Probabilistic model for the simulation of secondary electron emission

We provide a detailed description of a model and its computational algorithm for the secondary electron emission process. The model is based on a broad phenomenological fit to data for the secondary-emission yield and the emitted-energy spectrum. We provide two sets of values for the parameters by fitting our model to two particular data sets, one for copper and the other one for stainless steel.

Figure 1

A single electron with energy $E_0$ strikes a surface yielding $n$ secondary electrons with energies $E_1,E_2,\dots ,E_n$.

Figure 2

A sample of the measured energy spectrum $d\delta /dE$ for an unconditioned sample of stainless steel at $E_0=300\mathrm{eV}$, normal incidence. The three components of the secondary yield are given by the values of “area [$E_1,E_2$],” each of which represents the integrated spectrum between $E_1$ and $E_2$. Thus for this case, ${\delta }_{ts}=1.17$, ${\delta }_r=0.75$, and ${\delta }_e=0.12$, for a total SEY $\delta =2.04$. The upper energy cutoff for the true secondaries is somewhat arbitrarily, but conventionally, chosen to be 50 eV. Data courtesy of R. Kirby.

Figure 3

Sketch of the currents that are used to define the different components of secondary emission. The blob is meant to emphasize a nontrivial interaction yielding true-secondary electrons. This sketch embodies a mutual-exclusion property of the three types of emission processes, spelled out in the text.

Figure 4

The SEY for stainless steel for SLAC standard 304 rolled sheet, chemically etched and passivated but not conditioned. The parameters of the fit are listed in Table . Data courtesy of R. Kirby.

Figure 5

The emitted-energy spectrum for stainless steel at 300 eV incident energy and normal incidence, for SLAC standard 304 rolled sheet, chemically etched and passivated but not conditioned. Data courtesy of R. Kirby.

Figure 6

The SEY for copper. The parameters of the fit are listed in Table . Data courtesy of N. Hilleret for chemically cleaned but not in situ vacuum-baked samples.

Figure 7

The emitted-energy spectrum for copper at 295, 30, and 10 eV incident energy and normal incidence on chemically cleaned but not in situ vacuum-baked samples. Data courtesy of N. Hilleret.

Figure 8

The probability for emitting $n$ secondary electrons when a 300-eV electron strikes a surface at normal incidence, computed according to Eqs. (43, 44, 45, 46) using the fit parameters listed in Table  assuming a binomial form [Eq. (46)] for $P_{n,ts}^{\prime }$ with $M=10$. The large relative value of $P_1$ for stainless steel is due to the large value of ${\delta }_e+{\delta }_r$.

Figure 9

The two kinds of secondary energy spectra for normal-incident electrons of energy $E_0=300\mathrm{eV}$ for the parameter choices $p_n=3$, ${ϵ}_n=20\mathrm{eV}$, ${\delta }_e={\delta }_r=0$, and ${\delta }_{ts}=\delta =1.877$ assuming a Poisson distribution for $P_{n,ts}$ truncated at $n=10$. The two curves vanish identically beyond $E_0$ and are normalized so that the area under either of them in $0\le E\le E_0$ equals $\delta$. The analytic curves are given by Eqs. (52, B4), respectively. For the Monte Carlo method we used $N_0={10}^6$ incident electrons and we divided the energy interval $[0,E_0]$ into ${10}^3$ bins of width $\Delta E=0.3\mathrm{eV}$. The values chosen for ${ϵ}_n$ and $p_n$ are not meant to be realistic; we use them here for illustration purposes only. Note that $d\delta /dE$ peaks at $E\simeq (p-1)ϵ=40\mathrm{eV}$ in agreement with the discussion in Sec. 4, while $(d\delta /dE{)}_{\mathrm{jet}}$ is much broader, as each of its components peaks at $E=(np-1)ϵ$, as discussed in Appendix .

Figure 10

The probability for emitting $n$ electrons, $P_n$, for the same conditions as in Fig. 9. The “Poisson” curve is Eq. (37), extended analytically to continuous values of $n$. The analytic results were obtained by numerically integrating $d{\delta }_{ts}/dE$, given by Eq. (52), over $E$ in the range $0\le E\le E_0$ separately for each value of $n$. The Monte Carlo results were obtained concurrently with the energy spectra shown in Fig. 9, according to the computer algorithm described in Appendix .

Figure 11

The true-secondary emission yield scaling functions, Eqs. (32, F1), for $s=1.35$.