# A novel experimental system for the KDK measurement of the 40K decay scheme
relevant for rare event searches

M. Stukel  B.C. Rasco  N.T. Brewer  P.C.F. Di Stefano<EMAIL_ADDRESS>K.P. Rykaczewski  H. Davis  E.D. Lukosi  L. Hariasz  M. Constable  P. Davis
K. Dering  A. Fijałkowska  Z. Gai  K.C. Goetz  R.K. Grzywacz  J. Kostensalo
J. Ninkovic  P. Lechner  Y. Liu  M. Mancuso  C.L. Melcher  F. Petricca  C.
Rouleau  P. Squillari  L. Stand  D.W. Stracener  J. Suhonen  M. Wolińska-
Cichocka  I. Yavin  Department of Physics, Engineering Physics and Astronomy,
Queen’s University, Kingston, Ontario, Canada  Physics Division, Oak Ridge
National Laboratory, Oak Ridge, Tennessee, USA  Department of Physics and
Astronomy, University of Tennessee, Knoxville, Tennessee, USA  Joint Institute
for Nuclear Physics and Application, Oak Ridge, Tennessee, USA  Faculty of
Physics, University of Warsaw, Warsaw, Poland  Center for Nanophase Materials
Sciences, Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA  Department
of Nuclear Engineering, University of Tennessee, Knoxville, Tennessee, USA
MPG Semiconductor Laboratory, Munich, Germany  Joint Institute for Advanced
Materials,University of Tennessee, Knoxville, Tennessee, USA  Max-Planck-
Institut für Physik, Föhringer Ring 6, München, Germany  Scintillation
Materials Research Center, University of Tennessee, Knoxville, Tennessee, USA
Heavy Ion Laboratory, University of Warsaw, Warsaw, Poland  TRIUMF, Vancouver,
Canada  University of Alberta, Edmonton, Canada  Natural Resources Institute
Finland, Jokioinen, Finland  University of Jyvaskyla, Department of Physics,
P. O. Box 35, FI-40014, Finland

###### Abstract

Potassium-40 (40K) is a long-lived, naturally occurring radioactive isotope.
The decay products are prominent backgrounds for many rare event searches,
including those involving NaI-based scintillators. 40K also plays a role in
geochronological dating techniques. The branching ratio of the electron
capture directly to the ground state of 40Ar has never been measured, which
can cause difficulty in interpreting certain results or can lead to lack of
precision depending on the field and analysis technique. The KDK (Potassium
(K) Decay (DK)) collaboration is measuring this decay. A composite method has
a silicon drift detector with an enriched, thermally deposited 40K source
inside the Modular Total Absorption Spectrometer. This setup has been
characterized in terms of energy calibration, gamma tagging efficiency, live
time and false negatives and positives. A complementary, homogeneous, method
is also discussed; it employs a KSr2I5:Eu scintillator as source and detector.

###### keywords:

Potassium , Electron Capture , DAMA , Rare Decays , Geochronology , SDD , MTAS
, KSr2I5:Eu

††journal: Nuclear Instrument and Methods in Physics Research, A,

Journal : Nuclear Instruments and Methods in Physics Research, Section A:
Accelerators, Spectrometers, Detectors and Associated Equipment

##  1 Introduction

Natural potassium consists of three isotopes, of which potassium-40 (40K) is
the only one that is radioactive. 40K has a natural abundance of 0.0117(1)$\%$
[1] and is a main contributor to the radioactivity of the human body [2].
Trace amounts of 40K can also be found in a variety of minerals which, due to
its very long half-life ($\sim$109 years), makes it widely used for
geochronological dating [3, 4].

The decay scheme of 40K can be seen in Fig. 1. The most recent and complete
evaluations of the nuclear data for this isotope were performed in [1] and
[5]. As shown, the $\beta^{-}$ decay to 40Ca is the primary mode of decay with
a partial half-life of 1.407(7)$\times$109 years. There also exist two modes
of electron capture. The dominant one, with a partial half-life of
11.90(11)$\times$109 years, is to the excited state of 40Ar (EC∗). The
electron capture process potentially releases a cascade of X-rays or Auger
electrons, predominantly around 3 keV, from the shells of the argon daughter.
As the nucleus de-excites it releases an additional 1460 keV $\gamma$-ray, or
with a much smaller probability, another quantum like a conversion electron.
The electron capture directly to the ground state of 40Ar (EC) has never been
experimentally observed. It behaves exactly like the EC∗ decay but without the
associated de-excitation.

Figure 1: The decay scheme of 40K [1].

The decay scheme of 40K was built by three channel probability ratios and the
assumption that the decay scheme is complete. The first ratio is
$I_{EC^{*}}$/$I_{\beta-}$ = 0.1182(12) [5], which is determined from the ratio
of experimental half-lives, where $I_{x}$ represents the branching ratio of
the assigned channel. The second ratio is $I_{\beta+}$/$I_{\beta-}$ =
1.12(14)$\times 10^{-5}$, from a single experiment [6]. The final ratio is
$I_{EC}$/$I_{\beta+}$ = 200(100), from an extrapolation of the first and
second forbidden unique transitions determined by the LogFT [7, 8] calculator.

However, recent theoretical calculations have produced a large range of
$I_{EC}$/$I_{\beta+}$ values. The National Nuclear Data Center gives a
theoretical prediction of $I_{EC}$/$I_{\beta+}$ = 45.2(11) [9]. Using data
from [10], reference [11] calculates $I_{EC}$/$I_{\beta+}$ = 190, but this
analysis makes the approximation of K-shell only electron captures. By
performing higher order corrections, [12] calculates $I_{EC}$/$I_{\beta+}$ =
215.0(31). The range of $I_{EC}$/$I_{\beta+}$ causes almost an order of
magnitude variation from $I_{EC}$ = 0.045(12)–0.22(4)$\%$ in the branching
ratio of the electron capture directly to the ground state.

The lack of experimental data and the range of theoretical values for this
decay channel can pose issues for many fields. In nuclear physics, the EC
decay is a rare example of an experimentally observable unique third forbidden
electron capture decay (J${}^{~{}\pi}$(40K) = 4- $\rightarrow$
J${}^{~{}\pi}$(40Ar) = 0+). Knowledge of this branching ratio will inform and
test nuclear models. In geochronology, the K/Ar and derivative 40Ar/39Ar
dating methods are used extensively in a wide range of environments and time
periods. As discussed in [13, 14, 15], an outstanding issue of these dating
methods is verifying the relevance (or existence) of the EC decay.
Implications for K-Ar and Ar-Ar dating are reviewed in [16].

Finally, for many rare-event searches, the presence of 40K provides a
challenging radioactive background. Due to the chemical similarity between K
and Na, trace amounts of 40K can be found in ultra-radiopure scintillating
crystals grown from NaI powders. Experiments using such scintillators include
ANAIS [17], COSINE-100 [18], COSINUS [19], DM-Ice17 [20], PICO-LON [21], SABRE
[22, 23], and DAMA/LIBRA [24] (referred to as DAMA from here on).

Of the active NaI dark matter searches, only DAMA has claimed to observe a
dark matter signal. The effect of the 40K background on the interpretation of
the DAMA claim has been investigated [11, 25]. The EC decay presents an
unknown background directly in the 2–6 keV energy signal region which needs to
be understood.

Given the above, there is clear motivation to measure the direct to ground
state electron capture of 40K. In this paper we present the detector of the
KDK (potassium (K) decay (DK)) [26] experiment. This article is structured in
the following way: Section 2 will discuss the experimental method and a
technical description of two detector methods (homogeneous and composite) used
to obtain this measurement. Section 3 will detail the integration and
performance of the composite method detector, including data reduction and
energy calibration. Section 4 gives $\gamma$-ray efficiencies, live time
considerations and predicted sensitivity of the composite setup.

##  2 The detector and experimental setup

The KDK experiment consists of an inner detector with an energy sensitivity of
a few keV or better surrounded (close to 4$\pi$ sr coverage) by an outer
detector with higher energy (30–53000 keV) sensitivity. Ideally, the particles
emitted by the 40K electron capture process trigger the inner detector and
open a coincidence window with the outer detector. If a $\gamma$-ray is
detected during this period the event is classified as an EC∗. If not, the
event is classified as an EC. In practice, various factors complicate the
analysis, discussed further in A. By distinguishing between EC and EC∗ events,
the ratio $\rho=I_{EC}/I_{EC^{*}}$ can be determined and used in building the
40K decay scheme.

Two different methods are studied for the inner detector. The composite method
consists of a separate low energy detector and 40K source. This setup is
discussed in Sections 2.2, 3 and 4. The homogeneous method contains the 40K
source in a bulk scintillator and is discussed in Section 2.3. The outer
detector, used for both methods, is described in Section 2.1.

These methods were built in parallel with the founding design principles being
complementary. The homogeneous method benefits from the 40K being uniformly
distributed throughout the bulk of the scintillator. The X-rays and Auger
electrons produced during the decay of 40K have a near-zero probability of
escaping due to the dimensions of the scintillator being orders of magnitude
larger than their absorption length. Conversely the composite method has
excellent energy resolution which will successfully distinguish between the
different electron capture transitions (i.e. Kα and Kβ) and allow us to
investigate backgrounds. In addition, the energy threshold of the composite
inner detector will be considerably lower than that of the bulk scintillator
and the external source will allow for dedicated background and calibration
runs. At the start of development it was unclear whether each or both of these
methods would produce a successful $\rho=I_{EC}/I_{EC^{*}}$ measurement. As
such, both ideas were pursued with the details presented below.

###  2.1 Outer detector: Modular Total Absorption Spectrometer

The Modular Total Absorption Spectrometer (MTAS) [27] at Oak Ridge National
Laboratory (ORNL) is used as the outer detector for the KDK experiment. MTAS
was created to study complex $\beta$ decays of reactor fission products [28,
29, 30] and is composed of 19 NaI(Tl) hexagonal-shaped scintillators (53.34 cm
in length $\times$ 17.6 cm across), with a total mass of close to a metric
tonne. The centre module has a 63.5 mm-diameter through-hole, inside of which
the inner detectors are located. The crystals are arranged in a honeycomb
pattern with the different layers referred to as the central module, the inner
ring, the middle ring, and the outer ring, as shown in Fig. 2. These are
classified by their distance from the centre of MTAS. Each crystal is enclosed
by a carbon fiber housing, with thin layers of silicon putty, stainless steel,
and Teflon. Two 12.7 cm-diameter ETI9390 Photo-Multipliers (PMTs), one on each
end, are placed on every crystal in the Inner, Middle and Outer layers. The
centre crystal has six 3.81 cm-diameter ETI9102 PMTs on each end. All PMTs
were made by ET-Enterprises (Uxbridge, UK). In order to increase the
efficiency with which MTAS can capture escaping gammas in the composite
method, a 5.58 cm diameter $\times$ 25.4 cm length NaI(Tl) crystal plug from
Saint-Gobain can be placed inside the central module. The crystal is
surrounded by a 1.575 mm thick aluminum housing and has a ETI9266 PMT attached
to the back. The total configuration (MTAS + plug) leaves a $\sim 1.4\%$
numerical aperture for escaping particles. MTAS is surrounded by 2.5 cm of
lead shielding plus additional layers of lead blankets.

Figure 2: Schematic of the Modular Total Absorption Spectrometer. The
different layers (Central, Inner, Middle and Outer) are shown. Based on [31].

Signals from the PMTs are fed into Pixie-16 digital gamma finders from XIA LLC
(Hayward, CA) [32]. MTAS is connected to 49 Pixie-16 channels, one for each
PMT, and each is triggered independently. These signals are digitized by a
12-bit analogue-to-digital converter at 100 MHz. The digital signal processing
section of the Pixie-16 cards uses an optimized trapezoidal filtering
algorithm that determines the energy (pulse height) and time-stamp of the
signal [33].

Coincidence between different modules is enforced using a global rolling
window. After one channel is triggered, with a selected coincidence window of
$dt$, the algorithm checks if the subsequent triggered module time-stamp is
within $dt$. If it is, the event is classified as in coincidence and the
window is extended for another $dt$. This is repeated until no more events are
found within the extended coincidence window. Handling of pileup within the
coincidence window is explored in Section 4.4. The exact coincidence window
can be varied in the offline analysis in order to understand the impact of
different backgrounds.

Like typical NaI(Tl) scintillators, those that make up MTAS contain a certain
amount of 40K contamination. In addition, natural contamination from the
surrounding environment and cosmogenic particles are easily detected by the
massive device. The intrinsic background energy spectrum, when no external
source is present, is shown in Fig. 3. Thresholds of 30 keV for individual
MTAS modules arise from the combined effects of multiple photomultipliers
mounted symmetrically on each module and the NaI(Tl) signal processing. These
thresholds have been experimentally verified [27]. The energy resolution is 92
keV full width at half maximum (FWHM) at the 1460 keV peak. The total count
rate as a function of coincidence window is given in Table 1. This table shows
an approximately two percent variation in count rate between the largest and
smallest coincidence windows that were measured. The higher coincidence
windows have a decreased rate due to composite background events forming (i.e.
two background events are being added together). Performance of the MTAS
detector is further discussed elsewhere [27, 34].

Figure 3:  The MTAS background energy spectrum, triggered by MTAS itself. The three largest background peaks visible are the 40K: 1460 keV gamma peak, the 208Tl: 2614 keV gamma peak, and the overlapping 6800 keV 127I and 23Na neutron capture peaks. The insert shows a close-up of the spectrum in the 0 to 3000 keV range. Coincidence Window ($\mu$s) | Rate (counts/s)
---|---
1.0 | 2652.56(15)
2.0 | 2639.51(15)
4.0 | 2618.92(15)
Table 1: Background rate in the MTAS detector as a function of coincidence
window. Only events from 0 - 16000 keV are considered.

###  2.2 Composite inner detector: large area silicon drift detector

A large area silicon drift detector (SDD)111A large area avalanche photodiode
(APD) was also tested as a candidate for the KDK experiment. The results of
the APD testing can be seen in [25]. is used for the inner portion of the
composite method. This type of detector has been shown to successfully operate
above liquid nitrogen temperatures while maintaining good energy resolution
with greater than 90$\%$ quantum efficiency for 1-10 keV X-rays [35, 36]. Our
detector was fabricated by the Halbleiterlabor of the Max-Planck-Society in
Munich, Germany.

####  2.2.1 SDD operating principle

The SDD functions on the principle of sideward depletion designed by Gatti and
Rehak in 1983 [37]. In our configuration, a n-type silicon wafer is depleted
by a small n+ anode contact. Concentric p+ electrode rings with increasing
negative bias are placed around the anode. On the opposite end of the chip is
a large p+ planar cathode with a strong negative bias. In operating
conditions, this setup creates an electric potential with the anode at the
minimum. The potential energy distribution in the SDD can be seen in [36]. Any
electrons produced inside the depleted region are transported to this minimum
and are readout using a source-follower configuration [38]. The holes that are
generated are collected by the p+ implanted regions.

Integrated onto the silicon chip is a field effect transistor (FET) whose gate
is directly connected to the surrounding anode. The integrated FET avoids the
use of bond wires thus minimizing the capacitance of the system. A guard ring
structure is used to insulate the rest of the FET from the chip environment. A
schematic for the SDD substrate can be seen in Fig. 4.

Figure 4: Schematic tri-dimensional view of a SDD with integrated FET. Based
on figure in [38].

In order to avoid charge build-up in the anode, a self-reset mechanism is
implemented. As the electrons arrive at the anode/gate connection it becomes
more negatively charged. Eventually, the gate will reach a specific potential
difference causing a weak avalanche region to form in the channel between the
gate and the drain. The holes that are generated in this region are collected
by the gate and compensate for the electrons by increasing its potential.
Further details about the source-follower readout system and self-reset
operation can be found in [39] and [40].

####  2.2.2 SDD implementation and integration into MTAS

The SDD housing and vacuum setup is designed to minimize the amount of
material within close proximity of the source. This is to reduce the
scattering of the gammas in non-detecting areas. The SDD used in the KDK
experiment is a cylindrical silicon volume with an active surface area of 100
mm2. It is 450 $\mu$m thick with dead layers of silicon (p+, 30 nm), SiO2 (20
nm) and aluminum (30 nm) on the radiation facing side. The SDD is glued to a
printed (18 $\times$ 18 mm) ceramic board and connected together with aluminum
bond wires. Another connection is made from the board to the electronics via
feedthrough pins. Underneath the board is a thermoelectric cooler (TEC), Model
No. 1TML10-18$\times$18-15-000W from Thermion Company, Odessa, Ukraine. The
TEC is 3.6 mm thick, made out of Bi2Te3 with 0.5 mm Al2O3 coating on both
ends. The ensemble is housed in a custom designed Kovar unit that has 32
feedthrough pins and a thermal connection (M8 threading) to the heat sink.
Fig. 5 shows a schematic of the SDD housing setup alongside a front visual of
the actual detector.

(a)

(b)

Figure 5: (a) SDD detector in the Kovar housing, front facing. (b) Schematic
cross-section of the SDD assembly.

The output signal of the SDD is connected to a spectroscopy pre-amplifier
(Amptek-A275) residing on a printed circuit board (PCB) that is connected
directly to the Kovar housing. The signal is then passed through a Gaussian
shaping amplifier (Ortec 672, residing outside the SDD vacuum system) and fed
directly into the MTAS data acquisition system (DAQ).

To improve the signal-to-noise ratio of the SDD, it is cooled by means of an
aluminum rod threaded onto the thermal sink of the Kovar housing. The rod is
clamped to a copper piece that thermally connects it to a copper cooling loop.
The copper loop is a 4.76 mm inner diameter and 6.35 mm outer diameter tube
that extends for 1.05 m to a ISO-100 liquid feedthrough flange. The cooling
loop is powered by the RTE-140 refrigerated Bath/Circulator from Thermo Fisher
Scientific, USA. A 50/50 mixture of filtered tap water and laboratory grade
ethylene glycol is used as the cooling liquid. Temperatures of $-10^{\circ}$C
can be reached at the SDD using this method. Additional cooling is provided by
the TEC causing the SDD to reach the operating temperature of $-20^{\circ}$C.
The temperature is measured using an integrated diode that resides on the SDD.

The complete cooling setup is placed inside a custom designed vacuum tube,
shown in Fig. 6. The tube, and its internals, serve to centre the detector and
the source in MTAS reproducibly. The two flanges not shown are an electronic
feedthrough flange, containing a Sub-D 25 pin connector and three BNC
feedthrough connections, and an ISO-100 to KF-40 adapter that connects to a
turbo pump setup. The vacuum tube has a 30 cm-long aluminum cap with a
hemispherical end. The inner diameter of the cap is 4.85 cm with a wall
thickness of 0.63 mm. This cap was constructed by Wejay Machine Productions
Co. Ltd, Canada with welding performed by Laser Weld Creation, Canada. The cap
is attached to a larger (68.8 cm) aluminum tube with the same inner diameter
but a wall thickness of 2.31 mm. The resulting ensemble (known as the MTAS
insert) is wrapped with a plastic collar that ensures a source resting in
front of the SDD will consistently reside in the centre of MTAS. Small,
cylindrical copper spacers make certain that the source is 1 mm from the
surface of the SDD and an aluminum rod was used to align the SDD/source to the
midpoint of the hemisphere of the insert. Support triskels are also placed
evenly inside the vacuum chamber to manage the cables from the SDD and centre
it radially.

Figure 6: Full KDK SDD/MTAS Experiment. Close up on SDD housing, radioactive
source holder and cooling loop.

####  2.2.3 40K source development

Specifications for the 40K source for the composite method are that it should
contain $\sim 10^{18}$ atoms of 40K to ensure sufficient signal, and that it
be thin enough ($\lesssim 10\ \mu$m) to limit self-absorption of the 3 keV
X-rays. This is implemented as a disk of 1 cm diameter, similar to that of the
SDD. The source rests on a round graphite substrate of 23 mm in diameter and
0.23 mm thick.

Two approaches were explored for the creation of the source: ion implantation
and thermal deposition. The implantation of 40K ions into the graphite
substrate used the on-line test facility (OLTF) at ORNL [41]. A surface
ionization ion source was used to produce singly charged K ions from enriched
KCl with a 3.15$\%$ 40K content. The ions were extracted from the ion source
at 20 kV, transported through a 90-degree dipole magnet optimized for A=40
mass selection, and then focused toward the graphite substrate with an
implantation energy of 20 keV. The substrate was placed inside a shielded
Faraday Cup (FC) by which the total incident 40K ion current was measured. The
FC consisted of an entrance aperture of 1 cm diameter to limit the implanting
40K beam size to be about the same as the source geometry. The predicted
advantage of this method was that the only radioactive source would be 40K
atoms. However, it was found that only $\sim 10\%$ of the total implanted
activity was in the graphite substrate, the rest was on the interior of the
FC. This suggested that self-sputtering would prevent the source from reaching
the required activity. Additionally, this method produced a noticeable 125Sb
contamination in the source, the cause of which was unknown. For the above
reasons, ion implantation was not used for the creation of the experimental
source.

Instead, thermal deposition was used to place a thin KCl film on the graphite
substrate. Assuming KCl molecules are uniformly distributed in the source disk
of 1 cm diameter, the desired activity can be attained with a film thickness
of about about 25, 8, and 5 $\mu$m for 3.15$\%$, 10$\%$, and 16$\%$ enrichment
of 40K, respectively. Thermal deposition was carried out in a high vacuum
evaporation system with a specially designed thermal evaporator at the Center
for Nanophase Materials Sciences (CNMS) at ORNL. The special
evaporation/deposition assembly was constructed with the configuration as
illustrated in Fig. 7. The conical-shaped crucible, made from a thin Ta metal
sheet with base diameter of about 12 mm and cone height of 4 mm, centred the
KCl feed material. The crucible was heated with a tungsten wire basket heater
of 9 mm inner diameter and 13 mm height from Ted Pela, Inc. USA. The basket
heater was resistively heated by an electrical current, which directly
controlled the temperature. The graphite substrate was sandwiched between two
aluminum plates and held at approximately 2 to 3 mm above the crucible. The
confined space between the crucible and the substrate allowed a substantial
fraction of the evaporated material to hit the substrate, but due to this
close geometry the substrate was also heated to high temperatures, decreasing
deposition efficiency.

Figure 7:  Schematic drawing of the KCl deposition setup. For clarity, the
tantalum crucible is pictured outside of the tungsten basket heater in which
it resides.

To compromise between decreased efficiency due to slow heating and the risk of
sputtering due to fast heating, tests with natural KCl powder were carried out
to find the optimal deposition parameters. By varying the deposition time and
the heater current, we found the best condition was a heating time of 2
minutes at a heater current of about 18.9 A, which was the maximum heating
current of the evaporator, under a vacuum of 10-6 to 10-5 Torr. Under this
condition, feed materials up to 5 mg KCl powder in the Ta crucible could be
completely evaporated. After the 2-minute heating, the heater current was
immediately turned off to allow the substrate to cool naturally. We were
successful in depositing a smooth layer of KCl on graphite disks reproducibly,
as shown in Fig. 8. The substrates were weighed before and after deposition to
determine the deposition efficiency (the ratio of deposited KCl to feed KCl)
and it was in the range of 30-50$\%$.

Figure 8:  The 40K thin film source on graphite substrate in the KDK detector
holder, made from 16.1(6)$\%$ enriched KCl with $\sim 9\times 10^{17}$ 40K
atoms.

The thickness of the KCl films produced was characterized by scanning electron
microscopy (SEM) analysis. Two KCl films were analyzed, one with 1.5(1) mg of
natural KCl and the other with 1.45(12) mg of 3.15$\%$ enriched KCl. The
thickness of the natural KCl film was measured to be 11(1) $\mu$m (Fig. 9). It
is noted that the calculated thickness of 1.5 mg KCl uniformly distributed in
a circle of 1-cm diameter is 9.7(6) $\mu$m, which is statistically consistent
with the SEM measurement. The ratio of the calculated thickness to the SEM
measurement is 0.9(1).

Figure 9:  SEM image of the cross-section of the natural KCl film sample for
thickness estimation. 10 measurements were made per image. A total of 20
measurements gave an average thickness of 11(1) $\mu$m.

The final KCl thin film was made with 16.1(6)$\%$ enriched 40K. The deposition
used 1.76(7) mg of the enriched KCl as feed material and a net of 0.69(11) mg
KCl was deposited on the graphite substrate, corresponding to a deposition
efficiency of about 39$\%$. Based on how the source was prepared, the total
activity in the source should be about $9\times 10^{17}$ atoms of 40K. Using
the mass of deposited enriched KCl, and scaling by the ratio of SEM thickness
to calculated thickness for the natural KCl film, the final 40K source
thickness is estimated to be 5.1(9) $\mu$m.

Determining the actual activity of the source by measuring decays is
complicated. For instance, the gammas from EC∗ decays are negligible compared
to the background in MTAS, and measuring the lower energy quanta like betas or
X-rays depends on knowing the efficiency of the SDD and the self-absorption of
the source.

###  2.3 Homogeneous inner detector: KSI scintillator

We have also explored the use of the scintillator KSr2I5:Eu (KSI) [42] as a
combined 40K source and X-ray detector. The light yield of KSI varies
depending on the packaging and crystal size but has been reported to be as
high as 94,000 photons/MeV at 662 keV with good proportionality compared to
other scintillators [43]. Grown with natural potassium and with an estimated
density of 4.39 g/cm3, the intrinsic activity of KSI is $\sim$6.5 Bq/cm3.

In Fig. 10, the spectrum of an approximately 2.54 cm diameter by 2.54 cm long
cylindrical KSI scintillator is provided. The measurement was carried out in a
lead cabinet to shield the scintillator from external radiation, and is
dominated by the internal 40K. The 1.31 MeV end point beta continuum and 1.46
MeV $\gamma$-ray emitted during the decay of 40K are clearly observable in the
pulse height spectrum. Note that Sr would contribute to the beta continuum but
with a much lower activity.

Figure 10: Measurement of a shielded 2.54 cm diameter by 2.54 cm long KSI
scintillator. Main features are the 1.46 MeV gamma line as well as the
continuous beta spectrum, both from 40K.

The KSI sample used for the KDK measurements was grown using the vertical
Bridgman technique as described in [42]. The crystal was cut and polished into
a $7\times 7\times 19.9\mbox{ mm}^{3}$ rectangular parallelepiped. The sample
was then wrapped in an approximately 400 $\mu$m-thick layer of teflon inside
of a 2.5 mm thick aluminum housing with a nitrogen atmosphere. The aluminum
housing contained a small, 2 mm diameter hole covered with a 50 $\mu$m
aluminized mylar film to allow in low energy ionizing radiation for
calibration purposes with windows on both ends for a double readout, shown in
Fig. 11. The crystal was mounted to each silicon oxide window using the Eljen
optical cement EJ-500. Two R6231 Hamamatsu PMTs were used to readout each end
and were held together with a custom, 3D printed bracket. The combined system,
shown in Fig. 11, was placed inside the middle of MTAS and utilized the same
readout system. In order to process the information from the double window
readout, the signals from each PMT must be summed to collect all the light
generated during an ionizing radiation event within KSI, shown in Fig. 12.

Figure 11: $7\times 7\times 19.9\mbox{ mm}^{3}$ rectangular KSI sample wrapped
in 400 $\mu$m of teflon sealed inside an aluminum housing with a nitrogen
atmosphere placed in the centre with a custom 3D printed polyethylene bracket
holding the setup together. Figure 12: Analog sum of both PMTs when the KSI
scintillator is exposed to a 57Co source.

To confirm that the 3 keV X-ray/Auger peak for 40K is visible in KSI,
characteristic features from activation of Eu were used to verify the peak
location in this low energy range. The KSI sample was exposed to a 26.7
milliCurie PuBe source for 9 hours and then placed into MTAS so a spectrum
could be gathered from the activated Eu content while also tagging the
coincident higher energy gammas from the Eu decay. 151Eu ($n$,$\gamma$) 152Eu
produces photons and Auger electrons with a wide range of energies in the
region we are investigating. 153Eu ($n$,$\gamma$) 154Eu also produces the same
spectral features, however the nuclear cross section is orders of magnitude
lower. ${}^{128}I$ is also activated in this process, but with a 24.99(2)
minute half life, the contribution is negligible. This activation method
focused only on events from the 152Eu that were coincident in KSI and MTAS. By
gating the positive events in MTAS on the 1.46 MeV gamma from 40K the
contribution of the 5–6 keV X-ray/Auger electrons from Eu are removed leaving
only the 3 keV peak, as seen in Fig. 13. For comparison, the same is done on a
characteristic $\sim$970 keV gamma from Eu, reducing the intensity of the 3
keV 40K peak to those coincident with part of the Compton continuum from the
1.46 MeV. This corroborates that the peak seen in this low energy spectrum is
in fact the 3 keV X-ray/Auger peak.

Figure 13: Spectra from an activated KSI:Eu sample, in coincidence with MTAS.
Blue: strictly positive energy in MTAS, showing both $\approx 3$ keV 40K
X/Augers, and Eu X/Augers. Orange: energy in MTAS $\approx 1.46$ MeV, showing
mainly X-Auger from 40K. Green: energy in MTAS $\approx 0.97$ MeV, showing
mainly Eu where 40K is less visible.

Currently, for the purpose of the KDK 40K measurement, use of the KSI
scintillator is limited by our simulation model of the PMT structure, which
prevents us from determining the precise tagging efficiency for $\gamma$-rays,
as the simulations are sensitive to the distribution of material near the KSI
source. Variations in R6231 PMT materials that fall within design and
manufacturing parameters result in insufficient precision in the model. An
option under consideration is to replace the PMTs with smaller SiPMs. These
smaller SiPMs can be sized to cover only the window of the KSI sample, see
Fig. 11, leading to a large reduction in mass immediately near the windowed
sample. Additionally the design of SiPMs being mostly silicon, glass, and a
fiberglass PCB, along with their small size, greatly reduces the amount of
material and variation within the structure to be placed into MTAS.

##  3 Characterizing the SDD in MTAS

Of the two methods developed, the composite one is currently better
understood, and is the focus of the rest of this work.

###  3.1 Data reduction and SDD operation

When an event triggers the SDD it is digitized by the same DAQ as discussed in
Section 2.1. The time window of an SDD trace is 12 $\mu$s with 10 ns sampling
and a 40$\%$ pre-trigger length. Based on event rates for most sources in the
SDD (Table 5), the expected number of spurious coincidences in an SDD window
is $\lesssim 5\times 10^{-4}$. Baseline subtraction of the event is done
during the offline analysis by averaging the y-axis data of the pre-trigger
and subtracting the value from the trace. The amplitude of the event is
determined by fitting a gaussian to the trace. The amplitude corresponds to
the energy of the event and the energy calibration is discussed in Section
3.2. The standard deviation of the baseline is used as the uncertainty on each
sample of the trace for the fit. A sample trace for an 8.04 keV 65Zn event can
be seen in Fig. 14.

Figure 14: Shaped SDD pulse for a 8.04 keV X-ray from a 65Zn source. The trace
is baseline subtracted. There is a 40$\%$ pre-trigger length.

All traces produced from the SDD are fit in this manner and each run performed
was monitored for gain stability. The information is then combined with the
MTAS energy and timing data for further offline analysis. A global rolling
window (as described in Section 2.1) is used to determine coincidence between
the SDD and MTAS. A sample spectrum for the SDD using a 65Zn source is shown
in Fig. 15, a resolution of 198 eV FWHM at 8.04 keV, and a 5$\sigma$ noise
threshold of 370 eV was achieved. The decay scheme of 65Zn can be found in
[44]. The expected X-ray energy values quoted, can be found in Table 2.

In the spectrum the 65Zn L: 0.93 keV, Kα: 8.04 keV and Kβ: 8.94 keV X-ray
peaks are visible. Also visible are the 65Zn Auger electrons which are
identifiable by their characteristic wide continuous energy spectrum below the
X-ray peaks. Unlike X-rays, the exact energy deposit of Auger electrons
depends sensitively on the path out of the source and through the SDD dead
layer, thereby forming a continuous energy spectrum instead of a single peak.
The background for energies greater than 10 keV is formed by low-energy
interactions in the SDD from other 65Zn decay components.

Figure 15:  SDD spectra with no condition on MTAS. Blue: 65Zn. Black:
background (i.e. no source present). Energy resolution is 198 eV FWHM at the
65Zn Kα X-ray peak, and the 5$\sigma$ noise threshold is 370 eV. A description
of the different components of the 65Zn spectrum is offered in the text.

###  3.2 SDD energy calibration

The energy scale of the SDD was calibrated using four sources: 54Mn, 65Zn,
88Y, and 40K (the 40K signal region remained blinded). The 54Mn, 65Zn and 88Y
source were purchased from Spectrum Techniques, they are open sources with
$<$1 cm diameter. The corresponding run time, energies and shell capture for
each source can be found in Table 2. The SDD energy values are the weighted
average of the quantities found in the most recent Table of Radionuclides
publications in [1, 44, 45, 46].

The energy linearity of the SDD can be seen in Fig. 16. The multiple source
calibration shows that the SDD easily achieves its target for measuring the
40K X-rays.

Source | Activity | Energies | Energies | Run Length
---|---|---|---|---
| (kBq) | (X-Ray, eV) | (Gamma, keV) | (Days)
54Mn | 0.133(11) | Kα: 5411.68 | 835 | 4.7
| (2017/02) | Kβ: 5966.89 |  |
65Zn | 0.999(74) | L : 937.1 | 1115 | 1.4
| (2017/02) | Kα: 8041.1 |  |
|  | Kβ: 8941.25 |  |
88Y | 18.5 | L : 1827.84 | 3584 | 0.5
| (2017/05/26) | Kα: 14121.01 | 3218 |
|  |  | 2734 |
|  |  | 1836 |
40K | 0.015 | Kα: 2957.04 | 1460 | 44
| (2017/08) | Kβ: 3190.5 |  |
No Source |  | N/A | N/A | 1.3
Table 2: Overview of the KDK experimental data taking campaign. Figure 16:
Calibration of the SDD with 54Mn, 65Zn, 88Y and 40K sources. The slope is
229.8(1) Channel/keV and the y-intercept is 10.4(2) Channel.

###  3.3 SDD-MTAS coincidence spectrum

A coincidence plot can be made between the energy recorded in MTAS and the
energy recorded in the SDD. Fig. 17 shows the coincidence plot for 65Zn. There
are two sections to every coincident plot: the coincident and anti-coincident
region. Based on Section 2.1, if the event in the SDD has no coincident
partner it is assigned an MTAS energy of -1.0 keV and resides in the anti-
coincidence region. Any event with a coincident partner is given the
corresponding MTAS energy and resides in the coincident region.

The main feature of these plots is the coincidence peak. This is when the
X-ray and $\gamma$-ray are fully captured by their respective detectors. For
the Kα case of 65Zn this corresponds to a 8.04 keV X-ray and 1115 keV
$\gamma$-ray. The vertical line below the coincidence peak represents when the
$\gamma$-ray is not fully captured (whether due to Compton scattering or other
effects). A horizontal tail is created to the left when an Auger electron
interacts in the SDD simultaneously with a 1115 keV $\gamma$-ray in MTAS. A
$\gamma$-ray from the source can arrive at the same time as an external
background event creating the vertical tail above the coincidence peak, which
is further discussed in Section 4.2 and A. Finally, for the 65Zn-specific case
it is possible for the $\beta^{+}$ to annihilate inside the SDD, deposit a
small amount of energy, and release two 511 keV gammas which are captured by
MTAS as an 1022 keV event. This can be seen in the horizontal tail to the
right of the coincidence peak.

Figure 17: SDD/MTAS 65Zn coincidence spectrum (2 $\mu$s window). Color scale
represents counts per bin, with each bin having size 16 keV SDD/1000 $\times$
4 MeV MTAS/1000 (counts per MTAS energy bin per SDD energy bin). The Kα and Kβ
coincident peaks are clearly visible with the description of the horizontal
and vertical lines being found in the text.

##  4  Composite method tagging efficiency and sensitivity

In the isotopes considered in our study, the dominant decay mode from the
excited state is via $\gamma$-ray emission, therefore we focus on detecting
such emissions to tag EC∗.

###  4.1 54Mn experimental efficiency

The gamma tagging efficiency ($\varepsilon_{\gamma}$) is defined as the
probability that MTAS will detect the gamma when triggered by the X-ray (or
Auger) in the SDD. 54Mn is an ideal source to study this parameter because it
decays almost exclusively through the excited state, giving a $\rho$ (expected
ratio of EC to EC∗ events) that is very small: $\rho=3.0\times 10^{-6}$. This
value and the decay scheme of 54Mn are acquired from [45]. The small ground
state electron capture implies that there should be no false negatives (where
an EC event looks like a EC∗) during the exposure, which will simplify the
analysis. False positives and negatives are further discussed in A. In
addition, the energy of the released $\gamma$-ray from the excited state decay
is 835 keV which is within a factor of two of the 40K gamma, the ultimate goal
of the KDK experiment. Sec. 4.2 details how the efficiency determined for 835
keV gammas can be extrapolated to 1460 keV, and Sec. 4.4 details live time
considerations involved with the extrapolation. The 54Mn coincident plot can
be seen in Fig. 18 and the individual SDD spectra are shown in Fig. 19 for a 2
$\mu$s coincidence window.

Figure 18: SDD/MTAS 54Mn coincidence spectrum (2 $\mu$s window). Color scale
represents counts per bin, with each bin having size 16 keV SDD/1000 $\times$
4 MeV MTAS /1000 (counts per MTAS energy bin per SDD energy bin). Clearly
visible are the Kα and Kβ coincident peaks.

In order to extract the efficiency for detecting in MTAS the 835 keV gamma
from the 54Mn source when the SDD triggers on the X-ray, a likelihood fit was
performed with the efficiency as a shared parameter between the coincident and
anti-coincident spectra. As there should be very few EC events due to the
decay scheme of 54Mn, the dominant source of events in the anti-coincidence
spectrum would be from EC∗ decays whose gamma has not been tagged (i.e. a
false positive). The likelihood also accounts for coincidences with the
background which would otherwise cause the efficiency to be overestimated as
the coincidence window increases. More information about these coincidences
can be found in A. Only the fully collected X-ray energy region between 5.0
and 7.0 keV was considered for this analysis with no energy cuts being made on
the MTAS events. The shape of the coincidence spectrum was modelled with two
gaussians (for the Kα and Kβ X-ray peaks of the 54Mn), a gaussian containing
primarily an Auger contribution, and a flat line representing any background
contribution. The Auger gaussian was required to have the same shape for both
the coincidence and anti-coincidence spectrum. To within statistical
uncertainties, the result of this analysis does not depend on the precise
shape of this contribution as long as it is treated consistently in all
spectra.

It is possible for the EC∗ decay to produce a conversion electron instead of a
$\gamma$-ray and a respective term was included in the likelihood fit in order
to account for this effect. The internal conversion coefficient for excited
states is the ratio of internal conversion decays to decays via gamma ray
emission. If the transition is greater than 1022 keV, internal pair formation
is also possible. For a 54Mn K-shell excitation, the internal conversion
coefficient is $2.49(4)\times 10^{-4}$ [47, 48].

The anti-coincidence spectrum is modelled in an identical way except that two
additional gaussians were used to account for the Kα and Kβ of an 55Fe
contamination. Such contaminations are common in the production of 54Mn [45]
and would only be present in the anti-coincidence region as they have no
excited state partner. The result of the fit for the 54Mn source with a 2
$\mu$s coincidence window using all modules of MTAS can be seen in Fig. 19.
The efficiency at 835 keV was determined to be $0.9778(1)$ at 2 $\mu$s
coincidence window.

Figure 19:  A) SDD energy spectrum for the coincident events from the 54Mn
source. The experimental data is from a 2 $\mu$s coincidence window. The Kα
and Kβ peaks are clearly visible at 5.41 keV and 5.96 keV as well as the
silicon escape peak at 3.67 keV. B) Likelihood fit of the coincident data in
the 5.0–7.0 keV region. C) SDD energy spectrum for the anti-coincident events
from the 54Mn source. The 55Fe contamination is visible at 6.51 keV. D)
Likelihood fit of the anti-coincidence events in the 5.0–7.0 keV range. See
text for details of the fit components.

The same analysis was performed for three separate coincidence windows (1.0,
2.0 and 4.0 $\mu$s), with the results shown in Table 3.

| 1 $\mu$s | 2 $\mu$s | 4 $\mu$s
---|---|---|---
Total | 0.9775(1) | 0.9778(1) | 0.9778(1)
Table 3:  Efficiency for detecting in MTAS the 835 keV gamma from the 54Mn
source when the SDD triggers on the X-ray, as a function of coincidence
window.

###  4.2 Extrapolating the tagging efficiency to higher energies with Geant
simulations

In Section 4.1, we measured the tagging efficiency for an 835 keV gamma over a
selection of coincidence windows. However, for the KDK experiment the
efficiency at 1460 keV is required. This is achieved by extrapolating the
measured 54Mn efficiency with a 40K/54Mn efficiency ratio calculated using
Geant4 (version 10.2p01) [49].

The SDD/MTAS setup was modelled in the software and a 54Mn and 40K source were
individually simulated for ten million gamma events. The geometry of the
source was exactly as described in Section 2.2.3 and the initial position of
the events was uniformly distributed with isotropic directional vectors. The
number of gamma events from the source that deposit energy into any MTAS
module above selected thresholds (anywhere from 0–50 keV) was recorded. The
simulated efficiencies ($\varepsilon^{s}_{Mn}$ and $\varepsilon^{s}_{K}$) were
determined by taking that number and dividing by the initial number of events.
The simulated ratio between the 40K efficiency and 54Mn efficiency
($\frac{\varepsilon^{s}_{K}}{\varepsilon^{s}_{Mn}}$) was then calculated. The
procedure described above was repeated to extrapolate a tagging efficiency for
65Zn (1115 keV), and the results of both analyses are shown in Table 4.
Simulated ratios were found to be very robust as they were independent (within
one standard deviation) of energy threshold selections, different Geant4
physics lists (Penelope and Livermore) and large variations in material
thickness surrounding the source.

As can be seen in Table 4, the scaled efficiencies for both 65Zn and 40K are
higher than that of the lower gamma energy 54Mn source. This result, at the
one part in a thousand correction, is due to the near 100$\%$ efficiency (MTAS
radius $\sim$10 times the attenuation length of 40K in NaI) and modularity of
the MTAS detector.

Isotope | Measured $\varepsilon^{m}_{i}$ | Simulated $\varepsilon^{s}_{i}$ | Simulated Ratio $\frac{\varepsilon^{s}_{i}}{\varepsilon^{s}_{Mn}}$ | Scaled $\frac{\varepsilon^{s}_{i}}{\varepsilon^{s}_{Mn}}\varepsilon^{m}_{Mn}$
---|---|---|---|---
(2 $\mu$s CW) | (2 $\mu$s CW)
54Mn | 0.9778(1) | 0.9847(3) | - | -
40K | - | 0.9860(3) | 1.0013(5) | 0.9791(5)
65Zn | - | 0.9866(3) | 1.0019(5) | 0.9797(5)

Table 4:  Extrapolation of MTAS gamma-tagging efficiencies from 54Mn (835 keV)
to 40K (1460 keV) and 65Zn (1115 keV) using Geant4 simulations. The
experimental value is taken at 2 $\mu$s from Table 3. Due to the gammas
typically arriving within less than 100 $ns$ the simulated ratios calculated
above are valid for all coincidence windows, above that nominal threshold.

###  4.3 MTAS gamma spectrum components

The simulations allow us to study the different components that make up the
SDD-triggered MTAS data spectrum, shown in Fig. 20 for a 54Mn source. In this
spectrum the single deposition of the source gamma is clearly identifiable by
the photoelectric peak and Compton continuum. The tail ($>900$ keV) contains:
i) random coincidences with only the natural MTAS background; ii) random
coincidences convolving the MTAS background and the source gamma; and iii)
random coincidences convolving two source gammas (eg. 2$\times$835 keV line).
Fig. 20 shows the different levels that each one of these components
contribute to the overall spectrum. Other sources with more complicated decay
schemes could include contributions from the $\beta^{-}$ or $\beta^{+}$ decay
channels.

Figure 20:  Full 54Mn, 4 $\mu$s coincidence window, gamma spectrum in MTAS
when triggered by the SDD (dark blue) compared with Geant4 simulated data. All
modules of MTAS are used. Components of the simulated spectrum include: The
Geant4 single gamma deposition simulation (black), the MTAS natural background
spectrum (light blue), the source-background convolution spectrum (green) and
the source-source convolution spectrum (purple). These components are fit
(total simulated spectrum in red) to the data to show their individual
contributions to the experimental spectrum. Insert shows zoom of lower energy
region.

###  4.4 Live time correction

Due to the level of accuracy required for this project, corrections for data
losses need to be applied. There are four physical pathway groupings in the
digitizer: channel, channel group of four, Pixie module (16 channels), and the
crate. It is important to note that there are no sources of dead-time for the
channel group or for the Pixie Module. Our data rates in the crate were 2-3
orders of magnitude smaller than the level where this loss would be
considerable. Furthermore, this would be a global loss rate and could not
affect the ratio of EC/EC*. Channel level live times are the only factors to
consider.

There are three sources of acquisition dead-time and two sources of processing
dead-time to consider for this experiment. Acquisition dead times can be
incurred in the digital signal processing (DSP) pathway, by the length of the
trace, and from any shaping time. Acquisition dead-times only related to the
SDD do not impact the EC/EC∗ ratio because both EC and EC* require an SDD
trigger. This excludes the need to consider the trace and shaping related
dead-times. The DSP live time can be calculated from the count rate. Data
filtering from processing also affects our live time. Signal pile-up (when two
signals are detected by the digitizer in the range of the slow trapezoidal
filter) and event pile-up (when multiple signals are detected in the same
channel and are considered in coincidence) are accounted for. We also consider
the potential dead time associated with the multiplicity of events in the
centre module of MTAS.

All of these dead-times are modeled as the paralyzable type [32]. This means
that each occurrence of the situation leading to dead-time will prolong the
length of the dead-time. This model means that the Output Count Rate (OCR) is
related to the Input Count Rate ($ICR$) and the Dead Time ($dt$) by

$OCR=ICR\cdot e^{-ICR\cdot dt}.$ (1)

This function is invertable with the use of the Lambert-W function as

$ICR=W(-dt\cdot OCR)/-dt.$ (2)

The dead-time percentage can then be calculated as

$(1-OCR/ICR)\cdot 100.$ (3)

$OCR$ in the SDD and various rings of MTAS are shown in Table 5.

Isotope | Average Output Count Rates (OCR) (Hz)
---|---
| Plug | Central | Inner | Middle | Outer | SDD
88Y | 225 | 1545 | 381 | 219 | 211 | 239
65Zn | 27.6 | 309 | 174 | 198 | 204 | 46.5
54Mn | 18.4 | 217 | 166 | 200 | 206 | 29.9
40K | 12.7 | 160 | 159 | 196 | 202 | 11.9
Table 5:  Table of rates in various detector segments during measurements of
different sources. Errors are less than 0.01%.

The live time correction factor for the total MTAS efficiency is calculated as
a weighted average of the live time correction factors of the various rings of
MTAS (Centre, Inner, Middle or Outer). The weight is taken to be the
percentage that each ring contributes to the total efficiency, which is
acquired from the simulations discussed in 4.2. This yields the values listed
in Table 6. Live time correction is applied to the gamma-tagging efficiency
(shown in Table 4) via the ratio of the live time for the Mn isotope to that
of the pertinent isotope (shown in Table 6), which accounts for differing
activities across the sources. The final energy and live time corrected
efficiency value for the three coincidence windows are given in Table 7. This
correction is of the order of 0.01$\%$.

CW ($\mu$s) | Total Live Times
---|---
|  54Mn |  40K |  65Zn |  54Mn/40K |  54Mn/65Zn
1 | 0.9983(3) | 0.9982(4) | 0.9987(3) | 1.0001(4) | 0.9996(3)
2 | 0.9982(3) | 0.9980(4) | 0.9986(3) | 1.0001(4) | 0.9996(3)
4 | 0.9978(3) | 0.9977(4) | 0.9982(3) | 1.0001(4) | 0.9996(3)
Table 6: Total live times for each isotope and correction factors Mn/K and Mn/Zn for various coincidence windows (CWs). CW ($\mu$s) | Energy & Live Time Corrected Efficiency
---|---
|  54Mn |  40K |  65Zn
1 | 0.9775 (1) |  0.9789 (6) | 0.9790 (6)
2 | 0.9778 (1) |  0.9792 (6) | 0.9793 (6)
4 | 0.9778 (1) |  0.9792 (6) | 0.9793 (6)
Table 7: Energy and live time corrected efficiency for 40K and 65Zn at three
difference coincidence windows. Correction factors for energy and live time
can be found in Table 4 and Table 6 respectively.

###  4.5 Predicted sensitivity

The sensitivity of the composite method was studied using the blinded 44-day
40K data set. To reduce biases, the data in the EC signal region (2.0 – 3.8
keV in the anti-coincidence spectrum) and in the EC silicon escape peak (0.88
– 1.4 keV) are hidden until the analysis is finalized. This data set was then
used as a model to simulate the anti-coincidence and coincidence spectra over
the 2–3.85 keV energy range. The blinding on the anti-coincidence signal
region was maintained, forcing certain assumptions about the background. These
include the linearity of the $\beta^{-}$ background, and potential Cl and K
fluorescence (distributed as Gaussians around the specific energy values). The
signal (or $\rho$ value) is an input parameter of the simulation. The gamma
tagging efficiency from Table 7 and the probability of false negatives and
positives from A are included in the analysis.

By generating numerous simulations over a range of fixed $\rho$ input
parameters, a frequentist confidence belt is constructed using the Feldman and
Cousins ordering method [50]. This is shown in Fig. 21. This confidence belt
is purely statistical and does not contain any systematic errors. In addition,
a likelihood ratio analysis was performed on each simulation in order to
determine a $p$-value when comparing the null hypothesis ($\rho=0$ fixed, i.e.
no decay to ground state) versus the alternative hypothesis ($\rho$ free). The
$p$-value is defined as the probability of observing a test statistic (here,
twice the logarithm of the ratio of maximized likelihoods) at least as large
as the one calculated assuming the null hypothesis is true.

The original design goal of the KDK experiment was to measure a branching
ratio of 0.2$\%$ [11, 1]. Taking this value on the confidence belt in Fig. 21
would generate a measurement of (0.2 $\pm~{}0.03)\%$ at a 68.3% confidence
level. In addition, with 0.2$\%$ as the input parameter we expect to reject
the null hypothesis with a $p$-value of $1.84\times 10^{-13}$, corresponding
to a $7.27\sigma$ significance in terms of upwards fluctuations of a gaussian.
This result clearly indicates that the composite method can achieve the
initial design goal of the KDK experiment. The full 40K unblinding analysis
will be published in a future paper.

Figure 21:  Confidence belt of the statistical fluctuations of the simulated
44 day 40K data set. The belt is constructed using the Feldman and Cousins
ordering method. The 68.3, 95.4 and 99.7$\%$ confidence regions are shown in
blue, red and orange respectively. The x and y axes are labelled by decay
intensity, which is $\rho$$I_{EC^{*}}$.

##  5 Conclusion

This work has shown the design, commissioning and characterization of two
experimental setups to measure the electron-capture decay of 40K to its ground
state. This measurement will be relevant to the fields of rare-event searches,
nuclear physics, and geochronological dating.

For the composite method, the silicon drift detector (SDD) has been
successfully constructed and integrated into the large Modular Total
Absorption Spectrometer (MTAS). The SDD was calibrated with radioactive
isotopes (54Mn, 65Zn, 88Y and 40K) over the 0.9–12 keV energy range. The
energy threshold (370 eV) of the SDD was found to be well below the $\sim 3$
keV energy of the 40Ar K-shell X-rays and Auger electrons produced by the
electron capture. The $\sim 200$ eV resolution of the detector will also
resolve the Kα and Kβ lines from the 40K source. A 54Mn source was used to
determine the gamma ray tagging efficiency of the setup at 835 keV to a high
level of precision. A robust simulation and live time analysis were performed
to scale this efficiency to higher energies. This generated a final efficiency
of 97.89(6)$\%$ for 40K at a 1 $\mu$s coincidence window. The high value of
the efficiency and its precision limit the impact of untagged EC∗ events as
background. Possible cases of mis-identified events (see A) were studied and
characterized.

A complementary, homogeneous method was also presented using a KSr2I5:Eu
scintillator and dual PMT readout as the inner detector. The KSI crystal
benefits from a large 40K content and almost total X-ray and Auger detection
efficiency. The detector was calibrated using 137Cs, internal 40K, and Eu
internal activation. Due to limited technical detail of the PMTs attached to
the scintillator, work is being performed to replace them with better known
SiPMs, with plans for a future measurement.

With the composite method, a 44 day physics run has been completed using an
enriched, thermally deposited, KCl source. The sensitivity will allow a
successful branching ratio measurement if the true value is similar to the
predicted 0.2$\%$. The isotopes 65Zn and 88Y were also measured for 1.4 and
0.5 days respectively, for future analysis. Our well-characterized setup
should also allow the study of other isotopes.

##  6 Acknowledgements

Marek Karny provided thoughtful comments on an early version of this
manuscript. John Carter and Ryan Ickert provided stimulating input on the
topic of geochronology. Engineering support has been contributed by Miles
Constable and Fabrice Rétière of TRIUMF, as well as by Koby Dering through the
NSERC/Queen’s MRS. Paul Davis designed and supplied an earlier version of the
electronics through the NSERC/University of Alberta MRS.

Work was performed at Oak Ridge National Laboratory, managed by UT-Battelle,
LLC, for the U.S. Department of Energy under Contract DE-AC05-00OR22725.
Thermal deposition was conducted at the Center for Nanophase Materials
Sciences, which is a DOE Office of Science User Facility. This manuscript has
been authored by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725 with
the U.S. Department of Energy. The United States Government retains and the
publisher, by accepting the article for publication, acknowledges that the
United States Government retains a non-exclusive, paid-up, irrevocable, world-
wide license to publish or reproduce the published form of this manuscript, or
allow others to do so, for United States Government purposes. The Department
of Energy will provide public access to these results of federally sponsored
research in accordance with the DOE Public Access Plan
(http://energy.gov/downloads/doe-public-access-plan). Funding in Canada has
been provided by NSERC through SAPIN and SAP RTI grants, as well as by the
Faculty of Arts and Science of Queen’s University, and by the McDonald
Institute. US support has also been supplied by the Joint Institute for
Nuclear Physics and Applications. This material is based upon work supported
by the U.S. Department of Homeland Security under grant no.
2014-DN-077-ARI088-01. Disclaimer: The views and conclusions contained in this
document are those of the authors and should not be interpreted as necessarily
representing the official policies, either expressed or implied, of the U.S.
Department of Homeland Security.

##  Appendix A Classifying events

Our analysis involves determining the number of EC events relative to that of
EC∗ events, to obtain the ratio of the two branching ratios,
$\rho=\frac{I_{EC}}{I_{EC^{*}}}$. Experimentally, by fitting the spectra
obtained in the SDD, we have access to the number of uncoincident and
coincident events which ideally, would be proxies for the number of EC and EC∗
events. In reality, for various reasons, some EC events can be misidentified
as EC∗, and vice-versa. In addition, both types of events can be lost
completely while background events can look like either population. We discuss
these various events in the following text.

###  A.1 EC∗ misidentified as EC

For a source of activity $A$, and branching ratio $I_{EC^{*}}$, the expected
rate of SDD triggers is $A$ $I_{EC^{*}}$$\eta$, where $\eta$ is the fraction
of X-rays/Augers escaping the source, reaching the SDD, and triggering it.
Because the tagging efficiency ($\varepsilon$) of MTAS for $\gamma$s and
conversion electrons from the excited state is not quite 100%, an EC∗ can look
like an EC. Since $\gamma$-rays dominate the relaxation, the expected rate of
such events is $A$ $I_{EC^{*}}$$\eta$$(1-\varepsilon_{\gamma})$, while the
rate of properly identified EC* events is $A$
$I_{EC^{*}}$$\eta$$\varepsilon_{\gamma}$. For our $\sim 98$% efficiencies
(Sec. 4.4), assuming $\rho=0.02$ leads to as many events from this background
as from the signal.

###  A.2 EC misidentified as EC* because of coincidences with MTAS background

Spurious coincidences between EC triggers in the SDD and events in MTAS can
cause the EC to be misidentified as EC∗. Since the rate of MTAS background
$B_{M}$$\approx 2.4$ kHz (Sec. 2.1) is higher than the activity of the sources
used (Table 2), the MTAS background is the dominant factor.

The expected number of background events in an arbitrary time window $T\leq 4\
\mu$s is given by $B_{M}$$T\lesssim 0.01\ll 1$. In practice, though the rate
of background $B_{M}$ is well known, the effective coincidence window, $T$,
may differ from the nominal value due to the rolling coincidence window. We
can, however, use coincidences between calibration data and background to
determine the product $B_{M}$$T$. The Poisson probability of having no
spurious coincidences after an EC SDD trigger is $P_{0}=e^{-B_{M}T}$, and the
probability of having exactly one spurious coincidence is
$P_{1}=e^{-B_{M}T}B_{M}T$. If the rate of SDD events from the source is
$\sigma$, then the rate of source events with 0 background coincidence is
$\sigma P_{0}$, and the rate of source events with 1 background coincidence is
$\sigma P_{1}$. The ratio of these two populations is $P_{1}/P_{0}=B_{M}T$.
This number can be determined by considering the SDD triggered MTAS spectrum.
Neglecting higher-order source-background coincidences, the spectrum is the
sum of the 0-coincidence spectrum and of the 1-coincidence one, the latter
being the convolution of the 0-coincidence spectrum and of the background
spectrum. This is illustrated in Fig. 20 for 54Mn, and Fig. 22 for 65Zn. The
results are listed in Table 8, which shows a strong agreement between the
$B_{M}$$T$ values for both 54Mn and 65Zn, as expected from this parameter that
should be independent of the source isotope and activity.

Figure 22:  Full 65Zn, 4 $\mu$s coincidence window, gamma spectrum in MTAS when triggered by the SDD (dark blue) compared with Geant4 simulated data. All modules of MTAS are used. Components of the simulated spectrum include: The Geant4 single gamma deposition simulation (black, 0-background coincidence spectrum), the MTAS natural background spectrum (light blue), the source-background convolution spectrum (green, 1-background coincidence spectrum), the source-source convolution spectrum (purple) and the $\beta+$ annihilation spectrum (brown). These components are fit (total simulated spectrum in red) to the data to show their individual contributions to the experimental spectrum. Insert shows zoom of lower energy region.  Nominal Coincidence Window ($\mu$s) |  65Zn $B_{M}$$T$ |  54Mn $B_{M}$$T$ |  $B_{S}$$B_{M}$$T$ (Hz) |  $B_{S}$(1-$B_{M}$$T$) (Hz)
---|---|---|---|---
1.0 | 0.0075(1) | 0.0074(1) | 4.0(6)$\times 10^{-6}$ | 5.4(8)$\times 10^{-4}$
2.0 | 0.0126(1) | 0.0125(1) | 6.8(10)$\times 10^{-6}$ | 5.3(8)$\times 10^{-4}$
4.0 | 0.0230(2) | 0.0227(2) | 1.2(2)$\times 10^{-5}$ | 5.3(8)$\times 10^{-4}$
Table 8:  Measured $B_{M}$$T$ values for the different sources as a function
of coincidence windows. Also shown are the expected rate of SDD background
events that look like EC∗($B_{S}$$B_{M}$$T$) and EC ($B_{S}$(1-$B_{M}$$T$))
events as a function of coincidence window. The values of $B_{M}$$T$ are taken
from the 54Mn data.

###  A.3 SDD background

The SDD background, illustrated in Fig. 15, has a rate of
$B_{S}$$=5.4(8)~{}\times~{}10^{-4}$ Hz in the 2-6 keV energy region. The
expected rate of events that are not in coincidence with gamma background, and
that can therefore look like EC events, is $B_{S}$$(1-$$B_{M}$$T)$. The
expected rate of events that can look like an EC* because they are in
coincidence with the gamma background is $B_{S}$$B_{M}$$T$. Values are
provided in Table 8, taking the values of $B_{M}$T from 54Mn in the same
table.

###  A.4 Source tagging efficiency

It is possible for an EC event to be detected in the SDD in spurious
coincidence with a gamma from an EC∗ whose X-ray/Auger was missed. In this
case the EC event would be incorrectly classified as an EC∗. The expected rate
of this occurring is dependent on the X-ray/Auger tagging efficiency ($\eta$).
This efficiency is the probability that when an electron capture decay occurs,
the X-ray/Auger will make it out of the source, into the solid angle of the
SDD, past the dead layer of the SDD and deposit some energy into the detector.
If an energy selection is imposed on the X-rays/Augers that arrive on the SDD,
we label this efficiency $\eta_{E}$. Although simulations can provide a value
for $\eta_{E}$, an experimental measurement can be made by looking at the SDD-
triggered MTAS spectrum with X-ray/Auger selection cuts (seen in Fig. 23 and
Fig. 24). The area ($x$, counts/s) under the single gamma (black) spectrum in
Fig. 23 and Fig. 24 is given as the activity ($A$) of the source multiplied by
branching ratio and the probability of detecting the X-ray/Auger and the gamma
($I_{EC^{*}}$ $\varepsilon_{\gamma}~{}\eta$):

$x=A~{}I_{EC^{*}}\ \varepsilon_{\gamma}\ \eta.$ (4)

With an energy selection on the SDD keeping mainly X-rays, this becomes

$x=A~{}I_{EC^{*}}\ \varepsilon_{\gamma}\ \eta_{E}.$ (5)

In the same figures, the area ($y$, counts/s) under the purple curve
represents events when two EC∗ decays occur and either one or two X-rays from
the two decays are detected, and both gammas are seen in MTAS. The area can be
expressed as

$y=(A~{}I_{EC^{*}}\varepsilon_{\gamma})^{2}\ T\
\left[\underbrace{2\eta(1-\eta)}_{\text{1~{}X-ray}}+\underbrace{\eta^{2}}_{\text{both~{}X-rays}}\right],$
(6)

where $T$ is the true coincidence window. Due to the shape of the SDD
spectrum, if the selection $\eta_{E}$ is made on the K-shell lines, the
probability that two X-rays sum to this selection is negligible, leaving only
the term coming from single interactions:

$y=(A~{}I_{EC^{*}}\varepsilon_{\gamma})^{2}T\ 2\eta_{E}(1-\eta_{E}).$ (7)

Rearranging Eqn. 5 and Eqn. 7 solves for $\eta_{E}$:

$\eta_{E}=\frac{2Tx^{2}}{(y+2Tx^{2})}.$ (8)

The parameters $x$ and $y$ can be measured from the fit in Fig. 23 and Fig.
24. The true coincidence window value (T) can be determined from $B_{M}$,
Table 1 and, $B_{M}$$T$, Table 8. The measured $\eta_{E}$ values for 54Mn and
65Zn are shown in Table 9. A tight x-ray gate of 5.0 - 7.0 keV for the 54Mn
and 7.7 - 9.5 keV for the 65Zn was chosen for this analysis.

Nominal Coincidence Window ($\mu$s) |  54Mn $\eta_{E}$ |  65Zn $\eta_{E}$
---|---|---
1.0 | 31.1 $\pm$ 13.6 $\%$ | 20.2 $\pm$ 2.3 $\%$
2.0 | 26.0 $\pm$ 7.5 $\%$ | 22.2 $\pm$ 2.1 $\%$
4.0 | 24.0 $\pm$ 4.8 $\%$ | 21.8 $\pm$ 1.5 $\%$
Weighted Average | 25.9 $\pm$ 3.9% | 21.5 $\pm$ 1.1 %
Table 9:  Measured $\eta_{E}$ values for the 65Zn and 54Mn sources as a
function of coincidence windows. The values for $\eta_{E}$ agree within one
standard deviation as a function of coincidence window.

The values for $\eta_{E}$ shown in Table 9 agree within one standard deviation
when comparing different coincidence windows of the same source. The error on
the $\eta_{E}$ calculated for 54Mn is high due to the small integral of the
source-source convolution component (see Fig. 23). This is due to the lower
activity of the 54Mn source. Additionally, it is expected that the 65Zn source
will have a higher efficiency than the 54Mn source due to its higher X-ray
energy being more likely to penetrate past the dead layer of the detector.
However, this measurement is also dependent on the source geometry and
construction which could be different between our isotopes.

Figure 23:  54Mn, 4 $\mu$s coincidence window, gamma spectrum in MTAS when
triggered by the SDD (dark blue) with an energy gate of 5.0 - 7.0 keV. All
modules of MTAS are used. Components of the simulated spectrum include: The
Geant4 single gamma deposition simulation (black), the MTAS natural background
spectrum (light blue), the source-background convolution spectrum (green) and
the source-source convolution spectrum (purple). These components are fit
(total simulated spectrum in red) to the data to show their individual
contributions to the experimental spectrum. Insert shows zoom of lower energy
region. Figure 24:  65Zn, 4 $\mu$s coincidence window, gamma spectrum in MTAS
when triggered by the SDD (dark blue) with an energy gate of 7.7 - 9.5 keV.
All modules of MTAS are used. Components of the simulated spectrum include:
The Geant4 single gamma deposition simulation (black), the MTAS natural
background spectrum (light blue), the source-background convolution spectrum
(green), the source-source convolution spectrum (purple) and the $\beta+$
annihilation spectrum (brown). These components are fit (total simulated
spectrum in red) to the data to show their individual contributions to the
experimental spectrum. Insert shows zoom of lower energy region.

Other effects, due to coincidences with the various radioactive sources and
their decay channels, are possible, and will be addressed in publications
pertaining to those sources.

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