User's guide T-matrix program based on the null-field
Contents
1. There are three options in Intel Fortran Compiler which have an influence on how the variables are allocated during the execution on the call stack in static memory or in heap save causes all variables to be placed in static memory auto causes all local variables which do not have SAVE attribute to be allocated to the run time stack It does not affect variables that have the SAVE attribute or ALLOCATABLE attribute auto scalar causes scalar variables of intrinsic types INTEGER REAL COMPLEX and LOGICAL that do not have the SAVE attribute to be allocated to the run time stack All other variables are allocated statically If you are not using OpenMP Intel Fortran Compiler default uses the auto scalar But if you say openmp this changes to automatic and all local variables are allocated on the stack This is required to allow for thread safety 10 So if the program uses OpenMP then the size of run time stack might be not enough for the memory allocation of Q matrices for each threads and we need increase it Therefore it should be increased 8 1 on a Linux Unix system There are some utilities for Linux Unix systems which can change the stack size used by a program or set it unbounded We advise to use the ulimit utility which controls the resources available to a process started by the shell To change the maximum size of stack this utility has to be started once prior to the NFM DS program with the option2. s and the maximal size of call stack An example of the command line to start this utility ulimit s 10000000 8 2 on a Windows system For Windows there are no such utilities The stack size is defined by the compilation of a program exactly by linking the compiled object files in the executable file The necessary parameter in the Intel Fortran Compiler used by compilation in the command line is STACK reserve commit where reserve and commit are the total stack allocation sizes in virtual and physical memory respectively In an IDE there have to be also the corresponding options like Stack Reserve Size and Stack Commit Size Please see your compiler documentation for more detailed information Bibliography 1 P C Waterman Symmetry unitary and geometry in electromagnetic scattering Physical Review D 3 1971 825 839 2 M i Mishchenko G Videen N G Khlebtsov T Wriedt N T Zakharova Comprehensive T matrix reference database A 2006 07 update Journal of Quantitative Spectroscopy amp Radiative Transfer 109 2008 1447 1460 3 T Wriedt Editor Generalized Multipole Techniques for Electromagnetic and Light Scattering Elsevier Amsterdam 1999 4 Wriedt T Review of the null field method with discrete sources JQSRT 2007 106 535 545 5 A Doicu T Wriedt Y A Eremin Light Scattering by Systems of Particles Null field Method with Discrete Sources Theory an Programs Springer Berlin H
3. 1 7 0 0 The second relative refractive index of the uniaxial anisotropic particle AnSVWF alphaPR 120 Azimuthal angle specifying the orientation of the principal coordinate system with respect to the particle coordinate system AnSVWEF betaPR 90 Zenith angle specifying the orientation of the principal coordinate system with respect to the particle coordinate system AnSVWF Nbeta 181 Number of integration points for computing the vector quasispherical wave functions these variables must be provided if anisotropic true The angles alphaPR and betaPR identify the orientation of the optical axis of a crystal while the parameter ind_refRelZ defines the refractive index along the Z axis If alohaPR 0 and betaPR 0 the refractive index of a particle would have the components ind_refRel ind_refRel ind_refRelZ where ind_refRel is the parameter from the MatProp group see 5 The parameter Nbeta defines the number of points in a gauss integration rule for computing the vector quasispherical wave functions see 6 We don t advise to change its default value for non extreme refractive indices because a value of Nbeta 181 already gives the exact integration for polynomials of the degree up to 363 3 4 Summary To compute the electromagnetic scattering by an uniaxial anisotropic particle with the TMATRIX program the user has to e specify the optical properties geometry and error tolerances in the II
4. no value is defined outside The name is case sensitive The value of the variable can be provided in the command line During reading the input file by simulation this sentence will be automatically replaced with it or lt default_value gt if no value is provided in the command line Otherwise the sentence will be not replaced This feature is available for the following scattering problems AXSYM T matrix for a homogeneous isotropic chiral and perfectly conducting axisymmetric particle NONAXSYM T matrix for a homogeneous isotropic chiral perfectly conducting and uniaxial anisotropic non axisymmetric particle and SCT scattering characteristics using a previously calculated T matrix 6 Some examples of the usage of variables in input files are presented below in Table 3 Table 3 Exemplary usage of variables in the InputAXSYM dat file here the group OptProp properties of a particle and of the ambient medium 1 parameters OptProp without variables 0 628318530717959 1 0 1 5 0 Variables wavelength wavelength of the incident light in vacuo ind_refMed refractive index of the ambient medium ind_refRel relative refractive index of the particle 2 parameters OptProp with and without the wavel0 628318530717959 default value 1 0 ref_index Variables wavelength wavelength of the incident light in vacuo ind_refMed refractive index of the ambient medium ind_refRel relative re
5. the particle shape e particle with a high aspect ratio due to a large divergence of the radius vector over a the particle surface and therefore due to a large radial divergence of SVWFs under the integral e particle with a high refractive index due to a larger radial divergence of SWVFs for a larger refractive index e if nonsingularities of the scattered field continued inside the particle are placed close or outside the sphere inscribed in the particle This can be resolved by increasing of the number of integration points which leads to longer calculation time n Q Q a o Q Q J Y A V f Isotropic ni 1 5 ee Isotropic ni 1 5 Jd jot Or n AS iz 1 7 a ni 15 niz 1 7 NZ a ni 1 5 niz 1 3 Be amp ni 1 5 niz 1 3 a ni 1 5 niz 1 9 amp ni 1 5 niz 1 9 ni 1 5 niz 1 1 alpha beta 0 ni 1 5 niz 1 1 alpha beta 45 10 10 0 90 180 270 360 0 90 180 270 a Scattering angle deg b Scattering angle deg Fig 1 Normalized differential scattering cross sections for positive uniaxial anisotropic ellipsoids with n 1 5 n 1 7 and n 1 5 Nz 1 9 and negative uniaxial anisotropic ellipsods with n 1 5 nz 1 3 and nj 1 5 n 1 1 The orientation of the particle is a a 0 B 0 y 0 and b a 45 B 45 y 0 The dimensions of the ellipsoids are k a 1 0 k b 1 5 and k c 2 0 The maximum number of azimuthal modes is Mna 8 whi
6. NPUTFILES InputNONAXSYM dat e provide the scattering characteristics in the INPUTFILES InputSCT dat e set the model control parameters in the input file INPUTFILES Input dat e run the main program TMATRIX f90 and call the routine TNONAXSYM f90 e perform a convergence test and analyse the results written to the file OUTPUTFILES Output dat if convergence is achieved the program will write the T matrix to the file FileTmat in the directory TMATFILES and the differential scattering cross sections and the scattering characteristics to the files FileDSCS and FileScat in the directory OUTPUTFILES eto change the scattering characteristics calculation modify the input file INPUTFILES InputSCT dat and run the main program TMATRIX f90 by calling the routine SCT f90 An exemplary calculation of the normalized differential scattering cross section for an uniaxial anisotropic ellipsoid is presented in Fig 1 4 Parallelization using OpenMP The T matrix method is one of the most effective methods for accurate solving the light scattering problem But the numerical scheme for the calculation of a T matrix of a single particle consists of a huge number of integrals over the particle surface which contain products of the spherical vector wave 3 functions There are some numerical problems in so called difficult cases particle with a complex shape due to difficulties by approximation of
7. User s guide T matrix program based on the null field method with discrete sources NFM DS Update on 6 March 2009 Vladimir Schmidt Adrian Doicu Thomas Wriedt Universitat Bremen Germany e mail vschmidt iwt uni bremen de Institute f r Methodik der Fernerkundung Germany e mail adrian doicu dlr de Universitat Bremen Germany e mail thw iwt uni bremen de Abstract This is an update of the FORTRAN programs based on the Null Field Method with Discrete Source which originally was included on CD ROM with the book by A Doicu T Wriedt Y A Eremin Light scattering by systems of particles Springer Berlin 2006 1 Introduction The T matrix method or null field method is one of the most popular light scattering theories to accurately compute scattering by nonspherical particles 1 Recent reviews of the literature on this method have been published by Mishchenko et al 2 It is based on the expansion of incident internal and scattered field in the terms of suitable basis of vector wave functions which are classically spherical vector wave functions The T matrix relates the expansion coefficients of the incident field to the expansion coefficients of the scattered field Using a computed T matrix various light scattering problems multiple scattering scattering by a particle located near a plane interface scattering by a rotated or a translated particle or orientation averaged scattering can easily be calcula
8. e features 5 1 Variables in the input files The input parameters in the NFM DS program for each scattering problem are specified in the corresponding input files They all are called like InputXxXX dat where XXX denotes the scattering problem AXSYM NONAXSY M et all The structure of each input files is very simple and was explained in detail in the program description on the CD enclosed in the book 5 The input parameters in input files are divided between several groups each specified by a keyword Once the program have recognized the group through the corresponding keyword it expects a sequence of parameter values Each line consists only one value of the corresponding parameter The sentence of these parameters are strictly defined for each group in the program and described in comment lines in the input files An exemplary input text for the OptProp group optical properties of material is presented in the first line of Table 3 In the current NFM DS program values of parameters can be specified not only directly in the input file but also outside of them using so called a variable mechanism In this case instead of a explicit value the following sentence have to be written in the input file lt variable_name gt lt default_value gt or lt variable_name gt where lt variable_name gt is the name of the variable which value should be defined outside the file lt default_value gt is the default value in cases when
9. ed in the NFM DS program code since the book was published version 1 1 There are three important changes in the program Firstly the formulae of the T matrix for an uniaxial anisotropic nonaxisymmetric particle of arbitrary shape were obtained and implemented by A Doicu 6 Secondly the program was parallelized for the usage on multi core or multi processors computers Thirdly some improvements in usage of the program for multiple simulations were realized Here we would like to present detailed information about these additional features 3 Light scattering by uniaxialy anisotropic particles 3 1 T matrix method for uniaxialy anisotropic particles The T matrix method is based on the expansion of incident internal and scattered field in the terms of a suitable basis of vector wave functions For an isotropic medium the spherical vector wave functions SVWF are used classically A Doicu obtained the basis of the so called quasi spherical vector wave functions qSVWF for an uniaxial anisotropic medium by solving Maxwell s equations in Fourier s space 6 Using this basis the scattering problem for an uniaxial anisotropic non axisymmetric particle using the T matrix method is solved We refer the interested reader to the paper 6 for more detailed theoretical derivation 3 2 Description of program code The main program TMATRIX f90 calls a T matrix routine for solving a specific scattering problem For an uniaxial anisot
10. eidelberg New York 2006 6 A Doicu Null field method to electromagnetic scattering from uniaxial anisotropic particles Optics Communications 218 2003 11 17 7 OpenMP Architecture Review Board The OpenMP API specification for parallel programming http openmp org 8 University of Tennessee Message Passing Interface MPI Forum Home Page http www mpi forum org 9 P B Johnson R W Christy Optical constants of the noble metals Phys Rev B 6 12 1972 4370 4379 10 S Franzen Surface Plasmon Polaritons and Screened Plasma Absorption in Indium Tin Oxide Compared to Silver and Gold J Phys Chem C 112 2008 6027 6032 11 J Hellmers T Wriedt New approaches for a light scattering Internet information portal and categorization schemes for light scattering software doi 10 1016 j jqsrt 2009 01 023 11
11. es axisymmetric layered particles of arbitrary shapes e inhomogeneous dielectric axisymmetric particles with an arbitrarily shaped inclusion e inhomogeneous dielectric spheres with a spherical inclusion e inhomogeneous dielectric spheres with an arbitrarily shaped inclusion e inhomogeneous dielectric spheres with multiple spherical inclusions concentrically layered spheres multiple particles e clusters of arbitrarily shaped particles e two homogeneous dielectric spheres e clusters of homogeneous dielectric spheres particle on or near a plane e homogeneous dielectric or perfectly conducting axisymmetric particles on or near a plane surface Using the computed T matrix the scattering characteristics describing the scattered field in the far field region for the considered scatterer in a fixed or averaged orientation can be easily computed These include the far field pattern the differential scattering cross section DSCS the amplitude matrix the optical cross sections and the phase and extinction matrices To ensure convergence of the results corresponding routines are integrated into the software Additionally the effective wave number of a medium with randomly distributed spheroidal particles can be calculated 2 Update to the original code The original program code can be found on CD ROM enclosed in the monograph by Doicu et al 5 Here we would like to familiarize the reader with improvements implement
12. fractive index of the particle 3 complex parameters OptProp 0 628318530717959 1 0 ReRef lmRef Variables wavelength wavelength of the incident light in vacuo ind_refMed refractive index of the ambient medium ind_refRel relative refractive index of the particle The value of variables can be specified in the command line by starting of the program The format of the command line is the following lt program gt lt variable1 gt lt value1 gt lt variable2 gt lt value2 gt where lt program gt is the name of a executable file lt variablexX gt and lt valueX gt are names and values of corresponding variables There are some rules about possible values of variables The value of the variable must be not longer then 80 characters If the value contains a space character or some other special characters it must be to quoted There is an example of a command line double_par_TMATRIX wave 0 6283 size_a 1 2 ref_index 1 5 0 001 5 2 Refractive index for common materials The refractive index of some materials such as silver which are frequently used in simulations depends on the wavelength of the incident light We have included the dependences for some materials in the program For such material the user can write in the line responsible for the refractive index see the OptProp group in the Input dat InoutAXSYM dat or InoutSCT dat files instead of a explicit value the fol
13. ilers which helps to automatically change the program code before the compilation step For extended precision calculation the pre defined variable PRECISION QUAD must be set otherwise double precision calculation will be used If an IDE like the Microsoft Visual Studio is to be used the user must find in the project properties a option like Pre processor Definitions and enter PRECISION QUAD For a command line compiler eq the Intel Fortran Compiler on Linux System the corresponding option like DPRECISION QUAD must be included For more detailed information we advise the user to look his compiler documentation Of course the extended precision version will demand about twice as much memory as the double precision version The extended precision codes are also slower than the double precision codes by a factor of 5 6 but allow computations for larger scatterers 8 Some problems with memory allocation There are some problems which have been observed using the Intel Fortran Compiler For large problems large Q and T matrices a memory allocation error can arise during running the program Even when the computer should have enough memory for such simulation It has to be noted that this problem has not been occurred with other compilers Up to now the exact explanation of this error isn t known to us There is an excerpt from the Intel Fortran Documentation and internet forums devoted to high performance simulation using OpenMP
14. le the maximum expansion order is Nma 10 6 On modern computers an effective method to accelerate the algorithm is the usage of parallel computing Parallel computing is a form of computation in which many calculations are carried out simultaneously operating on the principle that large problems can often be divided into smaller ones which are then solved concurrently There are a lot of approaches for the parallelization of programs each of them is preferable for specific problems due to their advantages and disadvantages The well known are Open Multi Processing OpenMP 7 which is mostly used on shared memory systems multi processors computers and Message Passing Interface MPI 8 which is mostly used on distributed memory systems computer clusters 4 1 Parallelisation algorithm In a first step we choose the OpenMP paradigm to parallelize the NFM DS program Here the integration process for the Q and the RegQ matrixes elements was parallelized Using the gauss integration rule the calculation of integrals can be approximated as a weighted sum of function values at specified points within the domain of integration f dx Y wf A l Ng icA where S is particle surface x and w are points and weights of a particle surface element Nin is the number of series items For the parallelization on a computer with N processors we divide the particle 4 360 N surface into N areasA UA provide summation simul
15. logy was implemented see Table 2 It has also to be noted that the memory requirements are increased N 1 fold by using lt N gt computational cores 5 Automation of simulations The NFM DS is the good program code based on a good theoretical basis for simulations of light scattering by an ensemble of single particles or aggregates Practically a lot of simulations have to be provided before a suitable result will be obtained Firstly because there are no automatic built in convergence criteria in the NFM DS program the user has always to perform own convergence tests Secondly the practical interest exhibits not a single simulation but multiple simulations In the original NFM DS program the input parameters can be specified only through the keyboard and in three input files which are different for each kind of the scattering problem It can reduce the practical effect of the NFM DS program for multiple simulations To decrease this disadvantage we have developed some improvements in the program such as parameters for the scattering program can be provided not only through input file but also through the command line which is used to start the program e wavelength dependencies of the refractive index for common materials such as silver etc are included in the program parameters entered normally through the keyboard can be provided through the command line which is used to start the program Below we describe how to use thes
16. lowing sentence lt name_of_material gt where lt name_of_material gt is the name of the material During the calculation this sentence will be automatically replaced with the corresponding value of the refractive index for the current wavelength 7 In the program code the dependence of the refractive index m is defined through its values for the set of the wavelengths m m A i 1 N Between them the refractive index m A is calculated using an approximation to a line function int MN m A Aust In Table 4 materials included in the program are presented while on the Fig 3 the dependences are plotted For more detailed information we advise the interested user to check the ref_index f90 and interpretator f90 modules A A m Table 4 Materials which refractive index dependence is included in the program Material Command Range for Reference wavelength Silver silber 187 85 1937 25 nm Johnson and Christy 9 Ito Ito 100 2000 nm Franzen 10 Fig 3 The refractive index dependences of Silver left and Ito right which is included in the program refractive index refractive index 0 200 300 400 500 600 700 800 500 1000 1500 2000 wavelength nm wavelength nm 5 3 Input from console In the original NFM DS program some input parameters like Nann M ank has to be specified through console input Now it is also possible to provide these parameters thro
17. n the parallel mode it has to be compiled with a OpenMP compatible Fortran compiler Usually all modern both commercial eq Intel Fortran Compiler 11 0 and open source eq GNU Compiler Collection 4 3 3 Gfortran 4 3 compilers support this option The compilation instructions are described in the Sec 7 Further the compiled program can be started on a arbitrary multi processor computer without any special customization Table 2 The scattering problems for which parallelized subroutines can be used Routine Light scattering by TAXSYM f90 a homogeneous dielectric isotropic chiral and perfectly conducting axisymmetric particle TNONAXSYM f90 a homogeneous dielectric isotropic chiral and perfectly conducting nonaxisymmetric particle TINHOM f90 an inhomogeneous dielectric axisymmetric particle with an arbitrarily shaped inclusion TPARTSUB f90 a particle on or near a plane surface TPARTSUBFILM f90 The command line to start the program on Windows or Unix systems is lt program gt num_threads lt N gt where lt program gt is the name of a executable file lt N gt is the number of parallel threads to be started The parameter lt N gt can have any positive value but we advise to define a number which is equal or less than the number of computational cores on the computer This parameter is valid also only for the 5 scattering problems which are listed above and where the OpenMP techno
18. rmance advantage when using the NFM DS on multi cores or multi processors platforms To compile the developed program an OpenMP compatible Fortran compiler is required All modern both commercial eq Intel Fortran Compiler 11 0 and open source eq GNU Compiler Collection 4 3 3 Gfortran 4 3 compilers support OpenMP In the downloadable source file we also offer a makefile for easy compilation of the program using the Intel Fortran Compiler and Makefile utility 7 1 OpenMP If an Integrated Development Environment IDE like the Microsoft Visual Studio is to be used to 9 compile the program one must find in the project properties options like Parallelization and OpenMP Conditional Compilation and set them to True If the user uses the command line compiler eq the Intel Fortran Compiler on Linux System then the corresponding options like openmp and parallel must be included Please see your compiler documentation for more detailed information 7 2 Double vs Extended The NFM DS program is written to allow an easy generation of either double or extended precision versions of the executable file The precision control parameter O is defined in the file Parameters f90 For double precision calculation must be set O kind 1 d0 while for extended precision calculation the statement O kind 1 q0 This mode can be chosen also through pre processor definitions This is a typical feature of modern comp
19. ropic nonaxisymmetric particle the code is included in the file TNONAXSYM f90 which was used originally only for isotropic dielectric chiral or perfect conducting particles 3 3 Input parameters of the scattering problem For the scattering problem by an uniaxial anisotropic particle the input parameters are specified in three input files e INPUTFILES InputNONAXSYM dat provides the variables specifying the optical properties geometry and error tolerances e INPUTFILES InputSCT dat provides the variables specifying the scattering characteristics calculation e INPUTFILES Input dat specifies the model control parameters The input parameters are divided between several groups each specified by a keyword that is recognized by the program A detailed description of the parameters required by the input files can be found on the CD enclosed in the book 5 or in the comment lines of each T matrix routine Here we notice some changes in the structure of InputNONAXSYM dat relative to the original version only isotropic dielectric chiral or perfect conducting particles Five new parameters were added into the file InputNONAXSYM dat see Table 1 Table 1 The new input parameters for uniaxial anisotropic particles Group Parameter Examples for Comment name name the value MatProp anisotropic true If true then the particle is an uniaxial false anisotropic crystal AnSVWE ind_refRelZ
20. s Nrank and Ndgs 5 6 100 20 enter the type of convergence test 1 Nint 2 Nrank 3 Mrank 2 progress of main calculation ODIHNBW De ta e i NN RS CoRR Co CoCo CoE CoCo Convergence criterion for Nrank is satisfied 6 Obtaining and installing of the source code The source code can be downloaded from the internet portal http Avww scattport org which is provided by the our research group 11 The package must be downloaded and unzipped in any folder It contains the following directories e BIN e TMATSOURCES e TMATFILES e INPUTFILES e GEOMFILES and e OUTPUTFILES The executable program must be created in the directory BIN and must include the main program TMATRIX f90 and all F90 routines contained in the directory TMATSOURCES To compile the code on a Unix system use the makefile supplied in the directory BIN but edit the makefile to provide the required compiler options To compile the code using for instance Compaq Visual Fortran or Microsoft Developer Studio create a project in the directory TAATSOURCES and add all F90 files to the project The input parameters for each scattering problem are provided in text files in the directory INPUTFILES like as the geometry files are contained in the directory GEOMFILES The directory TMATFILES contains text files with calculated T matrices 7 Compiling OpenMP is a standard for the support of shared memory parallel programming and can provide a perfo
21. taneously for each area jal N I i w f k 1 N and then collect the results gt For example on a computer with six icA k icA l processors each face of the cube would be integrated separately Thereby the size of the total transferred data between the calculation kernels is minimal Because transfer time is normally the restricting factor for parallel programs here the efficiency of the developed program is relative high On the other hand this improvement increases the memory needed N 1 fold The memory is required to store the calculated Q matrices for each computing thread and for the master thread An exemplary results for the calculation time depending on the number of used cores is printed in Fig 2 Fig 2 An exemplary calculation time on computers with different number of processors computing scattering by a cube with size parameter k a 4 and refractive index m 1 1 1 1 1 5 3000 2500 2000 1500 500 Titre S246 6 Tes CPU number Time sec Q 4 2 Parallelized subroutines Using the OpenMP technology the following subroutines from the module Proces1 f90 which provides integration over the particle surface were parallelized matrix _Q matrix_Q_m matrix_Q_sym incident_matrix_m These subroutines are used for the calculation of the T matrix using localized and distributed sources for the scattering problems presented in Table 2 4 3 Usage of the program To start the developed program i
22. ted The NFM DS combines approaches of the Generalized Multipole Techniques GMT 3 with the T Matrix method For expansion of the internal field within a scatterer discrete sources are used This helps to compute the T Matrix of particles having a high aspect ratio up to 100 1 such as fibres and flat discs The full scope of the NFM DS has recently been presented in a review paper 4 For a full description of the theory we refer the interested reader to the published monograph by Doicu et al 5 which includes the original FORTRANSO program code on CD ROM The theory and corresponding program has been developed at Bremen University over the last 10 years It is intended for science professionals engineers and graduate students working in optics electromagnetics biomedical optics atmospheric radiation and remote sensing The program can be used to compute the scattering and absorption of electromagnetic waves by particles with arbitrary geometries using the NFM DS Various scattering problems are implemented in the program single particle homogeneous e homogeneous dielectric isotropic chiral and perfectly conducting axisymmetric particles of arbitrary shape incl having a high aspect ratio and concavities e homogeneous dielectric isotropic uniaxial anisotropic chiral and perfectly conducting nonaxisymmetric particles of arbitrary shape single particle inhomogeneous e axisymmetric composite particles of arbitrary shap
23. ugh the command line and moreover to save information printed during simulation on the screen to the specified file To use these possibilities the user must start the program with options both or one of them lt program gt input lt in_data gt output lt out_file gt where lt program gt is the name of a executable file lt in_data gt is the sequence of parameters to be entered into the program like they would be entered through keyboard lt out_file gt is the filename to save the output information of the simulation process The character of new line hex code 0x13 is given by writing the n in the sequence lt in_data gt An example of the usage for the calculation of light scattering by a dielectric axisymetric particle command line double_par_TMATRIX input 1 n100 20 n2 n screenshot T Matrix Code for Light Scattering Calculation enter an integer specifying the scattering problem 1 dielectric perfectly conducting or chiral axisymmetric particle dielectric perfectly conducting or chiral nonaxisymmetric particle axisymmetric composite particle axisymmetric layered particle inhomogeneous axisymmetric particle with an arbitrary inclusion inhomogeneous sphere with a spherical inclusion AuURWN 1 Convergence Test for an Axisymmetric Particle Nrank estimate the estimated value of Nrank from Wiscombe s criterion is 10 enter the estimated values of Nint and Nrank where Nint Ndg
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