using Origin software (Origin Lab Co., USA):

+∞ V(a, x)dx = π (2- 56) ∫−∞

35

Here, Aji is the spontaneous transition rate from upper energy state j to lower energy state

-1 i (in s ), nj is the atomic number density of j-th energy levels of species x and f is the oscillator strength. Assuming LTE condition, the population of j-th level of species x is given by

exp(−E j / kTe ) n = n g (2- 57) j x j U(T)

Here, nx is the total number density of species x, Te is the electron (excitation) temperature and U(T) is the atomic partition function of species x, which is defined as

m U(T) = g exp(−E / kT) (2- 58) ∑ j j j=0 where the energy state with energy Ej has degeneracy gj. The integrated emission line intensity is finally given by

hc exp(−E j / kTe ) I = B A ji n x g j (2- 59) λ U(T)

Notably, Ej is the energy of upper level for the transition and B is a constant.

2.11 Partition Function and Plasma Temperature

The atomic partition function (U(T)) was calculated using the polynomial approximation proposed by Irwin and Galan, which is valid up to 16000 K, 7000K.78, 79

Figure 2.13 shows the evaluation of the atomic partition function for Fe I.

36

90 80 Fe I 70 60 50

U(T) 40 30 Irwin 20 10 Galan 0 0 2000 4000 6000 8000 10000 12000 T/K

Figure 2.13

Partition function of Fe I based on Irwin’s and Galan’s polynomial approximation

By considering the relative intensity ratio and the natural logarithm of Equation

(2-53), the electron temperature can be calculated for the same species from80 81

E − E T = p m (2- 60) λ A g I k ln( pq mn m pq ) λ mn A pq g p I mn

I pqλ pq n E p ln = ln( x ) − (2- 61) A pq g p U(Te ) kTe

Here, m and p are indices for upper levels of two emission lines with wavelengths λmn and λpq.

37

0

-2

-4

-6 y = -1.82x - 5.6009 -8

/Ag) -10 λ

-12 ln (I

-14

-16

-18

-20 2 2.5 3 3.5 44.5 5 5.5 6 Energy (eV)

Figure 2.14

Boltzmann plots of Fe I obtained from tin (Sn) alloy (gate delay 1000 ns and gate width 4 μs)

For example, the electron temperature from Equation (2-61) was determined from the emission line intensities of Fe I observed in the laser-induced plasma of tin alloys.

(see Fig 2.14) The spectral wavelengths, energies of upper levels, statistical weights, and transition probabilities used for each element are obtained from the US National Institute of Standard Technology. (See Table 2-4).

38

Table 2.4

List of Fe I spectral information used for Boltzmann plot

E g A j j Element Line (nm) ji (upper (upper (108 s-1) energy level) level degeneracy) Fe (I) 367.991 0.0138 3.368256 9 Fe (I) 368.745 0.0801 4.220362 9 Fe (I) 370.924 0.156 4.256222 7 Fe (I) 372.761 0.225 4.283307 5 Fe (I) 373.331 0.062 3.43082 3 Fe (I) 374.336 0.26 4.301277 3 Fe (I) 379.501 0.115 4.256222 7 Fe (I) 381.584 1.3 4.73314 7 Fe (I) 382.043 0.668 4.103373 9 Fe (I) 382.444 0.0283 3.240969 7 Fe (I) 382.588 0.598 4.154353 7 Fe (I) 382.782 1.05 4.795465 5 Fe (I) 383.422 0.453 4.19086 5 Fe (I) 384.996 0.606 4.230536 1 Fe (I) 385.637 0.0464 3.265705 5 Fe (I) 385.991 0.097 3.211189 9 Fe (I) 386.552 0.155 4.217582 3 Fe (I) 387.25 0.105 4.190856 5 Fe (I) 389.576 0.094 3.291839 1 Fe (I) 389.97 0.0258 3.265705 5

Ionization temperature can be obtained by the combination of Boltzamann (Eq. 2-

59) and Saha equation along with knowledge of the electron density ne (Eq. 2- 40). Saha equation is given by11 ,82

3 n n 2(2πm kT ) 2 ⎛ U (T) ⎞ V+ e ion = e ion ⎜ ion ⎟ exp(− ) (2- 62) 3 ⎜ ⎟ n atom h ⎝ Uatom (T) ⎠ kTion

The combination of Boltzamann and Saha equation which can provide ionization

39 temperature is expressed by

3 I 2(2πm kT ) 2 ⎛ gA ⎞ ⎛ λ ⎞ V + + E − E ion = e ion ⎜ ⎟ exp(− ion atom ) (2- 63) 3 ⎜ ⎟ ⎟⎜ Iatom n e h ⎝ λ ⎠ion ⎝ gA ⎠atom kTion

+ Here, Iatom (Iion) is the integrated emission intensity of the atom (ion), V is the ion potential of the atom, Eion is the excitation energy of the ionic line, and Eatom is the excitation energy of the atomic line. natom (nion) is the number density of atom (ion),

Uatom(T) (Uion(T)) is the partition function of atom (ion) and Tion is ionization temperature.

2.12 Artificial Neural Networks

Artificial neural networks (ANNs) are computer models imitating the structure of the human brain. Pattern recognition and modeling capabilities obtained by network training are the main marks of ANN, which make it possible to predict the output from unseen input data. To detect a pattern between known input and output data, the neural network is trained by learning rules until obtaining good prediction ability. After training, unseen data is injected into a previously trained network to predict the output. This type of learning is known as “supervised learning.”83

2.12.1 Back-Propagation Learning Model

Back-Propagation (BP) learning model is a supervised learning technique of artificial neural network.84 A neural network consists of an input layer, an output layer and several hidden layers with several neurons (see Fig. 2.15). The training data are made up of the inputs and desired outputs presented at input and output layers.

40

Neurons (processing elements, nodes, or units) are the basic unit of ANNs. The neurons are inter-connected with weights (wij, i-th input and j-th output neuron) assigned to each connection. Each neuron performs multiplication of outputs from previous layer with connection weight matrix. The weighted sum is multiplied with an activation function (e.g. sigmoid function) to compute the output of a subsequent neuron (See Fig.

2.16). Continuing in this way, the output of each neuron in a layer is propagated forward to the next layer. The output from the the output layer of the ANN is compared with true values. In the event of large difference between the two values, the resulting error is beyond the tolerance level and the weights are dynamically according to learning rule so that the error is minimized. This process is repeated with the complete set of training data several times until the error function is within the tolerance limit.

Neuron

Input Hidden Output layer layer layer

Figure 2.15

Neural network architecture with three layers

41

The error function is given by the square of difference between the resultant output value and true value. Error function is mathematically described by

1 E = (T − O ) 2 (2- 64) ∑ i i 2 i=1

Here, E is error function, Ti is the desired output (target value) and Oi is the computed output from the network.85 After each training step the network is tested with test data set.

On successful completion of training validation data set is used for validating the network.

A well trained neural network is capable of predicting correct outputs using interpolation for new and unknown inputs that are within the range of input and parameter space. The utility of an ANN depends on the size of the training data set and also the various characteristics of the measurement environment that are included in the data set.

x 1 w 1 x 2 . w 2 .

∑ x w Input . i i Output

Neuron w x n n Connection Weight

Figure 2.16

An artificial neuron

42

2.12.2 Cascade-Correlation Learning Model

The cascade-correlation (CC) learning model introduced by Fahlman is a supervised learning architecture designed for overcoming the slow-pace learning performance of back-propagation.86 This constructive model technique initially begins with a minimal network (i.e., no hidden units, only input and output units).87 Hidden units are added to the net one at a time, resulting in formation of a multi-layer structure. Figure

2.17 shows cascade-correlation architecture after the addition of one new hidden unit to the network. The input-side weights (□) for the hidden neuron are frozen. Weights for the side of output neurons (×) are once again trained.24

output

Add Hidden Unit 1

Trained Frozen Weight Weight

input

+1

Figure 2.17

Cascade-correlation architecture after the addition of one hidden units.24 Square boxed connections (□) are frozen. × connections are trained

43

This iterative process is repeated until the correlation between the new unit’s output and the residual error is maximized. Notably, the CC learning model automatically determines the network size, depth, and topology. It is not necessary for the user to guess the network size and depth. 24

2.12.3 Software and Data Analysis

Data analysis was performed with an automated neural network tool Predict software (NeuralWare, Inc.) in this research. Predict interfaced with EXCEL (Microsoft.

Corp) is based on the Cascade-Correlation learning model. In our case, the input neurons consist of emission line intensity ratios at each different wavelength (λi) for the same species. Only one neuron of the output layer was used to estimate the sample concentration. 88 The calibration data set are successively substituted into the neural network, comparing the resultant output concentration with the true concentration, correspondingly. Neural network are trained until the optimized weights are obtained over the concentration range. The optimized weights obtained from the training are fixed, and the intensity ratios of an unknown sample are injected into the neural network. The elemental concentration of the unknown sample is predicted.

44

CHAPTER III

EXPERIMENTAL

Figure 3.1 shows a schematic diagram of the experimental setup used for recording LIBS spectra. A frequency-doubled, Q-switched Nd:YAG laser (Continuum

Surelite I or III) was incorporated into the LIBS system as an excitation source.

BD – Beam Dump DM – Diachronic Mirror OF – Optical Fiber HS – Harmonic Separators BD HS L – Lens

Nd: YAG Laser 2x 2x – KDP Doubling Crystal

DM ICCD – Intensified Prism Charge OF Coupled Device IDAD – Intensified ICCD or IDAD L L Diode Spectrometer Array Detector

Pulse Generator Controller Fan

Computer Sample

Figure 3.1

Schematic diagram of LIBS measurement apparatus

45

The 532-nm laser light is focused onto the sample surface using an ultraviolet

(UV) grade quartz lens of 300 or 500 mm focal length. Atomic emission from the laser- induced plasma was collected by an optical fiber bundle connected to the entrance slit of a spectrometer. The light is then focused onto an intensified charge coupled device

(ICCD) or an intensified diode array detector (IDAD). A pulse generator is used to trigger and synchronize the detector with laser operation to provide the desired gate delay and width for detection. Data acquisition and analysis were performed using a personal computer. Two different systems based on Czerny-Turner and Echelle spectrometers have been applied to record LIBS spectra in this study (see Table 3.1).

Table 3.1

Instrument information for three different LIBS systems

Spectrometer Czerny-Tuner Echelle Spectrometer Spex 500 HR 460 ESA 3000 LLA Model Spectral 0-750 nm 0-1300nm 200-780nm Coverage Simultaneous Spectral ~20nm ~20nm ~580nm range Grating 2400 lines/mm 1200 lines/mm 75 lines/mm

Pulse Princeton Inst. Princeton Inst. Fast pulse generator Generator PG-10 PG-10 (Model 3000 FP)

EG&E PARC ITE/CC ICCD or KAF-1000 Kodak Model 1456 Princeton Inst. IDAD ICCD IDAD ICCD

46

3.1 Spectrometers

A spectrometer is a class of well-known optical instruments whose main function is to separate light by use of optical dispersive elements, such as a diffraction grating or a prism. Light diffraction occurs according to the fundamental grating equation. The grating equation is

mλ = 2d(sin θi + sin θr ) (3- 1) where m is the diffraction order, θi and θr are the angles of incidence and diffraction, respectively, measured from the normal to the surface; d is the grating spacing; and λ is the wavelength of light. θi and θr are measured from the grating normal axis which is normal to the grating surface.

3.1.1 Czerny-Turner Spectrometer System

The Czerny-Turner (CT) optical configuration consists of two adjacent mirrors, entrance and exit silts and a diffraction grating. It has been widely used in spectroscopy.

CT mounting is a form of an Ebert-type mounting to reduce coma.2 Coma is an aberration where the image of a monochromatic point appears as a blurred pear shape. Figure 3.2 shows the general geometry of a CT optical configuration.

47

Grating normal axis Instrument axis Collimating Mirror (M1) Focusing Mirror θ (M2) r ρ

φ

θ i

Exit Slit Entrance Slit

Figure 3.2

Czerny-Turner optical configuration. θi is the incidence angle, θr is the diffraction angle, m is the diffraction order, f is the focal length of mirror, φ is the Ebert angle, and ρ is the grating rotation angle

The first mirror M1 transforms the focused incident wave from the entrance slit into nearly collimated plane waves which are incident upon the diffraction grating. The diffracted light is collected and focused onto the exit silt by the second mirror M2. Coma, one of the monochromatic aberrations, can be eliminated by adjustment of the tilt angle and focal length of the second mirror (M2). Because the simultaneous spectral range is limited (e.g. ~20 nm), the spectral range which we are interested in is obtained through the rotation of the grating. In Figure 3.2, the angles θi and θr are related to angles φ and ρ:

θi = ρ + φ (3- 2)

θr = ρ − φ (3- 3)

48

The rotation of the grating changes the incidence angle (θi) and diffraction angle

(θr) that can be modified by the Ebert angle (φ) and the grating rotation angle (ρ).

Transmission of the selected wavelength to the exit slit can be achieved by adjusting the grating rotation angle (ρ) for a fixed Ebert angle (φ).

The grating equation (Eq. 3- 1) becomes90 mλ = 2d cos(φ)sin(ρ) (3- 4)

Differentiating and multiplying the focal length of the focusing mirror (f) into the grating equation, the linear dispersion (dx/dλ) is obtained by89 , 90 dx / dλ = fm /(d cos(ρ − φ)) (3- 5)

However, the slit width (Δx) influences the resolution of the transmitted wave into adjacent spectral lines. This relation is described by the bandwidth (Δλ) and the reciprocal linear dispersion (dλ/dx). Bandwidth is an important index that represents a spectrometer’s ability to separate two adjacent spectral lines, which is defined in terms of the full-width-at-half maximum (FWHM) width of the wavelength distribution passing through the exit silt. The bandwidth (Δλ) is calculated from the silt width (Δx) and the reciprocal linear dispersion (i.e., Δλ = Δx(dx/dλ)-1). Examples of bandwidth to pass through exit silt are calculated with assumption of zero grating rotation angle (ρ=0). (see

Table 3.2) However, the bandwidth varies as a function of wavelength for a fixed slit width. Rosfjord et al. pointed out that constant bandwidth is needed to eliminate disparities in line-widths measured in different wavelength ranges of the spectrum.90

49

Table 3.2

Spectral parameters based on HR 460 spectrometer. d: grating spacing, f: focal length of focusing mirror, φ: Ebert angle, SW: slit width (μm), RLP: reciprocal linear dispersion, BW: bandwidth to pass through the instrument91

Grooves d f φ SW RLP BW

1200/mm 833 nm 451.89mm 16.5° 100 1.76 nm/mm 0.176nm

2400/mm 417 nm 451.89mm 16.5° 100 0.88 nm/mm 0.088nm

Calculation of the needed slit-width values to maintain constant bandwidth over broad spectral ranges was coded and performed in C computer language of visual C++ software (see Appendix A). The algorithm and notation describing physical quantities employed are from reference 90. An example of constant bandwidth is calculated with Δλ

= 0.176 nm, 1/d = 1200 mm-1, ƒ = 460mm, φ = 16.5°, and m = 1. The instrument parameters are based on an HR 460 spectrometer.91 The calculated slit width for constant bandwidth is shown in Figure 3.3 and Table 3.3.

The diffracted light from the exit slit enters into the detector (ICCD or IDAD) and is detected as a one-dimensional diode array. The detector transforms the diffracted light signal into an electric signal through several steps. The electric signal is proportional to the number of electrons induced on each of its diodes, and represents the intergrated emission line intensity.

50

180 f λΔ ,( λ max ,( ) )2sin(d φ )2sin(d 160

Δ x (λ j)(μm ) 140

fΔ λ 120 = φρλ ,)2(( ) d φ )cos( f Δ λ ),)(( = φρλ ),)(( 100 d

0 200 400 600 800 1000 1200 1400 Wavelength λ (nm )

Figure 3.3

The calibration curve of the needed slit width to achieve constant bandwidth based on HR460 spectrometer. Slit-width for constant bandwidth (Δλ = 0.176 nm). 1/d = 1200 mm-1, ƒ = 460 mm, φ = 16.5°, m = 1

Table 3.3

Look- up table to achieve the constant bandwidth (Δλ = 0.176 nm) based on HR 460 spectrometer. 1/d: 1200 mm-1, ƒ: 460 mm, φ: 16.5°, m: 1

j λ(nm) ∆xj (µm) ∆x(λj) (µm) (f∆λ)/(d∆x(λj)) 0 0 101 101.325 1 46.5786 99.9996 100.5 0.966691 2 112.286 98.9996 99.4996 0.976406 3 195.072 97.9996 98.4996 0.986319 4 322.986 96.9996 97.4996 0.996435 5 581.51 97.9996 97.4996 0.996435 6 700.241 98.9996 98.4996 0.986319 7 774.029 99.9996 99.4996 0.976406 8 830.917 101 100.5 0.966691

51

The number of diodes and the size of each diode (pixel) are also essential parameters to determine bandwidth per pixel and the spectral cover range (see Table 3.4).

This electric signal is converted into an image signal and then it is digitized and displayed on a PC (see Fig. 3.4).

Table 3.4

Spectral parameters based on HR 460 spectrometer system

Grooves RLP Bandwidth per pixel Spectral Coverage 1200/mm 1.76 nm/mm 0.04576nm 46.85nm 2400/mm 0.88 nm/mm 0.02288nm 23.42nm

RLP: Reciprocal linear dispersion, Pixel size: 26μm×26μm, 1024-element CCD array

2.E+05

2.E+05

1.E+05

1.E+05

1.E+05

8.E+04

Intensity(a.u.) Intensity(a.u.) 6.E+04

4.E+04

2.E+04

0.E+00 278.5 283.5 288.5 293.5 298.5 Wavelength(nm)

Figure 3.4

CCD image and LIBS spectrum of HR 460 spectrometer system

52

3.1.2 Echelle Spectrometer System

The Echelle Spectra Analyzer ESA 3000 manufactured by LLA Instruments is a prototype of echelle spectrometer that uses simultaneously a diffraction grating and a quartz prism to produce two-dimensional patterns. This spectrometer meets basic requirement needed for LIBS analysis, simultaneous multi-element detection capability of broadband (200~780nm) with high resolution spectra.92 The coarser grating with larger blaze angle, groove density between 20 and 350 grooves mm-1, and blazed angle from

26.5° up to 80°, is the major distinguishable aspect of a general Echelle grating spectrometer. The incident wave is normal to the long side of the groove and the resultant diffracted light is almost reflected back along the incidence angle (see Fig.3.5).

Grating Normal Axis

φ

θr

θi

∆ ~ d Sin(θi )

t s α

d

Figure 3.5

Configuration of echelle grating

53

The small angle deviation (φ) between the incidence angle and the reflection angle is determined from the simple grating equation by using the groove width (s) and depth (t).93 The equation for an Echelle grating is given by

mλ = 2d(sin θi + sin θr ) (3- 6)

= 2d(sin θi + sin(θi − φ)) (3- 7)

= 2t(1 + cos(φ)) − s sin(φ)) (3- 8)

= t (3- 9)

= 2d(sin(α)) (3- 10)

The final grating equation obtained from the optical path difference 2d(sin(α)) is given by mλ = 2d(sin(α)). The spectral resolving power (R) is given by

R = λ / Δλ min (3- 11)

Here, ∆λmin is the least resolvable wavelength difference between two adjacent lines. The use of higher spectral orders with this configuration compensates for the problem of lower spectral resolving power originating from too coarser grooves density. The actual spectral resolving power was estimated using a Hg-lamp spectrum (see Fig. 3.6). Two peak values of 313.158 nm and 313.185 nm from Hg lamp are compared with two systems based on ESA 3000 and HR 460 spectrometers. In the case of ESA 3000

Analyzer, peak values of the two emission lines are separated by 25 pm (4 pixels). The actual spectral resolving power of two emission lines with diffraction order of 76 was estimated as R = 12500.94 These two emission lines were not resolved by the HR 460 system with a slit width of 100 μm and a groove density of 2400 mm-1.

54

800

700

600

500

400

Intensity(a.u.) Intensity(a.u.) 300

200

100

0 313.000 313.100 313.200 313.300 313.400 313.500 313.600 Wavelength(nm)

(a)

14000

12000

10000

8000

6000 Intensity(a.u.) 4000

2000

0 313 313.1 313.2 313.3 313.4 313.5 313.6 Wavelength(nm) (b)

Figure 3.6

A part of the Hg spectrum obtained from echelle (a) and HR 460 (b) spectrometer systems, respectively

55

The reflected wave from the grating still overlaps multiple spectral orders. A quartz prism placed in front of the grating acts as a multiple spectral order sorter, generating a two-dimensional diffraction pattern, which it makes broadband95 (see Fig.

3.7). The resultant diffracted light from the two-dimensional pattern is collimated and directed by the camera mirror onto an ICCD detector. The linear dispersion per pixel, bandwidth per pixel, are given by

Δλ pixel = Δx pixel (dλ / dx) (3- 12)

= Δx pixeld(cos(α)) / mf (3- 13) where f is the focal length of the camera mirror, and Δxpixel is the pixel size. The larger blaze angle of Echelle gratings reduces the linear dispersion per pixel, which reflects high resolution (see Eq. (3- 13)). The spectrum parameters of the ESA 3000 Analyzer are listed in Table 3.511 , 96 By using equation (3- 10), equation (3- 13) can be simply modified:

λ cos(α) Δλ = Δx (3- 14) pixel pixel 2f sin(α)

λ 2f tan( α) = (3- 15) Δλ pixel Δx pixel

56

Figure 3.7

Experimental setup of ESA 3000 Analyzer (This figure is copied from the manual produced by LLA Instrument GmbH)

Table 3.5

Spectral parameters based on ESA 3000 Analyzer system

Spectrometer ESA 3000 LLA j Simultaneous Spectral range ~580nm Spectral coverage 200~780nm Groove density 75/mm Blazed angle 60° Focal length of camera mirror 250mm Pixel size 24×24μm2 Diffraction Bandwidth Free spectral wavelength (nm) order (pm/pixel) range(nm) 285.002 84 7.778 3.39 303.000 79 8.270 3.84 380.000 63 10.37 6.03 610.016 39 16.75 15.6

57

The intensified charged coupled devices (ICCD) in the ESA 3000 Analyzer consists of a photocathode, micro-channel plate (MCP), phosphor screen, and two- dimensional diode arrays.13 The diffracted light from the grating enters into the ICCD and induces an electron current, amplified electron current, and amplified light signal in three steps by photocathode, MCP, and phosphor screen, respectively.

The final amplified light signal from the phosphor screen produces electron-hole pairs in the semiconductor diode array of the CCD. These electrons and holes move in opposite directions due to the electric potential difference crossing the diode. These electrons and holes meet again through the external circuit. This current produces the final electric signal. This electric signal is digitized under 16-bit resolution (216), which indicates the count number displayed on the PC is from 0 to 65536. The image signal that represents the electric signal in the CCD array is displayed as a sequence of diffraction order (see Fig 3.8).

58

Mg II 279.553 nm

1024 pixel

Diffraction Order 86 Ca II 393.367nm

1024 pixel

2.E+04

2.E+04

2.E+04

1.E+04

1.E+04

1.E+04

8.E+03 Intensity(a.u.) Intensity(a.u.) 6.E+03

4.E+03

2.E+03

0.E+00 250 270 290 310 330 350 370 390 Wavelength(nm)

Figure 3.8

CCD image and LIBS spectrum for echelle spectrometer

59

Vertical pixel Mg ( I ) 285.222 nm

Wavelength

(a)

288.656 nm, Mg ( I ) 285.222 nm, diffraction order: 83 diffraction order: 84 Mg ( I ) 285.222 nm,

7000 diffraction order: 84 ) )

. 6500

u 281.868nm, .

a diffraction order: 85 (

6000

y y

t

i s

n 5500

e ) t

n 285.35 m

I n 5000 285.30 (t 285.25 h g 4500 285.20 n 14 le 12 285.15 e 10 8 v 6 285.10 a 4 2 Vert W ical P ixe l

(b)

Figure 3.9

(a), (b) and (c) indicate CCD image, digitalized spectrum, and averaged spectrum of Mg I 285.213 nm transition

60

7000

6500

6000

5500 Intensity (a.u.)

5000

4500 285.000 285.100 285.200 285.300 285.400 Wavelength (nm)

(c)

Figure 3.9 (continued)

The horizontal 60 pixels and vertical several pixels are taken to calculate the peak area intensity of atomic emission line (see Fig 3.9). The number of the vertical pixels depends on the wavelength. The peak area intensity is calculated from the spectrum area within the full-width-at-half-maximum averaged spectrum value of vertical pixels.

Signal saturation which indicates strong intensity above the threshold count, causes the overflow of the electrons on the pixel (see Fig 3.10). Some part of the overflow electrons can penetrate into nearby pixels with different orders and wavelengths. It creates an unwanted spectrum called a ghost line.11

61

Sn (I) 380.100nm 70000

60000

) )

. u

. 50000

a

(

y

Order 63 t 40000

i s

n 30000

e

t n I 20000 ) Order 62 380.4 m 380.3 n 10000 380.2 ( th 380.1 g 16 14 380.0 n 12 le 10 379.9 8 6 ve 4 379.8 a V 2 W erti 0 cal P ixel

Sn (I) 380.100nm

Figure 3.10

Blooming due to saturation of the Sn I 380.100 nm transition

The deuterium continuum spectrum shows that spectral response varies not only with diffraction order, but also with pixel position within one diffraction order (see Fig

3.11). In other words, emission line intensity which indicates the relative populations of excited species responds differently depending on the instrument (grating). The characteristic of nonlinearity may influence the evaluation of the plasma temperature from the relative line intensity. 11

62

Diffraction Order: 84

(a)

2.E+04

2.E+04

1.E+04

1.E+04

9.E+03 Intensity (a.u.)

7.E+03

5.E+03

3.E+03 280 290 300 310 320 330 340 Wavelength (nm)

(b)

Figure 3.11

Two dimensional echelle CCD image of a deuterium (a) and spectrum response of a deuterium continuum source from diffraction orders 85 to 71 (b)

63

CHAPTER IV

LIBS APPLICATION TO IN SITU ANALYSIS OF SLURRY SAMPLES PREPARED

FOR THE PURPOSE OF SIMULATING THE VITRIFICATION PROCESS OF

LIQUID RADIOACTIVE WASTES

4.1 Experimental Description

Laser induced breakdown spectroscopy (LIBS) was applied to primary liquid slurry sample prepared for the purpose of simulating the vitrification process of liquid radioactive wastes. A frequency-doubled pulsed Nd:YAG laser (Continuum Surelite I or

III) beam is focused onto the surface of a slurry sample filled in a shielded cell using a

50-cm (or 30-cm) focal length fused silica convex lens (see Figure 4.1). Spectroscopic analysis was performed by two different detection systems: Czerny-Turner and Echelle spectrometer systems. Data acquisition for the Echelle spectrometer was performed using the ESAWIN computer program from LLA Instruments GmbH. This program allows us to effectively control the laser firing during data acquisition; this reduces unnecessary laser firing during data processing and storing.

64

BD - Beam Dump DM - Diachronic Mirror OF – Optical Fiber

HS BD HS -Harmonic Separators L - Lens Nd: YAG Laser 2 x 2x - KDP Doubler DM IDAD - Intensified OF Prism Diode Array Detector ICCD - Intensified IDAD or ICCD L Computer L Charged Coupled Spectrometer Device

Pulse Generator Controller Fan

Glass Beaker

Figure 4.1

Schematic diagram of LIBS measurement apparatus

4.2 Results and Discussion

4.2.1 Sample Preparation

The slurry sample is mainly made up of 78.2% water, 8.3 % ferric oxide Fe2O3,

3.6 % alumina Al2O3, 5.18% sodium oxide Na2O, and small quantities of oxides of carbon, silica, chromium, manganese, magnesium, etc. The slurry sample was first acidified until the pH is 6 by the addition of strong nitric acid, glass frit was also added.

Glass frit is chemical solid composed of SiO2 (70.0%), B2O3 (12.0%), Na2O (11.0%),

Li2O (5.0%) and MgO (2.0 %). This mixture known as slurry mix evaporator (SME) products is finally fed into glass melter to make the solid stimulant low activity test

65 reference material (LRM) glass. The slurry composition was chosen as a surrogate for the radioactive slurry that is input into the Savannah River site’s Defense Wastes Processing

Facility (DWPF) glass melter.

4.2.2 Spectroscopic Comparison

The ESA 3000 analyzer with high resolution is more suitable for detecting the multiple elements than the Czerny-Turner spectrometer for application of LIBS to samples. Figure 4.2 shows a spectroscopic comparison of LIBS spectra recorded by a

SPEX 500M spectrometer with a grating of 2400 g/mm and the ESA 3000 analyzer, respectively.97 Intensity(a.u.) Intensity(a.u.)

381 383 385 387 389 391 393 Wavelengt(nm)

Figure 4.2

LIBS spectra of raw Sludge Receipt and Adjustment Tank (SRAT) slurry around 386 nm. The spectra of dashed line and solid line are recorded by SPEX 500M and echelle spectrometers, respectively

66

The SPEX 500M spectrometer has a constant linear dispersion value of 19.5 pm/pixel, which is not sufficient to resolve the multiple spectral lines in this line-rich spectral region. Meanwhile, the echelle spectrometer has a lower linear dispersion than the SPEX 500M spectrometer; this reflects the high resolution of echelle spectrometers.

4.3 Result Recorded by Czerny-Turner Spectrometer System

4.3.1 Reproducibility of Line Emission Intensity

Poor reproducibility and poor precision of signals attributed to shot-to-shot plasma fluctuations are the main disadvantage of the LIBS technique. The normalization method provided by the OMAVISION software program was introduced as an alternative way to improve reproducibility of LIBS signals.98 The normalized atomic emission line intensity is the ratio of the integrated atomic emission line intensity to the total integrated plasma emission line intensity. This normalization technique was applied to LIBS measurements over an extended period of time. Figure 4.3 shows the atomic line intensity of Fe 382.043 nm transition with and without plasma background normalization obtained from measurement repeated in 7 days.

67

9.00E+04 0.1

Mean Value of Absolute Intensity: 35728.71 8.00E+04 0.09

0.08 7.00E+04 Mean Value of Normailzed Intensity: 0.33 0.07 6.00E+04 0.06 5.00E+04 0.05 4.00E+04 0.04 3.00E+04 0.03 2.00E+04 0.02

1.00E+04 0.01

0.00E+00 0 09/07/2005 09/09/2005 09/12/2005 09/14/2005 10/17/2005 10/20/2005 10/24/2005

Figure 4.3

Reproducibility trend of the absolute and normalized intensity of Fe I 382.043 nm line for 7 days. Solid line and dashed line indicate mean value of absolute intensity and normalized intensity for Fe I 382.043 nm

68

4.3.2 Laser Power Effect and Calibration Curve of Normalized Intensity

The emission line intensities of Fe I 382.043 nm and Al I 394.403 nm as a function of laser energy are shown in Figure 4.4. The emission line intensity clearly increases when laser energy is increased. Meanwhile, the normalized intensity was insensitive to variation of laser energy. The RSD values of normalized intensities of Fe I

382.043 nm and Al I 394.403 nm were calculated to be 3.60% and 3.39%, respectively.

The signal normalization was applied to the calibration curves by adding Na2CO3 into the

SRAT slurry with pH value 6.0. Figure 4.5 shows the calibration curve data with and without the signal normalization.

69

1.0E+05 0.1 Intensity(a.u.): Fe(I) 382.043nm 9.0E+04 Intensity(a.u.): Al(I) 394.403nm 0.09 Normalized Intensity: Fe(I) 382.043nm, RSD = 3.6% 8.0E+04 0.08 Normalized Intensity: Al(I) 394.403nm, RSD = 3.39% 7.0E+04 0.07

6.0E+04 0.06

5.0E+04 0.05

4.0E+04 0.04 Intensity (a.u.)

3.0E+04 0.03 Intensity alized orm N

2.0E+04 0.02

1.0E+04 0.01

0.0E+00 0 60 70 80 90 100 110 120 Laser energy (mJ/pulse)

Figure 4.4

The absolute and normalized intensity variation of Fe I 382.043 nm line as a function of laser energy. The dashed line indicates mean value of normalized intensity for Fe I 387.857 nm

70

4.0E+04 0.45

0.4 3.5E+04 Intensity(a.u.)

0.35 3.0E+04 Normalized Intensity 0.3 2.5E+04 0.25 2.0E+04 0.2

Intensity(a.u.) 1.5E+04

0.15 Normalized Intensity

1.0E+04 0.1

5.0E+03 0.05

0.0E+00 0 0.0% 1.0% 2.0% 3.0% 4.0% 5.0% 6.0% 7.0% 8.0% 9.0% Na Concentration(%)

Figure 4.5

Calibration curve for Na I 330.237 nm line recorded by Czerny-Turner spectrometer system (SPEX 500M)

71

4.4 Result Recorded by Echelle Spectrometer System

4.4.1 Electron Density

We evaluated the electron density by using Equation (2- 40). Lorentz fitting using

Origin software was applied to determine the full-width-at-half-maximum (∆λobserved) of an observed atomic emission. ∆λ was obtained by subtracting the instrumental line

99,100 broadening width. The value of ∆λinstrumental estimated by reference 11 was used,

(∆λinstrumental = 0.01nm). The electron density of the plasma was estimated for Fe I spectral line 381.584 nm and for Al I spectral line 396.153 nm.53 ,101

9

) 8.5

-3 Fe

cm 8 Al +16 7.5

7

6.5

6

electron density (10 density electron 5.5

5 0 1000 2000 3000 4000 Gate delay (ns)

Figure 4.6

Electron density of Fe I 381.584 nm and Al I 396.271 nm as a function of gate delay time (gate width 2 μs)

72

4.4.2 Plasma Temperature

The excitation and ionization temperatures of Fe I were estimated by the equation

2- 61 and equation 2- 63.81, 102 The estimated electron density from Fe I 381.584 nm was applied to determine the ionization temperature. In the case of excitation temperature, the slope of the dashed line obtained from a linear fitting provides the excitation temperature

(see Fig. 4.7). The spectral line wavelengths, energies of the upper levels, statistical weights and transition probabilities used for each element were obtained from NIST.103

(see Table 4.1).

-4

-6

-8

y = -1.3561x - 8.6416 -10 )

/Ag -12 λ

ln (I ln -14

-16

-18

-20 2.5 3 3.5 4 4.5 5 Energy (eV)

Figure 4.7

Boltzmann plots of Fe I obtained from the SRAT slurry (gate delay 800 ns and gate width 2 μs)

73

Table 4.1

List of the Fe I spectral information used for Boltzmann plot

Ej gj 8 -1 Element Line (nm) Aji (10 s ) (upper (upper energy level) level degeneracy) Fe (I) 379.501 0.115 4.2562222 7

Fe (I) 379.955 0.0732 4.2203615 9

Fe (I) 381.584 1.3 4.7331339 7

Fe (I) 382.436 0.0283 3.2409687 7

Fe (I) 382.782 1.05 4.7954645 5

Fe (I) 383.422 0.453 4.1908599 5

Fe (I) 384.996 0.606 4.230536 1

Fe (I) 385.637 0.0464 3.265703 5

Fe (I) 385.991 0.097 3.211889 9

Fe (I) 386.552 0.155 4.2175824 3

Fe (I) 387.250 0.105 4.1908599 5

Fe (I) 389.576 0.094 3.2918393 1

Fe (I) 389.970 0.0258 3.2657053 5

74

The electron density obtained from the Fe I 381.584 nm line was used to calculate the ionization temperature. The ionic temperature was determined by using Fe I 381.584 nm and Fe (II) 275.574 nm. Figure 4.8 shows the temporal evolution of excitation and ionization temperature corresponding to a gate width of 2μs for various delay times ranging from 600 ns to 3500 ns. The largest energy difference (∆E = 3.27 eV) at plasma temperature of 8800 K (measured at gate delay 600 ns) was used to evaluate the validity

+15 -3 of the LTE assumption. The lower limit for ne was estimated to be 4.59×10 cm (see

Eq. 2- 9).

9500 Ionization Temperature 9000

Excitation Temperature 8500

8000

7500

7000 Plasma Temperature (K) 6500

6000 0 500 1000 1500 2000 2500 3000 3500 4000 Gate Delay (ns)

Figure 4.8

Excitation and ionization temperatures of SRAT slurry (gate width 2 μs)

75

4.4.3 Calibration Curve

Slurry mix evaporator (SME) products with different solid weights were prepared by adding glass frit into acidified SRAT slurry with a pH of 6. Five SME products were tested to collect the calibration curve. Li signal was found only to increase with the concentration of solids in the slurry (see Fig. 4.9). Despite high silica concentration in the glass frit, spectra response for Si I 288.158nm is particularly low (see Fig. 4.10). Glass frit of large and firm grains is rarely suspended on the surface of the sample due to its weight. Sedimentation in LIBS analysis can be a serious obstacle because the plasma is created on the surface. Raw SRAT (without adding glass frit) contains SiO2 (2.27%). A weaker atomic line intensity of Si I was observed compared to that of other elements with similar concentrations.

4500 4000 3500 3000 2500 2000 1500 Intensity (a.u.) (a.u.) Intensity 1000 500 0 0 0.1 0.2 0.3 0.4 Li (wt %)

Figure 4.9

Calibration curve for Li I 670.790 nm line using glass frit (70 mJ/pulse, gate delay 1 μs and gate width 4 μs)

76

350

300

250

200

150 Intensity(a.u.) Intensity(a.u.) 100

50

0 0 1 2 3 4 5 6 Si (wt %)

Figure 4.10

Calibration curve for Si I 288.158 nm line using glass frit (70 mJ/pulse, gate delay 1 μs and gate width 4 μs)

1.2E+04

1.0E+04

8.0E+03

6.0E+03

Intensity(a.u.) 4.0E+03

2.0E+03

0.0E+00 0 1 2 3 4 5 6 Si (wt %)

Figure 4.11

Calibration curve of Si I 288.158 nm by using silicon dioxide (SiO2) (80 mJ/pulse, gate delay 300 ns and gate width 4 μs)

77

Instead of glass frit, fine silicon dioxide powder (SiO2) and a short gate delay time were used to obtain the calibration curve of silicon. Figure 4.11 shows the calibration curve of Si I (288.158 nm) by using a fine powder of SiO2. The linearity and spectral response of Si I (288.158 nm) were dramatically improved. Meanwhile, a short gate delay and high laser power leads to observation of the saturation of the strong emission lines of other elements (see Fig. 4.12). Signal saturation causes an overflow of the electrons on the pixel. Some of the overflow electrons can penetrate into nearby pixels with different orders and wavelengths. Signal saturation should be avoided to keep the safety of the

ICCD equipment.

Diffraction order: 61 Diffraction order: 62 Peak wavelength: 393.365nm Peak wavelength: 387.023nm

70000

60000 ) )

. 50000 u

.

a

(

y y 40000

t

i s

n 30000

e

t n I 20000 393.55 ) 393.50 m n 10000 393.45 ( 393.40 th g 0 393.35 n 14 e 12 393.30 el 10 v 8 393.25 6 V 4 393.20 Wa erti 2 cal P ixel

Figure 4.12

Saturation of the Ca II 393.365 nm transition on spectrum (80 mJ/pulse, gate delay 300 ns and gate width 4 μs)

78

4.5 Conclusion

We performed the spectroscopic analysis of SRAT slurry using two different detection systems. In particular, the ESA 3000 Analyzer facilitates simultaneous measurement of the electron density and plasma temperature. However, experimental parameters, such as gate delay and laser power should be considered to avoid saturation of strong emission lines. Sample splashing might cause contamination of optics induced by the high-power pulsed-laser beam focused on liquid sample’s surface. Use of a lens with a long focal length is efficient to minimize contamination of the optics.

Sample preparation can be essential in LIBS analysis of liquid samples whose contents are homogenously distributed. The lower spectral response of Si I 288.158 nm for the glass frit was observed due to sedimentation.

79

CHAPTER V

COMPARATIVE STUDY OF LASER INDUCED BREAKDOWN SPECTROSCOPY

MEASUREMENT USING TWO SLURRY CIRCULATION SYSTEMS

5.1 Experimental Description

This chapter examines the experimental conditions associated with slurry measurements to achieve good precision using laser induced breakdown spectroscopy.

Two slurry circulation systems were devised to overcome major technical problems associated with LIBS measurements of slurry samples: namely sedimentation and change in the lens-to-sample distance (L.T.S.D) during measurement. 104 , 105 LIBS slurry measurements using both circulation systems are compared. The results show that the experimental configuration plays a crucial role in the on-line slurry analysis.

Figure 5.1 shows a schematic diagram of the LIBS experimental setup for slurry samples. A frequency-doubled, Q-switched Nd: YAG laser (Continuum Surelite I, 200 mJ max. at 532 nm, 10 ns pulse width) was incorporated with a Czerny-Turner spectrograph (SPEX 500M, 2400 lines/mm grating) to collect LIBS signal. The laser flashlamp discharge rate is 10 Hz, but the laser system has an option to allow laser output at lower repetition rates. So we can perform LIBS measurements at different laser pulse rates (< 10Hz). Laser light was focused onto the slurry surface using an ultraviolet (UV)-

80 grade quartz lens of 300-mm focal length. Data acquisition can be performed in either free running or external synchronization mode. Data acquisition and analysis were performed using a personal computer and EG&G OMAVISION PC software.

BD - Beam Dump BD DM - Diachronic Mirror HS OF – Optical Fiber HS -Harmonic Separators Nd: YAG Laser 2x L - Lens DM 2x - KDP Doubler OF IDPA - Intensified Prism Diode Array Detector IDPA L L

Spectrometer

Pulse Controller Fan Generator Computer Glass Beaker

L Circulation Circulation L system (I) Fan system (II) Fan

P S L: Lens P L:S: Lens Slurry L: Lens S:P: Slurry Peristaltic S: Slurry P: Peristaltic Pump S P: Peristaltic Pump Pump

Figure 5.1

Schematic diagram of LIBS and circulation systems for slurry sample

81

5.2 Slurry Handling System

The sample used in this study is a special waste compound in slurry form. This sample was prepared for the simulation tests by the Advanced Vitrification System.106

The slurry sample contains 85.4 % water, 2.5 % ferric oxide Fe2O3, 1.7 % alumina Al2O3, and small quantities of oxides of boron and chromium. The composition of the slurry samples used in the LIBS measurements has been independently determined with an inductively coupled plasma (ICP) system in our analytical laboratory. LIBS measurements with the slurry sample kept in a small beaker have shown a serious decline of line intensity with time caused by quick slurry sedimentation. Our two newly devised circulation systems with different geometric configurations were designed to avoid slurry sedimentation.

In the circulation system I (see Fig. 5.1 inset), a two-head peristaltic pump continuously circulated the slurry, and the slurry inside the cylindrical vessel was constantly refreshed. An optimum flow rate of ~ 340 mL/min was found for LIBS experiments with this slurry circulation system. Higher flow rates disturbed smooth slurry circulation, causing the slurry to overflow the cylindrical vessel. Lower flow rates provided insufficient slurry pumped by the peristaltic pump to the cylindrical vessel, which resulted in unstable slurry circulation.

This system can provide a new face of slurry for laser ablation and maintain a constant lens-to-sample distance (L.T.S.D). A plasma plume was created on the slurry surface of the cylindrical vessel by high-energy laser pulses. The vertical cylindrical

82 vessel, 20 mm in diameter and 69 mm in height, was manufactured to promote faster vapor dispersion of the plasma plume. A small fan was placed by the side of the slurry vessel to blow air over the sample surface to clear the mist generated by the laser. The mist generated by plasma dynamic heating affects both the incoming laser beam and the plasma emission light. The pulse repetition rate was set to 1 Hz in this experimental configuration to minimize the effects of laser produced mist on LIBS measurements. In this experiment, data acquisition is synchronized with the laser pulse. The homogeneity of the flowing slurry was confirmed by the reproducibility trend of LIBS signals which yield a relative standard deviation (R.S.D) value of 4.8% from 6 sets of data taken at different time, as shown in Fig. 5.2. Each data set in Figure 5.2 is an average of 120 laser shot LIBS spectra.

1.2E+05

1.0E+05

8.0E+04

6.0E+04

4.0E+04 Intensity(a.u.) Intensity(a.u.)

2.0E+04

0.0E+00 0 1 2 3 4 5 6 7 Data Set

Figure 5.2

Reproducibility trend of the intensity of Fe I 382.043 nm line obtained with circulation system I

83

In circulation system II (see Fig. 5.1 inset), a horizontal cylindrical tube of varying cross-sectional area, known as a Venturi tube, was connected by a rubber hose to the peristaltic pump and slurry reservoir. The center and the end of arm have a cross- sectional area of ~0.2 cm2 and ~3.1 cm2, respectively. The cylindrical nozzle of the

Venturi tube has a 10 mm diameter circular opening and is 32.5 mm in length from the center of Venturi throat. Laser light passes through the nozzle of the Venturi tube and strikes the flowing slurry in the Venturi tube. This configuration can minimize problems associated with slurry splashing. A fan was placed near the nozzle of the Venturi tube to remove the mist produced by the laser pulses. The laser was operated at 10Hz with this experimental configuration.

5.3 Results and Discussions

5.3.1 Effect of Laser Pulse Rate

To study the effect of laser pulse frequency on LIBS slurry measurements, a slurry sample kept in a small beaker was initially tested. The slurry sample used here was prepared by mixing dry slurry powder with water. An L-shape rod attached to a motor was used to continuously stir the slurry inside the glass beaker to avoid sedimentation

(see Fig. 5.3). The emission line intensities and the R.S.D values of Fe I (385.995 nm), Fe

I (382.043 nm), and Al I (394.407 nm) as a function of laser pulse rate are shown in Fig.

5.4. Data shown in Figure 5.4 were obtained from an average of over 400 laser shot LIBS spectra. It clearly shows that the data recorded at 1 Hz laser pulse rate gives the best

84 precision. The observed R.S.D values for 1 Hz laser pulse rate measurements were the lowest compared with 10 Hz and 5 Hz data. R.S.D values of 4.57% for Fe I (385.994 nm), 4.12% for Fe I (382.043nm) and 2.98% for Al I (394.047 nm) were obtained from the 1Hz data. The lower laser pulse rate is thought to have reduced the liquid surface disturbance by laser shockwave and hence to have resulted in better precision.107 The lower laser pulse rate (with external synchronization) would significantly increase the total sampling time. The desired laser pulse rate for practical applications will have to be based on practical experimental conditions.

Motor

o Θ = 19.188 Laser

Fan

Glass beaker

Slurry height: 3.5cm

Figure 5.3

Experimental arrangement of direct slurry sampling for studying the effect of laser pulse frequency

85

1.2E+05 Fe I (385.994nm) 1.0E+05 Fe I (382.042nm) Al I (394.407nm) 8.0E+04

6.0E+04

Intensity (a.u.) (a.u.) Intensity 4.0E+04

2.0E+04

0.0E+00 1Hz 5Hz 10Hz Laser Frequency(Hz)

a)

30.00% Fe I (385.994nm) 25.00% Fe I (382.042nm) Al I (394.407nm) 20.00%

15.00% RSD 10.00%

5.00%

0.00% 1Hz 5Hz 10Hz Laser Frequency (Hz)

b)

Figure 5.4

Comparison of intensity (a.u.) and RSD value (%) obtained from LIBS measurements with different laser pulse frequencies

86

5.3.2 Slurry Circulation Systems

To compare the performance of the two slurry circulation systems for LIBS measurements, LIBS slurry spectra with different experimental parameters (laser energy, gate delay, gate width, lens-to-sample distance) were recorded with each slurry circulation system. Typical LIBS spectra of slurry samples recorded with each circulation system are shown in Fig. 5.5.

Experimental data show that different optimum experimental parameters are needed for each circulation system. The detection window (i.e., gate delay and gate width) determines the LIBS signal-to-noise ratio and signal fluctuation. Therefore, it affects both the measurement precision and the detection limit. We found that the optimum detection windows for the two circulation systems are different. Note that a stronger signal with circulation system II was observed despite the low pulse energy.

The geometric feature of circulation system II accumulates and confines the vapor generated above the slurry surface. The mist generated above the slurry surface disturbs the laser light reaching the sample and affects the LIBS signal.105 For measurements with circulation system I, the gate delay and gate width were varied from 1 μs to 6 μs. In the circulation system II measurements, the gate delay was varied from 2 μs to 7 μs; gate width was varied from 2 μs to 10 μs; and laser energy was increased from 10 to 40 mJ/pulse. The optimum experimental parameters found for each circulation system are presented in Table 5.1.

87

8.0E+03 2 7.0E+03

6.0E+03 Al396.15 5 4 5.0E+03

4.0E+03 393.36 Al394. + 396.847 +

3.0E+03 Ca Ca Intensity (a.u.) (a.u.) Intensity Fe385.991 2.0E+03 Fe382.042 1.0E+03 0.0E+00 380 385 390 395 400 Wavelength (nm)

a)

4.0E+04 2 5 7 3.5E+04 2 393.39 Al396.15 3.0E+04 396.84 + + a 4 Ca 2.5E+04 C

2.0E+04 Fe382.04 Fe385.991 Al 394. 1.5E+04

Intensity (a.u.) (a.u.) Intensity 1.0E+04 5.0E+03 0.0E+00 380 385 390 395 400 Wavelength (nm)

b)

Figure 5.5

(a) LIBS spectrum with circulation system I and (b) LIBS spectrum with circulation system II

88

Table 5.1

Optimal experiment parameters employed for LIBS measurements

Experimental Circulation Circulation Parameter System (I) System (II) Gate Delay 1μs 3μs Gate Width 4μs 5μs Pulse Energy 80mJ 30mJ Laser Repetition 1Hz 10Hz Rate

LIBS signals were also evaluated as a function of the lens-to-sample distance

(L.S.T.D) for both systems (see Fig. 5.6). We found stronger line intensities with circulation system I when the focal spot was placed on the surface of the slurry sample

(see Fig. 5.6a). In contrast, the line intensity was greater when the focal spot was above the sample surface (i.e., in the air) for circulation system II (see Fig. 5.6b). When circulation system II was used, the laser was operated at the optimum pulse rate of 10 Hz.

This experimental configuration will preconcentrate the aerosol inside the nozzle (in the air). This results in stronger signals when the laser pulses’ focal spot is few millimeters above the sample surface. In the circulation system I measurements, the laser was operated at 1 Hz and the slurry container has a broader opening. This experimental configuration promotes the dispersion of laser-produced aerosols during measurement.

Hence the LIBS signal increases as the focal spot moves toward the slurry surface.

89

1.0E+05

8.0E+04

6.0E+04

4.0E+04 Intensity (a.u.) (a.u.) Intensity 2.0E+04

0.0E+00 -4 -2 0 2 4 6 Distance (mm) from focal point

a)

2.5E+05

2.0E+05

1.5E+05

1.0E+05 Intensity (a.u.) 5.0E+04

0.0E+00 -10-5 0 5 10 15 20 Distance(mm) from focal point

b)

Figure 5.6

The intensity variation of Fe I 385.991 nm line with changes in focal location above and inside the surface for (a) circulation system I and (b) circulation system II. The “0”, + value, and - value position indicate a focal location on, above and inside sample

90

5.3.3 Calibration

To compare the detection limits of the two slurry circulation systems for LIBS measurements, we prepared different composition slurry samples for calibration by adding Fe2O3 directly into the batch slurry sample. The data were collected under optimum experimental parameters for each system. LIBS’ poor signal reproducibility is mainly attributed to shot-to-shot plasma fluctuations. The use of a plasma background normalization technique shows promising results for reducing the signal fluctuation (see

Fig. 5.7).

6.0E+04 0.05 Intensity: RSD = 20.35% 0.045 5.0E+04 Normalized intensity: RSD = 8.97% 0.04 4.0E+04 0.035 0.03 3.0E+04 0.025 0.02 Intensity (a.u.) (a.u.) Intensity 2.0E+04 0.015

1.0E+04 0.01 Intensity Normalized 0.005 0.0E+00 0 0 5 10 15 20 25 30 35 Spectrum number

Figure 5.7

The intensity variation of Fe I 382.043 nm line during slurry sampling with circulation system I

In this method, the integrated atomic line area is normalized by the integrated plasma emission. It is a strict forward method and shows the best precision improvement

91 for the slurry samples compared with other post processing methods. The variation of the intensity of the Fe I 387.857 nm line at various Fe concentrations for both systems is shown in Fig. 8a and 8b.

1.2E+05 0.025 Intensity 1.0E+05 0.02 Normalized Intensity 8.0E+04 0.015 6.0E+04 0.01 4.0E+04 Intensity (a.u.) (a.u.) Intensity 0.005 2.0E+04 Intensity Normalized

0.0E+00 0 0 1 2 3 4 5 6 7 8 Fe (wt %)

a)

2.0E+05 Intensity 1.8E+05 0.02 Normalized Intensity 1.6E+05 1.4E+05 0.015 1.2E+05 1.0E+05 0.01 8.0E+04

Intensity (a.u.) (a.u.) Intensity 6.0E+04 4.0E+04 0.005 Normalized Intensity Intensity Normalized 2.0E+04 0.0E+00 0 0 1 2 3 4 5 6 7 8 Fe (wt %)

b)

Figure 5.8

Calibration curve for Fe I 387.857 nm line with (a) circulation system I and (b) circulation system II

92

The calibration curves for Fe from both systems show a linear increase as the Fe concentration in the slurry increase. Limit of Detection (L.O.D) values of Fe for both systems were estimated by using

CL = 3σB/S (5- 1) where σB is the standard deviation of the background and S is the slope of the calibration curve.13 The detection limits of Fe for circulation system I and II were estimated to be

37.8 and 278.7 ppm, respectively.

5.4 Conclusion

LIBS analysis of slurry samples that contain high water content faces the problems of poor detection limits and measurement precision. In this chapter, we evaluated two circulation systems for LIBS measurement of slurry samples. Circulation system II is capable of preventing the contamination of optics from slurry splashing which interferes with the passing of the laser pulse through the lens. However, excess mist or vapor created from the plasma plume remained about the sample. This will obstruct laser pulses from reaching the slurry sample, generating unrepresentative LIBS signals. In other words, the spark that should be generated on the surface of the slurry is created in the air because the vapor keeps the laser pulse from reaching the slurry surface.

Circulation system I, on the other hand, is a proper apparatus for removing the mist. Use of a lens with a long focal length is necessary to minimize contamination of the optics from slurry splashing. These findings led us to conclude that reliable experimental configurations are crucial for implementing the best approach to on-line slurry analysis.

93

CHAPTER VI

TIN ALLOY ANALYSIS COMBINED WITH ARTIFICIAL NEURAL NETWORK

PREDICTION

6.1 Experimental Description

In this chapter, LIBS technique was applied to tin alloy samples for the purpose of quantitative analysis. The LIBS system consists of a Nd:YAG laser (Continuum Surelite

III) and a broadband spectrometer (ESA 3000, LLA Instruments, GmbH) . The 532-nm laser light is focused onto the sample using an ultraviolet (UV) grade quartz lens of 300- mm focal length. The samples are mounted on a rotating platform at 0.25 rotation per minute (rpm) rotational speed. Tin (Sn) alloy samples present in a form of small fragments are mainly made up of tin and small quantities of other elements. LIBS measurement with raw sample without physical treatment was initially tested. The uneven sample surface influences the lens-to-sample distance and results in a relatively large relative standard deviation (RSD). To improve the measurement precision, LIBS data were then collected using pellets made from the raw samples. About 10 gram of raw sample is pressed into a pellet by putting it in a 25 mm bore to make a cylindrical shape pellet of 4-mm height under pressure of 3000 lb/in2. The pellet was placed on a rotating platform to minimize formation of deep craters made by laser pulses. Unknown samples

94 without information of elemental composition were tested to estimate several sample compositions simultaneously. An artificial neural network, calibration method and chemical analysis were applied to estimate the elemental concentrations of unknown samples.

6.2 Data Analysis

An artificial neural network (ANN) was applied to extract the three elemental impurity concentrations of tin (Sn) alloy. For application of the neural network, datasets are categorized into two groups for each tin (Sn) alloy: calibration set (known sample) for the training of neural network and validation set (unknown sample) for prediction of concentrations. The neural network was initially trained with known data sets of calibration phase over the concentration range. Inputs and outputs consist of atomic emission line intensity ratios and chemical elemental concentrations. In the case of input, two strong emission lines of each sample were selected and normalized by the intensity of the Sn I 333.062 nm line for analysis: Ag I 338.289 nm and 328.068 nm, Cu I 327.396 nm and 324.754 nm, and Pb I 405.782 nm and 368.347 nm.

Table 6.1

The composition of tin alloy (Sn) employed for LIBS measurement. All values are in wt (%)

Sample Ag Cu Pb Sn_0 0.0024 0.002 0.035 Sn_1 0.0086 0.009 0.03 Sn_2 0.025 0.054 0.127 Sn_3 0.007 0.101 0.29 Sn_4 0.016 0.029 0.012

95

The compositions of training samples are given by Table 6.1.During the training, the weights are dynamically adjusted until the sum-squared error (E) has been minimized

(see Eq. 2- 64). The root-mean-square (RMS) in the prediction software represents the square root of averaged sum-squared error (E). RMS is given by

N 1 * 2 RMS = ∑(ci − ci ) (6- 1) N C i=1

E = (6- 2) 2NC

* Here, ci and ci are the estimated and true concentrations. NC is the number of calibration spectra.

0.03

0.025

0.02

0.015

0.01 Computed Output Predicted Ag (wt %) (wt Ag Predicted

0.005 Chemical Anlaysis

0 0 0.005 0.01 0.015 0.02 0.025 0.03 Target Ag (wt %)

Figure 6.1

Computed output by neural network training of calibration set as a function of silver (Ag) concentration

96

Figure 6.1 shows the computed output (training result) as a function of target concentration (chemical analysis). Two intensity ratios (Ag I 338.289 nm and 328.068 nm) normalized by the intensity of the Sn I 333.062 nm line were used. After setting up the prediction model, validation data sets are injected into the network to predict elemental concentration.

6.3 Results

6.3.1 Effects of Crater Size

The pellet samples were mounted on a rotating platform to reduce fluctuations of the LIBS signals mainly attributed to craters produced by the laser pulse. Pellet position on a rotating platform was parametrically adjusted with the purpose of altering the range of sample surface which the laser pulse sweeps over (see Fig 6.2).

25mm L

S

M

Figure 6.2

L, M and S correspond to large, medium and small circles of ablation traces

97

The craters made by the laser pulse’s striking affects sample ablation so that it leads to a decrease of the emission intensity (see Fig 6.3 a, b).

8.E+04 Sn ( I ) 333.062 nm 7.E+04 6.E+04 5.E+04 4.E+04 3.E+04

Intensity (a.u.) (a.u.) Intensity 2.E+04 S, RSD: 57.86% M, RSD: 12.9% 1.E+04 L, RSD: 6.14% 0.E+00 0 100 200 300 400 Time (ns)

(a)

2.E+04 Cu ( I ) 324.754 nm 2.E+04 1.E+04 1.E+04 1.E+04 8.E+03 6.E+03 Intensity (a.u.) (a.u.) Intensity S, RSD: 54.24% 4.E+03 M, RSD: 14.06% 2.E+03 L, RSD: 5.25% 0.E+00 0 100 200 300 400 Time (ns)

(b)

Figure 6.3

The intensity variation of Sn I 333.062 nm (a) and Cu I 327.396 nm (b) line with time during the measurement. L, M and S correspond to large, medium and small circles of ablation traces

98

The deep craters can be an obstacle in forming a reproducible plasma-plume from subsequent striking of laser pluses. Meanwhile, the intensity ratio shows a good reproducible trend, independent of the data acquisition time (see Fig. 6.4). To ensure reasonably good measurements, the LIBS measurements for these samples are based on the calibration with intensity ratio.

0.4

0.35 Cu(324.754 nm)/Sn(333.062 nm) 0.3 0.25 0.2 Cu/Sn 0.15 S, RSD: 4.62% 0.1 M, RSD: 3.58% 0.05 L, RSD: 2.53% 0 0 100 200 300 400 Time (ns)

Figure 6.4

The intensity ratio of Cu I 327.396 nm to the reference line Sn I 333.062 nm during sampling. L, M and S correspond to large, medium and small circles of ablation traces in Fig. 6.2

6.3.2 Voigt Profile of Sn I 333.062 nm

Voigt profile fitting was applied to subtract a Lorentz component in the profile of the spectrum. The Sn I 333.062 nm emission line well isolated and symmetric line shape was identified as a candidate for determination of electron density. Figures 6.5 show the fitting of the neutral Sn I 333.062 nm emission line to Voigt profile using Origin software

99

(Origin Lab Co., USA). A non-linear least-squares fitting was performed by utilizing a

Levenberg-Marquardt algorithm installed in Origin software (Origin Lab Co., USA). The values of Lorentizian width and electron density were estimated (see Table 6.2).

2.5E+04

2.0E+04

Experiment 1.5E+04 Voigt Profile

1.0E+04 In te n s ity (a .u .) .u ity (a s n te In

5.0E+03

0.0E+00 332.79 332.89 332.99 333.09 333.19 333.29 Wavelength (nm)

Figure 6.5

Sn I 333.062 nm line profile. Theoretical Voigt profile fitted to the experiment data point using Levenberg-Marquardt algorithm installed in Origin software

The electron density can be estimated by measuring the Lorentzian half width of an observed atomic emission (see Eq. 2.40). The electron impact parameter given by reference 108 was employed to estimate the electron density.108

100

Table 6.2

The estimated spectroscopic information obtained by Voigt fitting of Sn I 333.062 nm by using Origin software. R is the correlation coefficient

Lorentzian ne 2 Transition λ(nm) 16 -3 R Width (pm) (10 cm ) 3p2 1D -6s3P 333.062 21.22 ± 0.99 2.51 0.9983

6.3.3 Reproducibility

We carried out the simultaneous multi-calibration of four tin Sn alloy samples for all elements of interest. Tin (Sn) is the main element for all the alloys (~ 99%), and is known as the internal standard.109 The intensity ratio normalized to the internal standard

(Sn) might improve the reproducibility of LIBS signals. The emission line from Sn I

333.062 nm is selected as the reference line in this study. The analyte lines used for LIBS calibration are given in Table 6.3. The calibration curves based on intensity ratio with reference line Sn I 333.062 nm are shown in Fig. 6.6.

Table 6.3

Selected analyte lines and RSD values from four samples (Sn_0 ~Sn_3)

Concentration RSD from Element Analyte Line (nm) (Wt %) Intensity Ratio

Ag 338.289 0.025 3.09% Bi 306.772 0.105 1.30% Pb 405.782 0.29 0.70% Cu 327.396 0.101 1.86%

101

In particular, non-linearity is pronounced in the calibration curve of Cu I at

327.396 nm due to self-absorption effect. Self-absorption appears at high elemental concentrations with resonance line.110 The resonance line is an atomic emission line whose lower level of the transition in the ground state.

0.25 0.25

0.2 Experiment I 0.2 Experiment I Experiment II Experiment II 0.15 0.15 Pb/Sn 0.1 Cu/Sn 0.1

0.05 0.05

0 0 0 0.1 0.2 0.3 0.4 0 0.02 0.04 0.06 0.08 0.1 0.12 Pb (wt %) Cu (wt %) 0.08 0.08

Experiment I Experiment I 0.06 0.06 Experiment II Experiment II

0.04 /Sn 0.04 Bi/Sn Ag

0.02 0.02

0 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0 0.02 0.04 0.06 0.08 0.1 0.12 Ag (wt %) Bi (wt %)

Figure 6.6

Calibration curves of Bi, Ag, Pb, Cu in reference to Sn I 333.062 nm line

102

6.4 Calibration Method and Artificial Neural Network

The calibration method and neural network were applied to extract individual concentrations of two unknown samples. Initially, individual 50 records with 5 different

Ag concentrations are injected into the neural network for training (5 × 10 spectra = 50 records). Predict software (Neuralware, Inc).internally picks up the training and test sets from input records that the user supplies (see Table 6.4). The selection is performed by mathematical procedures that generate the best outcome values.111 In our case, the neural network model was internally trained using 34 records and tested using 50 records (all inputs). The R value represents the correlation between the true values and the computed outputs.

Table 6.4

Statistical scores in neural training

Element R RMS Records

Train 0.9935 0.00094 34 Ag Test 0.9860 0.00132 50 Train 0.9920 0.00463 34 Cu Test 0.9888 0.00535 50

Train 0.9962 0.01004 34 Pb Test 0.9946 0.01131 50

After training, individual 20 records of validation data set were injected into the neural network for prediction. The curves of calibration and validation sets are presented in Fig 6.7.

103

elemental concentration(wt%) 405.782 nm linesasafunction ofelem Calibration curveandvalidati

Cu/Sn Ag/Sn 0.05 0.05 0.15 0.25 0.02 0.04 0.06 0.08 0.12 0.14 0.1 0.2 0.1 0 0 00 00 00 00 01 0.12 0.1 0.08 0.06 0.04 0.02 0 005 .1 .1 00 005 0.03 0.025 0.02 0.015 0.01 0.005 0 unknown 1 unknown areobtainedbycalibrationme on dataofAgI338.289nm, CuI327.396nmandPb chemical analysis unknown 2 Figure 6.7 ental concentration. Validaental 104 (b) (a) Cu (wt %) (wt Cu

Ag %) (wt

unknown 1, 2 Chemical Analysis Neural Prediction Calibration method Calibration Set Chemical Analysis Neural Prediction Calibration Method Calibration Set thod, neuralpredictionand tion dataextracting

0.25

0.2

0.15 Calibration Set

Pb/Sn 0.1 Unknown 1, 2 Calibration Method

Neural Prediction 0.05 Chemical Analysis

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Pb (wt %)

(c)

Figure 6.7 (continued)

Table 6.5

Comparison of neural predictions and calibration results of Ag, Cu, and Pb from the chemical analysis

Neural Network Chemical Calibration Method Predicted Value Analysis Prediction Relative Prediction Relative (wt %) (wt %) Error (%) (wt %) Error (%) unknown1 Ag 0.008 0.0105 31.25 0.0092 16.00 (wt %) unknown2 0.016 0.0165 3.13 0.0162 1.25 unknown1 Cu 0.083 0.0825 0.60 0.0690 16.81 (wt %) unknown2 0.076 0.0779 2.50 0.0683 10.17 unknown1 Pb 0.017 0.0153 10.00 0.0120 29.41 (wt %) unknown2 0.015 0.0190 26.67 0.0065 56.67

105

The extracted individual concentrations of the unknown samples are presented in

Table6.5. Predicted values of Cu concentrations from the neural network are closer to the true concentrations. Note that prediction error of sample unknown 1 for Ag (wt 0.008%) is larger than the calibration error. It might be due to the high fluctuation of silver (Ag) signals in sample unknown 1 resulting in lower prediction ability of the neural network.

When three spectra far from average values were omitted, precision and accuracy were clearly improved (see Table 6.6).

Table 6.6

Comparison of neural predictions for raw and optimized inputs

7 spectra Input 10 spectra (optimized input)

RSD for LIBS Signal 20.40 11.53 (Ag 328.068 nm)

Neural Network 0.0105 0.0081 Predicted Value (wt %)

Relative Error (%) 31.25 0.8055

The predicted Pb concentration of unknown 1 was lower than that of unknown 2 despite the higher concentration of unknown 1. Due to a small Pb concentration, background noise interfered with the weak signal of Pb at 368.347 nm, resulting in an inversion of LIBS signal strength (intensity) between unknown 1 and unknown 2.

106

6.5 Conclusion

LIBS is an almost non-destructive analytical technique allowing in suit and remote analysis. This technique can be widely applied in industrial areas if the reliability of quantitative analysis significantly improves. An artificial neural network can be an alternative way to improve the precision of LIBS analysis. In this study, we evaluated several elemental concentrations of tin (Sn) alloys without having information using the calibration method and an artificial neural network. In particular, LIBS analysis combined with an echelle spectrometer detection is highly efficient for simultaneously and quantitatively determining all elements of interest.

In the application of the echelle spectrometer, experimental parameters (e.g., gate delay, gate width, and laser power) should be adjusted to avoid signal saturation caused by strong emission lines of individual elements. Intensity ratio normalized by the Sn I

333.062 nm emission line intensity provides better reproducibility of LIBS signals.

Normalization by using tin (Sn) line as an internal standard can compensate for poor reproducibility attributed to shot-to-shot plasma fluctuations.

107

CHAPTER VII

CONCLUSION

Vitrification of liquid radioactive wastes is essential task in nuclear industrial areas. The ability of in-suit and remote analysis in LIBS technique promote good candidate for analytic tool of liquid radioactive wastes in vitrification process. However,

LIBS measurement in liquid sample includes various intrinsic challenges such as sedimentation, turbulence, and a short lifetime of the LIBS plasma. In particular, homogenous distribution of all contents in liquid sample is essential to getting accurate

LIBS signal. In the case of slurry sample with higher water concentration, data acquisition was collected form flowing slurry by using circulation apparatus. In the case of slurry sample with relatively lower water concentration, direct analysis was performed in shielded cell of slurry sample.

Spectroscopic analyses obtained by Czerny-Turner and echelle spectrometers were compared. Broadband (200~780nm) echelle spectrometer is more efficient for simultaneous multi-element detection. The ESAWIN program of the echelle detection system is efficient for reducing unnecessary operation time during data acquisition.

Moreover, the program is well organized to calculate required physical quantities, such as electron density and plasma temperature. We also applied the echelle detection system to

108 quantitatively determining elemental impurity concentrations in tin (Sn) alloy. Estimated values which were obtained from calibration method, artificial neural network (ANN), and chemical analysis were compared.

We observed that normalization by an internal standard provides better reproducibility of LIBS signals. However, it should be noted that strong emission lines might cause signal saturation and results non-linear calibration curve. Our results demonstrate that the ANN has a potential to improve the reliability of quantitative analysis. We think neural networks can be useful methodology in application of LIBS signal processing.

109

APPENDIX A

A. CALCULATION OF THE NEEDED SLIT WIDTH TO ACHIEVE CONSTANT

BANDWIDTH

110

/*An algorithm for determining the needed slit width to achieve constant bandwidth suggested by Rosfiord et al. Parameters for this calculation are chosen as follows: Groove density: 1200/mm Focal length of focusing mirror: 500mm Ebert angle of the instrument (φ): 5° Diffraction order (m): 1 Bandwidth (Δλ): 1nm outfile: d_w_1nm.txt d_x_w_init: slit width that corresponds to zero wavelength d_x_w_init1: setting new slit width without demical number that corresponds to zero wavelength. 0.292: demical number of slit width d_s: step size to update the new slit width. d_x_w[n]: the array of slit width with step size -0.5*d_s; w[n]: the array of the wavelength corresponding to the slit width a[n], a1, a2[n]: the array of multiplication with focal length, bandwidth, groove density, and inverse of slit width w1: the value of the next transition wavelength corresponding to the slit width angle: Ebert angle of the instrument f: focal length of focusing mirror d_w: bandwidth nm groove: groove density 1/mm */

#include #include #include #define d_s 1. #define angle 5.*4.*atan(1.)/180. #define pi 4.*atan(1.) #define weight 1000000. #define d_w 1. #define f 0.5 #define groove 1200. #define n 3000 #define m 10 #define f1(x,y) 2.*sin(x - y)*cos(x)*weight/groove #define f2(x,y) 2.*sin(x + y)*cos(x)*weight/groove #define g_wmax(x,y) 2.*sin(x + y)*cos(y)*weight/groove; void funct(double* n_d_x, double* n_d_x_w, double* a2, double* w, double w_max, int k,ofstream off){ for(int j=1;jw_max){ j = n+1;} n_d_x[j] = n_d_x_w[j] + 0.5*d_s; off< 1.){ n_d_x[0] = d_x_w[i] + 0.5*d_s; n_d_x_w[0] = n_d_x[0] + 0.5*d_s; a1 = f*d_w*groove/(n_d_x_w[0]); w1 = f2(angle, acos(a1)); w_max = g_wmax(pi/2., angle); off<

112

Table A.1

Look- up table to achieve the constant bandwidth (Δλ = 1nm) of Czerny- Turner spectrometer 1/d: 1200 mm-1, ƒ: 500mm, φ: 5°, m: 1

j λ(nm) ∆xj (µm) ∆x(λj) (µm) (df∆λ)/∆x(λj) 0 0 602 602.29 1 27.6126 601 601.5 0.997506 2 77.1106 600 600.5 0.999168 3 212.062 601 600.5 0.999168 4 261.079 602 601.5 0.997506 5 294.623 603 602.5 0.995851 6 321.741 604 603.5 0.994201 7 345.07 605 604.5 0.992556 8 365.818 606 605.5 0.990917 9 384.66 607 606.5 0.989283 10 402.017 608 607.5 0.987654 11 418.176 609 608.5 0.986031 12 433.34 610 609.5 0.984414 13 447.659 611 610.5 0.982801 14 461.249 612 611.5 0.981194 15 474.203 613 612.5 0.979592 16 486.593 614 613.5 0.977995 17 498.48 615 614.5 0.976404 18 509.914 616 615.5 0.974817 19 520.937 617 616.5 0.973236 20 531.584 618 617.5 0.97166 21 541.887 619 618.5 0.970089 22 551.873 620 619.5 0.968523 23 561.565 621 620.5 0.966962 24 570.983 622 621.5 0.965406 25 580.146 623 622.5 0.963856

113

APPENDIX B

VOIGT PROFILE

114

/* Voigt Profile calculation using the polynomial approximations proposed by Kuntz. The a-parameter value was chosen as 1.0 in this example. outfile: output.txt x[i]: distance from line center in units of Doppler halfwidths times sqrt(ln 2) y: a-parameter value k[i]: computed output of voigt profile at x[i] pi: π m: array number */

#include #include #include #define pi 4.*atan(1.) #define m 200 int i; void f_1(long double* x, long double* k, long double y){ long double a1, b1, a2, b2; long double a3,b3,c3,d3,a4,b4,c4,d4; long double a5,b5,c5,d5,e5,a6,b6,c6,d6,e6; long double a7,b7,c7,d7,e7,f7,g7,h7,o7,p7,q7,r7,s7,t7, a8,b8,c8,d8,e8,f8,g8,h8,o8,p8,q8,r8,s8,t8; long double sum1=0., sum2=0.; for(i=0;i 15){ a1 = 0.2820948 * y + 0.5641895 * pow (y, 3.); b1 = 0.5641895 * y; a2 = 0.25 + pow (y, 2.) + pow (y, 4.); b2 = -1. + 2 * pow (y, 2.); k[i] = (a1 + b1 * pow (x[i], 2.))/(a2 + b2 * pow (x[i], 2.) + pow (x[i], 4.));

} if( fabs (x[i]) + y < 15. && fabs (x[i]) + y >= 5.5){ a3 = 1.05786 * y + 4.65456 * pow (y, 3.) + 3.10304 * pow (y, 5.) + 0.56419 * pow (y, 7.); b3 = 2.962 * y + 0.56419 * pow (y, 3.) + 1.69257 * pow (y, 5.); c3 = 1.69257 * pow (y, 3.) - 2.53885 * y; d3 = 0.56419 * y; a4 = 0.5625+4.5 * pow (y, 2.) + 10.5 * pow (y, 4.) + 6. * pow (y, 6.) + pow (y, 8.);

115 b4 = -4.5+9. * pow (y, 2.) + 6. * pow (y, 4.) + 4. * pow (y, 6.); c4 = 10.5-6. * pow (y, 2.) + 6. * pow (y, 4.); d4 = -6.+ 4. * pow (y, 2.); k[i] = (a3 + b3 * pow (x[i], 2.) + c3 * pow (x[i], 4.) + d3 * pow (x[i], 6.))/(a4 + b4 * pow (x[i], 2.) + c4 * pow (x[i], 4.) + d4 * pow (x[i], 6) + pow (x[i], 8.));

} if( fabs (x[i]) + y < 5.5 && y - 0.195 * fabs (x[i]) > -0.176){ a5 = 272.102 + 973.778 * y + 1629.76 * pow (y, 2.) + 1678.33 * pow (y, 3.) + 1174.8 * pow (y, 4.) + 581.746 * pow (y, 5.) + 204.510 * pow (y, 6.) + 49.5213 * pow (y, 7.) + 7.55895 * pow (y, 8.) + 0.564224 * pow (y, 9.0); b5 = -60.5644-2.34403 * y + 220.843 * pow (y, 2.) + 336.364 * pow (y, 3.) + 247.198 * pow (y, 4.) + 100.705 * pow (y, 5.) + 26.6778 * pow (y, 6.) + 2.25689 * pow (y, 7.); c5 = 4.58029 + 18.546 * y + 42.5683 * pow (y, 2.) + 52.8454 * pow (y,3.) + 22.6798 * pow (y,4.) + 3.38534 * pow (y, 5.); d5 = -0.128922 + 1.66203 * y + 7.56186 * pow (y, 2.) + 2.25689 * pow (y, 3.); e5 = 0.000971457 + 0.564224 * y; a6 = 272.102 + 1280.83 * y + 2802.87 * pow (y, 2.) + 3764.97 * pow (y, 3.) + 3447.63 * pow (y, 4.) + 2256.98 * pow (y, 5.) + 1074.41 * pow (y, 6.) + 369.199 * pow (y, 7.) + 88.2674 * pow (y, 8.) + 13.3988 * pow (y, 9.) + pow (y, 10.); b6 = 211.102 + 902.306 * y + 1758.34 * pow (y, 2.) + 2037.31 * pow (y, 3.) + 1549.68 * pow (y, 4.) + 793.427 * pow (y, 5.) + 266.299 * pow (y, 6.) + 53.5932 * pow (y, 7.) + 5. * pow (y, 8.); c6 = 78.866 + 308.186 * y + 497.302 * pow (y, 2.) + 479.258 * pow (y, 3.) + 269.292 * pow (y, 4.) + 80.3928 * pow (y, 5.) + 10. * pow (y, 6.); d6 = 22.0353 + 55.0293 * y + 92.7586 * pow (y, 2.) + 55.5952 * pow (y, 3.) + 10. * pow (y, 4.); e6 = 1.49645 + 13.3988 * y + 5. * pow (y, 2.); k[i] = (a5 + b5 * pow (x[i], 2.) + c5 * pow (x[i], 4.) + d5 * pow (x[i], 6.) + e5 * pow (x[i], 8.))/(a6 + b6 * pow (x[i], 2.) + c6 * pow (x[i], 4.) + d6 * pow (x[i], 6.) + e6 * pow (x[i], 8.) + pow (x[i], 10.));

} if( fabs (x[i]) + y < 5.5 && y - (0.195 * fabs (x[i])) < -0.176){ a7 = -1.53575 * pow (10., 8.) + pow (y, 2.) * (4.08168 * pow (10., 7.) + pow (y, 2.) * (-9.69463 * pow (10., 6.) + pow (y, 2.) * (1.6841 * pow (10., 6.) + pow (y, 2.) * (-320772.0 + pow (y, 2.) * (40649.2 + pow (y, 2.) * (-5860.68 + pow (y, 2.) * (571.687 + pow (y, 2.) * (-72.9359 + pow (y, 2.) * (2.35944 - pow (y, 2.) * 0.56419))))))))); a7 = y * (1.16028e9 + pow (y, 2.) * (-9.86604 * pow (10., 8.) + pow (y, 2.) * (4.56662 * pow (10., 8.) + pow (y, 2.) * a7))); b7 = -2.91876 * pow (10., 8.) + pow (y, 2.) * (8.64829 * pow (10., 7.) + pow(y, 2.) * (-7.72359 * pow(10., 6.) + pow (y, 2.) * (3.59915*pow (10., 6.) + pow (y, 2.) * (-234417.0 + pow (y, 2.) * (45251.3 + pow (y, 2.) * (-2269.19 + pow (y, 2.) * (-234.143 + pow (y, 2.) * (23.0312 - pow (y, 2.)*7.33447)))))))); b7 = y * (-5.60505 * pow (10., 8.) + pow (y, 2.) * (-9.85386 * pow (10., 8.) + pow (y, 2.) * (8.06985 * pow (10., 8.) + pow (y, 2.) * b7))); c7 = -2.04467 * pow (10., 8.) + pow (y, 2.) * (2.29302 * pow (10., 7.) + pow (y, 2.) * (-2.3818 * pow (10., 7.) + pow (y, 2.) * (576054.0 + pow (y, 2.) * (98079.1 + pow (y, 2.) * (-25338.3 + pow (y, 2.) * (1097.77 + pow (y, 2.) * (97.6203 - pow (y, 2.) * 44.0068))))))); c7 = y * (-6.51523 * pow (10., 8.) + pow (y, 2.) * (2.47157 * pow (10., 8.) + pow (y, 2.) * (2.94262 * pow (10., 8.) + pow (y, 2.) * c7))); 116 d7 = -4.15013 * pow (10., 7.) + pow (y, 2.) * (3.83112 * pow (10., 7.) + pow (y, 2.) * (2.2404 * pow (10., 6.) + pow (y, 2.) * (-303569.0 + pow (y, 2.) * (-66431.2 + pow (y, 2.) * (8381.97 + pow (y, 2.) * (228.563 - pow (y, 2.) * 161.358)))))); d7 = y * (-2.63894 * pow (10., 8.) + pow (y, 2.) * (2.70167 * pow (10., 8.) + pow (y, 2.) * (-9.96224 * pow (10., 7.) + pow (y, 2.) * d7))); e7 = y * (-6.31771 * pow (10., 7.) + pow (y, 2.) * (1.40677 * pow (10., 8.) + pow (y, 2.) * (5.56965 * pow (10., 6.) + pow (y, 2.) * (2.46201 * pow (10., 7.) + pow (y, 2.) * (468142.0 + pow (y, 2.) * (-1.003 * pow (10., 6.) + pow (y, 2.) * (-66212.1 + pow (y, 2.) * (23507.6 + pow (y, 2.) * (296.38 - pow (y, 2.) * 403.396))))))))); f7 = y * (-1.69846 * pow (10., 7.) + pow (y, 2.) * (4.07382 * pow (10., 6.) + pow (y, 2.) * (-3.32896 * pow (10., 7.) + pow (y, 2.) * (-1.93114 * pow (10., 6.) + pow (y, 2.) * (-934717.0 + pow (y, 2.) * (8820.94 + pow (y, 2.) * (37544.8 + pow (y, 2.) * (125.591 - pow (y, 2.) * 726.113)))))))); g7 = y * (-1.23165 * pow (10., 6.) + pow (y, 2.) * (7.52883 * pow (10., 6.) + pow (y, 2.) * (-900010.0 + pow (y, 2.) * (-186682.0 + pow (y, 2.) * (79902.5 + pow (y, 2.) * (37371.9 + pow (y, 2.) * (-260.198-pow (y, 2.) * 968.15))))))); h7 = y * (-610622.0 + pow (y, 2.) * (86407.6 + pow (y, 2.) * (153468.0 + pow (y, 2.) * (72520.9 + pow (y, 2.) * (23137.1 + pow (y, 2.) * (-571.645 - pow (y, 2.) * 968.15)))))); o7 = y * (-23586.5 + pow (y, 2.) * (49883.8 + pow (y, 2.) * (26538.5 + pow (y, 2.) * (8073.15 + pow (y, 2.) * (-575.164 - pow (y, 2.) * 726.113))))); p7 = y * (-8009.1 + pow (y, 2.) * (2198.86 + pow (y, 2.) * (953.655 + pow (y, 2.) * (-352.467 - pow (y, 2.) * 403.396)))); q7 = y * (-622.056 + pow (y, 2.) * (-271.202 + pow (y, 2.) * (-134.792 - pow (y, 2.) * 161.358))); r7 = y * (-77.0535 + pow (y, 2.) * (-29.7896 - pow (y, 2.) * 44.0068)); s7 = y * (-2.92264-pow (y, 2.) * 7.33447); t7 = y * (-0.56419); a8 = 2.11107 * pow (10., 8.) + pow (y, 2.) * (-6.11148 * pow (10., 7.) + pow (y, 2.) * (1.44647 * pow (10.,7.) + pow (y, 2.) * (-2.85721 * pow (10.,6.) + pow (y,2.) * (483737.0 + pow (y, 2.) * (-70946.1 + pow (y,2.) * (9504.65 + pow (y, 2.) * (-955.194 + pow (y, 2.) * (126.532 + pow (y, 2.) * (-3.68288 + pow (y,2.)))))))))); a8 = 1.02827e9 + pow (y, 2.) * (-1.5599e9 + pow (y, 2.) * (1.17022e9 + pow (y, 2.) * (-5.79099 * pow (10., 8.) + pow (y, 2.) * a8))); b8 = 2.89676 * pow (10., 8.) + pow (y, 2.) * (-7.01358 * pow (10., 7.) + pow (y, 2.) * (1.39465 * pow (10., 7.) + pow (y, 2.) * (-2.84954 * pow (10., 6.) + pow (y, 2.) * (498334.0 + pow (y, 2.) * (-55600.0 + pow (y, 2.) * (3058.26 + pow (y, 2.) * (533.254 + pow (y, 2.) * (-40.5117 + pow (y, 2.) * 14.0)))))))); b8 = 1.5599e9 + pow (y, 2.) * (-2.28855e9 + pow (y, 2.) * (1.66421e9 + pow (y, 2.) * (-7.53828 * pow (10., 8.) + pow (y, 2.) * b8))); c8 = 6.33496 * pow (10., 7.) + pow (y, 2.) * (-4.60396 * pow (10., 7.) + pow (y, 2.) * (1.4841 * pow (10., 7.) + pow (y, 2.) * (-1.06352 * pow (10., 6.) + pow (y, 2.) * (-217801.0 + pow (y, 2.) * (48153.3 + pow (y, 2.) * (- 1500.17 + pow (y, 2.) * (-198.876 + pow (y, 2.) * 91.0))))))); c8 = 1.17022e9 + pow (y, 2.) * (-1.66421e9 + pow (y, 2.) * (1.06002e9 + pow (y, 2.) * (-6.60078 * pow (10., 8.) + pow (y, 2.) * c8))); d8 = 1.99846 * pow (10., 8.) + pow (y, 2.) * (-6.87656 * pow (10., 6.) + pow (y, 2.) * (-6.89002 * pow (10., 6.) + pow (y, 2.) * (280428.0 + pow (y, 2.) * (161461.0 + pow (y, 2.) * (-16493.7 + pow (y, 2.) * (-567.164 + pow (y, 2.) * 364.0)))))); d8 = 5.79099 * pow (10., 8.) + pow (y, 2.) * (-7.53828 * pow (10., 8.) + pow (y, 2.) * (6.60078 * pow (10., 8.) + pow (y, 2.) * (5.40367 * pow (10., 7.) + pow (y, 2.) * d8))); e8 = 2.11107 * pow (10., 8.) + pow (y, 2.) * (-2.89676 * pow (10., 8.) + pow (y, 2.) * (6.33496 * pow (10., 7.) + pow (y, 2.) * (-1.99846 * pow (10., 8.) + pow (y, 2.) * (-5.01017 * pow (10., 7.) + pow (y, 2.) * (- 5.25722 * pow (10., 6.) + pow (y, 2.) * (1.9547 * pow (10., 6.) + pow (y, 2.) * (240373.0 + pow (y, 2.) * (-55582.0 + pow (y, 2.) * (-1012.79 + pow (y, 2.) * 1001.0))))))))); f8 = 6.11148 * pow (10., 7.) + pow (y, 2.) * (-7.01358 * pow (10., 7.) + pow (y, 2.) * (4.60396 * pow (10., 7.) + pow (y, 2.) * (-6.87656 * pow (10., 6.) + pow (y, 2.) * (5.25722 * pow (10., 6.) + pow (y, 2.) *

117

(3.04316 * pow (10., 6.) + pow (y, 2.) * (123052.0 + pow (y, 2.) * (-106663.0 + pow (y, 2.) * (-1093.82 + pow (y, 2.) * 2002.0)))))))); g8 = 1.44647 * pow (10., 7.) + pow (y, 2.) * (-1.39465 * pow (10., 7.) + pow (y, 2.) * (1.4841 * pow (10., 7.) + pow (y, 2.) * (6.89002 * pow (10., 6.) + pow (y, 2.) * (1.9547 * pow (10., 6.) + pow (y, 2.) * (- 123052.0 +pow (y, 2.) * (-131337.0 + pow (y, 2.) * (-486.14 + pow (y, 2.) * 3003.0))))))); h8 = 2.85721 * pow (10., 6.) + pow (y, 2.) * (-2.84954 * pow (10., 6.) + pow (y, 2.) * (1.06352 * pow (10., 6.) + pow (y, 2.) * (280428.0 + pow (y, 2.) * (-240373.0 + pow (y, 2.) * (-106663.0 + pow (y, 2.) * (486.14 + pow (y, 2.) * 3432.0)))))); o8 = 483737.0 + pow (y, 2.) * (-498334.0 + pow (y, 2.) * (-217801.0 + pow (y, 2.) * (-161461.0 + pow (y, 2.) * (-55582.0 + pow (y, 2.) * (1093.82 + pow (y, 2.) * 3003.0))))); p8 = 70946.1 + pow (y, 2.) * (-55600.0 + pow (y, 2.) * (-48153.3 + pow (y, 2.) * (-16493.7 + pow (y, 2.) * (1012.79 + pow (y, 2.) * 2002.0)))); q8 = 9504.65 + pow (y, 2.) * ( - 3058.26 + pow (y, 2.) * (-1500.17 + pow (y, 2.) * (567.164 + pow (y, 2.) * 1001.0))); r8 = 955.194 + pow (y, 2.) * (533.254 + pow (y, 2.) * (198.876 + pow (y, 2.) * 364.0)); s8 = 126.532 + pow (y, 2.) * (40.5117 + pow (y, 2.) * 91.0); t8 = 3.68288 + pow (y, 2.) * 14.0; sum1 = a7 + b7 * pow (x[i], 2.) + c7 * pow (x[i], 4.) + d7 * pow (x[i], 6.) + e7 * pow (x[i], 8.) + f7 * pow (x[i], 10.) + g7 * pow (x[i], 12.) + h7 * pow (x[i], 14.) + o7 * pow (x[i], 16.) + p7 * pow (x[i], 18.) + q7 * pow (x[i], 20.) + r7 * pow (x[i], 22.) + s7 * pow (x[i], 24.) + t7 * pow (x[i], 26.); sum2 = a8 + b8 * pow (x[i], 2.) + c8 * pow (x[i], 4.) + d8 * pow (x[i], 6.) + e8 * pow (x[i], 8.) + f8 * pow (x[i], 10.) + g8 * pow (x[i], 12.) + h8 * pow (x[i], 14.) + o8 * pow (x[i], 16.) + p8 * pow (x[i], 18.) + q8 * pow (x[i], 20.) + r8 * pow (x[i], 22.) + s8 * pow (x[i], 24.) + t8 * pow (x[i], 26.) + pow (x[i], 28.); k[i] = exp(y * y - x[i] * x[i]) * cos(2 * x[i] * y * pi/180.) - (sum1/sum2);}}}

void main(){ long double y = 1.; long double x[m]; long double k[m]; ofstream off; off.open("output.txt"); for(i=0;i

118

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