Electron time-of-flight: A new tool in $\beta$-decay spectroscopy

We show that electron time-of-flight can be a useful tool in $\beta$-decay spectroscopy, even in the energy range up to 700 keV relevant for neutron $\beta$ decay. As a first application, we use electron time-of-flight measurements for the precise calibration of an electron detector.

Figure 1

Magnetic field profile $B\left(z\right)$ of our inverted magnetic mirror, with peak field $B_0=3\mathrm{T}$. The start detector and the source are at position $z_1$ where $B_1=2.5\mathrm{T}$, the stop detector at $z_2$ where $B_2=22\mathrm{mT}$.

Figure 2

Electron path length $\ell$ as a function of emission angle $\theta$. Dotted curve: for a uniform guiding field, the length of the electron trajectories $z_0/cos\theta$ reaches 14 m for $\theta ={70}^{\circ }$ and diverges for $\theta ={90}^{\circ }$. Dashed line: with no guiding field, for direct flight to a distant detector, all flight paths have the same length $\ell =z_0=4.452\mathrm{m}$. Solid line: for the inverse bottle field of Fig. 1, all path lengths from Eq. (8) are nearly the same, $\ell \approx z_0$, even for electron emission at $\theta ={90}^{\circ }$.

Figure 3

The size of the adiabatic parameter, Eq. (9), along the electron flight path within the superconducting magnet, Fig. 4, for $E=500\mathrm{keV}$.

Figure 4

Setup of the ToF apparatus. The ${}^{90}\mathrm{Sr}-{}^{90}\mathrm{Y}$ electron source is located inside the start detector on the left (small spot) at high field $B_1=2.5\mathrm{T}$. The stop detector is placed on the right at low field $B_2=0.022\mathrm{T}$. A light guide couples each detector to its photomultiplier (PM). In gray, the iron yoke of the magnet.

Figure 5

Close-up of the start scintillator (shown on the left of Fig. 4).

Figure 6

Simulated initial angular distribution at the start detector, for electrons that reach the stop detector. The line is drawn to guide the eye.

Figure 7

(a) Scatter plot derived from the simultaneous measurement of electron ToF $T$ and channel number $ch$ in the energy sensitive stop detector. The thin vertical line indicates the lower time limit $T_0=z_0/c$. (b) As (a), but with the abscissa changed to electron energy $E$, derived from $T$ via Eq. (12). (c) As (b), but with the ordinate changed to gain $g=ch/E$. For details on the dashed one-sigma error contours, see text.

Figure 8

Counts per bin as a function of gain $g$, obtained from vertical cuts in Fig. 7, for selected electron energies $E$. Solid curves: fits to the data. Dashed vertical lines: mean values of the distribution. Solid vertical lines: saturation gain $g_0=2.156\mathrm{ch}/\mathrm{MeV}$. At low energies, shown in the panels of the top row, the mean values steadily move into the direction of $g_0$. At high energies, shown in the panels of the bottom row, the distributions remain centered at $g_0$. Bin widths are $\mathrm{\Delta }E=25\mathrm{keV}$ and $\mathrm{\Delta }g=0.1\mathrm{ch}/\mathrm{keV}$.

Figure 9

The timing offset $\delta T$ and the saturation gain $g_0$ are fixed by the requirement that the normalized gain saturates to $g\left(E\right)=g_0$ for the higher electron energies $E$. Shown are the 1, 2, and 3 σ χ-square contours for deviations of the gain function $g\left(E\right)$ from a constant $g_0$ for the higher electron energies. The $\chi$-square minimum, ${\chi }^2=15.2$ with 9 dof, lies at $\delta T=-92\mathrm{ps}$ and $g_0=2.156\mathrm{ch}/\mathrm{keV}$.

Figure 10

The gain function $g\left(E\right)$ of the plastic scintillator, as derived from Fig. 8. Below 350 keV, a strong nonlinearity appears, with $g$ significantly deviating from its saturation value $g_0$. The solid curve is an empirical fit to Eq. (15). The dotted curve shows the expectation from theory, Eq. (16).

Figure 11

Result of a conventional energy calibration of a plastic scintillator with a set of conversion electron sources. The two entries at the lowest energies are from the Auger electrons accompanying the electron conversion process. Adapted from Ref. [34].