Neutrino mass sensitivity by MAC-E-Filter based time-of-flight spectroscopy with the example of KATRIN

An alternative idea is to use MAC-E-Filter time of flight (MAC-E-TOF) spectroscopy to measure the neutrino mass. The TOF of a β decay electron through a MAC-E-Filter like the main spectrometer of KATRIN is a function of the kinetic energy and the emission angle. The distribution of the kinetic energies is in first order governed by the beta spectrum (2) which contains the neutrino mass. By measuring the TOF distribution (TOF spectrum) of the electrons, one can reconstruct the parameters determining the beta spectrum, including m²_νe. Such a method would feature mainly two intrinsic advantages.

On the one hand the MAC-E-Filter slows down the electrons near the retarding energy. While the relative velocity differences between raw beta decay electrons near the endpoint are tiny, a TOF measurement of beta electrons passing through a MAC-E-Filter will be very sensitive to subtle energy differences just above the retarding energy. It can be seen (figure 4) that in principle electron energy differences even below the resolution of the MAC-E-Filter, which is ΔE = 0.93 eV for 18.5 keV electrons in case of KATRIN, can be resolved, given a sufficient time resolution.

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On the other hand, the standard MAC-E mode measures only the count rate for each retarding energy, as described above. In contrast, the TOF spectroscopy mode measures the TOF for each decay electron. Thus a full TOF spectrum, sensitive to m²_νe, is obtained for each retarding energy. For suitable measurement conditions, this gain of information improves the statistics.

The combination of these advantages allows useful optimizations. In principle it would be sufficient to measure only at a single retarding energy near the beta endpoint, though a small number of selected retarding energies might be more sensitive. Since the systematic uncertainty on m²_νe grows as the retarding energy is decreased, the goal is to minimize the amount of measurements far from the beta endpoint. The TOF method could in principle provide this and concentrate the measurement on a few retarding energies near the endpoint, each of them delivering a full TOF spectrum from which m²_νe can be disentangled.

The aim is to state the TOF spectrum as a function of certain fit parameters. These comprise the beta endpoint E₀ and the square of the neutrino mass m²_νe, as well as the relative signal amplitude S which depends on several factors. Thereby, E₀ is in principle known from the ³He–T mass difference measurements in penning traps to 1.2 eV precision [29]. However, this is not precise enough to fix it ab initio. Improvements on the beta endpoint precision are on the way [30]. For a full study also a constant background rate b needs to be fitted, which is however dependent on the implementation of the measurement method. In order to obtain an expression for the TOF spectrum, the TOF has to be known as a function of the kinetic energy E and the starting angle θ first and then has to be weighted by the corresponding distributions given by the windowless gaseous tritium source [31].

3.2.1. TOF as function of E and θ

Within the main spectrometer the principle of adiabatic motion, where the magnetic moment is constant (3), is valid to good approximation. Using a simplified geometry, we take only the field on the z-axis into account and neglect magnetron drifts. The B field then is only a function of the z coordinate. Then, the transverse momentum of an electron can be derived from equation (3) as a function of z,

$$\begin{equation} p_\bot ^2 (z) = p(z_{\mathrm{source}})^2 \sin^2 \theta (z_{\mathrm{source}}) \frac{B(z)}{B(z_{\mathrm{source}})}, \end{equation} \tag{ 8 }$$

where p(z_source), θ(z_source) and B(z_source) are the electron momentum, its emission angle and the total magnetic field at the electron's starting position z_source. The fraction reflects the role of the adiabatic magnetic field geometry of the MAC-E-filter: as the field B(z) decreases, the transverse momentum is converted continuously into longitudinal momentum. The relativistic energy of the electron is given by the energy–momentum-relation

$$\begin{equation} E_{\mathrm{rel}}^2(z) = p_\parallel^2(z)c^2 + p_\bot^2(z)c^2 + m_{\rm e}^2c^4. \end{equation} \tag{ 9 }$$

Since the total energy E_tot = E_rel + E_pot = E_kin + m_ec² + E_pot is conserved, we can express the relativistic energy as a function of z:

$$\begin{eqnarray} E_{\mathrm{rel}}(z) & = & E_{\mathrm{rel}} (z_{\mathrm{source}}) - E_{\mathrm{pot}} (z) + E_{\mathrm{pot}} (z_{\mathrm{source}}) \nonumber \\ & = & E_{\mathrm{kin}} (z_{\mathrm{source}}) + m_{\rm e} c^2 - q \Delta U(z), \end{eqnarray} \tag{ 10 }$$

where ΔU(z) is the difference of the retarding voltage at the source and at z,

$$\begin{equation} \Delta U(z) =|U(z) - U(z_{\mathrm{source}})| , \end{equation} \tag{ 11 }$$

and q is the magnitude of the electron charge. Combining equations (8)–(10), we derive an expression for the longitudinal momentum as a function of z only in terms of the field B and the potential difference ΔU:

$$\begin{eqnarray} &&\fl p_\parallel^2(z)c^2 = \left(E^2 + 2 E \ m_{\rm e} c^2 \right) \left( 1 - \sin^2 \theta \frac{B(z)}{B(z_{\mathrm{source}})} \right) + \ q^2 \Delta U^2(z) - 2 q \Delta U(z) \left( E + m_{\rm e} c^2 \right) \end{eqnarray} \tag{ 12 }$$

with the abbreviations E: = E_kin(z_source) and θ: = θ(z_source). The TOF is determined by integrating the reciprocal parallel velocity 1/v_∥ = γm/p_∥ = E_rel/p_∥c² over the measurement path:

$$\begin{eqnarray} &&\tau (E, \theta) = \int \mathrm{d} z \ \frac{1}{v_\parallel} = \int\nolimits_{z_{\mathrm{start}}}^{z_{\mathrm{stop}}} \mathrm{d} z \ \frac{E + m_{\rm e} c^2 - q \Delta U(z)}{\sqrt{p_\parallel^2(z) c^2}\, c}. \end{eqnarray} \tag{ 13 }$$

The lower bound z_start of the integration interval depends on where the start signal time is measured⁴ while z_stop corresponds to the z-position of the detector. As the adiabatic approximation (3) is valid through the whole transport section, this position is arbitrary. The starting angle θ = θ(z_source) is automatically transformed to its correct value at the start position θ(z_start) by (12) because only the ratio of local and source magnetic field B(z)/B(z_source) matters but not the field changes between B(z_source) and B(z_start).

The integral (13) is only correct for electrons emitted in the centre of the fluxtube r = 0 since the integration path is identical with the z-axis. As shown in figure 2, the electrons perform a cyclotron motion around the B field lines, where for the flight-times only the velocity component parallel to the B field lines v_∥ needs to be considered. Electrons emitted at r = 0 take a path different from the z-axis. This is, however, not a real shortcoming of our method because at KATRIN the starting position can be reconstructed by the point of arrival on the multipixel detector. Therefore we do not expect significant changes in sensitivity depending on the electron emission radius. For this in-principle study of the statistical sensitivity we consider it to be sufficient to use the central electron tracks only.

Equation (13) presumes a fixed kinetic starting energy and starting angle as arguments. For a real source, these parameters are not fixed but follow physical distributions. In order to be calculated numerically, the differential TOF spectrum $\frac {\mathrm {d} N}{\mathrm {d} t}$ is discretized into bins of constant length Δt and integrated over each bin j, leading to a binned spectrum

$$\begin{equation} F(t_j) := \int\nolimits_{t_j}^{t_{j+1} = t_j + \Delta t} \mathrm{d} t \ \frac{\mathrm{d} N}{\mathrm{d} t} . \end{equation} \tag{ 14 }$$

The number of events in a certain TOF bin depends on the distribution of starting energies and angles E and θ,

$$\begin{eqnarray} F(t_j) & = & { {\int\!\!\int} _{ \mbox{ $(E , \theta)$ with $t_j \leqslant \tau(E , \theta) \leqslant t_{j+1}$}}} \quad \frac{\mathrm{d}^2 N}{\mathrm{d}\theta\,\mathrm{d}E} \,\mathrm{d}\theta\,\mathrm{d}E \nonumber\\ & = & \int\nolimits _0 ^{\theta_{\mathrm{max}}} \ \int\nolimits_{E_j(\theta)}^{E_{j+1}(\theta)} \quad \frac{\mathrm{d}^2 N}{\mathrm{d}\theta\,\mathrm{d}E} \,\mathrm{d}\theta\,\mathrm{d}E, \end{eqnarray} \tag{ 15 }$$

where $\frac {\mathrm {d}^2 N}{\mathrm {d}\theta \,\mathrm {d}E}$ is the double differential event rate as function of E and θ. The integral limits E_j(θ) and E_j+1(θ) are defined in such way that τ(E_j,θ) = t_j and τ(E_j+1,θ) = t_j+1, respectively. At first order, $\frac {\mathrm {d}^2 N}{\mathrm {d}\theta \,\mathrm {d}E}$ is given by the double differential decay rate $\frac {\mathrm {d}^2 R}{\mathrm {d}\theta \,\mathrm {d}E}$ into the accepted solid angle $\Delta \Omega \over 4 \pi$,

$$\begin{equation} \frac{\mathrm{d}^2 N}{\mathrm{d}\theta\,\mathrm{d}E} \approx \frac{\mathrm{d}^2 R}{\mathrm{d}\theta\,\mathrm{d}E} . \end{equation} \tag{ 16 }$$

As the double differential is proportional to the joint probability distribution of emitting an electron with energy E at a polar angle of θ and, furthermore, the angle and the energy are uncorrelated in case of a non-oriented radioactive source, the quantity can be separated into a product of the single differential decay rate $\frac {\mathrm {d} R}{\mathrm {d} E}$, as given by (2), and the angular probability distribution g(θ) [32],

$$\begin{equation} \frac{\mathrm{d}^2 R}{\mathrm{d}\theta\,\mathrm{d}E} = \frac{\mathrm{d} R}{\mathrm{d} E} \, g(\theta). \end{equation} \tag{ 17 }$$

In case of an isotropic tritium source, a sine law applies for the angular distribution

$$\begin{equation} g(\theta) = \textstyle\frac{1}{2} \sin \theta . \end{equation} \tag{ 18 }$$

This angular distribution function is normalized to unity over the full solid angle 4π. Since for KATRIN the polar angle is restricted to θ_max = 50.77°, the signal rate is implicitly reduced by a factor

$$\begin{equation} \int\nolimits _0 ^{\theta_{\mathrm{max}}} \mathrm{d}\theta \, g(\theta) = \frac{\Delta \Omega}{4\pi} = \frac{(1 - \cos\theta_{\mathrm{max}})}{2}, \end{equation} \tag{ 19 }$$

which is enforced by the upper integral bound θ_max in (15).

The approximation (16) is only valid in case of an ideal tritium source. However, quite a few electrons lose energy in elastic and inelastic scattering processes with the tritium molecules. These losses are dependent on the emission angle since the path through the tritium source increases with 1/cos θ. Thus, for the differential rate of events which are actually analysed in the main spectrometer, given by $\frac {\mathrm {d}^2 N}{\mathrm {d}\theta \,\mathrm {d}E}$ in (15), starting energies and angles become correlated. Additionally, the signal rate decreases due to several losses inside the experiment. A factor _flux ≈ 0.83 applies since the flux tube transported through the whole system corresponds to a diameter of 82 mm w.r.t. the beam tube diameter of 90 mm, meaning that only a part of the windowless gaseous tritium source (WGTS) tube is imaged onto the detector. Furthermore, the detector efficiency gives an additional factor of _det ≈ 0.9.

In total, the true event rate can be calculated by applying the correction factors and convoluting the beta spectrum with an energy loss function, which gives

$$\begin{eqnarray} \frac{\mathrm{d}^2 N}{\mathrm{d}\theta\,\mathrm{d}E} & = & \epsilon_{\mathrm{flux}}\,\epsilon_{\mathrm{det}} \, g(\theta)\, \frac{\mathrm{d} R}{\mathrm{d} E} \otimes {f_{\mathrm{loss,}}}_\theta \nonumber\\ & = & \epsilon_{\mathrm{flux}}\,\epsilon_{\mathrm{det}}\, g(\theta) \left( p_0(\theta) \, \frac{\mathrm{d} R}{\mathrm{d} E} + \sum\limits_{n=1}^\infty p_n(\theta) \, \frac{\mathrm{d} R}{\mathrm{d} E} \otimes f_n \right), \end{eqnarray} \tag{ 20 }$$

where the f_n is the energy loss function of scattering order n which is defined recursively through the single scattering energy loss function f₁ as

$$\begin{equation} f_n = f_{n-1} \otimes f_1 \quad (n > 1) . \end{equation} \tag{ 21 }$$

The function f₁(ΔE) is the probability density of losing the energy ΔE in a singular scattering event [28]. The functions of f_n can then correspondingly be interpreted as the same for n-fold scattering. In this equation, all changes of the angle of the electron during scattering are neglected. p_n is the probability that an electron is scattered n times. If we again neglect changes of the angle, it is a function of the emission angle θ and given by a Poisson law

$$\begin{equation} p_n(\theta) = \frac{\lambda^n(\theta)}{n!}\mathrm{e}^{-\lambda^n(\theta)}~. \end{equation} \tag{ 22 }$$

Here, the expectation value λ is given in terms of the column density ρd, the mean free column density ρd_free and the scattering cross section σ_scat as

$$\begin{equation} \lambda(\theta) = \int_0^1 \mathrm{d} x \ \frac{\rho d\, x}{\rho d_{\mathrm{free}} \cos \theta} = \int_0^1 \mathrm{d} x \ \frac{\rho d \, x \sigma_{\mathrm{scat}}}{\cos \theta}~, \end{equation} \tag{ 23 }$$

where the integration factor x accounts for the fact that the starting position of the electron inside the WGTS is statistically distributed.

To calculate the TOF spectrum according to (15) the input parameters have to be obtained. A model for the one-dimensional field maps ΔU(z) and B(z) in (13) has been determined by the KATRIN simulation tools magfield and elcd3_2 [33] using a modestly simplified geometry that contains the most important coils and electrodes in the main spectrometer (figure 5). A model for the energy loss function (21) has been determined in the past by electron scattering experiments on hydrogen [28] and refined by using excitation and ionization data from hydrogen molecules [34]. The final-state excitation spectrum of the daughter molecules has been used from [24].

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A typical set of simulated TOF spectra for different neutrino mass squares is shown in figure 6. The following details and parameter-dependent behaviour can be observed:

• For each spectrum, there exists a minimal TOF t_min. This corresponds to the maximum kinetic emission energy that an electron can have, given by E_max = E₀ − m_νec².
• From t_min on, a steep slope begins, leading soon to a maximum somewhat above t_min, followed again by a long, slow fall. The maximum can be explained by the fact that the higher the energy becomes, the lower the number of electrons is, due to the shape of the end of the beta spectrum, whereas the 'TOF energy density', i.e. the interval size of the energy that corresponds to a certain TOF bin, increases. These effects balance each other, leading to a maximum somewhere in the middle.
• There is no maximal TOF. The closer the energy of an electron is to the retarding energy, the slower it will be. That means that electrons with an energy infinitesimally above the retarding energy will have an infinite TOF.
• If the neutrino mass square m²_νe is changed, the main signature is a change in count-rate and a change of the shape especially at the short-time end of the spectrum (figure 6).
• A higher retarding energy qU leads to a clearer distinction between spectra for different neutrino masses. The reason is that the neutrino mass is mainly visible in the last few eV of the beta spectrum. Therefore, it seems optimal for the TOF mode to measure with retarding energies near the endpoint. However, due to the lower count rate and the difficult decorrelation of neutrino mass square and endpoint, measurements from lower retarding energies should be added to the data.

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4.2.1. Method

In order to study the statistical uncertainty we used the spectra to fit Monte Carlo (MC) data. The MC data have been obtained by creating Poisson distributed random numbers based on the predictions from (15), where certain choices of the parameters m²_νe and E₀ as well as the retarding energy qU and the measurement time have been assumed. The data are fitted in this self-consistent method by the models (15) using a χ² minimization method. If multiple measurements with different qU are assumed, they can be fitted with a common chi square function by adding the chi square functions from each run. If the fit is performed correctly, the chosen parameters m²_νe and E₀ are reproduced. Additionally, estimates for the parameter errors can be determined as

$$\begin{equation} \chi^2 (\phi_0 \pm \Delta \phi_\pm) = \chi^2 (\phi_0) + 1 , \end{equation} \tag{ 24 }$$

where ϕ₀ is a parameter estimate and Δϕ_± are the requested, not necessarily symmetric parameter error bars [32]. To obtain a symmetric χ² parabola for neutrino masses near zero, there must also be an extension for a negative m²_νe that joins smoothly with the physical spectrum for m²_νe > 0. To accomplish this, to each term in the sum of the beta spectrum (2) a factor

$$\begin{equation} f_i = \left( 1 + \frac{m_{\mathrm{eff}}}{\epsilon_i} \mathrm{e}^{-(1+\epsilon_i/m_{\mathrm{eff}})} \right) \end{equation} \tag{ 25 }$$

is applied in case of m²_νe < 0 and _i + m_eff > 0. In this expression, the abbreviations _i = E₀ − V_i − E and $m_{\mathrm {eff}} = \sqrt {-m_{\nu _{\mathrm { e}}}^2}$ have been used [35]. This method allows a simple but realistic prediction of the statistical uncertainty of m²_νe.

4.2.2. Results

In order to determine the improvement potential by the TOF mode, an optimal choice of the measurement times of the runs with different retarding energies qU has to be made. For KATRIN a total on-line time of 3 years is planned, which has to be distributed among the retarding energies. We used a simple algorithm where we discretized the retarding potentials and the measurement time and determined the statistical uncertainty with the method above for all possible permutations. The results are shown in table 1. At the MC data creation, a neutrino mass of zero has been assumed. In this case, the average of the fit uncertainties m²_νe, as given by (24), describes the sensitivity on the neutrino mass squared.

Table 1. Average statistical uncertainty $\langle \sigma _{\mathrm {stat}}(m_{\nu _{\mathrm { e}}}^2)\rangle = \langle \frac {1}{2} (|{\Delta m_{\nu _{\mathrm { e}}}^2}_-| + | {\Delta m_{\nu _{\mathrm { e}}}^2}_+|)\rangle$ (arithmetic mean of positive and negative error of ten simulations and fits), average fit parameters $\langle \bar E_0\rangle$ and $\langle \bar m_{\nu _{\mathrm { e}}}^2\rangle$, as well as Pearson's correlation coefficient R(E₀,m²_νe) of uniform and optimized distributions and the KATRIN standard mode. The total assumed measurement time is the KATRIN standard of 3 years [18], distributed among four retarding energies qU = 18 550, 18 555, 18 560 and 18 565 V as well as for single retarding potentials for m²_νe = 0 eV²/c², E₀ = 18 575 eV and b = 0. The choice of retarding potentials for the TOF mode is motivated by the idea that a choice of a few potentials close to the endpoint will likely improve the systematics additionally to the statistical uncertainty.

Fraction of measurement time per retarding energy	Distribution type	Lowest retarding energy (eV)	Mean stat. error 〈σstat(m2νe)〉 (eV2/c4)	Mean fitted endpoint $\langle \bar E_0\rangle$ (eV)	Mean fitted ν mass squared $\langle \bar m_{\nu _{\mathrm { e}}}^2\rangle$ (eV2/c4)	Mean correlation coefficient R(E0,m2νe)
$(\frac {3}{12}, \frac {3}{12}, \frac {3}{12}, \frac {3}{12})$	Uniform	18 550	0.0033	18 574.9997	0.0004	0.65
$(\frac {1}{12}, \frac {0}{12}, \frac {3}{12}, \frac {8}{12})$	Optimized	18 550	0.0032	18 575.0002	0.0013	0.70
$(0, \frac {4}{12}, \frac {4}{12}, \frac {4}{12})$	Uniform	18 555	0.0034	18 575.0002	0.0015	0.73
$(0, \frac {2}{12}, \frac {1}{12}, \frac {9}{12})$	Optimized	18 555	0.0034	18 575.0002	0.0006	0.72
$(0, 0, \frac {6}{12}, \frac {6}{12})$	Uniform	18 560	0.0036	18 575.0002	0.0014	0.74
$(0, 0, \frac {4}{12}, \frac {8}{12})$	Optimized	18 560	0.0035	18 575.0007	0.0034	0.76
(1,0,0,0)	Single	18 550	0.0035	18 575.0000	0.0004	0.82
(0,1,0,0)	Single	18 555	0.0036	18 575.0000	0.0003	0.88
(0,0,1,0)	Single	18 560	0.0038	18 574.9999	−0.0015	0.79
(0,0,0,1)	Single	18 565	0.0039	18 574.9998	0.0007	0.66
–	Standard mode	18 555	0.020
–	Standard mode	18 550	0.019
–	Standard mode	18 545	0.018

The comparison shows that it is in principle sufficient to measure at only one retarding energy. If this single retarding energy is close to the endpoint, the correlation between the parameters E₀ and m²_νe becomes weaker at the cost of losing count-rate. It turns out that it is beneficial to combine measurements at more than one retarding energy, where this relation between lowest retarding energy and correlation coefficient does not necessarily hold true (see table 1). In almost all tested cases using multiple retarding potentials a solid decorrelation without suffering from too little count-rate has been possible.

The results in table 1 correspond to the optimum case since background, time uncertainty and other limitations have been neglected. They reflect the maximal improvement potential that can be achieved with a TOF mode. The motivation to neglect the background in the optimum case is based on the idea that a sensitive TOF measurement method may be able to reduce the background, too, depending on the implementation of the measurement. It can be shown that, compared with the statistical sensitivity for the reference configuration of KATRIN, σ_stat(m²_νe) = 0.018 eV²/c⁴, an improvement of up to a factor  ∼ 5–6 by TOF spectroscopy is possible. The actual improvement factor, however, depends on the method by which the TOF determination is implemented.

Furthermore, one can conclude that even a reduction of the systematic uncertainty might be possible with the TOF mode. This is due to the fact that the systematic uncertainty at KATRIN depends heavily on the measurement interval at which the spectrum is scanned [18]. That is mainly caused by the uncertainty of the parameters of the electron energy loss which becomes more important at lower retarding energies. An ideal TOF mode, in contrast, would allow one to measure solely at higher retarding energies.

For further analyses, the optimal distribution for the case of 18 560 eV lowest retarding energy has been taken which is likely a good compromise between statistical and systematic uncertainty. An example for a fit, based on this measurement time distribution, is shown in figure 7.

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For low-energy or mildly relativistic electrons moving in a magnetic field, the dominant energy-loss mechanism is cyclotron radiation [40]. The power radiated scales as the square of the magnetic field B, and depends on the pitch angle θ with respect to the static field. The cyclotron angular frequency is

$$\begin{eqnarray} \omega&=& \frac{qB}{\gamma m_{\rm e}} \equiv \frac{\omega_0}{\gamma} \end{eqnarray} \tag{ 28 }$$

and the radiated power is

$$\begin{eqnarray} P(\beta,\theta) &=& \frac{1}{4\pi \epsilon_0} \frac{2q^2 \omega_0^2}{3c} \frac{\beta^2 \sin^2{\theta}}{1-\beta^2}, \end{eqnarray} \tag{ 29 }$$

where q is the charge on the electron and m_e is the electron mass. At a field strength of 4.5 T, as is found between the KATRIN pre-spectrometer and the main spectrometer, the radiated power for 18.6 keV electrons having a 51° pitch angle is about 8 fW. If the observation region has a usable length x_m, the time spent by the electron in that region is

$$\begin{eqnarray} &&\delta \tau = \frac{x_m}{c\beta \cos \theta} \end{eqnarray} \tag{ 30 }$$

which, for x_m = 50 cm, ranges from 6 to 10 ns depending on the pitch angle. The energy lost by the electron in free radiation during its transit of the observation region is then ⩽10⁻³ eV, only 12k_BT at 1 K. The cyclotron frequency ω/2π is 126 GHz, still a technically challenging region to work in with non-bolometric amplifiers, and the antenna collection efficiency will necessarily be a compromise in order to allow the electrons to make an unobstructed transit.

More specifically, the noise power is k_BTΔν, where Δν is the bandwidth. There are two principal contributions to the bandwidth—the line broadening caused by the duration δτ of the signal, and the broadening caused by magnetic-field inhomogeneity. We assume the latter can be made smaller than the former, and neglect it. Only a fraction A of the radiated signal energy is absorbed in the receiver. The signal power is distributed over a bandwidth Δν = (2π δτ)⁻¹. Hence the energy from the signal W_s in the interval δτ may be compared with the noise energy W_n as follows:

$$\begin{eqnarray} W_{\mathrm{s}} &=& A P(\beta,\theta)\delta\tau, \end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray} W_{\mathrm{n}} &=& k_{\mathrm{B}}T\Delta\nu \, \delta\tau, \end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray} \frac{W_{\mathrm{s}}}{W_{\mathrm{n}}} &=& \frac{2\pi AP(\beta,\theta)\delta\tau}{k_{\mathrm{B}}T}. \end{eqnarray} \tag{ 33 }$$

Taking δτ = 10 ns, the bandwidth is 16 MHz and the signal-to-noise ratio in that interval is about 16 when θ = 51°, T = 1 K and A = 0.5. The magnetic-field inhomogeneity must be less than about 10⁻⁴. The signal-to-noise ratio drops rapidly below 51° pitch angle, which calls for a much longer magnet of high homogeneity.

Up to now we discussed classical electromagnetic radiation but we have to consider the quantization as well. The energy of a cyclotron radiation photon E_ph = ℏω amounts to 5 × 10⁻⁴ eV, which is comparable to the whole emitted radiation energy. This again calls for more radiated energy ΔE and thus for a longer magnetic field section.

Schottky pickups [36] encompass a broad class of devices used to detect particle beams without intercepting them. They have in common the induction of an image charge in a circuit connected to plates or a cavity. To estimate the energy that can be extracted from a single electron, we consider a conducting cylinder that the electron enters, passes through and emerges from. The image charge induced on the cylinder gives it a stored energy

$$\begin{eqnarray} W &=& \frac{q^2}{8\pi\epsilon_0 r}, \end{eqnarray} \tag{ 34 }$$

where r is the radius of the probe ring. For r = 5 cm, the stored energy is about 10⁻⁸ eV, which can be delivered to a suitably matched external circuit. Because the observation region x_m is considerably longer than the radius of the aperture, several probes can be cascaded to increase the energy extracted. If the probes are cascaded in a phased way, some further enhancement of the energy extraction can be achieved, but the dependence of electron transit times on pitch angle limits the usefulness of resonant structures.

The electron in flight produces a time-varying magnetic field that can be used to induce a transient current in a nearby inductively coupled element. The magnetic field can be obtained from a Lorentz transformation on a stationary electron

$$\begin{eqnarray} B(\zeta,t) & = & \frac{\mu_0}{4\pi}\frac{\gamma\beta c q \zeta}{(\zeta^2+\gamma^2\beta^2c^2t^2)^{3/2}}, \end{eqnarray} \tag{ 35 }$$

where ζ is the impact parameter, or distance of closest approach, and t the time since closest approach. For the present purposes, we neglect the cyclotron motion because its period is two orders of magnitude shorter than the characteristic time for establishing the magnetic field due to average linear motion. The magnetic energy in the field beyond a minimum radius ζ = r_i is

$$\begin{eqnarray} W &=& \pi \int_{r_i}^\infty \int_{-\infty}^\infty B^2 \beta \zeta c\, \mathrm{d}\zeta\, \mathrm{d}t \end{eqnarray} \tag{ 36 }$$

$$\begin{eqnarray} &=& \frac{3\pi^2}{4} \left(\frac{\mu_0}{4\pi}\right)^2 \frac{\gamma \beta^2 c^2 q^2}{r_i}. \end{eqnarray} \tag{ 37 }$$

The integral over time is equivalent to integrating over the third, axial, coordinate. Assuming all of the magnetic energy outside r_i can be delivered to an external circuit, the energy extracted from the electron is 1.6 × 10⁻¹⁵ eV for r_i = 5 cm. The estimate is somewhat pessimistic, because the spatial width of the instantaneous magnetic field distribution is considerably narrower than x_m, and so this amount of energy could be extracted several times with multiple inductors, but the available energy is so small that the details are not important.

This approach is standard and useful, but the energy extracted is not under control for any given collision, and sometimes may be much larger than desirable. Recently it has been proposed [37] to use highly excited Rydberg atoms as an interaction medium, since the cross section for modest energy losses is enhanced relative to more inelastic collisions.

In addition to the statistical advantage in measuring the signal, a substantial gain in background suppression can also be expected with electron tagging. The background suppression is based on the principle that a signal in the detector is placed in delayed coincidence with a signal from the tagger, within a time window of width t_o. Given a sufficiently low expected rate of tagger signals r_s, and neglecting pile-up, random coincidences would result in background reduction by a factor ϕ with

$$\begin{equation} \frac{1}{\phi} = 1 - \mathrm{e}^{-t_{\mathrm{o}} r_{\mathrm{s}}} . \end{equation} \tag{ 38 }$$

Too high a rate of tagger signals, either due to a high flux of incoming electrons or to a high noise level, would impair the measurement. However, in a dual-spectrometer setup like KATRIN, the electron flux through the tagger can be reduced by the pre-spectrometer down to $\mathcal {O}$(10³ Hz). On the other hand, electrons can be trapped between the pre- and main spectrometer, and would give rise to a high rate of tagger signals. Trapped electrons, however, could be reduced by an active measure such as a scanning wire [38] or suitably chosen time-dependent perturbations of the electric fields along the transport path.

The tagger must be placed before the main spectrometer, but should be placed in a low rate area. For best tagger functionality, it is also favourable to have the beam tube through the tagger fill a small area. The highest-field region of the 4.5 T magnet between the pre-spectrometer and main spectrometer is a good position; there magnetic reflection reduces the flux through the tagger even below the rate at the pre-spectrometer analysing plane. There are two major sources of electrons at this point in the beam line.

• Electrons from the source that make it through the pre-spectrometer, are reflected by the retarding potential at the main spectrometer analysing plane, and then go back through the pre-spectrometer and are absorbed in the source section. These will each pass the electron tagger twice.
• Electrons from the source that make it through the pre-spectrometer, are reflected by the main spectrometer, but, because of radiative losses or scattering inside the main spectrometer, lack the energy to make it back through the pre-spectrometer and become trapped. The rate of electrons becoming trapped is only $\mathcal {O}$(1 Hz), but, without any active removal method each trapped electron could pass the electron tagger millions of times before becoming undetectable. An active removal method only needs to remove trapped electrons in $\mathcal {O}$(10⁴ passes) before the rate from trapped electrons is lower than the rate from electrons reflected by the main spectrometer retarding potential.

Electrons from the source that pass both the pre-spectrometer retarding potential, and the magnet, each make two passes.

From equation (5) applied to the pre-spectrometer, we find the magnet at the end of the pre-spectrometer reflects all electrons with a starting angle at the source greater than

$$\begin{equation} \theta_{\rm pre}=\arcsin\left(\sqrt{\frac{B_{\rm source}}{B_{\rm max}}}\right), \end{equation} \tag{ 39 }$$

where B_max is the magnetic field in the magnet at the end of the pre-spectrometer.

Solving equation (12) for E and requiring that the parallel momentum p_∥ must be positive to make it past the retarding potential, we find the minimum initial kinetic energy for passage through the pre-spectrometer is given by

$$\begin{equation} E_{\rm min}(\theta,\Delta U,B)= \frac{\sqrt{k q^2 \Delta U^2 + (k - 1)^2 m_{\rm e}^2c^4 } +q \Delta U + (k - 1)mc^2}{1-k}, \end{equation} \tag{ 40 }$$

$$\begin{eqnarray*} &&k=\sin^{2}\theta\frac{B(z_{\mathrm{pre}})}{B(z_{\mathrm{source}})}, \end{eqnarray*}$$

where θ is the starting angle at the source, ΔU is the retarding potential of the pre-spectrometer and B(z_pre) is the magnetic field at the analysing plane of the pre-spectrometer.

Combining equations (39) and (40) we find the average rate of electrons from the pre-spectrometer passing the electron tagger is

$$\begin{equation} R_{\mathrm{pre}} = 2\int _{0}^{\theta_{\rm pre}}\frac{2\pi\, \sin\:\theta}{4\pi}\mathrm{d}\theta\int_{E_{\rm min}}^{E_{0}}\frac{\mathrm{d} N}{\mathrm{d} E}(\theta), \end{equation} \tag{ 41 }$$

where $\frac {\mathrm {d} N}{\mathrm {d} E}(\theta )$ is the rate at the source, given by equation (20). The factor 2 has been applied since each electron makes two passes. This assumes that all electrons that make it through the pre-spectrometer are reflected by the main spectrometer, of which the vast majority are. The rate as a function of pre-spectrometer potential is plotted in figure 8. In figure 9 the neutrino mass sensitivity as a function of the rate at the tagger is plotted. It can be seen that rates below  ∼ 10 kHz do not cause a significant loss of sensitivity.

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In summary, there is no fundamental obstacle to the detection of single electrons in flight, if the specification is only that energy perturbations be small enough to be acceptable for reliable energy determination. However, as a practical matter, no satisfactory method has yet been identified that allows a controlled and sufficiently large amount of energy to be extracted from the electron. The cyclotron-emission method may be adequately sensitive, but significantly longer superconducting magnets providing a very homogeneous field are needed at the tagger position, which are not available between the KATRIN spectrometers. With a successful tagger, there is the potential for suppression of backgrounds by virtue of the coincidence requirement between the electron tagger and the focal-plane detector, but there is also a random trigger rate for the tagger caused by electrons that do not pass through the main spectrometer.

Several parameters apply to most methods, chiefly the background rate, the time resolution and the efficiency. The dependence of the statistical uncertainty on the efficiency , i.e. the ratio of events whose TOF is correctly measured, follows a $1/\sqrt {\epsilon }$ law. This behaviour is theoretically predicted and has been verified by simulations. The dependence on the background rate and the time resolution found in the simulations is shown in figure 10.

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For the time resolution a Gaussian uncertainty has been assumed. Figure 10 shows that the timing is uncritical for resolutions within the order of magnitude of the KATRIN detector. For resolutions in the range up to 200 ns the error increases by about 20%. The scale of this behaviour is plausible as the scale of the neutrino-mass-sensitive part of the TOF spectrum is mainly contained in the first few μs after the onset (see figure 6) and becomes washed out if the time resolution of the TOF measurement method exceeds some 100 ns.

Assuming the background level in the TOF mode is the same as in the standard mode where it is specified for KATRIN as b < 10 mHz, it can be shown that the improvement by the TOF mode is still up to a factor 3 in terms of m²_νe. The behaviour follows a power law with approximately σ_stat(m²_νe) = 0.006 eV²/c⁴ × (b/Hz)^1/2.0 +  0.004 eV²/c⁴.⁵ In comparison, in the standard mode the background dependence can be determined to be approximately σ_stat(m²_νe) = 0.019 eV²/c⁴ × (b/mHz)^1/1.7 + 0.009 eV²/c⁴ [39], in reasonable agreement with the analytically approximated formula of σ_stat(m²_νe)∝b^1/3 [23].

A method that has been discussed and successfully applied for TOF in the past is to periodically cut off the electron flux [41]. While there it has been used as a band-pass filter, where all signals with a TOF outside a certain time window have been rejected and a classic, non-integrated beta spectrum has been measured, this technique might as well be applied for TOF spectroscopy. In the case of KATRIN, periodic filtering could be achieved by a high-frequency modulation of the source or the pre-spectrometer potential.

The principle in this case is to switch between two settings. In one setting a pre-spectrometer potential q(U_pre + ΔU_pre) > E₀ is chosen, to block completely the flux of β-electrons. In the other setting the retarding potential of the pre-spectrometer is set to qU_pre < E₀ − ΔE_i, leading to the full transmission of all electrons from the interesting energy region [E₀ − ΔE_i,E₀] and allowing to do TOF spectroscopy (see figure 11). The region of interest of width ΔE_i of a few 10 eV requires moderate switching voltages ΔU_pre ≈ −200 V, taking into account the energy resolution of the pre-spectrometer of about 100 eV. While pulsing the source potential is the traditional approach as in [41], in a dual spectrometer set-up like KATRIN it thus is more convenient to vary the retarding potential of the pre-spectrometer. This has the advantage that, in contrast to pulsing at the source, the potential for the 'on'-setting does not need to be precise as long as a transmission of the region of interest is guaranteed.

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6.2.1. Timing parameters

A periodically gated flux can in the simplest case be described by two timing parameters. The first one is the period t_r with which the flux is gated. After each period the detector clock is reset. The second one is the time t_s in which the gate is open in each period. The ratio of t_s and t_r gives the duty cycle

$$\begin{equation} \eta = \frac{t_s}{t_r}. \end{equation} \tag{ 42 }$$

This method uses no direct measurement of the starting times but restricts them to certain intervals of length t_s. That is equivalent to knowing the starting time with certain uncertainty. Thus, for t_s → 0 and sufficient period lengths t_r, infinitesimally sharp starting times with infinitesimally low luminosity are obtained. If t_s is extended, the luminosity increases and the time uncertainty grows. The uncertainty is given by a uniform probability distribution in the interval [0;t_s]. As the measured TOF spectrum is then given by the convolution with the detection time and the starting time distribution,

$$\begin{equation} \left(\frac{\mathrm{d} N}{\mathrm{d} t}\right)_{t_s} = \frac{\mathrm{d} N}{\mathrm{d} t} \otimes N(\sigma_{\mathrm{d}}) \otimes U(0, t_s) , \end{equation} \tag{ 43 }$$

where N(σ_d) is the Gaussian uncertainty profile of the detection time at the detector and U(0,t_s) is the uniform uncertainty due to the gate. If the detector clock is periodically reset with times t_r then some electrons with flight times >t_r might hit the detector in a later period. Therefore, the final measured spectrum is a superposition of all contributing time distributions (43), shifted by multiples of t_r and finally cut off at 0 and t_r:

$$\begin{eqnarray*} \left(\frac{\mathrm{d} N}{\mathrm{d} t} \right)_{t_s, t_r}(t) = \left\{\begin{array}{@{}ll@{}} 0, & t < 0, \\ \sum _{n=0}^\infty \frac{\mathrm{d} N}{\mathrm{d} t}_{t_s}(t + n \, t_r), & 0 \leqslant t \leqslant t_r, \\ 0, & t > t_r. \end{array} \right. \end{eqnarray*}$$

As >99.5% of the flight times lie within ≲50 μs,⁶ all contributions with nt_r ≳ t_s + 50 μs can be neglected. The resulting measured time spectrum (44) is strictly speaking not a spectrum of flight times, but rather of detection times for a periodically reset detector clock.

An illustrative TOF spectrum of simulated measurement data according to (44) is shown in figure 12. The two characteristics mentioned that describe a simple periodic gate show clear signatures in the curve. The uniform start time distribution within [0,t_s] imposed on the spectrum by the uniform uncertainty (43) leads to a clear broadening of the shape. Since the time precision is lower than in the tagger case the broadening is more pronounced. In addition, the smearing with the step function leads to steeper edges than the Gaussian one. A clear sign of the detector resets are the residuals from former gate periods at the beginning of the spectrum. The effects of the timing parameters on the spectral shape allow some preliminary predictions on the effects of the performance:

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• For constant t_s, reducing t_r will increase the duty cycle. However, more residuals from former periods contaminate the spectrum. That acts like a non-uniform background. Duty cycle and residual contributions need to be balanced.
• For constant t_r, reducing t_s will reduce the time uncertainty. In contrast, the duty cycle will be reduced, resulting in a lower count-rate. Here, the timing and the duty cycle need to be balanced.

Due to the trade-off between timing, duty cycle and residual background, the timing parameters need to be optimized. A global optimization of t_r and t_s has only shown a significant change in σ_stat(m²_νe) for extreme input values. However, an individual optimization of t_s together with the measurement time contribution for each retarding energy makes sense, as for higher retarding potentials the count-rate drops while the neutrino mass information grows, so a higher duty cycle is needed. As the gated filter is less TOF-sensitive than an ideal TOF measurement, a higher number of retarding energies as well as measurements at lower retarding energies are necessary.

6.2.2. Results

The results of a rough optimization run with six retarding potentials, giving 12 free parameters, are shown in table 2. The highest retarding energy is above the endpoint that was assumed, thus being sensitive to the background level. Starting with η = 0.5 and a uniform distribution, each parameter has been scanned successively and set to the position of the local minimum. This has been repeated until the improvements per iteration are sufficiently small. The optimum has been found after five iterations. It can be seen that the optimization of duty cycles and measurement times provides an improvement of  ∼ 30% compared to the uniform distribution with η = 0.5. The obtained result of σ_stat(m²_νe) = 0.021 eV²/c⁴ is nearly identical with the standard KATRIN value of σ_stat(m²_νe) = 0.020 eV²/c⁴ for the case of 20 eV difference between endpoint and lowest retarding energy. It remains open if a more detailed parameter optimization is able to increase the sensitivity.

Table 2. Average statistical uncertainty $\langle \sigma _{\mathrm {stat}}(m_{\nu _{\mathrm { e}}}^2)\rangle = \langle \frac {1}{2} (|{\Delta m_{\nu _{\mathrm { e}}}^2}_-| + | {\Delta m_{\nu _{\mathrm { e}}}^2}_+|)\rangle$ (arithmetic mean of positive and negative error of ten simulations and fits), average fit parameters $\langle \bar E_0\rangle$ and $\langle \bar m_{\nu _{\mathrm { e}}}^2\rangle$, as well as Pearson's correlation coefficient R(E₀,m²_νe) for uniform and optimized distribution of a gated filter setup. Assumed are 3 years measurement time with m²_νe = 0 eV²/c⁴, E₀ = 18 575.0 eV and qU =  18 555 eV as lowest retarding energy. The pulse period t_r was held constant at 40 μs.

(Duty cycle ts/tr, measurement time fraction) at qU=						σstat(m2νe)	$\langle \bar E_0\rangle$	$\langle \bar m_{\nu _{\mathrm { e}}}^2\rangle$	R(E0,m2νe)
18 555 V	18 560 V	18 565 V	18 567.5 V	18 570 V	18 575 V	(eV2/c4)	(eV2/c4)	(eV)	(eV2/c4)
(0.5, $\frac {1}{6}$)	(0.5, $\frac {1}{6}$)	(0.5, $\frac {1}{6}$)	(0.5, $\frac {1}{6}$)	(0.5, $\frac {1}{6}$)	(0.5, $\frac {1}{6}$)	0.025	18 574.9983	−0.0056	0.9230
(0.4, $\frac {1}{13}$)	(0.6, $\frac {1}{13}$)	(0.6, $\frac {2}{13}$)	(0.4, $\frac {1}{13}$)	(1.0, $\frac {4}{13}$)	(1.0, $\frac {4}{13}$)	0.021	18 575.0006	0.0049	0.8914