Monte Carlo methods in radiative transfer and electron-beam processing

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1 Journal of Quantitative Spectroscopy & Radiative Transfer 84 (2004) Monte Carlo methods in radiative transfer and electron-beam processing Basil T. Wong, M. Pnar Menguc 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-dierential equations. They are statistical in nature and can be easily adapted for simulation of the propagation of ensembles of quantum particles within absorbing, emitting, and scattering media. In this paper, we use MCM for the solution of the Boltzmann transport equation, which is the governing equation for both radiative transfer and electron-beam processing. We briey outline the methodology for the solution of MCMs, and discuss the similarities and dierences between the two dierent application areas. The focus of this paper is primarily on the treatment of dierent 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 particle transport phenomena within participating media. Even though they were once considered dicult to implement and computationally inecient, with the rapid development of fast and powerful computers they have become more ecient and accurate. Consequently, MC approaches have been used extensively in the last decade for photon, electron, phonon, neutron transport problems. Two of the important research areas for our interest where the MCMs have been fully exploited to study the 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 algorithms developed for radiative transfer can easily be adapted for modeling of electron transfer phenomena. Corresponding author. Tel.: /ext ; fax: address: menguc@engr.uky.edu (M.P. Menguc) /$ - see front matter? 2003 Elsevier Ltd. All rights reserved. doi: /s (03)

2 438 B.T. Wong, M.P. Menguc / Journal of Quantitative Spectroscopy & Radiative Transfer 84 (2004) Nomenclature A atomic weight (kg/mol) C scattering cross-section (m 2 ) 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 coecient (= + ) (m 1 ) absorption coecient (m 1 ) direction cosine (dimensionless) photon frequency (1/s)! single scattering albedo (==) phase function (sr 1 ) scattering coecient (m 1 ) optical thickness (=Z cr ) 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 scattered el elastic inel inelastic In the context of the electron transport inside solids, analytical solutions are virtually impossible to obtain due to the complicated electronic band structures [18 20] and the scattering probabilities

3 B.T. Wong, M.P. Menguc / Journal of Quantitative Spectroscopy & Radiative Transfer 84 (2004) [19,20]. With the introduction of MCMs, realistic simulations of propagating electrons are therefore easily carried out to better understand the electron transport in such systems. Indeed, electron beam scattering phenomena inside solids have been routinely analyzed via MCMs, leading to high resolution 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 radiative transfer equation and the electron transport equation. The main reason responsible for the striking similarities between the MCMs used in the two mentioned research areas is the fact that the governing equations are derived from the Boltzmann Transport Equation (BTE) [18 21], which asserts the balance between the rate of change of the particle distribution and the scattering rates. Here, we will introduce the BTE. After that we outline the application of MCMs to the solution of the BTE for both photon and electron transport. The similarity and dierences 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 time and space as particles undergo a number of scattering events. The general form of the BTE is written as [18 r; + ṽ f( r; k;t)= ( ; col where the collision term on the right-hand side is given as ( = r W ( k ; k)f( r ; k ;t) k W ( k; k )f( r; k;t): ; k col At the right hand side of Eq. (2), W denotes the scattering rate, the rst summation is the rate of change of f due to the in-scattering of particles while the second is the rate of change of f caused by the out-scattering of particles. Note that the particle distribution f in general depends on its location in space r, its wave vector k, and time t. In its general form, the BTE is virtually intractable owing to the seven independent variables and its integro-dierential form. Often, it is desirable to recast the BTE in terms of intensity as the solution of particle distributions in space and time may not be required Radiative transfer equation (RTE) In radiative transfer, the quantum particles are photons, each having energy of h for a given frequency (or wavelength = c=). To derive the RTE, the radiative intensity is dened in terms of the photon distribution f, the photon energy h, the density of states D, and the speed of

4 440 B.T. Wong, M.P. Menguc / Journal of Quantitative Spectroscopy & Radiative Transfer 84 (2004) light c: I ( r; ; ; t)= p f ( r; p; ; ; t)hd (p; ; )c: (3) The subscript is used to indicate that the equation is for a given specic frequency. The summation in Eq. (3) is to be performed over all polarization branches, which implies that the change of polarization of photons is ignored in this case. 1 Multiplying Eq. (1) by the photon energy, the density of states, and the speed of light, and then performing the summation over all the polarization branches, we cast the BTE in terms of the + ṽ I = ; W ( ; ; ; )I ( ; ;t) ; W (; ; ; )I ( r; ; ; t): (4) To further simplify the BTE we rewrite the in-scattering term as W ( ; ; ; )I ( ; ;t)= ( ; ; ; )I 4 ( ; ;t)d ; (5) ; where is the scattering coecient and the phase function of the medium. The in-scattering term contains all the contributions from the entire spherical solid angle. The out-scattering term is given as W (; ; ; )I ( r; ; ; t)=( + )I ( r; ; ; t); (6) ; with being the absorption coecient. After rearranging we obtain the familiar form of the RTE + ṽ I = I + ( ; ; ; 4 ( ; ;t)d : (7) Note that is the extinction coecient, which is the sum of and. Since we deal with mostly cold media in this paper, the emission term is dropped. The out-scattering term includes the absorptions and elastic out-scatterings of photons. One important thing to note for the radiative transfer is that the number of photons does not conserve; in other words, photons can be created and destroyed during inelastic scattering processes. Inelastic scatterings in general mean that the ensemble of photons is attenuated in terms of the population of photons (but not the frequency of the photons) which in turn reduces the energy of the entire ensemble Electron transport equation (ETE) In the eld of the electron-beam processing, free electrons are the energy carriers. The energy of 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 et al. (see this issue of the JQSRT; Eurotherm 73 paper #26).

5 B.T. Wong, M.P. Menguc / Journal of Quantitative Spectroscopy & Radiative Transfer 84 (2004) wavelength. Unlike photons, free (propagating) electrons do not undergo absorption by other particles, which implies that inelastic scatterings change the energy of these carriers but do not attenuate the number of carriers. Therefore the wave number or the wavelength of the electrons changes as they undergo inelastic scatterings [18 21]. From the computational standpoint, the scattering cross section changes once the energy of a propagating electron ensemble is altered [15] (see Eq. (20)). To derive the ETE, we rst dene intensity for electrons similar to that of Eq. (3): I e ( r; E; ; ; t)=f e ( r; E; ; ; t)e e D e (E; ; )v e (E): (8) After casting the BTE in terms of electron intensity, the ETE is given + ṽ e I e = [e inel (E)+e el (E)]I e + el e (E) 4 E e (E ; ; ; E; ; )I e(e; ; ;t)d : (9) The ETE resembles the RTE except that the scattering coecients 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 dicult to convert a MCM developed for radiative transfer into a MCM in electron-beam processing with correct implementations of the electronic scattering properties. However, computations of the ETE are usually more involved than those of the RTE. 3. MCM in radiative transfer Before attempting to derive the scattering probabilities and the distance of interactions for photons and electrons, it is informative to have an overview of the problem description and the list of assumptions. To better compare MCMs in radiative transfer and in electron-beam processing, we consider a simple homogeneous, absorbing and scattering medium. In this case, we assume that the geometry of the problem can be in any conguration. The procedures of the simulation remain unaltered for any given geometry except the denition of exiting boundaries for the particles (i.e., the exibility of a MCM). The boundaries are assumed to be transparent, i.e. non-emitting, non-absorbing or non-reecting. A monochromatic laser beam is incident on the boundary as shown in Fig. 1. The particle beam transparent boundary quantum particles participating medium Fig. 1. 2D-representation of the geometry considered in MC simulations.

6 442 B.T. Wong, M.P. Menguc / Journal of Quantitative Spectroscopy & Radiative Transfer 84 (2004) Φ ν (Θ) (sr -1 ) 10 2 Mie HG x = 1 g = x = 2 g = 0.5 x = 3 g = 0.5 x = 4 g = Θ (degrees) Fig. 2. Typical photon scattering phase functions for dierent size parameters (x = d=) in the Lorenz Mie (LM) theory and 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 contribution to the radiant energy distribution. The general procedures of constructing a MCM can be found in Refs. [5,6,10,11,15]; therefore, they are not repeated here. Below, we discuss only the most pertaining details Directions of propagation The direction of photon scattering determines the complexity of the problem. For the isotropic scattering 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 scattering polar and azimuthal angles in the simulations can be determined using random numbers (Ran and Ran )[10,11]: =2Ran ; (10) = cos 1 (1 2Ran ); where Ran and Ran may vary from 0 to 1. For spherical particles, the Lorenz Mie Theory predicts the scattering phase function at a given scattering polar angle. The phase function obtained from the Lorenz Mie theory is often expressed as a series in terms of orthogonal functions. The most common representation involves the Legendre polynomials; a typical phase function assuming azimuthal symmetry is expressed as [2,24] ()= a n P n (); (12) n=0 where P n are the Legendre polynomials of the nth order. Typical phase functions as obtained from the Lorenz Mie theory are plotted in Fig. 2 where x is the size parameter dened as d=. Here, d (11)

7 B.T. Wong, M.P. Menguc / Journal of Quantitative Spectroscopy & Radiative Transfer 84 (2004) refers to the diameter of the scatterers inside the medium. Use of the Legendre Polynomials in Monte Carlo techniques is quite involved as it is not easy to invert such a phase function to obtain. In order to overcome this setback, we build a table containing all the scattering data and interpolate as required during the simulation [25]. The Henyey Greenstein (HG) phase function can be used to simplify the anisotropic scattering phase function. The scattering polar angle as a function of random number is obtained as [10] { [ ( ) = cos g 2 1 g 2 2 ]} ; (13) 2g 1 g +2gRan where g is the asymmetry factor (i.e., the average value of the direction cosine in the propagating direction). When g 1 highly forward scattering is implied, while g 1 means highly backward scattering. Determination of the scattering azimuthal angle will follow that of the isotropic scattering as given in Eq. (10). Although the implementation of the HG phase function in a MCM is straightforward and convenient, it often compromises the correct physics representation of the complete prole Distance of interaction The interaction distance governs the distance a bundle travels without being scattered. Generally, there are three dierent approaches for determining the distance of interaction, which have been examined 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 ); and the random number, Ran is between 0 and 1 (i.e. 0 Ran 1). Note that the inverse of the scattering coecient is actually the mean free path of the photons between elastic scattering events Attenuation of photons The attenuation of photons for a given distance S depends on the absorption coecient. The fraction 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 attenuation of photons. Other approaches can be found in Ref. [11]. (14) 4. MCM in electron-beam processing In describing the MCM in the electron-beam processing, we will consider the same geometrical details as stated in Section 3. However, instead of impinging a laser beam upon a participating medium, an electron beam is assumed incident on a solid material. The medium is considered homogeneous and free of defects and cracks and it is not subjected to any other external forces imposed by the electric eld. The probability distributions needed for the MC simulations for the electron-beam processing are discussed below.

8 444 B.T. Wong, M.P. Menguc / Journal of Quantitative Spectroscopy & Radiative Transfer 84 (2004) Directions of propagation The HG and the Lorenz Mie theories provide the scattering phase functions for photons; counterparts for electrons are the screened Rutherford and the Mott scattering cross-sections, respectively [15,26]. The screened Rutherford dierential cross section for a given solid (or an atomic number Z) has the following form [15]: where dce el (; E) =5: Z 2 d E 2 ( ) E ( ( ) 2 sin 2 + ) ; (15) E :67 3 Z =3:4 10 E : Therefore, the phase function e for the electron scattering can be written as where e (; E)= 4 Ce; el total(e)= =4 dcel Ce;total el (E) (16) e (; E) ; (17) d dce el (; E) d: (18) d The scattering polar angle can be obtained as [15] ( = cos 1 1 2Ran ) ; (19) 1+ Ran where Ran is a random number. Note that depends the energy of the electron ensemble. Such an explicit expression for the scattering polar angle has a limitation. Similar to the HG phase function which is typically inaccurate in representing the backscattering, the screened Rutherford cross section is 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-energy electron beams, the Mott scattering cross section should be employed. The Mott scattering cross section for an unpolarized electron beam is typically given as [26] dce el (; E) = ; (20) d where 1 (; E)= 2i {(n + 1)[exp(2i n 1 ) 1] + n[exp(2i n ) 1]}P n (cos ); (21) E 2 1 n=0 1 (; E)= 2i ( exp(2i n 1 ) + exp(2i n ))P E 2 n (cos ): (22) 1 n=1 Here, n s are the Dirac phase shifts, P n s and P n s are the ordinary Legendre polynomials and the associated Legendre polynomials, respectively. Details of the Mott scattering cross section are

9 B.T. Wong, M.P. Menguc / Journal of Quantitative Spectroscopy & Radiative Transfer 84 (2004) Φ e (Θ) (sr -1 ) 10 3 Mott 0.02 kev 0.60 kev kev kev Rutherford 0.02 kev 0.60 kev 5.00 kev kev Θ (degrees) Fig. 3. The Rutherford and the Mott scattering phase functions for electron in gold for various electron energies; adapted from Refs. [15,26]. reported in Ref. [26]. Since azimuthal symmetry is always assumed, the azimuthal angle for scattering is obtained as in Eq. (10). The Rutherford and the Mott scattering phase functions in gold for several selected electron energies are illustrated in Fig Distance of interaction Using the same arguments given in Section 3.2, the interaction distance between elastic scattering events for electrons is determined as S = 1 ln(ran e el s ); (23) where e el = N ace; el total ; (24) A here e el is the elastic scattering coecient, which depends on the atomic number Z, the atomic weight A, the density of the solid target, and the electron energy E. Note that N a is the Avogadro number Inelastic scatterings In the electron-beam processing, the electron stopping power de=ds is used to determine the attenuation of the electron energy along the distance of interaction. The stopping power is basically the amount of electron energy lost per unit traveled distance and is dened based on the total inelastic scattering cross section as [27] de ds = ( ) Na A E inel e Ce; inel total; (25)

10 446 B.T. Wong, M.P. Menguc / Journal of Quantitative Spectroscopy & Radiative Transfer 84 (2004) where E inel e coecient can be obtained by rearranging the above equation as is the average energy loss per inelastic scattering event. Therefore, the inelastic scattering ( Na ) C inel e; total: (26) e inel = 1 de E inel e ds = A In principle, accounting the amount of energy lost in the MCM for electron-beam processing as the ensemble of electrons propagates through a distance of interaction follows the same procedures as those in Section 3.2; except, is to be replaced by e inel. However, the attenuation calculation is often performed by rst determining the stopping power and then multiplying it by the distance of interaction 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 modied Bethe relationship, which is expressed as [15] de ds = 78; 500 Z AE log e ( 1:166(E +0:85J ) J ) : (27) Here J, the mean ionization potential, is assumed to be available from the experiments [15]. The shortcoming of the Bethe relationship is that it incorrectly represents the stopping power at low electron energies (i.e., 1 kev). Recently, a compilation of the experimental data containing the stopping powers for most of the elements in the periodic table including compounds was presented by Joy [28]. These experimental data demonstrate a wider range of application in various electron energies, and they can be easily incorporated in a MCM which further improve the accuracy of MCM results. 5. Sample results from MCMs In the following sections a series of results obtained using MCMs are presented for plane-parallel media with a normal incident particle beam; all other assumptions are as stated in Section 3. The coordinate system is chosen such that the point of incidence of the beam is at the origin. The geometry is dened as a rectangular volume with X cr being the width and Y cr being the length. The upper boundary is considered to be at z = 0 while the lower boundary is located at z = Z cr, and r is dened as the radial distance from the origin. During the simulation, we assumed that X cr and Y cr approach innity MCM in radiative transfer The following results are for dierent incident beam proles and for a highly scattering medium (! =0:99) with an optical thickness of 2. The rst incident beam prole considered is an impulse function. The second one is a at beam prole with a radius of 1 mm (see [5,10]). The third case is a Gaussian beam prole with a 1=e 2 -radius of 1 mm (see [5,10]). Comparisons for these three cases reveal that their eects are primarily on the radial distributions of photons. (Note that several similar parametric study results can be presented; however, because of the space requirements we limit the discussion to only the phase function eects.) Absorptions of photons within media subject to three dierent beam proles are depicted in Fig. 4. The results are normalized with the total energy (or power) of the incident beam. The radiant energy

11 B.T. Wong, M.P. Menguc / Journal of Quantitative Spectroscopy & Radiative Transfer 84 (2004) Absorption x 10 3 (mm -3 ) (a) r (mm) z (mm) Absorption x 10 3 (mm -3 ) (b) 0 1 r (mm) z (mm) Absorption x 10 3 (mm -3 ) (c) 0 1 r (mm) z (mm) Fig. 4. Normalized absorption contours (with the total energy or power of the incident beam) in units of ( 10 3 =mm 3 ) for dierent incident beam proles: (a) impulse incident beam at r = 0, (b) at incident beam with a radius of 1 mm, and (c) Gaussian incident beam with a 1=e 2 radius of 1 mm, in isotropic scattering media with! =0:99, = 2, and Z cr = 2 mm.

12 448 B.T. Wong, M.P. Menguc / Journal of Quantitative Spectroscopy & Radiative Transfer 84 (2004) Deposition x 10 9 (nm -3 ) (a) r (nm) z (nm) Deposition x 10 9 (nm -3 ) (b) r (nm) z (nm) Fig. 5. The normalized deposition (with the total energy or power of the incident beam) of electron energies within gold in units of ( 10 9 =nm 3 ) for an applied voltage of 20 kv. An incident Gaussian beam of a 1=e 2 -radius of 25nm is considered in the MCM simulations. (a) Rutherford, and (b) Mott scattering cross sections. The solid in each case is innite in thickness. absorbed is in much concentrated area when impulse beam prole is considered. On the other hand, the at beam case has the largest diusion of the energy absorbed. Note that the energy absorbed as a function of z (i.e., after integrating over r) is the same for all three cases. If we were to evaluate the absorption in terms of temperature, the impulse case would have the highest temperature at the upper boundary, followed by the Gaussian case, and nally the at beam case MCM in electron-beam processing Fig. 5 depicts the depositions of electron energies due to the electron bombardments between propagating electrons and lattices of solid, as obtained using the Rutherford and the Mott scattering

13 B.T. Wong, M.P. Menguc / Journal of Quantitative Spectroscopy & Radiative Transfer 84 (2004) cross sections (see Section 4.1). The voltage applied in these cases is 20 kv. Both gures have the same level of contour divisions, indicating that the grey shadings have the same values, correspondingly. The energies of the incident electrons are 20 kev, which is considered to be suciently high for using the Rutherford cross section without signicant errors. This is evident from the gures as one can observe that the deposition contours in both cases are similar. An increase in the applied voltage denitely increases the validity of using the Rutherford cross section while a decrease certainly 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 many research areas. Due to the in-scattering nature of the particle transport, obtaining analytical solutions to the transport equation is considered to be extremely dicult; therefore, statistical methods such as Monte Carlo methods (MCM) are generally used. In this paper, the MCMs for the radiative transfer equation (RTE) and the electron transport equation (ETE) have been discussed. Since both governing equations are derived from the Boltzmann transport equation (BTE) they share the common simulation procedures except that each needs dierent scattering probabilities and properties. Details on how to obtain these scattering probabilities and properties are presented. The problem considered here 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. dierent refractive indices between the surroundings and the medium, is not discussed in this context. However, they can be found in Refs. [6,10]. Also, the electron scatterings at the interface between two dierent solids are not considered here. It should be noted that MCMs generally assume the wavelengths of the particles are small compared to the characteristic length of the object of interest. Should this condition be violated the problem is to be solved using wave theories. Acknowledgements This work is supported by an NSF Nanoscale Interdisciplinary Research Team (NIRT) award from the Nano Manufacturing program in Design, Manufacturing, and Industrial Innovation (DMI ). In addition, Basil T. Wong is supported by a TVA fellowship during this study. References [1] Howell JR. Application of Monte Carlo to heat transfer problems. In: Hartnett JP, Irvine TF, editors. Advances in heat transfer. vol. 5. New York: Academic Press; [2] Modest MF. Radiative heat transfer, 2nd ed. New York: Academic Press; [3] Flock ST, Patterson MS, Wilson BC, Wyman DR. Monte Carlo modeling of light propagation in highly scattering tissues I: model predictions and comparison with diusion theory. IEEE Trans Med Eng 1989;36: [4] Hasegawa Y, Yamada Y, Tamura M, Nomura Y. Monte Carlo simulation of light transmission through living tissues. Appl Opt 1991;30:4515. [5] Jacques SL, Wang L. Monte Carlo modeling of light transport in tissues. In: Welch Gemert V, editor. Optical-thermal response of laser-irradiated tissue. New York: Plenum Press; p

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