A pulsed, monoenergetic and angularselective UV photoelectron source for the commissioning of the KATRIN experiment
 497 Downloads
 5 Citations
Abstract
The KATRIN experiment aims to determine the neutrino mass scale with a sensitivity of 200 \({\mathrm{meV}/\mathrm{c}^2}\) (90% C. L.) by a precision measurement of the shape of the tritium \(\beta \)spectrum in the endpoint region. The energy analysis of the decay electrons is achieved by a MACE filter spectrometer. To determine the transmission properties of the KATRIN main spectrometer, a monoenergetic and angularselective electron source has been developed. In preparation for the second commissioning phase of the main spectrometer, a measurement phase was carried out at the KATRIN monitor spectrometer where the device was operated in a MACE filter setup for testing. The results of these measurements are compared with simulations using the particletracking software “Kassiopeia”, which was developed in the KATRIN collaboration over recent years.
1 Introduction
The outline of the KATRIN experiment is depicted in Fig. 1 [1, 2]. Molecular tritium is fed into the 10m long beam tube of the windowless gaseous tritium source (WGTS [6]). Superconducting magnets along the beam line create an adiabatic guiding field in a \({191}\,{\mathrm{T\,cm}^2}\) magnetic flux tube, and \(\beta \)decay electrons emitted in forward direction propagate towards the spectrometer section. The electrons then enter the transport and pumping section that reduces the tritium flow by a factor of \({10^{14}}\) in total [7], using a combination of a differential pumping section (DPS [8]) with turbomolecular pumps and a cryogenic pumping section (CPS [9]) where tritium is adsorbed by an argon frost layer. The kinetic energy of the decay electrons is analyzed in a tandem of MACE filter^{1} spectrometers [10, 11, 12]. The main spectrometer achieves an energy resolution of 0.93 eV at the tritium endpoint of \(E_0(\text {T}_2) = {18{,}571.8(12)}{\mathrm{eV}}\) [5, 13] by a combination of an electrostatic retarding potential and a magnetic guiding field. Electrons with sufficient kinetic energy pass the retarding potential and are counted at the focalplane detector, which uses a 148pixel PIN diode wafer for electron detection (FPD [14]). An integral energy spectrum is measured by varying the filter energy close to the tritium endpoint. The effective neutrino mass is determined by fitting the convolution of the theoretical \(\beta \)spectrum with the response function of the spectrometer to the data, taking into account important parameters such as the final states distribution and the energy loss spectrum and other systematic corrections [1, 15]. The spectrometer highvoltage is monitored by a pair of precision highvoltage dividers [16, 17] that support voltages up to 35 and 65 kV, respectively. An absolute voltage calibration is achieved by measuring the divider’s output voltage with ppm precision using a digital voltmeter. Additionally, the stability of the retarding potential is monitored continuously at the monitor spectrometer [18]. Like the main spectrometer, it is designed as a MACE filter with similar transmission characteristics and energy resolution. It uses a fivepixel PIN diode as a detector, which can detect electrons with kinetic energies \(E \gtrsim {10}\,{\mathrm{keV}}\). During normal operation, the monitor spectrometer is connected to the main spectrometer high voltage system and measures natural conversion lines of \({}^{\text {83m}}\text {Kr}\), where changes in the retarding potential are observed as shifts in the measured line position. It is also possible to use an independent power supply to operate the monitor spectrometer in standalone mode, which was used for the test measurements discussed in this article.
A precise knowledge of the transmission properties of the KATRIN main spectrometer is crucial to limit systematic uncertainties and reach the desired neutrino mass sensitivity. The transmission properties are affected by inhomogeneities of the electromagnetic fields in the main spectrometer. In addition to simulations, dedicated measurements are necessary to determine the spectrometer transmission function over the complete magnetic flux tube. Such measurements require a monoenergetic and angularselective electron source, which we present in this work. A pulsed electron beam allows us to access additional information from the electron timeofflight (ToF) [19].
This article is structured as follows: Sect. 2 discusses the revised technical design of the photoelectron source that was developed at WWU Münster over the recent years [20, 21, 22]. The design underwent many improvements for the second commissioning phase of the KATRIN main spectrometer. In Sect. 3 we show results from test measurements at the KATRIN monitor spectrometer. We determine important source characteristics such as the energy and angular spread of the produced electrons and the effective work function of the photocathode. Section 4 discusses simulation results that were produced by Kassiopeia, a particletracking software that has been developed as a joint effort in the KATRIN collaboration over recent years [23]. These simulations allow us to gain a detailed understanding of the electron acceleration and transport processes inside the electron source.
2 Setup and design
2.1 Principle of the MACE filter
The KATRIN beam line transports a maximum magnetic flux of \({191}\,{\mathrm{T\,cm}^2}\) from the source to the detector. Electrons that are created at the source follow different magnetic field lines, depending on their initial radial and azimuthal position. The transmission function for electrons is affected by inhomogeneities in the analyzing plane of the electric potential (\(\Delta U_\mathrm {ana} < {1.2}\,{\mathrm{V}}\)) and the magnetic field (\(\Delta B_\mathrm {min} < {50}\,{\upmu {\mathrm{T}}}\)). Because these variations are too large to be neglected, the detector features a pixelated wafer that can adequately resolve the position in the analyzing plane. This allows us to consider the electromagnetic inhomogeneities by determining transmission functions for individual detector pixels. The exact value of the retarding potential \(U_\mathrm {ana}\) and the magnetic field \(B_\mathrm {min}\) can be accessed through measurements with an electron source that generates electrons at defined kinetic energy and pitch angle. A source that fulfills these requirements has been developed at WWU Münster for the commissioning of the KATRIN main spectrometer.
2.2 Principle of the electron source
It was demonstrated in [21] that angular selectivity can be achieved by a combination of nonparallel electric and magnetic fields. An earlier design that used a goldplated quartz tip, which was illuminated by UV light from optical fibers on the inside of the tip was able to produce electrons with nonzero pitch angles. This setup achieved an insufficiently large angular spread of the electrons. The source was therefore not usable as a calibration source for a MACE filter. The design was further refined in [22] and the setup now resembles a plate capacitor that introduces a homogeneous electric acceleration field. The setup can be tilted against the magnetic field lines to imprint a welldefined pitch angle on the generated electrons. This design uses a planar photocathode, which is backilluminated by UV light from a single optical fiber.
The front plate (blue) with an aperture for electrons is mounted parallel to the back plate and placed in front of the emission spot. A potential difference \(U_\mathrm {acc} = U_\mathrm {front}  U_\mathrm {start} \le {5}\,{\mathrm{kV}}\) is applied between the plates to create an electric field perpendicular to the photocathode surface. The plates are mounted inside a grounded cage (yellow) to shield the electric field at the photocathode against outside influences. The whole setup can be mechanically tilted against the direction of the magnetic field. After passing the front plate, the electrons are accelerated adiabatically towards the ground potential at the spectrometer entrance where they achieve their maximum kinetic energy.
The pitch angle of the emitted electrons transforms adiabatically during propagation towards the spectrometer entrance magnet, where the electrons enter a higher magnetic field. According to (2) the transformation depends on the ratio of the magnetic fields at the emission spot, \(B_\mathrm {start}\), and the magnetic field in the entrance magnet, \(B_\mathrm {mag} \le B_\mathrm {max}\). The pitch angle increases because \(B_\mathrm {start} \ll B_\mathrm {mag}\). The electron source we present here produces defined pitch angles that cover the full range of \(\theta = {0^{\circ }}{}{90^{\circ }}\) in the entrance magnet with plate angles \(\alpha _\mathrm {p} \le {16^{\circ }}\) (Sect. 3.5). The energy spread in the entrance magnet is defined by the initial energy distribution of the emitted electrons, because the acceleration by the electric field does not deform the energy distribution. The kinetic energy is merely shifted by \(q U_\mathrm {start}\), while the spectral shape of the initial distribution is unaffected. A spectroscopic measurement of the electron energy, e. g. with a MACE filter, therefore allows us to determine the initial energy distribution of the emitted electrons. The method is also suited to determine the photocathode work function, which is discussed in Sect. 3.3.
2.3 Technical design
The technical design of the electron source is based on the platecapacitor setup depicted in Fig. 2. We use two stainless steel disks with radius \(r_\mathrm {p} = {30}\,{\mathrm{mm}}\) for the front and back plate, which are placed at a distance \(d = {10}\,{\mathrm{mm}}\). Both plates were electropolished before installation. The front plate has a thickness of \(d_\mathrm {fp} = {2}\,{\mathrm{mm}}\) and features an aperture with a radius \(r_\mathrm {afp} = {3}\,{\mathrm{mm}}\) for the emitted electrons. The back plate has a thickness of \(d_\mathrm {bp} = {3}\,{\mathrm{mm}}\) and allows mounting a photocathode holder at its center. The holder has an aperture to gluein an optical fiber with diameter \({200}\,{\upmu \mathrm{m}}\). The holder with the optical fiber is manually polished to create a flat surface, and the photocathode material is deposited on the surface by electron beam physical vapor deposition (EBPVD). For the measurements presented here we used a gold photocathode with a layer thickness of 20 nm; we also used silver with a thickness of 40 nm in other measurements. The plates are isolated against each other and the grounded cage by polyether ether ketone (PEEK) insulators. The grounded cage has an inner radius of \(r_\mathrm {c} = {50}\,{\mathrm{mm}}\) with an aperture \(r_\mathrm {ac} = {35}\,{\mathrm{mm}}\) at the front.
The grounded cage is gimbalmounted to allow tilting against two axes. The center of rotation is aligned with the emission spot on the back plate. This design ensures that the magnetic field line that the electron is following does not change when tilting the source cage. A precise readout of the plate angle is achieved by rotating piezoelectric motors (Attocube ANR240) that are installed at the pivot joints of the gimbal mount. These motors do not provide sufficient torque to tilt the electron source under vacuum conditions, but allow the relative tilt angle to be measured with a precision of \({0.05^{\circ }}\). To actuate the gimbal mount under vacuum conditions, our design uses two airpressure linear motors (Bibus Tesla 1620) that are mounted outside the vacuum chamber. The linear motion of the motors is transferred onto the chamber by Bowden cables that are attached to each axis of the gimbal mount. By operating the motors, each axis can be tilted separately. The motors are controlled with a LabView software, which also takes care of the transformation between the twoaxial and polar/azimuthal coordinate system for the plate angles.
The optical system to provide the UV light for the photocathode allows choosing between two light sources. A frequencyquadrupled \(\text {Nd:YVO}_4\) laser (InnoLas mosquito2660.1V) provides UV light at a wavelength of 266 nm (1 nm FWHM) at high intensity ( \({10}\,{\mathrm{mW}}\) output power). The intensity can be adjusted by an internal attenuator (\(\lambda /2\)plate with polarizing filter) and by a neutral density (ND) filter, which is placed in the laser beam. Behind the ND filter, a fraction of approximately 0.5% of the UV light is coupled out by a beam splitter to measure the UV light intensity with a photodiode. The laser light is focused by an aspheric lens into a \(\varnothing {200}\,{\upmu \mathrm{m}}\) optical fiber and guided into the source chamber. The laser is operated in pulsed mode with frequencies of 40–100 kHz at a pump diode current of 6–8 A. The current and frequency setting determines the output power, which can be tuned to produce a desired electron rate of several kcps (cps: counts per second) at the detector. The pulse width of \({20}\,{\mathrm{ns}}\) allows for timeofflight measurements with a precisely known starting time of the electrons.
Alternatively, an array of LEDs can be used as light source to provide UV light with \(\lambda = {260}{}{320\,\mathrm{nm}}\). Six balllens UV LEDs (Roithner UVTOP260–310) with peak wavelengths of 265, 275 nm etc. on are mounted on a revolver that is moved by a stepper motor. This allows us to automatically place the desired LED on the optical axis without manual adjustments. To achieve a sharp line width, a UV monochromator with 4 nm FWHM is used. The monochromator is operated by another stepper motor. The LED revolver in combination with the monochromator allows selecting arbitrary wavelengths in the available range. Like in the laser setup, a beam splitter with photodiode is used to monitor the light intensity. The divergent light beam of the LEDs is focused by an optical telescope consisting of two convex lenses, and guided into the electron source through an optical fiber. The current to operate the LEDs is provided by a function generator in pulse mode, using the internal \({50}\,{\Omega }\) resistor with an output voltage of 8.5 V. With this setting, the LEDs are driven by a peak current of 200 mA, which corresponds to a mean current of 20 mA at 10% duty cycle. Under nominal conditions, a pulse frequency of 100 kHz is used with a pulse length of \({1}\,{\upmu \mathrm{s}}\). Timeofflight measurements are thus also possible with LEDs as a light source. Depending on the LED and the monochromator setting, electron rates in the kcps range can be achieved.
The optical system (laser device, the stepper motors of the LED system and the two photodiodes), the actuation of the plate angle and the power supply for the dipole electrode are controlled and monitored by a LabView software that has been developed for use with the electron source. The photodiode readout allows us to monitor the stability of the UV light source, where intensity changes (e. g. because of warmup effects) could result in fluctuations of the observed electron rate.
2.4 Analytical transmission function
The measured transmission functions presented in this paper have been fitted by a Markov–Chain Monte Carlo (MCMC) method of minimizing the \(\chi ^2\) value, using a code that was implemented in Python. It utilizes emcee [31] for the MCMC fit process [29].
3 Measurements
3.1 Experimental setup
Figure 5 shows the high voltage scheme of the monitor spectrometer setup. The electron source is connected with a small difference voltage to the high voltage of the spectrometer in order to cancel out voltage fluctuations that would occur if two independent power supplies were used. The back plate voltage, \(U_\mathrm {start}\), can be varied against the spectrometer voltage \(U_\mathrm {spec}\) by combining a power supply that operates at 0 to −1.25 kV with a battery that delivers a voltage of about 90 V. By putting the two voltage sources in series, it is possible to vary the starting voltage to achieve a surplus energy \(q \Delta U = q (U_\mathrm {start}  U_\mathrm {spec}) = {90\mathrm{eV}}\, \mathrm{to}\, {1160\mathrm{eV}}\) without requiring a polarityswitching power supply. The voltage difference between electron source and spectrometer, \(U_\mathrm {start}  U_\mathrm {spec}\), is measured by a difference voltmeter (DVM) to monitor the electron surplus energy. Transmission functions can be measured by varying the starting voltage within a few V around zero while observing the electron rate at the detector. The high voltage system is mainly located inside a Faraday cage, which is operated on the spectrometer high voltage. This cage is put inside another grounded HV cabinet to allow safe operation. The acceleration voltage for the front plate, \(U_\mathrm {acc}\), is provided by an additional power supply that generates up to 5 kV w. r. t. the back plate voltage. The acceleration voltage is thus kept constant while varying \(U_\mathrm {start}\). This power supply is isolated for voltages up to 35 kV and can be placed outside the HV cabinet.
3.2 Electron rate
The dipole electrode in front of the electron source is intended to remove stored electrons from the beamline between source and spectrometer. The removal efficiency depends on the strength of the induced \(E \times B\) drift (9), and thus increases with a larger dipole voltage.^{4} In our setup, the magnetic field at the electrode is \(B_\mathrm {dip} = {78}\,{\mathrm{mT}}\) with \(E_\mathrm {dip} \approx {40}\,{\mathrm{kV}/\mathrm{m}}\) according to simulations. The removal efficiency of the dipole electrode was investigated by measuring transmission functions in direction of increasing and decreasing electron surplus energy.
Figure 6 shows that the observed transmission function is affected by a hysteresis effect that depends on the dipole voltage, which allows investigating the removal efficiency of the dipole electrode. The observed transmission functions show a similar behavior, except for the nominal electron rate that is reached at full transmission. The small rate drift that can be observed in the upper panel can be explained by fluctuations in UV light intensity. The hysteresis effect can be explained by the continuous filling of the trap from the beginning of the measurement when measuring in direction of increasing surplus energy, because the surplus energy at the beginning is too small for electrons to be transmitted. Electrons with a given energy stay trapped until they lost kinetic energy (e. g. through synchrotron radiation) or are removed by the dipole field. Scattering processes with electrons of higher kinetic energy that are generated at a later time during the measurement cause some of the trapped electrons to gain kinetic energy, thereby increasing transmission probability towards the detector. The effect does not occur when the measurement is performed in inverse direction, where the higherenergetic electrons are transmitted at the beginning of the measurement [34]. This leads to a hysteresis effect in the electron rate between the two scanning directions, which becomes smaller when the dipole voltage is increased and more electrons are removed from the trap. The observed rate difference is therefore a direct measure for the dipole efficiency. Our measurement indicates that a dipole voltage of \(U_\mathrm {dip} = {2}\,{\mathrm{kV}}\) is sufficient to avoid the hysteresis effect. With lower dipole voltages, the observed rate difference between the two scanning directions increases, indicating an insufficient removal of stored electrons.
3.3 Energy spread
The energy resolution of a MACE filter (4) depends on the retarding potential \(U_\mathrm {ana}\). At low voltages \(U_\mathrm {ana} \ll {18.6}\,{\mathrm{kV}}\) and low electron energies \(E \approx q U_{ana}\), the energy resolution improves because of the smaller amount of transversal energy left in the analyzing plane. A low voltage measurement with \(U_\mathrm {ana} \approx {200}{\mathrm{V}}\) allows us to directly determine the energy distribution of the produced electrons. Unfortunately, at the monitor spectrometer it is not possible to detect electrons with \(E \ll {10}\,{\mathrm{keV}}\) due to the energy threshold of the detector. Fortunately, the energy distribution can also be determined from a measurement performed at nominal high voltage.
If the source is operated at the socalled zero angle setting, it produces the smallest possible pitch angle in the spectrometer entrance magnet. The zero angle position has to be found manually by varying the plate angle \(\alpha _\mathrm {p}\) around \({0^{\circ }}\) independently for the vertical and horizontal axis. Such a calibration measurement was carried out before performing any other measurements. The determined zero angle position is automatically corrected by the slowcontrol software of the electron source, so that \(\alpha _\mathrm {p} = {0^{\circ }}\) always refers to the zero angle from here on. At an electron surplus energy \(E = q \Delta U \approx {0}\,{\mathrm{eV}}\) (6), the transmission probability is entirely dominated by the pitch angle of the emitted electrons. The observed electron rate is thus sensitive to small changes of the produced pitch angle, and the rate dependency w. r. t. the plate angle shows a maximum at zero angle \(\alpha _\mathrm {p} = {0^{\circ }}\). At the monitor spectrometer, the zero angle offset was found to be \(\alpha _\mathrm {hor} = {0.04(1)^{\circ }}\) and \(\alpha _\mathrm {ver} = {1.13(1)^{\circ }}\) at \(U_\mathrm {dip} = {2}\,{\mathrm{kV}}\). This offset is caused by mechanical imperfections, which result in a minor misalignment that can be easily corrected by such a measurement. The impact of the angular spread on the observed transmission function is marginal when the zero angle is applied. In this case the actual mean and width of the angular distribution are not relevant to the analytical transmission model as long as \(\theta < {5^{\circ }}\), and the energy spread dominates the shape of the resulting transmission function. It is thus possible to fit an (integrated) energy distribution to the measured transmission function while assuming a fixed angular distribution at a small pitch angle. For the case discussed here, an angular distribution with mean angle \({\hat{\theta }} = {2^{\circ }}\) and angular spread \(\sigma _\theta = {1^{\circ }}\) was used. These values are consistent with particletracking simulations (Sect. 4.4) and complementary measurements of the angular distribution that were performed at the monitor spectrometer (Sect. 3.5).
The measurements discussed in this section use the analytical transmission model (10) with five free parameters: the amplitude and background of the electron signal, as well as the mean, width and shape of the energy distribution. The statistical uncertainty at each data point is derived from the measured rate fluctuations by computing the median and \(1\sigma \)percentiles of the rate taken at 2 s intervals for each data run at a fixed value of \(U_\mathrm {start}\) (constant surplus energy). In most cases, the uncertainty determined by this method matches the \(\sqrt{N}\) expectation from Poisson statistics. However, the percentile method is believed to be more robust against asymmetric rate fluctuations, and is thus preferred. For the transmission function measurements, an uncertainty of \(\pm {60}\,{\mathrm{meV}}\) is assumed for the surplus energy, which is included in the fit as an additional term in the uncertainty of each data point. The value has been estimated from the contributions of the individual power supplies that are used in the setup [28].
Measured transmission functions at different wavelengths \(\lambda \) and fixed spectrometer voltage \(U_\mathrm {spec} = {18.6}\,{\mathrm{kV}}\) (Fig. 7). The table shows the upper limit of the energy distribution, \(E_\mathrm {lim}\), and the energy spread, \(\sigma _E\); both values are derived from the fit result. The measurement at 266 nm has been performed twice at −18.6 kV
\(\lambda \) (nm)  \(E_\mathrm {lim}\) (eV)  \(\sigma _E\) (eV)  \(\chi ^2/ndf\) 

266.0  0.82(2)  0.31(5)  1.39 
266.0  0.82(2)  0.28(4)  1.40 
272.4  0.74(1)  0.22(2)  1.18 
282.4  0.61(2)  0.19(3)  1.23 
292.4  0.47(2)  0.14(3)  3.38 
302.4  0.33(2)  0.09(7)  3.46 
Table 1 lists the parameters of the energy distribution, which are derived from the measured transmission functions. As an indicator for the upper limit of the energy distribution we use a value which we call \(E_\mathrm {lim}\). According to (7) it is possible to determine the photocathode work function by relating the measured value of \(E_\mathrm {lim}\) to the known wavelength \(\lambda \). For \(E_\mathrm {lim}\) we choose the energy where the distribution drops to 25% of its maximum, because this gave results compatible with the direct work function measurement (Fowler method, see Sect. 3.6 below). The range \([0; E_\mathrm {lim}]\) then includes at least 90% of the distribution’s integral, and we associate the width of the energy distribution with this range. The value \(\sigma _E\) in Table 1 refers to the width of a symmetric normal distribution, which can be derived from the generalized normal distribution. The transformation to \(\sigma _E\) takes into account the asymmetry of the distribution and allows comparing distributions with different asymmetry.
The results indicate that owing to the small angular spread in this setting, the width of the measured transmission function is fully dominated by the energy distribution of the electrons. This is true especially for measurements with zero angle and small wavelengths, where the angular distribution has only a minor effect on the transmission function and the energy spread is comparably large.
3.4 Magnetic reflection
At the monitor spectrometer, this measurement was performed at four different azimuthal directions of the plate angle to investigate possible asymmetries, \(\alpha _{az} = {0^{\circ }}, {90^{\circ }}, {180^{\circ }}, {270^{\circ }} \) in the global coordinate system.^{5} Unless stated otherwise in this publication, measurements are carried out with \(\alpha _{az} = {0^{\circ }}\). The results are shown in Fig. 8 (solid lines). The underlying gausscurves shown at the bottom of the figure allows a better comparison of the angular distributions.
Table 2 shows the fit results of these measurements. For nominal magnetic field at the electron source (\(B_\mathrm {start} = {27}\,{\mathrm{mT}}\)), reflection occurs at a plate angle \(\alpha _\mathrm {max} \approx {10^{\circ }}\). The width of the angular distribution is consistent over the four measurements, yielding a width of \(\sigma _\alpha = {0.40^{\circ }}\) for the underlying Gaussian distribution. The adiabatic transformation (15) converts the value \(\sigma _\alpha \) to an effective angular spread \(\sigma _\theta \) in the magnet. The conversion employs the constraint that magnetic reflection occurs at \(\alpha _\mathrm {p} = \alpha _\mathrm {max}\) with \(\theta = {90^{\circ }}\). This yields an average angular spread of \(\sigma _\theta = {16.2^{\circ }}\) at the maximal pitch angle of \({90^{\circ }}\). Note that the angular spread close to magnetic reflection increases because of the nonlinearity of (15), and is significantly lower at smaller pitch angles.
The discrepancy between the measurements in four azimuthal directions can be explained in two ways. Firstly, particletracking simulations indicate that misalignments of the emission spot relative to the plate setup of the electron source result in significant offsets of the produced pitch angles. Such misalignments can result from mechanical imperfections of the setup and are likely the explanation for the observed asymmetry [29]. Secondly, phase effects can affect the electron acceleration processes in the source. The cyclotron phase of the emitted electrons differs depending on the azimuthal direction into which the electron beam is collimated. This results in slight variations of the produced pitch angle, which depend on the azimuthal plate angle \(\alpha _\mathrm {az}\). The asymmetry in vertical direction (\(\alpha _\mathrm {az} = {0^{\circ }} , {180^{\circ }}\)) is further increased by the electric field of the dipole electrode.
Measured magnetic reflection curves at different azimuthal directions \(\alpha _\mathrm {az}\) of the plate angle (Fig. 8). The table shows the reflection angle \(\hat{\alpha } = \alpha _\mathrm {max}\) and the width \(\sigma _\alpha \) (in terms of plate angle) of the reflection curve that was determined by the fit. The angular spread \(\sigma _\theta \) (in terms of pitch angle) has been computed from the adiabatic transformation (15) with the known reflection angle
\(\alpha _\mathrm {az}\) (\(^{\circ }\))  \(\alpha _\mathrm {max}\) (\(^{\circ }\))  \(\sigma _\alpha \) (\(^{\circ }\))  \(\sigma _\theta \) (\(^{\circ }\))  \(\chi ^2/ndf\) 

0  10.06(2)  0.39(3)  16.0(10)  0.71 
180  11.13(3)  0.39(3)  15.2(12)  0.88 
90  9.56(2)  0.40(2)  16.7(9)  1.89 
270  9.73(2)  0.40(3)  16.6(10)  0.82 
Weighted average:  0.396(1)  16.20[SPAN](7) 
3.5 Angular selectivity
Figure 9 shows measured transmission functions at different plate angles \(\alpha _\mathrm {p}\). The zero angle setting \(\alpha _\mathrm {p} = {0^{\circ }}\) is used as a reference for the other measurements. The transmission functions are clearly separated and the expected shift to larger surplus energies is observed when increasing the plate angle. Table 3 shows the corresponding parameters of the derived angular distribution. Magnetic reflection occurs at \(\alpha _\mathrm {p} \ge {10^{\circ }}\) (as expected from the magnetic reflection measurement, which yields \(\alpha _\mathrm {max} = {10.1^{\circ }}\); cmp. Sect. 3.4). This results in a significantly deformed angular distribution, because reflected electrons are missing from the observed transmission function. Since the fit is based on a reference measurement at \(\alpha _\mathrm {p} = {0^{\circ }}\) to obtain the corresponding energy distribution, the deformation affects the fit result and explains the large \(\chi ^2\) value.
The transmission functions were fitted as explained above. However in this case, free parameters were the amplitude and background of the electron signal and the mean angle and the angular spread. This allows us to determine the produced pitch angle directly from the measurement, while assuming a known energy distribution of the electron source. In this case, a reference measurement at nominal settings (zero angle \(\alpha _\mathrm {p} = {0^{\circ }}\), \(\lambda = {266}\,{\mathrm{nm}}\), \(U_\mathrm {ana} = {18.6}\,{\mathrm{kV}}\)) was used for the energy distribution (Sect. 3.3). The fit using the analytical model of the transmission function is not very sensitive to the actual shape of the angular distribution for \(\theta \rightarrow {0^{\circ }}\) and \(\theta \rightarrow {90^{\circ }}\). The angular distribution determined from the measurements at \(\alpha _\mathrm {p} = {0^{\circ }}\) and \(\alpha _\mathrm {p} = \alpha _\mathrm {max}\) thus yield large uncertainties, and the angular spread is significantly smaller than at intermediate pitch angles. However, the fit results match expectations from an analytical calculation of the pitch angle based on the magnetic reflection limit discussed in Sect. 3.4). The observed pitch angles are also confirmed by simulation results (Sect. 4.4).
Measured transmission functions at different plate angles \(\alpha _\mathrm {p}\). The table shows the mean angle \({\hat{\theta }}\) and the angular spread \(\sigma _\theta \) in the spectrometer entrance magnet; the values have been determined by the fit. An expected pitch angle \({\hat{\theta }}_\mathrm {ana}\) is derived analytically from adiabatic transformation (5). At \(\alpha _\mathrm {p} \ge \alpha _\mathrm {max} = {10^{\circ }}\) magnetic reflection is observed, which leads to a significant deformation of the transmission function
\(\alpha _\mathrm {p}\) (\(^{\circ }\))  \({\hat{\theta }}\) (\(^{\circ }\))  \(\sigma _\theta \) (\(^{\circ }\))  \(\chi ^2/ndf\)  \({\hat{\theta }}_\mathrm {ana}\) (\(^{\circ }\)) 

0  1.7(1.3)  2.0(1.3)  1.09  2.0 
2  5.7(3.4)  9.3(2.6)  1.07  13.5 
4  23.2(0.3)  5.8(0.8)  1.12  25.4 
6  38.2(0.2)  4.3(0.5)  1.31  38.6 
8  55.2(0.3)  5.6(0.4)  1.50  54.7 
10  89.3(0.8)  0.8(0.7)  10.9  85.7 
Weighted average:  5.5[SPAN]0.3  (excluding \(\alpha _\mathrm {p} = {10}\)) 
3.6 Work function
The fit results in a work function of \(\Phi = 3.78\,\mathrm{eV} \pm 0.03\,\mathrm{eV}(\mathrm{sys})\pm 0.01\,\mathrm{eV}(\mathrm{stat})\) for the gold photocathode with 20 nm layer thickness at \(T = {300}\,{\mathrm{K}}\). The systematic uncertainty is estimated to 0.03 eV from the uncertainty of the wavelength caused by to the filter width of the monochromator (\({0.01}\,{\mathrm{eV}} \hat{=} {1}\,{\mathrm{nm}}\)), and the uncertainty of the LED peak wavelength (2 nm). The monochromator was calibrated beforehand using the known wavelength of the UV laser (266 nm). The determined work function is equivalent to a wavelength of \(\lambda _\mathrm {opt} = 328.2\,\mathrm{nm} \,\pm \, 2.3\,\mathrm{nm}(\mathrm{sys}) \pm 0.7\,\mathrm{nm} (\mathrm{stat})\). The energy spread of the electron source can be minimized by matching the UV wavelength to this value. Unfortunately, the available LEDs limit the usable wavelength range to about 320 nm, as the very low rate at larger wavelengths would require unfeasibly long measurement times. However, even at wavelengths well above 266 nm, the optimum for maximal intensity, the resulting energy spread of 0.3 eV or less is sufficiently small to determine the transmission properties of the spectrometer (Sect. 3.3).
Work functions determined from measured transmission functions at different wavelengths \(\lambda \) (Fig. 7). The work functions \(\Phi ^\dagger \) are derived from (19), with \(E_{\mathrm {lim}}\) the upper limit of the energy distribution (Table 1) and \(hc / \lambda \) the known photon energy. The results are compared with the work function \(\Phi = {3.78(4)}\,{\mathrm{eV}}\) that was determined in a Fowlertype measurement (Fig. 10)
\(\lambda \) (nm)  \(hc/\lambda \) (eV)  \(\Phi ^\dagger \) (eV)  \(\Phi ^\dagger \Phi \) (eV) 

266.0  4.66(2)  3.84(4)  0.05(7) 
266.0  4.66(2  3.84(4)  0.05(7) 
272.4  4.55(4)  3.81(5)  0.02(8) 
282.4  4.39(4)  3.79(5)  0.00(8) 
292.4  4.24(3)  3.77(5)  −0.02(8) 
302.4  4.10(3)  3.77(5)  −0.02(8) 
Weighted average:  3.810(1)  0.03(3) 
4 Simulations
The particletracking software Kassiopeia was developed as a joint effort from members of the KATRIN collaboration to simulate trajectories of charged particles such as electrons or ions in complex electromagnetic fields with very high precision [23]. Kassiopeia is embedded in the socalled KASPER framework, the overall KATRIN software package. The software is used to study the transmission properties of the KATRIN spectrometers and to investigate background processes, among other simulation tasks. For the development of the electron source presented in this paper, Kassiopeia simulations provided substantial input for optimizations of the existing design. Detailed simulations were performed to investigate the electron acceleration processes within the source and to understand how the welldefined pitch angles are produced.
4.1 Implementation into Kassiopeia
The electrode geometry of the electron source was implemented in Kassiopeia based on CAD drawings of the electron source design. The position of the electrodes w. r. t. the spectrometer setup was determined from measurements at the experimental site and from comparisons of simulated with measured magnetic fields (Sect. 3.1). Figure 11 shows the simulated magnetic field and electric potential between the photocathode of the electron source and the entrance magnet of the monitor spectrometer.
4.2 Energy and angular distributions
The simulations allow us to investigate the electron acceleration mechanisms inside the source. An important question is the effect of the electromagnetic fields on the energy and angular distributions achieved. Electrons were started from the emission spot on the back plate (radius \({100}\,{\upmu \mathrm{m}}\), according to the dimensions of the optical fiber in the experimental setup), where the starting voltage \(U_\mathrm {start} \approx {18.6}\,{\mathrm{kV}}\) is applied. The initial energy is normaldistributed in the range 0–0.6 eV (\(\upmu = \sigma = {0.2}\,{\mathrm{eV}}\)). The initial polar angle w. r. t. the back plate follows a \(\cos \theta \)distribution in the range \({0^{\circ }}{}{90^{\circ }}\). The parameters of the energy distribution were chosen according to measurement results, which yield an energy spread of up to 0.3 eV (Sect. 3.3), while the angular distribution matches the results from [26].
Figure 13 shows results of the same simulation, but here the correlation between the initial and final angular distributions is investigated w. r. t. the initial kinetic energy. The electron pitch angle is changed by the nonadiabatic acceleration and the subsequent adiabatic transport to the entrance magnet. While the initial pitch angles follow a \(\cos \theta \)distribution with angles up to \({90^{\circ }}\), the pitch angles in the magnet are narrowly distributed. As above, the distributions were analyzed by their median and width. The distribution in the magnet has a median pitch angle of \({\hat{\theta }} = {1.5^{\circ }}\) with an asymmetric width of \(\sigma _\theta ^ = {1.0^{\circ }}\) and \(\sigma _\theta ^+ = {1.4^{\circ }}\). Here the asymmetry is caused by the fact that the pitch angle is limited to the range \({0^{\circ }}\, \mathrm{to} \,{90^{\circ }}\) by definition. Whenever the pitch angle would assume negative values resulting from adiabatic transformation, it is instead mirrored to a positive value. The observed distribution is therefore “wrapped” into the positive regime at \(\theta = {0^{\circ }}\), and thus becomes asymmetric when this effect occurs. As indicated by the coloring in the figure, the kinetic energy of the electrons influences the produced angular distribution as well. Electrons with higher kinetic energies contribute more to the observed angular spread than lowenergetic electrons. The same effect is also observed at larger plate angles \(\alpha _\mathrm {p} > {0^{\circ }}\). This is explained by the efficiency of the nonadiabatic acceleration in the plate setup of the electron source, which is responsible for imprinting a welldefined pitch angle on the electrons. According to the Lorentz equation (8), the electrostatic acceleration becomes less effective as the electron energy increases (cmp. Sect. 2.2). Lowenergetic electrons are therefore more strongly collimated, while for electrons with higher initial energies the observed angular spread increases. It is thus possible to further reduce the angular spread by tuning the electron source to produce a small energy spread, which can be achieved by matching the UV wavelength to the photocathode work function.
4.3 Electron acceleration and transport
Electrons that pass the front plate are further accelerated to their full kinetic energy \(E = q U_\mathrm {start}\) inside the source cage and transported adiabatically towards the spectrometer magnet. Because transmitted electrons pass the dipole electrode only once, the electric dipole field has no significant influence on the pitch angle transformation. However, the stray electric field of the dipole electrode affects the electron acceleration process itself: because of the asymmetric dipole field, a vertical electric field gradient is generated inside the source cage. Depending on the cyclotron phase of the electrons (and thus, depending on the azimuthal plate angle \(\alpha _\mathrm {az}\)) the electrons are accelerated differently and the pitch angle changes accordingly. Simulations show that dipole voltages \(U_\mathrm {dip} = {0}{}{4\mathrm{kV}}\) lead to deviations of the pitch angle up to \({2^{\circ }}\), an observation also made by corresponding measurements. The deviations can be corrected by an empirical determination of the zero angle (cmp. Sect. 3). The pitch angle increases towards \(B_\mathrm {max}\) as a result of adiabatic transformation. When the pitch angle exceeds \(\theta _\mathrm {max} = {90^{\circ }}\), electrons are magnetically reflected. These electrons can get stored between the photocathode and the entrance magnet and need to be removed by the dipole electrode to avoid a possible Penning discharge.
4.4 Production of welldefined pitch angles
Simulated pitch angles in the spectrometer entrance magnet and derived effective starting angles. The table shows the median pitch angle, \({\hat{\theta }}\), and the angular spread, \(\sigma _\theta \), in the entrance magnet for different plate angles \(\alpha _\mathrm {p}\). The simulation results are compared with the pitch angle determined from corresponding measurements, \({\hat{\theta }}_\mathrm {meas}\). An effective initial pitch angle at the photocathode, \(\theta _\mathrm {start}^*\), can be computed from \({\hat{\theta }}\) using the adiabatic transformation (21) and the known magnetic fields at the setup (\(B_\mathrm {start} = {27}\,{\mathrm{mT}}\), \(B_\mathrm {max} = {6}\,\mathrm{T}\))
\(\alpha _\mathrm {p}\) (\(^{\circ }\))  \({\hat{\theta }}\) (\(^{\circ }\))  \(\sigma _\theta \) (\(^{\circ }\))  \({\hat{\theta }}_\mathrm {meas}\) (\(^{\circ }\))  \(\theta _\mathrm {start}^{*}\) (\(^{\circ }\)) 

0  1.5  1.2  1.7(13)  0.1 
2  11.5  1.3  5.7(34)  0.9 
4  23.6  1.3  23.2(3)  1.6 
6  36.8  1.5  38.2(2)  2.4 
8  52.5  1.9  55.2(3)  3.2 
10  78.9  4.2  89.3(8)  3.9 
12  Magnetically reflected  4.6 
Table 5 lists the corresponding pitch angles and angular spreads, which correspond to the median and the \(1\sigma \)width of the angular distributions. Again, the values were computed using percentiles. It should be noted that the pitch angle at \(\alpha _\mathrm {p} = {0^{\circ }}\) is systematically larger because of the asymmetric shape of the distribution, which shifts the median to larger values. Similarly, the median at \(\alpha _\mathrm {p} = {10^{\circ }}\) is systematically smaller due to the deformation of the angular distribution, which is caused by magnetic reflection. The angular spread is comparable over a wide range of plate angles with \(\sigma _\theta \approx {1.5^{\circ }}\). The spread becomes significantly larger for \(\theta \rightarrow {90^{\circ }}\) as a result of adiabatic transformation (21). The simulated pitch angles and the angular spread are in good agreement with the corresponding measurements (Sect. 3.5). Table 5 also lists the measured pitch angles \(\theta _\mathrm {meas}\) (cmp. tab. 3), and shows that both results are typically in agreement. An effective starting angle \(\theta _\mathrm {start}^*\) has been computed via (21), showing a strictly linear relation to the plate angle \(\alpha _\mathrm {p}\).
The measurements and simulations discussed in this work clearly show that the electron source achieves angular selectivity and can produce welldefined pitch angles with small angular spread. Figure 16 shows the produced pitch angle in the spectrometer entrance magnet (solid red line) and the pitch angle at the end of the source chamber (dashed blue line) according to simulations. At \(\alpha _\mathrm {max} = {10.1^{\circ }}\), the pitch angle reaches \({90^{\circ }}\) and magnetic reflection occurs. Resulting from the finite angular spread, at \(\alpha _\mathrm {p} = {10^{\circ }}\) a fraction of the produced electrons is already reflected and cut off from the observed angular distribution. The simulated reflection angle is in excellent agreement with the magnetic reflection measurement (Sect. 3.4), where \(\alpha _\mathrm {max} = {10.06(3)^{\circ }}\) was observed for \(\alpha _{az} = {0^{\circ }}\). The effective initial pitch angle shows a strictly linear dependency to the plate angle with a factor k and a constant angular spread \(\sigma _{\theta ,\mathrm {start}} = {0.1^{\circ }}\).
5 Conclusion

Angular selectivity: The source produces welldefined electron pitch angles in the spectrometer entrance magnet. Magnetic reflection occurs when the pitch angle exceeds \({90^{\circ }}\), which was observed at a plate angle of \({10.1^{\circ }}\) in measurements. This value is in excellent agreement with the corresponding simulations, which also yield a reflection angle of \({10.1^{\circ }}\).

Small energy spread: Depending on the wavelength of the used UV light source, an energy spread between 0.09(7) eV at 302 nm and 0.031(5) eV at 266 nm was observed in transmission function measurements at \(U_\mathrm {start} = {18.6}\,{\mathrm{kV}}\). A measurement at low voltage \(U_\mathrm {spec} \approx {200}{\mathrm{V}}\) allows us to determine the energy spread with much higher precision because of the improved energy resolution of the spectrometer. While this feature cannot be applied at the monitor spectrometer, it is of great use for the commissioning of the main spectrometer where lowenergetic electrons can be detected through the use of a postacceleration electrode.^{7}

Small angular spread: at the monitor spectrometer setup with \(B_\mathrm {max} = {6}\,{\mathrm{T}}\) and \(B_\mathrm {start} = {27}\,{\mathrm{mT}}\), an angular spread of \({5^{\circ }}\) or less was observed in transmission function measurements at different plate angles. Simulations indicate that the angular spread is typically even smaller (about \({2^{\circ }}\)) for pitch angles \(\theta \lesssim {70^{\circ }}\).

Electron rate: The electron source achieves a stable electron rate at the detector of 1500 cps with the laser, and up to 400 cps with the LEDs as light source. It is possible to regulate the rate by tuning the intensity of the UV photon system, e. g. varying the pulse width of the LED pulser or by adjusting the laser diode current.

Pulsed mode: The light sources were operated in pulsed mode during the monitor spectrometer measurements. The pulsed mode allows timeofflight (ToF) measurements to characterize several properties of the MACE filter. The ToF mode plays an important role in the commissioning measurements of the main spectrometer.
Particletracking simulations were performed with the Kassiopeia software, providing vital input for the analysis of the measurements, and allow us to get a precise understanding of the electron acceleration processes in the electron source. The simulation results are typically in good agreement with the measurements. We showed that the energy distribution of the electrons in the spectrometer entrance magnet corresponds to the initial energy distribution, while both distributions show the same width and shape in the simulations. It is thus possible to fully determine the electron energy spectrum by performing transmission function measurements with a MACE filter. The angular distribution in the spectrometer magnet results from the nonadiabatic acceleration of the emitted electrons in the plate setup of the electron source and the subsequent adiabatic transport towards the spectrometer entrance. The electron beam is collimated by the strong electric acceleration field at the photocathode and reaches an effective angular spread of roughly \({0.1^{\circ }}\) when leaving the nonadiabatic acceleration region. According to simulations, an angular spread of less than \({2^{\circ }}\) (increases to \({4.2^{\circ }}\) for \(\theta \rightarrow {90^{\circ }}\), see Table 5) is reached in the spectrometer magnet. The simulated angular spread of \({2^{\circ }}\) to \({4^{\circ }}\) is lower than the measured average of approximately \({5.5^{\circ }}\) (Table 3). The produced pitch angle and the angular spread in the magnet strongly depend on the magnetic fields at the setup. The differences between measurements and simulations can therefore be explained by undetected misalignments of the setup and entailing inaccuracies of the computed fields.
Our electron source allows us to investigate major characteristics of a MACE filter, such as the transmission properties and the effective energy resolution of the spectrometer. We studied key features of the electron source in measurements at the KATRIN monitor spectrometer and in a suite of accompanying simulations. We fully characterized our electron source and demonstrated a reliable operation in a MACE filter setup. The electron source can be utilized as a vital tool for the commissioning of the KATRIN main spectrometer and in preparation of the upcoming neutrino mass measurements.
Footnotes
 1.
Magnetic adiabatic collimation with electrostatic filter.
 2.
The work function \(\Phi \) affects the vacuum potential \(\phi \) of an electrode at a voltage U according to \(\phi = U  \Phi / e\) where e is the unit charge.
 3.
The photocathode holder surface is manually polished using sand paper and polishing paste with granularities down to \({0.1}\,{\upmu \mathrm{m}}\) before depositing the photocathode material. Impurities from the residual gas are expected in the given vacuum conditions with \(p \approx {10^{7}}\,\mathrm{mbar}\).
 4.
The alternative solution of reducing the global magnetic field would change the magnetic fields at the MACE filter and is thus disfavored.
 5.
When looking from the source towards the detector, \(\alpha _{az} = {0^{\circ }}\) points to the left and \(\alpha _{az} = {90^{\circ }}\) points upwards.
 6.
Note that the simulations use a 3 kV dipole setting instead of the 2 kV setting used in the measurements discussed above, because they are intended to be comparable with later measurements carried out at the KATRIN main spectrometer.
 7.
The postacceleration electrode (PAE) shifts the electron energy by up to 10 keV between pinch magnet and detector wafer [14].
Notes
Acknowledgements
This work has been supported by the Bundesministerium für Bildung und Forschung (BMBF) with project numbers 05A14PMA and 05A14VK2 and the Helmholtz Association (HGF).
References
 1.J. Angrik et al., KATRIN Design Report 2004, Wissenschaftliche Berichte FZKA 7090. Tech. rep., Forschungszentrum Karlsruhe (2005). http://bibliothek.fzk.de/zb/berichte/FZKA7090.pdf
 2.G. Drexlin et al., Adv. High. Energy Phys. 2013, 293986 (2013). doi: 10.1155/2013/293986 CrossRefGoogle Scholar
 3.C. Kraus et al., Eur. Phys. J. C 40(4), 447 (2005). doi: 10.1140/epjc/s2005021397 ADSCrossRefGoogle Scholar
 4.V.N. Aseev et al., Phys. Rev. D 84, 112003 (2011). doi: 10.1103/PhysRevD.84.112003 ADSCrossRefGoogle Scholar
 5.E.W. Otten, C. Weinheimer, Rep. Prog. Phys. 71(8), 086201 (2008). doi: 10.1088/00344885/71/8/086201 ADSCrossRefGoogle Scholar
 6.F. Priester et al., Vacuum 116, 42 (2015). doi: 10.1016/j.vacuum.2015.02.030 ADSCrossRefGoogle Scholar
 7.X. Luo et al., Vacuum 80(8), 864 (2006). doi: 10.1016/j.vacuum.2005.11.044 ADSCrossRefGoogle Scholar
 8.S. Lukić et al., Vacuum 86(8), 1126 (2012). doi: 10.1016/j.vacuum.2011.10.017 ADSCrossRefGoogle Scholar
 9.W. Gil et al., IEEE Trans. Appl. Supercond. 20(3), 316 (2010). doi: 10.1109/TASC.2009.2038581 ADSCrossRefGoogle Scholar
 10.A. Picard et al., Nucl. Instr. Meth. Phys. Res. B 63(3), 345 (1992). doi: 10.1016/0168583X(92)95119C ADSCrossRefGoogle Scholar
 11.V.M. Lobashev, P.E. Spivak, Nucl. Instr. Meth. Phys. Res. A 240(2), 305 (1985). doi: 10.1016/01689002(85)906400 ADSCrossRefGoogle Scholar
 12.G. Beamson et al., J. Phys. A 14(2), 256 (1981). doi: 10.1088/00223735/14/2/526 Google Scholar
 13.E.G. Myers et al., Phys. Rev. Lett. 114, 013003 (2015). doi: 10.1103/PhysRevLett.114.013003 ADSCrossRefGoogle Scholar
 14.J.F. Amsbaugh et al., Nucl. Instr. Meth. Phys. Res. A 778, 40 (2015). doi: 10.1016/j.nima.2014.12.116 ADSCrossRefGoogle Scholar
 15.M. Babutzka et al., N. J. Phys. 14(10), 103046 (2012). doi: 10.1088/13672630/14/10/103046 CrossRefGoogle Scholar
 16.S. Bauer et al., J. Instrum. 8(10), P10026 (2013). doi: 10.1088/17480221/8/10/P10026 CrossRefGoogle Scholar
 17.T. Thümmler et al., N. J. Phys. 11(10), 103007 (2009). doi: 10.1088/13672630/11/10/103007 CrossRefGoogle Scholar
 18.M. Erhard et al., J. Instrum. 9(06), P06022 (2014). doi: 10.1088/17480221/9/06/P06022 CrossRefGoogle Scholar
 19.N. Steinbrink et al., N. J. Phys. 15(11), 113020 (2013). doi: 10.1088/13672630/15/11/113020 CrossRefGoogle Scholar
 20.K. Valerius et al., N. J. Phys. 11(6), 063018 (2009). doi: 10.1088/13672630/11/6/063018 CrossRefGoogle Scholar
 21.K. Valerius et al., J. Instrum. 6(01), P01002 (2011). doi: 10.1088/17480221/6/01/P01002 CrossRefGoogle Scholar
 22.M. Beck et al., J. Instrum. 9(11), P11020 (2014). doi: 10.1088/17480221/9/11/P11020 CrossRefGoogle Scholar
 23.D. Furse et al. (2017). arXiv: 1612.00262
 24.G. Hechenblaikner et al., J. Appl. Phys. 111(12), 124914 (2012). doi: 10.1063/1.4730638 ADSCrossRefGoogle Scholar
 25.R.H. Fowler, Phys. Rev. 38, 45 (1931). doi: 10.1103/PhysRev.38.45 ADSCrossRefGoogle Scholar
 26.Z. Pei, C.N. Berglund, Jpn. J. Appl. Phys. 41(1A), L52 (2002). doi: 10.1143/JJAP.41.L52 ADSCrossRefGoogle Scholar
 27.S. Groh, Modeling of the response function and measurement of transmission properties of the KATRIN experiment. Ph.D. thesis, Karlsruher Institut für Technologie (2015)Google Scholar
 28.M.G. Erhard, Influence of the magnetic field on the transmission characteristics and neutrino mass systematic of the KATRIN experiment. Ph.D. thesis, Karlsruher Institut für Technologie (2016)Google Scholar
 29.J.D. Behrens, Design and commissioning of a monoenergetic photoelectron source and active background reduction by magnetic pulse at the KATRIN spectrometers. Ph.D. thesis, Westfälische WilhelmsUniversität Münster (2016)Google Scholar
 30.J.R.M. Hosking, J.R. Wallis, Regional frequency analysis: an approach based on Lmoments, chap. A.8 (Cambridge University Press, Cambridge, 1997)Google Scholar
 31.D. ForemanMackey et al., Publ. Astron. Soc. Pac. 125(925), 306 (2013). doi: 10.1086/670067 ADSCrossRefGoogle Scholar
 32.M. Kraus, Energyscale systematics at the KATRIN main spectrometer. Ph.D. thesis, Karlsruher Institut für Technologie (2016)Google Scholar
 33.J.P. Barrett, A spatially resolved study of the KATRIN main spectrometer using a novel fast multipole method. Ph.D. thesis, University of North Carolina at Chapel Hill (2016)Google Scholar
 34.K. Wierman, Charge accumulation in the KATRIN main spectrometer. Ph.D. thesis, University of North Carolina at Chapel Hill (2016)Google Scholar
 35.C.N. Berglund, W.E. Spicer, Phys. Rev. 136, A1030 (1964). doi: 10.1103/PhysRev.136.A1030 ADSCrossRefGoogle Scholar
 36.F. James, M. Roos, Comput. Phys. Commun. 10(6), 343 (1975). doi: 10.1016/00104655(75)900399 ADSCrossRefGoogle Scholar
 37.M. Nonnenmacher et al., Appl. Phys. Lett. 58(25), 2921 (1991). doi: 10.1063/1.105227
 38.C.N. Berglund, W.E. Spicer, Phys. Rev. 136, A1044 (1964). doi: 10.1103/PhysRev.136.A1044 ADSCrossRefGoogle Scholar
 39.C. Bandis, B.B. Pate, Phys. Rev. B 52, 12056 (1995). doi: 10.1103/PhysRevB.52.12056 ADSCrossRefGoogle Scholar
 40.D.E. Eastman, Phys. Rev. B 2, 1 (1970). doi: 10.1103/PhysRevB.2.1 ADSCrossRefGoogle Scholar
 41.G.F. Saville et al., J. Vac. Sci. Technol. B 13(6), 2184 (1995). doi: 10.1116/1.588101 CrossRefGoogle Scholar
 42.R. D’Arcy, N. Surplice, Surf. Sci. 34(2), 193 (1973). doi: 10.1016/00396028(73)901155 ADSCrossRefGoogle Scholar
 43.W. Li, D.Y. Li, J. Chem. Phys. 122(6), 064708 (2005). doi: 10.1063/1.1849135 ADSCrossRefGoogle Scholar
 44.M. Zacher, Highfield electrodes design and an angularselective photoelectron source for the KATRIN spectrometers. Ph.D. thesis, Westfälische WilhelmsUniversität Münster (2014)Google Scholar
 45.D. Winzen, Development of an angular selective electron gun for the KATRIN main spectrometer. Diploma thesis, Westfälische WilhelmsUniversität Münster (2014)Google Scholar
 46.J.A. Formaggio et al., Prog. Electromagn. Res. B 39, 1 (2012). doi: 10.2528/PIERB11112106 CrossRefGoogle Scholar
 47.F. Glück, Prog. Electromagn. Res. B 32, 319 (2011). doi: 10.2528/PIERB11042106 CrossRefGoogle Scholar
 48.F. Glück, Prog. Electromagn. Res. B 32, 351 (2011). doi: 10.2528/PIERB11042108 ADSCrossRefGoogle Scholar
 49.D.L. Furse, Techniques for direct neutrino mass measurement utilizing tritium \(\beta \)decay. Ph.D. thesis, Massachusetts Institute of Technology (2015)Google Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funded by SCOAP^{3}