Application of an adapted Fano cavity test for Monte Carlo simulations in the presence of B-fields

The classical Fano theorem was formulated as follows (Fano 1954):

'In a phantom of given composition exposed to a uniform fluence of primary radiation, the fluence of the secondary radiation field is also uniform and independent of the density of the phantom as well as of the density variations from point to point.'

The Fano theorem for applications in the presence of B-fields can be formulated in the following way (Fano 1954, Bouchard et al 2015):

'In a phantom of given composition exposed to a uniform and isotropic fluence of primary radiation in the presence of a B-field, the fluence of the secondary radiation is also uniform and isotropic, and independent of the density of the phantom and the B-field strength as well as of the density and B-field variations from point to point.'

The main difference with the classical Fano theorem is the extension with the condition for isotropy of the primary and secondary radiation fields, which we call charged particle isotropy (CPI). The proof for the extension of the conditions uses to a large extent the same arguments as for the classical formulation and starts from the general transport equation for secondary particle fields as given by (Fano 1954, Bouchard 2010) and illustrated in figure 1.

$$\begin{eqnarray}&&\frac{\text{d}{{\varphi}_{e}}\left(\vec{r},E,{\rm{\vec \Omega }}\right)}{\text{d}l}=\rho \left(\vec{r}\right){{S}_{e}}\left(\vec{r},E,{\rm{\vec \Omega }}\right)-\rho \left(\vec{r}\right){{\varphi}_{e}}\left(\vec{r},E,{\rm{\vec \Omega }}\right)\int_{0}^{\text{E}}\text{d}{{E}^{\prime}}{{\int\int}_{4\pi}}\text{d}\overset{{}}{\mathop{{{\Omega }^{\prime}}}}\,k\left({{E}^{\prime}},E,{{{\rm{\vec \Omega }}}^{\prime}}\centerdot {\rm{\vec \Omega }}\right)\\ &&+\rho \left(\vec{r}\right)\int_{\text{E}}^{{{\text{E}}_{0}}}\text{d}{{E}^{\prime}}{{\int\int}_{4\pi}}\text{d}\overset{{}}{\mathop{{{\Omega }^{\prime}}}}\,k\left(E,{{E}^{\prime}},{{{\rm{\vec \Omega }}}^{\prime}}\centerdot {\rm{\vec \Omega }}\right){{\varphi}_{e}}\left(\vec{r},{{E}^{\prime}},{{{\rm{\vec \Omega }}}^{\prime}}\right)\\ &&=\rho \left(\vec{r}\right)\left[{{S}_{e}}\left(\vec{r},E,{\rm{\vec \Omega }}\right)-I_{e}^{-}\left(\vec{r},E,{\rm{\vec \Omega }}\right)+I_{e}^{+}\left(\vec{r},E,{\rm{\vec \Omega }}\right)\right]\end{eqnarray} \tag{ 1 }$$

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Please note, for the remainder only the electrons of the secondary particle field will be considered. The formulation for positrons follows an analogous way.

With:

• $\frac{\text{d}{{\varphi}_{e}}\left(\vec{r},E,{\rm{\vec \Omega }}\right)}{\text{dl}}$ the left-hand side of equation (1) being the change in fluence for an infinitesimally small step in the direction of motion ${\rm{\vec \Omega }}$ for electrons with position $\,\vec{r}$ and energy E, which is identical to ${{\vec{\nabla}}_{r}}{{\varphi}_{e}}$ in the classical formulation (13), i.e. without the presence of the B-field (see figure 1).
• E₀ the maximum energy of the secondary particles in the system.
• S_e($\vec{r}$, E, ${\rm{\vec \Omega }}$): the number of secondary electrons generated per unit of mass due to interactions of the primary photons at position $\vec{r}$ with energy E and direction ${\rm{\vec \Omega }}=(u,v,w)$ (see figure 1).
• φ_e($\vec{r}$, E, ${\rm{\vec \Omega }}$): the fluence of the secondary electrons written as φ_e in the remainder for presentation purposes.
• $k\left({{E}^{'}},E,{{{\rm{\vec \Omega }}}^{\prime}}\centerdot {\rm{\vec \Omega }}\right)\text{d}{{E}^{\prime}}\text{d}{{\overset{{}}{\mathop{\Omega }}\,}^{\prime}}$: the probability per unit of mass traversed that the electrons with energy E and direction of motion ${\rm{\vec \Omega }}$ have an inelastic interaction with a resulting energy E' and a direction of motion ${{{\rm{\vec \Omega }}}^{\prime}}$
• $I_{e}^{-}\left(\vec{r},E,{\rm{\vec \Omega }}\right)=~{{\varphi}_{e}}\left(\vec{r},E,{\rm{\vec \Omega }}\right)\int_{0}^{E}\text{d}{{E}^{\prime}}{{\int\int}_{4\pi}}\text{d}{{\Omega }^{'}}k\left({{E}^{'}},E,{{{\rm{\vec \Omega }}}^{\prime}}\centerdot {\rm{\vec \Omega }}\right)$, describing the loss of electron fluence with direction ${\rm{\vec \Omega }}$ and energy E per unit of mass due to electron interactions (see figure 1).
• $I_{e}^{+}\left(\vec{r},E,{\rm{\vec \Omega }}\right)=\int_{E}^{{{E}_{0}}}\text{d}{{E}^{\prime}}{{\int\int}_{4\pi}}\text{d}{{\Omega }^{'}}k\left(E,{{E}^{'}},{{{\rm{\vec \Omega }}}^{\prime}}\centerdot {\rm{\vec \Omega }}\right){{\varphi}_{e}}\left(\vec{r},{{E}^{'}},{{{\rm{\vec \Omega }}}^{\prime}}\right)$, describing the increase of electron fluence with direction ${\rm{\vec \Omega }}$ and energy E per unit of mass due to electron interactions (see figure 1).

The main difference with the proof of the classical theorem lies in the expression of the left-hand side of equation (1), which changes due to the presence of the B-field. The left-hand side of equation (1) can be rewritten as (according to the derivation in the appendix with as the end result equation (A.9)):

$$\begin{eqnarray}&&\frac{\text{d}{{\varphi}_{e}}}{\text{d}l}={\rm{\vec \Omega }}\centerdot \left[{{\vec{\nabla}}_{r}}{{\varphi}_{e}}-{{\vec{\nabla}}_{\Omega }}{{\varphi}_{e}}\times {\rm{\vec{B}}}\left(\vec{r}\right)\frac{e}{{{m}_{e}}\gamma |\text{v}|}\right]\end{eqnarray} \tag{ 2 }$$

With ${{\vec{\nabla}}_{r}}{{\varphi}_{e}}=\left(\frac{\partial {{\varphi}_{e}}}{\partial x},\frac{\partial {{\varphi}_{e}}}{\partial y},\frac{\partial {{\varphi}_{e}}}{\partial z}\right)$ and ${{\vec{\nabla}}_{\Omega }}{{\varphi}_{e}}=\frac{\partial {{\varphi}_{e}}}{\partial \Theta}\rm{\hat \Theta}+\frac{1}{\sin \Theta}\frac{\partial {{\varphi}_{e}}}{\partial \phi}\hat{\phi}$. The proof consists of two steps. In the first step, it is shown that for a uniform and isotropic primary radiation field in a phantom of uniform composition and density in the presence of a uniform B-field, the fluence of the secondary field is uniform and isotropic as well. In the second step, it is shown that the solution of the first step is the same as the solution for a phantom with non-uniform density and B-field, with the same properties as the primary radiation field.

Step 1.

In a uniform and isotropic primary radiation field the production of the secondary electron, S_e($\vec{r}$, E, ${\rm{\vec \Omega }}$), is uniform and isotropic as well, yielding S_e($\vec{r}$, E, ${\rm{\vec \Omega }}$)  =  S_e(E). In a homogenous phantom with a homogenous B-field for any straight line l, $\frac{\text{d}{{\varphi}_{e}}}{\text{d}l}=0$. Therefore, from equation (1) it follows that,

$$\begin{eqnarray}&&\left[{{S}_{e}}(E)-I_{e}^{-}\left(E,{\rm{\vec \Omega }}\right)+I_{e}^{+}\left(E,{\rm{\vec \Omega }}\right)\right]=0\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\left[{{S}_{e}}(E)-\varphi \left(E,{\rm{\vec \Omega }}\right)\mu (E)+I_{e}^{+}\left(E,{\rm{\vec \Omega }}\right)\right]=0\end{eqnarray}$$

which can be written as,

$$\begin{eqnarray}&&\varphi \left(E,{\rm{\vec \Omega }}\right)=\frac{{{S}_{e}}(E)}{\mu (E)}+\frac{I_{e}^{+}\left(E,{\rm{\vec \Omega }}\right)}{\mu (E)}\end{eqnarray} \tag{ 4 }$$

From equation (1) it can be seen that, for the maximum energy in the system, E₀, $I_{e}^{+}\left({{E}_{0}},{\rm{\vec \Omega }}\right)=0$ and since the left term on the right-hand side of equation (4) does not depend on ${\rm{\vec \Omega }}$

$$\begin{eqnarray}&&\varphi \left({{E}_{0}},{\rm{\vec \Omega }}\right)=\varphi \left({{E}_{0}}\right)\end{eqnarray}$$

In addition, by definition,

$$\begin{eqnarray}&&I_{e}^{+}\left(E,{\rm{\vec \Omega }}\right)=I_{e}^{+}(E)\,\text{if}\,\varphi \left(E',{\rm{\vec \Omega }}\right)=\varphi \left({{E}^{\prime}}\right)\,\text{for} ~\text{all}\,{{E}^{\prime}}>E\end{eqnarray} \tag{ 6 }$$

Therefore, from equations (4), and (6),

$$\begin{eqnarray}&&\varphi \left({{E}_{0}}-\text{d}E,{\rm{\vec \Omega }}\right)=\frac{{{S}_{e}}\left({{E}_{0}}-\text{d}E\right)}{\mu \left({{E}_{0}}-\text{d}E\right)}+\frac{I_{e}^{+}\left({{E}_{0}}-\text{d}E,{\rm{\vec \Omega }}\right)}{\mu \left({{E}_{0}}-\text{d}E\right)}=\frac{{{S}_{e}}\left({{E}_{0}}-\text{d}E\right)}{\mu \left({{E}_{0}}-\text{d}E\right)}+\frac{I_{e}^{+}\left({{E}_{0}}-\text{d}E\right)}{\mu \left({{E}_{0}}-\text{d}E\right)}=\varphi \left({{E}_{0}}-\text{d}E\right)\end{eqnarray} \tag{ 7 }$$

for a choice of $\text{d}E\to 0$.

The steps made in equations (4), (6), and (7) can be repeated for i  =  1 ... n starting with $\varphi \left({{E}_{0}}-(i-1)\text{d}E,{\rm{\vec \Omega }}\right)$ in equation (4). From this, it is concluded that $\varphi \left({{E}_{0}}-(i-1)\text{d}E,{\rm{\vec \Omega }}\right)$  =  $\varphi \left({{E}_{0}}-(i-1)\text{d}E\right)$ for any i  ⩾  1, and therefore,

$$\begin{eqnarray}&&\varphi \left(E,{\rm{\vec \Omega }}\right)=\varphi (E)\end{eqnarray} \tag{ 8 }$$

the fluence of the secondary particle field is isotropic as well.

Step 2.

To prove the extension of the conditions for a non-uniform B-field and density, it has to be proven that the solution, $\varphi (E)$ (equation (8)) for the situation of a uniform isotropic primary radiation field in a phantom with a uniform composition and density is equal to the solution for equation (1) (i.e. for a phantom with a uniform composition with density variations in the presence of a non-uniform B-field). The left-hand side of equation (1) as given by equation (2) is only zero when the secondary electron field is uniform and isotropic. In that case equation (1) reduces to

$$\begin{eqnarray}&&0=\left[{{S}_{e}}(E)-I_{e}^{-}\left(E,{\rm{\vec \Omega }}\right)+I_{e}^{+}\left(E,{\rm{\vec \Omega }}\right)\right],\end{eqnarray} \tag{ 9 }$$

which is identical to equation (3). Hence, under the conditions of a uniform isotropic primary radiation field the solution for the homogeneous phantom, equation (8) is the solution for the phantom in which the B-field and the density varies as a function of the position (equations(1) and (2). Therefore the solution is unique.

The Fano test is applied to a Monte Carlo model using the Penmain routine from the PENELOPE 2011 code (Salvat et al 2011). This routine is modified to allow for simulation of the particle transport in the presence of a uniform B-field using the prescription in the Penfield package. Parameters are added to the input file to define the uniform B-field vector, and to set tolerances on the simulation of trajectories of charged particles between two interactions. Three parameters for the tolerances are used to put a maximum on the change in B-field, energy, and speed of the charged particle along the trajectory between two hard collisions. Note that the first tolerance parameter is not applicable in the case of a uniform B-field. If in a step movement of a single particle in the simulation the deviation on one of the above-mentioned quantities is higher than its tolerance parameter, the step length of the particle is reduced to a value for which the deviations obey the tolerance parameters. In case a maximum allowed step length is defined in the input file by the parameter DSMAX, the minimum of the two limitations on the step length is applied. In the study of Bouchard et al (2015) it is shown that the energy deposited under Fano conditions in the presence of an arbitrary B-field is the same as in the absence of the B-field, which implies that the validity of the Fano test can be investigated with a similar approach as used by Sempau and Andreo (2006), in which only the secondary electron field is simulated.

Geometry.

The simulations are performed for a simplified geometry of a Farmer-type ionization chamber. The geometry consists of three bodies: the cavity, the wall, and the phantom (see figure 2). The dimensions of the cavity are the same as for a Farmer-type ionization chamber with a cavity diameter of 3.15 mm and a cavity length of 24.1 mm. The thickness of the wall is 0.35 mm. To avoid boundary effects, the size of the phantom is such that the minimum distance between the cavity and edges of the phantom is larger than 1.2 times the CSDA range of the electrons. The material for the phantom is set to water. The material for the cavity is set to water vapour. To avoid influence of the density effect parameter, the same material data file as for water is used with the density modified to the density of water vapour, in this way keeping the density effect parameter constant. The orientation of the cavity with respect to the B-field is shown in figure 2.

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Implementation of the adapted Fano test and modelling of the secondary particle field.

For B  =  0 several approaches exist to implement the Fano test: the regeneration technique (Kawrakow 2000), the re-entrance technique (Yi et al 2006), and the technique in which only the secondary particle field is modelled as used by Sempau and Andreo (2006). Here the Fano test adapted for application in the presence of B-fields is implemented based on the Sempau approach by modelling only the secondary particle field.

In the section 'Extension of the Fano test conditions for the influence of B-fields' it is shown that a uniform and isotropic production per unit mass of secondary particles yields a uniform and isotropic secondary particle fluence, ${{\varphi}_{e}}$, regardless of spatial variations in density and magnetic field (strength and direction). This is a direct consequence of the cancelation of ${{\vec{\nabla}}_{\Omega }}{{\varphi}_{e}}$ in equation (2). As a result of the uniform ${{\varphi}_{e}}$, the absorbed dose in an infinite phantom of uniform atomic composition is uniform and equal to the energy of the secondary particles produced per unit mass, K. In photon fields the produced secondary electrons have in general a non-isotropic directional distribution due to the Compton process. This distribution varies as a function of the photon energy. Non-isotropic secondary particle production will not cancel the ${{\vec{\nabla}}_{\Omega }}{{\varphi}_{e}}$ term in equation (2). Therefore, it is expected that the spatial distribution of ${{\varphi}_{e}}$ will depend on spatial variations in density and magnetic field strength. The extent of these spatial variations in ${{\varphi}_{e}}$ for non-isotropic distributions is investigated by simulating two additional non-isotropic angular distributions. The three different angular distributions for the starting direction of the electrons that have been simulated are:

• Isotropic distribution, which obeys the CPI condition and which corresponds to an isotropic photon field with an artificial production process in which the total photon energy and momentum is transferred to the secondary electron.
• Compton distribution, which corresponds to the production of electrons due to the Compton interactions of a mono-directional uniform photon field.
• Mono-directional, which corresponds to a mono-directional photon field with an artificial production process in which the total photon energy and momentum is transferred to the secondary electron.

The angular distributions of the secondary electrons produced for the Compton distribution and the isotropic distribution are given in figure 3 as a function of the angle θ between the direction of the primary photon and the secondary electron. θ is the angle with the z-axis (figure 2). The Compton distribution was implemented in Penmain using the Source routine. The other two angular distributions can be generated in a straightforward way using the Penmain input file.

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Dose calculation.

The secondary electron field is modelled as a uniform source over the geometry scaled with the local mass density, in such a way that ${{S}_{e}}\left({\rm{\vec \Omega }},E\right)$ is constant. The result of the calculation is the ratio between the dose absorbed in the cavity, D_cav and the energy of the secondary particles produced per unit mass, K, in the source, which is given by,

$$\begin{eqnarray}&&K=\langle E\rangle {{S}_{e}}\end{eqnarray} \tag{ 10 }$$

The average energy of the produced secondary particles is calculated from

$$\begin{eqnarray}&&\langle E\rangle =h\nu \frac{{}^{{{\mu}_{\text{tr}}}} \diagup {}_{\rho}\;}{{}^{\mu} \diagup {}_{\rho}\;}\end{eqnarray} \tag{ 11 }$$

To keep the model consistent the ratio between ${}^{{{\mu}_{\text{tr}}}} \diagup {}_{\rho}\;$ and ${}^{\mu} \diagup {}_{\rho}\;$ was separately calculated for the Compton angular distribution with the same Monte Carlo code for each energy. For the other angular distributions (isotropic, mono-directional) this ratio is 1.

Since a source with a uniform number of particles per unit mass emitted cannot be modelled in Penmain using the input file, three separate simulations have been performed, one for each body. The final result is calculated from a summation over all the three simulations,

$$\begin{eqnarray}&&\frac{{{D}_{\text{cav}}}}{K}=\underset{i}{\mathop \sum}\,\frac{{{D}_{\text{cav},~i}}}{{{K}_{i}}}\end{eqnarray} \tag{ 12 }$$

K_i is calculated analytically from the number of simulated histories and the mass of the body. In the case where the conditions for the (reformulated) Fano theorem are violated, the ratios will deviate from 1.

The Monte Carlo calculations have been performed for photon energies between 0.5 MeV and 4.0 MeV with steps of 0.5 MeV and for a B-field strength B  =  1.5 T and B  =  0.0 T. The simulations have been repeated for a geometry in which the cavity consists of normal density water, i.e. the whole geometry has a uniform density.

Previous studies (Meijsing et al 2009, Reynolds et al 2013) on the response change of ionization chambers in the presence of a magnetic field have calculated the response change as a ratio between dose to the cavity with and without a B-field. A more solid approach is to calculate the ratios of calibration coefficients. Here the ratio, R_B, of relative calibration coefficients D_cav(ρ  =  ρ_water) / D_cav(ρ  =  ρ_vapour) with and without a B-field is calculated as an artificial response change of the cavity model (figure 2) in B-fields for a uniform radiation field. R_B is calculated from the D_cav/K results for the simulations with B  =  0.0 T and B  =  1.5 T, and with two cavity densities (vapour and water) by,

$$\begin{eqnarray}&&{{R}_{B}}=\frac{{}^{{{D}_{\text{cav}}}} \diagup {}_{K}\;\left(B=1.5~\text{T},{{\rho}_{\text{cav}}}={{\rho}_{\text{vapour}}}\right)}{{}^{{{D}_{\text{cav}}}} \diagup {}_{K}\;\left(B=0.0~\text{T},{{\rho}_{\text{cav}}}={{\rho}_{\text{vapour}}}\right)}\centerdot \frac{{}^{{{D}_{\text{cav}}}} \diagup {}_{K}\;\left(B=0.0~\text{T},{{\rho}_{\text{cav}}}={{\rho}_{\text{water}}}\right)}{{}^{{{D}_{\text{cav}}}} \diagup {}_{K}\;\left(B=1.5~\text{T},{{\rho}_{\text{cav}}}={{\rho}_{\text{water}}}\right)}\end{eqnarray} \tag{ 13 }$$

Input parameters.

It is expected that Monte Carlo simulations will play an important role in the determination of correction factors for ionization chambers and detectors for radiation dosimetry in the presence of external B-fields. Several examples (Meijsing et al 2009, Reynolds et al 2013) exist in the literature in which Monte Carlo simulations have been used to calculate the response of ionization chambers and detectors. The method presented here is the first example of applying the Fano test as a benchmark to Monte Carlo simulations in an external B-field. The adapted Fano test for radiation transport in the presence of an external B-field, as presented in this paper and in Bouchard et al (2015), was applied to a Monte Carlo model using the PENELOPE code. It was shown that the isotropy of the radiation field (CPI) is an essential condition to apply the Fano test radiation transport in the presence of a B-field.

The results from the Monte Carlo simulations for the isotropic distribution with B  =  1.5 T show significant deviations from 1 for E  ⩾  1.5 MeV with values up to 1%. Since these deviations correlate highly with the deviations seen for the simulation with B  =  0.0 T, it is unlikely that these deviations are the result of certain approximations made in PENELOPE to the simulation of the charged particle trajectories in the presence of B-fields. For simulation of the response change of ionization chambers due to the magnetic field the ratio of dose to the cavity with and without a magnetic field is calculated. For both calculations of the cavity dose (with and without a B-field) the same photon spectrum is used and therefore the same spectrum of secondary particle is produced. Since the deviations as a function of the secondary particle energy, E, for the Fano test with and without a B-field are highly correlated (figure 4(e)), it can be expected that the influence of these deviations on the response change calculation is lower than the statistical uncertainty on the ratio (⩽0.3%, k  =  2).

PENELOPE has been benchmarked for simulations with B  =  0.0 T (Sempau and Andreo 2006, Yi et al 2006). In this paper similar settings for the input parameters have been used as by Sempau and Andreo (2006). However, their cavity geometry is slab geometry with a thin cavity delimited by two parallel planes. Yi et al (2006) used parallel-plane geometry in which the majority of the electrons enter the cylindrical cavity through one of the parallel planes, which is more or less similar to Sempau and Andreo (2006). In the present study cylindrical cavity geometry was used for which the majority of the electrons enter through the cylindrical surface. A possible explanation for the observed deviations is that for the cylindrical geometry, some approximations in the electron transport become more pronounced than for slab geometry. Further investigations are needed on this aspect.

In analogy to the concept of charged particle equilibrium (CPE) we have introduced the concept of charged particle isotropy (CPI) as a key property of the secondary charged particle field in a certain point to allow charged particle equilibrium (CPE) in varying density media and arbitrary B-fields. If at a point in a phantom CPI occurs, the distribution of the angular directions of the secondary electron field is constant. Similar to CPE for which it holds that ${{\vec{\nabla}}_{r}}{{\varphi}_{e}}=0$ CPI can be defined mathematically by ${{\vec{\nabla}}_{r}}{{\varphi}_{e}}=0$.

The Monte Carlo model used in this study is meant for the validation and illustration of the adapted theorem in a practical situation rather than to give quantitative results. However the Compton angular distribution of the secondary electron field is a good approximation to the secondary particle field of a uniform mono-energetic photon beam. The response changes (R_B) for the Compton distribution (figure 5), which violates the CPI condition, are rather constant over the energy range and are of the same order of magnitude as published by (Meijsing et al 2009, Reynolds et al 2013) for the same geometrical orientations. This suggests that violation of the CPI condition is a major contribution to the response change of ionization chambers in B-fields.

Reynolds et al (2013) showed that the response of Farmer-type ionization chambers changes considerably when the orientation of the B-field with respect to the beam direction changes. For beams parallel to the B-field the response change due to the B-field is much smaller than for beams perpendicular to the B-field. The mechanism behind the small response change for the beam parallel to the magnetic field is not trivial and requires further analysis. Since the secondary electron field is distributed (non-isotropic) over all angles, they will be affected by the B-field, and therefore a response change can be expected.

The observed low response change (Reynolds et al 2013) can be understood from the second term on the right-hand side of equation (2), which can be written for a uniform field as,

$$\begin{eqnarray}&&{\rm{\vec \Omega }}\centerdot \left[{{\vec{\nabla}}_{\Omega }}{{\varphi}_{e}}\times \vec{B}\frac{e}{{{m}_{e}}\gamma |v|}\right]=\frac{e}{{{m}_{e}}\gamma |v|}\vec{B}\centerdot \left[{\rm{\vec \Omega }}\times {{\vec{\nabla}}_{\Omega }}{{\varphi}_{e}}\right]\end{eqnarray} \tag{ 14 }$$

Assuming a mono-directional photon field (in the z-direction), then ${{\vec{\nabla}}_{\Omega }}{{\varphi}_{e}}=\frac{\partial {{\varphi}_{e}}}{\partial \Theta}\rm{\hat \Theta}+\frac{1}{\sin \Theta}\frac{\partial {{\varphi}_{e}}}{\partial \phi}\hat{\phi}=\frac{\partial {{\varphi}_{e}}}{\partial \Theta}\rm{\hat \Theta}$, because the fluence of the Compton electrons only varies with θ. Therefore, ${\rm{\vec \Omega }}\times {{\vec{\nabla}}_{\Omega }}{{\varphi}_{e}}$ is a vector in the direction of $\hat{\phi}$. In the case of the B-field parallel to the beam the dot product $\vec{B}\centerdot \hat{\phi}=0$. Therefore, the fluence of the secondary particle field is not affected by the presence of the cavity, and no or small response changes can be expected. This suggests that for the special case of a photon field parallel to the B-field the CPI condition can be relaxed to the condition of isotropy only in $\hat{\phi}$ (i.e.$\,\frac{\partial {{\varphi}_{e}}}{\partial \Theta}$) or lateral CPI. This was confirmed by the results (not presented) for the Monte Carlo model of this study, when the mono-directional angular distribution was applied with a B-field parallel to the direction of the source electrons. In that case the response change is almost unity, in contrast to the situation when the starting direction was perpendicular to the B-field (figure 5).

In this study we used the generic implementation of the Lorentz force in PENELOPE, which was introduced by Bielajew (2001). In their approach, several approximations were made and can affect particle transport in two ways. One way involves electron transport at the core of regions where multiple scattering (MS) theory is used on the assumption of an infinite homogeneous media, and requires adapting a random hinge (RH) for the coupling between MS and B-fields to avoid bias in large step sizes. Another way affects transport near interfaces where the boundary crossing algorithm (BCA) must switch from MS to single scattering (SS) mode. In BCA mode, the coupling between the B-field and scattering does not exist and the accuracy of solving the trajectory relies on integration techniques. B-fields do not modify the closest distance of the particle from a boundary, but they modify the distance to the next boundary travelled by a free particle. Although the mean free path (MFP) between scattering events is small, it is more significant in low-density regions by allowing a sufficient distance for the B-field to bend the trajectory. In complex geometries, this can cause artefacts where the particle can be found in the wrong region or an object can be missed. While these approximations allow a simpler implementation, their effect on particle trajectories could be significant and show the potential for improving electron transport in B-fields. In the present study these effects have not been observed; however, further investigations are needed on the dependency of these effects on the used simulation parameters in PENELOPE for tolerances on the change in B-field, energy, and speed of the charged particle along the trajectory between two hard collisions.