Mie scattering in the X-ray regime

Mie Scattering in the X-Ray Regime

Diploma Thesis

vorgelegt von

Nandan Joshi

aus

Mumbai, Indien

angefertigt im

Institut fur Rontgenphysik

der Georg–August–Universitat zu Gottingen

2007

To my late father,

whose smile was delight in lucidity,whose presence was comfort with security,

whose advice was guidance with conformity,whose work was the inspiration of eternity,whose love was with no other similarity,

whose life was worth more than infinity....

Acknowledgements

Understanding the subject of electromagnetic scattering from its fundamental level to theapplication could not have been possible without the help of others.

First of all I would like to express my gratitude to Prof. Dr. Tim Salditt, for givingme an opportunity to explore such quasi virgin field, showing confidence in my success aswell as failure and, of course, giving me important inputs, whereby bringing me onto theright path to proper understanding.

My many thanks goes to the colleagues from the institute. I would like to show myspecial gratitude to my colleague Aram Giahi for timely discussion whenever I get stuckinto difficulties. I am very glad to have got prompt assistance in the correction of my thesisfrom Christian Fuhse and Simon Castorph. I am very thankful to them. On the verge ofsubmission, I have received great help from Michael Reese. His free nature has given megreat moral support working late and working very hard. I’m absolutely thankful to him.I would also like to thank Klaus Giewekemeyer, Sebastian Kalbfleich, Sebastian Pankninfor their efforts to guide me towards the understanding of X-ray physics. I acknowledgewith many thanks to all my colleagues from the institute.

My gratitude is also to my girlfriend Karolina Wagrodzka, who showed great supportand understanding to my work schedule. Without her presence and readiness, it wouldn’thave been possible to get moral boost to complete my thesis on time.

I owe my deepest gratitude to my parents, my mother and late father. It was their pri-mary intention, effort and support that I had a chance to study in Germany, subsequentlywrite this thesis.

5

Table of Contents

1 Introduction and motivation 1

1.1 Scattering by a single particle . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 X-ray scattering: fundamentals 9

2.1 Overview of scattering processes . . . . . . . . . . . . . . . . . . . . . . . . 102.2 X-ray interactions with matter . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Elementary scalar theory of scattering . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Outgoing Green function . . . . . . . . . . . . . . . . . . . . . . . . 132.3.2 Integral-equation formulation . . . . . . . . . . . . . . . . . . . . . . 142.3.3 First Born approximation . . . . . . . . . . . . . . . . . . . . . . . . 152.3.4 Scattering by a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.5 Scattering cross section . . . . . . . . . . . . . . . . . . . . . . . . . 18

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Mie Scattering 21

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Maxwell’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.2 Time-harmonic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.4 Solution of differential equations with boundary conditions . . . . . 253.2.5 Series expansions for the scalar potentials . . . . . . . . . . . . . . . 273.2.6 Mie coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.7 Field vectors of scattered wave . . . . . . . . . . . . . . . . . . . . . 323.2.8 Intensity and polarization of the scattered light . . . . . . . . . . . . 323.2.9 Amplitude functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.10 Scattering cross section . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.3.1 Debye potentials and preceeding attempts . . . . . . . . . . . . . . . 353.3.2 Vector representations in spherical polar coordinates . . . . . . . . . 373.3.3 Associated Legendre functions . . . . . . . . . . . . . . . . . . . . . 373.3.4 Cylindrical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3.5 Scattering matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

7

8 TABLE OF CONTENTS

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4 Approximation for X-rays 414.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Analytical Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.1 Limiting cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2.2 Anomalous diffraction theory . . . . . . . . . . . . . . . . . . . . . . 434.2.3 Diffraction patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3 Numerical approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3.1 Using Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.3.2 Using IDL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5 Comparison 535.1 Comparing theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.2 Errors and problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.2.1 Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.2.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.4.1 Term Rewriting System . . . . . . . . . . . . . . . . . . . . . . . . . 60

6 Outlook 65Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7 Numerical calculation 697.1 Mathematica code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707.2 IDL code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

List of Figures

1.1 Scattering by an obstacle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Interaction of light with a single particle . . . . . . . . . . . . . . . . . . . . 3

2.1 Monochromatic plane waves are incident from the left. Two ray paths areshown, one associated with unscattered beam, in the absence of the volumeand other with scattered beam, in the presence of the scatterer. . . . . . . . 12

2.2 Illustration to the derivation of (2.25) . . . . . . . . . . . . . . . . . . . . . 152.3 Illustration to the notation in the given Born approximation . . . . . . . . . 16

3.1 Scattering of a incident wave from a spherical target located at the origin. . 23

4.1 Limiting cases in which m→ 1 . . . . . . . . . . . . . . . . . . . . . . . . . 434.2 Ray passing through the sphere . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.1 Relative intensity graph for the sphere with radius 0.5 nm . . . . . . . . . . 555.2 Relative intensity graph for the sphere with radius 1.0 nm . . . . . . . . . . 555.3 Relative intensity graph for the sphere with radius 2.5 nm . . . . . . . . . . 565.4 Relative intensity graph for the sphere with radius 5.0 nm . . . . . . . . . . 575.5 Relative intensity graph for the sphere with radius 10.0 nm . . . . . . . . . 585.6 Scattering cross section graph for the sphere with radius 0.5 nm . . . . . . . 595.7 Scattering cross section graph for the sphere with radius 1.0 nm . . . . . . . 605.8 Scattering cross section graph for the sphere with radius 2.5 nm . . . . . . . 615.9 Scattering cross section graph for the sphere with radius 5.0 nm . . . . . . . 625.10 Scattering cross section graph for the sphere with radius 10.0 nm . . . . . . 625.11 Ratio of differential scattering cross sections of Mie theory to kinematical

theory at Q = 0 against radius . . . . . . . . . . . . . . . . . . . . . . . . . 63

9

Chapter 1

Introduction and motivation

1

2 Chapter 1. Introduction and motivation

No matter how old the subject of the scattering of plane waves by a sphere may look,it has not yet been fully explored for different wavelengths.

The phenomena of scattering of light are ubiquitous, and quite naturally one of themost important in investigative disciplines like science and engineering. Many optical phe-nomena, like colors of the sunset, rainbow, the glory around the sun, the corona observedin our atmospheres to the scattering by interstellar dust causing the starlight to scatterbefore reaching us, are subjected to thorough investigation using the theory of scattering.The strong dependence of the scattering interaction on particle size, shape, and refractiveindex makes measurements of electromagnetic scattering a powerful noninvasive means ofparticle characterization in different fields of science.

The optical properties of a medium is characterized by its refractive index. As long as itis uniform, the ray passing through the medium will traverse the distance undeflected. Butif any obstacle, which may be a single electron, an atom or molecule, a solid or even liquidparticle, is illuminated by an electromagnetic wave, electric charges in the obstacle are setinto oscillatory motion by the electric field of the incident wave, i.e. there is small scaledensity fluctuation, and the accelerated electric charges radiate electromagnetic energy inall directions, this secondary radiation is called radiation scattered by the very obstacle.Apart from reradiation of electromagnetic energy, the excited elementary charges maytransform part of the incident electromagnetic energy into other forms, a process calledabsorption. In my investigation in this thesis, absorption shall be playing an importantrole.

INCIDENT

OBSTACLE

SCATTERED

Figure 1.1: Scattering by an obstacle

The classic treatise in the subject has long been van de Hulst’s Light Scattering by SmallParticles[1], which was later supplemented in later years by more elaborative, chronicled,at the same time, more application oriented work of Kerker, The Scattering of Light andOther Electromagnetic Radiation[2], and much later by Bohren and Huffman, Absorptionand Scattering of Light by Small Particles[3]. But the scientific study of light scatteringmay be said to have commenced by Tyndall[4]. The problem is to relate the propertiesof the scatterer, like its size, shape, and refractive index, to the angular distribution ofthe scattered light. In our case, where the scatterer is absorptive, part of the light willbe absorbed within it as heat, another part will be scattered, and the remainder willbe transmitted unperturbed along the incident direction. A complete description of thescattered light entails a knowledge of wavelength, amplitude, phase and polarization of

1.1. Scattering by a single particle 3

the radiation from the scatterer. Scattering is hardly restricted to the optical part ofthe spectrum, i.e. the scattering laws apply with equal validity to all wavelengths. Itis a quite interesting fact that these depend upon the ratio of a characteristic dimensionof the particle to the wavelength rather than explicitly upon the size. Thus, there is abuilt-in scaling factor, called size parameter, which we shall come across in the up-comingchapters.

In this thesis, I shall be dealing with a single particle of the size of a sphere and elasticscattering. The complete mathematical solution to the problem of light scattering from asphere was obtained over a century ago as the well-known infinite series of particle waves.This series, which is generally known as the Mie solution, contains in principle all thephysics of the problem for any size of particle and all wavelengths. For particles withsmaller radii than the wavelength, the few terms of the series need to be retained and oneacquires the satisfying knowledge. As the particle size increases, the series becomes slowlyconvergent, and extracting the important physics from it becomes a daunting task. Forthese reasons much of the theoretical work concerning Mie solution was of computationalnature. Although it can provide insightful graphical representation, it is still less effectiveto uncover the physical origins of many of the interesting phenomena.

1.1 Scattering by a single particle

We can qualitatively understand the physics of scattering by a single particle without goinginto computation. Consider an arbitrary particle, which we can divide in our gedankenexperiment into small regions. As applied oscillating field, like in our case, e.g. an incidentelectromagnetic wave, induces a dipole moment in each region. These dipoles oscillate atthe frequency of the field and therefore scatter secondary radiation in all directions. Ina particular direction, the total scattered field is obtained by superposing the scatteredwavelets, like at the point P in the figure (1.2). The scattering by the dipoles is coherent.The phase relations change for a different scattering direction; we therefore expect thescattered field to vary with scattering direction. If the particle is small compared to thewavelength of the incident wave, all the secondary wavelets are approximately in phase,i.e. we don’t expect much variation in scattering with direction. But as the particlesize increases, the number of possibilities of interference effects of the scattered waveletsincreases. Shape is also important, if the particle is distorted, all the phase relations aredifferent.

INCIDENT

OBSTACLE

P

SCATTEREDWAVELETS

Figure 1.2: Interaction of light with a single particle

The phase relations among the scattered wavelets depend on geometrical factors, like

4 Chapter 1. Introduction and motivation

scattering direction, size and shape. But the amplitude and phase of the induced dipolemoment for a given wavelength depend on the material of which the particle is composedof. In such case, it is important to know how the matter responds to oscillatory electro-magnetic fields.

Although the methods for calculating scattering by a single particle is physically thesame as outlined above, but the mathematical form may cast a shadow on the underlyingphysics. For certain classes of particles, the scattered field may be approximated bysubdividing the particle into dipole scatterers and superposing the scattered wavelets,which is normally done in known Rayleigh-Gans theory, where interaction among thedipoles are ignored. But in more general Mie theory, the interactions among the dipolesare accounted for, which shall be explained mathematically in Chapter 3.

1.2 Resume

In this thesis, I shall be describing the scattering of waves, both scalar and electromagnetic,from dielectric spheres in the X-ray regime. As mentioned above, we are treating here onlysingle particle scattering and elastic scattering. The latter condition means that there is noshift of frequency between the incident and the scattered radiation. This excludes quantummechanical phenomena such as Raman effect and fluorescence or Brillouin scattering whicharises from the Doppler shifts associated with the motion of the scattering particles. Therestriction to single particle scattering implies that the scattering particle is unaffected bythe presence of neighboring particles.

Chapter 2 is a review of X-ray scattering. Here, the overall scattering process alongwith the fundamentals of X-ray interactions with the matter is briefly explained. Thedetailed discussion is available in the popular literature like Attwood[5], Jens-Nielson[6].The X-ray scalar theory of scattering is developed by starting with the monochromaticwave field. The whole development is restricted to the single particle scattering, whichleads to the solution for a dielectric sphere.

Chapter 3 deals with the theory of scattering by spheres. We are very fortunate inhaving this exact theory in the sense of classical physics. The only restrictions are that thesubstance of which the particle is composed of is isotropic and that any variations in therefractive index are radially symmetric. This theory, as the title of this thesis is designatedafter it, is frequently called the Mie theory, owing to the work of Gustav Mie[7]. Mie’spaper was preceded by the independent solutions of a number of workers, starting with theelegant work of Lorenz[8] to the work of Debye[9], which can more historically be studiedin the appendix of this chapter. This chapter gives the rigorous solution on the basis ofthe explanation by Born & Wolf[10] with more of a inclination towards the results derivedby van de Hulst[1]. These are the results on which the modern computing technique ismostly based on. The alternative rigorous derivation is due to Bohren & Huffman[3].

Chapter 4 gives attempts towards the rationalization of the Mie theory for X-rays.The more adventurous, speculative, but yet possible analytical explanation towards theapproximation for X-ray region has been put forward, known as anomalous diffractiontheory. It is formerly due to the work of van de Hulst[1]. Since the elementary scalartheory is pretty analogous to the Rayleigh-Gans-Theory, which is valid for particles ofsmaller sizes, there is yet no proper analytical theory for the larger particle sizes, thoughnot large enough to be considered in geometrical optics region. Anomalous DiffractionTheory may fill up this gap, maybe qualifying for the problems we are considering here.The other half of the chapter deals with numerical solution, first with Mathematica, andlater with IDL. The respective algorithmic approach has been discussed.

Chapter 5 compares the result produced by Mathematica, which compares the Mietheory with scalar kinematical theory. The results are discussed extensively in the chapter.

Chapter 6 gives the outlook in studying the exact theory for the X-ray region. And inthe last chapter the source code of the written program is attached.

1.3. Notation 5

1.3 Notation

The selection of notation is always a big problem, when it comes to the writings onelectromagnetic scattering. The attempt is made to make this thesis self-consistent, I haveused right-handed coordinate systems and the angle of rotation is positive if the rotation isin clockwise direction when one is looking in the positive direction of the rotation axis, andI have adapted the time-harmonic factor exp(−iωt), which is the most preferred choicein various literature and publications, and implies a non-negative imaginary part of therelative refractive index. The primes on a function always denote differentiation, whereasprimes on variables distinguish them from each other. An asterisk on a variable or functionalways denotes complex conjugation and vectors are always denoted by boldface type. Thechoices made for Bessel, spherical Bessel, and Ricatti-Bessel functions are described in theappendix of Chapter 3. The comprehensive list of conventions, notations and results onMie theory prior to 1970 can be found in Kerker[11]. Whenever necessary, alternativeconvention is given, but I have shown a reluctance to abandon widely used and recognizedsymbols.

Bibliography

[1] H.C. van de Hulst. Light Scattering by Small Particles. Dover Publications Inc., NewYork, 1981.

[2] Milton Kerker. The Scattering of Light. Academic Press, 1969.

[3] Craig F. Bohren and Donald R. Huffman. Absorption and Scattering of Light bySmall Particles. Wiley-VCH, 2004.

[4] J. Tyndall. On the blue color of the sky, the polarization of skylight and the polar-ization of light by cloudy matter generally. Phil. Mag., 37:384–394, 1869.

[5] David Attwood. Soft X-Rays and Extreme Ultraviolet Radiation: Principles andApplications. Cambridge University Press, 1999.

[6] Jens Als-Nielson and Des McMorrow. Elements of Modern X-Ray Physics. JohnWiley & Sons, 2001.

[7] Gustav Mie. Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen.Ann. d. Phys., 25:377–445, 1908.

[8] L. V. Lorenz. Upon the light reflected and refracted by a transparent sphere. Videknsk.Selesk. Shrifter, 6:1–62, 1890.

[9] P. Debye. Der Lichtdruck auf Kugeln von beliebigem Material. Ann. d. Phys., 30:57–136, 1909.

[10] Max Born and Emil Wolf. Principles of Optics. Cambridge University Press, 7 edition,2002.

[11] Milton Kerker. The Scattering of Light, chapter 3, page 60. Academic Press, 1969.

7

Chapter 2

X-ray Scattering

9

10 Chapter 2. X-ray scattering: fundamentals

Diffraction by X-rays has been of great success in crystal structure determination, e.g.structures of DNA, reconstructed silicon surfaces, etc. For a perfectly ordered crystal,diffraction results in arrays of sharp Bragg reflection spots periodically arranged in re-ciprocal space. Analysis of the Bragg peak locations and their intensities gives variousinformation of the crystal lattice, like lattice type, symmetry group, unit cell dimensions,etc. On the other hand by crystals containing lattice defects such as dislocations, diffuseintensities are produced in addition to Bragg peaks. The distribution and magnitude ofdiffuse intensities are dependent on the type of imperfection present and the X-ray energyused in a diffraction experiment. Diffuse scattering is usually weak, and thus more difficultto measure, but it is rich in structure information that often cannot be obtained by otherexperimental means.

In this chapter we are going to address the fundamental principles of diffraction basedupon the kinematic diffraction theory for X-rays. We are not going to discuss dynamicaldiffraction since we are concentrating on single particle scattering. Dynamical diffractionis applied to diffraction from single crystals of high quality so that multiple scatteringbecomes significant and kinematic diffraction theory becomes invalid. But in practice,most X-ray diffraction experiments are carried out on crystals containing a sufficientlylarge number of defects that kinematic theory is generally applicable.

2.1 Overview of scattering processes

When a stream of radiation strikes matter, various interactions can take place, one ofwhich is the scattering process, that may well be described using the wave properties ofradiation. Depending on the wavelength of the incident radiation, scattering may occuron different levels like atomic, molecular or microscopic level. Although some scatteringevents are noticeable in daily life, others are more difficult to be taken true by naked eyes,esp. those involving X-rays or neutrons.

X-rays are electromagnetic waves with wavelengths ranging from a few hundredths ofan angstrom (A) to a few hundred angstroms. The conversion from wavelength to energyfor all photons is given in the following equation with wavelength λ in angstroms andenergy in kilo-electron volts (keV ):

λ(A) =c

ν=

12.40 keVE(keV )

A (2.1)

where c is the speed of light and ν the frequency. It is important to classify X-rays witha wavelength longer than a few angstroms as ”soft X-rays” whereas ”hard X-rays” withshorter wavelengths (≤ 1A) and higher energies (≥ 1keV ).

It is however important to understand what it means by ”scattering” and ”diffrac-tion”. The word ”scattering” refers to a deflection of a beam from its original directionby the scattering centers that could be electrons, atoms, molecules, voids, dislocations,etc. Whereas, the word ”diffraction” is generally defined as the constructive interferenceof coherently scattered radiation from regularly arranged scattering centers such as grat-ings, crystals, superlattices, and so on. Diffraction generally results in strong intensity inspecific, fixed directions in reciprocal space, which depends on the translational symmetryof the diffracting system. Scattering, on the other hand, often generates weak and diffuseintensities that are widely distributed in reciprocal space. E.g., interaction of radiationwith an amorphous substance is a ”scattering” process that shows broad and diffuse in-tensity maxima, whereas with a crystal it is a ”diffraction” event, as sharp and distinctpeaks appear.

2.2. X-ray interactions with matter 11

2.2 X-ray interactions with matter

X-rays are electromagnetic waves with wavelengths in the region of an A. It is a well-known fact that when X-rays traverse through the medium, photons are absorbed andintensity of the beam decreases exponentially over the distance x

I(r) = I0e−µx (2.2)

where µ is the linear absorption coefficient. This expression is known as Lambert-Beerlaw. In this process, an X-ray photon is absorbed by the atom, and the excess energy istransferred to an electron, which is expelled from the atom, leaving the atom ionized. Thisprocess is usually known as photoelectric absorption. Apart from that, the phase velocityof the electromagnetic wave changes. The real term of refractive index n of a materialis given as Re(n) = c0/cm, where c0 is the velocity in the vacuum and cm is the phasevelocity in the medium. Likewise the wave vector in the medium changes with the similarrelationship Re(n) = km/k0. Then the monochromatic plane electromagnetic wave in themedium can be written as

E(x, t) = E0 e−i(ωt−nk0x) (2.3)

where ω is the frequency of the light. Since we are here concerned only with the amplitudesas function of x, the time-dependent term e−iωt will be skipped. So we can write

E(x) = E0 eink0x (2.4)

The refractive index for electromagnetic waves displays resonant behavior at frequen-cies corresponding to electronic transitions in atoms and molecules. The binding energyof most of the electrons in an atom lies way below than that of 10keV . The electric vectorof the plane X-ray wave excites the electrons way above the resonance to small forcedoscillations. This way the phase velocity of the X-ray wave is a bit more than the velocityof light, hence the refractive index is less than 1. X-ray frequencies are usually higher thanall transition frequencies, except those involving the inner K- or maybe L-shell electrons.In the end the X-ray region Re(n) turns out to be less than unity.

The refractive index is usually given in the following form

n = 1− δ + iβ (2.5)

where δ is the dispersive term and β is the absorption term.The dispersive term δ, in fact, the deviation of Re(n) from unity is related to the

scattering properties of the medium. Each electron scatters a X-ray beam with Thomsonscattering amplitude r0, which is, in fact, the classical electron radius of e2/(4πε0mc2) ≈2.8 · 10−15m. It turns out that δ is proportional to the product of r0 and the electrondensity ρ. Hence, it can be given as

δ =2πρr0k2

0

(2.6)

δ is normally of the order 10−6 . . . 10−5.To get the imaginary term, we substitute the definition of the refractive index in (2.4),

then we getE(x) = E0 e

i(1−δ+iβ)k0x = E0 ei(1−δ)k0xe−βk0x (2.7)

For the intensity we getI(x) = |E(x)|2 = I0 e

−2βk0x (2.8)

Comparing this equation with (2.2), we get for the absorbing term of the refractive index

β =µ

2k0(2.9)

β is normally of the order less than of δ. These optical constants varies from material tomaterial.∗

∗the optical constants for X-ray interactions with matter can be obtained here:[1]


2.3 Elementary scalar theory of scattering

We consider a monochromatic electromagnetic plane wave incident on a linear, isotropic,nonmagnetic scattering medium of volume V , as shown in figure (2.1) as a series of parallelarrows. Although knowing that we are strictly working within the context of wave optics,nevertheless ’ray paths’ such as the one which is shown in the figure can be visualized. Ifwe were working within the framework of geometric optics, these lines would represent thetrajectory of a certain ray in the absence of the scatterer. However, within the framework ofscalar wave optics, this path may be defined such that the phase gradient of the unscatteredfield is everywhere parallel to the trajectory. Here we assume that the scatterers aresufficiently weak so as to negligibly perturb the ray paths which would have existed in thevolume occupied by the scatterer had the scatterer been absent. In such kind of situationthe phase and amplitude of the disturbances at the surface, where rays are exiting, canbe expressed in terms of the phase and amplitude shifts accumulated as the disturbancetraverses a given ray path connecting the entry and exit surfaces, with the ray pathscorresponding to those that would have existed if the scattering volume were replaced bythe surrounding medium.

INCIDENT

V

Figure 2.1: Monochromatic plane waves are incident from the left. Two ray paths areshown, one associated with unscattered beam, in the absence of the volume and otherwith scattered beam, in the presence of the scatterer.

We assume that there are no sources in V and that the space-dependent part of thecomplex electric field will satisfy the following equation, specialized to a monochromaticwavefield[2]:

∇2E(r, ω) + k2ε(r, ω)E(r, ω) + ~∇ [E(r, ω) · ~∇ ln ε(r, ω)] = 0 (2.10)

where k = ω/c, E is the electric vector and ε(r, ω) is the dielectric constant (permittivity).We note that the last term on the left of the equation couples the Cartesian components

of the electric field, which makes the treatment of scattering based on this equation quitecomplicated. It can be simplified to the inhomogeneous Helmholtz equation with thefollowing assumptions:

• the medium is non-magnetic

• the electrical permittivity, hence the refractive index, is time-independent and slowlyvarying over spatial length scales comparable with the wavelength of the radiationwith which it interacts

• we may work with a single complex quantity to quantify the disturbance due to themonochromatic electromagnetic field

• the effects of inelastic scattering may be ignored

2.3. Elementary scalar theory of scattering 13

Under these circumstances the last term of the left-hand side of the equation may beneglected, delivering the reduced form of the equation:

∇2E(r, ω) + k2n2(r, ω)E(r, ω) = 0 (2.11)

with Maxwell formula ε(r, ω) = n2(r, ω), where n(r, ω) denotes the refractive index ofthe medium. Better understanding of the general behaviour of the scattered field maybe obtained by studying the behaviour of the solution of the equation (2.10) for a singleCartesian component of E(r, ω), i.e. the problem concerning the single point scatterer.We shall denote this component by U(r, ω), there we obtain

∇2U(r, ω) + k2n2(r, ω)U(r, ω) = 0 (2.12)

The same equation can be rewritten in the following way

∇2U(r, ω) + k2U(r, ω) = −4πF (r, ω)U(r, ω), (2.13)

whereF (r, ω) =

14πk2[n2(r, ω)− 1] (2.14)

The function F (r, ω) is usually called the scattering potential of the medium. We shallexpress U(r, ω) as the sum of the incident field and the scattered field

U(r, ω) = U (i)(r, ω) + U (s)(r, ω) (2.15)

The incident field, being the plane wave, satisfies the Helmholtz equation throughoutthe space

(∇2 + k2)U (i)(r, ω) = 0 (2.16)

Whereas we get for the scattered field

(∇2 + k2)U (s)(r, ω) = −4πF (r, ω)U(r, ω) (2.17)

Our ultimate aim is to convert this differential equation into an integral equation, whichwe shall do after introducing Green functions in the next section.

2.3.1 Outgoing Green function

Green functions are a powerful and widely applicable tool in mathematical physics. Herewe shall use them to study the single Cartesian component of the field defining the Greenfunction to be the field that is scattered from the volume. We can decompose the totalfield, in the presence of the scatterer, as the sum of the two fields: the field which wouldhave existed in the absence of the scatterer, and the scattered field. The outgoing Greenfunction G(r − r′) is by definition equal to the scattered field due to a point scattererlocated at r′, as a function of the position coordinate r in three-dimensional space. So letG(r−r′) be the Green function of the Helmholtz operator, obeying the following equation:

(∇2 + k2)G(r− r′, ω) = −4πδ(3)(r− r′) (2.18)

where δ(3)(r− r′) is the three-dimensional Dirac delta function.Now we have to obtain free-space Green function of the Helmholtz operator. Since free

space is homogeneous and isotropic, the scattered field observed at a given point dependsonly on the relative position of the observation point with respect to the scatterer∗. weshall now introduce polar coordinates centered at r, with radial distance from the origin

∗For rigorous method of obtaining Green function using complex analysis can be found in [3]


denoted by r. Using Laplacian in spherical polar coordinates and assuming Green functionbeing rotationally symmetric, we can get for equation (2.18)

1r2

d

dr

[r2d

drG(r)

]+ k2G(r) = −4πδ(r) (2.19)

After multiplying both sides by r, thereby eliminating Dirac delta function using therelation

∫∞−∞ f(x)δ(x− a)dx = f(a), we get

1r2

d

dr

[r2d

drG(r)

]= −k2G(r) (2.20)

d2

dr2[rG(r)] = −k2G(r) (2.21)

It shows that rG(r) is the oscillatory eigenfunction of d2/dr2, so that rG(r) = exp(±ikr).Which yields the explicit form of the free-space Green function:

G(r) =exp(±ikr)

r(2.22)

In our context, we should choose positive sign in the exponent, since we are concerned witha outgoing spherical wave. So we can not define the outgoing free-space Green function as

G(r− r′, ω) =exp(ik|r− r′|)

|r− r′|(2.23)

2.3.2 Integral-equation formulation

In this section, we shall rewrite the differential equation (2.17) as an integral equation. Itis mandatory to note that the integral equation is more restrictive than the differentialequation from which it is obtained, in the sense that the latter must be supplemented byappropriate boundary conditions, whereas the former has those conditions within itself.This integral equation will be used to derive approximate expressions later in the chapter.

Let us multiply the equation (2.17) by G(r−r′, ω) and the equation (2.18) by U (s)(r, ω)and subtract them from each other, which gives[4]

U (s)(r, ω)∇2G(r− r′, ω)−G(r− r′, ω)∇2U (s)(r, ω) =

4πF (r, ω)U(r, ω)G(r− r′, ω)− 4πU (s)(r, ω)δ(3)(r− r′, ω)(2.24)

We now integrate both sides of (2.24) with respect to r′ throughout a volume VR,bounded by a large sphere SR of radius R, centered on the origin O in the region of thescatterer. After applying Green’s theorem to convert the volume integral on the left intoa surface integral, we obtain

U (s)(r, ω) =∫

VF (r′, ω)U(r′, ω)G(r− r′, ω)− 1

4π∫SR

[U (s)(r′, ω)

δG(r− r′, ω)δn′

−G(r− r′, ω)δU (s)(r′, ω)

δn′

]d SR

(2.25)

where δ/δn′ denotes the differentiation along the normal n′ to SR. The first integralis taken only over the scattering volume V , since the scattering potential F vanishesthroughout the exterior of V , as can be seen from the equation (2.14)

Now we shall substitute the G(r − r′, ω) with the free-space Green function obtainedin the last section. It seems plausible that sufficiently far away from the scatterer, thescattered field will behave like an outgoing spherical wave. So we can conclude that in the


O

V

R

VR SR

n'

Figure 2.2: Illustration to the derivation of (2.25)

limit for R → ∞, the surface integral will not contribute to the total field. So we get forscattering field

U (s)(r, ω) =∫F (r′, ω)U(r′, ω)

exp(ik|r− r′|)|r− r′|

d3 r′ (2.26)

Supposing that the incident field on the scatterer is a monochromatic plane wave ofunit amplitude and frequency ω, the time-independent part of the incident field is givenas

U (i)(r, ω) = eik0·r, (2.27)

so we get the final integral equation

U(r, ω) = eik0·r +∫

VF (r′, ω)U(r′, ω)


d3 r′ (2.28)

This equation determines the total field U(r, ω), including the incident and scattered fieldand is usually called as the integral equation of potential scattering. Note that the desiredfield U(r, ω) appears on the both sides of the equation, which is why we call it as anintegral equation rather than an integral expression for U(r, ω).

2.3.3 First Born approximation

It is not easy to obtain solutions of the integral equation of potential scattering in a closedform. These, then, should be solved by some approximate technique. The most commonamong different approximate techniques is perturbation. In this technique, the successiveterms in the perturbation expansions are obtained by iteration from the previous ones,provided the medium scatters weakly.

From expression (2.14) it can said that a medium will scatter weakly if its refractiveindex differs only slightly from unity, which is exactly our case. If one can assume thatthe X-ray perturbation inside the scattering volume is only slightly different from theperturbation that would have existed at each point r′ in that volume in the absence of thescatterer, then one can replace U(r′, ω) by the unscattered perturbation U (i)(r′, ω), insidethe integral. From now on we shall not display the dependancy of the various quantitieson the frequency ω for convenience. The expression looks like,

U(r) = eik0·r +∫

VF (r′)eik0·r′ exp(ik|r− r′|)

|r− r′|d3 r′ (2.29)

Now this is no longer an integral equation, but rather an approximate expression for thetotal wave-field, which is popularly known as the first Born approximation.


It can be noticed that the first Born approximation corresponds to a single-scatteringtheory in which the incident wave-field is either not scattered at all, or scattered only onceby a single point within the sample. Such single-scattering approximations are knownas kinematical theories. If the incident wave-field may be scattered multiple times, thedevised theories on this scenario are known as dynamical theories.

NO

V

r'

|r-r'|

r

P

Q

Figure 2.3: Illustration to the notation in the given Born approximation

Now we treat the special case of the first Born approximation which corresponds tothe scattered radiation very far from all scatterers. The treatment given here is in directmathematical correspondence to the treatment of non-relativistic potential scattering inquantum mechanics. From the figure (2.3), let Q be a typical point in the scatteringvolume V and P a point far away from it. Let r′ be the position vector of Q and r = rrbe the position vector of P , where r is the unit vector in the direction of r. Let N be thefoot of the perpendicular dropped from Q onto the line OP . Like in our case, when r islarge enough, then

|r− r′| ∼ r − r · r′ (2.30)

Hence we obtain the following approximation form for the spherical wave scattered froma point r′ within the scattering volume:


=exp[ik

√(r− r′) · (r− r′)]|r− r′|

(2.31)

=exp[ik

√(|r|2 − 2r · r′ + |r′|2]|r− r′|

(2.32)

≈exp[ik

√(|r|2 − 2r · r′]r

(2.33)

=exp[ikr

√1− 2r−2r · r′]r

(2.34)

After making the binomial approximation√

1− 2r−2r · r′ ≈ 1− r−2r · r′, we get

exp(ik|r− r′|)|r− r′

∼ exp[ikr]r

exp[−ikr · r′] (2.35)

Substituting it in the equation (2.29), we get the approximation

U(r) ∼ eik0·r + f(Q)exp[ikr]

r(2.36)


where,

f(Q) ≡∫F (r′)e−iQ·r′dr′ (2.37)

Q ≡ kr− k0. (2.38)

The function f(Q) is known as scattering amplitude and it has important physical impli-cations. Because the scattered radiation is given by a distorted spherical wave originatingfrom the point, but still behaving like an outgoing spherical wave at very far distance, andthe form of distortion is quantified by the envelope f . f multiplies the undistorted outgo-ing spherical wave exp(ikr)/r in order to form the distorted spherical wave emerging fromthe scatterer. The value of f is the three-dimensional Fourier transformation of F (r′). Soit can be stated that within the accuracy of the first Born approximation, the scatteringamplitude f in the far zone of the scatterer depends entirely on one and only one Fouriercomponent of the scattering potential, namely the vector

Q = k− k0 (2.39)

2.3.4 Scattering by a sphere

In order to find a differential scattering cross section of the sphere, we shall calculate thescattering amplitude of a sphere using the equation (2.37). We can rewrite the equationfor a sphere as

f(Q) =∫

Vsph

F (r)e−iQ·r (2.40)

Here is the integration over the volume of the scatterer. The scattering potential F (r) isdefined like before in (2.14). Let the sphere be homogeneous, then we can redefine thescattering potential as

F (r) =

14πk

2(n2 − 1) r ≤ R ∵ n(r ≤ R) = n,

0 r ≥ R ∵ n(r ≥ R) = 1.(2.41)

where R is the radius of the sphere.Thus for f(Q) in spherical coordinates we get

f(Q) =k2

4π

∫ ∞

0r2(n2(r)− 1)dr

∫ 2π

0dφ

∫ +1

−1d(cos θ)ei|Q||r| cos θ (2.42)

=k2

2iQ

∫ ∞

0

r2

r(n2(r)− 1)(eiQr − e−iQr)dr (2.43)

=k2

Q(n2 − 1)

∫ R

0r sin(Qr)dr (2.44)

= k2(n2 − 1)sin(QR)−QR cos(QR)

Q3(2.45)

=34πk2(n2 − 1)V

sin(QR)−QR cos(QR)(QR)3

(2.46)

Now we can approximate this expression for the refractive index in X-ray region. Withthe help of the discussion in the section sec. 2.2, we can write

n2 ≈ 1− 2δ + δ2 ' 1− 2δ (2.47)

Substituting it in (2.46), we get

f(Q) =34πk2 · 2δV sin(QR)−QR cos(QR)

(QR)3(2.48)

Replacing δ with (2.6) from the section sec. 2.2, we finally get for scattering amplitude

f(Q) = 3ρr0Vsin(QR)−QR cos(QR)

(QR)3(2.49)


2.3.5 Scattering cross section

The power radiated in the scattered direction, per unit solid angle, per unit incident flux(power per unit area) in the incident direction, is a quantity with dimensions of areaper unit solid angle, which is known as differential scattering cross section. The properdefinition is the following way:

dσ

dΩ=

Energy radiated/unit time/unit solid angleIncident energy flux/unit area

(2.50)

So in this example of scattering by a sphere, the differential scattering cross section isgiven by

dσ

dΩ= |f(Q)|2 (2.51)

After substituting (2.49) in the equation above, we get

dσ

dΩ= r20ρ

2V 29(sin(QR)−QR cos(QR))2

(QR)6(2.52)

Bibliography

[1] Eric Gullikson. X-ray interactions with matter. http://www-cxro.lbl.gov/.

[2] Max Born and Emil Wolf. Principles of Optics, chapter 1, page 11. Cambridge Uni-versity Press, 7 edition, 2002.

[3] David Paganin. Coherent X-Ray Optics, chapter 2, page 80. Oxford University Press,1 edition, 2006.

[4] Max Born and Emil Wolf. Principles of Optics, chapter 13, page 697. CambridgeUniversity Press, 7 edition, 2002.

19

Chapter 3

Mie Scattering

21

22 Chapter 3. Mie Scattering

3.1 Introduction

In 1902, Gustav Mie moved to Greifswald, where he assumed a special professorship post(Extraordinarius) at the University of Greifswald, where, in fact, he wrote the famous69-page paper on particle light scattering, published in 1908 in the Annalen der Physik,bearing the title ”Beitrage zur Optik Truber Medien, speziell kolloidaller Metallosungen”.It appears that Mie’s 1908 paper represents a single major involvement with the subjectof particle light scattering and absorption. In these years in Greifswald, having beenconfounded with the idea of developing a comprehensive Theory of Matter, it was the ex-perimental investigations on colloidal gold suspensions by a student (Walter Steubing) atthe Greifswald Institut, which actually triggered his interest to develop a rigorous theoret-ical interpretation of the empirical results of the student. This theoretical interpretationwas based on Maxwell’s equations, on whose ramification Mie had concentrated his at-tention since at least 1896. He proceeded further beyond the original intention of merelyexplaining the colors of colloidal gold observed by Steubing. The thorough understandingof the theory using the behavior of ellipsoidal particles, as mentioned and wished by Miein conclusion, was unfortunately left behind, having been entangled in his other theoret-ical works. The reason was that this great work was underestimated by its author andcontemporary scientists. The study lies in the trap of various cylindrical functions andLegendre polynomials. It was not until the dawn of computational machines, when thesubject was taken for serious investigation. In this chapter we shall be voyaging throughthe original analytical work done on light scattering by small particles.

3.2 Derivation

The Mie solution to the scattering of light at particles of any size is a classical electro-dynamics problem. As will be seen in the following sections, the scattering of light by ahomogeneous sphere is treated in a general way by the formal solution of Maxwell’s equa-tion with the appropriate boundary conditions. Although I shall be, to certain extent,employing the procedure followed by Gustav Mie[1], I shall also be applying the similartechnique developed quite independently by Debye[2] a year later. It is, in fact, the so-lution of Maxwell’s equation with certain boundary conditions using Debye potentials forthe electromagnetic field[3]. The definition of the Debye potential and historical perspec-tive to arrive at this idea, subsequently the solution given by Gustav Mie, is explainedexplicitly in the appendix (see 3.3.1).

3.2.1 Maxwell’s equation

The treatment of scattering of light by small particles is a problem in electromagnetictheory. We are adopting here a macroscopic approach to the problem, i.e. the Maxwellequations for the macroscopic electromagnetic field at interior points in matter, which canbe written as,

curl H =4πc

I +1c

dDdt

(3.1)

curl E = −1c

dHdt

(3.2)

div D = 4πρ (3.3)div H = 0 (3.4)

where E is the electric field, H is the magnetic field and D is the electric displacement.The charge density ρ and current density I are associated with free charges. Although itis shown convincingly by Purcell[4] that it is not possible to unambiguously distinguishbetween free and bound charges in matter, we shall assume that the ambiguity in the

3.2. Derivation 23

meanings of free and bound leads to no observable consequences in the concerned problemin the following sections.

The Maxwell’s equations must be supplemented with constitutive relations, which hasthe form in our disscussion

D = εE (3.5)I = σE (3.6)

where ε is the permittivity of free space and σ is the conductivity. These coefficients ε, σdepend on the medium under consideration, but here it will be assumed to be independentof the fields, position and direction. Now we have complete set of Maxwell’s equations.As per the method of Mie, we are going to write them in polar coordinates, but the littlemore discussion of periodic phenomena, i.e. nature of time-harmonic fields is required.

Es1

Es2

Ei1Ei2

θ

Φ

X

Y

Z

k

E

H

Figure 3.1: Scattering of a incident wave from a spherical target located at the origin.

3.2.2 Time-harmonic fields

Let A be the general time-harmonic field, then it takes the form

A = X cosωt+ Y sinωt, (3.7)

where ω is the angular frequency. But in this chapter, we shall be dealing with complexrepresentation of the this field and it is convenient to work with complex representation.Then, we can say A = ReAc, where

Ac = Ze−iωt, Z = X + iY (3.8)

It might be quite important to notice that the same real field A can also be writtenas A = ReA?

c, where A?c = Z?eiωt is, in fact, the complex conjugate of Ac. It says that


there are two different choices of representing time-harmonic fields, namely e−iωt and eiωt.We take the usual convention, seen in many popular books, i.e. that of e−iωt.

Now let us assume here the time-dependency is taken care by the term exp(−iωt), andwe substitute the constitutive relations (3.5) in the Maxwell’s equations (3.1). Then wecan say that the both time independent parts of the electric and magnetic vectors outsideand inside the scattering medium satisfy the following relationship

curl H = −k1E (3.9a)curl E = k2H (3.9b)

where

k1 =iω

c

(ε+ i

4πσω

)(3.10a)

k2 =iω

c(3.10b)

Now we write the same equations in polar coordinates using the relations explained in3.3.2, and we get

−k1Er =1

r2 sin θ

[∂(r sin θHφ)

∂θ− ∂(rHθ)

∂φ

](3.11a)

−k1Eθ =1

r sin θ

[∂Hr

∂φ−∂(r sin θHφ)

∂r

](3.11b)

−k1Eφ =1r

[∂(rHθ)∂r

− ∂Hr

∂θ

](3.11c)

k2Hr =1

r2 sin θ

[∂(r sin θEφ)

∂θ− ∂(rEθ)

∂φ

](3.12a)

k2Hθ =1

r sin θ

[∂Er

∂φ−∂(r sin θEφ)

∂r

](3.12b)

k2Hφ =1r

[∂(rEθ)∂r

− ∂Er

∂θ

](3.12c)

These equations will now be our point of departure in scattering problems.

3.2.3 Boundary conditions

The scattering medium we are considering is a sphere. Now let us give prefix I to themedium surrounding the sphere and prefix II to the sphere. If the refractive index of boththe media is infinite, then surface current exist, else there shall be no surface current toconsider. So we can put the boundary condition that the tangential component of E andH shall be continuous across the surface of the sphere:

IEtang = IIEtang (3.13a)IHtang = IIHtang (3.13b)

which can be interpreted in spherical polar coordinates as

IEθ = IIEθ,IEφ = IIEφ (3.14a)

IHθ = IIHθ,IHφ = IIHφ (3.14b)

3.2. Derivation 25

3.2.4 Solution of differential equations with boundary conditions

The equations (3.11) and (3.12) with boundary conditions (3.14) are to be solved. What weare trying to solve is the vector wave equation. So we will start with the some explanation,which will help us determine Debye Potentials for our problem.

Let a complex-valued vector field Π is defined in a domain D and satisfies

∇2Π + k2Π = ∇φ (3.15)

where φ is an arbitrary function, normally called a Hertz vector for D. Now the equation(3.15) may be written as

∇× (∇×Π)− k2Π = ∇(∇ ·Π− φ) (3.16)

from which each Hertz vector gives rise to a pair of associated vector wave functions,

M = k(∇×Π), N = ∇× (∇×Π). (3.17)

These functions M and N have all the required properties of an electromagnetic field.They satisfy the vector wave equation, they are divergence-free, the curls are proportionalto each other. This way the problem of finding solutions to the field equations reducesto the comparatively simpler problem of finding solutions to the scalar wave equation.Now, we shall consider a scalar function ψ, a generating function for the vector harmonicsM and N for the domain D and r a position or guiding vector relative to an origin butoutside D, then we can write

Π = ψr (3.18)

which defines a radial Hertz vector for D

M = k(∇× ψr) (3.19)

Now, M is a solution to the vector wave equation in spherical polar coordinates. Inproblems involving spherical symmetry, we shall take M given in (3.19) and the associatedN as the fundamental solution to the field equations. The scalar wave equation in sphericalpolar coordinates is

1r

∂2 (rψ)∂r2

+1

r2 sin θ∂

∂θ

(sin θ

∂ψ

∂θ

)+

1r2 sin2 θ

∂2ψ

∂φ2+ k2ψ = 0 (3.20)

But before that, we must define these vector wave functions M and N. From (3.18),we can write

∇2Π + k2Π = 2∇ψ (3.21)

The scalar wave function ψ is, in fact, called a Debye potential for the vector wave functionsdefined by (3.17). To be more precise, ψ is called an electric Debye potential for M anda magnetic Debye potential for N.

These vector wave functions can be represented the following way

M ≡ (eE,eH) and N ≡ (mE,mH) (3.22)

One can divide all the solutions of the equations in three groups:

• The first one represents the waves, which are produced through electrical oscillationof the sphere, i.e.

eEr = Er,eHr = 0 (3.23)

• The second group represent waves, which are produced through magnetic oscillationof the sphere, i.e.

mEr = 0, eHr = Hr (3.24)


• The third group contains the integrals of Maxwell’s equations, which shows theregular periodic oscillations, which, in fact, can be attained through addition ofintegrals of first two groups, i.e. addition of all polar components respectively

Let us consider a first case. For Hr = 0 equations (3.11b) and (3.11c) become

ke1Eθ =

1r

∂

∂r(r eHφ)

ke1Eφ = −1

r

∂

∂r(r eHθ)

(3.25)

Substituting it back in (3.12b) and (3.12c) we get(∂2

∂ r2

)(r eHθ) = − k1

sin θ∂ eEr

∂φ(∂2

∂ r2

)(r eHφ) = +k1

∂ eEr

∂θ

(3.26)

Equation (3.26) along with the equation (3.11a) constitute a system of equations foreEr,

eHθ,eHφ. The only solutions satisfying the condition ’div eH = 0’ represent physical

fields. In spherical polar coordinates, taking into consideration the assumption eHr = 0,this condition becomes

∂

∂θ(sin θeHθ) +

∂

∂φ(eHφ) = 0 (3.27)

On the substitution from (3.25), the equation (3.12a) becomes

0 =1

k21r

2 sin θ∂

∂r

[∂

∂θ(r sin θeHθ) +

∂

∂φ(eHφ)

](3.28)

It is well satisfied from the relation (3.27). Analog to the method above, one can obtainthe solutions to the second case. Now, I shall write the generating function ψ with respectto vector wave functions M and N as Debye potentials eΠ and mΠ, respectively.

Like done above, we shall calculate for the scalar potential eΠ in the first case ofeEr = Er and eHr = 0. Since eHr = 0, eEφ and eEθ can be represented as

eEφ =1

r sin θ∂2r eΠ∂r∂φ

, eEθ =1r

∂2r eΠ∂r∂θ

(3.29)

So the equation (3.25) may be satisfied by

eHφ = k1∂ eΠ∂θ

=k1

r

r∂ eΠ∂θ

eHθ = − k1

sin θ∂ eΠ∂φ

= − k1

r sin θr∂ eΠ∂φ

(3.30)

After substituting this equation in (3.11a) we get

eEr = − 1r sin θ

∂

∂θ

(sin θ

∂ eΠ∂θ

)+

1sin θ

∂2 eΠ∂φ2

(3.31)

Substitution from (3.30) and (3.31) into (3.26) gives two equations, the first of whichexpresses the vanishing of the φ derivative, the second the vanishing of the θ derivative ofthe same expression. So the best way is to equate them to zero which leaves us with thefollowing expression,

1r

∂2 (reΠ)∂r2

+1

r2 sin θ∂

∂θ

(sin θ

∂ eΠ∂θ

)+

1r2 sin2 θ

∂2 eΠ∂φ2

+ k2 eΠ = 0 (3.32)

3.2. Derivation 27

That means the equation (3.31) becomes

eEr =∂2 (reΠ)∂r2

+ k2r eΠ (3.33)

After substituting all these equations in main Maxwell’s equations with polar coordi-nates, in (3.11) and in (3.12), we obtain a solution of our set of equations.

In a similar way, one can find that (3.32) is also valid in case of potential mΠ. Thecomplete solution of the field equations can be obtained by adding the two fields.

As we have seen, both the potentials are solutions of the differential equation (3.32),which, in fact, is the following wave equation, but written in polar coordinates,

∇2Π + k2Π = 0 (3.34)

For the components Eθ, Eφ,Hθ,Hφ to be continuous over the spherical surface r = a,it is sufficient that the following four quantities shall be continuous over the same surface,

k1reΠ, k2r

mΠ,∂

∂r(r eΠ),

∂

∂r(r mΠ) (3.35)

So the boundary condition splits into examining independent condition for the potentialseΠ and mΠ. That means, the whole problem is reduced to the problem of finding twomutually independent solution of the wave equation with boundary conditions given above.

3.2.5 Series expansions for the scalar potentials

Let us consider that the solution of the wave equation comprises of undetermined coeffi-cients, which can be determined by using boundary conditions. So the solution takes aform

Π = R(r)Θ(θ)Φ(φ) (3.36)

After substituting in (3.32), one can see that these coefficients must satisfy the followingordinary differential equations

d2(rR)dr2

+(k2 − α

r2

)rR = 0 (3.37a)

1sin θ

d

dθ

(sin θ

dΘdθ

)+(α− β

sin2 θ

)Θ = 0 (3.37b)

d2Φdφ2

+ βΦ = 0 (3.37c)

where α and β are integration constants.Taking into account that the field E, H is a single valued function of position, Π must

also be single valued, imposing conditions on Θ and Φ. E.g., one can write down thegeneral equation for the last ODE in (3.37c)

a cos(√βφ) + b sin(

√βφ) (3.38)

But due to the condition of single-valuedness demanding

β = m2, (m = integer) (3.39)

we get the solutionΦ = am cos(mφ) + bm sin(mφ) (3.40)

The equation (3.37b) is the equation of spherial harmonics. In this case, a necessaryand sufficient condition for a single valued solution is

α = l(l + 1), (l > |m|, integer) (3.41)


with the introduction of the following variable ‡

ξ = cos θ (3.42)

The equation then takes the form

d

dξ

(1− ξ2)

dΘdξ

+l(l + 1)− m2

1− ξ2

Θ = 0 (3.43)

The solution to this differential equation is the well-known associated Legendre func-tions

Θ = Pml (ξ) = Pm

l (cos θ) (3.44)

In an attempt to find a solution to the remaining first equation (3.37a) with thefollowing substitutions

kr = ρ, R(r) =1√ρZ(ρ) (3.45)

we obtain the Bessel equation §

d2Z

dρ2+

1ρ

dZ

dρ+

1−

(l + 12)2

ρ2

Z = 0 (3.46)

The solution of this equation is the general cylindrical function Z = Zl+ 12(ρ) of the

order l + 12 , which gives us the complete solution to be

R =1√krZl+ 1

2(kr) (3.47)

Each cylindrical function may be expressed as a linear combination of two cylindricalfunctions of standard type, e.g. the Bessel function Jl+ 1

2and the Neumann function Nl+ 1

2.

Let us introduce the convenient form of functions

ψl(ρ) =√πρ

2Jl+ 1

2and χl(ρ) = −

√πρ

2Nl+ 1

2(3.48)

The functions χl(ρ) have singularities at the origin ρ = 0, whereas the functions ψl(ρ)are regular in every finite domain of the ρ-plane, including that of origin. That is why wewill use the functions ψl(ρ) for representing wave inside the sphere.

So the general integral of the equation (3.37a) may be given as

rR = clψl(kr) + dlχl(kr) (3.49)

Taking cl = 1, dl = −i, we have

rR = ζ(1)l (ρ) = ψl(ρ)− iχl(ρ) =

√πρ

2H

(1)

l+ 12

(ρ) (3.50)

where H(1) is the Hankel function. This has been acquired through immediate con-sequences of the well known relationship between the Bessel, Neumann and Hankel func-tions, that Jp + iNp = H

(1)p . These Hankel functions have distinguishing property among

cylindrical functions that they vanish at infinity in the complex plane. This very Hankelfunction obtained vanishes in the half-plane of the positive imaginary part of ρ, hencesuitable for the representation of the scattered wave.

‡Cf. [5]§Cf. [6]

3.2. Derivation 29

Now we put all the solutions of the equations (3.37) in the equation (3.36), to obtainthe solution of the wave equation:

rΠ = r

∞∑l=0

l∑m=−l

Π(m)l

=∞∑l=0

l∑m=−l

clψl(kr) + dlχl(kr)P

(m)l (cos θ)

am cos(mφ) + bm sin(mφ)

(3.51)

where am, bm, cl, dl are arbitrary constants, commonly known as Mie Coefficients.

3.2.6 Mie coefficients

Definitely we would like to know how the various observable quantities vary with the sizeand optical properties of the sphere and the nature of surrounding medium. To have somenumerical examples in hand, we should obtain explicit expressions for these coefficientsam, bm, cl, dl. These are four unknown coefficients, i.e. pure mathematically speaking, weneed four independent equations, which are best obtained from four boundary conditions(3.14). These four expressions have to have equal values at either side of the boundarysurface r = a. These boundary conditions can be written as:

∂

∂rr(eΠ(i) + eΠ(s))(r=a) =

∂

∂rr eΠ(w)(r=a) (3.52a)

∂

∂rr(mΠ(i) + mΠ(s))(r=a) =

∂

∂rr mΠ(w)(r=a) (3.52b)

1k1r(eΠ(i) + eΠ(s))(r=a) = IIk1reΠ(w)(r=a) (3.52c)1k2r(mΠ(i) + mΠ(s))(r=a) = IIk2rmΠ(w)(r=a) (3.52d)

where the superscripts (i), (s), and (w) represents incident wave, scattered wave, andinside (within sphere) wave, respectively.

We must express the potentials of incident, scattered and inside wave in a series formof (3.51). We shall first start with a incident wave, where we write the incident wave intospherical polar coordinates with help of 3.3.2:

E(i)r = ei

Ikr cos θ sin θ cosφ, H(i)r =

i Ik1k2

eiIkr cos θ sin θ sinφ, (3.53a)

E(i)θ = ei

Ikr cos θ cos θ cosφ, H(i)θ =

i Ik1k2

eiIkr cos θ cos θ sinφ, (3.53b)

E(i)θ = −ei Ikr cos θ sinφ, H

(i)θ =

i Ik1k2

eiIkr cos θ cosφ, (3.53c)

(3.53d)

From the previous section we have Er = eEr + mEr, and with the use of (3.33) wecan write:

eiIkr cos θ sin θ cosφ =

∂2 (r eΠ(i))∂r2

+ Ik2r eΠ(i) (3.54)

Using Bauer’s formula

eiIkr cos θ =

∞∑l=o

il(2l + 1)ψl( Ikr)

IkrPl(cos θ) (3.55)

and using the following identities

eiIkr cos θ sin θ ≡ − 1

i Ikr

∂

∂θ(ei

Ikr cos θ) (3.56)

∂

∂θPl(cos θ) ≡ −P (1)

l (cos θ); P(1)0 (cos θ) ≡ 0 (3.57)


we can finally write the left hand side of the equation (3.54)

eiIkr cos θ sin θ cosφ =

1( Ikr)2

∞∑l=o

il−1(2l + 1)ψl( Ikr)Pl(cos θ) cosφ. (3.58)

Let us take similar series as a supposed solution of the equation (3.54)

r eΠ(i) =1

Ik2

∞∑l=o

αlψl( Ikr)Pl(cos θ) cosφ. (3.59)

Now substituting these two series back in the equation (3.54), we obtain the followingrelation

αl

Ik2ψl( Ikr) +

∂2 ψl( Ikr)∂r2

= il−1(2l + 1)

ψl( Ikr)r2

(3.60)

Interestingly we have encountered a similar kind of equation in the previous section,which we can write appropriately for this situation with the help of the equation (3.49)and with cl = 1, d = 0, which yields

ψl( Ikr) = rR, (3.61)

a solution of the equation:

d2ψl

dr2+(

Ik2 − α

r2

)ψl( Ikr) = 0, (3.62)

provided that α = l(l + 1). Now we shall compare (3.62) with (3.60), giving

αl = il−1 2l + 1l(l + 1)

. (3.63)

The similar calculation follows for the magnetic potential of the incident wave mΠ(i).Thus, we can write the two potentials of the incident wave as

r eΠ(i) =1

Ik2

∞∑l=o

il−1 2l + 1l(l + 1)

ψl( Ikr)Pl(cos θ) cosφ (3.64a)

r mΠ(i) =1

Ik2

∞∑l=o

il1k1k2

2l + 1l(l + 1)

ψl( Ikr)Pl(cos θ) sinφ (3.64b)

As we can see from the equation above that the equations (3.52) can only be satisfiedif the terms with m = 1 occurs in the expansion (3.51) for the remaining potentials Π(s)

and Π(w) and if a1 = 0 for the magnetic and b1 = 0 for the electric potential.As explained in the previous section, only functions ψl are appropriate for Π(w), then

we get

r eΠ(w) =1

IIk2

∞∑l=o

il−1 2l + 1l(l + 1)

eAlψl( IIkr)Pl(cos θ) cosφ (3.65a)

r mΠ(w) =1

IIk IIk2

∞∑l=o

il1k1k2

2l + 1l(l + 1)

mAlψl( IIkr)Pl(cos θ) sinφ (3.65b)

We have also seen in the previous section that the function ζ(1) is appropriate torepresent the potential for a scattered wave. It can be seen from the equation (3.50) thatfor large ρ, H(1) behaves as eiρ/

√ρ, which says ζ(1)

l behaves as eiρ, i.e. R = ζ(1)l /r as

eiIkr/r. It says simply that at larger distances from the sphere, the scattered wave is

3.2. Derivation 31

spherical. For the simplicity in the following discussion, we skip the superscript (1) fromζ(1)l . So we set the potentials for scattered wave

r eΠ(s) =1

Ik2

∞∑l=o

il−1 2l + 1l(l + 1)

eBlζl( Ikr)Pl(cos θ) cosφ (3.66a)

r mΠ(s) =1

Ik Ik2

∞∑l=o

il1k1k2

2l + 1l(l + 1)

mBlζl( Ikr)Pl(cos θ) sinφ (3.66b)

Now we substitute (3.64), (3.65) and (3.66) into the boundary conditions (3.52),

ψl(1ka)− eBlζl(1ka) =1k2

2k2

eAlψl(2ka), (3.67a)

ψ′l(

1ka)− eBlζ′l(

1ka) =1k2k

eAlψ′l(

2ka), (3.67b)

ψl(1ka)− mBlζl(1ka) =1k2k

mAlψl(2ka), (3.67c)

ψ′l(

1ka)− mBlζ′l(

1ka) =1k2

2k2

mAlψ′l(

2ka). (3.67d)

In fact, these are linear independent equations between the coefficients, exactly whatwe were searching for. But we are only interested in the coefficients eBl and mBl, since theycharacterize the scattered wave. With the elimination of other unimportant coefficientswe can get the required one:

eBl =IIk2

Ikψ′l(

IIka)ψl( Ika)− Ik2IIkψ′

l(Ika)ψl( IIka)

IIk2Ikψ′

l(IIka)ζl( Ika)− Ik2

IIkζ ′l(Ika)ψl( IIka)

(3.68a)

mBl =Ik2

IIkψl( Ika)ψ′l(

IIka)− IIk2Ikψ′

l(Ika)ψl( IIka)

Ik2IIkζl( Ika)ψ′

l(IIka)− Ik2

Ikψl( IIka)ζ ′l(Ika)

(3.68b)

Let us simplify the expression given above. In this case, we have again to recallthe meaning of the constants. We are assuming that the surrounding medium is non-conducting, i.e. Iσ = 0. Let us write only σ instead of Iσ. Let λ0 be the wavelengthof the light in the vacuum and Iλ in the surrounding medium. Now we have

Ik1 =iω

cIε = i

2πλ0

Iε, Ik2 =iω

c= i

2πλ0

(3.69a)

Ik =√− Ik1

Ik2 =2πλ0

√Iε = frac2π Iλ (3.69b)

IIk1 =iω

c

(IIε+ i

4πσω

)= i

2πλ0

(IIε+ i

4πσω

), IIk2 =

iω

c= i

2πλ0

(3.69c)

IIk =√− IIk1

IIk2 =2πλ0

√IIε+

4πσω

(3.69d)

Let us introduce now a complex refractive index relative to each medium

m2 =IIn2

In2=

IIk2

Ik2=

IIεIε

+ i4πσω Iε

=IIk1

Ik1, (3.70)

and a dimensionless parameter x and argument y

x =2πIλa, y = mx. (3.71)

It is interesting to see that x equals to the ratio of the circumference of the sphereto the wavelength, which is, in fact, very important factor in further calculations. Itcan also reduce the argument to Rayleigh scattering, which is valid for the diameter ofthe object is one tenth of the wavelength.


Now with the given simplification, we can rewrite the Mie coefficients as follows∗

eBl =ψ′

l(y)ψl(x)−mψ′l(x)ψl(y)

ψ′l(y)ζl(x)−mζ ′l(x)ψl(y)

(3.72a)

mBl =mψl(x)ψ′

l(y)− ψ′l(x)ψl(y)

mζl(x)ψ′l(y)− ψl(y)ζ ′l(x)

(3.72b)

3.2.7 Field vectors of scattered wave

The components of the field vectors of the scattered wave can now be obtained by usingMie coefficients from (3.68) as follows

E(s)r =

1Ik2

cosφr2

∞∑l=1

l(l + 1) eBlζl( Ikr)P (1)l (cos θ), (3.73a)

E(s)θ = − 1

Ik

cosφr

∞∑l=1

eBlζ

′l(

Ikr)P (1)′

l (cos θ) sin θ − i mBlζl( Ikr)P (1)l (cos θ)

1sin θ

,

(3.73b)

E(s)φ = − 1

Ik

sinφr

∞∑l=1

eBlζ

′l(

Ikr)P (1)l (cos θ)

1sin θ

− i mBlζl( Ikr)P (1)′

l (cos θ) sin θ,

(3.73c)

H(s)r =

iIk IIk2

sinφr2

∞∑l=1

l(l + 1) eBlζl( Ikr)P (1)l (cos θ), (3.74a)

H(s)θ = − 1

Ik2

sinφr

∞∑l=1

eBlζl( Ikr)P (1)

l (cos θ)1

sin θ+ i mBlζ

′l(

Ikr)P (1)′

l (cos θ) sin θ,

(3.74b)

H(s)φ =

1Ik

sinφr

∞∑l=1

eBlζl( Ikr)P (1)′

l (cos θ) sin θ + i mBlζ′l(

Ikr)P (1)l (cos θ)

1sin θ

,

(3.74c)

This is the final solution of the boundary value problem.

3.2.8 Intensity and polarization of the scattered light

We shall now examine the intensity and the polarization of the scattered light. As beingonly interested in relative values of the intensity, we may take as a measure of the intensitythe square of the real amplitude of the electric vector. Here we shall only consider thedistant field, i.e. r λ, which allows us to replace the functions ζl and ζ ′l by theirasymptotic approximations

ζl(x) ∼ (−i)l+1eix and ζ ′l(x) ∼ (−i)leix (3.75)

∗Mie coefficients eBl and mBl are denoted as an and bn respectively in the various literature on thistopic: Cf. H.C. van de Hulst, Light scattering by small particles (New York, Dover publications, 1981),p.123 and Cf. Bohren and Huffman, Absorption and scattering of light by small particles (Weinheim,Wiley-VCH Verlag, 2004), p.101

3.2. Derivation 33

we obtain for intensity

Ipar =Iλ2

4π2r2|∞∑l=1

(−i)l

(eBlP

′(1)l (cos θ) sin θ − mBl

P(1)l (cos θ)

sin θ

)|2, (3.76a)

Iperp =Iλ2

4π2r2|∞∑l=1

(−i)l

(eBl

P(1)l (cos θ)

sin θ− mBlP

′(1)l (cos θ) sin θ

)|2 (3.76b)

Then|E(s)

θ |2 = Ipar cos2 φ, |E(s)φ |2 = Iperp sin2 φ (3.77)

We define here the plane of observation as the plane that contains the direction ofpropagation of the incident light and the direction (θ, φ) of observation. φ representsthe angle between this plane and the direction of vibration of the electric vector of theincident wave. Since either E(s)

θ or E(s)φ vanishes when φ = 0 or φ = π/2, the scattered

light is linearly polarized when the plane of observation is parallel or perpendicular to theprimary vibrations, whereas for any other direction (θ, φ) the light is normally ellipticallypolarized, since the ratio of E(s)

θ /E(s)φ is complex. But in case of Rayleigh scattering, it

is always real, so that the scattered light is then linearly polarized for all directions ofobservation.

3.2.9 Amplitude functions

To simplify the numerical calculations of intensity, we shall look into the amplitude func-tions, which for our approximation gives clear understanding of the phenomena. For theamplitude functions, we have to analyze again field potentials for scattered wave given as(3.66). The problem we are studying is about the very large distances from the sphere tothe observer compared to l, i.e. l | Ikr|. In this case, we can take asymptotic expressionfor ζl( Ikr)

ζl( Ikr) ∼ (−i)l+1eiIkr (3.78)

so we get for the scattered field

r eΠ(s) = − iIk2r2

eiIkr cosφ

∞∑l=o

2l + 1l(l + 1)

eBlPl(cos θ) (3.79a)

r mΠ(s) = − iIk Ik2r2

eiIkr sinφ

∞∑l=o

2l + 1l(l + 1)

mBlPl(cos θ) (3.79b)

Let us introduce the following expressions

πl(cos θ) =dPl(cos θ)d cos θ

(3.80a)

τl(cos θ) = cos θπl(cos θ)− sin2 θdπl(cos θ)d cos θ

(3.80b)

Resulting field components can be put simply as

Eθ = Hφ = − iIk2r2

eiIkr cosφS2(θ) (3.81a)

−Eφ = Hθ = − iIk2r2

eiIkr sinφS1(θ) (3.81b)

where

S1(θ) =∞∑l=o

il−1 2l + 1l(l + 1)

eBlπl(cos θ) + mBlτl(cos θ) (3.82a)

S2(θ) =∞∑l=o

il−1 2l + 1l(l + 1)

mBlπl(cos θ) + eBlτl(cos θ) (3.82b)


S1(θ) and S2(θ) are the elements of the scattering matrix (see sec. 3.3.5).The radical components Er and Hr tend to zero with higher powers of 1/r. With these

amplitude functions we can get intensity and the state of polarization of the scattered wave.The scattered light is generally elliptically polarized, even if the incident light has linearpolarization, since S1(θ) and S2(θ) are complex numbers with different phase.

Due to spherical symmetry, S3(θ) and S4(θ) are zero (see sec. 3.3.5), one can obtainthe solution to Mie problem with known i1 = |S1(θ)|2 and i2 = |S2(θ)|2, which is describedin the Chapter 04.

If Iin is the initial flux, then in terms of different polarization, the scattered fluxes are:

Iperp(θ) =i1k2r2

Iin (3.83)

Ipar(θ) =i2k2r2

Iin (3.84)

Iunpol(θ) =i1 + i22k2r2

Iin. (3.85)

3.2.10 Scattering cross section

We can recall from the last chapter, that the differential scattering cross section is definedas the ratio of the scattered flux to the incident flux per unit area:

dσ

dΩ=IscIin

. (3.86)

Following from the last subsection, one can write the differential cross section:

dσ

dΩ=

1k2

(|S2(θ)|2 cos2 φ+ |S1(θ)|2 sin2 φ

)(3.87)

And the total scattering cross section is found by integrating over all solid angles, but theintegral is rather difficult to solve. There is another easier expression for the differentialcross section:

dσ

dΩ=

12k2

[|S+(θ)|2 + |S−(θ)|2 + (S+(θ)S∗−(θ) + S−(θ)S∗+(θ))(cos2 φ− sin2 φ)

](3.88)

where

S+(θ) ≡ S1(θ) + S2(θ) (3.89)S−(θ) ≡ S1(θ)− S2(θ) (3.90)

Then one can write for the total scattering cross section:

σsca =∫ (

dσ

dΩ

)dΩ =

π

2k2

∫ +1

−1(|S+(θ)|2 + |S−(θ)|2) d(cos θ) (3.91)

As one can see, σsca does not provide intensity and phase information like the differentialscattering cross section.

3.3. Appendix 35

3.3 Appendix

3.3.1 Debye potentials and preceeding attempts

Def. If u and v are functions that satisfy the Helmholtz equation

∇2u+ k2u = 0 (3.92)

in some domain D and if r is a position vector from the origin, but lying outside D, thenthe vector fields

A = ∇× (∇× ur) + k∇× vr, B = ∇× (∇× vr) + k∇× ur (3.93)

satisfy the following Maxwell’s equations

∇×A = kB, ∇×B = kA (3.94)

for a time-harmonic electromagnetic field in D. The functions u and v are called Debyepotentials for the electromagnetic field (A,B). Of course, these potentials were inventedby Debye[2] to solve Maxwell’s equation. The procedure is quite peculiar, in the sense, byassuming that the unknown electromagnetic field has a representation by Debye potentials,using the condition of the problem to determine what u and v must be, and then verifyingthat the field A and B generated by this pair of potentials actually solve the originalproblem.

To understand the preceding endeavor to solve such Maxwell’s explanation, it makessense to start with an example of the theorem that every electromagnetic field defined ina spherical domain can be expanded in a series of electromagnetic multipoles. The field isgiven as

An = ∇× (∇× unr), Bn = k∇× unr (3.95)

where

un(r) =Zn+ 1

2(kr)

√r

Sn(θ, φ) (3.96)

Sn is a surface spherical harmonic, and Zn+ 12

is a Bessel function of order n+ 12 .

The first such attempt to do the expansions in spherical harmonics was due to A.Clebsch(1863),where he showed that the vector field A can be expanded into a series A =

∑∞n=0 An where

An(r) = αn(r)∇un(r) + βn(r)r2n+1∇ vn(r)r2n−1

+ γn(r)r×∇wn(r) (3.97)

where αn, βn, and, γn(r) are functions of the spherical coordinate r alone, while un, vn, and, wn(r)are solid harmonics of degree n. If A is the electric or magnetic part of a time-harmonicelectric field then the expansion A =

∑∞n=0 An may be shown to be equivalent to a

multipole expansion of A.A very similar method was developed by C.W.Borchardt(1873), where he considered

a vector fields A satisfying

∇× (∇×A) = 0, ∇ ·A = 0 (3.98)

showing that every field could be given as

A = ∇u+ r×∇v (3.99)

where u and v are solutions of Laplace’s equation. In fact, these solutions, expanded insolid harmonics, are substituted in (3.99) to obtain the quite similar expansion of A asthat of Clebsch expansion.


Another closely related method was due to H.Lamb(1881). He considered the followingequations

(∇2 + h2)u = 0, (∇2 + h2)v = 0, (∇2 + h2)w = 0,∂u

∂x+∂v

∂y+∂w

∂z= 0, (3.100)

which are, in fact, equivalent to the vector equation

∇× (∇×A)− h2A = 0 (3.101)

for the vector field A with u, v, and w as rectangular components. After seeing carefully,it is nothing but the more generalized case of (3.98) devised by Brochardt, and actually itwas so, when Lamb calculated it in his way. He said: every solution of A may be expandedas A =

∑∞n=0(An + A′

n) where

An(r) = ψn−1(hr)∇φn(r) + ψn+1(hr)r2n+3∇φn(r)r2n+1

,

A′n(r) = ψn(hr)r×∇χn(r)

(3.102)

χn and φn are solid harmonics whereas ψn(x) is a solution of

d2ψn

dx2+

2(n+ 1)x

dψn

dx+ ψn = 0. (3.103)

This expansion is equivalent to the Clebsch expansion for solutions of (3.101). He remarkedon the fundamental role of radial components

φ = r ·A, ψ = r · (∇×A) (3.104)

He discovered that φ and ψ determine A uniquely, are solutions of the Helmholtz equationand the harmonics are simply multiples of the harmonics occurring in expansions of φ andψ. Which leads to the conclusion that any problem involving the vector equations (3.101)can be reduced to a pair of scalar problems.

J.J.Thomson(1893) was the first to use this method in the context of electromagneticproblems to find the wave scattered by a perfectly conducting sphere when it is illuminatedby a plane electromagnetic wave. It was in 1908, when G.Mie extended this problem tospheres having arbitrary conductivity and dielectric constant. Unlike the earlier writers,Mie used the unknown functions as the components of the field A relative to a system ofspherical coordinates rather than the rectangular components. Like Lamb, Mie had takenthe radial components r ·A and r ·B (where kB = ∇×A) as the principal unknown ofhis problem. Of course, in such cases, the spherical components of A and A′ are simplerto describe than rectangular ones.

Borchardt’s theorem differs from that of Clebsch and Lamb in the sense that it repre-sents the general solution of his system of differential equations (3.98) in terms of a pairof scalar functions, with absolutely no explicit reference to spherical harmonics. And itwas in 1909, that the analogous representation for solutions of Maxwell’s equations wasobtained by P.Debye in a paper concerning the pressure exerted by electromagnetic waveson spheres of arbitrary material. But he, like G.Mie, used the spherical components of theelectromagnetic fields. On adding the fields A and A′ one obtains Debye’s representationof an electromagnetic fields in terms of a pair of potentials u and v. In fact, these fields Aand A′ can be written in the simpler vector representation in relations to the potentials∗

A = ∇× (∇× ur), A′ = k∇× vr (3.105)

in compliance with the definition mentioned at the beginning. This is how the historypromoted Debye at the helm of idea of using the scalar potentials to simply the quiteoften problems involving Maxwell’s equations.

∗thanks to the work of Sommerfeld[7]

3.3. Appendix 37

3.3.2 Vector representations in spherical polar coordinates

The curvilinear coordinates can be represented in spherical polar coordinates as

x = r sin θ cosφ (3.106a)y = r sin θ sinφ (3.106b)z = r cos θ (3.106c)

The components of vector A are transformed from cartesian to polar coordinates as

Ar = Ax sin θ cosφ+Ay sin θ sinφ+Az cos θ (3.107a)Aθ = Ax cos θ cosφ+Ay cos θ sinφ−Az sin θ (3.107b)Aφ = −Ax sinφ+Ay cosφ (3.107c)

Applying these formulae to the vector curl A

(curl A)r =1

r2 sin θ

[∂(r sin θAφ)

∂θ− ∂(rAθ)

∂φ

](3.108a)

(curl A)θ =1

r sin θ

[∂Ar

∂φ−∂(r sin θAφ)

∂r

](3.108b)

(curl A)φ =1r

[∂(rAθ)∂r

− ∂Ar

∂θ

](3.108c)

3.3.3 Associated Legendre functions

The Legendre polynomials are the polynomials

Pl(x) =l/2∑

m=0

(−1)m (2l − 2m)!2lm!(l −m)!(l − 2m)!

xl−2m (3.109)

and the associated Legendre functions of the first kind are defined by

Pml (x) = (1− x2)m/2 d

m

dxmPl(x) (3.110)

Notice that there is an arbitrariness in phase associated with the definition and someauthors may insert a factor (−1)m on the right-hand side of equation (3.110). We shalladopt the convention to insert the phase factor into the definition of the spherical har-monics below, in fact, either way the same phase factor will arise in the latter. We shallneed the following relations

P(1)l (cos θ) =

1sin θ

[Pl−1(cos θ)− cos θPl(cos θ)], (3.111)

P(1)l (cos θ) =

cos θsin2 θ

[P

(1)l (cos θ)− l(l + 1)

Pl(cos θ)sin θ

](3.112)

3.3.4 Cylindrical functions

Small values of argument x For small values of x, one can write for ψl(x)

ψl(x) =xl+1

1 · 3 · . . . · (2l + 1)fl(x) (3.113)

wherefl(x) = 1− 2

2l + 3

(x2

)2+ . . . (3.114)


For ζ(1)l (x), one has

ζ(1)l (x) = −i1 · 3 · . . . · (2l − 1)

xleix[hl(x)− ixgl(x)], (3.115)

where hl(x) and gl(x) are power series of which the first term is unity and the second termis quadratic. Analogous, one can express the functions ψ′

l(x) and ζ ′l(x):

ψ′l(x) =

(l + 1)xl

1 · 3 · . . . · (2l + 1)f+

l (x) (3.116)

ζ ′l(x) = il1 · 3 · . . . · (2l − 1)

xl+1eix[h+

l (x)− ixg+l (x)] (3.117)

where f+l (x), h+

l (x) and g+l (x) are power series of the same kind as before.

Large values of argument x For larger values of x, provided that l is small comparedwith |x|, then the following asymptotic relations can be used:

ψl(x) ∼ 12[il+1e−ix + (−i)l+1eix], (3.118)

ζ(1)l (x) ∼ (−i)l+1eix, (3.119)

and

ψ′l(x) ∼ 1

2[ile−ix + (−i)leix], (3.120)

ζ′(1)l (x) ∼ (−i)leix, (3.121)

3.3.5 Scattering matrix

The linearity of Maxwell’s equations implies that the scattering process mixes the compo-nents E1 and E2 linearly (see figure 3.1). Such a process is described in a matrix context,allowing us to write for the far field amplitudes(

E2

E1

)sc

=eikr

ikr

(S2 S3

S4 S1

)(E2

E1

)in

(3.122)

which defines the scattering amplitude matrix S.†

But a simplification for spherical symmetry gives the scattering-amplitude matrix thefollowing way: (

E2

E1

)sc

=eikr

ikr

(S2 00 S1

)(E2

E1

)in

(3.123)

It can be explained as follows: the reflection of the scatterer through the scattering planeis equivalent to reflecting everything but the scatterer; then (E1)in and (E1)sc change sign,but (E2)in and (E2)sc do not; hence, S3 and S4 must change sign. But, if the scattereris completely symmetric under reflection, S3 and S4 can not change sign, and this way,S3 = S4 = 0.

†S is not what is usually termed the ’scattering matrix’, but rather a matrix of scattering amplitudes.

Bibliography

[1] Gustav Mie. Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen.Ann. d. Phys., 25:377–445, 1908.


[3] Calvin H. Wilcox. Debye potentials. 1957.

[4] E.M.Purcell. Electricity and Magnetism. McGraw-Hill, New York, 1963.

[5] Arnold Sommerfeld. Partial Differential Equations of Physics, page 127. AcademicPress, 1949.

[6] Arnold Sommerfeld. Partial Differential Equations of Physics, page 86. AcademicPress, 1949.

[7] Arnold Sommerfeld. Elektromagnetische Schwingungen, volume Die Differential- undIntegralgleichungen der Mechanik und Physik. Vieweg, Braunschweig, 1927.

39

Chapter 4

Approximation for X-rays

41

42 Chapter 4. Approximation for X-rays

4.1 Introduction

The practical importance of approximate theories of electromagnetic scattering by smallparticles fades away as various exact techniques mature and become even more powerful.This is absolutely true in case of Mie theory, which provides the exact solution to particlesof any size. The computational power has developed in last few decades and will be the besttool to analyze the theory in the future. Nonetheless, approximate theories still remaina valuable source of physical insight into the processes of scattering of electromagneticradiation. In this chapter, we are going to devote ourselves to both of these methods.

The last two chapters have given comprehensive overview of two different scatteringexplanations available on the ground for small particles. What we are going to do hereis to develop those theories in the context of X-ray region. The analytical approximationmatching to our prerequisites will be mainly dealt with in the next section. In the latterpart of this chapter, the computational methods will be elaborated, where the powerfulmathematical tools like Mathematica and IDL are used.

4.2 Analytical Approximation

In the context of Mie theory, which is an exact theory in itself, various approximationshave been proposed, because the numerical techniques consume a lot of computationaltime. Mainly they were divided with respect to the refractive index and size parameter.Traditionally X-ray scattering can be confined to the idea of Rayleigh-Gans scattering,since this approximation in optical problems is nothing but the Born approximation, whichwe have studied extensively in kinematic scattering theory. But in this subsection, we willwork with a different scattering approximation which gives more accurate result withrespect to the exact Mie theory.

In the last chapter we have derived amplitude functions (see sec. 3.2.9), from whichthe intensities are calculated. The best way to attain approximation is to approximatethese amplitude functions to find solutions for intensities in X-ray regions, which we shallbe doing in the subsequent discussion.

4.2.1 Limiting cases

As discussed above, we are going to consider a complex refractive index with m → 1.In this case the scattering is small. If m − 1 is very small, it makes no difference in thescattering pattern whether m− 1 is < 0 or > 0. Involving m→ 1, there are two limitingcases, one is Rayleigh-Gans scattering and other the anomalous diffraction[1]. The limitingcases for m→ 1 can very well be illustrated in the figure 4.1

Let us write ρ = 2x(m−1). For the very small values of x and ρ we get Rayleigh-Ganstheory, where x is kept fixed and transition is made for m → 1. On the other hand,the theory of anomalous diffraction is based on keeping ρ fixed and making the sametransition. Here x→∞ gives definite values for cross-sections and the scattering patternsruns through a set of ”homologous diagrams”, which will be discussed shortly.

As could be seen in the figure 4.1, Rayleigh-Gans domain can be described by inter-ference of light scattered independently by all volume elements, where the anomalous-diffraction domain can be described as straight transmission and subsequent diffractionaccording to Huygen’s principle.

Although for x→∞ the anomalous diffraction is the limiting case, there could be twodifferent limiting cases within, namely ρ → 0 and ρ → ∞. What we are interested in isthe case for ρ → 0, so-named intermediate case, whereas ρ → ∞ describes geometricaloptics.

4.2. Analytical Approximation 43

Rayleigh-Gans scattering

x small x large x large

ρ small ρ large

Rayleigh scattering

Intermediate case

Geometrical optics plus diffraction

Anomalous diffraction

ρ small

Figure 4.1: Limiting cases in which m→ 1

4.2.2 Anomalous diffraction theory

The anomalous diffraction theory is a brain-child of van de Hulst [1], which was primarilydeveloped to obtain the extinction of particles in interstellar space. Later the theory hasbeen applied to describe the scattering properties of many different types of particles.The anomalous diffraction theory is based on the premise that the extinction of light bya particle is primarily a result of the interference between the rays that pass throughthe particle with those that do not. This interference takes place at some point very farfrom the particle and gives rise to extinction. It is based on two assumption as describedin previous sub-subsection. The assumption |m − 1| 1 states that the refraction andreflection of the ray passing through the particle are negligible. And the assumption x 1allows one to trace the wave, in the form of ray, through the particle.

These assumptions imply that the ray cannot be traced through the sphere nor onecan observe deviations at two boundaries it crosses, i.e. the ray is virtually straight. Theenergy reflected at the boundaries is negligible, since the Fresnel reflection coefficient goesto 0, when m→ 1. That means the field is not changed in amplitude, but only in phase.From the figure 4.2, it can be seen that the path traveled in the medium wih refractiveindex m is 2a sin τ . The phase lag observed at the point Q is

2a sin τ · (m− 1) · 2π/λ = ρ sin τ (4.1)

where ρ = 2x(m− 1). The physical meaning of ρ is the phase lag by the central ray thatpasses through the sphere along a diameter.

Now the field in the entire plane V is known. If we put it to 1 in all points outside thegeometrical shadow circle, it is e−iρ sin τ at a point, which is at a distance of a cos τ fromthe center of the circle, for phase lag corresponding to a negative imaginary exponent. Sothe field added to the field of the original wave only inside the shadow circle is e−iρ sin τ−1.This added field determines the scattered wave. For an opaque body only the term −1 ispresent and gives according to [2]

S(0) =k2

2πG (4.2)

whereG =

∫ ∫area

dx dy = πa2 (4.3)


τ

τ

a

n+½

Q

Incident Ray

Figure 4.2: Ray passing through the sphere

G is the area of geometrical shadow and generalizing this term for our case it gives

S(0) =k2

2π

∫ ∫circle

(1− e−iρ sin τ )dx dy (4.4)

The same result can be obtained using Babinet’s principleNow we use polar coordinated inside the shadow circle.

−2πa cos τd(a cos τ) = 2πa2 cos τ sin τ dτ (4.5)

S(0) = k2a2

∫ π/2

0(1− e−iρ sin τ ) cos τ sin τ dτ (4.6)

with

K(w) =∫ π/2

0(1− e−iρ sin τ ) cos τ sin τ dτ (4.7)

=12

+e−w

w+e−w − 1w2

(4.8)

So the result is[2]

S(0) = x2K(iρ) (4.9)

and∗

Qext =4x2Re[S(0)] = 4Re[K(iρ)] (4.10)

∗Qext is the extinction factor, i.e. how much energy has been lost from the incident flux to the area ofthe surface. It is widely used tool to analyze Mie scattering. More about this and the same absorptionfactor Qabs and scattering factor Qsca can be found in the book from van de Hulst[2]

4.2. Analytical Approximation 45

Complex refractive index

Let us denote the refractive index by m = n− in′. Since m→ 1, both n− 1 and n′ mustbe 1. We write,

n′

n− 1= tanβ (4.11)

Now the phase shift of a ray passing through the center of the sphere is

ρ∗ = 2x(m− 1) = ρ(1− i tanβ) (4.12)

Here, the real part denotes an actual phase shift and imaginary the decay of amplitude.From the above discussion, we can rewrite for the efficiency factor for extinction

Qext = 4Re[K(iρ+ ρ tanβ)] (4.13)

The phase shift factor e(−iρ∗ sin τ) contains a factor e−2xn′ sin τ for the decrease in am-plitude, which consequently leads to decrease in intensity by e−4xn′ sin τ and the fractionof 1− e−4xn′ sin τ of the original ray is absorbed within the sphere. The absorbed fractionof the total energy incident on the sphere is found to be

Qabs = 2K(4xn′) (4.14)

where the argument can also be written as 4xn′ = 2ρ tanβ = 2aγ with radius a and γ asthe absorption coefficient of its material per unit length.

When Qabs approaches 1, hardly any light gets through the sphere and nothing inter-feres with the ordinary diffraction and we have simply Qext = 2, which holds for any largeopaque body.

The scattered fraction of the total incident energy, i.e. Qsca can be easily calculatedtaking subtraction of Qabs from Qext. So here we get

Qsca = 2(2Re[K(iρ+ ρ tanβ)]−K(2ρ tanβ)) (4.15)

But in our case, we are considering small values of ρ, then we can use the seriesexpansion of K(w) for small w

K(w) =w

3− w2

8+w3

30− . . . (4.16)

So we get

Qext =43ρ tanβ +

12ρ2(1− tan2 β) (4.17)

Qabs =43ρ tanβ + ρ2 tan2 β (4.18)

and for scattered fractionQsca =

12ρ2(1 + tan2 β) (4.19)

4.2.3 Diffraction patterns

Not only the amplitude function S(0) is affected due to the interference between diffractedand transmitted light, but also the function S(θ) for which the diffraction is appreciablefor all values of θ. To derive this function, we shall make use of rigorous Mie theory andwe shall know in the end that the result can well be acquired from Huygen’s principle.

We know the Mie coefficients eBl and mBl from last chapter (3.72), which we shalldenote as an and bn in the upcoming discussion for simplicity. These coefficients can bewritten as

an =12(1− e−2iαn), bn =

12(1− e−2iβn) (4.20)


They consist of two terms: one independent of the nature of particle and another dependenton it. The term 1 gives the Fraunhofer diffraction pattern and the term e−2iαn and e−2iβn

the scattering by reflection and refraction. This separation can be useful only if αn andβn are large and should be made only for the terms with n+ 1

2 < x (localization principle[3]).

In the limit of x→∞ and mx→∞, αn can be approximated as[4]

exp(−2iαn) = exp[2i(xf − x′f ′)]1− ir2exp(2ix′f ′)

1 + ir2exp(−2ix′f ′)(4.21)

where

r1 =sin τ −m sin τ ′

sin τ +m sin τ ′, r2 =

m sin τ − sin τ ′

m sin τ + sin τ ′(4.22)

andf = sin τ − τ cos τ, f ′ = sin τ ′ − τ ′ cos τ ′, x′ = mx (4.23)

The angle of incidence τ is related to τ ′ and n through the equations

x cos τ = x′ cos τ ′ = n+12

(4.24)

The phase (xf − x′f ′) may the be expressed as

xf − x′f ′ = −xη + x(τ ′ − τ) cos τ, (4.25)

where η = m sin τ ′ − sin τ .If u = (n + 1

2)θ is fixed and n goes to infinity, the asymptotic formulae for sphericalharmonics appearing in amplitude functions in the last chapter are

πn(cos θ) ' 12n(n+ 1)[J0(u) + J2(u)], (4.26)

τn(cos θ) ' 12n(n+ 1)[J0(u)− J2(u)], (4.27)

which for u 1 or θ 1/x may further be approximated as

πn(cos θ) ' τn(cos θ) ' 12n(n+ 1)J0(u) (4.28)

For very large x one can make the following substitution

∑n

= x

∫ π/2

0sin τdτ (4.29)

After all the calculation the scattering function for near forward scattering assumesthe following form

S(θ) = x2

∫ 1−

[exp(−2iα(τ)) + exp(−2iβ(τ))

2

]J0(xθ cos τ) cos τ sin τdτ (4.30)

where S1(θ) ' S2(θ) ≡ S(θ).Again here the term 1 give the Fraunhofer diffraction and the remaining terms give

the scattering by reflection and refraction. The later part of S(θ) can also be expressed as

A(θ) =− x2

2

∫exp[2i(xf − x′f ′)]

(r1 + r2)ν

− (1− r21)∞∑

p=1

(−r1ν)p−1 − (1− r22)

∞∑p=1

(−r2ν)p−1

J0(xθ cos τ) cos τ sin τdτ

(4.31)

4.3. Numerical approximation 47

with ν = iexp(−2ix′f ′).Although it looks more complicated, it is one of the best approximations available

yet. The original idea is from Debye[5]. The extension of which provides interestinginsight into Rainbow theory, which investigates particles of very large size. The validityof anomalous diffraction theory with Mie theory has been scrutinized by Farone[6] andSharma[7]. The algorithms are developed in this direction for numerical calculations,mostly in the applications in environmental physics or astrophysics[8]. Direct calculationof anomalous diffraction theory in the context of X-ray scattering, to my knowledge, hasnot been done yet.

4.3 Numerical approximation

4.3.1 Using Mathematica

Mathematica is a computer algebra system of high-level programming language emulatingmultiple paradigms on the top of term-rewriting (more in sec. 5.4.1). In our calcula-tion of the exact Mie theory, it comes very handy, knowing that the solution containsBessel, Ricatti-Bessel functions and Legendre polynomials. Mathematica has these func-tions built-in, which makes writing the algorithm presumably very straight forward.

Since, in the X-ray, the analysis is confined to reciprocal space, there was a need torewrite the all those Ricatti-Bessel functions and Legendre polynomials in reciprocal space.Although these functions are selected functions of the family of hypergeometric functions,it was unfortunately not possible to write them in simple readable forms. For this reason,they were kept dependent on the scattering angle θ, whereas θ is defined as the functionof the momentum-transfer vector Q, which is itself defined in sec. 2.3.3. So the relationbetween θ and Q is given as:

θ = 2arcsin(Q

2k

)(4.32)

where k is the wave vector in the vacuum.The relative intensity that we are interested in are squares of the amplitude functions

(3.82), which can be obtained from Mie coefficients (3.72) and angle-dependent functions(3.80). Mie coefficients are composed of Ricatti-Bessel functions, which can be givensymbolically in Mathematica, so is the case with angle-dependent functions πn and τn,which are nothing but Legendre polynomials and its derivatives, respectively. Legendrepolynomials are in-built functions in Mathematica as well. The detailed algorithm withthe accompanying theory has been reproduced extensively in Chapter 7, precisely in thesection sec. 7.1.

4.3.2 Using IDL

The algorithm used to calculate with IDL is due to Bohren & Huffman[9]. In this pro-gramme, unlike in Mathematica, the convergence of series has to be justified. After theextensive study by Wiscombe[10], the stopping term for the series has been introduced,which is, in fact, the integer closest to

x+ 4x1/3 + 2 (4.33)

where x is the largest size parameter among the size parameters calculated for all themedium coming into questions. This stopping term has been used for all the series calcu-lations concerned.

Fortunately, recursive techniques are available for the calculations of Mie coefficients(3.72) and angle-dependent functions (3.80), required to calculate the amplitude functions(3.82). In next sub-subsections the respective recursive techniques will be put into details.


Recursion for Mie coefficients

Like in the section before, here I shall stay attached to the popular convention of repre-senting Mie coefficients of an and bn for eBl and mBl respectively detailed in the lastchapter(3.72). Mie coefficients are rather complicated functions of the spherical Besselfunctions and their derivatives, the argument of which are generally complex. But thespehrical Bessel function satisfy the recurrence relations

zn−1(ρ) + zn+1(ρ) =2n+ 1ρ

zn(ρ) (4.34)

(2n+ 1)d

dρzn(ρ) = nzn−1(ρ)− (n+ 1)zn+1(ρ) (4.35)

where zn is either a Bessel function of first or second kind. So in this algorithm, we aretempted to start with n = 2. Moreover, these functions have simple trigonometric firstorders, which makes the calculation easier. The logarithmic derivative was first introducedby Aden[11] in the context of computing coefficients for a sphere.

Dn(ρ) =d

dρlnψn(ρ) (4.36)

Now, we may rewrite the Mie coefficients as

an =[Dn(mx)/m+ n/x]ψn(x)− ψn−1(x)[Dn(mx)/m+ n/x]ξn(x)− ξn−1(x)

(4.37)

bn =[mDn(mx) + n/x]ψn(x)− ψn−1(x)[mDn(mx) + n/x]ξn(x)− ξn−1(x)

, (4.38)

using the recurrence relation

ψ′n(x) = ψn−1(x)−

nψn(x)x

, ξ′n(x) = ξn−1(x)−nξn(x)x

(4.39)

And the logarithmic derivative in itself satisfies the recurrence relation

Dn−1 =n

ρ− 1Dn + n/ρ

(4.40)

which are the consequences of recurrence relation (4.34). There are two possible schemesfor calculating Dn(mx), but in this programme we shall concentrate only on downwardrecurrence, i.e. lower order generated from higher order. Beginning with an estimate forDn, where n is larger than number of terms required for convergence, successively moreaccurate lower-order logarithmic derivatives can be generated. So here we shall begin withDNMX. Let NMX be sufficiently greater than stopping term and |mx|. Thus, NMX istaken to be Max(NSTOP, |mx|) + 15 and the recurrence begins with DNMX = 0.0 + i0.0.Both ψn and ξn satisfy

ψn+1(x) =2n+ 1x

ψn(x)− ψn−1(x) (4.41)

where ξn = ψn − iχn, with the initial values for recurrence of

ψ−1(x) = cosx, ψ0(x) = sinx, (4.42)χ−1(x) = − sinx, χ0(x) = cosx. (4.43)

The downward recurrence is computationally stable method than forward recurrence,which was shown by Kattawar and Plass[12].

4.3. Numerical approximation 49

Recursion for angle-dependent functions

The angle-dependent functions πn and τn can as well be calculated using recurrence rela-tionship, given as

πn =2n− 1n− 1

µπn−1 −n

n− 1πn−2 (4.44)

τn = nµπn − (n+ 1)πn−1, (4.45)

where µ = cos θ. In case of angle-dependent functions, the upward recurrence is enoughfor calculation, unlike in case of Mie coefficients, where downward recurrence was the timesaving solution. The initial values for recurrence will be

π0 = 0, π1 = 1. (4.46)

Due to the following relationship, these functions should only be calculated between 0

and 90,πn(−µ) = (−1)n−1πn(µ), τn(−µ) = (−1)nτn(µ). (4.47)

Bibliography

[1] H.C. van de Hulst. Light Scattering by Small Particles, chapter 11, page 172. DoverPublications Inc., New York, 1981.





[6] W.A.Farone and M.J.Robinson III. The range of validity of the anomalous diffractionapproximation of electromagnetic scattering by a sphere. Applied Optics, 7(643-645),1968.

[7] S.K.Sharma. On the validity of the anomalous diffraction approximation. Journal ofmodern optics, 39(11):2355–2361, 1992.

[8] B.T.Draine and Khosrow Allaf-Akbari. X-ray scattering by nonspherical grains.i.oblate spheroids. The Astrophysical Journal, (652):1318–1330, 2006.

[9] Craig F. Bohren and Donald R. Huffman. Absorption and Scattering of Light bySmall Particles. Wiley-VCH, 2004.

[10] W. J. Wiscombe. Improved Mie scattering algorithms. Applied Optics, 19:1505–1506,1980.

[11] A.L. Aden. Electromagnetic scattering from spheres with sizes comparable to thewavelength. Journal of Applied Physics, 22:601–605, 1951.

[12] G. W. Kattwar and G. N. Plass. Electromagnetic scattering from absorbing spheres.Applied Optics, 6:1377–1382, 1967.

51

Chapter 5

Comparison

53

54 Chapter 5. Comparison

In this up-coming discussion, the exact Mie theory and the scalar kinematical theorywill be compared with the help of graphs generated by Mathematica, using the code givenin sec. 7.1. This is the direct comparison between the relative intensity graphs generatedby the Mie solution and the solution given by the scalar kinematical theory. For the sake ofuniformity, only the relative intensity with perpendicular polarization has been consideredin graphs concerning Mie theory.

As we have seen in the preceding chapters, the size parameter x is the most crucialfactor in Mie theory. The size parameter is given as

x =2πaλ

(5.1)

where a is the radius of the sphere and λ is the wavelength in the vacuum. For achievingclearer understanding, I shall be analyzing only with fixed wavelength and only one typeof material, but with varying radii. It shall be interesting to know how far the scalartheory matches the exact vector theory. It is always useful to keep in mind that scalartheory, i.e. kinematical theory, views a sphere as a disk, whereas the vector theory, i.e.Mie theory, encompasses three-dimensional structure. So it is interesting to know to whatsize of matter the assumption of taking sphere as a disc is plausible.

5.1 Comparing theories

In this analysis, the wavelength in vacuum is kept constant at 1A. We shall be consideringthe gold spheres of various radii, giving us different size parameters to compare. Therefractive index n = 1− δ + iβ, we have for the gold:

δ = 1.9 · 10−5

β = 2.5 · 10−6

The density appearing in the equation (2.49) is the electron density. It can be calcu-lated using the relation between δ and ρ given in (2.6). It turns out to be ρ = 4233.351/nm3

for Gold.We shall compare here the relative intensity given by the kinematical theory to the

relative intensity given by Mie theory. In case of graphs with Mie theory, only the perpen-dicular polarization has been taken into account. It is usual in X-ray theories to compareintensities against the scattering wave vector Q, and we shall stick to this tradition. Therelative intensity for the kinematical theory is given by the relation

ikin(Q) = |f(Q)|2 (5.2)

where f(Q) is the scattering amplitude, in our case, of a sphere (see the equation (2.49)).Whereas the relative intensity for the Mie theory is given by

imie(Q) = |S(Q)|2 (5.3)

where S(Q) is the amplitude function described in sec. 3.2.9∗.We shall also compare the differential scattering cross section given by both theories

versus Q. We shall be taking the azimuthal angle φ = π2† for a convenience. To put it in

a simple form, the differential scattering cross section by the kinematical theory given by,

dσ

dΩ= r20|f(Q)|2 (5.4)

∗In the subsection of sec. 3.2.9, the amplitude function has been given in the dependancy of scatteringangle θ, and θ relates to the scattering wave vector Q with Q = 4π

λsin(θ/2), where λ is the wavelength in

the medium.†see sec. 3.2.10 for more information on azimuthal angle in the scattering

5.1. Comparing theories 55

and by the Mie theory,dσ

dΩ=

1k2|S(Q)|2. (5.5)

The reference to graphs are tabulated with respect to taken radius in the following:

Radius [nm] Intensity graphs Scattering cross section graphs0.5 fig. 5.1 fig. 5.61.0 fig. 5.2 fig. 5.72.5 fig. 5.3 fig. 5.85.0 fig. 5.4 fig. 5.910.0 fig. 5.5 fig. 5.10

1e-15

1e-10

1e-05

1

1e+05

1e+10

Q [1/nm]0 50 100

Ikin

Imie

Figure 5.1: Relative intensity graph for the sphere with radius 0.5 nm

1e-15

1e-10

1e-05

1

1e+05

1e+10

Q [1/nm]0 20 40 60 80 100 120

Ikin

Imie



1e-10

1e-05

1

1e+05

1e+10

1e+151e+15

Q [1/nm]0 20 40 60 80 100

Ikin

Imie


It can be seen from the differential scattering cross section graphs that they matchalmost on each other, except for the graphs with size parameter of 100, where someinternal calculation error occurs in Mathematica (see sec. 5.2.2). Although, the graphslooks quite the same, there is small difference between maxima, which is almost of the sameorder for each graph. It was tested taking the differential cross section for kinematical andMie graphs for the lowest possible value of Q. It means that we have to find the functionImie(Q→ 0) and Ikin(Q→ 0) which gives us the differential cross section for Q = 0.

Imie(Q→ 0) For Q to be 0, θ should be 0. We have to find amplitude function S1(θ = 0).It is known from the last chapter (see 3.2.9) that for cos θ = 1, we get

πn(1) = τn(1) =n(n+ 1)

2(5.6)

Then we can write simply for S1(0)

S1(0) =12

∞∑l=o

(2l + 1)(an + bn) (5.7)

It gives us the differential scattering cross section for Q = 0 as

dσ

dΩ=

1k2

∣∣∣∣∣12∞∑l=o

(2l + 1)(an + bn)

∣∣∣∣∣2

(5.8)

It solves the problem with long calculation time required in Mathematica, since the mosttime-consuming term has been angle-dependent functions, which are absent in the expres-sion above.

Ikin(Q→ 0) In case of Ikin(Q), the lim has been applied on the Q-dependent term. Itgives

limQ→0

(sin(QR)−QR cos(QR))2

(QR)6= 0.1111 =

19

(5.9)

The final expression for the differential scattering cross section looks like

dσ

dΩ= r20ρ

2

(43πr3)2

(5.10)

5.2. Errors and problems 57

1e-10

1e-05

1

1e+05

1e+10

1e+151e+15

Q [1/nm]0 10 20 30 40 50

Ikin

Imie


The figure 5.11 gives the ratio between(

dσdΩ

)kin

and(

dσdΩ

)mie

. It can be seen from thefigure 5.11 that the ratio decreases with increasing radius, i.e increasing size parameter.But the linear regression on it shows that it has slope of 2.5 · 10−5, i.e. hardly anydifference between

(dσdΩ

)kin

and(

dσdΩ

)mie

. The absolute values of the differential scatteringcross section of lower radii are way smaller than for higher radii, which means that thisslope is the result of linearly increasing computational anomaly as the radius increases.So we can say that (

dσdΩ

)mie(

dσdΩ

)kin

' 1 (5.11)

We can substitute (5.5) and (5.4) in the ratio given above, giving us

imie(Q)ikin(Q)

' k2r20 (5.12)

where r0 is the classical electron radius and k is the wave vector in the surroundingmedium. It gives us interesting result that the ratio between relative intensities of twodifferent theory is (kr0)2. Analogous to the size parameter, where x = kR, the same way,kr0 can be termed as the classical electron size parameter. It should be noticed that thisdiscussion is based on the results acquired upto the size parameter as large as 500. It seemsto be valid in this range of size parameter. Unfortunately, due to problems discussed later,it was not possible to calculate for larger size parameters than of 500.

In fact, the most interesting part of the Mie theory is the intensity graphs for the higherQ, which was unfortunately not possible. Mie theory takes care of all the phenomena likeinternal reflection, backscattering, etc. It might have been possible to see some differenceswith the kinematical theory. In the up-coming sections, it will be explained why thecalculation for single value of Q for larger particles takes enormous time.

5.2 Errors and problems

5.2.1 Errors

The possible error in the calculation can be observed in the graph with the size parameterof 100. No direct statement can be made, regarding the error observed. The most possible


1e-10

1e-05

1

1e+05

1e+10

1e+151e+15

Q [1/nm]0 5 10 15 20 25

Ikin

Imie


explanation would be the limit of the algorithms used lying beneath the infinite seriescontaining Ricatti-Bessel functions and Legendre polynomials. The problem underlyingthe internal algorithm has been discussed to some extent in sec. 5.2.2

5.2.2 Problems

With the given algorithm, there are constraints for Mathematica, which doesn’t allowcalculating spheres with larger size parameters. To my personal knowledge, there has beenno attempt to calculate the intensity graphs for such a large parameter for a wavelength sosmall as considered here. The problems encountered in Mathematica have been shortlistedin the following discussion.

Runtime

The runtime for calculating intensity graphs for kinematical theory was a fraction of asecond, whereas the runtime for calculating intensity graphs for Mie theory increaseswith increasing size parameter and decreasing wavelength. The following table gives theexact time required to calculate particular intensity graphs for Mie theory for particularQ-domain. It can be determined using the command Timing[expr]Timing[expr]Timing[expr]

Radius [nm] Q-Range Time required in seconds0.5 0 . . . 100 382.491.0 0 . . . 110 1648.882.5 0 . . . 100 9541.15.0 0 . . . 50 41102.610.0 0 . . . 25 206513.0

After step-by-step investigation it has been found that the calculation for the Legendrepolynomials takes enormous time. Since Mathematica selects randomly its x-value inplotting a graph for f(x), it has not been possible to temporarily save the values ofLegendre polynomials, which are evoked as many times as the termination term, describedin sec. 4.3.2. The termination term is larger for large size parameters. Nonetheless, thereis a possibility of using an alternative algorithm, though with problems, which shall beexplained later.

5.2. Errors and problems 59

GRAPH

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140Q [nm]

1e−16

1e−15

1e−14

1e−13

1e−12

1e−11

1e−10

1e−9

1e−8

1e−7

1e−6

1e−5

Diffe

rent

ial s

catte

ring

cros

s se

ctio

n [n

m*n

m]

I_kinI_mie

Figure 5.6: Scattering cross section graph for the sphere with radius 0.5 nm

Internal algorithm

Mathematica does have internal built-in functions, which may have eased the calcula-tion time. The one, which is widely used in the command like Plot[expr]Plot[expr]Plot[expr], is calledEvaluate[expr]Evaluate[expr]Evaluate[expr]. Evaluate[expr]Evaluate[expr]Evaluate[expr] causes exprexprexpr to be evaluated even if it appears as the ar-gument of a function whose attributes specify that it should be held unevaluated. Thiscommand is good, when the graphs cannot be calculated for particular values, which stopsthe further calculation(see sec. 5.4.1 for details). It speeds up the whole process, but un-fortunately, it hasn’t been of use in our Mathematica programme. For the size parametersgreater than 40, the initial values started showing greater discrepancies, which is not tobe seen, when using Plot[expr]Plot[expr]Plot[expr] without using this command. The internal algorithm isnot accessible, so no further comments can be made on its functioning.

Problems with alternative algorithm

There is a possibility of using an alternative algorithm used for IDL and described inthe last chapter in sec. 4.3.2. These algorithms are based on the recurrence relationgiven in (4.34) and (4.41). The attempt was made to introduce recursive algorithmsin Mathematica. Unfortunately, it couldn’t fructify, since for the larger parameters therecursion depth is larger too. The default Mathematica RecursionLimit is 1000, which canbe increased by setting it using the command $RecursionLimit$RecursionLimit$RecursionLimit. But Mathematica can’tmanage larger values. The current Mathematica kernel occupies stack space proportionalto the number of recursive calls, limited by $RecursionLimit$RecursionLimit$RecursionLimit. Moreover, every stack frameis about 10 times the size of equivalent stack frame in a popular compiled language like C,FORTRAN, etc. So the $RecursionLimit$RecursionLimit$RecursionLimit must be kept low to avoid kernel segmentationfault, causing it to stop functioning when the operating system runs out of stack space.This is not an inherent or theoretical limitation, just a practical limitation of the currentimplementation that may well be fixed at a future date.∗

∗The theory that underpins this problem can be marginalized by understanding tail recursion in func-tional languages and continuation passing style. It is covered in the book, OCaml for scientists, by Dr.JonD. Harrop


GRAPH

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115Q [nm]

1e−16

1e−15

1e−14

1e−13

1e−12

1e−11

1e−10

1e−9

1e−8

1e−7

1e−6

1e−5

1e−4

1e−3

Diffe

rent

ial s

catte

ring

cros

s se

ctio

n [n

m*n

m]

I_kinI_mie


5.3 Conclusion

Apart from marginal exception of the last graph with size parameter of 100, the theoryseems to be in greater accordance with the scalar kinematical theory. Unfortunately, it hasnot been possible, due to reasons mentioned above, to see the breaking point where onecan differentiate the theory better. The technical difficulties encountered for this endeavorhave been provided with technical discussion in subsection sec. 5.2.2. The possible solu-tion would be to other programming language in entirety using the algorithm techniquesmentioned in sec. 4.3.2.

It is true that the Mie theory gives the explanation based on only two variables, thesize parameter x and the refractive index m. But it should be noticed that refractive indexdepends on the material and wavelength. And the size parameter is just the ratio betweenthe radius and the wavelength. That means there is implicit dependency on the variableson these variables of radius, wavelength, and the refractive index. One can write pairsof radius and wavelength giving the same size parameter, but graphs containing thesepairs cannot show the same nature. The point to be made is that it is not possible tomake graphs of one pair from the knowledge of the other, though both have the same sizeparameter. Otherwise, it would have been possible to make graphs for higher radii. Thismay explain why it takes comparatively more time for shorter wavelength than the longerones, irrespective of the size parameter.

The same argumentation can be made to the interesting parameter QR, which appearsin the equation for the kinematical scattering, but also contains size parameter in it (asQR = 2x sin(θ/2)). Nonetheless, investigation with this parameter may be of greater helpin one way or the other.

5.4 Appendix

5.4.1 Term Rewriting System

Mathematica is a term rewriting system. Whenever an expression is entered, it is evaluatedby term rewriting using rewrite rules. These rules consist of two parts: a pattern on theleft-hand side(lhs) and a replacement text on the right-hand side(rhs). When the lhs ofa rewrite rule is found to pattern-match part of the expression, that part is replaced by

5.4. Appendix 61

GRAPH

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100Q [nm]

1e−15

1e−14

1e−13

1e−12

1e−11

1e−10

1e−9

1e−8

1e−7

1e−6

1e−5

1e−4

1e−3

1e−2

1e−1

Diffe

rent

ial s

catte

ring

cros

s se

ctio

n [n

m*n

m]

I_kinI_mie


the rhs of the rule, after substituting values in the expression which match labeled blanksin the pattern into the rhs of the rule. Evaluation then proceeds by searching for furthermatching rules until no more are found.

The evaluation process in Mathematica can be understood with a simple analogy todaily experiences. One can recount the experience of using handbook of mathematicalformulas, such as the integral tables. In order to solve an integral, one consults thehandbook which contains formulas consisting of a lhs and rhs, separated by an ’equals’sign. And then one looks for an integration formula in the handbook whose lhs has thesame form as the questioned integral.

While no two formulas in the handbook have the identical lhs, there may be severalwhose lhs have the same form as the required integral. When such thing happens, one usethe formula whose lhs gives the closest fit to the integral. Then, one replaces the integralwith the rhs of the matching lhs and one substitutes the specific values in an integral forthe corresponding variable symbols in the rhs. Finally, one looks through the handbookfor formulas, like trigonometric identities, etc., that can be used to change the answerfurther.

This depiction provides a better description of the Mathematica evaluation process,but not entirely. It can be absolutely understood in Mathematica Documentation.


GRAPH

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0 42.5 45.0 47.5 50.0Q [nm]

1e−13

1e−12

1e−11

1e−10

1e−9

1e−8

1e−7

1e−6

1e−5

1e−4

1e−3

1e−2

1e−1

1e0

1e1

Diffe

rent

ial s

catte

ring

cros

s se

ctio

n [n

m*n

m]

I_kinI_mie


GRAPH

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0Q [nm]

1e−11

1e−10

1e−9

1e−8

1e−7

1e−6

1e−5

1e−4

1e−3

1e−2

1e−1

1e0

1e1

1e2

1e3

Diffe

rent

ial s

catte

ring

cros

s se

ctio

n [n

m*n

m]

I_kinI_mie


5.4. Appendix 63

1

1.005

1.01

1.015

Radius [Ang]0 100 200 300 400 500

Figure 5.11: Ratio of differential scattering cross sections of Mie theory to kinematicaltheory at Q = 0 against radius

Chapter 6

Outlook

65

66 Chapter 6. Outlook

In this thesis, the whole attempt was to understand the scattering by a sphere in theX-ray region with more insight. The two theories taken for comparison have their ownmight in the respective expression. This comparison was of the scalar scattering theoryand vector scattering theory. The scalar theory was reduced to the discussion of the firstBorn approximation, mostly due to its great proven applicability in the X-ray physics. Onthe other hand, the vector theory, first developed by G.Mie, was tried in X-ray region. Thelatter theory is the exact theory to the scattering by a sphere, valid for all wavelengths.This theory has won the ground on applicability quite recently, mostly due to increasingmachine powers for calculations. The physics represented by this theory is very muchembedded in the mathematical jargon its solution carries. Nonetheless, in this thesis, it wastried to see this theory approximated in X-ray region, analytically as well as numerically.The aim was to comprehend the scattering for the particles, not so small to be in the rangeof Rayleigh-Gans theory, and not so big to be in the region of geometrical optics. To theexplanation of such spheres, the aforementioned anomalous diffraction theory has beenintroduced (see 4.1). But more effort was laid on the numerical calculation. Due to itssymbolic and clear representative nature, the mathematical tool Mathematica was used.The whole straight-forward algorithm was written for the calculation. It provided verywonderful graphs for comparison, but again the machine calculation power has becomethe obstacle. Due to the problems mentioned in sec. 5.2.2, the direct calculation for theparticles larger than size parameter 100 has not been possible. Looking at the graphsof differential scattering cross section, it can be stated that the scalar theory seemedin accordance with the vector theory, at least, up to this size parameter. The possibledeviation from the scalar theory in the X-ray region may lie in the particles larger thanthat of with size parameter 100.

There are several outlooks that can be mentioned in this subject for the future inves-tigation. They vary from the computational level to promising analytical level. Perhaps,the work in other fields where Mie theory has been used successfully, like Meteorology,Astrophysics, could be of very much use.

Another programming language Using different programming languages for the de-velopment of Mie code could be very helpful, especially those languages which disallowsthe problems encountered in Mathematica. The low-level language is the primary choicefor its reusability and compatibility with different platforms. Other commercially avail-able languages like OCaml, IDL, etc. can be used extensively, which avoids exactly thesame problem shown in Mathematica. The algorithm provided in the last chapter (seesec. 4.3.2) is the most widely used till now.

Use of Anomalous Diffraction Theory The possible analytical approximation forthe X-rays, proposed here as anomalous diffraction theory is also the great candidate,when it comes to understanding the physics behind this phenomenon. There is also acomputational method available for calculating anomalous diffraction theory, and inter-estingly, for the X-ray scattering by interstellar dust. It is explained extensively in thepaper from Draine and Allaf-Akbari[1]. Although there is less literature available on thistopic, it is a virgin field in the X-ray physics, where its validity can be explored.

Patterns in Mie scattering The novel effort of evaluating Mie scattering intensityversus the scattering wave vector Q has been due to C.M.Sorensen[2]. To my knowledge, itwas the first such effort. He has used the FORTRAN code given by Bohren and Huffman[3],where the idea was to create graphs versus QR, where R is the radius of the sphere, ratherthan rudimentary against the scattering wave vector Q. With the calculation for largerQR for different size parameter, certain patterns have been observed. He gives certaininterpretations to the results. His analysis was restricted to a higher refractive index. Itwould be interesting to analyze it for the X-ray Mie scattering.

Bibliography

[1] B.T.Draine and Khosrow Allaf-Akbari. X-ray scattering by nonspherical grains.i.oblate spheroids. The Astrophysical Journal, (652):1318–1330, 2006.

[2] C. M. Sorensen and D. J. Fischbach. Patterns in Mie scattering. Optics Communica-tions, 173:145–153, 2000.

[3] Craig F. Bohren and Donald R. Huffman. Absorption and Scattering of Light by SmallParticles. Wiley-VCH, 2004.

67

Chapter 7

Numerical calculation

69

70 Chapter 7. Numerical calculation

7.1 Mathematica code

Mie Scattering - X-Ray - Gold

Packages required.

Some of the plot routines require certain graphics packages be loaded. A working directory may be

set for subsequent input/output operations. For example, this directory could contain the optical

dispersion data, n(l) = n(l)+ik(l), required for any spectral analysis.

Needs@"Graphics`Graphics`"D H* Needed for the polar plotting routine. *LNeeds@"Graphics`ParametricPlot3D`"D H* Needed for the spherical plotting routine. *LNeeds@"Graphics`Legend`"D H* Needed for plot legends *L

Off@General::spell1D;

SetDirectory@"êUsersênandan"D;

Parameters

This section is meant for the known parameters for the problem (i.e., complex indices of refraction, wavelength of the

illuminating radiation, sphere radius, etc.). Subscripts of 1 correspond to the embedding medium and subscripts of 2

represent the sphere. The permeability is put to one, as described in the theory of the thesis. It is important to notice

that the imaginary part of the refractive index is represented by a lower case k, and that capital K represents the wave

number. Input parameters are integers (which have infinite precision in Mathematica) whenever possible. If real num-

bers are required, I would be better to use increased precision (via the backtick mark, `) if high precision output is

required. The increased precision will slow down the code execution. Finally, it would be more efficient to assign the

parameter values using rewrite rules in the actual function calls but I've presented them as constants because this format

saves valuable runtime.

The calculations are extensible to the general case of a sphere of arbitrary radius and known optical

properties, within a medium of known optical properties. In order to end up with a prefered size parameter

as in the reference example in Bohren & Huffman, the chosen wavelength is entered.

n1 = 1; H* Real part of refractive index of medium *Ld2 = 1.9 * 10-5; H* Delta part of refractive index of scatterer *Ln2 = 1 - d2; H* Real part of refractive index of scatterer *L

k1 = 0 * 10-2 ; H* Imaginary part of refractive index of medium *Lk2 = 2.5 *10-6; H* Imaginary part of refractive index of scatterer *L

l0 = 10 * 10-2; H* V acuum wavelength in nm *Lr0 := 2.82 * 10^H-6L; H* The classic electron radius in nm *Lr = 100 * 10-2; H* Sphere radius in nm *L

Required derived quantities for given parameter set.

m1 = n1 + I k1; H* Complex refractive index of medium *Lm2 = n2 + I k2; H* Complex refractive index of scatterer *L

mRel =n2 + I k2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅn1 + I k1

; H* Relative refractive index between scatterer and medium *L

l1 =l0

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅn1 + I k1

; H*W avelength in external medium in nm *L

l2 =l0

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅn2 + I k2

; H*W avelength inside sphere in nm *L

K0 =2 pÅÅÅÅÅÅÅÅl0

; H* Wave number in vacuum in nm-1 *L


; H* Wave number in external medium in nm-1 *L


; H* Wave number inside the sphere in nm-1*L

rho = Hd2 * K02L ê H2* p * r0L;H* Electron density in 1êcubic nm *L;

Now create the size parameter. Note there are actually three size parameters defined (as opposed to Bohren and Huff-

man's single size parameter), two of which are now complex and due to the fact that the external medium may be absorp-

tive. They come from the paper by Fu and Sun (Applied Optics, Vol. 40, No. 9, 2001,pp. 1354-1361) which explains

how to treat a particle in an absorbing medium. They serve as arguments to the scattering coefficient equations. They

are set up as functions of particle radius so that this may be varied.

s@rD := K0 r

s1@rD := K1 r

s2@rD := K2 r

I use the equation suggested by Wiscombe (Applied Optics, Vol. 19, 1980, pp. 1505-1506) for obtaining the series

solution termination point for a given size parameter. Since there are now three (two possibly complex) size parameters,

I choose the largest of the absolute values of the three as the argument to the equation. The equation was found to be

valid for the range 8 < s < 4200. For smaller s values, the suggestion is LastTerm = s + 4 s1ÅÅÅÅÅ3 + 1. It is not suggested to

try to run the program for very large size parameters due to the resulting long computation times (in fact, anything over

s=100 becomes tediously long).

Largest@rD := Max@s@rD, Abs@s1@rDD, Abs@s2@rDDD

LastTerm@rD := CeilingAAbsALargest@rD + 4.05 HLargest@rDL1ÅÅÅÅ3 + 2EE ;

Required function definitions

We are going to need spherical Bessel, Neumann and Hankel functions to represent the radial field dependence. First

introduced by Debye, the mean was to simply the notations. These can be given as follow:

yr1 = "#######prÅÅÅÅÅÅÅ

2 Jn+1ê2 HrL, with Bessel function of 1st kind

cn1 = -"#######prÅÅÅÅÅÅÅ

2 Nn+1ê2 HrL, with Bessel function of 2nd kind

hn1 = "#######prÅÅÅÅÅÅÅ

2 H H2L

n+1ê2 HrL = yr1 + i cn

1 with Hankel function

Boundary conditions at r = 0 and r = ¶ require finite fields and hence the need for all three types of Bessel functions.

We go straight to the Ricatti-Bessel functions here by simply multiplying through by r.

Printed by Mathematica for Students

7.1. Mathematica code 71

y@l_, r_D := r $%%%%%%%%%%pÅÅÅÅÅÅÅÅ2 r

BesselJAikjjjl +

1ÅÅÅÅ2

yzzz, r E

H* Ricatti-Bessel function of 1 st kind. *L

yPrime@l_, r_D := Evaluate@D@ y@l, rD, rDD ;H* Partial derivative of Ricatti-Bessel function w.r.t. r. *L

z@l_, r_D := y@l, rD + I r $%%%%%%%%%%pÅÅÅÅÅÅÅÅ2 r

BesselYAikjjjl +

1ÅÅÅÅ2

yzzz, r E ; H* Ricatti-Hankel function *L

zPrime@l_, r_D := Evaluate@D@ z@l, rD, rDD;H* Partial derivative of Ricatti-Hankel function w.r.t. r *L

We also need the Associated Legendre functions for the azimuthal components of the field. These are built into Mathe-

matica. Note that the Mathematica version has a H-1Lm in front of the function definition, which Born and Wolf do not

have (but make mention of). Here, m is the separation constant that arises from the separation of variables technique

used to solve the wave equation and must equal one for this problem. Since m = 1, we get a negative sign out front so I

place another one in my definition of the function to conform with what we actually want. Once that's done, we can

construct the "angle function" coefficients pn and tn . They derive from the Legendre functions

as,

pn :=Pl

1HcosqLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

sinq , and tn :=d Pl

1HcosqLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅdHcosqL

Since, we are interested in writing the Phase function in the dependency of scattering transfer vector q”÷ these "angle

function" are represented as follows,

pn := Pl1@cosH2 sin-1 gLDÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

2 g è!!!!!!!!!!!

1-g2 and

tn :=K1

è!!!!!!!!!!!1-g2 d Pl

1@cosH2 sin-1 gLDÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

dq, where g := qÅÅÅÅÅÅÅÅÅÅ

2 K1

pi@i_, q_D :=LegendrePAi, 1, CosA2 ArcSinA q

ÅÅÅÅÅÅÅÅ2 K1

EEEÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

2 q $%%%%%%%%%%%%%%%%%%%%%%%%%%%1-I qÅÅÅÅÅÅÅÅÅÅÅ2 K1

M2

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ2 K1

;

t@i_, q_D :=1

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ"##########################

H2 K1L2 - q2

i

k

jjjjjjjK1$%%%%%%%%%%%%%%%%%%%%%%%%%%1 -

ikjj

qÅÅÅÅÅÅÅÅÅÅÅ2 K1

yzz2

ikjj-2 CscA2 ArcSinA


EE ikjjH1 + iL LegendrePA-1 + i, 1, CosA2 ArcSinA


EEE -

i CosA2 ArcSinAq

ÅÅÅÅÅÅÅÅÅÅÅ2 K1

EE LegendrePAi, 1, CosA2 ArcSinAq

ÅÅÅÅÅÅÅÅÅÅÅ2 K1

EEEyzzyzzy

zzzzzzz;



ü We now have enough information to construct the Mie coefficients.

an@l_D := Evaluate@Hm2 yPrime @ l, s1@rDD y@l, s2@rDD - m1 y@l, s1@rDD yPrime @ l, s2@rDDLêHm2 zPrime@l, s1@rDD y@l, s2@rDD - m1 z@l, s1@rDD yPrime@l, s2@rDD LD

bn@l_D := Evaluate@Hm2 y@l, s1@rDD yPrime @ l, s2@rDD - m1 yPrime @ l, s1@rDD y@l, s2@rDD L êHm2 z@l, s1@rDD yPrime@l, s2@rDD - m1 zPrime@l, s1@rDD y@l, s2@rDDLD

cn@l_D := Evaluate@ Hm2 z@l, s1@rDD yPrime @ l, s1@rDD - m2 zPrime@l, s1@rDD y@l, s1@rDDLêHm2 z@l, s1@rDD yPrime@l, s2@rDD - m1 zPrime@l, s1@rDD y@l, s2@rDDLD

dn@l_D := Evaluate@Hm2 zPrime@l, s1@rDD y@l, s1@rDD - m2 z @ l, s1@rDD yPrime@l, s1@rDD L êHm2 zPrime@l, s1@rDD y@l, s2@rDD - m1 z@l, s1@rDD yPrime@l, s2@rDD LD

Calculate and plot the relative intensities

Next, we construct the formulae for intensity as a function of scattering transfer vector q”÷ polarized in either of the two

linear orthogonal polarization states. Begin by defining the amplitude matrix elements, S1(r, q) and S2(r, q) which

simplify the form of the resulting phase function equations a little

S1@q_D := ‚l=1

LastTerm@rDikjjjH2 l + 1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅl Hl + 1L

yzzz Han@lD pi@l, qD + bn@lD t@l, qDL;

S2@q_D := ‚l=1

LastTerm@rDikjjjH2 l + 1LÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅl Hl + 1L

yzzz H an@lD t@l, qD + bn@lD pi@l, qD L;

Here are the relative intensities and differential scattering cross sections. The normalization factor (to insure

Ÿ PHcosHqLL „ W = 1) is sometime used in the literature (e.g., Fu and Sun) and sometimes neglected (e.g Bohren and

Huffman). We shall consider the latter. Although the formula for both polarization has been given, hereafter, we shall be

considering only the perpendicular polarization. The kinematical theory can be given as a straightforward formula

ikin HqL := r2 I 4 pr3ÅÅÅÅÅÅÅÅÅÅ3

M2

9 HSin HqrL-qrCos Hq*rLL2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅHqrL6

And the differential scattering cross section for the kinematical theory isdsÅÅÅÅÅÅdW

= r02 ikin HqL

whereas for the Mie theory isdsÅÅÅÅÅÅdW

= imie HqLÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅk0

2 for the perpendicular polarization.

Iperp@q_D := HAbs@S1@qDDL2; H* Relative intensity for perpendicular polarization *LIpar@q_D := HAbs@S2@qDDL2; H* Relative intensity for parallel polarization *L

Ikin@q_D := HrhoL2 i

kjjjj4 pr3ÅÅÅÅÅÅÅÅÅÅÅÅÅ3

y

zzzz

2

9 HSin@qrD - qrCos@qrDL2ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

HqrL6;

H* Relative intensity for kinematical theory *L

SQperp@q_D :=Iperp@qDÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ

K02

; H* Differential scattering cross section for Mie theory in perpendicular polarization *L

SQkin@q_D := r02 Ikin@qD; H* Differential scattering cross section for kinematical theory *L

If the calculation of whole graphs becomes time-consuming, only the values for q=0 can be considered. The relative

intensity and differential scattering cross section can be reduced to the following expression is calculated at q=0,



S10 :=1ÅÅÅÅ2

‚l=1

LastTerm@rD

H2 l + 1L H an@lD + bn@lD L; H* The amplitude function for q=0 *L

Ikin0 := HrhoL2 i

kjjjj4 pr3ÅÅÅÅÅÅÅÅÅÅÅÅÅ3

y

zzzz

2

; H* Relative intensity for kinematical theory at q=0 *L

Iperp0:= HAbs@S0DL2; H* Relative intensity for perpendicular polarization at q=0 *L

SQperp0:=

Iperp0

ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅK0

2; H* Differential scattering cross section for Mie theory at q=0 *L

SQkin0 := r02 Ikin0; H* Differential scattering cross section for kinematical theory at q=0*L

ü Create plots of the relative intensities.

ü Log-linear plots of scattered relative intensity vs. q.

This first cell generates some graphics that make the plots look better.

MyTickLabels = Table@8i, ToString@iD<, 8i, 0.0, 100.0, 10.0<D;XGridLines = Transpose@MyTickLabelsDP1T;

SetOptions@LogPlot, Frame Ø True, PlotRange Ø All,

ImageSize Ø 600, GridLines Ø 8XGridLines, Automatic<,FrameTicks -> 8MyTickLabels, Automatic, None, None<, PlotRange Ø All D;

Here are the plots.

Ikin vs.q

Timing@IKinLogPlot = LogPlot@Ikin@qD, 8q, 0, 100<, PlotStyle Ø RGBColor@0, 0, 1D,FrameLabel Ø 8"q H1ênmL", "Ikin Relative Intensity"<D;D

ListIKinLog = Flatten@Cases@IKinLogPlot, Line@x__D Ø x, InfinityD, 1D;Export@"dataêMie_Roentgen_Ikin.dat", ListIKinLog, "Table"D;Export@"dataêMie_Roentgen_Ikin.eps", IKinLogPlot, "EPS"D

Imie vs. q

Timing@IMieLogPlot =

LogPlot@Evaluate@Iperp@qDD, 8q, 0, 100<, PlotStyle Ø RGBColor@0, 0, 1D,FrameLabel Ø 8"q H1ênmL", "Imie Relative Intensity"<D;D

ListIMieLog = Flatten@Cases@IMieLogPlot, Line@x__D Ø x, InfinityD, 1D;Export@"dataêMie_Roentgen_Imie.dat", ListIMieLog, "Table"D;Export@"dataêMie_Roentgen_Imie.eps", IMieLogPlot, "EPS"D

All graphs on the same plot. The first expression creates a legend for the plot.

ILegend = 888RGBColor@1, 0, 0D, "Imie "<, 8RGBColor@1, 0, 1D, "Ikin "<<,LegendPosition Ø 8.5, 0.3<, LegendSize Ø 80.25, 0.25<, LegendShadow Ø 80, 0<<;

ICompPlot = ShowLegend@Show@IMieLogPlot, IKinLogPlot, PlotRange Ø All,

DisplayFunction Ø Identity, FrameLabel Ø 8"q H1ênmL", "Relative intensities"< D,ILegend, DisplayFunction Ø $DisplayFunction D;

Export@"dataêMie_Roentgen_IComp.eps", ICompPlot, "EPS"D



ü Create plots of the differential scattering cross sections.

ü Log-linear plots of differential scattering cross sections vs. q.

This first cell generates some graphics that make the plots look better.

MyTickLabels = Table@8i, ToString@iD<, 8i, 0.0, 100.0, 10.0<D;XGridLines = Transpose@MyTickLabelsDP1T;

SetOptions@LogPlot, Frame Ø True, PlotRange Ø All,

ImageSize Ø 600, GridLines Ø 8XGridLines, Automatic<,FrameTicks -> 8MyTickLabels, Automatic, None, None<, PlotRange Ø All D;

Here are the plots.

J dsÅÅÅÅÅÅÅÅÅÅÅdW

Nkin

vs.q

TimingASQKinLogPlot = LogPlotASQkin@qD, 8q, 0, 100<, PlotStyle Ø RGBColor@0, 0, 1D,

FrameLabel Ø 9"q H1ênmL", "ikjjjdsÅÅÅÅÅÅÅdW

yzzzkin

Differential scattering cross section"=E;E

ListSQKinLog = Flatten@Cases@SQKinLogPlot, Line@x__D Ø x, InfinityD, 1D;Export@"dataêMie_Roentgen_SQkin.dat", ListSQKinLog, "Table"D;Export@"dataêMie_Roentgen_SQkin.eps", SQKinLogPlot, "EPS"D

J dsÅÅÅÅÅÅÅÅÅÅÅdW

Nmie

vs.q

TimingASQMieLogPlot =

LogPlotAEvaluate@SQperp@qDD, 8q, 0, 100<, PlotStyle Ø RGBColor@0, 0, 1D,

FrameLabel Ø 9"q H1ênmL", "ikjjjdsÅÅÅÅÅÅÅdW

yzzzmie

Differential scattering cross section"=E;E

ListSQMieLog = Flatten@Cases@SQMieLogPlot, Line@x__D Ø x, InfinityD, 1D;Export@"dataêMie_Roentgen_SQmie.dat", ListSQMieLog, "Table"D;Export@"dataêMie_Roentgen_SQmie.eps", SQMieLogPlot, "EPS"D

All graphs on the same plot. The first expression creates a legend for the plot.

SQLegend = 999RGBColor@1, 0, 0D, "ikjjjdsÅÅÅÅÅÅÅdW

yzzzmie

"=, 9RGBColor@1, 0, 1D, "ikjjjdsÅÅÅÅÅÅÅdW

yzzzkin

"==,

LegendPosition Ø 8.5, 0.3<, LegendSize Ø 80.25, 0.25<, LegendShadow Ø 80, 0<=;SQCompPlot = ShowLegend@Show@IMieLogPlot, IKinLogPlot,

PlotRange Ø All, DisplayFunction Ø Identity,

FrameLabel Ø 8"q H1ênmL", "Differential scattering cross section"< D,SQLegend, DisplayFunction Ø $DisplayFunction D;

Export@"dataêMie_Roentgen_SQComp.eps", SQCompPlot, "EPS"D




7.2 IDL code

pro MyMie

; Obtain pi:pii = 4.D0*atan(1.D0)

; nang = number of angles between 0 and 90 degreesnang = 1000

n1 = 1 ; Real part of refractive index of mediumdelta = 0.000019 ; Delta part of the refractive index of scatterern2 = 1 - delta ; Real part of refractive index of scatterer

k1 = 0 ; Imaginary part of refractive index of mediumk2 = 0.00001 ; Imaginary part of refractive index of scatterer

lambda0 = 0.05 ; Vacuum wavelength in "nm"

r = 04.0 ; Sphere radius in "nm"r0 = 0.00000282 ; The classic electron radius

; Derived quantities for given parameter

m1 = n1 ; Complex refractive index of mediumm2 = n2 ; Complex refractive index of scatterer

mRel = m2/m1 ; Relative refractive index between scatterer and medium

lambda1 = lambda0/m1 ; Wavelength in external medium in nmlambda2 = lambda0/m2 ; Wavelength inside sphere in nm

K0 = 2*pii/lambda0 ; Wave number in vacuum in nm^-1K1 = 2*pii/lambda1 ; Wave number in the medium in nm^-1K2 = 2*pii/lambda2 ; Wave number inside the sphere in nm^-1rho = (delta*K0^2)/(2*pii*r0) ; The electron density of the scatterer

; According to the paper of Fu and Sun, the size parameters can be given as follows

sz = K0*r ; size parametersz1 = K1*r ; external size parametersz2 = K2*r ; internal size parameter

LargestTerm = MAX([sz, sz1, sz2]) ;LastTerm = FLOOR(ABS(LargestTerm + 4.05 * (LargestTerm)^(1/3) + 2))

; ..Parameters..mxnang = 2100nmxx= 150000

; Scalar Argumentss1 = dcomplexarr(2*mxnang-1)s2 = dcomplexarr(2*mxnang-1)

7.2. IDL code 77

; Local Arraysmie = fltarr((nang/4)+1)d = dcomplexarr(nmxx)amu = dblarr(mxnang)kin = fltarr((nang/4)+1)pi=dblarr(mxnang)pi0=dblarr(mxnang)pi1=dblarr(mxnang)tau=dblarr(mxnang)

IF (nang GT mxnang) THEN BEGINprint, ’ error: nang > mxnang in bhmie’STOP

ENDIF

IF (nang LT 2) THEN nang = 2

dx = sz1 ; Size parameter as x

y = LastTerm*mRelymod = abs(y)

;; *** Series expansion terminated after NSTOP terms; Logarithmic derivatives calculated from NMX on down

nmx = max(LastTerm,ymod) + 15nmx = fix(nmx)nstop = LastTerm

IF (nmx GT nmxx) THEN BEGINprint, ’error: nmx > nmxx=’, nmxx, ’ for |m|x=’, ymodSTOP

ENDIF

; *** Require NANG.GE.1 in order to calculate scattering intensitiesdang = 0.D0q = 0.D0IF (nang GT 1) then dang = .5D0*pii/double(nang-1)

FOR j=1, nang DO BEGIN ; DO I j = 1, nangtheta = double(j-1)*dangamu(j) = cos(theta)

ENDFOR ; I CONTINUE

FOR j=1, nang DO BEGIN ;DO II j = 1, nangpi0(j) = 0.D0pi1(j) = 1.D0

ENDFOR ; II CONTINUE

nn = 2*nang - 1FOR j=1, nn DO BEGIN ;DO III j = 1, nn


s1(j) = dcomplex(0.D0,0.D0)s2(j) = dcomplex(0.D0,0.D0)

ENDFOR ; III CONTINUE

; Calculate relative intensity for the kinematical theoryFOR j=1, 2*nang-1 DO BEGIN ; DO IV n = 1, 180

theta = double(j-1)*dangq = 2.D0*K0*sin((theta)/2)kin(j) = rho^2*(4.D0*pii*r^3)*(((sin(q*r)-q*r*cos(q*r))^2)/((q*r)^6))

ENDFOR ; IV CONTINUE

; Calculating Mie coefficients using methods of logarithmic derivation of; Bessel Function as explained in Bohren & Hoffman, p.127; *** Logarithmic derivative D(J) calculated by downward recurrence; beginning with initial value (0.,0.) at J=NMX

d(nmx) = DCOMPLEX(0.D0,0.D0)nn = nmx - 1

FOR n=1, nn DO BEGIN ; DO V n = 1, nnen = nmx - n + 1d(nmx-n) = (en/y) - (1.D0/ (d(nmx-n+1)+en/y))

ENDFOR ; V CONTINUE

;; *** Riccati-Bessel functions with real argument X; calculated by upward recurrence;psi0 = COS(dx)psi1 = SIN(dx)chi0 = -SIN(dx)chi1 = COS(dx)xi1 = DCOMPLEX(psi1,-chi1)p = -1.D0FOR n=1, nstop DO BEGIN ; DO VIII n = 1, nstop

en = nfn = (2.D0*en+1.D0)/ (en* (en+1.D0))

; FOR given N, PSI = psi_n CHI = chi_n; PSI1 = psi_n-1 CHI1 = chi_n-1; PSI0 = psi_n-2 CHI0 = chi_n-2; Calculate psi_n and chi_n;

psi = (2.D0*en-1.D0)*psi1/dx - psi0chi = (2.D0*en-1.D0)*chi1/dx - chi0xi = DCOMPLEX(psi,-chi)

;; *** Store previous values of AN and BN for use; in computation of g=<cos(theta)>

IF (n GT 1) THEN BEGINan1 = an

7.2. IDL code 79

bn1 = bnENDIF

;; *** Compute AN and BN:

an = (d(n)/mRel+en/dx)*psi - psi1an = an/ ((d(n)/mRel+en/dx)*xi-xi1)bn = (mRel*d(n)+en/dx)*psi - psi1bn = bn/ ((mRel*d(n)+en/dx)*xi-xi1)

; *** Now calculate scattering intensity pattern; First do angles from 0 to 90

FOR j=1, nang DO BEGIN ; DO VI j = 1, nangjj = 2*nang - jpi(j) = pi1(j)tau(j) = en*amu(j)*pi(j) - (en+1.D0)*pi0(j)p = (-1.D0)^(n-1)s1(j) = s1(j) + fn* (an*pi(j)+bn*tau(j))t = (-1.D0)^n2(j) = s2(j) + fn* (an*tau(j)+bn*pi(j))

;; *** Now do angles greater than 90 using PI and TAU from; angles less than 90.

IF (j NE jj) THEN BEGINs1(jj) = s1(jj) + fn* (an*pi(j)*p+bn*tau(j)*t)s2(jj) = s2(jj) + fn* (bn*pi(j)*t+an*tau(j)*p)

ENDIFENDFOR ; VI CONTINUE

psi0 = psi1psi1 = psichi0 = chi1chi1 = chixi1 = dcomplex(psi1,-chi1)

;; *** Compute pi_n for next value of n; For each angle J, compute pi_n+1; from PI = pi_n , PI0 = pi_n-1;

FOR j=1, nang DO BEGIN ; DO VII j = 1, nangpi1(j) = ((2.D0*en+1.D0)*amu(j)*pi(j)- (en+1.D0)*pi0(j))/$enpi0(j) = pi(j)

ENDFOR ; VII CONTINUEENDFOR ; VIII CONTINUE

; Relative intensity for the Mie theorymie = ABS(s1)^2

; Plot the relative intensities given by both theoriesSET_PLOT, ’X’


DEVICE, DECOMPOSED=0DEVICE, RETAIN=2LOADCT, 39PLOT, mie, /YLOG, COLOR=255, /NOERASEOPLOT, kin, COLOR=240

END

Date post:	13-Nov-2014
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Mie scattering in the X-ray regime

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