The small momentum transfer approximation

Kawrakow [18] performed a Wentzel-type analysis using the momentum transfer instead of the scattering angle as the angular variable. The connection between the momentum transfer $q$ and the scattering angle $\theta$ is given by:

$$q^2 = 2 p^2 (1 - \cos \theta )~,$$ (5)

$p$ being the electron's momentum. The resulting multiple-scattering distribution in the small-angle approximation is:

$$F_{{\rm SA}q}(q,\lambda) q\,{\rm d}q = {\rm d}q\,q~ \int... ...\prime~ \tilde{\sigma}(q^\prime)[1-J_0(q^\prime b)] \right).$$ (6)

For small scattering angles $q \approx p \theta$ and the above expression is equivalent to Eq. (1). It was concluded in Ref. [18] that Eq. (6) has advantages compared to Eq. (1) due to the fact that the any-angle form of the single-scattering cross section can be used to evaluate the integral in the exponential. In this paper we will show that Eq. (6) yields an exact description of the multiple-scattering distribution at all angles when the small-angle approximation is justified (see the condition (7) below).

Two approximations are required to obtain Eq. (6) from the GS-series: (a) the replacement of the summation by an integral and (b) the approximation of the Legendre polynomials by the Bessel function of zeroth order $J_0$. Concerning (a), Euler's summation formula [19] may be used. Higher order corrections will be small when a large number of terms is required to obtain convergence of the GS-series. Convergence of the GS-series is obtained for $l > (\lambda Q_1)^{-1/2}$ (see Ref. [20] and the discussion below) where $Q_1$ is defined in Eq. (4). That is, the first condition for the applicability of the small-angle approximation is

$$\lambda Q_1 = \lambda \langle 1 - \cos \chi \rangle \ll 1~.$$ (7)

Here, $\langle \cdots \rangle$ means averaging with respect to the single-scattering law. The above condition will be satisfied when both the average angle in a single-scattering event and the average multiple-scattering angle are small.

The approximation (b) affects the evaluation of the GS-moments $Q_l$ in the exponent of Eq. (3) and the summation itself. To investigate the errors introduced by (b) we first note that Eq. (6) can be transformed to

$$F_{{\rm SA}q}(\cos \theta,\lambda) = \int_0^{\infty} {\r... ...)} \right) \exp \left( - \lambda Q_{{\rm SA}q}(z) \right)~,$$ (8)

where we have defined

$$Q_{{\rm SA}q}(z) = \int {\rm d}\chi\,\sin\chi~ \tilde{\si... ...\left[ 1-J_0 \left(z \sqrt{2(1 - \cos \chi)} \right) \right]~.$$ (9)

We can then compare the power series expansion of the zeroth order Bessel function to the expansion of the Legendre polynomials:

$$J_0\left(z \sqrt{2(1 - \cos \theta)} \right) \approx 1 - {l (l+1) \over 4} \theta^2 + \left( {l (l+1) \over 48} + {l^2 (l+1)^2 \over 64} \right) \theta^4$$

$$- \left( {l (l+1) \over 1440} + {l^2 (l+1)^2 \over 384} + {l^3 (l+1)^3 \over 2304} \right) \theta^6 \pm \cdots$$ (10)

and

$$P_l(\cos \theta) \approx 1 - {l (l+1) \over 4} \theta^2 + \left( {l (l+1) \over 48} + {(l-1) l (l+1) (l+2) \over 64} \right) \theta^4$$

$$- \left( {l (l+1) \over 1440} + {(l-1) l (l+1) (l+2) \over 384} + \right.$$

$$\left. { (l-2) (l-1) l (l+1) (l+2) (l+3) \over 2304} \right) \theta^6 \pm \cdots ~.$$ (11)

In the first equation we have made use of $z^2 = l (l+1)$. With this formulas we have

$$Q_l \approx {l (l+1) \over 4} \langle \chi^2 \rangle - \left( {l (l+1) \over 48} + {(l-1) l (l+1) (l+2) \over 64} \right) \langle \chi^4 \rangle \pm \cdots$$

$$Q_{{\rm SA}q} \approx {l (l+1) \over 4} \langle \chi^2 \rangle - \left( {l (l+1) \over 48} + {l^2 (l+1)^2 \over 64} \right) \langle \chi^4 \rangle \pm \cdots~.$$ (12)

Here, $\langle \cdots \rangle$ means again averaging over the single scattering cross section. For small-angle scattering, $\langle \chi^4 \rangle \ll \langle \chi^2 \rangle$ and therefore the terms proportional to $\langle \chi^4 \rangle$ are only small corrections for small $l$. On the other hand, the difference between $l (l+1)$ and $(l-1) (l+2)$ is negligible for large $l$ (this is of course true also for the higher order terms, not shown in Eq. (12) for the sake of brevity). We can therefore conclude that $Q_{{\rm SA}q}$ represents a sufficiently accurate approximation to $Q_l$ for any $l$ when the small-angle approximation is justified (see also Eq. (18) and the subsequent discussion below).

Concerning the summation, it is important to realize that terms with large $l$ are more important when condition (7) is satisfied. To see this, we write Eq. (3) in the form $F_{\rm GS}(\cos \theta) = \sum_{l=0}^\infty w_l P_l(\cos \theta)$ with

$$w_l = \left( l + {1 \over 2} \right) \exp \Big( - \lambda Q_l \Big)$$

$$\approx \left( l + {1 \over 2} \right) \exp \left[ - {\lambda \langle \ch... ...ver 2} \right)^2 \right] \exp \Big( - \lambda \langle \chi^2 \rangle/16 \Big)~.$$ (13)

Here, we have made use of Eq. (12). It is obvious that Legendre polynomials around $l_{max}$,

$$l_{max} = \sqrt{2 \over \lambda \langle \chi^2 \rangle} - {1 \over 2} \approx \sqrt{ 1 \over \lambda Q_1}~,$$ (14)

will give the most important contribution to the summation since there $w_l$ has a maximum ³. If condition (7) is satisfied, $l_{max}$ is large and therefore the most important contributions to the GS-series will come from terms with large $l$ where we can neglect the difference between $l (l+1)$ and $(l-1) (l+2)$, $(l-2) (l+3)$, etc. In this case, the power series expansion of $J_0$ almost exactly matches the power series expansion of the Legendre polynomials (this holds also for higher order terms not shown in Eq. (10) and (11)). Therefore, the replacement of the Legendre polynomials by the Bessel function will be sufficiently accurate and we expect good agreement between the exact multiple-scattering distribution and the small-angle approximation when the condition expressed by Eq. (7) is satisfied.

To be more concrete, let us consider the case where the screened Rutherford cross section is used to describe single scattering, i.e.

$$\tilde{\sigma}( \cos \chi) = {2 \eta (1 + \eta) \over \Big( 1 - \cos \chi + 2 \eta \Big)^2 }~,$$ (15)

where $\eta$ is the screening parameter (we use the notation of Berger and Wang [10]). In this case the GS-moments $Q_l$ are given by

$$Q_l = 1 - {1 + \eta \over \eta^{l + 1}}~ {\Gamma(l+1) \Gamma... ... + 2)}~_2 F_1 \left( l+1,l+2,2 l + 2,- {1 \over \eta} \right)$$ (16)

where $_2 F_1$ is the hypergeometric function. In comparison, the small-angle moments $Q_{{\rm SA}q}$ read

$$Q_{{\rm SA}q}(y) = 1 - y K_1(y),~~~~~~~y = 2 \sqrt{l (l+1) \eta}~,$$ (17)

where $K_1$ a modified Bessel function. To investigate the differences between both expressions, we have expanded the ratio $Q_l/Q_{{\rm SA}q}$ in a power series of the screening parameter $\eta$. The result is

$$Q_l = 1 - y K_1(y) \left[1 + 2 \eta l (l+1) \left( \Psi(l) - {1 \over 2} \ln[l (l+1)] \right) \pm \cdots \right]$$ (18)

where $\Psi(l)$ is the logarithmic derivative of the gamma function,

$$\Psi(l) = {{\rm d}~ \ln \Gamma(l+1) \over {\rm d} l} = -\gamma + 1 + {1 \over 2} + \cdots + {1 \over l}~.$$ (19)

( $\gamma = 0.5772...$ is Euler's constant.) The term in the square brackets proportional to $\eta$ is a monotonically increasing function of $l$ which approaches $1/6$ for $l \to \infty$. Because for most physically relevant situations $\eta \ll 1$, the first order correction to the small angle approximation is always small. The ratio $(Q_{{\rm SA}q}-Q_l)/Q_{{\rm SA}q}$ which expresses the relative error has a maximum for $l=1$ given by

$${Q_{{\rm SA}q}(l=1)-Q_1 \over Q_{{\rm SA}q}(l=1) } = { 4 \eta \Big(1 - \gamma - \ln(2)/2 \Big) \sqrt{8 \eta} K_1\left(... ...\eta}\right) \over 1 - \sqrt{8 \eta} K_1\left(\sqrt{8 \eta}\right) } \pm \cdots$$

$$\approx {2 - 2 \gamma - \ln(2) \over 1 - 2 \gamma - \ln(2 \eta)}$$ (20)

For $\eta$ as large as 0.01 ( e.g. 2 keV electrons in water) the error is of the order of 4% being much smaller at higher energies.

Figure 1: The ratio of the exact MS-distribution to the MS-distribution in a small-angle approximation using the momentum transfer as the angular variable, $F_{{\rm GS}}/F_{{\rm SA}q}$, calculated from equations (3) and (8) for $\xi=0.001$ (top figure) and $\xi=0.2$ (bottom figure). For comparison, the same ratio calculated with the scattering angle as the angular variable with (short-dash line) and without (long-dash line) Bethe correction is shown.

Finally, we have calculated the multiple-scattering distributions $F_{{\rm GS}}$ according to Eq. (3) and $F_{{\rm SA}q}$ from Eq. (8) respectively, using the screened Rutherford cross section and different values of $\xi = \lambda Q_1$. The ratio $F_{{\rm GS}}/F_{{\rm SA}q}$ is shown in Fig. 1a for $\xi=0.001$ and in Fig. 1b for $\xi=0.2$. The later value of $\xi$ corresponds to a typical condensed-history path-length for the simulation of electron transport in low Z materials (e.g. graphite) at low energies (a few keV to a few hundred keV). The former has been chosen in order to demonstrate that $F_{\rm GS}$ in fact converges to $F_{{\rm SA}q}$ when the small-angle approximation is justified ($\xi \ll 1$). To show the strong improvement when using the momentum transfer as the angular variable, also the small-angle results from the scattering angle formulation with and without Bethe correction are shown in these figures. For $\xi=0.001$ the maximum deviation between $F_{{\rm GS}}$ and $F_{{\rm SA}q}$ is $0.2\%$. It is very interesting to observe that the deviations between $F_{{\rm GS}}$ and $F_{{\rm SA}q}$ grow to approximately 10% when changing the screening parameter by three order of magnitude! This fact will allow us to construct a method for sampling the multiple-scattering angle from the exact distribution for arbitrary path-lengths with a relatively small amount of pre-calculated data. The procedure is described in the next section.