In the KATRIN experiment, the energy of the \(\upbeta \)-electrons is analyzed using the MAC-E filter technique as described in Sect. 3. For a specific electrostatic retardation potential U, the count rate of electrons at the detector can be calculated, given the probability of an electron with a starting energy E to traverse the whole apparatus and hit the detector. This probability is described by the so-called transmission function T(E, U). Additional modifications arise from energy loss and scattering in the source, and reflection of signal electrons propagating from their point of origin until detection. These effects are incorporated together with the transmission function into the response function R(E, U), which is vital for the neutrino mass analysis as it describes the propagation of signal electrons that contribute to the integrated \(\upbeta \)-spectrum.
For illustrative purposes, we first consider a source containing a given number of tritium nuclei (\(N_\mathrm {T}\)) that decay with an isotropic angular distribution.Footnote 4 The emitted electrons are guided by magnetic fields through the spectrometer. The detection rate at the detector for a given spectrometer potential U can be expressed as:
$$\begin{aligned} \dot{N}(U) = \frac{1}{2}\,N_\mathrm {T}\int \limits _{qU}^{E_0} \frac{\,\mathrm {d}\varGamma }{\,\mathrm {d}E}(E_0, m_\upnu ^2) \cdot R(E, U) \, \,\mathrm {d}E, \end{aligned}$$
(14)
where the factor of \(\frac{1}{2}\) incorporates the fact that the response function R(E, U) only considers electrons emitted in the forward direction.
In the following, an analytical description of the response function of the KATRIN experiment will be laid out. At first, we derive the transmission function of the MAC-E filter that is implemented by the main spectrometer (Sect. 4.1). In Sect. 4.2 we consider energy loss in the source and develop a first description of the response function. Inhomogeneities in the MAC-E filter (Sect. 4.3) and the source (Sect. 4.4) requires extension of the model by a segmentation of the source and spectrometer volume. Further modifications to the response function arise from considering the effective source column density which an individual \(\upbeta \)-electron traverses (Sect. 4.5), changes to the electron angular distribution (Sect. 4.6), thermal motion of the source gas (Sect. 4.7), and energy loss by cyclotron radiation (Sect. 4.8). After discussing these contributions, in Sect. 4.9 we arrive at a description of the integrated spectrum that is measured by the KATRIN experiment. We close the discussion with a general note on experimental energy uncertainties (Sect. 4.11) and give a quantitative overview of theoretical corrections and systematic effects (Sect. 4.12) on the neutrino mass analysis.
Transmission function of the MAC-E filter
The transmission of \(\upbeta \)-electrons through the MAC-E filter is an important characteristic of the measurement and a significant part of the response function. In the simplest case, one can assume that electrons enter the MAC-E filter with an isotropic angular distribution and propagate adiabatically towards the detector. In the discussion here we apply the adiabatic approximation (see Eq. (15) below), which is fulfilled in the case of KATRIN.
In general, an electron from the source will reach the detector if the momentum \(p_\parallel \) parallel to the magnetic field lines (or the corresponding fraction \(E_\parallel \) of the kinetic energy) is always positive. The transformation of transverse to parallel momentum and back in a slowly varying magnetic field B is governed by the following adiabatic invariant (which corresponds to the conserved orbital momentum \(\mu = E_\perp / B\) in the non-relativistic limit):
$$\begin{aligned} \frac{p^2_{\perp }}{B} = \text {const.} \end{aligned}$$
(15)
In the following discussion we use the general relation between the transverse momentum \(p_\perp \) of an electron with its transverse kinetic energy \(E_\perp \):
$$\begin{aligned} p_\perp ^2 = E_\perp \; (\gamma + 1) \cdot {m_\text {e}} \end{aligned}$$
(16)
with the relativistic gamma factor \(\gamma = \frac{E}{ {m_\text {e}} } + 1\), and thereby define the transverse kinetic energy as:
$$\begin{aligned} E_\perp = E \; \sin ^2 \theta . \end{aligned}$$
(17)
Similarly, we define the longitudinal kinetic energy as \(E_\parallel = E \; \cos ^2 \theta \). The polar angle \(\theta = \angle (\mathbf {p},\mathbf {B})\) of an electron momentum to the magnetic field is called the pitch angle.
We can now define the adiabatic transmission condition for an electron starting at the position \(z_\text {S}\) with a magnetic field \(B_\text {S} = B(z_\text {S})\), an electrostatic potential \(U_\text {S} = U(z_\text {S})\), a kinetic energy \(E = E(z_\text {S})\) with a corresponding gamma factor \(\gamma \), and a pitch angle \(\theta = \theta (z_\text {S})\). The transmission condition then reads for all longitudinal positions z:
$$\begin{aligned} 0&\le E_\parallel (z) \nonumber \\&= E + q U_\text {S} - E_\perp (z) - q U(z) \nonumber \\&= E + q U_\text {S} - E \; \sin ^2 \theta \cdot \frac{B(z)}{B_\text {S}} \; \frac{\gamma +1}{\gamma (z)+1} - qU(z), \end{aligned}$$
(18)
where \(\gamma (z)\) corresponds to the gamma factor at an arbitrary position z along the beam line where the electron has a kinetic energy \(E(z) = E_\parallel (z) + E_\perp (z)\) at a magnetic field B(z) and an electrostatic potential U(z).
Usually in a MAC-E filter the highest retarding potential U and at the same time the smallest magnetic field \(B_\text {A}\) is reached in the analyzing plane (located at \(z_\text {ap} = 0\) in our definition). Secondly we can assume the electrical potential \(U_\text {S}\) at the start to be zero and the relativistic factor in the analyzing plane at the largest retardation (minimum kinetic energy) to equal one, \(\gamma (z_\text {ap}) = 1\). Therefore the transmission condition in Eq. (18) simplifies to
$$\begin{aligned} 0 \le E - E \; \sin ^2 \theta \cdot \frac{B_\text {A}}{B_\text {S}} \; \frac{\gamma +1}{2} - qU. \end{aligned}$$
(19)
For a given electric potential and magnetic field configuration of the MAC-E filter, the transmission condition \({\mathcal {T}}\) is thus just governed by the starting energy E, the starting angle \(\theta \) and the retarding voltage U.
$$\begin{aligned} {\mathcal {T}}(E,\theta ,U) =\left\{ \begin{array}{ll} 1 &{}\quad \text {if}\quad \displaystyle E \, \left( 1 - \sin ^2 \theta \cdot \frac{B_\text {A}}{B_\text {S}} \cdot \frac{\gamma +1}{2} \right) \\ &{}\qquad \qquad - qU > 0\\ 0 &{}\quad \text {else} \end{array} \right. \, . \end{aligned}$$
(20)
For an isotropically emitting electron source with angular distribution \(\omega (\theta ) \, \,\mathrm {d}\theta = \sin \theta \, \,\mathrm {d}\theta \), we can integrate \({\mathcal {T}}(E,\theta ,U)\) over the angle \(\theta \) and define a response or transmission function. From here on we associate the remaining energy in the analyzing plane of the MAC-E filter – the surplus energy – with the expression \({\mathcal {E}}= E - qU\).
In the KATRIN setup the maximum magnetic field \(B_\text {max}\) is larger than \(B_\text {S}\), so that \(\upbeta \)-electrons emitted at large pitch angles in the source are reflected magnetically before reaching the detector. The magnetic reflection occurs at the pinch magnet (with \(B = B_\text {max}\) and zero potential), and in the source the electric potential is zero. The maximum pitch angle of the transmitted electrons is therefore independent of the electron energy and given by:
$$\begin{aligned} \theta _\text {max} = \text {arcsin}\left( \sqrt{\frac{B_\text {S}}{B_\text {max}}}\right) \; , \end{aligned}$$
(21)
For the standard operating parameters of KATRIN (see Table 2), \(\theta _\text {max}\) evaluates to about \(50.8^{\circ }\). This reflection is desired by design, since \(\upbeta \)-electrons emitted with larger pitch angles have to traverse a longer effective column of source gas and are therefore more likely to scatter and undergo energy loss, as detailed in the following sections.
With this additional magnetic reflection after the analyzing plane, the transmission function is given by:
$$\begin{aligned} T(E,U)&= \int \limits _{\theta =0}^{\theta _\text {max}} \; {\mathcal {T}}(E,\theta ,U) \cdot \sin \theta \, \,\mathrm {d}\theta \nonumber \\&= \left\{ \begin{array}{ll} 0 &{}\quad \;\;{\mathcal {E}}<0 \\ 1 - \sqrt{1-\frac{{\mathcal {E}}}{E} \frac{B_\text {S}}{B_\text {A}} \frac{2}{\gamma +1}} &{}\quad 0\le {\mathcal {E}}\le \varDelta E \\ 1 - \sqrt{1-\frac{B_\text {S}}{B_\text {max}}} &{}\quad \;\;{\mathcal {E}}>\varDelta E \end{array} \right. \, , \end{aligned}$$
(22)
with the filter width \(\varDelta E\) from Eq. (13). In Fig. 7, the transmission function is shown for the nominal KATRIN operating parameters and for the case \(B_\text {S} = B_\text {max}\). The magnetic reflection imposes an upper limit on the pitch angle, which reduces the effective width of the transmission function. As indicated in Fig. 7, this improves the filter width of the spectrometer to 0.93 eV, compared with 1.55 eV for \(\theta _\mathrm {max} = 90 ^{\circ }\) without magnetic reflection.
Response function and energy loss
In the next step we consider the energy loss when the electron traverses the gaseous source. The dominant energy loss process is the scattering of electrons on gas molecules within the source. Because the pressure decreases rapidly outside the source, scattering processes in the transport section or thereafter are of no concern.
Two ingredients are required to appropriately treat electron scattering in the source. First, the energy loss function \({\tilde{f}}(\epsilon , \delta \vartheta )\) describes the probability for a certain energy loss \(\epsilon \) and scattering angle \(\delta \vartheta \) of the \(\upbeta \)-electrons to occur in a scattering process. Because the scattering angles \(\delta \vartheta \) are small,Footnote 5 we will neglect them in the following formulae and describe the scattering energy losses by the function \(f(\epsilon )\). Here we do not consider a dependence of f or \(P_s\) on the incident kinetic energy E of the electrons, since for the KATRIN experiment the energy range of interest amounts to a very narrow interval of a few times 10 eV below the tritium endpoint only, where these functions can be considered as independent of E. The other important ingredients are the scattering probability functions \(P_s(\theta )\) for an electron with pitch angle \(\theta \) to scatter s times before leaving the source. These scattering probabilities depend on \(\theta \), since electrons with a larger pitch angle must traverse a longer path, meaning a larger effective column density, and are thus likely to scatter more often.
With these considerations, the response function no longer comprises only the transmission function, but is modified as follows:
$$\begin{aligned} R(E, U)&= \int \limits _{\epsilon = 0}^{E-qU} \; \int \limits _{\theta =0}^{\theta _\text {max}} \; {\mathcal {T}}(E-\epsilon ,\theta ,U) \cdot \sin \theta \nonumber \\&\qquad \cdot \biggl [ \; P_0(\theta ) \, \delta (\epsilon ) \, + \, P_1(\theta ) \; f(\epsilon ) \; \biggr . \nonumber \\&\qquad \biggl . \; + \, P_2(\theta ) \; (f \otimes f)(\epsilon ) \, + \, \cdots \; \biggr ] \, \,\mathrm {d}\theta \, \,\mathrm {d}\epsilon \end{aligned}$$
(23)
$$\begin{aligned}&= \int \limits _{\epsilon = 0}^{E-qU} \; \int \limits _{\theta =0}^{\theta _\text {max}} \; {\mathcal {T}}(E-\epsilon ,\theta ,U) \cdot \sin \theta \nonumber \\&\qquad \cdot \, \sum _{s} \; P_s(\theta ) \; f_s(\epsilon ) \, \,\mathrm {d}\theta \, \,\mathrm {d}\epsilon . \end{aligned}$$
(24)
Electrons leaving the source without scattering \((s = 0)\) do not lose any energy, hence \(f_0(\epsilon ) = \delta (\epsilon )\). For s-fold scattering, \(f_s(\epsilon )\) is obtained by convolving the energy loss function \(f(\epsilon )\) s times with itself.
The scattering cross section can be divided into an elastic and an inelastic component. The inelastic cross section and the energy loss function for electrons with kinetic energies of \(\approx 18.6\hbox { keV}\) scattering from tritium molecules have both been measured in [40, 41]. In this work, the inelastic scattering cross section was determined to be \(\sigma _\text {inel} = (3.40 \pm 0.07) \times 10^{-18}\,\hbox {cm}^2\) and an empirical model was fit to the energy loss spectrum.
The latter is parameterized by a low-energy Gaussian and a high-energy Lorentzian part:
$$\begin{aligned} f(\epsilon ) = \left\{ \begin{array}{ll} A_1 \cdot \, \exp \left( -2 \, \left( \dfrac{\epsilon - \epsilon _1}{\omega _1}\right) ^2 \right) &{}\;\;\epsilon < \epsilon _c \\ A_2 \cdot \, \dfrac{\omega _2^2}{\omega _2^2 + 4 (\epsilon - \epsilon _2)^2} &{}\;\;\epsilon \ge \epsilon _c \\ \end{array} \right. \, , \end{aligned}$$
(25)
with \(A_1 = (0.204 \pm 0.001)\,\text {eV}^{-1}\), \(A_2 = (0.0556 \pm 0.0003)\,\text {eV}^{-1}\), \(\omega _1 = (1.85 \pm 0.02)\,\text {eV}\), \(\omega _2 = (12.5 \pm 0.1)\,\text {eV}\), \(\epsilon _2 = (14.30 \pm 0.02)\,\text {eV}\) and a fixed \(\epsilon _1 = 12.6\,\text {eV}\). To obtain a continuous transition between the two parts of \(f(\epsilon )\), a value \(\epsilon _c = 14.09\,\text {eV}\) was chosen. The Gaussian part summarizes the energy loss due to (discrete) excitation processes, while the Lorentzian part describes the energy loss due to ionization of tritium molecules.
This parameterization of the energy loss function is used for the response model presented in this paper. However, the parameters are not precise enough for KATRIN to meet its physics goals. Dedicated electron gun measurements with the full experimental KATRIN setup have been planned for the determination of the inelastic scattering cross section and the energy loss function with higher precision; the analysis of these data will involve a sophisticated deconvolution technique [42].
At \(\sigma _\mathrm {el} = 0.29 \times 10^{-18}\,\mathrm {cm}^{2}\), the total cross section of elastic scattering of 18.6 keV electrons with molecular hydrogen isotopologues is smaller than that for inelastic scattering by an order of magnitude [43, 44]. In addition, the elastically scattered electrons are strongly forward peaked with a median scattering angle of \({{\overline{\theta }}}_\text {scat} = 2.1 ^{\circ }\) near the tritium endpoint energy. The energy loss due to elastic scattering is given by the relation
$$\begin{aligned} \varDelta E_\text {scat} = 2 \; \frac{ {m_\text {e}} }{M_\mathrm {T_2}} \; E \cdot \left( 1-\cos \theta _\text {scat} \right) . \end{aligned}$$
(26)
With an angular distribution for elastic scattering of molecular hydrogen by electron impact based on [45], the corresponding median energy loss amounts to \(\overline{\varDelta E} = 2.3\hbox { meV}\). The energy loss function, containing the elastic and inelastic components weighted by their individual cross section, is shown in Fig. 8.
The elastic energy loss component can be accurately calculated. Due to its narrow width and steep slope, \(\sim \hbox {meV}\) binning is required for incorporating it accurately in the response function, thereby increasing computational cost considerably. We will neglect the elastic scattering component in neutrino mass measurements as the associated systematic error on an \( {m_{\upnu }^2} \) is minute (\(\sim 5.10^{-5}\hbox {eV}^2\), see Table 1).
Radial inhomogeneity of the electromagnetic field
To calculate the transmission and response functions of the KATRIN setup as explained in Sects. 4.1 and 4.2, it is in principle sufficient to only consider the axial position of an electron to identify the initial conditions such as electromagnetic fields or scattering probabilities. In the case of the main spectrometer, radial dependencies must be incorporated in the description of the magnetic field and the electrostatic potential in the analyzing plane. Additional radial dependencies in the source are discussed in Sect. 4.4; these are then incorporated into the model together with the spectrometer effects.
In order to achieve a MAC-E filter width in the eV-regime, a reduction of the magnetic field strength in the analyzing plane on the order of \(\frac{B_\text {A}}{B_\text {max}} \approx \frac{\varDelta E}{E} \approx 10^{-4}\) is required (see Eq. (13)). Consequently the diameter of the flux-tube area A is drastically increased due to the conservation of magnetic flux \(\varPhi = \mathrm {const} \approx B \cdot A\). When nominal field settings are applied (see Table 2), the projection of the detector surface with radius \(r_\mathrm {det} = 4.5\hbox { cm}\) has a radius of about 4 m in the analyzing plane. A larger (smaller) magnetic field in the analyzing plane \(B_\mathrm {A}\) shifts the transmission edge to a larger (lower) energy, see Eq. (20). This effect is even more pronounced for larger electron pitch angles. Consequently, the transmission function (see Eq. (22)) is also widened or narrowed. Utilizing a set of magnetic field compensation coils, operated with an optimal current distribution, around the spectrometer vessel, the spread of the radial inhomogeneity of the magnetic field is minimized to a few \({\upmu \hbox { T}}\) when an optimized current distribution is applied [37, 38]. The resulting variation in the filter width in the analyzing plane due to the magnetic field inhomogeneity is thus reduced to about 10 meV [46].
Table 1 Impact of individual theoretical and experimental model corrections on the measured squared neutrino mass \( {m_{\upnu }^2} \), if neglected or approximated. The analysis energy window is restricted to \([E_0 - 30\hbox { eV}; E_0 + 5\hbox { eV}]\). For \( {m_{\upnu }} \) a true value of 200 meV is assumed
In the case of the electrostatic potential, unavoidable radial variation arises from the design of the spectrometer. To fulfill the transmission condition in Eq. (19), the electrode segments at the entrance and exit are operated on a more positive potential than in the central region close to the analyzing plane.Footnote 6 Depending on the final potential setting, the radial potential variation in the analyzing plane is expected to be of order 1 V [39]. In comparison, azimuthal variations are negligible. It is possible to considerably reduce the radial potential inhomogeneity by operating the MAC-E filter at larger \(B_\mathrm {A}\). However, this would require better knowledge of the magnetic field in the analyzing plane [46] and also increase the filter width.
Table 2 Key operational and derived parameters of KATRIN as defined in the technical design report [4]
Even with these optimizations of the setup, the small radial variations in the electromagnetic fields at the analyzing plane, as shown in Fig. 9, cannot be neglected. The segmentation of the KATRIN main detector into annuli of pixels allows us to incorporate such radial variations in the response function model for each individual detector pixel. Because the tritium source also features radial variations of certain parameters, this segmentation is combined with a full segmentation of the source volume as described in Sect. 4.4. Dependencies of the electromagnetic field are typically averaged over the surface area of a pixel. The specific detector geometry with thinner annuli towards outer radii (each with equal surface area) helps minimize the potential variation within individual annuli, despite the increasing steepness of the potential.
Source volume segmentation and effects
In addition to radial dependencies of the analyzing plane parameters that govern the energy analysis of the \(\upbeta \)-electrons (Sect. 4.3), the tritium source also features radial and axial dependencies of its parameters. In the following, we will briefly outline the most relevant source parameters that are required to accurately model the differential \(\upbeta \) spectrum and the response function. These parameters include the beam tube temperature \(T_\text {bt}\), the magnetic field strength \(B_\text {S}\), plasma potentials \(U_\text {P}\), the particle density \(\rho \) and the bulk velocity u of the gas, all of which may vary slightly in longitudinal, radial and azimuthal directions. The complex gas dynamic simulations, which are needed to calculate these local source parameters, are described in comprehensive detail in [47, 48].
In order to model accurately these effects for each individual detector pixel, the simulation source model is partitioned to match the detector geometry. It is partitioned longitudinally into \(N_L\) slices and segmented radially into \(N_R\) annuli (rings) of \(N_S\) segments each, resulting in a total of \(N_L \cdot N_R \cdot N_S\) segments (see Fig. 10). The geometry of these segments is chosen in such a way, that a longitudinal stack of segments is magnetically projectedFootnote 7 onto a corresponding detector pixel. Note that all detector pixels have identical surface area, which leads to broader annuli at the center and thinner annuli towards larger radii. In the following, we index the longitudinal slices by the subscript i and radial/azimuthal segments with their corresponding detector pixel by the subscript j.
At a retarding potential U, the detection rate for a specific detector pixel j can then be stated as
$$\begin{aligned} \dot{N}_j(U) = \frac{1}{2} \, \sum _{i=0}^{N_L-1} \; N_{\text {T},i} \, \int \limits _{qU}^{E_0} \; \frac{\,\mathrm {d}\varGamma }{\,\mathrm {d}E}(E_0, m_\upnu ^2) \; R_{i,j}(E, U) \, \,\mathrm {d}E, \end{aligned}$$
(27)
where \(N_{\text {T},i}\) is the number of tritium nuclei (assuming that the gas density has no radial or azimuthal dependence). The response function \(R_{i,j}(E, U)\) depends on the index i (i.e. the axial position) and the index j (i.e. the radial/azimuthal position) of the source segment. With the indices i, j we can describe the dependence on local source parameters such as the magnetic field. The most significant effect on the response is caused by the scattering probabilities, as detailed in Sect. 4.2. The index j further describes non-uniformities of the retarding potential U and the magnetic field \(B_\text {A}\) in the spectrometer (see Fig. 9).
Scattering probabilities
As discussed in Sect. 4.2, inelastic scattering results in an energy loss that directly affects the energy analysis of the signal electrons, and needs to be incorporated accurately into the analytical description. Changes to the angular distribution of the emitted electrons due to scattering processes, which also modify the response function, are discussed in Sect. 4.6.
The scattering probability for \(\upbeta \)-electrons is considerably different depending on their starting position in the 10 m long source beam tube, as visualized in Fig. 11. The longitudinal segmentation of the source volume in our model allows us to incorporate this behavior. The probability \(P_s\) for an electron to leave the source after scattering exactly s times depends on the total cross section \(\sigma \) and the effective column density \(\mathcal {N}_\text {eff}\) that the electron traverses. This effective column density depends not only on the electron’s starting position z inside the source and the axial density distribution \(\rho (z)\), but also on the starting pitch angle \(\theta \) in the source (Eq. (21)):
$$\begin{aligned} \mathcal {N}_\text {eff}(z,\theta ) = \frac{1}{\cos (\theta )} \cdot \int \limits _{z}^{L/2} \; \rho (z') \, \,\mathrm {d}z'. \end{aligned}$$
(28)
L denotes the length of the source beam tube with \(-L/2 \le z \le L/2\). The nominal column density is then given by \(\mathcal {N}= \mathcal {N}_\text {eff}(z=-L/2,\ \theta =0)\).
Because of the low probability to scatter off a single tritium molecule, the number of scatterings during propagation can be calculated according to a Poisson distribution:
$$\begin{aligned} P_s(z, \theta ) = \frac{(\, \mathcal {N}_\text {eff}(z,\theta ) \cdot \sigma \,)^s}{s!} \, \cdot \, \exp ( -\mathcal {N}_\text {eff}(z,\theta )\cdot \sigma ). \end{aligned}$$
(29)
The mean scattering probabilities for a specific position z can be calculated using the isotropic angular distribution \(\omega (\theta ) = \sin {\theta }\) and the maximum pitch angle \(\theta _\text {max}\):
$$\begin{aligned} P_s(z) = \frac{1}{1-\cos (\theta _\mathrm {max})} \; \int \limits _{\theta =0}^{\theta _\mathrm {max}} \; \sin (\theta ) \; P_s(z,\theta ) \, \,\mathrm {d}\theta . \end{aligned}$$
(30)
This integration assumes that the angular distribution is not significantly affected by the small angular change in the discussed scattering processes. A higher total column density \(\mathcal {N}\), as well as a larger \(\theta _\text {max}\), would provide a larger number of \(\upbeta \)-electrons at the exit of the source and at the detector. However, they also raise the proportion of scattered over unscattered electrons, thereby increasing the systematic uncertainties due to energy loss, and at some point, limiting the \(\upbeta \)-electron detection rate close to the endpoint. The optimal design values of \(\mathcal {N}= 5.10^{17}\hbox { cm}^{-2}\) and \(\theta _\text {max} = 50.8 ^{\circ }\) [4] balance these effects.
Response function for non-scattered electrons
The transmission function in Eq. (22) describes the transmission probability of isotropically emitted electrons. Even if we consider only non-scattered electrons, the \(\upbeta \)-electrons do not follow an isotropic angular distribution before entering the spectrometer due to the pitch angle dependence of the s-fold scattering probabilities \(P_s(z,\theta )\) in the source (see Sect. 4.5).
Following the description in [39], the zero-scattering (\(s=0\)) transmission function needs to be modified to the form:
$$\begin{aligned} T^\star _{s=0}(E, U)&= \biggl . R(E,U) \biggr |_{\; {\mathcal {E}}\, < \, 10\hbox { eV}} \nonumber \\&= \int \limits _{\theta =0}^{\theta _\text {max}} \; {\mathcal {T}}(E,\theta ,U) \cdot \sin \theta \; P_0(\theta ) \, \,\mathrm {d}\theta . \end{aligned}$$
(31)
The zero-scattering probability \(P_0(\theta )\) is computed by averaging \(P_0(z,\theta )\) over z. Figure 12 illustrates the resulting difference in the response function. The surplus energy range \({\mathcal {E}}< 10\hbox { eV}\) corresponds to the steep increase in the response function at low energies as shown in Fig. 11, where energy loss from inelastic scattering does not contribute.
Doppler effect
The thermal translational motion and the bulk gas flow of the \(\upbeta \)-emitting tritium molecules in the WGTS lead to a Doppler broadening of the electron energy spectrum, which further modifies the response function model that was derived in Sect. 4.2 and thereafter. These two effects can be expressed as a convolution of the differential spectrum \(\frac{\,\mathrm {d}\varGamma }{\,\mathrm {d}E}\) with a broadening kernel g, denoted by the subscript \(\text {D}\):
$$\begin{aligned} \left( \frac{\,\mathrm {d}\varGamma }{\,\mathrm {d}E}\right) _\text {D}&= \left( g \otimes \frac{\,\mathrm {d}\varGamma }{\,\mathrm {d}E} \right) (E_\text {lab}) \end{aligned}$$
(32)
$$\begin{aligned}&= \int \limits _{-\infty }^{+\infty } \; g(E_\text {cms}, E_\text {lab}) \; \frac{\,\mathrm {d}\varGamma }{\,\mathrm {d}E}(E_\text {cms}) \, \,\mathrm {d}E_\text {cms} \, , \end{aligned}$$
(33)
with \(E_\text {cms}\) being the electron kinetic energy in the \(\upbeta \)-emitter’s rest frame (which is approximately the center-of-mass system), and \(E_\text {lab}\) the electron energy in the laboratory frame.
The magnitude of the thermal tritium gas velocity follows a Maxwell-Boltzmann distribution. However, considering only the velocity component \(v_\text {M}\) that is parallel to the electron emission direction, the thermal velocity distribution of the tritium isotopologue mass M is described by a Gaussian
$$\begin{aligned} g(v_\text {M}) = \frac{1}{\sqrt{2\pi }\sigma _v} \cdot \text {e}^{ -\frac{1}{2} \left( \frac{v_\text {M}}{\sigma _v}\right) ^2 }, \end{aligned}$$
(34)
which centers around \(v_\text {M}= 0\) with a standard deviation \(\sigma _v = \sqrt{k_\text {B}T_\text {bt} / M}\). For the component of the bulk gas velocity u that is parallel to the electron emission direction with pitch angle \(\theta \), the mean \(v_\text {M}\) is shifted by \(\cos \theta \cdot u\). Integrating over all emission directions up to \(\theta _\text {max}\), the expression expands to
$$\begin{aligned} g(v_\text {M})&= \frac{1}{(1-\cos \theta _\text {max})} \nonumber \\&\quad \cdot \! \int \limits _{\cos \theta _\text {max}}^1 \! \frac{1}{\sqrt{2\pi }\sigma _v} \cdot \text {e}^{ -\frac{1}{2} \left( \frac{v_\text {M}- \cos \theta \cdot u}{\sigma _v}\right) ^2 } \, \,\mathrm {d}\cos \theta . \end{aligned}$$
(35)
Using the Gaussian error function this expression can be rewritten as
$$\begin{aligned} g(v_\text {M})&= \frac{1}{(1-\cos \theta _\text {max})\cdot 2u} \nonumber \\&\quad \cdot \text {erf}\left( \frac{v_\text {M}- \cos \theta _\text {max} \cdot u}{\sqrt{2}\,\sigma _v}, \frac{v_\text {M}- u}{\sqrt{2}\,\sigma _v}\right) \; . \end{aligned}$$
(36)
Finally, the tritium gas velocity distribution \(g(v_\text {M})\) can be translated into an electron energy distribution \(g(E_\text {cms}, E_\text {lab})\). Using the Lorentz factors and the electron velocities defined in the CMS and lab frames, we can write
$$\begin{aligned} g(E_\text {cms}, E_\text {lab}) = \frac{g(v_\text {M})}{\gamma _\text {cms}\, {m_\text {e}} \, v_{\text {e},\text {cms}}} \end{aligned}$$
(37)
with
$$\begin{aligned} v_\text {M}\approx \frac{v_{\text {e},\text {lab}}- v_{\text {e},\text {cms}}}{1 - v_{\text {e},\text {lab}}\cdot v_{\text {e},\text {cms}}/ c^2} \; . \end{aligned}$$
The standard deviation of this convolution kernel evaluates to
$$\begin{aligned} \sigma _E&= \sigma _v \; \gamma _\text {cms}\; {m_\text {e}} \; v_{\text {e},\text {cms}}\nonumber \\&= \sqrt{(E_\text {cms}+ 2 {m_\text {e}} ) \, E_\text {cms}\cdot k_\text {B}T_\text {bt} / M}. \end{aligned}$$
(38)
With \(\sigma _v \approx 203\hbox { m/s}\) for \(\mathrm {T_2}\) molecules at \(T_\text {bt}=30\hbox { K}\) and the weighted mean bulk velocity at nominal source conditions being \({\bar{u}} \approx 13\hbox { m/s}\), thermal Doppler broadening clearly is a dominating effect. The standard deviation of the broadening function \(g(E_\text {cms}, E_\text {lab})\) at a fixed bulk velocity \(u = 0\) for \(T_\text {bt} = 30\hbox { K}\) and \(E \approx E_0\) evaluates to \(\sigma _E \approx 94\hbox { meV}\) (also see Fig. 13). This value can be interpreted as a significant smearing of the energy scale. Its implication for the neutrino mass measurement is shown in Table 1.
Cyclotron radiation
As electrons move from the source to the spectrometer section in KATRIN, they lose energy through cyclotron radiation. In contrast to energy loss due to scattering with tritium gas (Sect. 4.5), this energy loss process applies to the entire trajectory of an electron as it traverses the experimental beamline [49].
For a particle with kinetic energy E spending a time \(\varDelta t\) in a fixed magnetic field B, the cyclotron energy loss is (in SI units):
$$\begin{aligned} \varDelta E_\perp ^\text {cycl} = -\frac{q^4}{3 \pi c^3 \varepsilon _0 {m_\text {e}} ^3} \cdot B^2 \cdot E_\perp \; \frac{\gamma +1}{2} \cdot \varDelta t. \end{aligned}$$
(39)
In general, cyclotron radiation reduces the transverse momentum component of the particle.Footnote 8 Consequently, the losses are maximal for large pitch angles and vanish completely at \(\theta = 0 ^{\circ }\).
For complex geometric and magnetic field configurations as in the KATRIN experiment, the overall cyclotron energy loss can be computed using a particle tracking simulation framework such as Kassiopeia [50]. By this means, the cyclotron energy loss from the source to the analyzing point in the main spectrometer can be obtained as a function of the electron’s starting position z and pitch angle \(\theta \). Particles starting in the rear of the source will lose more energy due to their longer path through the whole setup. The total cyclotron energy loss can be up to 85 meV for electrons with the maximum pitch angle \(\theta _\text {max} = 50.8 ^{\circ }\).
Because the resulting decrease in the angle \(\varDelta \theta \) due to the loss of transverse momentum is of order \(10^{-6}\) or less, it can be neglected. We thus consider the loss of cyclotron energy \(\varDelta E^\text {cycl}(\theta ,z)\) to be a decrease in the total electron kinetic energy E. Essentially, this effect causes a shift of the electron transmission condition (see Eq. (20))
$$\begin{aligned} {\mathcal {T}}^\text {cycl}_i(E,\theta ,U) = {\mathcal {T}}(E - \varDelta E^\text {cycl}(\theta ,z),\theta ,U) \end{aligned}$$
(40)
with the index i denoting the longitudinal slice where the electron starts from the source position z (see Fig. 10).
The influence of the cyclotron energy loss on the averaged response function is shown in Fig. 14.
Expected integrated spectrum signal rate
Earlier in this section we have laid out the different contributions to the response function of the experiment, which describes the probability for \(\upbeta \)-electrons to arrive at the detector where they contribute to the measured integrated spectrum. The response function describes the energy analysis at the spectrometer (Sects. 4.1 and 4.3), energy loss caused by scattering in the tritium source (Sects. 4.2 and 4.5), and additional corrections (Sect. 4.6 and following).
Combining the response function with the description of the differential spectrum that was developed in Sect. 2, the integrated spectrum signal rate observed on a single detector pixel j for a retarding potential setting U can finally be expressed as
$$\begin{aligned} {\dot{N}}_j^\text {sig}(U)&= \frac{1}{2} \, \epsilon _{\text {det},j} \cdot \sum _{i=0}^{N_L-1} \; N_{\text {T,}i} \nonumber \\&\quad \cdot \, \int \limits _{qU}^{\infty } \left( \frac{\,\mathrm {d}\varGamma }{\,\mathrm {d}E}\right) _\text {C,D}(m_\upnu ^2, E_0) \cdot R_{i,j}(E, U) \, \,\mathrm {d}E. \nonumber \\ \end{aligned}$$
(41)
This expression incorporates all theoretical corrections (see Eq. (12) with subscript C) and the Doppler broadening (see Eq. (33) with subscript D) of the differential spectrum \(\frac{\,\mathrm {d}\varGamma }{\,\mathrm {d}E}\) (see Eq. (10)), and the full response function which incorporates the energy loss as a result of source scattering and cyclotron radiation:
$$\begin{aligned} R_{i,j}(E, U)&= \int \limits _{\epsilon =0}^{{\mathcal {E}}} \, \int \limits _{\theta =0}^{\theta _\mathrm {max}} \, \sum _{s} \; {\mathcal {T}}_{s,i,j}^\text {cycl}(E - \epsilon , \theta , U) \nonumber \\&\quad \cdot \, P_{s,i}(\theta ) \; f_s(\epsilon ) \, \,\mathrm {d}\epsilon \, \sin \theta \,\mathrm {d}\theta . \end{aligned}$$
(42)
The response function depends on the path traversed by the \(\upbeta \)-electron between its origin in source segment (i, j) and the target detector pixel j (see Fig. 10 for the segmentation schema). The detection efficiency \(\epsilon _{\text {det},j}\) is an energy-dependent quantity, which needs to be measured for each pixel j. Its value is between \(\approx 90 \%\) and \(95 \%\) [36].
To first order (due to nearly constant magnetic field and tritium concentration in the source), the integrated signal rate in Eq. (41) depends on \(\mathcal {N}\sigma \) – which can be accurately determined by calibration measurements with a photoelectron source – but is independent of the longitudinal gas density profile \(\rho (z)\) which cannot be measured directly (see [47, 48] for simulation results).
Scan of the integrated spectrum
A scan of the integrated \(\upbeta \) spectrum comprises a set of detector pixel event counts \(N_j(U_k)\), observed at various retarding potential settings \(U_k\) for the duration of \(\varDelta t_k\) each, with \(k \in \{1 \ldots n_k\}\). In the following, the indices j and k are condensed by writing \(N_{jk} = N_j(U_k)\), with \(N_{jk}\) denoting the event count on a single detector pixel j for a specific retarding potential setting k.
The observed event count \(N^\text {obs}_{jk}\) is a Poisson-distributed quantity with the expectation value given by
$$\begin{aligned} \text {E}[N^\text {obs}_{jk}] = \varDelta t_k \cdot \left( \, \dot{N}_j^\text {sig}(U_k) + \dot{N}_j^\text {bg} \, \right) , \end{aligned}$$
(43)
where \(\dot{N}^\text {bg}_j\) is an energy-independent background rate component (possibly with a radial dependency indicated by the index j).
KATRIN will be operated for a duration of 5 calendar years in order to collect 3 live years of spectrum data over multiple runs.
Energy uncertainties
At the end of this section we will briefly discuss the influence of energy uncertainties on the neutrino mass measurement. In general, any fluctuation with variance \(\sigma ^2\) induces a spectrum shape deformation which – if not considered in the analysis – is indistinguishable to first order from a shift of the measured value of \( {m_{\upnu }^2} \) in the negative direction with \(\varDelta {m_{\upnu }^2} = -2 \sigma ^2\) [11]. This shift of \(\varDelta {m_{\upnu }^2} \) also holds if an accounted fluctuation or distribution of true variance \(\sigma ^2_\mathrm {true}\) is described wrongly in the analysis by the variance \(\sigma ^2_\mathrm {ana} = \sigma ^2_\mathrm {true} - \sigma ^2\).
Different sources of fluctuations and distributions with uncertainties can be distinguished. One group comprises \(\upbeta \)-decay and source physics, such as molecular final states, scattering processes and the Doppler effect (all discussed in this work). Others are experimental systematics originating in the energy measurement, which have to be studied during commissioning of the setup and then incorporated into the model. An example is the distortion of the spectrometer transmission function due to retarding-voltage fluctuations [51, 52].
Impact of theoretical and experimental corrections
In Table 1 we review and quantify the impact of theoretical corrections to the differential \(\upbeta \)-spectrum, discussed in Sect. 2, and of experimental corrections which have been introduced above. Many individual model components can be safely neglected, while others need to be considered more accurately, such as the radial dependence of retarding potentials (Sect. 4.3), energy loss due to cyclotron radiation (Sect. 4.8) or the Doppler effect (Sect. 4.7).