Monte Carlo methods in radiative transfer and electron ... · Monte Carlo methods (MCMs) are quite...

Journal of Quantitative Spectroscopy &Radiative Transfer 84 (2004) 437–450

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Monte Carlo methods in radiative transfer andelectron-beam processing

Basil T. Wong, M. P1nar Meng2u3c∗

Radiative Transfer Laboratory, Department of Mechanical Engineering, University of Kentucky, 322 RGAN Building,Lexington, KY 40506, USA

Received 6 March 2003

Abstract

Monte Carlo methods (MCMs) are the most versatile approaches in solving the integro-di5erential equations.They are statistical in nature and can be easily adapted for simulation of the propagation of ensembles ofquantum particles within absorbing, emitting, and scattering media. In this paper, we use MCM for thesolution of the Boltzmann transport equation, which is the governing equation for both radiative transfer andelectron-beam processing. We brie8y outline the methodology for the solution of MCMs, and discuss thesimilarities and di5erences between the two di5erent application areas. The focus of this paper is primarilyon the treatment of di5erent scattering phase functions.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Monte Carlo; Radiative transfer; Electron-beam processing

1. Introduction

Monte Carlo methods (MCMs) are quite versatile in handling the transient and steady particletransport phenomena within participating media. Even though they were once considered di=cult toimplement and computationally ine=cient, with the rapid development of fast and powerful com-puters they have become more e=cient and accurate. Consequently, MC approaches have been usedextensively in the last decade for photon, electron, phonon, neutron transport problems. Two ofthe important research areas for our interest where the MCMs have been fully exploited to studythe transport phenomena are the radiative transfer [1–11] and the electron-beam processing [12–17].Radiative transfer applications of MCM are well-documented in the literature [1,2]. The algorithmsdeveloped for radiative transfer can easily be adapted for modeling of electron transfer phenomena.

∗ Corresponding author. Tel.: +1-859-257-6336/ext. 80658; fax: +1-859-257-3304.E-mail address: [email protected] (M.P. Meng2u3c).

0022-4073/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0022-4073(03)00261-9

mailto:[email protected]

438 B.T. Wong, M.P. Meng2uc3 / Journal of Quantitative Spectroscopy & Radiative Transfer 84 (2004) 437–450

Nomenclature

A atomic weight (kg/mol)C scattering cross-section (m2)d diameter of scatterer (m)E electron energy (keV)f particle distribution function (dimensionless)g average direction cosine in s-direction (dimensionless)h Planck constant (J-s)k wave number (m−1)p polarization branch (dimensionless)R cumulative probability distribution function (dimensionless)Ran A random number (dimensionless)s axis of propagation (dimensionless)W scattering rates (s−1)x size parameter (=�d=�) (dimensionless)Z atomic number (dimensionless)

Greek symbols

� extinction coe=cient (=� + �) (m−1)� absorption coe=cient (m−1)� direction cosine (dimensionless)� photon frequency (1/s)! single scattering albedo (=�=�)� phase function (sr−1)� scattering coe=cient (m−1)� optical thickness (=�Zcr)� scattering polar angle measured from s-axis (rad)’ azimuthal angle measured from an axis normal to s-axis (rad)� wavelength (�m)

Subscripts

e for electrons� wavelength dependent� for photons

Superscripts′ scatteredel elasticinel inelastic

In the context of the electron transport inside solids, analytical solutions are virtually impossibleto obtain due to the complicated electronic band structures [18–20] and the scattering probabilities

B.T. Wong, M.P. Meng2uc3 / Journal of Quantitative Spectroscopy & Radiative Transfer 84 (2004) 437–450 439

[19,20]. With the introduction of MCMs, realistic simulations of propagating electrons are there-fore easily carried out to better understand the electron transport in such systems. Indeed, electronbeam scattering phenomena inside solids have been routinely analyzed via MCMs, leading to highresolution recognition and visualization of nanostructures as done by scanning electron microscopy(SEM), transmission electron microscopy (TEM), and electron energy loss spectroscopy (EELS).

In this paper, our objective is to show the similarities between the MC simulations for the radiativetransfer equation and the electron transport equation. The main reason responsible for the strikingsimilarities between the MCMs used in the two mentioned research areas is the fact that the governingequations are derived from the Boltzmann Transport Equation (BTE) [18–21], which asserts thebalance between the rate of change of the particle distribution and the scattering rates. Here, wewill introduce the BTE. After that we outline the application of MCMs to the solution of the BTEfor both photon and electron transport. The similarity and di5erences between these approaches,particularly the corresponding scattering phase functions are discussed.

2. Boltzmann transport equation (BTE)

The propagation of quantum particles or energy carriers such as electrons, photons, phonons,neutrons, obeys the BTE, which describes the evolution of a particle distribution (say, f) over timeand space as particles undergo a number of scattering events. The general form of the BTE is writtenas [18–21]

@f(r; k ; t)@t

+ v · ∇f(r; k ; t) =(

@f@t

)col

; (1)

where the collision term on the right-hand side is given as(@f@t

)col

=∑r′ ;k′

W (k ′; k)f(r′; k ′; t)−∑

k′

W (k; k ′)f(r; k ; t): (2)

At the right hand side of Eq. (2), W denotes the scattering rate, the Krst summation is the rateof change of f due to the in-scattering of particles while the second is the rate of change of fcaused by the out-scattering of particles. Note that the particle distribution f in general dependson its location in space r, its wave vector k, and time t. In its general form, the BTE is virtuallyintractable owing to the seven independent variables and its integro-di5erential form. Often, it isdesirable to recast the BTE in terms of intensity as the solution of particle distributions in spaceand time may not be required.

2.1. Radiative transfer equation (RTE)

In radiative transfer, the quantum particles are photons, each having energy of h� for a givenfrequency � (or wavelength � = c=�). To derive the RTE, the radiative intensity is deKned in termsof the photon distribution f�, the photon energy h�, the density of states D�, and the speed of


light c:

I�(r; &; '; t) =∑

p

f�(r; p; &; '; t)h�D�(p; &; ')c: (3)

The subscript � is used to indicate that the equation is for a given speciKc frequency. The summationin Eq. (3) is to be performed over all polarization branches, which implies that the change ofpolarization of photons is ignored in this case. 1

Multiplying Eq. (1) by the photon energy, the density of states, and the speed of light, andthen performing the summation over all the polarization branches, we cast the BTE in terms of theradiative intensity:

@I�

@t+ v� · ∇I� =

∑&′ ;'′

W�(&′; '′; &; ')I ′�(&′; '′; t)−

∑&′ ;'′

W�(&; '; &′; '′)I�(r; &; '; t): (4)

To further simplify the BTE we rewrite the in-scattering term as∑&′ ;'′

W�(&′; '′; &; ')I ′�(&′; '′; t) =

��

4�

∫(′

��(&′; '′; &; ')I ′�(&′; '′; t) d(′; (5)

where �� is the scattering coe=cient and �� the phase function of the medium. The in-scatteringterm contains all the contributions from the entire spherical solid angle (′. The out-scattering termis given as∑

&′ ;'′

W�(&; '; &′; '′)I�(r; &; '; t) = (�� + ��)I�(r; &; '; t); (6)

with �� being the absorption coe=cient. After rearranging we obtain the familiar form of the RTE[2,22,23]:

@I�

@t+ v� · ∇I� =−��I� +

��

4�

∫(′

��(&′; '′; &; ')I ′�(&′; '′; t) d(′: (7)

Note that �� is the extinction coe=cient, which is the sum of �� and ��. Since we deal withmostly cold media in this paper, the emission term is dropped. The out-scattering term includesthe absorptions and elastic out-scatterings of photons. One important thing to note for the radiativetransfer is that the number of photons does not conserve; in other words, photons can be createdand destroyed during inelastic scattering processes. Inelastic scatterings in general mean that theensemble of photons is attenuated in terms of the population of photons (but not the frequency ofthe photons) which in turn reduces the energy of the entire ensemble.

2.2. Electron transport equation (ETE)

In the Keld of the electron-beam processing, free electrons are the energy carriers. The energyof an electron is always characterized by its wave number k (k = 1=�) instead of its frequency or

1 Note that polarization can be considered in the radiative transfer formulation as discussed in the paper by Vaillon etal. (see this issue of the JQSRT; Eurotherm 73 paper #26).


wavelength. Unlike photons, free (propagating) electrons do not undergo absorption by other parti-cles, which implies that inelastic scatterings change the energy of these carriers but do not attenuatethe number of carriers. Therefore the wave number or the wavelength of the electrons changes asthey undergo inelastic scatterings [18–21]. From the computational standpoint, the scattering crosssection changes once the energy of a propagating electron ensemble is altered [15] (see Eq. (20)).

To derive the ETE, we Krst deKne intensity for electrons similar to that of Eq. (3):

Ie(r; E; &; '; t) = fe(r; E; &; '; t)EeDe(E; &; ')ve(E): (8)

After casting the BTE in terms of electron intensity, the ETE is given as

@Ie@t

+ ve · ∇Ie =−[�inele (E) + �el

e (E)]Ie +�ele (E)4�

×∑E′

∫(′

�e(E′; &′; '′;E; &; ')I ′e(E; &′; '′; t) d(′: (9)

The ETE resembles the RTE except that the scattering coe=cients �e’s, as well as the phase function�e, depend on the energy (i.e., wave number k) of the electron beam. It is not di=cult to converta MCM developed for radiative transfer into a MCM in electron-beam processing with correctimplementations of the electronic scattering properties. However, computations of the ETE are usuallymore involved than those of the RTE.

3. MCM in radiative transfer

Before attempting to derive the scattering probabilities and the distance of interactions for photonsand electrons, it is informative to have an overview of the problem description and the list ofassumptions. To better compare MCMs in radiative transfer and in electron-beam processing, weconsider a simple homogeneous, absorbing and scattering medium. In this case, we assume thatthe geometry of the problem can be in any conKguration. The procedures of the simulation remainunaltered for any given geometry except the deKnition of exiting boundaries for the particles (i.e., the8exibility of a MCM). The boundaries are assumed to be transparent, i.e. non-emitting, non-absorbingor non-re8ecting. A monochromatic laser beam is incident on the boundary as shown in Fig. 1. The

particle beam

quantumparticles

transparent boundary

participatingmedium

Fig. 1. 2D-representation of the geometry considered in MC simulations.


0 20 40 60 80 100 120 140 160 18010-3

10-2

10-1

100

101

102

HG g = 0.9 g = 0.5 g = −0.5 g = −0.9

Miex = 1x = 2x = 3x = 4

Φν(

Θ)

(sr-1

)

Θ (degrees)

Fig. 2. Typical photon scattering phase functions for di5erent size parameters (x = �d=�) in the Lorenz–Mie (LM) theoryand for various asymmetry factors (g) in the Henyey–Greenstein (HG) phase function.

medium is considered to be at zero temperature, which helps to eliminate the emission contributionto the radiant energy distribution. The general procedures of constructing a MCM can be found inRefs. [5,6,10,11,15]; therefore, they are not repeated here. Below, we discuss only the most pertainingdetails.

3.1. Directions of propagation

The direction of photon scattering determines the complexity of the problem. For the isotropicscattering phase function, the probability that a photon is scattered in any given direction is uniform.The direction of scattering is sampled from the expression of all possible solid angles. The scatteringpolar and azimuthal angles in the simulations can be determined using random numbers (Ran’ andRan�) [10,11]:

’ = 2�Ran’; (10)

� = cos−1(1− 2Ran�); (11)

where Ran’ and Ran� may vary from 0 to 1. For spherical particles, the Lorenz–Mie Theory predictsthe scattering phase function at a given scattering polar angle �. The phase function obtainedfrom the Lorenz–Mie theory is often expressed as a series in terms of orthogonal functions. Themost common representation involves the Legendre polynomials; a typical phase function assumingazimuthal symmetry is expressed as [2,24]

��(�) =∞∑

n=0

anPn(�); (12)

where Pn are the Legendre polynomials of the nth order. Typical phase functions as obtained fromthe Lorenz–Mie theory are plotted in Fig. 2 where x is the size parameter deKned as �d=�. Here, d


refers to the diameter of the scatterers inside the medium. Use of the Legendre Polynomials in MonteCarlo techniques is quite involved as it is not easy to invert such a phase function to obtain �. Inorder to overcome this setback, we build a table containing all the scattering data and interpolate asrequired during the simulation [25].

The Henyey–Greenstein (HG) phase function can be used to simplify the anisotropic scatteringphase function. The scattering polar angle � as a function of random number is obtained as [10]

� = cos−1

{12g

[1 + g2 −

(1− g2

1− g + 2gRan�

)2]}

; (13)

where g is the asymmetry factor (i.e., the average value of the direction cosine in the propagatingdirection). When g ∼ 1 highly forward scattering is implied, while g ∼ −1 means highly back-ward scattering. Determination of the scattering azimuthal angle ’ will follow that of the isotropicscattering as given in Eq. (10). Although the implementation of the HG phase function in a MCMis straightforward and convenient, it often compromises the correct physics representation of thecomplete proKle.

3.2. Distance of interaction

The interaction distance governs the distance a bundle travels without being scattered. Generally,there are three di5erent approaches for determining the distance of interaction, which have beenexamined and reported before [11]. Here, we consider only one of them (referred to as M2 in Ref.[11]) where the distance of interaction between elastic scattering events S� is sampled as

S� =−1�ln(Ran�); (14)

and the random number, Ran� is between 0 and 1 (i.e. 0¡Ran� ¡ 1). Note that the inverse ofthe scattering coe=cient � is actually the mean free path of the photons between elastic scatteringevents.

3.3. Attenuation of photons

The attenuation of photons for a given distance S depends on the absorption coe=cient �. Thefraction of photons in an ensemble recovered after travelling a distance S is given as e−�S while(1− e−�S) is the fraction absorbed. Note that this is not the only method for treating the attenuationof photons. Other approaches can be found in Ref. [11].

4. MCM in electron-beam processing

In describing the MCM in the electron-beam processing, we will consider the same geometricaldetails as stated in Section 3. However, instead of impinging a laser beam upon a participatingmedium, an electron beam is assumed incident on a solid material. The medium is consideredhomogeneous and free of defects and cracks and it is not subjected to any other external forcesimposed by the electric Keld. The probability distributions needed for the MC simulations for theelectron-beam processing are discussed below.


4.1. Directions of propagation

The HG and the Lorenz–Mie theories provide the scattering phase functions for photons; counter-parts for electrons are the screened Rutherford and the Mott scattering cross-sections, respectively[15,26]. The screened Rutherford di5erential cross section for a given solid (or an atomic numberZ) has the following form [15]:

dCele (�; E)d(

= 5:21× 10−21 Z2

E2

(E + 511E + 1024

)2 (sin2

(�2

)+ .

)−2

; (15)

where

. = 3:4× 10−3 Z0:67

E: (16)

Therefore, the phase function �e for the electron scattering can be written as

�e(�; E) =4�

Cele; total(E)

dCele (�; E)d(

; (17)

where

Cele; total(E) =

∫(=4�

dCele (�; E)d(

d(: (18)

The scattering polar angle can be obtained as [15]

� = cos−1

(1− 2.Ran�

1 + . − Ran�

); (19)

where Ran� is a random number. Note that � depends the energy of the electron ensemble. Such anexplicit expression for the scattering polar angle has a limitation. Similar to the HG phase functionwhich is typically inaccurate in representing the backscattering, the screened Rutherford cross sectionis inaccurate when it comes to low-energy (i.e., ¡ 10 keV) electron beams.

In order to correctly represent the scattering phase functions for both the low- and high-energyelectron beams, the Mott scattering cross section should be employed. The Mott scattering crosssection for an unpolarized electron beam is typically given as [26]

dCele (�; E)d(

= |/|2 + |0|2; (20)

where

/(�; E) =1

2i√

E2 − 1

∞∑n=0

{(n + 1)[exp(2i1−n−1)− 1] + n[exp(2i1n)− 1]}Pn(cos�); (21)

0(�; E) =1

2i√

E2 − 1

∞∑n=1

(−exp(2i1−n−1) + exp(2i1n))P∗n (cos�): (22)

Here, 1n’s are the Dirac phase shifts, Pn’s and P∗n ’s are the ordinary Legendre polynomials and

the associated Legendre polynomials, respectively. Details of the Mott scattering cross section are


0 20 40 60 80 100 120 140 160 18010-3

10-2

10-1

100

101

102

103 Mott 0.02 keV 0.60 keV 5.00 keV 25.00 keV

Rutherford 0.02 keV 0.60 keV 5.00 keV

25.00 keV

Φe(

Θ)

(sr-1

)

Θ (degrees)

Fig. 3. The Rutherford and the Mott scattering phase functions for electron in gold for various electron energies; adaptedfrom Refs. [15,26].

reported in Ref. [26]. Since azimuthal symmetry is always assumed, the azimuthal angle for scatteringis obtained as in Eq. (10). The Rutherford and the Mott scattering phase functions in gold for severalselected electron energies are illustrated in Fig. 3.

4.2. Distance of interaction

Using the same arguments given in Section 3.2, the interaction distance between elastic scatteringevents for electrons is determined as

S =− 1�eleln(Rans); (23)

where

�ele =

Na3Cele; total

A; (24)

here �ele is the elastic scattering coe=cient, which depends on the atomic number Z , the atomic

weight A, the density 3 of the solid target, and the electron energy E. Note that Na is the Avogadronumber.

4.3. Inelastic scatterings

In the electron-beam processing, the electron stopping power dE=ds is used to determine theattenuation of the electron energy along the distance of interaction. The stopping power is basicallythe amount of electron energy lost per unit traveled distance and is deKned based on the totalinelastic scattering cross section as [27]

dEds

=(

Na3A

)TEinele C inel

e; total; (25)


where TEinele is the average energy loss per inelastic scattering event. Therefore, the inelastic scattering

coe=cient can be obtained by rearranging the above equation as

�inele =

1TEinele

− dEds

=(

Na3A

)C inele; total: (26)

In principle, accounting the amount of energy lost in the MCM for electron-beam processing as theensemble of electrons propagates through a distance of interaction follows the same procedures asthose in Section 3.2; except, � is to be replaced by �inel

e . However, the attenuation calculation isoften performed by Krst determining the stopping power and then multiplying it by the distance ofinteraction to determine the amount of electron energy lost within the interval.

One of the commonly used expressions for the stopping power is given by the modiKed Betherelationship, which is expressed as [15]

dEds

=−78; 500Z3AE

loge

(1:166(E + 0:85J )

J

): (27)

Here J , the mean ionization potential, is assumed to be available from the experiments [15]. Theshortcoming of the Bethe relationship is that it incorrectly represents the stopping power at lowelectron energies (i.e., ¡ 1 keV). Recently, a compilation of the experimental data containing thestopping powers for most of the elements in the periodic table including compounds was presentedby Joy [28]. These experimental data demonstrate a wider range of application in various electronenergies, and they can be easily incorporated in a MCM which further improve the accuracy ofMCM results.

5. Sample results from MCMs

In the following sections a series of results obtained using MCMs are presented for plane-parallelmedia with a normal incident particle beam; all other assumptions are as stated in Section 3. Thecoordinate system is chosen such that the point of incidence of the beam is at the origin. Thegeometry is deKned as a rectangular volume with Xcr being the width and Ycr being the length. Theupper boundary is considered to be at z =0 while the lower boundary is located at z = Zcr, and r isdeKned as the radial distance from the origin. During the simulation, we assumed that Xcr and Ycr

approach inKnity.

5.1. MCM in radiative transfer

The following results are for di5erent incident beam proKles and for a highly scattering medium(! = 0:99) with an optical thickness of 2. The Krst incident beam proKle considered is an impulsefunction. The second one is a 8at beam proKle with a radius of 1 mm (see [5,10]). The third caseis a Gaussian beam proKle with a 1=e2-radius of 1 mm (see [5,10]). Comparisons for these threecases reveal that their e5ects are primarily on the radial distributions of photons. (Note that severalsimilar parametric study results can be presented; however, because of the space requirements welimit the discussion to only the phase function e5ects.)

Absorptions of photons within media subject to three di5erent beam proKles are depicted in Fig. 4.The results are normalized with the total energy (or power) of the incident beam. The radiant energy


0.1

0.2

0.3 0.0

0.5

1.0

1.5

2.00

50

100

150

200

250

300

350

Abs

orpt

ion

x 10

3 (m

m-3

)

z (mm)r (mm)

1

2

3

4 0.0

0.5

1.0

1.5

2.00

1

2

3

4

5

Abs

orpt

ion

x 10

3 (m

m-3

)

z (mm)r (mm)

1

2

3

4 0.0

0.5

1.0

1.5

2.00

2

4

6

8

Abs

orpt

ion

x 10

3 (m

m-3

)

z (mm)r (mm)

(a)

(b)

(c)

Fig. 4. Normalized absorption contours (with the total energy or power of the incident beam) in units of (×103=mm3) fordi5erent incident beam proKles: (a) impulse incident beam at r=0, (b) 8at incident beam with a radius of 1 mm, and (c)Gaussian incident beam with a 1=e2 radius of 1 mm, in isotropic scattering media with ! = 0:99, � = 2, and Zcr = 2 mm.


2040

6080

10030

6090

120150

0

100

200

300

400

500

600

Dep

ositi

on x

109 (

nm-3

)

z (nm)r (nm)

2040

6080

10030

6090

120150

0

100

200

300

400

500

600

Dep

ositi

on x

109 (

nm-3

)

z (nm)r (nm)

(a)

(b)

Fig. 5. The normalized deposition (with the total energy or power of the incident beam) of electron energies withingold in units of (×109=nm3) for an applied voltage of 20 kV. An incident Gaussian beam of a 1=e2-radius of 25 nm isconsidered in the MCM simulations. (a) Rutherford, and (b) Mott scattering cross sections. The solid in each case isinKnite in thickness.

absorbed is in much concentrated area when impulse beam proKle is considered. On the other hand,the 8at beam case has the largest di5usion of the energy absorbed. Note that the energy absorbed asa function of z (i.e., after integrating over r) is the same for all three cases. If we were to evaluatethe absorption in terms of temperature, the impulse case would have the highest temperature at theupper boundary, followed by the Gaussian case, and Knally the 8at beam case.

5.2. MCM in electron-beam processing

Fig. 5 depicts the depositions of electron energies due to the electron bombardments betweenpropagating electrons and lattices of solid, as obtained using the Rutherford and the Mott scattering


cross sections (see Section 4.1). The voltage applied in these cases is 20 kV. Both Kgures have thesame level of contour divisions, indicating that the grey shadings have the same values, correspond-ingly. The energies of the incident electrons are 20 keV, which is considered to be su=ciently highfor using the Rutherford cross section without signiKcant errors. This is evident from the Kguresas one can observe that the deposition contours in both cases are similar. An increase in the ap-plied voltage deKnitely increases the validity of using the Rutherford cross section while a decreasecertainly requires the use of the Mott cross section in the MC simulation.

6. Conclusions

The use of the laser and electron beams as diagnostic tools has been the focal point in manyresearch areas. Due to the in-scattering nature of the particle transport, obtaining analytical solutionsto the transport equation is considered to be extremely di=cult; therefore, statistical methods suchas Monte Carlo methods (MCM) are generally used. In this paper, the MCMs for the radiativetransfer equation (RTE) and the electron transport equation (ETE) have been discussed. Since bothgoverning equations are derived from the Boltzmann transport equation (BTE) they share the commonsimulation procedures except that each needs di5erent scattering probabilities and properties. Detailson how to obtain these scattering probabilities and properties are presented. The problem consideredhere is relatively simple where the boundaries are transparent and the medium is homogeneous.The treatment of the mismatched boundaries for the radiative transfer, i.e. di5erent refractive indicesbetween the surroundings and the medium, is not discussed in this context. However, they can befound in Refs. [6,10]. Also, the electron scatterings at the interface between two di5erent solids arenot considered here. It should be noted that MCMs generally assume the wavelengths of the particlesare small compared to the characteristic length of the object of interest. Should this condition beviolated the problem is to be solved using wave theories.

Acknowledgements

This work is supported by an NSF Nanoscale Interdisciplinary Research Team (NIRT) award fromthe Nano Manufacturing program in Design, Manufacturing, and Industrial Innovation (DMI-0210559).In addition, Basil T. Wong is supported by a TVA fellowship during this study.

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