Numerical simulation and experimental characterization of the TRIUMF-FEBIAD cathode temperature for optimizing the ion source performance

The FEBIAD ion source is routinely used to produce radioactive ions of halogens, molecules, and noble gases in several ISOL facilities worldwide. At TRIUMF, an extensive numerical and experimental campaign has been performed to fundamentally understand the source while improving its reliability and overall performance. Particularly, the cathode temperature has been studied by pyrometric measurements, Schottky analysis and numerical simulations to properly understand the electron emission driving the ionization. The temperature values found are consistent within the error bars and confirm the equivalence of the methodologies used. The findings can be used as part of a numerical ionization model for more realistic electron emission and the benchmarked thermal model can be used to propose novel and more robust geometries.


Introduction
In the Isotope Separation OnLine (ISOL) method [1,2], a driver beam impinges on a target material where nuclear reactions take place.The nuclear reaction products diffuse thermally out of the target material and effuse via a transferline into an ion source for ionization.The ions are extracted towards a dipole magnet for mass separation and are finally sent to an experiment station.
The ion source used to produce Radioactive Ion Beams (RIBs) of molecules, halogens, or noble gases is the Forced Electron Beam Induced Arc Discharge (FEBIAD) ion source [3].The FEBIAD consists of a resistively heated cathode that produces electrons which are accelerated towards an anode volume with a voltage difference (see Fig. 1).An electromagnetic coil generates a tunable axial magnetic field that changes the ionization rate by confining the electrons and effectively increasing the current density inside the anode volume.
A key parameter for the FEBIAD operation is the cathode temperature because it defines the number of electrons emitted.The combined transferline and hot cathode must be maintained at high temperatures (ideally ≥2000 • C) to maximize the electron emission and also to reduce the sticking time of neutrals, thus preventing losses due to radioactive decay.Accurate numerical models of the FEBIAD ionization process [4,5] require an understanding of the realistic electron emission profile.To accomplish this for the Isotope Separator and ACcelerator (ISAC) facility at Canada's particle accelerator center (TRIUMF), a combined numerical and experimental campaign has been performed to characterize the FEBIAD cathode temperature.

Joule heating simulations
The AC/DC and heat transfer modules from the software COMSOL Multiphysics v. 6.0 [6] have been used to simulate the Joule heating of the Target-and-Ion-Source (TIS) unit and the surrounding elements (see Fig. 1).By including imperfect electrical contacts and temperaturedependent material properties, accurate temperature profiles are obtained.The cathode temperature simulation comprises Joule heating with conductive, convective, and radiative heat transfer, as well as heat exchange between surfaces.In this type of simulation, electric currents flow into the TIS, generating a heat load in the system.Together with the imposed cooling boundary conditions, the program solves for the steady state temperature as a function of the electrical input currents.
Figure 1: FEBIAD ion source (no electromagnetic coil shown) alongside boundary conditions imposed to study heat transfer from Joule heating, radiative heat transfer, conductive cooling.Imperfect electrical contacts exist between the copper feedthroughs and the Target-and-Ion-Source assembly.

Electron emission theory
In the thermionic emission, a heated metal provides the electrons with enough energy to overcome the potential barrier to the vacuum.A lower barrier-or material work functionincreases the probability of an electron escaping the metal.The electron current emission per unit area is described by the Richardson-Dushman equation where A 0 is a material dependent coefficient (A m −2 K −2 ), T the cathode temperature (K), W the material dependent work function (eV), and k B the Boltzmann constant (eV K −1 ) [7].
The electron emission current is further increased by the presence of an electric field, such as the one between the FEBIAD's cathode and anode.Specifically, the exponential factor in equation 1 increases because the Schottky effect lowers the work function by where e is the elementary charge (C), ϵ 0 the vacuum permittivity (C V −1 m −1 ), and E the electric field at the cathode (V m −1 ) [8].The enhanced electron current is described with a correction on equation 1: where ).The Schottky equation can also be expressed as the product of the thermionic emission and an enhancement factor as where in the last expression, all the constants and the temperature have combined into a(T ).For a voltage of 0 V, the Richardson's equation is recovered.By further taking the natural logarithm, the following expression is obtained: With equation 5 it is possible to experimentally determine, at a constant temperature, the onset of the Schottky regime as it shows a linear dependence with √ V .For a measured electron current, a fit allows to numerically solve for the temperature via J R (T ) = e b , where b is the intercept of the linear fit.

Pyrometry and electron current measurements
The non-actinide ISAC vacuum furnace has been used to study the cathode temperature by pyrometry measurements thanks to the visual access to the FEBIAD's cathode from a viewport installed on the vacuum chamber (see Fig. 2a).Non-contact measurements were performed with a 2-color pyrometer (METIS 311 by Sensotherm) that requires a so-called emissivity slope value, which has been found through extensive calibrations on the tantalum material used on ISAC's ion sources and has allowed to established systematic errors in the temperature region study here [9,5].A modified ion source consisting of an increased anode outlet has been used to allow visual access to the cathode (see Fig. 2b), for which preliminary simulations indicated no change in the cathode temperature.Despite the cathode being partially blocked by the grid, the latter was not removed during the pyrometric measurements not only because is necessary for electron extraction but also because defines a boundary condition for heat transfer between surfaces.The measured average cathode surface temperature follows the well-known Stefan-Boltzman law, which indicates that P ∝ T 4 , or more conveniently, that T 4 ∝ P .The temperature and input heating power relation is fitted with the following equation: where y = T 4 is the surface temperature to the fourth power (K 4 ), k 1 (K 4 W −1 ) and k 2 (K 4 ) fitting parameters, and P the electrical input heating power (W).The electrical power P into the assembly is obtained by subtracting the previously characterized losses along the feedthroughs [10] to the total power provided by the power supply.
The vacuum furnace used has been upgraded by installing a voltage power supply to enable an operational anode while investigating the cathode temperature.This upgrade allows both extracting and recording the electron emission current from the cathode for each voltage sweep at different input heating currents.The electron current was recorded from the readback on the newly installed voltage power supply (PS/FL0.7F2.0 by Glassmaan).Due to the manual operation of the power supply, error bars are assigned as the smallest increment in both the voltage and current.

Result and Discussion
The resulting simulated cathode emitting face presents an inhomogeneous temperature for all the electrical currents used in the simulations.The general result for heating currents between 290 A to 320 A (or 560 W to 700 W), shows a maximum temperature at the inner radius that decreases toward the outer radius with a roughly constant gradient of ≈300 • C (see Fig. 3).A similar area to the pyrometer viewfinder (see Fig. 3a) is used on the simulations to obtain a comparable surface average temperature (see Fig. 3b).The simulated cathode average temperature is plotted as a function of heating current and is compared to measurements in Fig. 6.The average temperature measured by the pyrometer, when raised to the fourth power, shows a linear trend as a function of input power with the parameters k 1 = 3.42 × 10 10 K 4 W −1 ± 3 K 4 W −1 and k 2 = −3.23 × 10 12 K 4 ± 1933 K 4 confirming that the thermal equilibrium was reached when measuring the cathode temperature (see Fig. 4).From the statistical error δk i of the estimated parameters, a 95 % Confidence Interval (CI) is also plotted as the area between the lines defined by the parameters k i ± = k i ± 2δk i .To obtain an average measurement over the viewfinder area, the manufacturer recommends defocusing the pyrometer lens to allow a homogeneous contribution from the objects inside the area.Furthermore, the pyrometer was calibrated over different temperatures and different emissivity slopes have been found [9].The error bars associated to the pyrometer measurements correspond to an up to 1 % emissivity slope change that corresponds to ≈50 • C [4].The measured temperatures from the pyrometer are plotted against heating current in Fig. 6.
Using the Schottky analysis, a linear fit is possible by letting x = √ V and y = ln J S (x, T ), with J S being the measured enhanced thermionic emission.A Schottky plot exhibits two regimes: for low voltages the space-charge limited regime dominates; but as the voltage increases, as explained above, a linear trend indicates the onset of the Schottky regime (see Fig. 5).
The Schottky temperature estimation is also plotted in Fig. 6 alongside with the pyrometer measurements, and the average simulated temperature.The Schottky calculated temperature   confidence interval originates from the statistical error on the fitted parameter b, whereas the error bar on the measured temperature corresponds to the dominant systematic error of the pyrometer [5].The error bars on the simulation originate from the lack of knowledge of the insulators' true temperature because they are in contact to the center of an edge-cooled plate that was not modelled to save computational resources.However, to quantify their effect on the cathode temperature, the temperature of the insulators were varied from 15 • C to 60 • C (see Fig. 1).The simulated average temperature shows good agreement with the experimental values for all the heating currents investigated.Within the error bars, both direct and indirect temperature measurements are consistent with each other and indirectly confirm that thermal deformations in this temperature range are negligible.
The validated thermal numerical model has provided insight into different experimentally observed failure modes due to thermal stress at heating currents >325 A [4], and has therefore already been used to generate novel geometries with robust performance [11].

Conclusion
A new methodology has been established to characterize the temperature of the FEBIAD cathode.Systematic numerical and experimental data demonstrate the consistency of the different approaches and will allow to estimate the cathode temperature in the most convenient way.For instance, by using the Schottky analysis during an online run where optical access to the cathode is difficult.
The benchmarked thermal model can be further used to propose geometries with more robust thermal performance while maintaining similar if not better electron emission.Finally, by using a realistic electron emission, the TRIUMF ionization models are being used to propose novel geometries to improve the overall FEBIAD ionization efficiency.
is the effective work function.Equation 3 shows how increasing E and T increases the electron emission.With a fixed cathode-anode distance d, the effective work function depends only on the voltage V by the ideal relation E = V /d by which the electron emission function becomes J S (V, T

Figure 2 :
Figure 2: (a) Experimental setup depiction.(b) Anode outlet modification.The diameter is ≈8 mm.(c) Nominal geometry for comparison purposes.The diameter is 3 mm.
Figure 3: (a) Photograph of the cathode emitting face with a blue circle indicating the pyrometer viewfinder.(b) Simulated cathode face temperature for an input heating current of 320 A ( ∼700 W) showing the ≈300 • C gradient (see text).Average temperature taken inside blue circle.

Figure 4 :
Figure 4: Cathode temperature as a function of electrical input heating power.The linear fit describes well the data and serves as a potential calibration for predicting the temperature in other assemblies by using the fitting parameters k 1 = 3.42 × 10 10 K 4 W −1 ± 3 K 4 W −1 and k 2 = −3.23 × 10 12 K 4 ± 1933 K 4 (see text).

Figure 5 :
Figure 5: Natural logarithm of the measured electron current as a function of the square root of the applied voltage for several cathode heating currents.By plotting the mentioned quantities, the cathode temperature is estimated from a Schottky analysis (see text).

Figure 6 :
Figure 6: Cathode temperature as a function of input heating current showing good agreement between the methods investigated.