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The concept of mass angular scattering power and its relation to the diffusion constant
PERGAMON
Radiation Physics and Chemistry Radiation Physics and Chemistry 53 (1998) 295±304
The concept of mass angular scattering power and its relation to the diusion constant George A. Sandison a, c, *, Lech S. Papiez b, c a
Department of Medical Physics, Tom Baker Cancer Centre, and Department of Oncology, University of Calgary, 1331-29 Street N.W., Calgary, Alberta, Canada T2N 4N2 b Department of Radiation Oncology, Indiana University, 535 Barnhill Drive, Indianapolis, IN 46202-5289, USA c School of Health Sciences, Purdue University, West Lafayette, IN 47907, USA Received 21 November 1997; accepted 21 November 1997
Abstract An understanding of the scattering of high energy charged particle beams by tissue is required in radiotherapy since the particle trajectories determine the pattern of radiation dose deposition in patients. Numerical calculations of radiation dose often utilize energy dependent values of the angular scattering power. However, the physics literature is replete with confused interpretations of the concept of angular scattering power and its relation to the single scattering cross section for the medium or the diusion constant in the diusional limit. The purpose of this article is to clarify these notions. # 1998 Elsevier Science Ltd. All rights reserved.
1. Introduction The mass angular scattering power, T/r, has been de®ned in Report 35 of the International Commission on Radiation Units and Measurements (ICRU, 1984) as the increase in mean square angle of scattering per unit of mass thickness of medium traversed by electrons of given energy. An expression for its calculation is given in Report 35 in terms of the screened Rutherford single scattering cross section under a small angle scattering approximation. It is a macroscopic quantity in the sense that the average number of scatterings experienced by the charged particles traversing the unit mass thickness of medium must be large. However, the average energy loss experienced by the incident monoenergetic particles over this same thickness should not be large enough to aect each angular single scattering distribution signi®cantly. The origin of the ICRU approach may be traced back to the work of Rossi (Rossi and Greisen, 1941; Rossi, 1952). It should be noted well that the calculation of
* Corresponding author. Fax: +1-403-6702327; e-mail: [email protected].
mass angular scattering power is not necessarily limited to the small angle scattering approximation nor is it restricted to the use of the screened Rutherford single scattering cross section. McParland (McParland, 1989) showed that evaluation using the Mott cross section (Mott, 1929) without using the small angle approximation leads to values of T/r 6% less than those provided by ICRU (ICRU, 1984). Small corrections may also be made to account for electron energy loss eects by using Monte Carlo simulations to compute the mass angular scattering power (Li and Rogers, 1995). A common misinterpretation of the mass angular scattering power, as de®ned by ICRU, is that it is based on the scalar characteristic of a 2-dimensional multiple scattering probability density. This is not a good de®nition because 2-dimensional distributions are characterized by correlation or covariance matrices not scalar parameters. Another common misconception is that the relation between the multiple scattering variance and the single scattering variance quoted by ICRU is a consequence of the Gaussian form of the multiple scattering distribution in the diusional limit. In fact, the relation is a consequence of the independence of each scattering event and is a general relation including also multiple scattering distributions that are
0969-806X/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 9 - 8 0 6 X ( 9 8 ) 0 0 1 1 1 - X
296
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non-Gaussian in form. Finally, in the diusional limit, mass angular scattering power is closely related to the diusion constant for the scattering process but is not equal to it due to a peculiarity of the de®nition adopted by ICRU (ICRU, 1984). The purpose of this article is a pedagogical one to clarify the notion of mass angular scattering power, its appropriate mathematical interpretation and its relation to the diusion constant in the diusional limit for scattering.
2. The multiple scattering distribution At its most basic level multiple scattering of charged particles is a compound Poisson process (CPP) and the angular distribution of the particles may be constructed from individual scattering events (Fig. 1) that occur along the path of a charged particle from the point it enters the scattering medium to the point it exits or stops. The number of scattering events on a ®xed path length t is a random variable N(t) which has a Poisson probability distribution. Equivalently, it is assumed that the path length between successive scatterings is independent and of random length but this length is exponentially distributed with the common
parameter l. This parameter l is to be interpreted as the average number of scattering events per unit path length experienced by the charged particle. It is assumed to be constant for a given energy and equal to Ns where N is the number of scattering atoms per unit volume in the scattering medium and s is the total single scattering cross section at the energy of the charged particle, here considered constant. The ®nal angle of travel O(t) for the charged particle after it has travelled the path length t is a compound Poisson process whose probability density f(O, t) is given by (Ning et al., 1995) N t 1 n X X ÿlt lt n e DOi , t > 0, 1 p O f O,t n! n0 i0 where DOi are the individual changes in angular direction caused by each single scattering event along the pathlength (Fig. 2) and pn (O) is the normalized angular distribution of charged particles that have experienced exactly n collisions on the path length t. Explicitly, pn (O) is given by p0 O d O,
n 0,
1 ds , s dO
n 1,
p1 O
Fig. 1. Spherical coordinate system on the surface of the unit radius sphere and the planar coordinate systems on the in®nite plane tangent to the sphere at O (0,0). Each direction O, mapped as a point on the surface of a unit radius sphere, is dependent on angles (f, y) with f $ (0, 2p] and y $ (0, p]. Small angle scattering may be approximately treated on the plane. The direction O on this ~ whose magnitude is the radius y from the origin to the point and plane is then identi®ed as a point parameterized by the vector O f is its direction. In this case the domain of (f, y) is f $ (0, 2p] and y $ (0, 1].
G.A. Sandison, L.S. Papiez / Radiation Physics and Chemistry 53 (1998) 295±304
297
Fig. 2. Tip of directional vector On=aN(t) i=0 DOi describing a random walk motion on the surface of the unit sphere with single scattering events.
pn O
O
pnÿ1 O 0 p1 O ÿ O 0 dO 0 ,
nr2,
where the integration for pn(O) may be performed numerically. The multiple scattering probability density distribution f(O, t) is rotationally invariant with respect to the z-axis if the single scattering probability density exhibits this symmetry (i.e. f(O, t) = f(y, t) if pn(O) = pn(y)). The probability density f(O, t) on the surface of the unit sphere given by Eq. (1) is exact and can be reformulated to show that it is equivalent to the Goudsmit± Saunderson probability density distribution (Goudsmit and Saunderson, 1940a,b; Ning et al., 1995). Under the small angle scattering approximation, and following coordinate transformation to the in®nite plane (Fig. 1), Eq. (1) may also be used to recover the Moliere probability density distribution (Moliere, 1947, 1948; Bethe, 1953; Ning et al., 1995) below
~ fplane y,t fplane O,t 1 1 1 1 ÿlt 1ÿ2p pplane y 0 J0 y 0 uy 0 0 J0 yue 2p 0
dy 0
u du, 2
where J0(yu) is the zeroth order Bessel function. In this article we will be mostly concerned with scattering on the in®nite plane under the small angle scattering approximation. The 2-dimensional distribution fplane(y, t) may be integrated over the circle of radius y on the plane (Fig. 1) to produce a 1-dimensional marginal distribution Fplane(y, t). Fplane y,t
2p 0
fplane y,ty df 2py fplane y,t,
3
The second moment or variance of this marginal distribution may then be computed as follows
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s2F var plane
1 2p 01 0
0
4. The Fermi equation
y2 fplane y,ty dy df
y2 Fplane y,t dy:
4
3. Diusional limit of the multiple scattering distribution The multiple scattering probability density distribution given by Eq. (1) includes single scattering events that produce a large change in the direction of travel of the charged particle. Such events are rare but they signi®cantly aect the multiple scattering probability density distribution. In contrast, angular scattering diffusion is a phenomenon that occurs in the limit that each scattering event results in an in®nitesimally small angular change in direction and that there are an in®nite number of collisions on the path length. Thus, the diusional multiple scattering probability density distribution f di(y, t) is given by f
diff
y,t
lim f O,t ltÿ ÿ41 ÿ ÿ40 ÿ DOi ÿ 1 X
n
lt n p O eÿlt lim n! ltÿ ÿ41 ÿ n0 ÿ40 ÿ DOi ÿ
N t X
5
i0
Under the small angle approximation on the plane f di plane(y, t) may be expressed as a rotationally symmetric 2-dimensional Gaussian probability density with zero mean, as follows f
diff plane y,t
2 1 eÿy =2Dplane t , 2pDplane t
6
where Dplane is the diusion constant on the plane. The covariance matrix, M, for this 2-dimensional Gaussian distribution on the plane is 0 Dplane t : 7 M 0 Dplane t We may also integrate f di plane(y, t) over all f to obtain the 1-dimensional marginal probability density, Fdi plane(y, t), for the diusional multiple scattering marginal distribution on the plane. 2p diff f diff 8 F diff plane y,t plane y,ty df 2pyf plane y,t: 0
The variance of this 1-dimensional marginal probability density is given by 1 diff 2 var sF plane y2 F diff 9 plane y,t dy 2Dplane t: 0
and under the initial condition lim f
tÿ 40
diff plane yx ,yy ,t
d yx d yy
it has a 2-dimensional Gaussian solution (equivalent to Eq. (6) where y2=y2x+y2y) f
diff plane yx ,yy ,t
1 diff
2p
diff s2 var 1=2 s2yx var plane yy plane
2 diff 2 var ÿ y2x =2diff s2yx var syy plane plane yy =2
e
,
whose covariance matrix, M, is " diff 2 var # syx plane 0 M , diff 2 var syy plane 0
DOi :
The Fermi angular diusion transport equation (Rossi and Greisen, 1941; Eyges, 1948) is commonly used in radiotherapy dose calculations. Expressed in variables yx and yy (Fig. 1) it is given by " # 2 diff 2 diff @ f diff Dplane @ f plane @ f plane plane yx ,yy ,t , 10 @t 2 @ y2x @ y2y
11
12
where the variances of the diusional 1-dimensional marginal distributions projected on to the (z, yx) and (z, yy) planes (Fig. 3) may be explicitly expressed as 1 1 diff 2 var syx plane y2x f diff plane yx ,yy t dyx dyy ÿ1 ÿ1 1 y2x F diff 13 plane yx ,t dyx , ÿ1
diff
s2yy var plane
1 1
y2y f diff plane yx ,yy ,t ÿ1 ÿ1 1 y2y F diff plane yy ,t dyy : ÿ1
dyy dyx 14
di (F di plane(yx, t) is the projection of f plane(yx, yy, t) on the di (z, yx) plane and F plane(yy, t) is the projection of f di plane(yx, yy, t) on the (z, yy) plane). Comparing Eq. (7) with Eq. (12) we note
diff
diff s2yx var s2yy var plane plane Dplane t:
15
Also, comparing Eq. (15) with Eq. (9) we ®nd that the diusional multiple scattering variance for the 1-dimensional marginal probability distribution di F di plane(y, t) (which is an integral of f plane(y, t) over the circle for ®xed y) is related to corresponding projected variances as follows diff
diff 2 var s2F var syx plane 2diff s2yy var plane 2Dplane t 2 plane :
16
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299
Fig. 3. Elements of integration on the in®nite plane for non-projected variables (y, f) and projected variables (yx, yy). The direction on the plane may be parameterized by the Cartesian coordinates yx and yy, where the transformation between (f, y) and (yx, yy) is yx=y cosf and yy=y sinf, where yx, yy$(ÿ1, 1], f$(0, 2p] and y$0, 1].
5. The single scattering distribution
where ymin is the atomic screening parameter of minimum angle of scattering. s h Za 2 : 1:13 3:76 ymin pa b
There are various explicit forms of the dierential single scattering cross section in use. The simplest is the classical Rutherford scattering formula for a massive bare point nucleus scattering the charged particle (Rutherford, 1911). Corrections to this formula for relativistic velocities, screening eects due to the atomic electrons, and ®nite size of the nucleus, lead to the so called screened Rutherford single scattering cross section (e.g. Rossi, 1952; Jackson, 1975). Quantum-mechanical considerations for relativistic velocities, spin of the incident particle and indistinguishability between target and projectile charged particles lead to more accurate re®nements of the Rutherford scattering formula. The cross sectional formulas considered most exact for spherically symmetric scattering potentials are based on quantum mechanical partial wave analysis of the single scattering process (e.g. Riley et al., 1975). Following ICRU, we deal with the screened Rutherford cross section for an electron. Due to the symmetry of the scattering process, p1plane(O) of Eq. (1) is invariant with respect to rotations of the plane about the z-axis and is given by
e0 is the relative permittivity of free space, p and v are the momentum and velocity of the projectile charged particle respectively, e is the elementary unit of electrical charge, Z, A and r are the atomic number, gram atomic weight and density of the scattering medium, N0 is Avogadro's number, h is the reduced Planck constant, a the ®ne structure constant, b is the ratio v/c, c is the velocity of light, a = 0.885a0Zÿ1/3 is the Fermi radius of the atom and a0 is the Bohr radius. The parameter ymin we adopt is the atomic electron screening parameter or minimum angle of scattering used by Moliere (Moliere, 1947; Moliere, 1948). The normalized probability density p1plane(y) is a 2dimensional function in the plane (y, f). We may easily integrate this function over all f for a ®xed value of y since it is constant for any f. Thus this integration is also over a circle on the plane (Fig. 1) leading to the 1-dimensional marginal probability density, F single plane (y), for the single scattering distribution. 2p y p1plane yy df 2pyp1plane y: 18 F single plane
y2min , 2 p y y2min 2
The variance single s2F]var plane of the 1-dimensional marginal probability density F single plane (y) is then
It is important to note that the diusion constant, Dplane, is not a function of position on the plane.
p1plane O
p1plane y
17
0
300
single
G.A. Sandison, L.S. Papiez / Radiation Physics and Chemistry 53 (1998) 295±304
1 1 s2F var y2 p1plane yy dy y2 F plane 2p 0
0
single plane y
dy: 19
single
s2F]var plane
The evaluation of the variance in analytic closed form using the screened Rutherford cross section is problematic unless a maximum scattering angle, ymax, is speci®ed on the plane. The value of ymax is determined by the nuclear size and is given approximately by (Jackson, 1975; ICRU, 1984) ymax 1
h , pR
where R is the size of the nuclear radius, R = 0.5 reA1/3, and re is the classical electron radius. Substituting Eq. (17) into Eq. (19) and performing the integration from 0 to ymax we obtain "
single
!ÿ1 y2max 1 2 ymin ! # 2 ymax ÿ1 : ln 1 2 ymin y2min
s2F var plane
20
The integrations given by Eqs. (18) and (19) are over the plane. Referring to the captions of Figs. 1 and 3 we may also de®ne p1plane y p1plane yx ,yy : Then obtain the 1-dimensional projected marginal probability distributions F
single plane yx
F
single plane yy
1 ÿ1 ymax
p1plane yx ,yy
ÿymax
1 ÿ1 ymax
dyy
s2yx
1
p1plane yx ,yy dyy ,
p1plane yx ,yy dyx :
ÿ1
y2x F
single plane yx
ymax y2x F ÿymax
dyx
single plane yx
dyx ,
1 ÿ1 ymax
y2y F
ÿymax
single plane yy
y2y F
dyy
single plane yy
dyy :
22
6. Relationship between single scattering and multiple scattering In the diusional limit the relationship between the variance of the 1-dimensional marginal single scattering probability density and the variance of the 1dimensional marginal multiple scattering distribution is particularly simple. In fact the relationship is a consequence of the central limit theorem that states, regardless of the form of the 1-dimensional single scattering probability density F single plane (yx) for the projected stochastic variable yx, after a large number of soft scatters (Ning et al., 1995), lsoftt, the multiple scattering distribution F di plane(yx, t) will be Gaussian with a variance lsoftt times the variance single s2yx of F single plane (yx). The only restriction is that the probability density for single scattering must have ®nite moments and individual scattering events must be independent. Since we comply with this restriction we have from Eqs. (6) and (7). 0 Dplane t M 0 Dplane t l tsingle s2 0 soft yx 23a 0 lsoft tsingle s2yy and also from Eq. (16) diff s2yy var diff s2yx var plane plane :
The corresponding variances for these 1-dimensional projected single scattering distributions are single
s2yy
Dplane t lsoft tsingle s2yx lsoft tsingle s2yy
p1plane yx ,yy dyx
ÿymax
single
If the scattering process is not diusional then the probability distribution for multiple scattering will, in general, be non-Gaussian. In our example this nonGaussian distribution is the probability density fplane(O, t) given by Eq. (2). However the fact that the underlying feature of multiple scattering is the compound Poisson process for O(t) allows us to interpret each scattering event to be independent and hence the relationship between the variance of the distribution ~ t) and the single scattering variance continues fplane(O, to hold. That is, the parameters de®ned by Eqs. (4) and (20) are related as follows whether the diusion limit is considered or not single 2 var sF plane , s2F var plane lt
21
23b
24
where lt is equal to the sum of lsoftt and lhardt (Ning et al., 1995).
G.A. Sandison, L.S. Papiez / Radiation Physics and Chemistry 53 (1998) 295±304
7. Mass angular scattering power The ICRU (ICRU, 1984) de®nition of the mass angular scattering power, T/r, is T 1d l fs2 var g r r dt F plane r
single
s2F var plane ,
25
where [s2F]var plane is given by Eq. (4). However, the variance [s2F]var plane is not, for a general model of multiple scattering, a statistically meaningful parameter for the 2-dimensional multiple scattering probability density distribution fplane(O, t), given by Eq. (2). This is because, in general, for non-Gaussian distributions such as fplane(O, t) an in®nite number of moments are required to describe the distribution not just the second moment. The mass angular scattering power only acts as a statistically meaningful parameter in the diusional limit where fplane(O, t) becomes the Gaussian multiple scattering probability density f di plane(y, t) given by Eq. (6), or in (yx, yy) variables by Eq. (11). In this case Eq. (16) leads to T diff 1 d diff 2 var 2Dplane , f sF plane g r r r dt
26
where we have expressed T di in terms of the diusion constant Dplane by use of the relations in Eq. (9). In other words, in the diusional limit, the linear angular scattering power, T di, is twice the value of the diusion constant, Dplane. The parameter Dplane de®nes completely the Gaussian distribution of f di plane(y, t) through the matrix, M, (Eq. (7)) and therefore, in the diusional limit, so does T since 2 diff 3 T t 0 6 2 7 0 Dplane t 7, M 6 27 4 diff 5 0 Dplane t T t 0 2 where the elements of M indicate that fdi plane(y, t) is rotationally symmetric and scattering in the yx and yy directions are uncorrelated. We may also note from Eqs. (23a)±(b) and (26) that diff 2 var the variances, diff s2yx var syy plane of the plane and projected 1-dimensional marginal probability density f di plane(y, t) are proportional to the thickness, t. This linear dependence on t and the Gaussian nature of f di plane(y, t) is characteristic of the diusion process. Values of mass angular scattering power computed using Eq. (25) may dier signi®cantly from those computed in the diusional limit using Eq. (26) since the distributions Fplane(y, t) may be greatly dierent from the corresponding distributions in the diusional limit F di plane(y, t). This dierence is because Fplane(y, t) includes all scattering events, both large and small
301
angle, while F di plane(y, t) considers only small angle soft collisions. 8. Discussion The compound Poisson process approach to multiple scattering (Eq. (1)) helps us to understand that the multiple scattering probability density distribution is built up from the cumulative eect of independent single scattering events. Therefore, the multiple scattering probability density distribution must be sensitive to the single scattering cross section chosen to describe the scattering process. This sensitivity relates to mass angular scattering power through the variance single 2 var sF]plane. Approximating the true multiple scattering probability density distribution fplane(y, t), Eq. (2) by its Gaussian form in the diusional limit, f di plane(y, t), Eq. (6) is a poor assumption. This is because large angle single scattering events, although relatively rare, have a great eect on the multiple scattering probability density distribution fplane(y, t) and the corresponding 1-dimensional marginal probability distribution Fplane(y, t). Fig. 4 shows this dierence for a 15.7 MeV electron incident upon 0.01 mm of gold where calculations have been performed on the in®nite plane. These dierences translate to dierences between the variance, [s2F]var plane on the plane for di Fplane(y, t) and the variance diff s2F]var plane for F plane(y, t) obtained in the diusional limit. Thus, mass angular scattering power, T, computed using Eq. (25) is, in general, quite dierent from the value, T di, obtained from Eq. (26) in the diusional limit. Therefore, to maintain an interpretation of the diusion process that is consistent, the constant T di=2Dplane must be used in the Fermi Eq. (10). Unfortunately, it is customary in the Medical Physics literature to use the ICRU (1984) value of T given by Eq. (25) instead of Tdi given by Eq. (26) and this leads to the large discrepancies observed in Fig. 4. The diusional limit and central limit theorem dictate that the multiple scattering probability density distribution f di plane(y, t) is of a 2-dimensional Gaussian form and that the projected probability distribution F di plane(yx, t) on the (yx, t) plane has a variance that is equal to the product of the diusion constant and thickness, Dt. The diusion constant is a local parameter that has the same value whether diusion is considered on the plane or on the sphere (Ning et al., 1995). However its value is dependent upon the single scattering cross section chosen (Eqs. (23a)±(b)). Obviously, therefore, we will have dierent 2-dimensional Gaussian distributions describing the multiple scattering probability density distribution in the diusional limit for dierent single scattering cross sections.
302
G.A. Sandison, L.S. Papiez / Radiation Physics and Chemistry 53 (1998) 295±304
Fig. 4. Angular probability densities on the plane for a 15.7 MeV electron after penetrating a 0.01 mm gold foil. Calculations performed using Eq. (2) for fplane(y, t) and for the diusional limit form, fdi plane(y, t) calculated from Fermi Eq. (10) with Dplane erroneously substituted by T/2 Eq. (25). Medium dashed line and circles indicate fplane(y, t) and Fplane(y, t) while the small dashed line di and crosses indicate f di plane(y, t) and F plane(y, t).
The diusional limit is interesting in that all scattering events are considered small angle and the limit requires that the projected variances, single syx and single syy , approach zero as lt approaches in®nity. Thus balance must be maintained so that the matrix, M, remains well de®ned (Eqs. (23a)±(b)).
The diusional limit is a gross approximation for the multiple scattering probability density (Fig. 4). This has tempted some investigators (e.g. Jette, 1996) to improve the accuracy of their calculations by using a Gaussian ®t to the Moliere distribution (Eq. (2)) or adopting the Gaussian zeroth term of Moliere's in®nite
G.A. Sandison, L.S. Papiez / Radiation Physics and Chemistry 53 (1998) 295±304
303
Fig. 5. Angular probability densities fsphere(y, t), fplane(y, t), Fsphere(y, t) and Fplane(y, t) for a 0.128 MeV electron after penetrating a 1.33 mm gold foil. Calculations performed using Eq. (1). The small angle scattering approximation is invoked for the plane calculations. Dashed line indicates results on the sphere and the full line indicates results on the plane. A screened Rutherford single scattering cross section is used in both cases with the small angle scattering form adopted on the plane.
series approximation for fplane(y, t) (Moliere, 1947). Such approaches are inconsistent with diusional descriptions of multiple scattering since they result in variances for the 1-dimensional marginal distributions that are non-linearly dependent upon thickness t and hence are not characteristic of a diusional process.
Advocates of the ICRU de®nition of linear scattering power may claim that this de®nition is adequate for most radiotherapy applications. However, this is not universally so and investigators should be clear as to the interpretation and approximations inherent in the ICRU approach. Finally, it should be noted that
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the diusion transport equation given by Eq. (10) is of the standard form for probabilistic interpretation of Brownian motion. This means that the diusion constant Dplane is twice the value of the diusional limit for scattering power Tdi de®ned by Eq. (26). Although we have devoted this article to calculations on the in®nite plane there may be dierences between f(y, t) and F(y, t) computed on the surface of the unit sphere and the corresponding distributions on the in®nite plane even when the same single scattering cross section is used. This dierence is demonstrated in Fig. 5, without quali®cation, for a 0.128 MeV electron beam after penetrating 1.33 mm of gold (Ning, 1994). The percentage dierences are greater at large angles where the plane approximation is less accurate. Acknowledgements We would like to thank Dr. X. Ning for the use of data presented in some of the ®gures. Thanks are due also to Mr. J. Rowcastle and Dr P. Brasher. References Bethe, H.A., 1953. Moliere's theory of multiple scattering. Phys. Rev. B 89, 1256±1266. Eyges, L., 1948. Multiple scattering with energy loss. Phys. Rev. 74, 1534±1535. Goudsmit, S., Saunderson, J.L., 1940a. Multiple scattering of electrons. Phys. Rev. 57, 24±29. Goudsmit, S., Saunderson, J.L., 1940b. Multiple scattering of electrons II. Phys. Rev. 58, 36±42. ICRU (International Commission on Radiation Units and Measurements), 1984. Radiation dosimetry: electron beams
with energies between 1 and 50 MeV. ICRU Report 35, Bethesda, Maryland, USA. Jackson, J.D., 1975. Classical Electrodynamics, 2nd ed. Wiley, New York, pp. 643±651. Jette, D., 1996. Electron dose calculations using multiple scattering theory: a new theory of multiple scattering. Med. Phys. 23 (4), 459±477. Li, X.A., Rogers, D.W.O., 1995. Electron mass scattering powers: Monte Carlo and analytical calculations. Med. Phys. 22 (5), 531±541. McParland, B.J., 1989. A derivation of the electron mass scattering power for electron dose calculations. Nucl. Instr. Meth. Phys. Res. A 274, 592±596. Moliere, G., 1947. Theorie der streuuing schneller geladener teilchen I. Z. Naturforsch. a 2, 133±145. Moliere, G., 1948. Theorie der streuuing schneller geladener teilchen II. Z. Naturforsch. a 3, 78±97. Mott, N.F., 1929. The scattering of fast electrons by atomic nuclei. Proc. R. Soc. 124, 425±442. Ning, X., 1994. Multiple scattering of electrons as a compound Poisson process. Ph.D. thesis, Purdue University, IN. Ning, X., Papiez, L., Sandison, G., 1995. Compound Poisson process method for the multiple scattering of charged particles. Phys. Rev. E 52 (5), 5621±5633. Riley, M.E., MacCallum, C.J., Bigg, F., 1975. Theoretical electron-atom elastic scattering cross sections. Atomic and Nuclear Data Tables 15, pp. 443±476 (also see Riley, M.E., MacCallum, C.J., Bigg, F., 1983. Erratum Atomic and Nuclear Data Tables 27, p. 379). Rossi, B., Greisen, K., 1941. Cosmic-ray theory. Rev. Modern Phys. 13, 240±281. Rossi, B., 1952. High-Energy Particles. Prentice-Hall, Englewood Clis, NJ, pp. 63±79. Rutherford, E., 1911. The scattering of a and b particles by matter and the structure of the atom. Proc. of the R. Soc. LXXIX, pp. 669±688.
Radiation Physics and Chemistry Radiation Physics and Chemistry 53 (1998) 295±304
The concept of mass angular scattering power and its relation to the diusion constant George A. Sandison a, c, *, Lech S. Papiez b, c a
Department of Medical Physics, Tom Baker Cancer Centre, and Department of Oncology, University of Calgary, 1331-29 Street N.W., Calgary, Alberta, Canada T2N 4N2 b Department of Radiation Oncology, Indiana University, 535 Barnhill Drive, Indianapolis, IN 46202-5289, USA c School of Health Sciences, Purdue University, West Lafayette, IN 47907, USA Received 21 November 1997; accepted 21 November 1997
Abstract An understanding of the scattering of high energy charged particle beams by tissue is required in radiotherapy since the particle trajectories determine the pattern of radiation dose deposition in patients. Numerical calculations of radiation dose often utilize energy dependent values of the angular scattering power. However, the physics literature is replete with confused interpretations of the concept of angular scattering power and its relation to the single scattering cross section for the medium or the diusion constant in the diusional limit. The purpose of this article is to clarify these notions. # 1998 Elsevier Science Ltd. All rights reserved.
1. Introduction The mass angular scattering power, T/r, has been de®ned in Report 35 of the International Commission on Radiation Units and Measurements (ICRU, 1984) as the increase in mean square angle of scattering per unit of mass thickness of medium traversed by electrons of given energy. An expression for its calculation is given in Report 35 in terms of the screened Rutherford single scattering cross section under a small angle scattering approximation. It is a macroscopic quantity in the sense that the average number of scatterings experienced by the charged particles traversing the unit mass thickness of medium must be large. However, the average energy loss experienced by the incident monoenergetic particles over this same thickness should not be large enough to aect each angular single scattering distribution signi®cantly. The origin of the ICRU approach may be traced back to the work of Rossi (Rossi and Greisen, 1941; Rossi, 1952). It should be noted well that the calculation of
* Corresponding author. Fax: +1-403-6702327; e-mail: [email protected].
mass angular scattering power is not necessarily limited to the small angle scattering approximation nor is it restricted to the use of the screened Rutherford single scattering cross section. McParland (McParland, 1989) showed that evaluation using the Mott cross section (Mott, 1929) without using the small angle approximation leads to values of T/r 6% less than those provided by ICRU (ICRU, 1984). Small corrections may also be made to account for electron energy loss eects by using Monte Carlo simulations to compute the mass angular scattering power (Li and Rogers, 1995). A common misinterpretation of the mass angular scattering power, as de®ned by ICRU, is that it is based on the scalar characteristic of a 2-dimensional multiple scattering probability density. This is not a good de®nition because 2-dimensional distributions are characterized by correlation or covariance matrices not scalar parameters. Another common misconception is that the relation between the multiple scattering variance and the single scattering variance quoted by ICRU is a consequence of the Gaussian form of the multiple scattering distribution in the diusional limit. In fact, the relation is a consequence of the independence of each scattering event and is a general relation including also multiple scattering distributions that are
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non-Gaussian in form. Finally, in the diusional limit, mass angular scattering power is closely related to the diusion constant for the scattering process but is not equal to it due to a peculiarity of the de®nition adopted by ICRU (ICRU, 1984). The purpose of this article is a pedagogical one to clarify the notion of mass angular scattering power, its appropriate mathematical interpretation and its relation to the diusion constant in the diusional limit for scattering.
2. The multiple scattering distribution At its most basic level multiple scattering of charged particles is a compound Poisson process (CPP) and the angular distribution of the particles may be constructed from individual scattering events (Fig. 1) that occur along the path of a charged particle from the point it enters the scattering medium to the point it exits or stops. The number of scattering events on a ®xed path length t is a random variable N(t) which has a Poisson probability distribution. Equivalently, it is assumed that the path length between successive scatterings is independent and of random length but this length is exponentially distributed with the common
parameter l. This parameter l is to be interpreted as the average number of scattering events per unit path length experienced by the charged particle. It is assumed to be constant for a given energy and equal to Ns where N is the number of scattering atoms per unit volume in the scattering medium and s is the total single scattering cross section at the energy of the charged particle, here considered constant. The ®nal angle of travel O(t) for the charged particle after it has travelled the path length t is a compound Poisson process whose probability density f(O, t) is given by (Ning et al., 1995) N t 1 n X X ÿlt lt n e DOi , t > 0, 1 p O f O,t n! n0 i0 where DOi are the individual changes in angular direction caused by each single scattering event along the pathlength (Fig. 2) and pn (O) is the normalized angular distribution of charged particles that have experienced exactly n collisions on the path length t. Explicitly, pn (O) is given by p0 O d O,
n 0,
1 ds , s dO
n 1,
p1 O
Fig. 1. Spherical coordinate system on the surface of the unit radius sphere and the planar coordinate systems on the in®nite plane tangent to the sphere at O (0,0). Each direction O, mapped as a point on the surface of a unit radius sphere, is dependent on angles (f, y) with f $ (0, 2p] and y $ (0, p]. Small angle scattering may be approximately treated on the plane. The direction O on this ~ whose magnitude is the radius y from the origin to the point and plane is then identi®ed as a point parameterized by the vector O f is its direction. In this case the domain of (f, y) is f $ (0, 2p] and y $ (0, 1].
G.A. Sandison, L.S. Papiez / Radiation Physics and Chemistry 53 (1998) 295±304
297
Fig. 2. Tip of directional vector On=aN(t) i=0 DOi describing a random walk motion on the surface of the unit sphere with single scattering events.
pn O
O
pnÿ1 O 0 p1 O ÿ O 0 dO 0 ,
nr2,
where the integration for pn(O) may be performed numerically. The multiple scattering probability density distribution f(O, t) is rotationally invariant with respect to the z-axis if the single scattering probability density exhibits this symmetry (i.e. f(O, t) = f(y, t) if pn(O) = pn(y)). The probability density f(O, t) on the surface of the unit sphere given by Eq. (1) is exact and can be reformulated to show that it is equivalent to the Goudsmit± Saunderson probability density distribution (Goudsmit and Saunderson, 1940a,b; Ning et al., 1995). Under the small angle scattering approximation, and following coordinate transformation to the in®nite plane (Fig. 1), Eq. (1) may also be used to recover the Moliere probability density distribution (Moliere, 1947, 1948; Bethe, 1953; Ning et al., 1995) below
~ fplane y,t fplane O,t 1 1 1 1 ÿlt 1ÿ2p pplane y 0 J0 y 0 uy 0 0 J0 yue 2p 0
dy 0
u du, 2
where J0(yu) is the zeroth order Bessel function. In this article we will be mostly concerned with scattering on the in®nite plane under the small angle scattering approximation. The 2-dimensional distribution fplane(y, t) may be integrated over the circle of radius y on the plane (Fig. 1) to produce a 1-dimensional marginal distribution Fplane(y, t). Fplane y,t
2p 0
fplane y,ty df 2py fplane y,t,
3
The second moment or variance of this marginal distribution may then be computed as follows
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s2F var plane
1 2p 01 0
0
4. The Fermi equation
y2 fplane y,ty dy df
y2 Fplane y,t dy:
4
3. Diusional limit of the multiple scattering distribution The multiple scattering probability density distribution given by Eq. (1) includes single scattering events that produce a large change in the direction of travel of the charged particle. Such events are rare but they signi®cantly aect the multiple scattering probability density distribution. In contrast, angular scattering diffusion is a phenomenon that occurs in the limit that each scattering event results in an in®nitesimally small angular change in direction and that there are an in®nite number of collisions on the path length. Thus, the diusional multiple scattering probability density distribution f di(y, t) is given by f
diff
y,t
lim f O,t ltÿ ÿ41 ÿ ÿ40 ÿ DOi ÿ 1 X
n
lt n p O eÿlt lim n! ltÿ ÿ41 ÿ n0 ÿ40 ÿ DOi ÿ
N t X
5
i0
Under the small angle approximation on the plane f di plane(y, t) may be expressed as a rotationally symmetric 2-dimensional Gaussian probability density with zero mean, as follows f
diff plane y,t
2 1 eÿy =2Dplane t , 2pDplane t
6
where Dplane is the diusion constant on the plane. The covariance matrix, M, for this 2-dimensional Gaussian distribution on the plane is 0 Dplane t : 7 M 0 Dplane t We may also integrate f di plane(y, t) over all f to obtain the 1-dimensional marginal probability density, Fdi plane(y, t), for the diusional multiple scattering marginal distribution on the plane. 2p diff f diff 8 F diff plane y,t plane y,ty df 2pyf plane y,t: 0
The variance of this 1-dimensional marginal probability density is given by 1 diff 2 var sF plane y2 F diff 9 plane y,t dy 2Dplane t: 0
and under the initial condition lim f
tÿ 40
diff plane yx ,yy ,t
d yx d yy
it has a 2-dimensional Gaussian solution (equivalent to Eq. (6) where y2=y2x+y2y) f
diff plane yx ,yy ,t
1 diff
2p
diff s2 var 1=2 s2yx var plane yy plane
2 diff 2 var ÿ y2x =2diff s2yx var syy plane plane yy =2
e
,
whose covariance matrix, M, is " diff 2 var # syx plane 0 M , diff 2 var syy plane 0
DOi :
The Fermi angular diusion transport equation (Rossi and Greisen, 1941; Eyges, 1948) is commonly used in radiotherapy dose calculations. Expressed in variables yx and yy (Fig. 1) it is given by " # 2 diff 2 diff @ f diff Dplane @ f plane @ f plane plane yx ,yy ,t , 10 @t 2 @ y2x @ y2y
11
12
where the variances of the diusional 1-dimensional marginal distributions projected on to the (z, yx) and (z, yy) planes (Fig. 3) may be explicitly expressed as 1 1 diff 2 var syx plane y2x f diff plane yx ,yy t dyx dyy ÿ1 ÿ1 1 y2x F diff 13 plane yx ,t dyx , ÿ1
diff
s2yy var plane
1 1
y2y f diff plane yx ,yy ,t ÿ1 ÿ1 1 y2y F diff plane yy ,t dyy : ÿ1
dyy dyx 14
di (F di plane(yx, t) is the projection of f plane(yx, yy, t) on the di (z, yx) plane and F plane(yy, t) is the projection of f di plane(yx, yy, t) on the (z, yy) plane). Comparing Eq. (7) with Eq. (12) we note
diff
diff s2yx var s2yy var plane plane Dplane t:
15
Also, comparing Eq. (15) with Eq. (9) we ®nd that the diusional multiple scattering variance for the 1-dimensional marginal probability distribution di F di plane(y, t) (which is an integral of f plane(y, t) over the circle for ®xed y) is related to corresponding projected variances as follows diff
diff 2 var s2F var syx plane 2diff s2yy var plane 2Dplane t 2 plane :
16
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299
Fig. 3. Elements of integration on the in®nite plane for non-projected variables (y, f) and projected variables (yx, yy). The direction on the plane may be parameterized by the Cartesian coordinates yx and yy, where the transformation between (f, y) and (yx, yy) is yx=y cosf and yy=y sinf, where yx, yy$(ÿ1, 1], f$(0, 2p] and y$0, 1].
5. The single scattering distribution
where ymin is the atomic screening parameter of minimum angle of scattering. s h Za 2 : 1:13 3:76 ymin pa b
There are various explicit forms of the dierential single scattering cross section in use. The simplest is the classical Rutherford scattering formula for a massive bare point nucleus scattering the charged particle (Rutherford, 1911). Corrections to this formula for relativistic velocities, screening eects due to the atomic electrons, and ®nite size of the nucleus, lead to the so called screened Rutherford single scattering cross section (e.g. Rossi, 1952; Jackson, 1975). Quantum-mechanical considerations for relativistic velocities, spin of the incident particle and indistinguishability between target and projectile charged particles lead to more accurate re®nements of the Rutherford scattering formula. The cross sectional formulas considered most exact for spherically symmetric scattering potentials are based on quantum mechanical partial wave analysis of the single scattering process (e.g. Riley et al., 1975). Following ICRU, we deal with the screened Rutherford cross section for an electron. Due to the symmetry of the scattering process, p1plane(O) of Eq. (1) is invariant with respect to rotations of the plane about the z-axis and is given by
e0 is the relative permittivity of free space, p and v are the momentum and velocity of the projectile charged particle respectively, e is the elementary unit of electrical charge, Z, A and r are the atomic number, gram atomic weight and density of the scattering medium, N0 is Avogadro's number, h is the reduced Planck constant, a the ®ne structure constant, b is the ratio v/c, c is the velocity of light, a = 0.885a0Zÿ1/3 is the Fermi radius of the atom and a0 is the Bohr radius. The parameter ymin we adopt is the atomic electron screening parameter or minimum angle of scattering used by Moliere (Moliere, 1947; Moliere, 1948). The normalized probability density p1plane(y) is a 2dimensional function in the plane (y, f). We may easily integrate this function over all f for a ®xed value of y since it is constant for any f. Thus this integration is also over a circle on the plane (Fig. 1) leading to the 1-dimensional marginal probability density, F single plane (y), for the single scattering distribution. 2p y p1plane yy df 2pyp1plane y: 18 F single plane
y2min , 2 p y y2min 2
The variance single s2F]var plane of the 1-dimensional marginal probability density F single plane (y) is then
It is important to note that the diusion constant, Dplane, is not a function of position on the plane.
p1plane O
p1plane y
17
0
300
single
G.A. Sandison, L.S. Papiez / Radiation Physics and Chemistry 53 (1998) 295±304
1 1 s2F var y2 p1plane yy dy y2 F plane 2p 0
0
single plane y
dy: 19
single
s2F]var plane
The evaluation of the variance in analytic closed form using the screened Rutherford cross section is problematic unless a maximum scattering angle, ymax, is speci®ed on the plane. The value of ymax is determined by the nuclear size and is given approximately by (Jackson, 1975; ICRU, 1984) ymax 1
h , pR
where R is the size of the nuclear radius, R = 0.5 reA1/3, and re is the classical electron radius. Substituting Eq. (17) into Eq. (19) and performing the integration from 0 to ymax we obtain "
single
!ÿ1 y2max 1 2 ymin ! # 2 ymax ÿ1 : ln 1 2 ymin y2min
s2F var plane
20
The integrations given by Eqs. (18) and (19) are over the plane. Referring to the captions of Figs. 1 and 3 we may also de®ne p1plane y p1plane yx ,yy : Then obtain the 1-dimensional projected marginal probability distributions F
single plane yx
F
single plane yy
1 ÿ1 ymax
p1plane yx ,yy
ÿymax
1 ÿ1 ymax
dyy
s2yx
1
p1plane yx ,yy dyy ,
p1plane yx ,yy dyx :
ÿ1
y2x F
single plane yx
ymax y2x F ÿymax
dyx
single plane yx
dyx ,
1 ÿ1 ymax
y2y F
ÿymax
single plane yy
y2y F
dyy
single plane yy
dyy :
22
6. Relationship between single scattering and multiple scattering In the diusional limit the relationship between the variance of the 1-dimensional marginal single scattering probability density and the variance of the 1dimensional marginal multiple scattering distribution is particularly simple. In fact the relationship is a consequence of the central limit theorem that states, regardless of the form of the 1-dimensional single scattering probability density F single plane (yx) for the projected stochastic variable yx, after a large number of soft scatters (Ning et al., 1995), lsoftt, the multiple scattering distribution F di plane(yx, t) will be Gaussian with a variance lsoftt times the variance single s2yx of F single plane (yx). The only restriction is that the probability density for single scattering must have ®nite moments and individual scattering events must be independent. Since we comply with this restriction we have from Eqs. (6) and (7). 0 Dplane t M 0 Dplane t l tsingle s2 0 soft yx 23a 0 lsoft tsingle s2yy and also from Eq. (16) diff s2yy var diff s2yx var plane plane :
The corresponding variances for these 1-dimensional projected single scattering distributions are single
s2yy
Dplane t lsoft tsingle s2yx lsoft tsingle s2yy
p1plane yx ,yy dyx
ÿymax
single
If the scattering process is not diusional then the probability distribution for multiple scattering will, in general, be non-Gaussian. In our example this nonGaussian distribution is the probability density fplane(O, t) given by Eq. (2). However the fact that the underlying feature of multiple scattering is the compound Poisson process for O(t) allows us to interpret each scattering event to be independent and hence the relationship between the variance of the distribution ~ t) and the single scattering variance continues fplane(O, to hold. That is, the parameters de®ned by Eqs. (4) and (20) are related as follows whether the diusion limit is considered or not single 2 var sF plane , s2F var plane lt
21
23b
24
where lt is equal to the sum of lsoftt and lhardt (Ning et al., 1995).
G.A. Sandison, L.S. Papiez / Radiation Physics and Chemistry 53 (1998) 295±304
7. Mass angular scattering power The ICRU (ICRU, 1984) de®nition of the mass angular scattering power, T/r, is T 1d l fs2 var g r r dt F plane r
single
s2F var plane ,
25
where [s2F]var plane is given by Eq. (4). However, the variance [s2F]var plane is not, for a general model of multiple scattering, a statistically meaningful parameter for the 2-dimensional multiple scattering probability density distribution fplane(O, t), given by Eq. (2). This is because, in general, for non-Gaussian distributions such as fplane(O, t) an in®nite number of moments are required to describe the distribution not just the second moment. The mass angular scattering power only acts as a statistically meaningful parameter in the diusional limit where fplane(O, t) becomes the Gaussian multiple scattering probability density f di plane(y, t) given by Eq. (6), or in (yx, yy) variables by Eq. (11). In this case Eq. (16) leads to T diff 1 d diff 2 var 2Dplane , f sF plane g r r r dt
26
where we have expressed T di in terms of the diusion constant Dplane by use of the relations in Eq. (9). In other words, in the diusional limit, the linear angular scattering power, T di, is twice the value of the diusion constant, Dplane. The parameter Dplane de®nes completely the Gaussian distribution of f di plane(y, t) through the matrix, M, (Eq. (7)) and therefore, in the diusional limit, so does T since 2 diff 3 T t 0 6 2 7 0 Dplane t 7, M 6 27 4 diff 5 0 Dplane t T t 0 2 where the elements of M indicate that fdi plane(y, t) is rotationally symmetric and scattering in the yx and yy directions are uncorrelated. We may also note from Eqs. (23a)±(b) and (26) that diff 2 var the variances, diff s2yx var syy plane of the plane and projected 1-dimensional marginal probability density f di plane(y, t) are proportional to the thickness, t. This linear dependence on t and the Gaussian nature of f di plane(y, t) is characteristic of the diusion process. Values of mass angular scattering power computed using Eq. (25) may dier signi®cantly from those computed in the diusional limit using Eq. (26) since the distributions Fplane(y, t) may be greatly dierent from the corresponding distributions in the diusional limit F di plane(y, t). This dierence is because Fplane(y, t) includes all scattering events, both large and small
301
angle, while F di plane(y, t) considers only small angle soft collisions. 8. Discussion The compound Poisson process approach to multiple scattering (Eq. (1)) helps us to understand that the multiple scattering probability density distribution is built up from the cumulative eect of independent single scattering events. Therefore, the multiple scattering probability density distribution must be sensitive to the single scattering cross section chosen to describe the scattering process. This sensitivity relates to mass angular scattering power through the variance single 2 var sF]plane. Approximating the true multiple scattering probability density distribution fplane(y, t), Eq. (2) by its Gaussian form in the diusional limit, f di plane(y, t), Eq. (6) is a poor assumption. This is because large angle single scattering events, although relatively rare, have a great eect on the multiple scattering probability density distribution fplane(y, t) and the corresponding 1-dimensional marginal probability distribution Fplane(y, t). Fig. 4 shows this dierence for a 15.7 MeV electron incident upon 0.01 mm of gold where calculations have been performed on the in®nite plane. These dierences translate to dierences between the variance, [s2F]var plane on the plane for di Fplane(y, t) and the variance diff s2F]var plane for F plane(y, t) obtained in the diusional limit. Thus, mass angular scattering power, T, computed using Eq. (25) is, in general, quite dierent from the value, T di, obtained from Eq. (26) in the diusional limit. Therefore, to maintain an interpretation of the diusion process that is consistent, the constant T di=2Dplane must be used in the Fermi Eq. (10). Unfortunately, it is customary in the Medical Physics literature to use the ICRU (1984) value of T given by Eq. (25) instead of Tdi given by Eq. (26) and this leads to the large discrepancies observed in Fig. 4. The diusional limit and central limit theorem dictate that the multiple scattering probability density distribution f di plane(y, t) is of a 2-dimensional Gaussian form and that the projected probability distribution F di plane(yx, t) on the (yx, t) plane has a variance that is equal to the product of the diusion constant and thickness, Dt. The diusion constant is a local parameter that has the same value whether diusion is considered on the plane or on the sphere (Ning et al., 1995). However its value is dependent upon the single scattering cross section chosen (Eqs. (23a)±(b)). Obviously, therefore, we will have dierent 2-dimensional Gaussian distributions describing the multiple scattering probability density distribution in the diusional limit for dierent single scattering cross sections.
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Fig. 4. Angular probability densities on the plane for a 15.7 MeV electron after penetrating a 0.01 mm gold foil. Calculations performed using Eq. (2) for fplane(y, t) and for the diusional limit form, fdi plane(y, t) calculated from Fermi Eq. (10) with Dplane erroneously substituted by T/2 Eq. (25). Medium dashed line and circles indicate fplane(y, t) and Fplane(y, t) while the small dashed line di and crosses indicate f di plane(y, t) and F plane(y, t).
The diusional limit is interesting in that all scattering events are considered small angle and the limit requires that the projected variances, single syx and single syy , approach zero as lt approaches in®nity. Thus balance must be maintained so that the matrix, M, remains well de®ned (Eqs. (23a)±(b)).
The diusional limit is a gross approximation for the multiple scattering probability density (Fig. 4). This has tempted some investigators (e.g. Jette, 1996) to improve the accuracy of their calculations by using a Gaussian ®t to the Moliere distribution (Eq. (2)) or adopting the Gaussian zeroth term of Moliere's in®nite
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303
Fig. 5. Angular probability densities fsphere(y, t), fplane(y, t), Fsphere(y, t) and Fplane(y, t) for a 0.128 MeV electron after penetrating a 1.33 mm gold foil. Calculations performed using Eq. (1). The small angle scattering approximation is invoked for the plane calculations. Dashed line indicates results on the sphere and the full line indicates results on the plane. A screened Rutherford single scattering cross section is used in both cases with the small angle scattering form adopted on the plane.
series approximation for fplane(y, t) (Moliere, 1947). Such approaches are inconsistent with diusional descriptions of multiple scattering since they result in variances for the 1-dimensional marginal distributions that are non-linearly dependent upon thickness t and hence are not characteristic of a diusional process.
Advocates of the ICRU de®nition of linear scattering power may claim that this de®nition is adequate for most radiotherapy applications. However, this is not universally so and investigators should be clear as to the interpretation and approximations inherent in the ICRU approach. Finally, it should be noted that
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the diusion transport equation given by Eq. (10) is of the standard form for probabilistic interpretation of Brownian motion. This means that the diusion constant Dplane is twice the value of the diusional limit for scattering power Tdi de®ned by Eq. (26). Although we have devoted this article to calculations on the in®nite plane there may be dierences between f(y, t) and F(y, t) computed on the surface of the unit sphere and the corresponding distributions on the in®nite plane even when the same single scattering cross section is used. This dierence is demonstrated in Fig. 5, without quali®cation, for a 0.128 MeV electron beam after penetrating 1.33 mm of gold (Ning, 1994). The percentage dierences are greater at large angles where the plane approximation is less accurate. Acknowledgements We would like to thank Dr. X. Ning for the use of data presented in some of the ®gures. Thanks are due also to Mr. J. Rowcastle and Dr P. Brasher. References Bethe, H.A., 1953. Moliere's theory of multiple scattering. Phys. Rev. B 89, 1256±1266. Eyges, L., 1948. Multiple scattering with energy loss. Phys. Rev. 74, 1534±1535. Goudsmit, S., Saunderson, J.L., 1940a. Multiple scattering of electrons. Phys. Rev. 57, 24±29. Goudsmit, S., Saunderson, J.L., 1940b. Multiple scattering of electrons II. Phys. Rev. 58, 36±42. ICRU (International Commission on Radiation Units and Measurements), 1984. Radiation dosimetry: electron beams
with energies between 1 and 50 MeV. ICRU Report 35, Bethesda, Maryland, USA. Jackson, J.D., 1975. Classical Electrodynamics, 2nd ed. Wiley, New York, pp. 643±651. Jette, D., 1996. Electron dose calculations using multiple scattering theory: a new theory of multiple scattering. Med. Phys. 23 (4), 459±477. Li, X.A., Rogers, D.W.O., 1995. Electron mass scattering powers: Monte Carlo and analytical calculations. Med. Phys. 22 (5), 531±541. McParland, B.J., 1989. A derivation of the electron mass scattering power for electron dose calculations. Nucl. Instr. Meth. Phys. Res. A 274, 592±596. Moliere, G., 1947. Theorie der streuuing schneller geladener teilchen I. Z. Naturforsch. a 2, 133±145. Moliere, G., 1948. Theorie der streuuing schneller geladener teilchen II. Z. Naturforsch. a 3, 78±97. Mott, N.F., 1929. The scattering of fast electrons by atomic nuclei. Proc. R. Soc. 124, 425±442. Ning, X., 1994. Multiple scattering of electrons as a compound Poisson process. Ph.D. thesis, Purdue University, IN. Ning, X., Papiez, L., Sandison, G., 1995. Compound Poisson process method for the multiple scattering of charged particles. Phys. Rev. E 52 (5), 5621±5633. Riley, M.E., MacCallum, C.J., Bigg, F., 1975. Theoretical electron-atom elastic scattering cross sections. Atomic and Nuclear Data Tables 15, pp. 443±476 (also see Riley, M.E., MacCallum, C.J., Bigg, F., 1983. Erratum Atomic and Nuclear Data Tables 27, p. 379). Rossi, B., Greisen, K., 1941. Cosmic-ray theory. Rev. Modern Phys. 13, 240±281. Rossi, B., 1952. High-Energy Particles. Prentice-Hall, Englewood Clis, NJ, pp. 63±79. Rutherford, E., 1911. The scattering of a and b particles by matter and the structure of the atom. Proc. of the R. Soc. LXXIX, pp. 669±688.