Abstract
The KATRIN experiment is designed for a direct and modelindependent determination of the effective electron antineutrino mass via a highprecision measurement of the tritium \(\upbeta \)decay endpoint region with a sensitivity on \(m_\nu \) of 0.2 \(\hbox {eV}/\hbox {c}^2\) (90% CL). For this purpose, the \(\upbeta \)electrons from a highluminosity windowless gaseous tritium source traversing an electrostatic retarding spectrometer are counted to obtain an integral spectrum around the endpoint energy of 18.6 keV. A dominant systematic effect of the response of the experimental setup is the energy loss of \(\upbeta \)electrons from elastic and inelastic scattering off tritium molecules within the source. We determined the energyloss function insitu with a pulsed angularselective and monoenergetic photoelectron source at various tritiumsource densities. The data was recorded in integral and differential modes; the latter was achieved by using a novel timeofflight technique. We developed a semiempirical parametrization for the energyloss function for the scattering of 18.6keV electrons from hydrogen isotopologs. This model was fit to measurement data with a 95% \(\hbox {T}_2\) gas mixture at 30 K, as used in the first KATRIN neutrinomass analyses, as well as a \(\hbox {D}_2\) gas mixture of 96% purity used in KATRIN commissioning runs. The achieved precision on the energyloss function has abated the corresponding uncertainty of \(\sigma (m_\nu ^2)< {{10}^{2}}{\hbox {eV}^{2}}\) [1] in the KATRIN neutrinomass measurement to a subdominant level.
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1 Introduction
The KArlsruhe TRItium Neutrino (KATRIN) experiment aims to determine the effective electron antineutrino mass in a modelindependent way by examining the kinematics of tritium \(\upbeta \)decays. The observable \(m^2_\nu = \sum _{i}{ \left U_{\mathrm {e}i}\right ^2 m^2_{i}}\) is the squared incoherent sum of neutrinomass eigenstates \(m_{i}\) weighted by their contribution \(U_{\mathrm {e}i}\) to the electron antineutrino. The target sensitivity for the neutrinomass measurement in KATRIN is 0.2 eV/c\(^2\) (at 90% CL) with three liveyears of data [2]. The \(5\sigma \) discovery potential is 0.35 eV/c\(^2\). This requires a precise control of all systematic effects. The experiment is designed for a highprecision spectral shape measurement of \(\hbox {T}_2\) \(\upbeta \)decay electrons around the endpoint of 18.6 keV. An overview of the KATRIN experiment is shown in Fig. 1. The setup [3] includes a highactivity Windowless Gaseous Tritium Source (WGTS) and a highresolution electrostatic retarding spectrometer of the MACE (Magnetic Adiabatic Collimation with an Electrostatic filter) type [4,5,6]. Molecular tritium gas at 30 K is continuously injected through the capillaries at the center of the WGTS and pumped out at both ends. This allows a nominal steadystate column density (i.e. the integrated nominal source density \(\rho _0(z)\) along the length d of the source cryostat) \(\rho _0 d= 5 \times {10}^{17}\hbox {cm}^{2}\) resulting in an activity of \(1.7\times {10}^{11}\)Bq with a stability better than 0.1%\(\hbox {h}^{1}\) [3].
In order to prevent tritium from entering the spectrometer section which would induce background in the measurement, the transport section reduces the tritium flow by at least 14 orders of magnitude [7]. This is achieved with a differential pumping section [3, 8], which comprises turbomolecular pumps followed by a cryogenic pumping section that makes use of an argon frost layer to adsorb tritium cryogenically [3, 9]. The spectrometer section consists of the pre and the main spectrometer. The prespectrometer rejects lowenergy electrons, which reduces the electron flux into the main spectrometer. The final precision discrimination of the electron energy is performed in the analyzing plane at the center of the main spectrometer with a resolution of 2.77 eV [3] for 18.6keV electrons with isotropic angular distribution.
The pre and main spectrometer are MACE type highpass filters, which can only be traversed by electrons with longitudinal kinetic energy higher than the preset potential. The isotropically emitted \(\upbeta \)electrons are adiabatically collimated to a longitudinal motion inside the spectrometer. This is achieved by a gradual decrease of the magnetic field strength B from the entrance of the spectrometer towards its center, conserving the magnitude of the \(\upbeta \)electron’s magnetic moment in the cyclotron motion \(\mu =E_\perp /B\) [4], with \(E_\perp \) being the transverse component of the electron’s kinetic energy with respect to the magnetic field lines. Varying the electric potential of the spectrometer allows the energy region around the endpoint of the tritium \(\upbeta \)decay to be scanned as an integral spectrum, i.e. the rate of electrons with kinetic energy above the set filter potential [1].
Electrons passing the main spectrometer are reaccelerated by the main spectrometer potential and a postacceleration of 10 kV at the focalplane detector (FPD) system and are then counted by a 148pixel silicon PIN detector [10] shown at the far right in Fig. 1. An 18keVwide selection window (14 keV to 32 keV) around the 28keV electron energy peak is chosen to minimize systematic effects in counting efficiencies [1].
The observable \(m_\nu ^2\) is determined by fitting the recorded integral spectrum with a model that comprises four parameters: the normalization, the endpoint energy, the background rate, and \(m_\nu ^2\) [11]. The model is constructed from the shape of the \(\upbeta \)decay spectrum and the response of the experimental setup. The main components of the response are the transmission function of the main spectrometer and the energy loss of electrons from elastic and inelastic scatterings in the \(\hbox {T}_2\) source. The latter is the focus of this work.
At the nominal source density, approximately 60% of all electrons scatter inelastically and lose energies between \(\approx \,\)11 eV and 9.3 keV. The upper limit of this energy transfer arises due to the fact that the primary and secondary electrons from the ionization process are indistinguishable in the measurement and always the higher energetic electron is measured. Minuscule energy losses can result in electrons with energies close to the endpoint downgraded to lower energies in the spectrum fit window. Therefore, the energyloss function needs to be known with high precision in order to meet the systematic uncertainty budget of \(\sigma (m_\nu ^2)< {7.5} \times {{10}^{3}}{\hbox {eV}^{2}}\) [2] reserved for this individual systematic.
Theoretical differential cross sections for 18.6keV electrons scattering off molecular tritium are not available at the required precision for the \(m_\nu ^2\) measurements. While data from energyloss measurements for gaseous tritium or deuterium from the former neutrino mass experiments in Troitsk and Mainz [12, 13] exist, the precision is not sufficient to achieve the KATRIN design sensitivities. Other more precise experimental data on the energy losses of electrons with energies near the tritium \(\upbeta \)decay endpoint energy are only available for molecular hydrogen as the target gas [13,14,15]. In this paper we report the results of the insitu measurements of the energyloss function in the KATRIN experiment.
We used a monoenergetic and angularselective electron gun, of the type described in [16], mounted in the rear section (far left in Fig. 1), which allowed us to probe the response of the entire KATRIN setup, including the energy loss in tritium gas.
We begin this paper in Sect. 2 with a brief introduction to existing energyloss function models and continue with the description of the novel semiempirical parametrization developed in this work. In Sect. 3, the measurement approaches of the integral as well as the novel differential timeofflight measurements are explained, including a description of the working principle of the electron gun used for these measurements. The analysis of the tritium data using a combined fit is presented in Sect. 4 including a detailed discussion of the systematic uncertainties of the measurements. Additional measurement results for the energyloss function in deuterium gas are provided in Sect. 4.3. We conclude this paper in Sect. 5 by summarizing and discussing our results in the context of the neutrinomasssensitivity goal of KATRIN.
2 Energyloss function
Multiple processes contribute to the energy loss of electrons traversing molecular tritium gas. The median energy loss from elastic scattering amounts to \(\overline{\varDelta E}_{\mathrm {el}}={2.3}\,\hbox {meV}\) [11], which is negligible in the KATRIN measurement. The predominant processes for the KATRIN experiment are inelastic scatterings, resulting in electronic excitations in combination with rotational and vibrational excitations of the molecule, ionization, and molecular dissociation.
Data from detailed measurements is only available for the scattering of 25keV electrons on molecular hydrogen gas [14, 15]; these direct measurements of the energyloss function were made with energy resolutions down to 40 meV. In these measurements, the contribution of three different groups of lines can be discerned, which are created from the excitations of the \((\mathrm {2p}\sigma \ ^1\varSigma ^+_\mathrm {u})\), \((\mathrm {2p}\pi \ ^1\varPi _\mathrm {u})\), and \((\mathrm {3p}\pi \ ^1\varPi _\mathrm {u})\) molecular states around 12.6 eV and 15 eV, respectively.
Aseev et al. [12] and Abdurashitov et al. [13] report on the measurements of energy losses of electrons in gaseous molecular hydrogen, deuterium, and tritium. The shape of the energyloss function was evaluated by fitting an empirical model to the integral energy spectra obtained with a monoenergetic electron source which generated a beam of electrons with kinetic energies near the endpoint energy of the tritium \(\upbeta \)decay. Because of the low energy resolution of several eV, the shape of the energyloss function was coarsely approximated by a Gaussian to represent electronic excitations and dissociation, and a onesided Lorentzian to represent the continuum caused by ionization of the molecules [12].
2.1 New parametrization
The highquality data from the first KATRIN energyloss measurements described in Sect. 3 allows us to improve the parametrization used in Aseev et al. [12] and Abdurashitov et al. [13]. While the experimental energy resolution is not sufficient to resolve individual molecular states, the combined contribution of each of the three groups of states can clearly be discerned in the KATRIN data.
A new parametrization of the energyloss function was developed to describe the inelastic scattering region between about 11 eV and 15 eV using three Gaussians, each of which is approximating one group of molecular states. The ionization continuum beyond this energy region is described by the relativistic binaryencounterdipole (BED) model developed by Kim et al. [17]. While the parameters required by this model are only available for the ionization of \(\hbox {H}_2\)molecules [18], by taking into account the ionization thresholds for the different isotopologs [19]
the shape of the BED model is a good representation for the tritium data, as can be seen from the fit result in Sect. 4.1. The new parametrization of the full energyloss function is written as:
where \(\varDelta E\) is the energy loss and \(a_j\), \(m_j\), and \(\sigma _j\) are the amplitude, the mean, and the width of the three Gaussians, respectively. \(f_{\mathrm{BED}}(\varDelta E)\) is the functional form of the BED model as given in [17] and \(E_\mathrm {i}\) is the junction point between the two regions given by the ionization threshold. For a smooth continuation of the model at the junction, the BED function \(f_{\mathrm{BED}}(\varDelta E)\) is normalized to the local value \(f(E_\mathrm {i})\) of the Gaussian components at that position.
3 Measurements
The energyloss function \(f(\varDelta E)\) Eq. (2) describes the electron energy losses \(\varDelta E\) from scattering inside the source, which distort the shape of the response function. By measuring the response function, it is possible to determine \(f(\varDelta E)\). For this, a quasimonoenergetic and angularselective photoelectron source (“electron gun”), located at the end of the rear section (see Fig. 1), is used. Guiding the quasimonoenergetic beam – at a pitch angle of approx. \(\theta ={0}^{\circ }\) between the magnetic field lines and the electrons’ momentum vector – through the WGTS allows the investigation of the energy loss from scatterings with the source gas molecules stabilized at 30 K. Measuring the electron rate at the focalplane detector as a function of the electron surplus energy \(E_\mathrm {s}\) at the analyzing plane (see Eq. (3)) yields the response function of the setup.
The working principle of the electron gun and a general description of the measurement strategy are provided in the following. This is followed by a discussion of the measurement data taken in the two different measurement modes (integral and differential) as well as two important systematic effects in the measurements (pileup and background).
Electron gun A schematic drawing of the electron gun is provided in Fig. 2. The electrons are generated by photoelectric emission when ultraviolet light is shone through an approximately 30nmthick gold photocathode, which is installed inside two electrically charged parallel plates. The photoelectrons are accelerated by a potential difference of 4 kV between the plates separated by 10 mm; the electrons exit the setup through a hole in the front plate (see Fig. 2). This first nonadiabatic acceleration collimates the beam of photoelectrons in a cosine distribution [20] initially. By tilting the plates by the angle \(\alpha \), welldefined pitch angles \(\theta \) can be obtained. A pitch angle of \(\theta ={0}{^{\circ }}\), which is reached by aligning the plates with the magnetic field lines, is used in the measurements. The generated electrons are further accelerated by a cascade of cylinder electrodes to the desired kinetic energy. The working principle is explained in more detail in [16]. The energy profile of the generated beam depends on the work function \(\varPhi \) of the photocathode and the wavelength \(\lambda \) of the light source.
For the measurements in this work, a 266 nm pulsed UV laser^{Footnote 1} with pulse widths of less than 18 ns (FWHM) is used. The Qswitch of the laser can be externally triggered, which allows the synchronization of the creation time of the electron pulses with the detector system. This allows the timeofflight (TOF) of the signal electrons to be measured. The TOF is used for a differential analysis of the data (see Sect. 3.2).
The photon energy of the monochromatic laser light (\({h\nu ={4.66}\,\hbox {eV}}\)) is only 0.22 eV above the work function \(\varPhi ={4.44}\,\hbox {eV}\) [21] of the gold photocathode, which results in a measured energy spread of \(\sigma _\mathrm {E}< {90}\,\hbox {meV}\).
To generate electrons with well defined kinetic energies close to the tritium endpoint, voltages down to \(\)21 kV can be applied to the photocathode and cylinder electrodes. The photocathode potential \(U_{\mathrm {ph}}\) is varied to produce electrons with different surplus energies \(E_\mathrm {s}\) with respect to the negative main spectrometer retarding potential \(U_0\):
taking into account the additional initial energy of the electrons given by the difference of the photon energy \(h\nu \) and the work function \(\varPhi _i\) of the electrons populating different energy levels in the solid (neglecting further solidstate effects). The total initial kinetic energy of the electrons is given as \(E_{\mathrm {kin}}=q\cdot U_{\mathrm {ph}}\).
Measurement approach To resolve the fine structures of the response function, small voltage steps on the order of 0.1 eV are required over the analysis interval of \(E_\mathrm {s}={5\,\hbox {eV}~\hbox {to}~60}\) eV. Multiple fast voltage sweeps in alternating directions are preferred to compensate for systematic uncertainties associated with the scan direction and longterm instabilities of the setup. A single highvoltage setpoint adjustment of the main spectrometer requires more than 10 s to stabilize, which does not allow repeated measurements within a reasonable time. For faster measurements, the surplus energy of the electron beam is modified by performing voltage sweeps of \(U_{\mathrm {ph}}\) while the filter potential of the main spectrometer is kept fixed at \(U_0={18575}\,\hbox {V}\). The electron energy is chosen to be slightly above the tritium endpoint energy to avoid \(\upbeta \)electron backgrounds but close to the region of interest to minimize effects from the energy dependence of the scattering cross section.
Changing the kinetic energy of the electrons results in a small change of the total inelastic scattering cross section \(\sigma ^\mathrm {tot}_\mathrm {inel}\) of up to 0.27% over the scanned energy range. This is considered later in the data analysis.
Each sweep (called “scan” in the following) took 30 min and was repeated in alternating scanning directions for approximately 12 h. The obtained rates as a function of the continuous voltage ramp were binned to obtain discrete energy values for the analysis. The data taking was performed in integral and differential modes, which are described in more detail in the following.
3.1 Integral measurements
In the standard KATRIN measurement mode, only electrons with high enough surplus energies to overcome the main spectrometer retarding potential reach the detector. By changing the kinetic energy of the electrons and keeping the retarding potential at a fixed value, the integral response function was measured. A set of integral measurements at three different nonzero column densities as well as one reference measurement at zero column density (see Table 1) were performed. The pulse frequency of the laser was set to 100 kHz, which results in an estimated mean value of 0.05 generated electrons per light pulse.
The individual scans were both corrected for rate intensity fluctuations and detector pileup (see Sect. 3.3). The former are caused by fluctuations of the laser intensity, which is stable to 1.2%\(\hbox {h}^{1}\). The light intensity is continuously monitored by a photodiode connected to a fiber splitter, which is installed just before the light is coupled into the vacuum system of the electron gun (see Fig. 2). The light intensity correction is done by dividing the measured FPD rate by the relative deviation of the light intensity to its mean intensity. The precision of the measured light intensity with this monitoring system is 0.4% and is propagated into the uncertainties of the correction. Data from scans at the same column density are accumulated (Fig. 3). The resulting integral response functions are superpositions of nfold scattering functions, as indicated in the figure by arrows above the measurement data.
3.2 Differential (timeofflight) measurements
The time of each trigger pulse for the laser is saved in the detector data stream and used to define the electronemission time at the electron gun. For each event at the detector, its time difference to the laser pulse is calculated. The time difference corresponds to the timeofflight (TOF) of the electron through the KATRIN beamline from the electron gun to the detector, including delays for the signal propagation and processing on the order of \({1}\,\upmu {\hbox {s}}\). The knowledge of the electron’s timeofflight can be used as additional information on its kinetic energy.
The negative retarding potential in the main spectrometer \(U_0\) acts as a barrier for the electrons, slowing them down and only allowing electrons with surplus energies \(E_\mathrm {s}> 0\) to pass through (highpass filter). The higher the electrons’ surplus energy, the less they are slowed down inside the main spectrometer; connecting their flight time through the main spectrometer \(\tau \) to their surplus energy by \(\tau \sim \frac{1}{\sqrt{E_\mathrm {s}}}\) [22].
Selecting only electrons with \(\tau > \tau _{\mathrm {cut}}\) is equivalent to a lowpass filter on \(E_{\mathrm {s}}\) [23]. Applying this TOF selection, the highpass filter main spectrometer is transformed into a narrow bandpass filter for measuring the differential energy spectrum.
For the differential measurements, the laser was pulsed at 20 kHz to be able to distinguish flight times up to 50 \(\upmu \)s between the pulses (see Fig. 4). In this mode, an estimated 0.35 electron per pulse are emitted. Measurements at four different column densities were performed, which are listed in Table 1. Figure 4 shows the measurements at 86% nominal column density \(\rho _0 d\) as an example. The top panel shows the timeofflight versus surplus energy. Here the unscattered electrons as well as onefold and twofold scattered electrons are prominently visible as hyperbolic structures.
A TOF selection of events with flight times longer than \(\tau _{\mathrm {cut}} = {35}\,{\upmu \hbox {s}}\) is applied to obtain a differential spectrum, which is projected on \(E_\mathrm {s}\) and shown in the bottom panel. \(\tau _{\mathrm {cut}}\) is chosen such that an energy resolution of \(\approx \,\)0.02 eV is achieved. Higher \(\tau _{\mathrm {cut}}\) allows for a higher energy resolution but results in significantly lower statistics. The vertical features – at 0 eV, 12.5 eV, and 25 eV – for \({\tau < {25}\,{\upmu {\hbox {s}}}}\) are electrons with flight times \(>\,\) 50 \({\upmu \hbox {s}}\) from a previous laser pulse. These events are neglected in the analysis.
All events with \(\tau \) in the range of 35–50 \({\upmu \hbox {s}}\) are selected and corrected for laser intensity fluctuations analogous to the integral analysis. The energy scale for each measurement is constructed using the measured ramping speed of the high voltage and the position of the peak of unscattered electrons set to \(E_\mathrm {s} = 0\).
3.3 Pileup correction
The focalplane detector is optimized to count singleelectron events with an energy resolution of \(\varDelta E_\mathrm {FPD}\approx {2}\,\hbox {keV}\). Due to the high electron rate of the electron gun (\(\approx \, {10}^4\)cps) and the use of a single detector pixel, pileup effects become relevant. Furthermore, the pulsed electron beam with \(<\,{18}\,\hbox {ns}\) FWHM windows creates a nonPoisson time distribution compared to a constant wave light source.
The electron flight time depends on the retarding potential and the energy loss from scatterings inside the WGTS. The time difference of the electrons from the same pulse arriving at the detector is thus modified as a function of the surplus energy. For arrivaltime differences shorter than the shaping time (\(L={1.6}\) \(\upmu \)s) of the trapezoidal filter used for pulse shaping of the detector signal, the electrons are counted as one single event with correspondingly higher event energy \(E_\mathrm {FPD}\) (Fig. 5). The number of electrons within the same detector event is denoted as event multiplicity \({\mathcal {M}}\). As the peaks for different multiplicity \({\mathcal {M}}\) events overlap in the \(E_\mathrm {FPD}\) histogram, a simple estimation of \({\mathcal {M}}\) based on \(E_\mathrm {FPD}\) is not possible. Processing the event signal with two additional stages of trapezoidal filters allows more information on the signal shape, such as the bipolar width \({\mathcal {W}}\) (i.e. the time difference of two consecutive zero crossings of the third trapezoidalfilter output), to be obtained [3]. Electrons with arrivaltime differences close to the shaping time distort the trapezoidal output of the first filter stage and thus change the determined bipolar width as a function of the arrivaltime difference. With the additional information on the pulse shape, these ambiguities can be resolved and \({\mathcal {M}}\) can be estimated. The multiplicity estimate \(\hat{{\mathcal {M}}}(E_\mathrm {FPD}, {\mathcal {W}})\) is obtained from Monte Carlo simulations of the detector response for random combinations of \({\mathcal {M}}\) electrons arriving within the shaping time L. The estimate \(\hat{{\mathcal {M}}}(E_\mathrm {FPD}, {\mathcal {W}})\) is not necessarily identical to \({\mathcal {M}}\) as there are still remaining ambiguities, which are considered in the uncertainty propagation (see Sect. 4.2). In the case of the integral measurement data, the correction is made by weighting each event with the estimator value. For the differential measurements, no pileup correction is required, but a \(\hat{{\mathcal {M}}}(E_\mathrm {FPD}, {\mathcal {W}})>1\) cut is applied for background suppression (see Sect. 3.4).
A comparison of the integral response function before and after pileup correction is provided in Fig. 6 to demonstrate its dependence on the surplus energy at two different values of \(\rho d\). The dependence of \(\hat{{\mathcal {M}}}(E_\mathrm {FPD}, {\mathcal {W}})\) on the kinetic energy of the electrons over the measurement range of 60 eV is neglected in the correction and an average estimate is used instead. The uncertainty due to the correction method was evaluated with a full simulation of the detector response for each of the response functions measured in integral mode. This yields a correction stability at \(5\times 10^{4}\), which is considered as a systematic uncertainty for the energyloss function determination.
3.4 Backgrounds
As the rear section is directly connected to the WGTS, tritium migration upstream towards the electron gun cannot be completely prevented. Tritium can decay within the acceleration fields of the electron gun. Ions created from the \(\upbeta \)decays are accelerated towards the photocathode, where their impact can generate multiple secondary electrons simultaneously. Those electrons are accelerated to the same energy as the signal photoelectrons. The kinetic energy of the background electrons changes along with the change of the photocathode voltage \(U_\mathrm {ph}\) in a scan. This results in a background spectrum following the shape of an integral response function, as it is shown in Fig. 7. The background electrons only differ in their initial energy distribution and the emission multiplicity (i.e. the number of electrons generated from an ion impact). The mean energy \(m_\mathrm {Bg}\) and the Gaussian width \(w_\mathrm {Bg}\) of the initial energy distribution of the secondary electrons can be obtained by performing a combined fit to the three background measurements using the same integral responsefunction model as described in Sect. 4. The initial energy distribution dominates the spectral shape of the transmission function \(T(E_\mathrm {s})\), which describes the transmission probability of the electrons inside the main spectrometer as a function of the surplus energy \(E_\mathrm {s}\). The transmission function can be approximated with an error function using \(m_\mathrm {Bg}\) and \(w_\mathrm {Bg}\) as free parameters. The nine energyloss function parameters were fixed to preliminary evaluated values during the fit. The bestfit result yields
The electron multiplicity distribution of the ioninduced events follows a Poisson distribution (including ioninduced events with no electrons being emitted) with the mean value
Background events cause a larger detector pileup effect compared to the signal electrons generated by the pulsed laser, especially in the differential data. The remaining events after the TOF selection are nearly unaffected by detector pileup, since only the scattered electrons survive. As the arrival time of scattered electrons is delayed compared to other unscattered electrons from the same light pulse, they do not arrive at the detector in time coincidence with other electrons. This allows the multiplicity estimator \(\hat{{\mathcal {M}}}(E_\mathrm {FPD}, {\mathcal {W}})\) to discriminate background events from signal electrons. By excluding events with \(\hat{{\mathcal {M}}}>1\) in the analysis, the background component can be reduced by about a factor of two without any significant distortion of the signal component. A comparison of the differential response function (at 15% \(\rho _0 d\)) before and after applying the multiplicity cut is provided in Fig. 6, showing the reduction of the background component. However, the multiplicity \(\hat{{\mathcal {M}}}>1\) cut causes a distortion of the shape of the background component, which is determined from simulations. The resulting response functions of the background component after an event multiplicity cut are displayed in Fig. 8. The four simulated spectra of the background components for the individual column densities are included in the fit model.
4 Analysis
The energyloss parameters in Eq. (2) are extracted with a \(\chi ^2\)fit to multiple datasets in integral and differential mode at different column densities. The systematic uncertainties in the energyloss function (for example, those due to the measurement conditions, pileup and background effects) are determined with Monte Carlo simulations (cf. Sect. 4.2). Results are given for molecular tritium and deuterium source gases below.
4.1 Combined fit of the datasets
The fit model is constructed with the energyloss function, effects of multiple scatterings in the source, energy smearing in the experimental setup, and the described background component. The energyloss function describes a single electron scattering. The probability for nfold scattering follows a Poisson distribution and is given by
with the expected mean number of scatterings \(\mu \) given by
\(\rho d\) is the column density during the individual measurements and \(\sigma ^\mathrm {tot}_\mathrm {inel}\) is the total inelastic scattering cross section. To correct for the inelastic scattering cross section at different kinetic energies, the parameter \(\mu \) is scaled by the ratio \({\sigma ^\mathrm {tot}_\mathrm {inel}(E_\mathrm {kin})/\sigma ^\mathrm {tot}_\mathrm {inel}(qU_0)}\), which gives \(P_n(\mu , E_\mathrm {s})\). The effects of elastic scattering off tritium can be neglected since the amount of energy transferred in these scattering processes (\({\overline{\varDelta E}_{\mathrm {el}}={2.3}\,\hbox {meV}}\) [11]) is negligible compared to the energy smearing caused, among others, by the width of the kinetic energy distribution of the electrons produced with the electron gun or the finite energy resolution of the KATRIN main spectrometer. The experimental response to electrons that have been scattered n times in the source gas is given by the nfold convolution of the energyloss function \(f(\varDelta E)\) with itself and convolved one time with the experimental transmission function \(T(E_\mathrm {s})\), leading to the following definition of the corresponding scattering functions \(\epsilon _n(E_\mathrm {s})\)
with \(E_\mathrm {s}\) being the surplus energy of the electrons (see Eq. (3)) and \(\varDelta E\) being the energy loss resulting from an inelastic scattering. The shape of the ionization tail of the energyloss function is corrected for the shape distortion (\(<\, 10^{2}\)%) caused by the change of the kinetic energy.
The model \(R(E_\mathrm {s},\mu )\), which is fit to data, is the sum of the scattering functions \(\epsilon _n(E_\mathrm {s})\) weighted by the corresponding Poissonian probabilities
Given that the surplus energies considered in the energyloss analysis are limited to \({E_\mathrm {s} \le {56}\,\hbox {eV}}\), the highest scattering order that needs to be considered is \(n=4\).
In the integral measurement, the shape of the experimental transmission function \(T_{\mathrm{int}}(E_\mathrm {s})\) is obtained from the response function with an empty source volume; Eq. (9) collapses to \(R(E_\mathrm {s}, 0)= T(E_\mathrm {s})\). \(T(E_\mathrm {s})\) is modeled with an error function. Similarly, the transmission function for the differential data \(T_{\mathrm{dif}}(E_\mathrm {s})\) could be obtained from a TOF measurement with an empty source. However, it is simply given by the shape of the peak of unscattered electrons observed at nonzero column densities; no additional measurement is required in this case. Thus, we directly use the measurement data to construct the fit model. Figure 9 shows the scattering functions constructed for the differential (\(\epsilon _{n}^{\mathrm{dif}} (E_\mathrm {s})\)) and the integral (\(\epsilon _{n}^{\mathrm{int}} (E_\mathrm {s})\)) measurement modes for the first four scattering orders.
In addition to the nine parameters in the energyloss model in Eq. (2) (amplitude, mean and width of the three Gaussians contained in the model), several nuisance parameters are included in the combined fit to differential and integral datasets taken at different column densities. These nuisance parameters include normalization factors \(c^{\mathrm{{dif(int)}}}_{i}\), mean scattering probabilities \(\mu ^{\mathrm{dif(int)}}_{ i}\), and background amplitudes \(b^{\mathrm{dif(int)}}_{i}\) for each differential (integral) dataset that is added to the fit. In the fit, we minimize the following \(\chi ^2\) function for the vector of free fit parameters \(\mathbf {{\mathcal {P}}}\)
where \(N_{\mathrm{dif(int)}}\) are the number of differential (integral) datasets considered. \(y^{\mathrm{dif(int)}}\) and \(dy^{\mathrm{dif(int)}}\) represent the individual data points and their uncertainties. The index of summation j denotes the data points of the individual datasets. The first summand of Eq. (10) describes the contribution of the differential datasets to the \(\chi ^2\) value. The fit range for the differential datasets extends from 10 eV to 56 eV, excluding the zeroscatter peak and the adjacent background region, which do not contain information on the energyloss function. The second summand describes the contribution of integral datasets with the fit range of − 1 to 56 eV.^{Footnote 2} The ioninduced background component (see Sect. 3.4) is considered in both summands. For the integral measurements, the shape of the background component \(B^{\mathrm{int}}(E_{\mathrm {s}, j},\mu ^{\mathrm{int}}_{ i})\) is described by an integral response function (see Fig. 7), but with a different initial energy distribution than the signal electrons. For the differential measurement, \(B^{\mathrm{dif}}(E_{\mathrm {s}, j},\mu ^{\mathrm{dif}}_{ i})\) is more complex and is obtained from simulations described in Sect. 3.4 and depicted in Fig. 8. The third summand is a pull term that ensures a proper normalization of the fitted energyloss function up to \(E_\mathrm {max}=(E_\mathrm{kin}E_\mathrm {i})/2\) with a desired precision of \(\delta = 10^{4}\).
With the definition of the \(\chi ^2\) given in Eq. (10), a combined fit to four differential datasets and three integral datasets taken at different column densities (see Table 1) was performed. The results are displayed in Fig. 10 for each of the differential and integral datasets included in the fit.
The corresponding bestfit parameters of the energyloss function are given in Table 2. The fit has a reduced \(\chi ^2\) value of 1.13(2). A deviation from \(\chi ^2/N_\mathrm {dof}=1\) can arise from an imperfect semiempirical parametrization of the energyloss function or an underestimation of uncertainties. We do not observe significant structures in the fit residuals in Fig. 10 and thus inflate the uncertainties of the data points by \(\sqrt{\chi ^2/N_\mathrm {dof}}\) to achieve a \(\chi ^2/N_\mathrm {dof}=1\) [24]. The statistical uncertainties from the fit are included in the third column of Table 2 with the covariance matrix shown in Table 5 in the Appendix. Compared to the empirical energyloss models of Aseev et al. and Abdurashitov et al. superimposed on our results in Fig. 11, the KATRIN result provides a better energy resolution and reduced uncertainties. As a consistency check, we extrapolate the energyloss function (fitted up to 56 eV) to \({E_\mathrm {max} = {9.280}\,\hbox {keV}}\) yielding a mean energy loss of \(\overline{\varDelta E}(\mathrm {T}_2)=30.79(1)_\mathrm {fit}\) eV, which agrees well with the value of 29.9(10) eV reported by Aseev et al. [12].
4.2 Systematic uncertainties
Systematic uncertainties are not included in the combined fit; they are determined separately by a Monte Carlo (MC) simulation framework. The framework generates many MC samples, each composed of a detailed simulation of all integral and differential datasets. The systematic effects under investigations can be folded into these MC sets individually, or combined, with or without statistical fluctuation of the count rates included. The underlying response function, on which the MC generation is based, is taken from the bestfit values given in Table 2.
The considered systematic uncertainties cover known effects that arise from the measurement conditions and effects specific to the integral or differential analysis. All systematic effects are shown in Table 3. Their implementation in MC generation is described in the following.

Transmissionfunction model In order to obtain an analytical description of the integral transmission function \(T(E_\mathrm {s})\) for the construction of the integral responsefunction model, an error function is fit to a reference measurement with an empty WGTS. The error function models the electron’s surplus energy threshold needed for transmission in the main spectrometer \(m_\mathrm {E}={0.2(29)}\,\hbox {meV}\) and the energy spread
\(w_\mathrm {E}=\hbox {90(2)}\,{\hbox {meV}}\) due to the angular and energy distribution of the electron gun and the energy resolution of the main spectrometer. To investigate the uncertainty of this analytical model, MC samples of the measurements at different column densities were generated with \(m_\mathrm {E}\) and \(w_\mathrm {E}\) drawn from a multivariate normal distribution according to the bestfit values above with the correlation between them taken into account. No uncertainty on the transmissionfunction model was considered for the differential data, since the peak of the unscattered electrons from the measurement data is directly used as the transmission function.

Columndensity drift As the scattering probability \(P_n\) depends on the column density, drifts in the column density during the measurements can cause a distortion of the response function. During the measurements at 41% of the nominal column density, drifts on the order of 0.2%h\(^{1}\) were visible. The reduced stability was caused by CO and tritiated methane freezing inside the injection capillaries. The CO and the methane were generated from radiochemical reactions with the stainlesssteel surface during the burnin period of the first tritium operation [1]. The column density is constantly monitored with a throughput sensor, which allows the drift to be modeled precisely in the simulations. To do so, a linear function \(\rho (t)\) is fit to the sensor data, yielding the slope of the drift and the corresponding parameter uncertainty. This linear function is used to model the rate drift due to the column density drift with the slope sampled according to its uncertainty.

Rate drift The electronproduction rate of the electron gun can drift due to changes in the work function or a possible degradation of the photocathode (e.g. by ion impacts). The number of unscattered electrons is analyzed for each run after correcting for drifts in the light intensity and the column density to monitor for intrinsic longterm rate drifts. Although the rate drift is very small at \({\mathcal {O}}({0.1}\,{ \% \hbox {h}^{1}})\), the resulting drift is used to modulate the response functions accordingly.

Background A background component created from secondary electrons by ion impact on the photocathode (cf. Sect. 3.4) adds to the response functions. In the MC simulations, a background component is added with the parameters of the initial energy distribution of the background electrons (see Eq. (4)) sampled according to their uncertainties.

Multiplicity cut The event multiplicity \({\hat{{\mathcal {M}}}(E_\mathrm {FPD}, w)>1}\) cut distorts the background shape in the differential measurements (see Fig. 8) depending on the initial electron multiplicity \({{\hat{S}}}\) (see Eq. (5)) of the ion impact. In the MC simulations, the value of \({{\hat{S}}}\) is sampled according to its Gaussian uncertainty determined from the measurement and the resulting background component is added to the differential data. Distortions on the signal component from the photoelectrons due to the multiplicity cut were investigated by dedicated detector simulations and added to the differential response functions.

Pileup correction Detector pileup is a dominant systematic effect for the integral measurements and is corrected with the pileup reconstruction method described in Sect. 3.3. The efficiency \(\zeta (E_{s})\) of this pileup correction method is determined with detector simulations for each data point. The simulated response functions are multiplied by \(\zeta (E_{s})\) to include the remaining distortions after applying the pileup correction. The efficiency \(\zeta (E_{s})\) is varied according to the Gaussian uncertainty determined in detector simulations.

Binning The response functions are measured by continuously ramping the emission energy of the electron gun. For the data analysis, the continuous data stream is binned into 50meV bins. This binning effect is included in the MC simulations.
A total of 10,000 MC datasets are generated from the distributions of the systematic effects. Every MC dataset is fit and the bestfit values are taken to construct the probability distribution for each of the nine parameters of interest. From these distributions, the parameter uncertainties are determined from the standard deviations. In addition, systematic parameter shifts are determined from the difference between the median of the distribution and the initial input value from the underlying energyloss function. The results of this evaluation are shown in Fig. 12. The total uncertainty is dominated by the statistics in the data and the widths of the distributions agree well with the parameter uncertainties of the bestfit result provided in Table 2. In order to condense the information of the nine parameter uncertainties for easier interpretation, two metrics are defined. They are shown in the last two columns of Table 3. The first metric, \(\int \sigma _\mathrm {sys}/\int \sigma _\mathrm {all}\), is the area of the error band in the energyloss function caused by the specific systematic (\(\int \sigma _\mathrm {sys}\)) with respect to the area of the error band caused by all systematic effects (\(\int \sigma _\mathrm {all}\)). The error bands originate from the combination of all nine parameter uncertainties. The areas of the error bands are estimates for the uncertainty of the scattering probability over the whole energy range. The second metric, \(\int f_0  f_\mathrm {sys}/\int \sigma _\mathrm {all}\), is the area of the difference between the nominal energyloss function (\(f_0\)) and the energyloss function (\(f_\mathrm {sys}\)) obtained from the simulations including the individual systematic uncertainties. This difference is normalized to \(\int \sigma _\mathrm {all}\). A difference can be created by shifts of the nine parameter values caused by a given systematic effect. The impact of parameter shifts on the functional form of the energy loss is found to be smaller than the impact of the parameter uncertainties. The dominant contribution to the systematic uncertainty originates from the transmissionfunction model. Since the total uncertainty of the energyloss function is dominated by statistical uncertainties in the data and no significant parameter shifts are found, the considered systematic effects are negligible and not further considered in this study.
4.3 Deuterium results
Measurements, similar to the ones described in Sect. 3, were performed with molecular deuterium as source gas in an early commissioning run of the KATRIN experiment. Four integral measurements at 0%, 5%, 35%, and 87% of the nominal source density and a single differential measurement at 5% were made. The data were processed and fit in the same manner as described in Sects. 3 and 4. For the combined \(\chi ^2\)fit of the deuterium measurements, the bestfit result is obtained at a reduced \(\chi ^2={1.57(2)}\). Similar to the tritium data, the uncertainties of the data points are rescaled by \(\sqrt{\chi ^2/N_\mathrm {dof}}\) to obtain a reduced \(\chi ^2=1\). The parameter values as well as the covariance matrix are provided in Tables 4 and 6. The slightly increased \(\chi ^2\) value and the larger model uncertainties (cf. Fig. 11) can be explained by the presence of a stronger detector pileup in the integral data due to an electron rate that was twice as high as that of the tritium measurements combined with the availability of only one differential dataset. A full propagation of the systematic uncertainties was not performed for the deuterium measurements as the simulations for tritium showed that the measurements are strongly dominated by the statistical uncertainty. Furthermore, neither the systematic uncertainty due to methane freezing causing columndensity drift nor the background generated from tritium ions is present in the absence of tritium.
Figure 11 shows the minor differences of the energyloss models for deuterium and tritium, as the electronic excitation states are shifted to lower energies on the order of 100 \({\hbox {meV}}\).^{Footnote 4} Extrapolating again to \(E_\mathrm {max}\) the energy loss function results in a mean energy loss of \(\overline{\varDelta E}(\mathrm {D}_2)=30.64(1)_\mathrm {fit}\) eV, for the dominant deuterium isotopologs. This mean energyloss value is 0.15 eV smaller than for tritium isotopologs, but we should not forget that we extrapolate the energyloss function in energy by a factor 200 and we do not account for systematic uncertainties here for this consistency check.^{Footnote 5}
5 Summary and outlook
A series of precision measurements of the energyloss function of 18.6keV electrons scattering off molecular tritium and deuterium gas was performed. The measurements were carried out in the KATRIN setup by using a pulsed beam of monoenergetic and angular selected electrons from a photoelectron source. The measurements were made in integral and differential timeofflight measurement modes.
A new semiempirical parametrization of the energyloss function was developed, which describes the set of electronic states in combination with molecular excitations, dissociation, and ionization better than previous models. This new model is described by nine parameters, which were determined by performing a combined \(\chi ^2\)fit to both integral and differential measurement data. The measurements and analyses performed in this work achieved a significant improvement over existing empirical energyloss models in terms of energy resolution and uncertainties. A detailed investigation of the systematic effects shows that the parameter uncertainties are dominated by statistical uncertainties. This allows further improvement in precision in future measurements.
The obtained electron energyloss function in tritium was used in the analysis of the first KATRIN dataset, which led to an improved upper limit of the effective neutrino mass \({m_\nu <{1.1}\,\hbox {eV}}\) (90% CL) [26]. For this dataset, recorded at reduced source strength, the uncertainty of the energyloss model contributes to the systematic uncertainty of the observable \(m_\nu ^2\) with \(\sigma (m_\nu ^2)<{10^{2}}{\hbox {eV}^{2}}\) and is inconsequential compared to other effects [1]. The achieved precision of the energyloss function is close to the target effect of \({\sigma (m_\nu ^2)< {7.5 \times {10}^{3}}{\hbox {eV}^2}}\) [2] that is necessary for reaching the final KATRIN sensitivity of \({m_\nu ={0.2}\,\hbox {eV}}\) (90% CL).
Availability of data and material
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.
Notes
InnoLas Mosquitoo Nd:\(\hbox {YVO}_4\) 1064 nm (frequency quadrupled).
This extended fit range is required to determine the amplitude of the background component, which is only accessible below the transmission edge at \(E_\mathrm {s}={0}\,\hbox {eV}\).
Such a difference between the different hydrogen isotopologs is theoretically expected. The observed difference of \({\mathcal {O}}({100}\,\mathrm{meV})\) is in agreement with preliminary calculations in dipole approximation in which the peak positions of the rovibrationally resolved spectra for the 2p\(\sigma \) \(^1\varSigma _\mathrm {u}\) and the 2p\(\pi \) \(^1\varPi _\mathrm {u}\) states were compared for the isotopes \(\hbox {D}_2\) and \(\hbox {T}_2\) [25].
Just to get an order of magnitude estimate of the systematic uncertainties of the mean energy loss, we have left the junction point \(E_\mathrm {i}\) between the three Gaussians and the BED tail in Eq. (2) free in our fits, yielding already an additional systematic uncertainty on the mean energy loss as big as the discrepancy. We want to add that the systematics of this extrapolation is not critical for the determination of the energyloss function in our interval of interest \([{0}\,\hbox {eV}, {54}\,\hbox {eV}]\).
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Acknowledgements
We acknowledge the support of Helmholtz Association, Ministry for Education and Research BMBF (5A17PDA, 05A17PM3, 05A17PX3, 05A17VK2, and 05A17WO3), Helmholtz Alliance for Astroparticle Physics (HAP), Helmholtz Young Investigator Group (VHNG1055), and Deutsche Forschungsgemeinschaft DFG (Research Training Groups GRK 1694 and GRK 2149, and Graduate School GSC 1085  KSETA) in Germany; Ministry of Education, Youth and Sport (CANAMLM2015056, LTT19005) in the Czech Republic; Ministry of Science and Higher Education of the Russian Federation under contract 075152020778; and the United States Department of Energy through grants DEFG0297ER41020, DEFG0294ER40818, DESC0004036, DEFG0297ER41033, DEFG0297ER41041, DESC0011091 and DESC0019304, Federal Prime Agreement DEAC0205CH11231, and the National Energy Research Scientific Computing Center. Data Availability Statement This manuscript has no associated data or the data will not be deposited. [Authors’ comment: Further analyses of the measurement data are subject of a dissertation in preparation.]
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Appendix
See Table 5.
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Aker, M., Beglarian, A., Behrens, J. et al. Precision measurement of the electron energyloss function in tritium and deuterium gas for the KATRIN experiment. Eur. Phys. J. C 81, 579 (2021). https://doi.org/10.1140/epjc/s1005202109325z
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DOI: https://doi.org/10.1140/epjc/s1005202109325z