The half-life of 212Po

P. Bellia,b, R. Bernabeia,b,111Corresponding author. e-mail address:
<EMAIL_ADDRESS>(R. Bernabei)., R.S. Boikoc,d, F. Cappellae,f, V.
Caraccioloa,b,g, R. Cerullia,b, F.A. Danevichc, A. Incicchittie,f, D.V.
Kasperovychc, V.V. Kobychevc, O.G. Polischukc, N.V. Sokurc, V.I. Tretyakc

aINFN sezione Roma “Tor Vergata”, I-00133 Rome, Italy

bDipartimento di Fisica, Universit$\grave{a}$ di Roma “Tor Vergata”, I-00133
Rome, Italy

cInstitute for Nuclear Research of NASU, 03028 Kyiv, Ukraine

dNational University of Life and Environmental Sciences of Ukraine, 03041
Kyiv, Ukraine

eINFN sezione Roma, I-00185 Rome, Italy

fDipartimento di Fisica, Universit$\grave{a}$ di Roma “La Sapienza”, I-00185
Rome, Italy

gINFN, Laboratori Nazionali del Gran Sasso, I-67100 Assergi (AQ), Italy

###### Abstract

The half-life of 212Po was measured with the highest up-to-date accuracy as
$T_{1/2}=295.1(4)$ ns by using thorium-loaded liquid scintillator.

Keywords: 212Po, Half-life, $\alpha$ decay, Loaded liquid scintillator

##  1 Introduction

The 212Po nuclide is the $\alpha$ active daughter of the 212Bi from the 232Th
decay chain with the shortest decay time among the naturally occurring
radioactive nuclides. The decay scheme of the
212Bi$\rightarrow$212Po$\rightarrow$208Pb (BiPo) and 212Bi$\rightarrow$208Tl
sub-chains is shown in Fig. 1. The current recommended value of the 212Po
half-life is $T_{1/2}=294.3(8)$ ns [1]. The history of the 212Po half-life
measurements is presented in Table 1.

Figure 1: The decay chain 212Bi$\rightarrow$212Po$\rightarrow$208Pb [1]. The $\alpha$ transition of 212Bi to 208Tl is shown too [2]. The half-life of 212Po is from the present study. Energies of $\beta$ and $\alpha$ decays are given in keV. Table 1: Historical view of the measured half-lives of 212Po. Reference | Experimental technique / Compilation | Half-life, ns
---|---|---
(year) |  |
[3] (1939) | Geiger-Müller counters, external source | $300(100)$
[4] (1943) | Geiger-Müller counters, external source | $260(40)$
[5] (1948) | Geiger-Müller counters, external source | $300(15)$
[6] (1949) | Geiger-Müller and proportional counters, |
| external source | $304(4)$
[7] (1949) | Geiger-Müller counters, external source | $220(10)$
[8] (1953) | Geiger-Müller and proportional counters, |
| external source | $290(10)$
[9] (1962) | Plastic scintillators, external source | $305(25)$
[10] (1963) | CsI(Tl) and plastic scintillators, external source | $305(5)$
[11] (1972) | Source in liquid scintillator |  $302(6)$1
[12] (1975) | Plastic scintillator, surface barrier Au-Si detector, |
| external source | $296(2)$
[13] (1981) | Surface barrier Au-Si and HP-planar Ge detectors, |
| external source | $309(11)$
[14] (2013) | Source in liquid scintillator | $294.7(10)$
[15] (2014) |  Source in BaF2 scintillator | $298.8(16)$
[16] (2017) | Source in liquid/gas Xe time projection chamber | $293.9(12)$
[17] (2018) | Source in liquid scintillator |
| (1st stage of the current experiment) | $294.8(19)$
[1] (2020) | Nuclear Data Sheets compilation | $294.3(8)$
Present study | Source in liquid scintillator | $295.1(4)$
1 Average of the two values $300(8)$ ns and $304(8)$ ns reported in Ref. [11].

Gaseous counters have been used in the early experiments [3, 4, 5, 6, 7, 8] to
detect the $\beta$ particle emitted in the 212Bi decay and the subsequent
$\alpha$ particle of 212Po. Combinations of scintillation and semiconductor
detectors were utilized in the experiments [9, 10, 12, 13], with $\beta$ and
$\alpha$ particles from an external source (where the BiPo sequence of decays
occurred) registered by the detectors.

A different approach has been exploited by using a liquid scintillator
enriched with a source containing the BiPo chain [11]. A similar approach was
exploited by the Borexino collaboration: quartz vials with thorium and 220Rn-
loaded liquid scintillators were inserted into the Borexino Counting Test
Facility (CTF) detector [14]. The half-life of 212Po was also measured in the
experiment [15] with the help of a BaF2 crystal scintillator contaminated by
radium. However, the accuracy of the experiment was limited by the
comparatively slow scintillation response of the BaF2 scintillator (effective
scintillation decay time $\tau_{eff}\approx 0.6$ $\mu$s) and a poor signal-to-
noise ratio due to a rather modest light yield of the scintillator ($\simeq
10^{3}$ photons/MeVγ) [18]. In the XENON100 detector (a xenon liquid/gas time
projection chamber) a 220Rn source was used to calibrate it; using such
calibration data, a further measurement of the 212Po half-life was obtained
[16].

It should also be noted that the half-life of 212Po was measured in Refs. [19,
20, 21, 22, 23, 24, 25] as a by-product of the BiPo sub-chain analysis for
different purposes (estimation of low-counting detectors’ internal
contamination by thorium, calibration of detectors, study of low-lying states
in $\alpha$-decaying nuclei). Typically the results are affected by large
statistical uncertainties up to 17%222It should be noted, however, a rather
small uncertainty of the half-life value obtained in work [21]:
$T_{1/2}=297(1)$ ns., and the systematic effects were not evaluated. Much
larger statistics was gathered in recent measurement [26] where an external Th
source was placed between plastic scintillators viewed by a photomultiplier
(PMT) obtaining $T_{1/2}=294.09(7)$ ns. Unfortunately, systematic effects in
the experiment were not estimated. In Ref. [27] a radon gaseous source was
used to calibrate the xenon liquid/gas time-projection chamber of the PandaX-
II experiment obtaining $T_{1/2}=297(6)$ ns with quite big uncertainty
comparing with those in the last experiments in Table 1.

Considering the data in Table 1, the liquid scintillator loaded with Th
radionuclides appears a very promising way to reduce the 212Po half-life
uncertainty. However, although the measurements performed by the Borexino
collaboration have profited of a fast scintillation signal, the measurement’s
precision was limited by the following aspects: the time jitter of the 100
photomultipliers array, scattering of the scintillation photons, their
absorption and re-emission inside the CTF volume, and the readout electronics
bandwidth. Moreover, a substantial difference ($\sim 40$ cm) in the photon
paths to reach the PMTs has induced an additional time-spread distribution
(average value of 2 ns) due to the large volume of the CTF detector (4.8 m3
liquid scintillator).

According to all these observations, in order to improve the accuracy on the
212Po half-life measurements, one should minimize the liquid scintillator
volume and use a fast response PMT/electronics. A thorium-loaded liquid
scintillator (LS(Th)) was developed for the present study. Preliminary results
of the measurements were published in [17]. Here we report the final result of
the experiment with $\approx 64$ times larger statistics.

##  2 Experiment

###  2.1 Thorium-loaded liquid scintillator

####  2.1.1 Production of thorium-loaded liquid scintillator

Thorium nitrate pentahydrate Th(NO${}_{3})_{4}\cdot 5$H2O was used as initial
Th compound to prepare the LS(Th). A 20% solution of trioctylphosphine oxide
(TOPO) in toluene was taken as complexing organophosphorous agent to bind
thorium in organic phase. The mixture was stirred with thorium nitrate
pentahydrate salt to obtain a solution containing 2 mg of Th in 1 mL of TOPO
solution:

Th${}^{4+}~{}+~{}4$NO${}^{-}_{3}$ \+
3$\overline{\mathrm{TOPO}}\rightleftharpoons\overline{\mathrm{Th(NO}_{3})_{4}(\mathrm{TOPO})_{3}}$.

The obtained Th-containing organic solution was diluted 20-fold with the
liquid scintillator based on toluene with the addition of 0.1% 2.5-diphenyl
oxazole (PPO) and 0.01% 1,4-bis(5-phenyloxazol-2-yl) benzene (POPOP). Taking
into account the preparation procedure, the liquid scintillator contains
$\approx 0.1$ wt% of thorium (232Th and 228Th with daughters), while the
activity of 228Ra is expected to be rather low. Moreover, while 228Ra has too
small energy of $\beta$ decay ($Q_{\beta}=45.5(6)$ keV [28]) well below the
hardware threshold (see Sec. 3.1), presence of some amount of 228Ac ($\beta$
active daughter of 228Ra with the decay energy $Q_{\beta}=2123.8(26)$ keV
[28]) leads to a mild contribution to the random-pairs background. Thus
presence of some amount of 228Ra in the scintillator almost does not affect
the half-life of 212Po determination.

####  2.1.2 Scintillation properties and activity of 232Th and its daughters
in the LS(Th)

A 7-mL sample of the LS(Th) was sealed inside a quartz vial with inner sizes
$(\oslash 33\times 14)$ mm to test its scintillation properties and the
activities of 232Th and 228Th with daughters. The vial was optically connected
to a PMT Philips XP2412 and covered by 3M reflector foil to improve the
scintillation-light collection. The signals from the PMT after a
preamplification stage entered a shaping amplifier with 0.5 $\mu$s shaping
time and then were processed by a peak sensitive analog-to-digital converter.

The relative light yield of the scintillator was estimated by using 137Cs and
207Bi $\gamma$ sources to be 42(3)% in comparison to a polystyrene based
plastic scintillator $\oslash 30\times 15$ mm measured in the same conditions.

The energy spectrum measured with the LS(Th) is presented in Fig. 2. The
energy scale of the detector was determined by analysis of the Compton
electron spectra of the 137Cs and 207Bi $\gamma$ sources as suggested in [29].
The peaks, in the (0.25–0.7) MeV energy range (see Fig. 2), can be attributed
to $\alpha$ decays of 232Th and 228Th with daughters. The broad peak at $\sim
1.1$ MeV is due to the $\alpha$ decays of 212Po when the energy of the
previously emitted $\beta$ particle is low. The overlap of $\beta$ events of
212Bi and $\alpha$ events of 212Po produces the broad energy distribution up
to $\sim 3.3$ MeV because the used acquisition did not separate these two
events occurred in too short time in comparison to the amplifier shaping time.

Figure 2: Energy spectrum taken with 7 mL LS(Th) scintillator over 5910 s. The
main features of the spectrum are shown. (Inset) Fit of the energy spectrum
accumulated with the LS(Th) over 2050 s by a model which includes $\alpha$
peaks of 232Th and 228Th with daughters.

To estimate the activity of 232Th and 228Th with daughters, the energy
spectrum accumulated with the LS(Th) has been fitted by a model built from the
$\alpha$ peaks of 232Th, 228Th, 224Ra, 220Rn, 216Po and 212Bi. The activities
of the radionuclides, the energy resolution and the $\alpha/\gamma$
ratio333Defined as position of $\alpha$ peak in the energy scale measured with
$\gamma$ quanta. Here we neglect the contribution of nuclear recoils because
of a much higher quenching [30]. were free parameters of the fit. A linear
function has been used to describe the distribution of $\beta$ particles and
$\gamma$-ray quanta. Furthermore, taking into account the radon escape
possibility from the LS(Th), we introduce a coefficient to consider a lower
activity of 220Rn and its daughters. The result of the fit in the energy
interval $165-785$ keV ($\chi^{2}$/n.d.f. = 1.45, where n.d.f. is the number
of degrees of freedom) is presented in the Inset of Fig. 2. The obtained
behavior of the $\alpha/\gamma$ ratio in the energy range (4–9) MeV is
described by the following formula:
$\alpha/\gamma=0.02149(14)+0.01104(3)\times E_{\alpha}$ (where $E_{\alpha}$ is
energy of the $\alpha$ particles in MeV). The activities of 232Th and 228Th
have been measured as 4.61(2) Bq/mL and 3.82(7) Bq/mL, respectively. The lower
activity of 228Th is due to its decay after the thorium compound preparation
in January 2016 (the measurements were performed on July 8th, 2016). The
activity of the 220Rn and its daughters is $92(2)\%$ of the 228Th activity;
this can be explained by radon escape from the LS(Th). The properties of the
LS(Th) are summarized in Table 2.

Table 2: Properties of the thorium-loaded liquid scintillator. Property | Value | Note
---|---|---
Light yield | 42(3)% | Relatively to polystyrene based
|  | plastic scintillator
Activity of 232Th | 4.61(2) Bq/mL |
Activity of 228Th | 3.82(7) Bq/mL | Reference date July 8th, 2016
Total $\alpha$ activity | 20.7(10) Bq/mL | Reference date July 8th, 2016
Concentration of thorium | 0.113(1) wt% |

###  2.2 Recording of BiPo waveforms

A sample of the LS(Th) in the same quartz vial was viewed by a fast-time-
response PMT Hamamatsu R13089-100-11 with rise time 2.0 ns, transit time 20 ns
and transit time spread 170 ps (full width at half maximum, FWHM). The quartz
vial was covered by polytetrafluoroethylene tape to improve the light
collection. The signals waveforms were recorded by a LeCroy WavePro 735Zi-A
oscilloscope with a sampling frequency 20 GSa/s and a 3.5 GHz bandwidth.

The experiment was performed in two stages. In the first stage, carried out in
June 2017, a 4.4 g (5 mL) sample of the liquid scintillator was utilized (the
results of the first stage are published in [17]). A 10.6 g (12 mL) sample of
the LS(Th) was used in the second stage of the experiment in April-May 2018\.
In total 785548 events were recorded in the first stage, while 50340611 events
(larger by a factor of $\approx 64$) were recorded in the second stage of the
experiment during 216.67 hours in order to improve the half-life value
precision. The oscilloscope energy threshold in the second stage was set above
the 232Th $\alpha$ peak in order to reduce the counting rate.

##  3 Data analysis and results

###  3.1 Time intervals between $\beta$ and $\alpha$ pulses in BiPo events

An example of a pair event in the LS(Th) classified as a BiPo event is shown
in Fig. 3. After-pulses both for $\beta$ and $\alpha$ pulses of approximately
$12-14$ ns after the main pulses arise in the PMT because of the elastic
scattering of the accelerated photoelectrons on the 1st dynode. The scattered
electrons return to the photocathode and then are multiplied again [31].

Figure 3: Example of $\beta$ pulse of 212Bi and subsequent $\alpha$ pulse of
212Po in the liquid scintillator loaded by thorium. Dashed red line shows the
software threshold used to identify pairs of events in the data. The time
interval between the signals is denoted as $\Delta t$.

Amplitude spectra of the first and second events are shown in Fig. 4. The
spectra were built by calculating the signals area, after baseline
subtraction, in a time interval 11 ns (the time interval was chosen to avoid
the effect of after-pulses). A constant was taken to describe the baseline of
the first pulse, while a slow component of $\beta$ pulse (see Fig. 5 and
discussion of the scintillation signals pulse shapes below) was added to the
constant to reconstruct the second pulse shape. The detector energy scale was
determined by comparison of the first events distribution with the Monte Carlo
simulated $\beta$ spectrum of 212Bi. The response of the detector to the
$\beta$ decay of 212Bi was simulated using the Geant4 package version 10.4.p02
(Shielding PEN physics list) [32, 33, 34] with initial kinematics given by the
Decay0 event generator [35, 36]. The simulated-distribution shape reasonably
agrees with the experimental data (see Fig. 4). The energy resolution of the
LS(Th) detector was estimated for the $\alpha$ peak of 212Po as
FWHM$=15.5\%$444The difference in the 212Po $\alpha$-peak position in the
spectra presented in Figures 2 and 4 can be explained by different methods to
build the spectra: a shaping amplifier and a peak sensitive analog-to-digital
converter (the spectrum Fig. 2), and by using area of pulses for $<11$ ns in
the recorded waveforms (Fig. 4).. The obtained energy spectra confirm that the
recognized pairs of events are mainly BiPo events.

Figure 4: (Color online) Energy spectra of the 1st events ($\beta$ particles
of 212Bi, solid histogram) and 2nd events ($\alpha$ particles of 212Po, dots).
The Monte Carlo simulated response function of the detector to the $\beta$
decay of 212Bi is shown by dashed histogram. The energy interval of $\beta$
events used in the analysis of the 212Po half-life is shown by vertical dashed
lines labelled $E^{\beta}_{1}$ and $E^{\beta}_{2}$. The fit of the $\alpha$
peak by Gaussian function is shown by solid line. The $\alpha$ peak is divided
by a factor of 2 to fit the figure.

In addition to the after-pulses observed $12-14$ ns after the main pulses, a
rather long sequence of pulses is visible in a sum of a big number of signals
(see Fig. 5 where sums of about ten thousands $\beta$ pulses of 212Bi and
$\alpha$ pulses of 212Po are presented). Explanation of the multiple after-
pulses is problematic. However, we assume that this effect is somehow related
to the operation of the photomultiplier. While the after-pulses have no impact
on the determination of the 1st signal origin ($\beta$-particle of 212Bi),
they may affect the determination of the start time of the 2nd signal
($\alpha$-particle of 212Po), and thus of the time interval between the two
signals, especially for small $\Delta t<(30-60)$ ns.

Figure 5: Shape of scintillation pulses in the LS(Th) for $\beta$ particles of
212Bi (a) and $\alpha$ particles of 212Po (b). Fitting functions for $\beta$
and $\alpha$ pulses (solid lines), with four components of the scintillation
signals (dashed, dotted and dash-dot lines) are shown. Zoomed $\beta$ and
$\alpha$ pulses are shown in Insets. $\tau_{R}$ denotes the rise time of the
pulses (see text for explanation).

The pulse shapes of $\beta$ and $\alpha$ scintillation signals are slightly
different in the LS(Th). Fits of the pulses presented in Fig. 5 were done by
the function:

$f(t)=\sum A_{i}(e^{-t/\tau_{i}}-e^{-t/\tau_{0}})/(\tau_{i}-\tau_{0}),\qquad
t>0,$ (1)

where $A_{i}$ are the relative intensities, $\tau_{i}$ are the decay constants
for different light-emission components, and $\tau_{0}$ is the time constant
that describes the scintillation signal rise and integration of the
photomultiplier and electronics ($\tau_{0}\approx 1.6(2)$ ns both for $\beta$
and $\alpha$ pulses). Four decay components were observed with
$\tau_{1}\approx 1.8$ ns, $\tau_{2}\approx 7$ ns, $\tau_{3}\approx 0.05$
$\mu$s and $\tau_{4}\approx 0.4$ $\mu$s with different intensities for $\beta$
and $\alpha$ particles (see Table 3). It should be noted that the present
investigation cannot be considered as analysis of the LS(Th) pulse-shape, in
particular since the after-pulses were not excluded from the analysis. The
estimations of the pulse-shape decay profile were performed to take into
account the slow components of the $\beta$ pulses in the determination of the
time intervals between $\beta$ and $\alpha$ pulses in BiPo events.

Table 3: Decay time of the LS(Th) scintillator for $\beta$ and $\alpha$ particles. The decay constants and their relative intensities are denoted as $\tau_{i}$ and Ai, respectively. Type of events | Decay constants (ns) and relative intensities
---|---
|  $\tau_{1}$, A1 |  $\tau_{2}$, A2 |  $\tau_{3}$, A3 |  $\tau_{4}$, A4
$\beta$ particles | 1.8(4), | 6.7(5), | 47(3), | 353(28),
| $56(10)\%$ | $32(4)\%$ | $2.7(2)\%$ | $9.3(4)\%$
$\alpha$ particles | 1.8(4), | 7.7(6), | 51(3), | 409(134),
| $52(9)\%$ | $37(7)\%$ | $7.7(4)\%$ | $3.3(2)\%$

The following method was developed to find pairs of events in the data and
determine the time intervals $\Delta t$ between the pulses:

1) A simple low-level-discriminator algorithm with a high threshold (see Fig.
3) was applied to find pairs of events in the data.

2) The recognized pairs of events were then analyzed by using the method of
digital constant-fraction discrimination illustrated in Fig. 6. In the method
the pulse-origin time (denoted in Fig. 6 as “Zero crossing time”) was
determined by analyzing the sum of two pulses produced from the recorded
signal after the baseline subtraction: the first pulse was inverted and
shifted in time by 11 ns, and the second one integrated and multiplied by a
factor 0.003. The delay 11 ns was chosen to minimize effect of the undelayed
pulse fluctuations on the timing (however, the delay was taken small enough to
avoid possible after-pulses effect on the integrated pulse). The zero crossing
time was found by fitting the sum pulse by exponential function around zero
value. An example of the fit is shown in Fig. 6 too. The approach allows to
eliminate an amplitude dependence of the pulse-time origin, which appears in
the simple low-level-discriminator algorithm555We have checked how the
obtained half-life value (see below Section 3.2) depends on the constant-
fraction discrimination method parameters. The data production was performed
by using a constant-fraction discrimination method with the delay 1.8 ns and
the multiplication factor 0.4 for the undelayed pulse (without integration). A
fit of the obtained time distribution returns the half-life value
$T_{1/2}=295.09(26)$ ns in agreement with the value $T_{1/2}=295.10(26)$ ns
obtained with the method described above..

Figure 6: Illustration of the digital constant-fraction discrimination method
used to determine the signals starting time and to calculate $\Delta t$. (a) A
scintillation pulse (after the baseline subtraction) inverted and shifted by
11 ns (1); the pulse integrated and multiplied by a factor 0.003 (2). (b) Sum
of the pulses 1 and 2. The solid line shows the fit of the data by an
exponential function. The arrow shows the zero crossing time accepted as the
signal starting time.

The $\beta$ and $\alpha$ pulses have slightly different rise time ($\tau_{R}$,
defined as a time interval of the $(10-90)\%$ rising edge):
$\tau_{R}=2.44(16)$ ns for $\beta$ particles and $\tau_{R}=2.31(14)$ ns for
$\alpha$ particles (see Insets in Fig. 5)666Rise time of scintillation
detector depends on several factors: photodetector sensitivity and time
properties, the readout electronics bandwidth, scintillation material, size
and geometry of the scintillator and reflector, energy and ionization density
of particle (see, e.g., [37, 38, 39]). The difference in the rise time for
$\alpha$ and $\beta$ particles observed in the present study can be explained
by their different energy distributions and ionization densities.. The
$\tau_{R}$ values were calculated for $\beta$ and $\alpha$ signals with
amplitudes in the energy interval $700-1050$ keV (in the energy scale of
$\beta$ particles). This difference produces a systematic shift of $\Delta t$
that was estimated by using Monte Carlo simulations. The experimental data on
the rise and trailing time distributions, and the energy spectra for $\beta$
and $\alpha$ events were taken as input parameters for the simulations. Noise
was superimposed on the generated pulses. The noise was generated taking into
account the baseline fluctuations of the recorded waveforms. In total $10^{4}$
double pulses were generated with $\Delta t=100$ ns. Analysis of the generated
data by the constant-fraction-discrimination algorithm returned $\Delta
t=99.74$ ns. The difference 0.26 ns was added to the $\Delta t$ values for
each BiPo event in the data production process.

Taking into account the difference of scintillation pulse shapes for $\beta$
and $\alpha$ particles, the mean-time method was applied to analyze the pulse
profiles of the events. For each signal, the numerical characteristic of its
shape (mean time, $\zeta$) was defined by using the following equation:

$\zeta=\sum f(t_{k})\times t_{k}/\sum f(t_{k}),$ (2)

where the sum is taken over the time channels $k$, starting from the origin of
signal up to 60 ns; $f(t_{k})$ is the digitized amplitude (at the time
$t_{k}$) of a given signal.

Figure 7: Mean time (a) and rise time (b) vs energy distributions accumulated
over 16.7 h with the LS(Th) (see text for explanation of the parameters). The
$\pm 3\sigma$ intervals for the mean-time and rise-time values for $\beta$ and
$\alpha$ events selection are depicted by dashed lines (for $\beta$ events)
and by solid curves in form of ellipses (for $\alpha$ particles).

The mean time vs energy distributions for the 1st ($\beta$ particles) and 2nd
($\alpha$ particles) events selected from the data accumulated with the LS(Th)
for 16.7 h are presented in Fig. 7 (a). We have used $\pm 3\sigma$ regions
both for $\beta$ particles of 212Bi and for the $\alpha$ peak of 212Po to
select the BiPo events for the further analysis. The intervals to select
$\beta$ and $\alpha$ events are shown in Fig. 7. Despite a rather poor
particle discrimination ability, the filter reduces the contribution of random
pairs of events and discards irregular-shape pulses of different origin.

Figure 8: Examples of $\alpha$ pulses with the rise-time values beyond the
region of the accepted values for $\alpha$ particles ((a) and (b)). A pulse
with a normal rise time also shown (c).

The rise-time parameters for $\beta$ and $\alpha$ events were also analyzed
(see Fig. 7 (b)). All the pairs with $\beta$ or $\alpha$ pulses having the
rise time outside the $\pm 3\sigma$ regions were discarded from the further
analysis. Two examples of $\alpha$ pulses with abnormal rise-time values are
shown in Fig. 8 (a) and (b) together with a normal pulse (c). The pulses (a)
and (b) with irregular shape can be explained by overlap of scintillation
pulses or overlap of scintillation pulses with noise of different origin.

###  3.2 Half-life of 212Po

The distribution of the time intervals between the first and second signals in
the recognized pairs of events is presented in Fig. 9. The pairs of events
were selected with the first events amplitudes inside the energy interval
$500-1500$ keV and with the mean-time and rise-time values inside the $\pm
3\sigma$ bands shown in Fig. 7. The energy interval to select the first events
($\beta$ particles of 212Bi) for the analysis is also shown in Fig. 4 (the
choice of the energy interval for the 1st pulses selection will be explained
below). The second events ($\alpha$ particles of 212Po) were accepted inside
the ellipses shown in Fig. 7.

Figure 9:  (a) Distribution of the time intervals between the 1st and 2nd
signals in the selected pairs of events ($\Delta t$) and its fit by an
exponential function with the half-life $T_{1/2}=295.1$ ns plus background due
to random pairs of events. The distribution of random pairs of events is shown
too. (b) Residuals of the fit. The interval of fit is shown by vertical dashed
lines.

The time distribution was fitted by a sum of two exponential functions: the
first one describes the $\alpha$ decay of 212Po and the second one is added to
take into account random pairs of events [40]:

$N(t)=N_{1}~{}e^{(-t~{}\ln 2/T_{1/2})}+N_{2}~{}e^{(-t~{}b)},$ (3)

where $N_{1}$ is proportional to the number of 212Po $\alpha$ decays selected,
$T_{1/2}$ is the half-life of 212Po, $N_{2}$ is proportional to the number of
random pairs of events, and $b$ is an average rate of random pairs. The
random-pairs component (the parameters $N_{2}$ and $b$) was bounded in the
energy intervals chosen for the analysis from the counting rate of the events
taking into account the selection criteria for the mean-time and rise-time
parameters. It should be stressed that the background due to the random pairs
of events is very low in the present experiment: the parameters $N_{2}$ and
$b$ were estimated to be $N_{2}\approx 3.3\times 10^{-6}~{}N_{1}$ and
$b\approx 2.4$ s-1. The distribution of random pairs of events is shown in
Fig. 9 (a).

A fit of the $\Delta t$ distribution in the time interval $80-1600$ ns by the
maximum-likelihood method was performed with the help of the PAW [41] package
that uses MINUIT software [42] for function minimization and uncertainty
analysis. The fit returns the half-life of 212Po 295.10(26) ns with
$\chi^{2}=77.4$ for 73 degrees of freedom. The results of the fit are shown in
Fig. 9. According to the recommendations [43], taking into account that the
uncertainty of the half-life value was obtained by statistical methods, it is
standard uncertainty obtained by type A evaluation method. Uncertainties due
to possible systematic effects (Type B evaluation of standard uncertainty
[43]) and a combined standard uncertainty of the result are reported in the
next Section.

###  3.3 Combined standard uncertainty

The $T_{1/2}$ values obtained by the fit of the data with the bin widths from
0.05 ns (the oscilloscope time bin) to 100 ns lie between 295.06 ns and 295.12
ns (see Fig. 10). Assuming a uniform distribution of possible $T_{1/2}$ values
inside the interval, the standard deviation due to time-bin width, 0.02 ns,
was calculated as the upper value minus the lower value divided by the square
root of 12 [43].

Figure 10: The half-life of 212Po as a function of the $\Delta t$ distribution
time-bin width. The value of the 212Po half-life 295.1 ns is shown by solid
line, while the upper and lower values are shown by dashed lines.

The result of fit depends on the low and high time bounds (see Fig. 11). The
interval of fit ($80-1600$ ns) was chosen to minimize the combined uncertainty
of the half-life value, taking into account that a bigger time interval is
preferable to obtain smaller uncertainty due to statistical fluctuations.
However, the point 60 ns was excluded from the analysis to reduce possible
effect of after-pulses and slow components of the scintillation decay on the
determination of the second pulse zero-crossing time777It should be noted that
a much bigger effect of the fitted half-life increase at low time bounds was
observed in the experiment [16].. Assuming a normal distribution of possible
$T_{1/2}$ values, the standard deviations of the 15 half-life values in the
time interval 80–360 ns were taken as uncertainties: $s_{1}=_{-0.22}^{+0.09}$
ns (the 7 points above the value $T_{1/2}=295.1$ ns were taken to calculate
the upper uncertainty, while the lower uncertainty was calculated by using the
7 points below the value). Similarly the standard deviations of the 15 fit
results in the time interval 1400–1680 ns were taken as uncertainties due to
the upper bound of the fit interval: $s_{2}=_{-0.06}^{+0.07}$ ns. The
intervals where the uncertainties were evaluated, as well as the lower and
upper uncertainties, are shown in Fig. 11.

Figure 11: The half-life of 212Po as a function of the lower (a) and higher
bound (b) of the fit. 1600 ns was chosen as the end point for the fits
presented in the panel (a), while the starting point was 80 ns for the fits
shown in the panel (b). The standard deviations (shown by dashed lines) were
calculated for 15 values in the time intervals 80–360 ns (denoted as $s_{1}$)
and 1400–1680 ns ($s_{2}$) to estimate the uncertainties of the lower and
upper bounds of the fit. The value of the 212Po half-life $T_{1/2}=295.1$ ns
is shown by solid line.

There is an indication of the half-life dependence on amplitude of the $\beta$
and $\alpha$ pulses. To estimate uncertainties due to the $\beta$-events
amplitude, the lower energy threshold of $\beta$ events was varied within the
energy interval 330–1500 keV with the steps from 170 keV to 500 keV (see Fig.
12). The dependence of the half-life on the $\alpha$-signals amplitude was
analyzed for the BiPo events with $\alpha$ events in the first and second
halves of the $\alpha$ peak, while the $\beta$ events were in the energy
interval 500–1500 keV. The effect could be explained by dependence of the PMT
transit time on the pulse amplitude. The related uncertainties were estimated
as standard deviations within the energy interval 500–1500 keV for $\beta$
particles ($s_{\beta}=_{-0.16}^{+0.13}$ ns), and as the differences
($d_{\alpha}=_{-0.15}^{+0.16}$ ns) between the half-life value $T_{1/2}=295.1$
ns and the half-life values obtained for the two halves of the $\alpha$ peak.
Eventually the energy interval 500–1500 keV for the energy of $\beta$
particles was chosen for the analysis to reduce a possible effect of the half-
life dependence on the pulses amplitude.

Figure 12: The half-life of 212Po as a function of the amplitude of the 1st
($\beta$ particles of 212Bi, dots) and 2nd events ($\alpha$ particles of
212Po, squares). Dashed lines show the uncertainties due to the $\beta$ pulses
amplitude (denoted $s_{\beta}$), dotted lines show the uncertainties due to
the $\alpha$ pulses amplitude ($d_{\alpha}$).

A possible effect of temperature variation during the data taking was checked
by analyzing the two data sets gathered in the time intervals during the day
with low (on the average 18∘ C for 3.9 hours around 4:42 AM) and high (21∘ C
for 3.9 hours around 1:36 PM) temperatures in the room where the experimental
set-up was installed. The analysis of the data sets returned the half-life
values $T_{1/2}=295.66(64)$ ns and $T_{1/2}=295.43(62)$ ns for the “low” and
“high” temperature periods, respectively, that is no evidence for the effect.
Nevertheless, a systematic uncertainty due to a possible temperature effects
was estimated as the half of the difference between the two values: $\pm 0.12$
ns.

Finally, uncertainty of the oscilloscope to measure time interval between two
pulses was estimated as $\sqrt{2}\times SCJ+CAR$ [44], where $SCJ$ is Sample
Clock Jitter ($SCJ=0.0001$ ns) and $CAR$ is the product of the Clock Accuracy
(1 ppm) and the time stamp of the second pulse (conservatively we took the
maximal time interval 1600 ns and got the value $CAR=0.0016$). Thus, we accept
$0.0017$ ns as an uncertainty due to a possible oscilloscope inaccuracy.

No other sources of uncertainty were observed. A summary of the uncertainties
is given in Table 4.

Table 4: Uncertainty evaluation of the half-life of 212Po (ns). Standard deviation by statistical methods | $\pm 0.26$
---|---
Lower bound of the fit | ${}_{-0.22}^{+0.09}$
Upper bound of the fit | ${}_{-0.06}^{+0.07}$
Amplitude of $\beta$ events | ${}_{-0.16}^{+0.13}$
Amplitude of $\alpha$ events | ${}_{-0.15}^{+0.16}$
Time bin | $\pm 0.02$
Variations of temperature | $\pm 0.12$
Uncertainty of the oscilloscope | $\pm 0.0017$
Combined standard uncertainty | ${}_{-0.43}^{+0.37}$

Treating all the listed in Table 4 uncertainties as independent and adding
them in quadrature, we obtain the following half-life of 212Po:

$T_{1/2}=295.1(4)$ ns.

The present measurement is the most accurate determination of the 212Po half-
life. The value is in agreement with the preliminary result of the previous
stage of the present experiment $T_{1/2}=294.8(19)$ ns [17] and with the
recommended value $T_{1/2}=294.3(8)$ ns [1]. The result agrees with the recent
experiments: $T_{1/2}=(294.7\pm 0.6(\mathrm{stat.})\pm 0.8(\mathrm{syst.}))$
ns [14] and $T_{1/2}=(293.9\pm 1.0(\mathrm{stat.})\pm 0.6(\mathrm{syst.}))$ ns
[16], but it is smaller than the one obtained in the measurement with BaF2
scintillation detector: $T_{1/2}=(298.8\pm 0.8(\mathrm{stat.})\pm
1.4(\mathrm{syst.}))$ ns [15]. A historical perspective of the half-life of
212Po as a function of the publication date is presented in Fig. 13.

Figure 13: A historical perspective of the half-life of 212Po as a function of
the publication date. References to the publications are as follows: Dunworth
1939: [3], Bradt 1943: [4], Hill 1948: [5], Bunyan 1949: [6], Van Name 1949:
[7], Hayashi 1953: [8], Flack 1962: [9], Astner 1963: [10], McBeth 1972: [11],
Sanyal 1975: [12], Bohn 1981: [13], Bellini 2013: [14], Belli 2014: [15],
Aprile 2017: [16]. The recommended value Auranen 2020 [1] is shown by square;
the result of the present study is shown by triangle. In the Inset the most
recent measurements are reported.

One can calculate a weighted average of the last half-life values [14], [15],
[16], and of the present study as $295.1(4)$ ns, with the uncertainties
combined in quadrature. If the value [15] is excluded, the weighted average
becomes $294.9(4)$ ns.

##  4 Conclusions

The half-life of 212Po relative to $\alpha$ decay to the ground state of 208Pb
(the only known channel of 212Po decay) was measured with thorium-loaded
liquid scintillator as $T_{1/2}=295.1(4)$ ns. This result is the most accurate
up-to-date value (relative uncertainty: 0.14%). It has been achieved thanks to
the utilization of the fast liquid scintillator (rise time does not exceed
$\sim 1$ ns), its small volume (12 mL), the use of the fast photomultiplier
with $\sim 2$ ns rise time and of the fast oscilloscope with a sampling
frequency of 20 GSa/s and a 3.5 GHz bandwidth, the high statistics of the
acquired data, the developed algorithm to determine with a high accuracy the
time intervals between the $\beta$ events of 212Bi and the $\alpha$ events of
212Po.

##  5 Acknowledgements

The group from the Institute for Nuclear Research of NASU (Kyiv, Ukraine) was
supported in part by the National Research Foundation of Ukraine Grant No.
2020.02/0011. D.V. Kasperovych, O.G. Polischuk and N.V. Sokur were supported
in part by the project “Investigations of rare nuclear processes” of the
program of the National Academy of Sciences of Ukraine “Laboratory of young
scientists”.

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