ABSTRACT: The area of particle characterization is expansive; it contains many technologies and methods of analysis. Light spectroscopy techniques yield information on the joint property distribution of particles, comprising the chemical composition, size, shape, and orientation of the particles. The objective of this dissertation is to develop a hybrid scattering-absorption model incorporating Mie and Rayleigh-Debye-Gans theory to characterize submicron particles in suspension with multiwavelength spectroscopy.Rayleigh-Debye-Gans theory (RDG) was chosen as a model to relate the particles joint property distribution to the light scattering and absorption phenomena for submicron particles. A correction model to instrument parameters of relevance was implemented to Rayleigh-Debye-Gans theory for spheres. Behavior of nonspherical particles using RDG theory was compared with Mie theory (as a reference).A multiwavelength assessment of Rayleigh-Debye-Gans theory for spheres was conducted where strict adherence to the limits could not be followed. Reported corrections to the refractive indices were implemented to RDG to try and achieve Mies spectral prediction for spheres.The results of studies conducted for RDG concluded the following. The angle of acceptance plays an important role in being able to assess and interpret spectral differences. Multiwavelength transmission spectra contains qualitative information on shape and orientation of non-spherical particles, and it should be possible to extract this information from carefully measured spectra. There is disagreement between Rayleigh-Debye-Gans and Mie theory for transmission simulations with spherical scatterers of different sizes and refractive indices.

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Thesis (Ph.D.)--University of South Florida, 2005.

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ABSTRACT: The area of particle characterization is expansive; it contains many technologies and methods of analysis. Light spectroscopy techniques yield information on the joint property distribution of particles, comprising the chemical composition, size, shape, and orientation of the particles. The objective of this dissertation is to develop a hybrid scattering-absorption model incorporating Mie and Rayleigh-Debye-Gans theory to characterize submicron particles in suspension with multiwavelength spectroscopy.Rayleigh-Debye-Gans theory (RDG) was chosen as a model to relate the particles joint property distribution to the light scattering and absorption phenomena for submicron particles. A correction model to instrument parameters of relevance was implemented to Rayleigh-Debye-Gans theory for spheres. Behavior of nonspherical particles using RDG theory was compared with Mie theory (as a reference).A multiwavelength assessment of Rayleigh-Debye-Gans theory for spheres was conducted where strict adherence to the limits could not be followed. Reported corrections to the refractive indices were implemented to RDG to try and achieve Mies spectral prediction for spheres.The results of studies conducted for RDG concluded the following. The angle of acceptance plays an important role in being able to assess and interpret spectral differences. Multiwavelength transmission spectra contains qualitative information on shape and orientation of non-spherical particles, and it should be possible to extract this information from carefully measured spectra. There is disagreement between Rayleigh-Debye-Gans and Mie theory for transmission simulations with spherical scatterers of different sizes and refractive indices.
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Hybrid Model for Characterization of Submicron Particles Using Multiwavelength Spectroscopy by Alicia C. Garcia-Lopez A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Electrical Engineering College of Engineering University of South Florida Major Professor: Arthur David Snider, Ph.D. Wilfrido Moreno, Ph.D. Kenneth Buckle, Ph.D. Stanley Deans, Ph.D. Oscar D. Crisalle, Ph.D. Date of Approval: March 29, 2005 Keywords: Particle Analysis, Spherical Partic les, Non-spherical Par ticles, In suspension Particles, Rayleigh-Debye-Gans Copyright 2005, Alicia C. Garcia-Lopez

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Dedication I would like to dedicate this dissertation to my younger br other Rodrigo and my family.

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Acknowledgments I would like to acknowledge Prof. L. H. Garcia-Rubio for the permission to use his ideas for this research project, the vast amount of consultation time, and his expertise in the area of particle characterization. I would also lik e to acknowledge Prof. A. D. Snider for giving me a wonderful graduate experience in pursuing my Ph.D. I have greatly enjoyed working together and your guidance and mathematics expertise have greatly enhanced my life. Finally, I would like to acknowle dge the PERC located at the University of Florida for their financial support.

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Table of Contents List of Tables iii List of Figures iv List of Symbols ix Abstract xii Chapter One: Introduction and Methods 1 1.1 Introduction 1 1.2 Materials and Methods 2 1.3 Overview of Chapters 3 Chapter Two: Scattering Theories and Models 4 2.1 Background 6 2.2 General Concepts and Equations 7 2.3 Mie Theory and Model 12 2.4 Rayleigh-Debye-Gans Theory and Model 13 2.5 Hybrid Theory and Model 15 2.6 Methods Review 16 Chapter Three: Instrumentation Correction Model for Transmission 20 3.1 Instrumentation Correction Model 20 3.2 Implementation of Instrument Corrections 23 Chapter Four: Nonspherical Particles 26 4.1 Geometry and Notation for Ellipsoids 26 4.2 Ellipsoid Simulations 30 Chapter Five: Validation Study of Rayleigh-Debye-Gans Theory 34 5.1 Exploration of Theoretical Limits 34 5.2 Particle Diameter << Wavelength 35 5.3 Particle Diameter ~ Wavelength, No Absorption 38 5.4 Particle Diameter ~ Wavelength, Absorption > 0 42 5.5 Conclusion 45 Chapter Six: Corrections to the Refractive Index 46 6.1 Refractive Index 46 i

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6.2 Hypochromic Effect 47 6.3 Implementation of Optical Correction for Absorption 48 6.4 Effective Refractive Index Estimation 53 6.5 Conclusion 54 Chapter Seven: New Hybrid Theory 55 7.1 Geometry and Notation 55 7.2 Internal Field 58 7.3 Dipole Scattering Approach 60 7.4 Hybrid Theory 62 7.5 Determining the Transmission 65 7.6 Scattering Intensity Ratio and Turbidity 66 Chapter Eight: Validation and Sensitivity of Hybrid Theory 67 8.1 Validation of Hybrid Theory Implementation 67 8.2 Case 1: Relative Refractive index 0nn ~1 and Absorption = 0 73 8.3 Case 2: Relative Refractive index 0nn 1 and Absorption > 0 76 8.4 Case 3: Relative Refractive index 0nn ~1 and Absorption > 0 79 8.5 Conclusions 82 Chapter Nine: Contributions and Future Work 83 9.1 Contributions 83 9.2 Recommendations and Future Work 84 References 85 Appendices 87 Appendix A: Intensity Ratio and Turbidity Model 88 Appendix B: Optical Properties 92 Appendix C: Validation for Rayleigh-Debye-Gans Theory 98 C.1 Validation of Rayleigh-Debye-Gans Theory 98 Appendix D: Estimation of Absorption Coefficient and Hypochromism Model 102 D.1 Hypochromism Model 104 About the Author End Page ii

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List of Tables Table 1.1 Simulation Parameters 3 Table 4.1 Table of Form Factors 29 iii

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List of Figures Figure 2.1 Diagram for Complete Particle Characterization Model 7 Figure 2.2 Diagram of Coordinate System used in the Mie and RDG Models 12 Figure 3.1 Transmission System (a) Open Detector (b)Pinhole Detector 21 Figure 3.2 Calculated Transmission of Rayleigh-Debye-Gans for 1 m Hemoglobin Spheres with Pinhole Detector Setup 25 Figure 3.3 Calculated Transmission of Rayleigh-Debye-Gans for 1 m Hemoglobin Spheres with Open Detector Setup 25 Figure 4.1 Light Scattering in Laboratory Frame 27 Figure 4.2 Light Scattering in Particle Frame 27 Figure 4.3 Fixed Orientations A, B, and C for an Ellipsoid 31 Figure 4.4 Calculated Transmission of Soft Body Prolate Ellipsoid =0.3 with a Volume Equivalent to a 1 m Sphere 33 Figure 4.5 Calculated Transmission of Soft Body Prolate Ellipsoid =0.8 with a Volume Equivalent to a 1 m Sphere 33 Figure 5.1 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 25 nm AgBr Spheres 36 Figure 5.2 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 25 nm AgCl Spheres 36 Figure 5.3 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 50 nm AgBr Spheres 37 iv

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Figure 5.4 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 50 nm AgCl Spheres 37 Figure 5.5 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 500 nm Soft Body Spheres 38 Figure 5.6 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 1 m Soft Body Spheres 39 Figure 5.7 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 5.5 m Soft Body Spheres 39 Figure 5.8 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 500 nm Hemoglobin Spheres with =0 41 Figure 5.9 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 1 m Hemoglobin Spheres with =0 41 Figure 5.10 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 5.5 m Hemoglobin Spheres with =0 42 Figure 5.11 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 100 nm Hemoglobin Spheres 43 Figure 5.12 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 500 nm Hemoglobin Spheres 44 Figure 5.13 Calculated Transmission of Mie and Rayleigh-Debye-Gans for a Suspension of 1 m Hemoglobin Spheres 44 Figure 6.1 Calculated Transmission of Mie and Rayleigh-Debye-Gans for 1 m Hemoglobin Sphere with Veshkin Correction 50 Figure 6.2 Calculated Transmission of Mie and Rayleigh-Debye-Gans for 1 m Hemoglobin Sphere with 0% and 100% Hypochromocity 50 Figure 6.3 Calculated Transmission of Mie and Rayleigh-Debye-Gans for 1 m Hemoglobin Sphere with Veshkin Correction to k c and an Effective n eff Calculated through Kramer-Kronig Transform 51 v

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Figure 6.4 Zoom in of Figure 6.4 of Calculated Transmission of Rayleigh-Debye-Gans for 1 m Hemoglobin Sphere with Veshkin Correction to k c and an Effective n Calculated through Kramer-Kronig Transform 52 Figure 7.1 Diagram of Scatterer Point and Detector Location 55 Figure 7.2 Local Unit Vectors with Respect to the Detector 56 Figure 7.3 Local Unit Vectors with Respect to the Scatterer 57 Figure 7.4 Diagram of the Volume at Height z for a Sphere 64 Figure 8.1 Coefficients c 1 ,d 1 and d 2 Versus at the Limit when k 1 =k 68 Figure 8.2 Form Factor f 1 for Hybrid Theory at k 1 =k 69 Figure 8.3 Form Factor f for Rayleigh-Debye-Gans Theory 70 Figure 8.4 Real Part of Form Factor f 1 for Hybrid Theory using Polystyrene 70 Figure 8.5 Imaginary Part of Form Factor f 1 for Hybrid Theory using Polystyrene 71 Figure 8.6 Real Part of Form Factor f 2 for Hybrid Theory using Polystyrene 71 Figure 8.7 Imaginary Part of Form Factor f 2 for Hybrid Theory using Polystyrene 72 Figure 8.8 Form Factor f for Rayleigh-Debye-Gans Theory using Polystyrene 72 Figure 8.9 Comparision of Calculated Transmission for 50 nm Soft Body Spheres using RDG, Mie, and Hybrid Theories 74 Figure 8.10 Comparision of Calculated Transmission for 100 nm Soft Body Spheres using RDG, Mie, and Hybrid Theories 74 Figure 8.11 Comparision of Calculated Transmission for 250 nm Soft Body Spheres using RDG, Mie, and Hybrid Theories 75 vi

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Figure 8.12 Comparision of Calculated Transmission for 500 nm Soft Body Spheres using RDG, Mie, and Hybrid Theories 75 Figure 8.13 Comparision of Calculated Transmission for 50 nm Polystyrene Spheres using RDG, Mie, and Hybrid Theories 77 Figure 8.14 Comparision of Calculated Transmission for 100 nm Polystyrene Spheres using RDG, Mie, and Hybrid Theories 78 Figure 8.15 Comparision of Calculated Transmission for 250 nm Polystyrene Spheres using RDG, Mie, and Hybrid Theories 78 Figure 8.16 Comparision of Calculated Transmission for 500 nm Polystyrene Spheres using RDG, Mie, and Hybrid Theories 79 Figure 8.17 Comparision of Calculated Transmission for 100 nm Hemoglobin Spheres using RDG, Mie, and Hybrid Theories 80 Figure 8.18 Comparision of Calculated Transmission for 250 nm Hemoglobin Spheres using RDG, Mie, and Hybrid Theories 80 Figure 8.19 Comparision of Calculated Transmission for 500 nm Hemoglobin Spheres using RDG, Mie, and Hybrid Theories 81 Figure 8.20 Comparison of Calculated Transmission for 500 nm Hemoglobin Spheres using RDG, Mie, and Hybrid Theories 81 Figure B.1 Optical Properties for Water 92 Figure B.2 Optical Properties for Soft Body 92 Figure B.3 Optical Properties for Hemoglobin 93 Figure B.4 Optical Properties for Polystyrene 93 Figure B.5 Optical Properties of AgCl 94 Figure B.6 Optical Properties of AgBr 94 Figure B.7 Relative Refractive Index of Soft Body in Water 95 Figure B.8 Relative Refractive Index of Hemoglobin in Water 95 Figure B.9 Relative Refractive Index of Polystyrene in Water 96 vii

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viii Figure B.10 Relative Refractive Index AgCl in Water 96 Figure B.11 Relative Refractive Index of AgBr in Water 97 Figure C.1 Calculated Transmissi on of Mie and Rayleigh-Debye-Gans for a Suspension of 25 nm Polystyrene Spheres 99 Figure C.2 Calculated Transmissi on of Mie and Rayleigh-Debye-Gans for a Suspension of 50 nm Polystyrene Spheres 100 Figure C.3 Calculated Transmissi on of Mie and Rayleigh-Debye-Gans for a Suspension of 100 nm Polystyrene Spheres 100 Figure C.4 Calculated Transmissi on of Mie and Rayleigh-Debye-Gans for a Suspension of 500 nm Polystyrene Spheres 101 Figure D.1 Stack Arrangement of Chromophores along Chain Axis 104

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List of Symbols Angle of directional cosine Angle of directional cosine Angle of directional cosine Extinction coefficient for a single chromophore Average extinction coefficient for a single chromophore m Molar extinction coefficient for a single chromophore c Correction imaginary part of the refractive index Angle between scattering and incident beam Wavelength o Wavelength in vacuo n Angle dependent function Solid angle Transmission/Turbidity n Angle dependent function Rotational Euler angle around Z axis Rotational Euler angle around Y axis Rotational Euler angle around new Z axis Size parameter ix

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a Radius a n Mie coefficient b n Mie coefficient C abs Absorption cross section C ext Extinction cross section C sca Scattering cross section D Particle diameter E Extinction coefficient Average extinction coefficient E Calculated Extinction coefficient f Form factor G Cross sectional area h Hypochromicity h n Hankel functions I o Incident intensity I s Scattering intensity k Wave number k Quantity of chromophores j n Bessel functions l Pathlength of sample M w Molecular weight m Relative refractive index x

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N Complex refractive index of particle N 2 Complex refractive index of medium N A Avogadros number N p Number of particles n Real part of the refractive index of particle P Probability of absorption by a photon P n Legendre polynomial Q abs Absorption efficiency factor Q ext Extinction efficiency factor Q sca Scattering efficiency factor Q ext Apparent extinction efficiency factor Q sca Apparent scattering efficiency factor R Correction factor R Path averaged correction factor r Distance between Middle of Sample and Detector S Scattering Amplitude Function s Effective Geometric Area of Chromophore V Volume of Particle vf Volume fraction of xi

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Hybrid Model for Characterization of Submicron Particles Using Multiwavelength Spectroscopy Alicia C. Garcia-Lopez ABSTRACT The area of particle characterization is expansive; it contains many technologies and methods of analysis. Light spectroscopy techniques yield information on the joint property distribution of particles, comprising the chemical composition, size, shape, and orientation of the particles. The objective of this dissertation is to develop a hybrid scattering-absorption model incorporating Mie and Rayleigh-Debye-Gans theory to characterize submicron particles in suspension with multiwavelength spectroscopy. Rayleigh-Debye-Gans theory (RDG) was chosen as a model to relate the particles joint property distribution to the light scattering and absorption phenomena for submicron particles. A correction model to instrument parameters of relevance was implemented to Rayleigh-Debye-Gans theory for spheres. Behavior of nonspherical particles using RDG theory was compared with Mie theory (as a reference). A multiwavelength assessment of Rayleigh-Debye-Gans theory for spheres was conducted where strict adherence to the limits could not be followed. Reported corrections to the refractive indices were implemented to RDG to try and achieve Mies spectral prediction for spheres. xii

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The results of studies conducted for RDG concluded the following. The angle of acceptance plays an important role in being able to assess and interpret spectral differences. Multiwavelength transmission spectra contains qualitative information on shape and orientation of non-spherical particles, and it should be possible to extract this information from carefully measured spectra. There is disagreement between Rayleigh-Debye-Gans and Mie theory for transmission simulations with spherical scatterers of different sizes and refractive indices. Finally, it is not possible to adequately or realistically compensate for the differences between Mie and RDG through the use of hypochromicity models and/or effective refractive indices. A hybrid model combining RDG and Mie theories was developed and tested for spheres of different sizes and refractive indices. The results of hybrid model is that it approximates Mie theory much better than Rayleigh-Debye-Gans for particle sizes smaller than the wavelength and for a broader range of optical properties in the context of multiwavelength spectroscopy. Overall, this new model is an improvement over Rayleigh-Debye-Gans theory in approximating Mie theory for submicron particles and is computationally more effective over other methods. The development of the hybrid spherical model constitutes a platform for developing nonspherical models. xiii

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Chapter One Introduction and Methods 1.1 Introduction Characterization of particles entails obtaining information about size, shape, orientation and chemical composition. Particle characterization is a broad area of undertaking which encompasses many technologies, among them light spectroscopy techniques. Light spectroscopy typically involves scattering and absorption methods. Scattering measurements are performed at a single wavelength but measured as a function of the direction of observation. For absorption, the light is measured in the forward direction as a function of wavelength. Light scattering techniques typically use highly collimated sources (lasers), whereas absorption spectrophotometric techniques use broadband sources to produce multiwavelength spectra. In either case the resulting spectra can be interpreted with the theory of electromagnetic radiation, which describes interaction of light with matter (Maxwells Equations). Mie and Rayleigh-Debye-Gans theories are solutions to Maxwells Equations that relate the particles joint property distribution to the light scattering and absorption phenomena. This connection is made through the optical properties that are characteristic of the materials contained in the particle. The objective of this study is to develop a hybrid scattering-absorption model incorporating both theories to characterize submicron particles with multiwavelength spectroscopy. To accomplish this objective Mie theory and Rayleigh-Debye-Gans theory 1

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are revisited and extended to account for the field alteration predicted by Mie, and for the dipole radiation mechanism employed by Rayleigh-Debye-Gans. Throughout this dissertation Mie theory and Rayleigh-Debye-Gans theories are emphasized because at this point they enable real time particle characterization for industrial and biomedical applications. The largest area of application that would profit from this study is in the biological and biomedical field in the subject of microbial and disease detection in tissue and bodily fluids. 1.2 Materials and Methods The programs for Mie theory, Rayleigh-Debye-Gans theory, instrument models and hypochromicity were developed in Matlab v6.5.1. Computations for these programs were conducted using a Dell Inspiron 4100 with 1GHz Pentium III processor and 512 MB RAM. The optical properties (refractive indices) utilized were provided by Dr. Garcia-Rubio and the SAPD laboratory through the College of Marine Science at the University of South Florida [17]. The computer codes developed for the analysis of Rayleigh-Debye-Gans and Mie particles were tested against published values of the scattering functions [1, 14]. In testing and exploring the algorithms for Rayleigh-Debye-Gans the refractive indices selected were those of soft bodies and hemoglobin, where soft bodies are defined here as particles whose relative refractive index is close to one with no absorption component. The values of the index of refraction n+i for biological particles commonly used are soft bodies (1) and hemoglobin (1 04.145.n 6.148. n 15.001.0 ) [17]. Polystyrene (1, 2. 25.n 82 .001.0 ), silver bromide ( 5 .36.2 n 2

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6.1001.0 ) and silver chloride ( 7.22 n 0 85.0001. ) are materials found in industrial applications whose properties are used as standards for optical instruments [17]. Water (1) was used as the suspending medium. The refractive indices as function of wavelength are reported in Appendix B. 4.13n The ranges of particle volumes were chosen between 12700 nm 3 and 87 m 3 The spherical diameter equivalents to the volume range are between 25 nm -5.5 m. The table below gives the simulation parameters used to define the suspensions for the analyses conducted in this dissertation. Table 1.1: Simulation Parameters Light Source Wavelength Particle Concentration Particle Density 200-900 nm 1E-4 g/cc 1 g/cc 1.3 Overview of Chapters This dissertation is divided into nine chapters. Chapter two presents a review of Mie and Rayleigh-Debye-Gans theory, citing the resulting formulas for the scattered field and their matrix formulations. For each theory the scattering intensity ratio that governs scattering measurements/simulations and the turbidity formula that governs transmission measurements/simulations are displayed. The latter formula contains a term proportional to the scattering cross section, which takes different forms for the two models, and a term proportional to the absorption cross section, which is the same for both models. Only simulations of transmission are reported, at multiwavelength. 3 Chapter three describes aperture correction models that account for the fact that actual transmission measurements are inevitably polluted by the presence of some near-forward scattered radiation. The simulations of this effect are for the RDG model only, for 1 m hemoglobin spheres.

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Chapter four reports a study of nonspherical scatterers, simulating soft body ellipsoids. Transmission curves are compared for two eccentricities, each with three different body orientations, using the RDG model. Chapter five compares RDG theory and Mie theory for transmission with simulations for spherical scatterers of different sizes and refraction indices. Disagreement between the two theories is demonstrated. The rest of the dissertation is concerned with attempts to modify RDG to bring the transmission simulations into closer agreement with Mie (the exact solution for spheres). In chapter six, two approaches are described to increase the computed RDG turbidity to that of Mie. The first approach is to use hypochromicity as a correction to RDG to account for absorption. The second approach is based on the observation that the RDG formula for the extinction cross section is (very nearly) a simple quadratic function of n and ; therefore one can invert this function and find "effective" values of or n that will result in turbidity values calculated by RDG in agreement with those computed by Mie. Reasons for rejecting these approaches are cited. Chapter seven presents the new hybrid model, based on the rigorous Mie calculation of the internal field and the Rayleigh-Debye-Gans approach for the scattering radiation. This theory is developed in full for spheres. In chapter eight the hybrid model is tested by simulated comparisons with Mie and Rayleigh-Debye-Gans theories, employing the span of optical properties of interest. Finally, conclusions, contributions, and recommendations are covered in chapter nine. 4

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Chapter Two Scattering Theories and Models This chapter describes light scattering theories and measurements of submicron particles in suspension. The first section provides information on scattering and turbidity measurements used to characterize particles. This section also provides a description of the models used to describe the observed measurement. The subsequent sections provide an outline of Mie theory and RDG theory. These theories describe the scattering phenomena observed in transmission and scattering measurements. Specifically, the scattering intensity ratio as a function of wavelength (in our case a broad wavelength range) and angle of observation, and the turbidity as a function of wavelength (again broad wavelength range) are quantified. Other more computationally intensive, techniques for solving light scattering and absorption problems are discussed. These techniques include the T-Matrix and the Purcell-Pennypacker methods. 2.1 Background There are many types of spectroscopy measurement used to characterize particles in suspension. Most interest focuses on transmission and scattering measurements. In the former the electromagnetic energy of an incident wave is measured after interaction with a particle or suspension as it leaves the system in the forward direction. In contrast, scattering measurements capture the light after interacting with a particle as it leaves the 5

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system at any angle of observation. They differ because transmission measurements capture both the (forward) scattered light and the unscattered portion of the incident beam. Information concerning the properties of the scattering and absorbing particle is contained in the measured spectra, which are plots of the power intensity versus frequency or wavelength (and, direction). Through the uses of the appropriate theories and models, it is possible to obtain estimates of the size, shape, chemical composition, internal structure, and surface charge from spectroscopic measurements [1]. A complete scheme for particle characterization must take into account various experimental conditions occurring in the lab system when spectroscopy measurements are conducted. These include the type of measurement, instrumentation setup, particle-light interaction, and other optical phenomena. Figure 2.1 illustrates how these components relate to one another. The scattering intensity ratio equation and the turbidity equation are energy balance equations that are developed from the scattering theories studied. A detailed description of the transmission measurement and analysis is provided in chapter three. Refractive indices and corrections are discussed in chapter six. The desire to characterize particulate systems for real-time continuous monitoring has led to the selection of Rayleigh-Debye-Gans (RDG) theory and Mie theory. Computation time being the restricting factor, these theories provide light scattering solutions in a suitable time. The rest of this chapter is dedicated to the description of Mie and RDG theories along with the development of the corresponding scattering intensity ratio and turbidity formulas. 6

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Figure 2.1: Diagram for Complete Particle Characterization Model 2.2 General Concepts and Equations The emphasis of this section is the utilization of the scattering matrix formalism to evaluate the extinction of light predicted by the scattering intensity and turbidity equations. Throughout the recent course of light scattering history, the terms turbidity (or optical depth) and optical density have caused much confusion [4]. Turbidity has been traditionally defined as an attenuation coefficient due to scattering (only) for the transmission of the incident beam. Herein, turbidity is described as the total attenuation observed due to scattering and absorption. The term optical density (O.D.) was originally used synonymously with absorption; the units of O.D. are absorption unit per pathlength (Au/cm). Turbidity will be described in the units of optical density. 7

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The amplitude scattering matrix is used to relate the incident and scattered fields. iizrkissEESSSSikreEE||1432)(|| 2.1 where are the asymptotic incident and scattering fields parallel and perpendicular to the scattering plane; r is the distance from the scattering center to the detector, z is distance along the axis of propagation of the incoming wave, and k is the propagation constant or wave number in the medium surrounding the particle; ssiiEEEE,,,|||| oonk 2 where o is the wavelength in vacuo and n o is the refractive index of the medium. Van de Hulst [2] describes in detail various assumptions made for simplifying the scattering functions with regard to rotation and symmetry of particles. For a spherical particle S 3 and S 4 are equal to zero. S 1 and S 2 are complex amplitude scattering elements; they depend on the indices of refraction, particle size, and the scattering direction. S 1 and S 2 are given in the Mie model by formidable series expansions involving Bessel, Neumann, and Legendre functions. The expressions for S 1 and S 2 predicted by Mie and RDG theory are provided in subsequent sections. If the detector is not situated in the forward direction, it is illuminated only by light that is scattered by the particle; its construction shields it from the incident beam. The term scattering measurement refers to this configuration. This scattered intensity I s is given by 8

PAGE 25

2,12||,222,2||,2Re121Re21Re21iisssssESESrkEEEI 2.2 where is the permittivity and is the permeability. If the incoming light is unpolarized, 22,2||,21iiEE oE so ooosI r k SSErkSSESSrkI2221222222122221222221Re21212Re121 2.3 I s /I o is known as the scattering intensity ratio. On the other hand, if the detector is aligned with the incoming beam, it measures the forward scattered wave together with the transmitted incident wave. The analysis of such a transmission measurement is most easily conducted by accounting for the energy loss suffered by the original incident beam. There are two loss mechanisms which attenuate the incident beam, scattering and absorption. The power scattered out of the beam by a particle is evaluated by the integrating the scattered intensity over an enclosing sphere at infinity (in spherical coordinates). ddrrIpowerscattereds0202sin,,, 2.4 9

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It is convenient to define a scattering cross section C sca as the area over which one would integrate the incoming intensity I o to balance the scattered power; thus powerscatteredCIscao 2.5 or ddrIrICosscasin,,,2020 2.6 The scattering efficiency Q sca is the ratio of the C sca to the actual cross section G that the particle presents to the incoming beam: GCQscasca 2.7 The power absorbed by the particle will be discussed in detail in chapter five; it too can be expressed using an equivalent area C abs : powerabsorbedCIabso 2.8 According to Van de Hulst [2], the absorption cross section for a particle of volume V and relative refractive index of refraction oninm is given (in both theories) by 21Im322mmkVCabs 2.9 The absorption efficiency is expressed analogously: GCQabsabs 2.10 Thus one can mathematically characterized the power loss in the incident light as if an area (C sca + C abs ) were blocked out of the incident beam by each particle. The extinction cross section and efficiency then, are given by 10

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absscaextCCC GCextext Q 2.11 If the particle density in the medium is N p then in an infinitesimal length dz there are N p dz particles per unit area; each effectively depletes the beam power by C ext I. Therefore the attenuation of the beam intensity I(z) as a function of pathlength z is governed by )()()(zICdzNzIdzzIextp 2.12 or )(zICNdzdIextpz 2.13 The solution to the differential equation is zCNextpeoIzI)()( 2.14 The optical theorem [1] states that the extinction cross section can be expressed in terms of the elements of the scattering amplitude matrix in the forward direction 0Re40Re42212 S k S k Cext 2.15 The terms turbidity, or optical density are used to characterize the total attenuation in a sample of length l. lGQNlCNextpextp 2.16 In practice, a detector placed in the forward direction will have a finite aperture, and thus capture some of the radiation scattered at small angles; corrections for this effect are discussed in chapter three. 11

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2.3 Mie Theory and Model The exact solution to the boundary value problem for light scattering by a sphere is generally referred to as Mie Theory [1]. Mie theory assumes that the spherical scattering object is composed of a homogeneous, possibly absorbing isotropic and optically linear material irradiated by an infinitely extending plane wave. Figure 2.2: Diagram of Coordinate System used in the Mie and RDG Models In figure 2.2 (x,y,z) refer to Cartesian coordinates and (,, r ) refer to spherical coordinates. The amplitude scattering matrix elements S 1 and S 2 are expressed explicitly as 1211coscos112coscos112nnnnnnnnnnabnnnSbannnS 2.17 12

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where n (cos) and n (cos) are defined in terms of the associated Legendre polynomial ; 1nP coscoscossin1cos11nnnnPddP 2.18 Note that there is no dependence in the Mie model. To calculate the scattering intensity ratio using Mie theory, the equations in 2.17 for the amplitude scattering matrix elements can be substituted into equation 2.2 The amplitude scattering matrix elements for Mie theory at 0 are given by: 1211221000nnnbanSSS 2.19 The turbidity formula for Mie theory is calculated by substituting 2.l9, into equations 2.15 and 2.16. Through the direct calculation of the turbidity and scattering intensity ratio equations, Mie theory has been shown to be an effective tool for determining particle size distributions of nonspherical shapes, internal structures, and optical properties [4]. 2.4 Rayleigh-Debye-Gans Theory and Model The basis of the theory for Rayleigh-Debye-Gans scattering is Rayleigh scattering. Rayleigh presented an approximate theory for particles of any shape and size having a relative refractive index near unity, Debye and Gans later added refinements. Kerker [5] states that the fundamental approximation in the Rayleigh-Debye-Gans approach is that the phase shift, the change of the phase of a light ray that passes through the sphere is 13

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negligible. A restriction is therefore put upon the particle size, the wavelength, and the refractive index. For a particle of radius a the restriction is 11mka 2. The physical assumption of Rayleigh-Debye-Gans scattering is that each infinitesimal volume element of the particle gives rise to Rayleigh scattering and does so independently of the other volume elements. The waves scattered in a given direction by these elements interfere due to the different positions of the volume elements in space. For spherical and non-spherical particles, a form factor dVeVfi1 2.20 is introduced that averages the phase difference throughout the volumes V of the (spherical and nonspherical) particle; the scattering amplitude elements S 1 and S 2 then take the form of fVmmikS21432231 2.21 cos21432232fVmmikS 2.22 were Kerker lists form factors from various shapes, of which some are quoted in this dissertation. In subsequent sections the specific derivation of the form of f is given, as well as a discussion for utilizing f to deduce orientation. A detailed derivation of the scattering intensity ratio equation and the turbidity equation for RDG is developed in Appendix A. The scattering intensity ratio is easily obtained by substituting equations 2.20 and 2.21 for the amplitude scattering elements into equation 2.2. For unpolarized light, 2.3 becomes 043SS 14

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222222224cos12132k9,fVmmrIIos 2.23 The scattering cross section C sca is expressed from 2.6 as 02222224sincos121169dfmmVkCsca 2.24 The general expression for the absorption cross section is, from equation 2.9, 21Im322mmkVCabs 2.25 The extinction cross section is the sum of these two (equation 2.24 and 2.25): 21ImV3sincos1211692202222224mmkdfmmVkCext 2.26 The expression for the turbidity for a monodispersed system can then be explicitly expressed by the following equation. 21Im3sincos1211692202222224mmkVdfmmVkNp 2.27 2.5 Hybrid Theory and Model The new scattering model proposed in this dissertation is a hybrid combination of Mie theory and Rayleigh-Debye-Gans theory. Chapter six is dedicated to the description and development of the hybrid theory and model. 15

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2.6 Methods Review In sections 2.3 and 2.4 the models presented for characterizing particles have been Mie theory for spherical particles and Rayleigh-Debye-Gans theory for arbitrary shaped particles. To summarize, Mie theory is an exact mathematical solution to Maxwells Equations for light scattered by spheres. It has been used extensively to characterize non-spherical particles approximately for a broad range of sizes and optical properties. Although Mie theory provides good estimates of the size and chemical composition it does not posses the ability to estimate the actual shape and orientation of non-spherical particles. Rayleigh-Debye-Gans theory provides information on shape and orientation for spherical and nonspherical particles; however, its applicability has limitations with regards to size and optical properties of systems. For the case of nonspherical particles, many exact and approximate methods have been developed. Singham and Bohren discuss advantages and disadvantages of several [6, 7]. Two methods considered significant to this investigation are the T-Matrix and Purcell-Pennypacker method. The T-Matrix is the linear transformation connecting the coefficients of the eigenfunctions in the scattered field and those for the incident field. It is through the linearity of Maxwells equations and the boundary condition satisfied by the electromagnetic field that these coefficients are linearly related. Bohren states that in principle the coefficients of the T-matrix are obtainable by integration; however, computational difficulties arise if the particles are highly absorbing or their shapes extreme. Recent progress has been made for computing the T-matrix as discussed by Mishchenko et. al [8]. Mishchenko provides a review and description of several 16

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numerical techniques developed for single and aggregated particles. The scope of this dissertation is on single particles and as such a brief overview follows. The standard approach for computing the T-matrix for single scatterers is based on the extended boundary condition method developed by Waterman for homogenous particles. Nearly all numerical results computed by the T-matrix relates to bodies of revolution. The first computational advance of the T-matrix is using nonspherical particles of fixed orientation using the extended boundary condition method. This method proved to be faster than the conventional separation of variables method for spheroids and discrete dipole approximation integral equation formulation. The disadvantage to using the extended boundary condition method is its poor numerical stability for particles with very large real and/or imaginary parts of the refractive index, large size compared to the wavelength, and/or large extreme geometries. Mishchenko discusses the iterative extended boundary condition as another approach for overcoming the problem of numerical instability in computing the T-matrix for highly elongated spheroids. The disadvantage of this technique is that the numerical stability comes at the expense of considerable increase in computation code, complexity, and CPU time. The last computational approach discussed by Mishchenko is that of using extended precision instead of double precision floating point variables. This technique provides an increase in size parameter for spheroids and better accuracy. The use of the extended precision variable requires only a negligibly small additional memory and its approach is simple with little additional programming. The disadvantage to the extended precision is the CPU time. 17

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In the Purcell-Pennypacker method a particle is approximated by a lattice of dipoles, each small compared with the wavelength but still large enough to contain many atoms. Each dipole is excited by the incident field and by the fields of all the other dipoles. Two methods are used to solve for these equations; either iteration or matrix inversion. The iterative method is slow for large parameters, dipoles and values of the refractive index. The matrix inversion method, although useful for calculating scattering by a particle in more than one orientation and for orientational averaging over an ensemble of particles, is limited to small number of dipoles. An embedding method using a scattering-orders approach was developed by Singham and Bohren [6]. The scattering-orders perturbation method is an extension of the series for the perturbation method which looks upon a nonspherical particle as a sphere, the boundary of which is distorted or perturbed by different amounts at different points [1]. In the case of Singham and Bohren, the scattering-order perturbation method is used to formulate the coupled dipole method. This method uses the dipolar interactions as infinite series in scattering-orders. Two interactions exist, those between dipoles within the sphere and all others. The scattering fields resulting in the dipole interaction within the sphere are described by Mie theory or any other equivalent theory. It was demonstrated that this method worked and shortened computation time. However, it was limited by the scattering-order series, which can diverge. The greater the refractive index, the smaller the particle size for which it diverges. 18

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19 The endeavor to characterize biopartic ulates in the context of engineering application requires identifying the problem, finding solution s, analyzing, designing and testing of the solution all taking place in real-time. Mie, Rayleigh-Debye-Gans, and the new hybrid theory provide a more palatable m eans to characterize bi ological systems as compared to those aforementioned. The rest of this dissertation describes how these theories are utilized and explored.

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Chapter Three Instrumentation Correction Model for Transmission The research presented in this document utilizes many models to represent those conditions present when taking experimental measurements. As stated in the previous chapter and illustrated in figure 2.1, light scattering, instrumentation and optical formulas are integrated to make up a complete model for experimental conditions observed when taking turbidity measurements. The previous chapter described the light scattering equations for the intensity ratio and transmission spectroscopy measurements. This chapter is dedicated to discussing the instrument formulas available for simulating transmission measurements and implementation of the formulas to RDG theory. 3.1 Instrumentation Correction Model Deepak et al. [9] defines the expression forward scattering for a single scattering phenomenon as scattered radiation reaching the detector after being scattered only once by a scatterer situated within the path of the direct radiation. Ideally, the conditions for turbidity measurements entail a non-divergent beam illuminating a homogeneous medium and a detector directly, opposite the light source, possessing an acceptance angle close to zero (to capture light solely in the forward direction). The turbidity equation presented in the previous chapter does not account for the instrument setup, therefore an instrument model or correction to the turbidity is introduced in this chapter. On the basis 20

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of the scattering theories presented and the instrument setup used for transmission measurements, the correction factor provided by Deepak et al. was chosen. The correction factor is dependent upon the geometry of the transmission measurement through the angle subtended at the scatterer by the detector window (figure 3.1). Therefore the instrument design has to be taken into account. For completeness the two designs provided by Deepak et al. were implemented [9]: the open detector and pinhole detector. The geometries are illustrated in figure 3.1 (a) and (b). Figure 3.1: Transmission System (a) Open Detector (b)Pinhole Detector Deepak et al. states that the true optical depth from transmission measurements cannot be obtained directly. Deepak et al. notes that the finite aperture of the detector, in a transmission measurement, picks up some additional light that has been scattered into the aperture. As seen in chapter two the turbidity formula (equation 2.16) 21

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extpabsscapabsscaplQNQQlGNCClN 3.1 accounts for light that has been scattered out of, or absorbed from, the incident beam. Recall that the power scattered by a particle equals (equation 2.4) oscasICddrrIpowerscattered0202sin,,, 3.2 However, if the finite aperture captures scattered light within a cone of half-angle in the forward direction, then the beam depletion due to scattering only ddrrIs202sin,,, 3.3 giving an effective scattering cross section of ddrIrICCosscasca202sin,,, 3.4 The apparent extinction cross section of the particle can be written ddrIrICCCCCosabsscaabsscaext202sin,,, 3.5 or PddrIrICCCosextextext202sin,,,11 3.6 In the open detector system (figure 3.1a) a radiation source is placed at the focal length of the transmitter lens L 1 which transmits a parallel beam through the medium of thickness l, which is in turn measured by an open detector of radius R 2 The path averaged correction factor P for particles on axis is defined as 22

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22csc,tandRP 3.7 where the angle for the scatters satisfies 12tanLR The pinhole detector system (Figure 3.1b) consists of a radiation source placed at the focal point of a lens L 1 A second lens L 2, with focal length f, focuses the light through an aperture of radius r in the focal plane and onto the detector. The path averaged correction factor for the pinhole detector is simply P PP 3.8 The result of equation 3.2 stems from the fact that is determined by fr tan which stays constant. The correction factor corresponding to the design set up can be introduced into the turbidity equation, equation 2.11, by multiplying through by P : PmQalNextp),()(2 3.9 3.2 Implementation of Instrument Corrections The corrections for scattering developed by Deepak [9] have been evaluated for RDG theory for hemoglobin spheres. Hemoglobin spheres are hypothetical spheres whose refractive indices are those of hemoglobin. The refractive indices of hemoglobin were used to test these effects for strong scattering and absorption relative refractive indices ( 1.0001.0,2.1 1 oonnn ). The acceptance angle for spectrometer detectors are typically less than or equal to 2 degrees. The acceptance angles studied for both detector 23

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configurations were 1, 2, and 5 degrees for a spherical particle of 1 m. The parameters used to conduct the simulations are provided in table 1.1. Figures 3.2 and 3.3 show the simulated transmissions using Rayleigh-Debye-Gans theory with and without acceptance angle corrections for each detector configuration. The (uncorrected) Mie theory simulation is included as a reference. The pinhole detector setup (figure 3.2) shows smaller subtle changes in the corrected spectra for all angles (RDG 1 RDG 2 RDG 5 ) compared to uncorrected (RDG) spectra. The open detector setup (figure 3.3) shows dramatic changes in the spectra as the aperture angle increases. It is evident that the angle of acceptance plays an important role in being able to assess and interpret spectral differences. Bohren has made similar statements concerning the angle of acceptance though here calculations have been provided. Having demonstrated the methodology for correcting for finite detector apertures in transmission measurements, we next turn to a study of the effects of particle shape and orientation. Aperture corrections are omitted in the simulations reported in the remainder of the dissertation, except where noted. 24

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200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 3.5 4 Wavelength (nm)Optical Density RDGRDG1RDG2RDG5 Figure 3.2: Calculated Transmission of Rayleigh-Debye-Gans for 1 m Hemoglobin Spheres with Pinhole Detector Setup 200 300 400 500 600 700 800 900 0 0.5 1 1.5 2 2.5 3 3.5 4 Wavelength (nm)Optical Density RDGRDG1RDG2RDG5 Figure 3.3: Calculated Transmission of Rayleigh-Debye-Gans for 1 m Hemoglobin Spheres with Open Detector Setup 25

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Chapter Four Nonspherical Particles Form factors have been developed for many shapes such as cylinders, disks, rods, spheres, and ellipsoids. We are interested in biological systems, whose particles are best represented by ellipsoids. 4.1 Geometry and Notation for Ellipsoids It is convenient to carry out the present analysis in two non-standard, left handed coordinate systems, the laboratory and particle frame. The laboratory frame establishes the position and orientation of the particle relative to the source and detector. The particle frame exploits the convenience of particle coordinates in describing the scattering. Therefore scattering calculations are in the particle frame and related back to the laboratory frame. Unit vectors along the x L y L and z L axes in the lab frame are denoted by i, and LLj, Lk ; and similarly for the particle frame. 26

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Figure 4.1: Light Scattering in Laboratory Frame Figure 4.2: Light Scattering in Particle Frame The behavior of light scattered from the particle is dependent upon its size, shape, orientation and chemical composition. Figure 4.1 illustrates the incident beam approaching the particle in the z L -direction in the laboratory frame. The light scattered from the particle can radiate in all directions. The line in the scattering plane that bisects the angle between the incident and scattered beam is called the bisectrix [2], denoted 27

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by BIS in figures 4.1 and 4.2. The plane through the bisectrix ( BIS ) and perpendicular to the plane of the scattering ( SCA ) will be called the bisectrix plane. Equation 4.1 describes the directions of incident light, scattered light and the bisectrix from the laboratory frame. LLLLLkiINSCABISkiSCAkIN 1cossincossin 4.1 The description can also be expressed in the particle frame. Figure 4.2 shows the angles used to relate the bisectrix to the particle axes. Calculation of the scattered light intensity at different angles in the laboratory frame can be reconstructed through its relation to the particle frame. pppLLLkjiEkji 4.2 where i p =|1 0 0|, j p =|0 1 0|, k p =|0 0 1| in the particle frame, the Euler angles , are used to relate the frames using the zyz Euler rotation sequence, and E is given by cossinsinsincossinsincoscossincossincossinsincoscoscossinsincoscoscossinsinsincoscoscosE 4.3 Using this relationship, the directional cosines of angles can be found in the laboratory frame to mathematically describe the bisectrix (equation 4.1). LLLkBISBISjBISBISiBISBIScos,cos,cos 4.4 28

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In the case of an ellipsoid with a fixed orientation, we can see the use of the directional cosines in the form factor computed by Kerker: 22222232232coscoscos29cbaAhAuuuJf 4.5 where A is a vector described by the directional cosines and a, b, c are the semiaxes for ellipsoid. The variable h equals 2sin4h The following is a short table of various form factors, pertinent to this work, taken from Kerker [4]. The equations presented in this section were programmed; simulations conducted are reported in the next section. Table 4.1: Table of Form Factors Shapes Definition of variables Form factor f 2 () Sphere Radius=a u=ha Concentric Sphere with spherical shell Inner radius=a Outer radius=b u=ha v=hb Inner refractive index=m1 Inner refractive index=m2 Ellipsoid of Revolution Semi axes are a,b, is angle between figure axis & bisectrix u=h(a 2 cos 2 b 2 sin 2 Ellipsoid Semi axes are a,b,c u=hA A 2 =a 2 cos 2 b 2 cos 2 c 2 cos 2 are the directional cosines of the bisectrix 32232)(9uuJ 232333212323)(12)(29uuJbammmvvJ 32232)(9uuJ 32232)(9uuJ 29

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4.2 Ellipsoid Simulations For the simulation of the effects of shape and orientation on the multiwavelength transmission spectra, prolate ellipsoids with refractive indices of soft particles have been selected. Ellipsoids have been used to model the scattering behavior of a large variety of biological systems such as microorganisms and red blood cells, and offer the possibility of exploring geometrical extremes between spheres and needle-like particles. The semimajor and semiminor axes for prolate ellipsoid were determined for a volume equivalent to 1 m diameter sphere. For absorption it is apparent from equation 2.9 that the pathlength for the particle is not relevant since the absorbed power is proportional to the volume. Nevertheless, scattering is proportional to the cross sectional area and therefore orientation. For a prolate ellipsoid with length a along its semimajor axis and lengths b=c (
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