Secondary electron emission under magnetic constraint: from Monte Carlo simulations to analytical solution

The secondary electron emission process is essential for the optimal operation of a wide range of applications, including fusion reactors, high-energy accelerators, or spacecraft. The process can be influenced and controlled by the use of a magnetic field. An analytical solution is proposed to describe the secondary electron emission process in an oblique magnetic field. It was derived from Monte Carlo simulations. The analytical formula captures the influence of the magnetic field magnitude and tilt, electron emission energy, electron reflection on the surface, and electric field intensity on the secondary emission process. The last two parameters increase the effective emission while the others act the opposite. The electric field effect is equivalent to a reduction of the magnetic field tilt. A very good agreement is shown between the analytical and numerical results for a wide range of parameters. The analytical solution is a convenient tool for the theoretical study and design of magnetically assisted applications, providing realistic input for subsequent simulations.

and maintaining the discharge 1,16 . Secondary electrons are used in scanning electron microscopy to form images 17 or in photomultiplier devices to amplify signals 18 . Also, the SEY can be used as an indicator for the surface cleanliness 19 .
Whether it is treated as an issue or not, the control of the SEE process and the accurate knowledge of the SEY are of great importance for both applications and numerical simulations. Surface morphology has been shown to play an important role in the emission process. Natural or induced surface roughness may diminish or increase the SEY. For example, fuzz surfaces obtained by exposing W targets to He + ion bombardment can reduce the SEY by more than 50% 20 . Such effect is expected to influence the operation of ITER, in which W is the leading candidate material for the divertor region and He will result from the fusion reaction 21 . The roughness of a fuzz surface is not manageable by the user when it is produced inside a fusion reactor. In contrast, a featured surface allows controlling the SEY by adjusting the shape and the aspect ratio of the pattern [22][23][24] . The pattern can be regular, as in the case of continuously shaped surfaces with different groove profiles 22,23 , or irregular, as in the case of velvet surfaces (lattices of normally-oriented fibers) 24 .
The SEE process can also be influenced and controlled by the presence of a magnetic field to the emitting surface. By changing the electronic and magnetic state of materials 25,26 , the magnetic field affects the internal mechanism of electron emission and, consequently, modifies the SEY. Moreover, the magnetic field has a major effect on the electrons that already escaped from the surface. The magnetic field guides the secondary electrons on helical trajectories. It forces a certain number of electrons to return to the emitting surface where they might be reflected or recaptured. If recaptured, the electrons will not generate any relevant effect in plasma/vacuum and they should not be considered for further calculations. The direct consequence of electron recapture is a decrease in the number of secondary electrons able to contribute to the evolution of the investigated system. Therefore, the SEY itself is no longer significant for such a case and it should be replaced by a new parameter which is often referred to as the effective SEY, δ eff or γ eff .
The effective SEY is obtained by subtracting the number of recaptured electrons from the number of secondary electrons released by a single primary particle. It is a crucial parameter in various applications assisted by a magnetic field, such as: magnetically confined plasma in fusion devices 27 , Hall thrusters 8,9 , hollow cathode discharges 28,29 , or magnetron sputtering reactors 16,30,31 ; electron guidance by magnetic immersion lenses in scanning electron microscopy 32 ; magnetic suppression of the SEE from the beam screen of a high-energy accelerator 33 or from the negative electrode of a beam direct energy converter 34 . Therefore, the effective secondary electron emission under magnetic field influence has been investigated, independently or in connection with the mentioned applications.
The change of the SEY due to a magnetic field has been measured 22,27,[33][34][35] and calculated, both analytically 31,36-38 and numerically 22,30,[37][38][39][40] . Calculation results are in line with the experimental findings. To ensure a generally valid description of the SEE process under magnetic field influence, the physical quantity which is often calculated by numerical simulations is the relative SEY, f. It is defined as the effective SEY normalized to the SEY, f = δ eff /δ or f = γ eff /γ . The relative SEY does not describe the internal mechanism of electron emission. It is a fraction of the total number of secondary electrons emitted by the surface (ranging between 0 and 1), which contains only the electrons that are relevant for the investigated system. The relative SEY is independent of all parameters that influence the SEY. However, it depends on the magnetic field magnitude and tilt, the velocity (speed and orientation) distribution function of the emitted electrons, surface recapture probability, surface morphology, and the presence of an electric field to the surface.
For open magnetic field lines, the influence of surface morphology on the SEY has been investigated in 22 with respect to the energy of incident particles. Self-consistent analysis of the secondary electron emission process and sheath formation is reported in 36,38 . In the self-consistent approach, the electric field in the sheath and the effective SEY are dependent on each other. The dependence of the relative SEY on the energy of the secondary electrons has been analyzed in 37 . A detailed analysis of the relative SEY is reported in 39 , considering most of the depending parameters (electron reflection on the surface is not included). Analytical investigations of the effective SEY 36-38 resulted in integral based formulas, which have to be estimated for each specific condition. For the particular case of a magnetron discharge, a detailed numerical analysis is reported in 30 . Also, a simple formula of the relative SEY has been derived from a fluid model 31 , but with only a few parameters and truncated reflection. However, a consistent analytical solution valid for a wide range of parameters, summing up the previous findings and offering an easy-to-use tool is still required.
This study uses a three-dimensional Monte Carlo (MC) simulation method to investigate the SEE process in an oblique magnetic field B. It explores the effects produced on the relative SEY f by the magnetic field magnitude and tilt, electron reflection, electron emission energy, and electric field in front of the surface. The role of the angular distribution of the secondary electrons is also investigated. An analytical formula is derived for the relative SEY, based on the analysis of the numerical results. It is specific for a cosine angular distribution of the secondary electrons. With respect to the other parameters, the analytical formula has a wide range of applications. Simulations are made for a flat surface. The electrons trapped by the asperities of a rough surface, due to the presence of the magnetic field 22 , are not considered to escape from the surface. In this regard, the surface roughness affects the SEY and not the relative SEY. The effect of the electric field on the electron binding energy, which generates the so-called field emission process, also affects the SEY and not the relative SEY. Consequently, the latter two phenomena are not captured by the analytical formula. Because the relative SEY does not describe the internal mechanism of electron emission, the proposed analytical formula also applies to secondary electrons emitted in transmission configuration.

Monte Carlo method
In the MC simulation, N 0 secondary electrons are randomly released from the surface obeying a cosine angular distribution, which is generally accepted to define the secondary electron emission process, regardless of the nature of the primary particle 2,41 . A correct angular distribution has to be defined with respect to the solid angle, resulting in the following formula for the cosine distribution 42 : with dN(θ) being the number of electrons emitted in the solid angle d� = sin θdθdϕ defined by the polar angle θ and the azimuthal angle ϕ . The polar angle θ is measured with respect to the surface normal, varying from zero to π/2 , since electrons are emitted on a single side of a planar surface. To assure a cosine angular distribution, the angle θ is generated as 42 : with r being a random number between 0 and 1. The azimuthal angle ϕ = 2πr ′ is uniformly generated between zero and 2π , using a new random number r ′ ranging between 0 and 1. The energy ǫ of the secondary electrons is randomly sampled according to a Maxwell-Boltzmann like distribution: using the acceptance/rejection method 43 . In Eq. (3), C is a normalization constant and ǫ S is the energy corresponding to the most probable speed of the secondary electrons. According to 44 the energy distribution function (EDF) of the secondary electrons has basically the same form for all metals, independent of the work function. Unlike other distributions in the literature [45][46][47][48] , Eq. (3) is independent of the surface material and has the advantage of a single fitting parameter. It can also be seen as a simplified form of the EDF derived in 46 . With the right choice of ǫ S , usually below 10-15 eV 1 , Eq. (3) is a good approximation of different secondary electron distributions reported in the literature 1,2,10,41,49 .
The secondary electrons are moving in a low background pressure, on collisionless trajectories (the collision frequency is much lower than the cyclotron frequency). Each trajectory is integrated using the leap-frog algorithm coupled with the Boris scheme, since it is known to achieve a good balance between accuracy, efficiency and stability for an imposed time step limit ω c �t ≤ 0.2 , where ω c = eB m e is the electron cyclotron frequency 50 . Secondary electrons that return to the surface can be either reflected or recaptured, a process described by the reflection coefficient R. If reflected, the electron is returned in the simulation space having the same speed as the incident one, angularly distributed according to (1). Some of the electrons may experience multiple reflections.
In the present simulation, the time step is 1% of the electron cyclotron period, which corresponds to ω c �t ≈ 0.06 . Each electron is tracked either until it is recaptured by the surface or a total integration time of 20 electron cyclotron periods. The latter allows treating a large number of successive reflections, assuring the convergence of Eq. (5) to Eq. (6) for all investigated conditions. N 0 is 10 6 for each computation.
The magnetic field is homogeneous, tilted by a polar angle θ B relative to the surface normal. Magnetic field lines are open, leaving the surface and closing to infinity. An electrostatic sheath is considered in front of the surface, with a constant electric field E pointing perpendicularly towards the surface. The electric field repels the secondary electrons from the surface. It acts along the entire trajectory of the secondary electrons, assuming that the sheath thickness is twice as large as the Larmor radius of the secondary electrons. This is a valid assumption for magnetic fields of the order of 0.1−1 T , as in magnetron sputtering devices 30 and tokamaks 39 , but it may fail for lower magnetic fields ( ∼ 0.01 T ), as in Hall thrusters 9 or hollow cathode discharges 28,29 . The schematic representation of magnetic field B, electric field E and secondary electron velocity v e vectors is plotted in Fig. 1.

Analytical solution and results
The first step in obtaining an accurate analytical expression for the relative SEY f is the analysis of the reflection process, schematically shown in Fig. 2. Without a magnetic field, the relative SEY is f = 1 . With a magnetic field, a certain fraction of the secondary electrons ξ returns to the surface. This fraction is more important as the angle of the magnetic field θ B increases. From the returned fraction ξ , a sub-fraction ξ R is reflected back into the simulation space. Thus, after the first reflection, the relative SEY loses the fraction ξ(1 − R) . The reflected sub-fraction ξ R will experience the same cycle. After n successive reflections on the surface, the relative SEY can be written as: which is a power series having the sum: Since both ξ and R are smaller than 1, the term (ξ R) n tends to zero for an infinite number of reflections (n → ∞) and the relation (5) converges to: Equation (6) is a generally valid formula that defines the relative SEY in the case of multiple reflections of secondary electrons on the emissive surface. It can be customized for particular cases by explicitly including the returning fraction ξ . In the limit case of R = 1 , the relative SEY is 1, regardless of the value of any other parameter. In the absence of an electrostatic sheath to the surface ( E = 0 V/m ) and without reflection ( R = 0 ), MC calculations show (Fig. 3a) that the relative SEY is described by:   (6) and (7), the fraction of secondary electrons that return to the surface due to the magnetic field is: Introducing (8) in (6) we obtain: Equation (9) is the analytical formula of the relative SEY in an oblique magnetic field, without an electric field to the surface, for a flat surface that emits and reflects electrons with a cosine angular distribution. It also applies when the electron Larmor radius is much larger than the sheath thickness (usually at lower magnetic fields) and the effect of the electric field becomes negligible. Figure 3a shows a perfect agreement between the analytical expression (9)  The dependence of the relative SEY on R (Fig. 3a) has not been investigated in the works cited for R = 0 . It is compared with the results reported for magnetron discharges 30,31 , showing the same trend: the relative SEY increases with the increase of R. For higher reflection coefficients, more of the returned secondary electrons have the chance to be reflected back into plasma/vacuum, increasing thus the value of f. In the present study, the magnetic field has straight lines, having one end to the surface and one end to infinity (open lines). In magnetron discharges, most of the magnetic field lines are curved, having both ends to the surface (closed lines). Such configuration changes the interaction of secondary electrons with the surface. The interaction is non-local in magnetrons because an electron that escapes from the surface at one end of the magnetic field line can return to the surface at the other end. The comparison with magnetron discharge results should remain at the trend stage, unless the collision mean free path is shorter than the magnetic field lines.
Simulations show that, in the absence of an electric field, the relative SEY does not depend on the magnetic flux density B or the EDF of the secondary electrons, in accordance with the results reported in 39 . However, the relative SEY depends on the angular distribution of the secondary electrons, hence f | E=0 = f (θ B , R, g � ) . The current results are in disagreement with the statements made in 36,37 , according to which the relative SEY has a weak dependence on the angular distribution. The dependence is quite important since the angular distribution sets the emission angle of each secondary electron. The emission angle determines the ratio of the two parameters that define the helical trajectory: the Larmor radius and the pitch. The mentioned ratio coupled with the magnetic field tilt is responsible for the return of the electron to the surface. Figure 3b shows the relative SEY calculated for three angular distributions: isotropic, cosine and over-cosine ( g � (θ) ∼ cos 2 θ ). The reflection coefficient has been set to zero. Thus, the results reflect only the influence of the angular distribution on the relative SEY. All distributions have been generated according to ref. 42 . With respect to the isotropic distribution, cosine-type distributions exhibit an enhanced secondary electron emission in quasiperpendicular direction to the surface 42 . For cosine-type distributions and low values of θ B , more electrons have a higher velocity component parallel to the magnetic field line. This results in a higher pitch of the helical trajectory and a faster electron drift from the surface. Thus, more electrons can escape from the surface and the relative SEY becomes higher with respect to the isotropic distribution. With the increase of θ B , the velocity component parallel to the magnetic field line decreases for most of the electrons emitted with a cosine-type distribution and more of them return to the surface. Thus, for θ B larger than 70 • , the relative SEY is higher for the isotropic distribution than for cosine-type distributions. The effect of an over-cosine distribution on the relative SEY is stronger than that of the cosine distribution (Fig. 3b).
Even if the EDF of the secondary electrons does not explicitly appear in (9), the reflection coefficient R might depend on the energy of the incident electrons 10,11,14 . The secondary electrons become incident electrons when they are returned to the surface by the magnetic field. So, there might be an indirect dependence of f | E=0 on the EDF of the secondary electrons. This aspect was not analyzed due to the large dispersion of values reported in the literature on the reflection coefficient R. Not only does R depend on the surface material 48-51 and the chemical state of the surface 14,52 , but even for the same material (e.g. Cu) the reported results are scattered 10,14,52,53 . Consequently, each case with variable R should be treated separately.
In a previous work, a simple analytical formula of the relative SEY has been obtained from a fluid model 31 . The result is based on the difference between the secondary electron flux with and without a magnetic field. In a low-pressure approximation (the collision frequency is much lower than the cyclotron frequency), without electron reflection, the formula proposed in 31 is reduced to f = ( B n B ) 2 = cos 2 θ B . B n is the component of the magnetic field B normal to the surface. The difference with respect to Eq. (7) suggests an analogy with the difference between the classical cross-field transport, which is proportional to B −2 and the empirical cross-field transport, which is proportional to B −1 (the so-called Bohm diffusion). At this point, the analogy is just an assumption that deserves further investigation. Regarding the electron reflection, the analytical formula proposed in 31 considers a single reflection, obtaining different results with respect to this study. In the presence of an electrostatic sheath to the surface ( E = 0 V/m ), MC calculations show that the relative SEY depends on more parameters than in Eq. (9), namely f = f (θ B , R, E, B, ǫ S , g � ) . All further calculations are made for a cosine angular distribution, so g is fixed. Analysing the emission angles of the recaptured electrons shows that the presence of the electric field is equivalent to a reduction of the magnetic field inclination with respect to the surface normal. The simulation results show that the reduced angle θ BE can be expressed as: where and v S is the most probable speed of the secondary electrons: The fraction ξ that returns to the surface due to the combined action of E and B is obtained by replacing the angle θ B with θ BE in (8). Introducing ξ in (6), the simplified form of f is written as: with θ BE given by (10). Including (10)- (12) in (13), the simple dependence f = f (θ BE , R) turns into the more general f = f (θ B , R, E, B, ǫ S ) . The analytical formula (13) describes the SEE process under the combined action of E and B, for a flat surface that emits and reflects electrons with a cosine angular distribution. It provides a straightforward solution for the calculation of the relative SEY. Equation (9) is a particular case of (13) for E = 0 V/m . Simulation results with different combinations of E, B and ǫ S reveal that f depends only on the value of A, regardless of the individual values of the three parameters. Therefore, the relative SEY can be expressed as f = f (θ B , R, A) . The consistency of formula (13) is validated in Fig. 4

by comparison with the results of the MC simulation. The relative SEY is plotted for different values of A and different reflection coefficients.
A very good agreement is found between the MC calculations and the analytical solution (13). For very small values of the relative SEY ( θ B angles close to 90 • ), the difference between numerical and analytical results may reach 10% . Excepting this irrelevant case, the largest deviations are below 3% and they correspond to small values of R, A, and θ B (e.g. R = 0 , A = 0.5 , and θ B around 30 • in Fig. 4a). The relative SEY increases with A, while the general dependence on θ B and R remains as discussed for E = 0 V/m . Figure 4d shows that a higher value of R reduces the influence of A on the relative SEY. Also, a higher value of A reduces the influence of R (see Fig. 4a-d for A = 3 ). Individual influences of E, B and ǫ S on f are reflected in the dependence of f on A. They were also discussed in 39 , suggesting the dependence of f on the ratio over the parameter range in which f changes rapidly. As shown in Fig. 4, the rapid change of f is characteristic to large magnetic field angles θ B . The fraction (14), which is included in A, was inappropriately associated with the ratio of the E × B drift speed to the emission speed of the secondary electrons 39 . In fact, the variation of f is more complicated than (14). From Fig. 4, it can be observed that the curves corresponding to different A values can be described by the curve corresponding to A = 0 (Eq. (9)) by a translation of the magnetic field angle from θ B to a smaller angle θ BE . The angle translation is proportional to A and it also depends on θ B . It is smaller for θ B close to 0 • and 90 • and larger for intermediate θ B values. Analytically, the angle translation has been found to be expressed by Eq. (10). Thus, f does not depend only on A but on the product A cos θ B . This indicates that the parameter that counts for the increase of f in the presence of an electric field is the electric field component parallel to the magnetic field E || = E cos θ B and not the E × B drift velocity. The drift velocity causes electrons to move along the surface, while E || is responsible for the acceleration of electrons along the magnetic field lines 40 . The two fields E and B act opposite. A higher magnetic flux density enforces a smaller gyration radius and a shorter cyclotron period for the secondary electrons. The shorter the cyclotron period, the more likely the electron is to return to the surface, which is reflected in a reduction of the relative SEY. A higher electric field, i.e. a higher E || , increases the pitch of the helical trajectory, allowing secondary electrons to move away from the surface even in a short cyclotron period. As a result, the relative SEY increases with E. The dependence of f on ǫ S reflects the dependence of f on the EDF of the secondary electrons. A higher electron emission energy reduces the relative SEY 37,39,40 . A higher velocity component along B, directed to the surface, allows secondary electrons to return to the surface even in the presence of a repelling electric field. Thus, the increase of ǫ S diminishes the effect of the electric field. Figure 4 also shows that the curves corresponding to A = 0 (no electric field to the surface) set the minimum values of the relative SEY imposed by a tilted magnetic field. These curves, described by Eq. (9), have already been discussed on Fig. 3. For all possible combinations of E, B and ǫ S , the value of the relative SEY is between the minimum curve f | E=0 and the maximum curve f = 1 . Results similar to those plotted in Fig. 4a have been reported in 37,39 , but for mono-energetic secondary electrons. The results of the current MC simulations become identical to those in 37,39 if mono-energetic secondary electrons are implemented in the numerical code.  Fig. 5, for the energy ǫ S = 5 eV of the secondary electrons. The influence of ǫ S is illustrated by plotting the curve A = 3 for two more values of ǫ S (2 and 10 eV). If ǫ S increases by a factor of 2, the electric field has to increase twice in order to preserve the value of A. In other words, for a fixed electric field, an increase of ǫ S by a factor of 2 results in reducing A by half. In such a case, the relative SEY will decrease, as shown in Fig. 4. Typical magnetic field values are indicated in Fig. 5 for different applications: 0.01−0.03 T for Hall thrusters (HT) and hollow cathode discharges (HC), 0.03−0.1 T for magnetron sputtering devices, and 1−5 T for tokamaks. The magnetic field range between magnetrons and tokamaks, 0.1−1 T , is covered by magnetically confined linear plasma generators (LPG) 54,55 . Such devices are specially designed to investigate plasma-surface interactions which are relevant for edge regions of fusion reactors 56 . Whilst the magnetic field is externally imposed, the electric field in front of the surface depends on plasma parameters. Customized electric field values can be obtained if plasma density, electron temperature and surface bias with respect to plasma are known. For an electric field of 10 5 V/m (a relatively common value for the mentioned applications) and ǫ S = 5 eV , the values of A are: A > 3 for HT, A ≈ 3 for magnetrons, A ≈ 1 for linear devices and A ≈ 0.1 for tokamaks. The relative SEYs plotted in Fig. 4 correspond very well to this example. Thus, for tokamaks, an electric field of 10 5 V/m is sufficiently weak not to influence the relative SEY. However, it is very strong for Hall thrusters and magnetrons, completely cancelling the influence of the magnetic field for tilt angles below 70 • −75 • . The latter statement is valid only if the sheath thickness is twice as large as the Larmor radius of the secondary electrons. Otherwise, a thinner sheath results in a lower relative SEY. This is because the electric field influences only a part of the electron trajectory. A sheath contraction is equivalent to a reduction of the electric field in large sheaths. The sheath thickness is usually a few Debye lengths. The Debye length is approximately 7.4 Te n 0 , where T e is the temperature of plasma electrons, expressed in energy units eV, and n 0 is the plasma density in cm −3 . The Larmor radius can be roughly estimated as    Fig. 4b, f = 1 for A = 2 and θ B < 60 • or A = 3 and θ B < 70.5 • . Figure 5 shows that A larger than 1 is obtained for electric fields larger than ∼ 10 4 V/m in HT and ∼ 10 6 V/m in tokamaks. Including (11)- (12) in (15), an electric field limit E * can be calculated: Above E * , the effect of the magnetic field on the SEE is suppressed. The value of E * is plotted in Fig. 6 as a function of θ B angle, for different magnetic flux densities (the same orders of magnitude as for HT, magnetrons and tokamaks) and for ǫ S = 5 eV . As in Fig. 5, the influence of ǫ S is illustrated by plotting the curve B = 0.1 T for  www.nature.com/scientificreports/ two more values of ǫ S (2 and 10 eV). At high magnetic field angles θ B > 80 • , the electric field limit E * increases by an order of magnitude. E * is not defined when the magnetic field is parallel to the surface ( θ B = 90 • ) but, in this case, the relative SEY is zero for all conditions. Reminder: results were obtained assuming non-collisional electron trajectories (very low pressure) and an electrostatic sheath thickness larger than twice the Larmor radius.

Conclusion
The relative secondary electron emission yield in an oblique magnetic field can be calculated with formula (13) which has been derived based on the results of Monte Carlo simulations. The analytical formula is valid for a flat surface, open magnetic field lines, a constant electric field in the sheath and a cosine angular distribution of the secondary electrons. The magnetic flux density and the emission energy of the secondary electrons contribute to the reduction of the relative SEY. Electron reflection coefficient on the surface acts the opposite. The magnetic field tilt with respect to the surface normal has a major influence on the effective emission. An electric field reduces the magnetic field effect, equivalent to a reduction of the magnetic field tilt. Without electric field, the relative SEY depends on the magnetic field angle, reflection coefficient, and the angular distribution of the secondary electrons. Formula (13) is a reliable tool for studying the implications of an effective SEE in magnetically assisted devices (tokamaks, magnetrons, Hall thrusters), in scanning electron microscopy, in electron cloud mitigation etc, helping the design of such applications. It also provides realistic input for simulations of already mentioned applications, especially for 0D and 1D codes that are not able to describe the effective SEE process. A very good agreement has been found between the MC simulations and the analytical formula. Further studies should aim investigating the influence of the background pressure on the relative SEY. A variable electric field within the electrostatic sheath or a self-consistent model of the sheath may also be considered. www.nature.com/scientificreports/ Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.