Improved determination of the $\beta \text{−}{\overline{\nu }}_e$ angular correlation coefficient $a$ in free neutron decay with the $a\mathrm{SPECT}$ spectrometer

We report on a precise measurement of the electron-antineutrino angular correlation ($a$ coefficient) in free neutron beta decay from the $a\mathrm{SPECT}$ experiment. The $a$ coefficient is inferred from the recoil energy spectrum of the protons which are detected in $4\pi$ by the $a\mathrm{SPECT}$ spectrometer using magnetic adiabatic collimation with an electrostatic filter. Data are presented from a 100-day run at the Institut Laue Langevin in 2013. The sources of systematic errors are considered and included in the final result. We obtain $a=-0.10430\left(84\right)$ which is the most precise measurement of the neutron $a$ coefficient to date. From this, the ratio of axial vector to vector coupling constants is derived giving $\left|\lambda \right|=1.2677\left(28\right)$.

Figure 1

(a) Expected proton recoil spectrum for $a=-0.103$ (blue solid line) which we use in the following as reference value ($a_{\text{ref}}$) and for an extreme value of $a=+0.3$ (red dashed line). The decay proton has its maximum energy at $T_{\text{max}}=751\mathrm{eV}$. Also shown are the respective transmission functions $F_{\text{tr}}$ for the retardation potentials $U_{\text{AP}}=50\mathrm{V}$ and $U_{\text{AP}}=400\mathrm{V}$ and $r_{\text{B}}=0.203$ (magenta lines). (b) The corresponding (normalized) integral proton spectra. Normalization means [see Eqs. (8)–(10)] that the area under the respective curve is 1, i.e., does not depend on $a$. The derivative $dy/da$ (black solid curve) expresses the sensitivity of the yield $(y_{\text{theo(n)}}/N_0)$ to changes in $a$ which (in absolute numbers) is maximal at $U_{\text{AP}}$ voltage settings of $\approx 50$ V and $\approx 400$ V (top horizontal axis). We do not use a lower voltage setting ($U_{\text{AP}}<50$ V), as it is potentially sensitive to background from the ionization of residual gas.

Figure 2

Schematic of $a\mathrm{SPECT}$. Only the most important electrodes are shown. The magnetic field is oriented in vertical direction (blue lines). The whole setup is under ultrahigh vacuum conditions.

Figure 3

The $a\mathrm{SPECT}$ setup at the cold neutron beam line of PF1b at the ILL in 2013. Clearly visible are (1) the detector electronics on top, (2) the vertically aligned $a\mathrm{SPECT}$ cryostat system, (3) the massive magnetic field return yoke for the magnetic shielding, and (4) the beam dump in front.

Figure 4

Fields inside $a\mathrm{SPECT}$ along the vertical direction ($z$ axis). The magnetic field is shown in blue (dotted curve), the electric potential in red. The position of the electrodes as mentioned in the text and listed in Table 1 is indicated by horizontal bars (in blue). The most important Penning-like traps for positively ($+$) and negatively ($-$) charged particles inside $a\mathrm{SPECT}$ are indicated as ellipses.

Figure 5

Photograph of the DV electrode (left) and the main, octagonally shaped AP electrode (right) as used during the 2013 beam time. The sketch shows the collimated neutron beam (in gray) as it passes through the DV electrode. The dark area indicates the projection (to $z=0$) of the fiducial decay volume in which the protons are magnetically focused along the flux tube onto the two pads (2, 3) of the SDD. The side ports of the DV electrode are used for pumping and lateral access, e.g., beam profile measurements.

Figure 6

Measured neutron beam profiles (projection onto the horizontal $y$ axis) for standard (blue) and reduced (red) beam size. The latter was used to investigate systematic effects. The shaded area indicates the magnetic projection of the central pad (2) of the SDD onto the $y$ axis in the DV [cf. Eq. (5)].

Figure 7

The SDD with its three detector pads is mounted inside the detector cup electrode featuring a wall thickness of 3 cm, which is at the reacceleration potential of $-15\mathrm{kV}$. The alignment with respect to the $a\mathrm{SPECT}$ coordinate system was determined by dedicated measurements [38] and is shown in the left plot. Data from detector pad 1 could not be used for the analysis of $a$ (for details see text).

Figure 8

Pulse height spectrum of protons and electrons from neutron decay (in red) measured at ILL in 2013 (config 1, cf. Table 2). The proton peak is well separated from the noise. A background measurement at $U_{\text{AP}}=780\mathrm{V}$ is shown in blue. The small peak visible in the 780-V spectrum is caused by ionized rest gas and reduced in later configurations; see Sec. 4e. The two vertical lines denote the chosen lower (ADC channel: 29) and upper (ADC channel: 120) integration limits for the proton region. Demonstration of the logarithmic amplification of the SDD electronic (black solid curve) using characteristic x rays of energy $E$ (black circles). The right axis indicates the x-ray energy.

Figure 9

Measurement sequence of the 2013 beam time which repeats after 300 min until data taking is stopped. Each bar in the diagram corresponds to a measurement cycle of $\approx 5$ min duration for the respective value of $U_{\text{AP}}$ and has a time structure as shown in Fig. 16 of Sec. 4e.

Figure 10

Temporal sequence of 50 V runs for config 1. Plotted are the average count rates of the central detector pad for the individual measurement runs of 200 s duration. The distribution of data points around their common mean (solid line) corresponds to the drawn error bars resulting from pure counting statistics. A constant fit to the data gives ${\chi }^2/\nu =1.07$.

Figure 11

NMR measurements of the magnetic field on axis around the center position of the DV and the AP electrodes. The uncertainties in the field measurement correspond to the symbol size and mainly reflect the measurement reproducibility. The positioning error of the NMR probe was estimated to be $\pm 1$ mm. The solid lines are the results from KEMField field simulations based on the known $a\mathrm{SPECT}$ coil configuration as well as the current settings used in the 2013 beam time. (a) The $B$ field inside the DV exhibits a small axial gradient of $\approx 2\times {10}^{-4}$ T/cm to ensure that no decay protons get trapped by the magnetic mirror effect between the DV and EM. Inset: Sketch of NMR probe used to measure the fields. (b) In the AP region the $B$ field has a tiny, local maximum for optimal transmission condition [2, 34]. Deviations of the NMR field measurements from the KEMField simulations are mainly caused by the positioning error of the NMR probe. A total offset error common to all $\langle r_{\text{B}}\rangle$ values ($<2.4\times {10}^{-5}$, relative) takes this matter into account (see text).

Figure 12

Simulation of the potential distributions along the $z$ axis in the DV region. The blue curve (left axis) shows the potential for configurations 1–6 based on the measured work functions of the DV electrode and surrounding materials. The red curve (right axis) shows the resulting potential for config 7 where the electrodes below and above the DV were set to $+4$ V and $-4$ V, respectively. Inset $I\left(z\right)$: Measured beam profile along the $z$ axis (intensity in arbitrary units).

Figure 13

Simulated potential distribution along the $z$ axis in the AP region with the retardation voltage set to 400 V. A symmetric distribution around $z\approx 131$ cm (solid curve) is the result if only field leakages are considered. The inclusion of the electrodes' WF leads to an asymmetric shape with somewhat lower potential values and a slightly shifted position of the maximum (dashed curve).

Figure 14

Deviation $\mathrm{\Delta }{U}_{\text{AP}}=\langle U_{\text{A}}\rangle -U_{\text{AP}}$ extracted from particle tracking simulations for $U_{\text{AP}}=50$, 150, 400, and 600 V. We find that $\mathrm{\Delta }{U}_{\text{AP}}$ depends linearly on $U_{\text{AP}}$. However, there are differences in slope and intercept for the respective detector pad 2/3 (upper/lower pair of curves) and symmetric/asymmetric settings of the E15 electrode. The error bars are dominated by the statistics of the particle tracking simulation. Further drawn are the global fit results of $g_{\text{sys}}^{\langle U_{\text{A}}\rangle }$ [cf. Eqs. (12) and (13)] for the overall dataset.

Figure 15

Evolution of the background count rate in the proton region as a function of the time after opening the fast neutron shutter for $U_{\text{AP}}=780$ V. The red solid curve is a fit to the data: $y=p_0+p_1\left[1-exp(-t/{\tau }_s)\right]$. The constant part of the background, $p_0$, is calculated to be ($4.67\pm 0.06$) count/s; the nonconstant part shows an exponential saturation behavior with ${\tau }_s=(51\pm 10)$ s being the characteristic time constant and $p_1=(0.66\pm 0.06)$ count/s the resulting count rate after saturation is reached. These background investigations were carried out in the commissioning phase before the runs config 1–config 7 used in the analysis. During commissioning, the somewhat higher residual gas pressure produced a higher nonconstant background ($\approx$ factor of 2) as compared to config 1 (cf. Fig. 17).

Figure 16

Temporal sequence of a measurement cycle showing the different background contributions (not to scale). The $U_{\text{AP}}$-dependent background (${\text{bg}}_{U_{\text{AP}}}$) can be extracted from the counting rate difference of two measurement intervals ($\mathrm{\Delta }{t}_{\text{int}}$) displaced in time, one before shutter opening ($t_{\text{beam,}\text{on}}$), the other immediately after closing the shutter ($t_{\text{beam,}\text{off}}$), i.e., the regions enclosed by vertical dashed lines. For $t\ge t_{\text{beam,}\text{off}}$, the trap empties again with the time constant ${\tau }_{\text{s}}$ allowing us to monitor the yield of the rest gas ions (${\text{bg}}_{U_{\text{AP}}}$).

Figure 17

Pulse height spectra measured at $U_{\text{AP}}=780$ V within the proton integration window for config 1. Red squares: Spectrum of ${\text{bg}}_{\text{780}\text{V}}^{II}$ extracted from the difference measurement (cf. Fig. 16) with $\langle {\text{bg}}_{\text{780}\text{V}}^{II}\rangle \approx 0.34$ count/s. Black squares: Spectrum measured with beam on (shutter opened) with instrumental/environmental background (green stars) already subtracted. Subtraction of ${\text{bg}}_{\text{780}\text{V}}^I={\text{bg}}_{\text{780}\text{V}}^{II}R$ according to the background model yields the blue data points (essentially electrons from neutron decay) with an integral count rate of $\approx 5.75$ count/s.

Figure 18

Measured retardation voltage-dependent background in the proton region with pad 2. Shown is the count rate in the measurement interval immediately after $t_{\text{beam,}\text{off}}$ (cf. Fig. 16) in dependence of the applied retardation voltage $U_{\text{AP}}$ for config 1 (a) and config 3 (b). A clear increase of the count rate is the finding for config 1, whereas a voltage dependency is no longer observed (dashed horizontal line to guide the eyes) for config 3 (as well as for the subsequent configuration runs). The constant instrumental/environmental background contributes with $\langle {\text{bg}}_{\text{env}}\rangle \approx 0.14$ count/s. Further drawn is the fit result of $g_{\text{sys}}^{\text{bg}}$ for the global fit to the config 1 dataset.

Figure 19

(a) Radial spread $r\le 2r_{\text{g}}^{\text{max}}$ of decay protons at the height of the detector plane emitted from a point source in the DV. The relative count rate losses due to the edge effect are illustrated for two radial probability distributions of gyrating protons at mean distance $\mathrm{\Delta }y<2r_{\text{g}}^{\text{max}}$ left and right from the detector edge in case of $|dI/dy|>0$. (b) Ratio ${\left(y_{\text{sys}}^{\text{ee(re)}}/y_{\text{sys}}^{\text{ee(st)}}\right)}_{U_{\text{AP}}}$ of relative count rate losses for the reduced and standard beam profile from particle tracking simulations. Within the error bars, no dependence on the chosen retardation voltage settings (7 in total) can be observed (black horizontal line and grey shaded area represent the mean and its standard error). This finding coincides with the simple expression $\langle {\varepsilon }_{\text{re}}\rangle /\langle {\varepsilon }_{\text{st}}\rangle$ from Eq. (23) which gives ($6.9\pm 1.4$) (mean and $\sigma$ error shown in blue).

Figure 20

Simulation of the retardation voltage dependence of the relative edge-effect losses $y_{\text{sys}}^{\text{ee(st)}}$ and $y_{\text{sys}}^{\text{ee(re)}}$ for the standard and reduced beam profile, respectively. Further drawn is the fit result of $g_{\text{sys}}^{\text{ee(re)}}\left(U_{\text{AP}}\right)$ and $g_{\text{sys}}^{\text{ee(rst)}}\left(U_{\text{AP}}\right)$ for the global fit to the overall data set.

Figure 21

Measured pulse height spectra (pad 2) at different acceleration voltages $U_{\text{acc}}$ and retardation voltages ($U_{\text{AP}}$). The data were taken with a linear shaper at the end of the 2013 beam time to avoid spectral distortions, which would otherwise make the combined fitting cumbersome. The corresponding histograms of the calculated pulse heights $\propto E_{\text{eff}\text{ion.}}$ are shown as continuous lines to improve readability. From the fit, the parameters of the charge-collection efficiency could be deduced: $S=0.5562\left(10\right)$, $L=38.01\left(27\right)$ nm, $c=3.171\left(23\right)$, and $\tau =84.39\left(37\right)$ nm. For $\mathrm{\Delta }z$ we obtained $\mathrm{\Delta }z=6.54\left(4\right)\mathrm{nm}$ resulting in $d_{\mathrm{eff}}=36.54\left(4\right)\mathrm{nm}$. From the comparison with pulse height data taken with the logarithmic shaper (see text), we determined the proper threshold (ADC channel: 37.6; dashed line) which was needed to calculate the below-threshold losses including the events with no energy deposition inside the detector.

Figure 22

Fraction of undetected protons of the integral proton spectrum whose corresponding pulse heights fall below the threshold of the DAQ system. The $U_{\text{AP}}$ dependence of the fractional losses for pad 2 and pad 3 can be described by Eq. (30). The solid lines are the fit functions $g_{\text{sys}}^{\text{blt}}\left(U_{\text{AP}}\right)$ from the global fit to the overall data set.

Figure 23

Example of two individual proton events within one event window. The signal as recorded by the ADC is shown in blue, the spline interpolation in red. The position of the two events is indicated by the vertical green lines. The trigger algorithm is based on the comparison of two windows (w1, w2) within the shift register the data from the ADC is continuously shifted through. If the mean values of those two windows differ by more than an externally set threshold, the trigger condition is fulfilled. Window w1 is used to determine the baseline (first 15 time bins of $0.75\mu \mathrm{s}$), whereas w2 is separated from w2 by $0.8\mu \mathrm{s}$. For the subsequent signal analysis, the baseline is subtracted in each case (cf. Fig. 8).

Figure 24

Quantitative determination of pileup events (blue area) within the event window. The counts per time bin (50 ns) of the separated events (crosses) which reach a plateau at $1\mu \mathrm{s}\le \mathrm{\Delta }{t}_{\text{s}}\le 2\mu \mathrm{s}$ are extrapolated to $\mathrm{\Delta }{t}_{\text{s}}\to 0$. The integral number of pileup events divided by the measurement time is then the rate of pileup events. The data shown are from config 1 (pad 2) at $U_{\text{AP}}=50$ V with the total measuring time of 38 600 s. The reduction of the distribution above 2 $\mu \mathrm{s}$ is caused by the finite length of the event window in which the maximum of the second pulse no longer falls.

Figure 25

Pileup rate $y_{\text{sys}}^{\text{pileup}}$ as function of the event rate $y_{\text{exp}}$ in the proton region. Since $y_{\text{exp}}$ depends, a.o., on the retardation voltage, the pileup results in a retardation voltage-dependent loss. Data taken from config 1, 3, 7, pad 2, and 3 are bundled in clusters for a given retardation voltage setting. Further drawn is the global fit result of $g_{\text{sys}}^{\text{pileup}}$ to the overall data set.

Figure 26

Relative loss $y_{\text{sys}}^{\text{tr}}$ due to trapped protons in the DV region as a function of the retardation voltage. The black data points are from particle tracking simulations whereas the red solid line is the result of an analytical calculation $\left(y_{\text{sys}}^{\text{tr,}\text{cal}}\right)$ under simplified assumptions (proton trajectories: on axis). Further drawn is the global fit result of $g_{\text{sys}}^{\text{tr}}$. Inset: Conditions for protons to be stored in the DV region. Shown is the color-coded yield (arb. units) for the parameter space ${\theta }_0$ and $T_0$, the proton emission angle ${\theta }_0={\theta }_{\text{em}}-{90}^{\circ }$, and its kinetic energy $T_0$ at the decay point.

Figure 27

Ideogram of $a$ values for each configuration fitted separately without any systematic correction. The blue data points are derived from configuration runs with standard parameter settings. Forced enhancement of the edge effect by a reduced beam profile leads to more negative $a$ values (green data points), whereas config 2b (EM off) shows the trend from systematic shifts to positive $a$ values. The total error from the fit is shown together with the uncorrelated error which reflects the count rate statistics and which essentially constitutes the total error. The error bars of the extracted $a$ values are scaled with $\sqrt{{\chi }^2/\nu }$ whenever the $p$ value is less than the conventionally accepted significance level of $\alpha =0.05$ (cf. Table 3) which is indicated by an asterisk (*). The degrees of freedom are $\nu =15$ for each configuration.

Figure 28

Global fit results on $a$ with systematic corrections included. Ideograms (black/red) of extracted $a$ values where independent fit parameters $a_c$ for the $\beta -{\overline{\nu }}_e$ angular correlation coefficient have been assigned to each configuration run ($c$) (Global-Config-$a_c$). The red one includes the outlier value of $a$ from config 2b (red data point), where $a\mathrm{SPECT}$ operated as a $2\pi$ spectrometer (EM off). The global fit (Global-$a$) to the overall dataset (except config 2b) with only one common fit parameter for $a$ yields $\langle a\rangle =-0.10430\left(84\right)$ for the production beam time in 2013 (black data point with vertical line and error band indicated by gray bar). The respective error bars are scaled with $\sqrt{{\chi }^2/\nu }$, i.e., the ${\chi }^2/\nu$ values listed on the right-hand side, since the associated $p$ values are in all cases less than the significance level (cf. Table 3) for $\nu >100$. For details, see text.

Figure 29

Linear relationship (correlation) coefficient of $a$ with various fit parameters (67 in total) as result of the global fit to the overall data set. Most of the parameters show correlation coefficients $<\left|0.2\right|$ which is weak and likely insignificant. Strong correlations, i.e., values that surpass $\left|0.8\right|$ are not found. Moderate correlations ($+0.5$) are found for $c_{\text{AP,}\text{offset}}^{\langle U_{\text{A}}\rangle }$ (data point highlighted in red) to describe the offset error common to all $\langle U_{\text{A}}\rangle$ values and for the parameters $c_{2,4}^{\text{ee(st)}}$ ($\approx \left|0.4\right|$; data points highlighted in blue) to correct for the edge effect (standard beam profile). A further breakdown according to detector pads ($p$) and configuration run ($c$) was not made in this correlation plot.

Figure 30

Published results on $\lambda$ derived from $\beta -{\overline{\nu }}_e$ angular correlation measurements (red data points: Grigoriev [91], Stratowa [27], Byrne [28], Darius [29], our work), $\beta$-asymmetry measurements (blue data points: Bopp [90], Yerozolimsky [89], Liaud [88], Abele [11, 83], Liu [84], Mund [8], Mendenhall [85], Brown [10], and Märkisch [9]), and other measurements (black data points: Mostovoi [87] and Schumann [86]).

Figure 31

WF scan of a flat electrode segment (AP-83) of size $48\times 108$ ${\mathrm{mm}}^2$. The false color map shows the ${\text{WF}}_{\text{rel}}$ distribution (meV) across the surface with ${\overline{\text{WF}}}_{\text{rel}}=92.76$ meV and rms fluctuation of 24.73 meV. Scan time: $\approx 1.5$ h. (b) Repeated ${\overline{\text{WF}}}_{\text{rel}}$ measurements for segment AP-83 using a time span overnight ($\approx 12$ h) for the system to stabilize. To a good approximation the ${\overline{\text{WF}}}_{\text{rel}}$ distribution can be described by a Gaussian function (solid curve) with mean $\langle {\overline{\text{WF}}}_{\text{rel}}\rangle =96.2$ meV, $\sigma =26.1$ meV, and $\delta \langle {\overline{\text{WF}}}_{\text{rel}}\rangle \approx 4.2$ meV (error on mean value determined on a statistical basis).

Figure 32

Distribution of ${\overline{\text{WF}}}_{\text{rel}}$ for each segment of the DV (red) and the AP electrode (blue). The bars indicated in dashed red are from the bottom part (BP) of the DV electrode which was re-machined and led to slightly different surface properties. Otherwise the ${\text{WF}}_{\text{rel}}$ averages of both electrodes overlap to quite some extent. Inset: Distribution of the rms fluctuation of ${\text{WF}}_{\text{rel}}$ across each electrode. All electrodes show a remarkably similar behavior of their WF fluctuations.

Figure 33

Distribution of the WF differences ${\overline{\text{WF}}}_{i,\text{diff}}$ used to extract temporal WF changes within a time span of about one year.

Figure 34

Time sequence of line scans showing the extracted means of the relative WF during evacuation of a UHV Kelvin probe. The two electrode samples were measured alternately.

Figure 35

$U_{\text{AP}}$ dependence of the integral count rate (weighted average over all configuration runs) of low-energetic $\beta \text{−}\text{decay}$ electrons in the background window between ADC channels 150 and 170. The straight line fit to the data (red curve) gives $m_{\beta }=(0.32\pm 1.81)\times {10}^{-5}$ count/s ${\mathrm{V}}^{-1}$ for the slope and $b=(4.078\pm 0.008)$ count/s for the intercept. Both numbers are used to extract a possible voltage dependence of the $\beta \text{-electron}$ rate in the proton region [cf. Eq. (B1)].

Figure 36

Spin-independent ${\omega }_{\text{p}}^{*}(T,a)$ and the spin-dependent component ${\omega }_{\text{ps}}^{*}(T,a)$ of the differential proton recoil spectrum $W^{*}(T,a,c)$ from Eq. (C1) for $a_{\text{ref}}=-0.103$ and for an extreme value of $a=+0.3$. Note that ${\omega }_{\text{ps}}(T,a)$ is negative, but for the yield we have $W^{*}(T,a,c)>0$ for all $cos\vartheta$ values since $|{\omega }_{\text{ps}}^{*}(T,a)|<|{\omega }_{\text{p}}^{*}(T,a)|$. The yield is given in (a.u.). To get prefactors for the absolute numbers, see [99].

Figure 37

PDF of $a$ determined by maximum likelihood profile (blue points), also called projection method, and by a histogram (yellow) of a MCMC data set which is also called marginalization method. Normal distribution using our final fit result for $a$ (based on the scaled error; see Fig. 28).