Energy loss

To take the energy loss of electrons during the step into consideration, we have to replace $\lambda Q_l$ in the exponent of Eq. (3) by

$$G_l = 2 \pi N \int_0^t {\rm d}t^\prime\,\int_{-1}^1 {\rm d}... ...,\sigma(\cos \chi,t^\prime) \Big[ 1 - P_l (\cos \chi) \Big]~.$$ (39)

Here, $N$ is the number of atoms per unit volume and $t$ the path-length. The dependence of the single-scattering cross section $\sigma(\cos \chi,t^\prime)$ on the energy is expressed through the dependence on the path-length $t^\prime$. Although in principle it is possible to use $G_l$ instead of $\lambda Q_l$ to calculate the $q^{(2+)}$-surface, this approach would not be efficient on a present-day computer due to the large increase of memory required to score the pre-calculated data. In fact, when we take energy loss into account, the dependence of the multiple-scattering distribution on the screening parameter $\eta$ alone is replaced by a dependence on the energy and the material in which the transport takes place. Therefore, one additional dimension will be needed to store the pre-calculated data. We will therefore approximate $G_l$ in such a way that the application of the method presented in the last section becomes possible.

We rewrite Eq. (39)

$$G_l = 2 \pi N \int_{E_{\rm f}}^{E_{\rm i}} {{\rm d}E \over S(E)} \sigma... ... {\rm d}(\cos\chi)\,\tilde{\sigma}(\cos \chi,E) \Big[ 1 - P_l (\cos \chi) \Big]$$

$$= b_c \int_{E_{\rm f}}^{E_{\rm i}} {{\rm d}E \over S(E) \beta^2} Q_l(E)~.$$ (40)

Here, $E_{\rm i}$ and $E_{\rm f}$ are the initial and final kinetic energies of the electron, $S(E)$ the restricted collision stopping power⁴, $\sigma_{\rm tot}(E)$ the total elastic cross section and $\tilde{\sigma}$ is again normalized to unity. To arrive at the second equation, we have made use of the fact that $\sigma_{\rm tot}$ is proportional to $1/\beta^2$ where $\beta$ is the electron's velocity in units of the velocity of light and introduced the short hand notation $b_c$ for the product of all constants in the total elastic cross section times $2 \pi N$. If we now neglect the very weak (logarithmic) energy dependence of $S \beta^2$, Eq. (40) becomes

$$G_l \approx {b_c \Delta E \over \tilde{\beta}^2 S(\tilde{E})... ...over \Delta E} \int_{E_{\rm f}}^{E_{\rm i}} {\rm d}E\,Q_l(E)~.$$ (41)

Here, $\Delta E = E_{\rm i} - E_{\rm f}$ is the energy loss during the step under consideration, $\tilde{E} = (E_{\rm i}+E_{\rm f})/2$ the average energy of the electron and $\tilde{\beta}$ the velocity calculated from $\tilde{E}$. To carry out the $E$-integration we can perform a power series expansion in $\Delta E$,

$$\int_{E_{\rm f}}^{E_{\rm i}} {\rm d}E\,Q_l(E) \approx \Delta... ...\prime}(\tilde{E}) \over Q_l(\tilde{E})} \pm \cdots \right]~,$$ (42)

where $Q_l^{\prime \prime}$ is the second derivative of $Q_l$ with respect to $E$. Using Eq. (18) for $Q_l$ and neglecting terms of the order of $\eta$ and terms small compared to $\ln(1/\eta)$, we arrive at the result

$$G_l \approx \lambda_{{\rm eff}} Q_l(\tilde{E})~,$$

$$\lambda_{{\rm eff}} = {b_c \Delta E \over \tilde{\beta}^2 S(\tilde{E}) }~ \left[1 + {4 ... ...u}^2 \over 3 (2 + \tilde{\tau})^2} ~{\epsilon^2 \over (2 - \epsilon)^2} \right]$$ (43)

where $\tilde{\tau}$ is the ratio of the average electron kinetic energy to it's rest mass energy and $\epsilon=\Delta E/E_{\rm i}$ the energy loss fraction. That means, when energy loss is taken into account, the multiple-scattering distribution is to a good approximation equivalent to the multiple-scattering distribution without energy loss resulting from $\lambda_{{\rm eff}}$ elastic collisions of electrons with the energy $\tilde{E}$. With this observation we can easily apply the theory developed in the previous sections to realistic calculations with electron energy loss taken into account.

Figure 4: Ratio of the exact-MS distribution to the MS-distribution with energy loss taken into account according to Eq. (43) for 100 keV and 10 MeV electrons in water and energy loss fractions $\epsilon$ of 10, 25 and 33%. For other energies and materials, similar dependence on the energy-loss fraction $\epsilon$ is observed.

To test the accuracy of the approximations leading to Eq. (43), we have calculated the multiple-scattering distribution resulting from the exact GS-moments $G_l$ and compared it to the distribution obtained with the approximated $G_l$'s given in Eq. (43) for various energies, materials and energy-loss fractions. The energy integration in (40) was done by a 32-point Gauss-Legendre quadrature. The disagreement between the exact and approximated distributions increases with increasing $\epsilon$. The maximum deviation found for $\epsilon=25$% was of the order of 1%. For $\epsilon \le 10$% the agreement was almost perfect. The ratio of the approximated to the exact distribution for $\epsilon=10$, 25 and 33% is shown for typical cases in Fig. 4.