An unfolding algorithm for high resolution microcalorimetric beta spectrometry

Nuclear Inst. and Methods in Physics Research, A xxx (xxxx) xxx

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An unfolding algorithm for high resolution microcalorimetric beta spectrometry Michael Paulsen a,b ,∗, Karsten Kossert c , Jörn Beyer a a

Physikalisch-Technische Bundesanstalt, Abbestrasse 2-12, D-10587 Berlin, Germany Kirchhoff-Institute for Physics, Im Neuenheimer Feld 227, D-69120 Heidelberg, Germany c Physikalisch-Technische Bundesanstalt, Bundesallee 100, D-38116 Braunschweig, Germany b

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Keywords: Beta spectrometry Unfolding Deconvolution Microcalorimeter Monte Carlo simulation Bremsstrahlung

ABSTRACT In this work, a matrix inversion algorithm for beta spectrometers is proposed that relies on Monte Carlo simulations of single-energy bin distributions. The algorithm allows for high resolution statistical correction of measured beta spectra and does not lead to underdetermined equation systems. An advantage of the method is that a prior knowledge of the true spectrum is not needed. The algorithm is consistent for unfolding beta spectra measured with Metallic Magnetic Calorimeters that employ 4𝜋 solid angle absorbers. It also shows considerable potential for being applied to microcalorimeters with other absorber geometries.

1. Introduction The precise knowledge of beta spectrum shapes is relevant in various fields such as radionuclide metrology, e.g. when determining the activity of beta emitting isotopes by means of liquid scintillation counting (Broda et al. [1]), Čerenkov counting (Kossert et al. [2]) or classical microcalorimetry (Collé [3]). Beta spectra are also relevant for fundamental research, e.g. due to the relation between beta spectra and neutrino spectra as shown in Mougeot [4] and for consideration in radioactive waste management as well as for applications in nuclear medicine. Beta spectra are continuous and can be measured by methods that employ semiconductor detectors, scintillation devices or magnetic spectrometers. These methods suffer from rather low energy resolutions and the measurement of low-energy electrons is difficult due to absorption and attenuation effects within sources and dead layers of detectors. However, accurate measurements of beta spectra with low maximum energy or the low-energy part of beta spectra with higher end-point energy is of highest importance for several measurement techniques, see e.g. Kossert et al. [5], Kossert and Mougeot [6] and Kossert et al. [7]. Such measurements are also indispensable for sound validations of theoretical models and corresponding improvements of calculation methods as in Mougeot and Bisch [8] and Mougeot [9], respectively. A relatively new approach for beta spectra measurements is based on Metallic Magnetic Calorimeters (MMC) [10–12] which has proven to be among the best techniques in terms of energy resolution, especially for low energy beta transitions and is of increasing interest for radionuclide metrology [13,14]. The radionuclide to be measured is embedded

in a 4𝜋 absorber, dimensioned to absorb the complete energy of the emitted beta particle. The absorbed energy increases the temperature of the absorber, which is thermally connected to the paramagnetic sensor of the MMC. The temperature increase causes the paramagnet, which is situated in an induced magnetic field, to change its magnetization and the corresponding magnetic flux change can be read out very precisely using a Superconducting Quantum Interference Device (SQUID) [15]. Thus, the energies of single beta decays can be detected by the MMC and is proportional to the output signal of the SQUID. To achieve a large signal to noise ratio, an MMC beta spectrometer typically operates at temperatures below 50 mK. When aiming at a high energy resolution it is important to keep the combined sensor and absorber heat capacity as low as possible. Hence, the absorber dimensions are often designed to be rather small but large enough to ensure that all beta electrons are completely stopped within the absorber. Considering a high-Z material for the absorber (e.g. gold, with Z = 79) and beta emitters with end-point energies of a few hundred keV or more, fractions of the initial energies of emitted beta particles may not be detected due to bremsstrahlung escaping from the absorber. In this case, the measured beta spectrum features distortions and needs to be corrected. Fig. 1 illustrates simulated decay events of beta electrons starting in the center of a 4𝜋 absorber. In all cases, the electrons are stopped well before reaching the border of the absorber whereas bremsstrahlung photons often escape. The effect causes some electrons with high initial energies to be detected with a reduced energy. From such a simulation it is rather straightforward to predict the measured spectrum provided that the true beta spectrum and the absorber geometry are known. Fig. 1

∗ Correspondence to: Physikalisch-Technische Bundesanstalt (AG.7.61), Abbestrasse 2-12, D-10587 Berlin, Germany. E-mail address: [email protected] (M. Paulsen).

https://doi.org/10.1016/j.nima.2019.163128 Received 1 April 2019; Received in revised form 11 October 2019; Accepted 12 November 2019 Available online xxxx 0168-9002/© 2019 Physikalisch-Technische Bundesanstalt (PTB). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Please cite this article as: M. Paulsen, K. Kossert and J. Beyer, An unfolding algorithm for high resolution microcalorimetric beta spectrometry, Nuclear Inst. and Methods in Physics Research, A (2019) 163128, https://doi.org/10.1016/j.nima.2019.163128.

M. Paulsen, K. Kossert and J. Beyer

Nuclear Inst. and Methods in Physics Research, A xxx (xxxx) xxx

Fig. 1. Plot of 1000 simulated particle tracks from a 36 Cl point source embedded in a 4𝜋 Au absorber (left) with dimensions 0.6 × 0.6 × 0.6 mm3 together with the simulated true and measured spectra (right) having bin width 10 keV. The total energy loss is approximately 0.8%.

2. The statistical discrete unfolding problem

also shows that the measured spectrum underestimates the probabilities for electron emission with high energies whereas low-energy emissions are overestimated. This leads to a systematic skew in the measured spectrum. To retrieve the true beta spectrum from the measured spectrum requires unfolding (or deconvolution) procedures [16,17]. In this work, we use a divide and conquer approach and propose to divide the problem of unfolding such measured data (which is typically in the form of a histogram i.e. a discretized spectrum) from high resolution microcalorimeters into two successive parts:

)𝑇 ( Let 𝒉𝑁,𝑛 ∶ = ℎ𝑁,𝑛,1 , … , ℎ𝑁,𝑛,𝑁 denote the vector of all true histogram heights for n beta decays over N energy bins and 𝒉𝑚𝑒𝑎𝑠 𝑁,𝑛 ∶ = ( )𝑇 𝑚𝑒𝑎𝑠 ℎ𝑚𝑒𝑎𝑠 , … , ℎ denote the corresponding vector for the measured 𝑁,𝑛,𝑁 𝑁,𝑛,1 histogram. It is obvious that these objects are 𝑁 × 1 random vectors in 1 𝑛 which each component can take a value in {0, 𝑛𝛥𝐸 , 2 , … , 𝑛𝛥𝐸 }. 𝑁 𝑛𝛥𝐸𝑁 𝑁 They correspond to discrete samples of the corresponding continuous true and measured spectra, respectively [29]. The statistical discrete unfolding problem is defined as 𝒉𝑚𝑒𝑎𝑠 𝑁,𝑛 = 𝑹 𝑁 𝒉𝑁,𝑛

(i) Reducing the noise, pile-up and background of the raw pulse data e.g. via optimal filtering which is a standard procedure when using MMCs [18–20] or using other methods such as a resolution correction [21]. The resulting pulse height distribution is referred to as the measured histogram in this work. (ii) Applying a matrix inversion unfolding algorithm to correct for systematic energy losses within well-defined absorbers. This amounts to estimating a response matrix for the physical properties of the absorber using Monte Carlo simulations. The unfolded histogram is then retrieved from the measured histogram using the calculated absorber response matrix.

[𝑠𝑡𝑎𝑡𝑖𝑠𝑡𝑖𝑐𝑎𝑙 𝑑𝑖𝑠𝑐𝑟𝑒𝑡𝑒 𝑢𝑛𝑓 𝑜𝑙𝑑𝑖𝑛𝑔 𝑝𝑟𝑜𝑏𝑙𝑒𝑚]

(1)

where 𝑹𝑁 denotes the discrete response (forward) operator given by the 𝑁 × 𝑁 matrix ⎛ 𝑅11 𝑹𝑁 ∶ = ⎜ ⋮ ⎜ ⎝ 𝑅𝑁1

⋯ ⋱ ⋯

𝑅1𝑁 ⋮ 𝑅𝑁𝑁

⎞ ⎟, ⎟ ⎠

[𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒 𝑚𝑎𝑡𝑟𝑖𝑥 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑟] (2)

and 𝑅𝑖𝑗 denotes the response coefficient coupling the 𝑖th energy bin of the measured histogram with the 𝑗th energy bin of the true histogram [16]. To solve the discrete unfolding problem, one needs to estimate the forward response matrix 𝑹𝑁 and then calculate 𝒉𝑁,𝑛 . A direct inspection indicates that the system is heavily underdetermined as there are (𝑁 2 + 𝑁) unknowns and only N equations. It may be considered as an inverse problem [30–32]. As will be shown below, the system of equations simplifies considerably using the proposed Monte Carlo simulation approach such that the response matrix can be estimated, and from this, the true histogram may be calculated.

The main contribution of this work is describing the regularity, implementation and performance of the matrix inversion method of part (ii). While extensive Monte Carlo simulations are needed for the second part, the procedure is highly practical and applicable to various absorber geometries as calculated examples indicate. Similar, somewhat simpler matrix inversion methods have been used in several works e.g. for unfolding gamma spectra [22–24]. However, to the best of our knowledge, a matrix inversion approach of this kind has not been applied to unfold high resolution beta spectra, thus far. An advantage of the proposed unfolding method is that the true spectrum shape is not needed as an input. The presented method will be required in the analysis of experimentally determined beta spectra of 36 Cl (𝐸max = 709.5 keV) and other radionuclides which are being measured within the scope of the EMPIR MetroBeta project [25]. The paper is organized as follows: the general and statistical unfolding problems are formulated in Section 2 along with mathematical definitions. Section 3, where the unfolding algorithm is presented in detail, constitutes the main part of the paper. The consistency of the method is demonstrated in Section 4 using simulated data. In Section 5, the accuracy of the algorithm is tested by: (a) correcting external bremsstrahlung effects for independently simulated measurement data and (b) unfolding a high resolution 36 Cl spectrum that was measured with a Si(Li) detector [26, 27] and compared to a reference spectrum [12,28] measured with an MMC detector, respectively. A conclusion and a discussion are provided in Section 6. The mathematical proofs are found in the supplementary material section.

3. The unfolding algorithm The proposed unfolding algorithm is based on the following assumptions: (A1) The extensive utilization of Monte Carlo simulations is a sufficiently accurate method to simulate particle and radiation transport in matter [33–35]. (A2) The detector dynamics are assumed to be slower than the particle material dynamics: if there are several energy depositions for a single decay, the sum of these are detected as a single energy deposition. This is a typical property of microcalorimetric detectors. (A3) It is assumed that for at least one of the simulated decay events in every single-energy bin, the entire decay energy is deposited in the absorber. (A4) The measured decays are assumed to be independent of each other i.e. the detector and the radionuclide source have no memory and pile-up issues are not considered. 2

Please cite this article as: M. Paulsen, K. Kossert and J. Beyer, An unfolding algorithm for high resolution microcalorimetric beta spectrometry, Nuclear Inst. and Methods in Physics Research, A (2019) 163128, https://doi.org/10.1016/j.nima.2019.163128.

M. Paulsen, K. Kossert and J. Beyer

Nuclear Inst. and Methods in Physics Research, A xxx (xxxx) xxx

is operated upon:

3.1. Single-energy bin Monte Carlo simulations covering the entire energy range of the measured spectrum

⎛ 0 ⎜ ⎜ ⋮ ⎜ 0 ⎝

A key challenge in the statistical discrete unfolding problem consists in handling the underdetermined system of equations described above. To resolve this issue, N single-energy bin pulse height distributions are simulated for the given absorber geometry. The bin energies cover the entire range of the measured spectrum. This may seem as an overly laborious method, however, a multi-energy bin simulation and correction approach will in general lead to an underdetermined system of linear equations, see Proposition 8.4 in the supplementary material. The single-energy bin approach will allow us to consistently estimate the forward response matrix, column by column, and thus solve the unfolding problem, statistically. The (forward) input in the algorithm are m ∼ 105 simulated (pseudo)-random variables i.e. beta particle energies, uniformly distributed over the 𝑗th energy bin of length 𝛥𝐸𝑁 . Thus, the response of a single energy bin is simulated at a time and the corresponding histogram is very simple, as depicted in Fig. 2. The corresponding histogram heights are given by 𝒉𝑠𝑖𝑚_𝑖𝑛𝑝𝑢𝑡,𝑗 𝑁,𝑚

( )𝑇 = 0, … , 0, ℎ𝑠𝑖𝑚_𝑖𝑛𝑝𝑢𝑡,𝑗 ,0…,0 , 𝑁,𝑚,𝑗

where

ℎ𝑠𝑖𝑚_𝑖𝑛𝑝𝑢𝑡,𝑗 𝑁,𝑚,𝑗

⎛ 𝑆 𝑗,1←𝑗 ⎜ 𝑁,𝑚 ⋮ ⎜ ⎜ 𝑆 𝑗,𝑁←𝑗 ⎝ 𝑁,𝑚

𝑺 𝑁,𝑚 =

⎞⎛ 0 ⎟ ⎜ 𝑠𝑖𝑚_𝑖𝑛𝑝𝑢𝑡,𝑗 ℎ ⎟ ⎜ 𝑁,𝑚,𝑗 ⎟⎝ 0 ⎠

⎞ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎟ ⎠

⎞ ⎛ ℎ𝑠𝑖𝑚_𝑜𝑢𝑡𝑝𝑢𝑡,𝑗 ⎟ 𝑠𝑖𝑚_𝑖𝑛𝑝𝑢𝑡,𝑗 ⎜ 𝑁,𝑚,1 =⎜ ⋮ ⎟ ℎ𝑁,𝑚,𝑗 ⎟ ⎜ ℎ𝑠𝑖𝑚_𝑜𝑢𝑡𝑝𝑢𝑡,𝑗 ⎠ ⎝ 𝑁,𝑚,𝑁

𝑁 ∑

(7)

⎞ ⎟ ⎟, ⎟ ⎠

(8)

𝑺 𝑗𝑁,𝑚

𝑗=1

⎛ 𝑆 1,1←1 ⎜ 𝑁,𝑚 =⎜ ⋮ ⎜ 0 ⎝

⋯ ⋱ ⋯

𝑁,1←𝑁 𝑆𝑁,𝑚 ⋮ 𝑁,𝑁←𝑁 𝑆𝑁,𝑚

⎞ ⎟ ⎟ ⎟ ⎠

(9) [𝑡𝑜𝑡𝑎𝑙 𝑓 𝑜𝑟𝑤𝑎𝑟𝑑 𝑚𝑎𝑡𝑟𝑖𝑥].

By construction and conservation of energy (see Lemma 8.1) it follows that the total forward matrix is an 𝑁 × 𝑁-dimensional random variable which is upper triangular. The matrix operator 𝑺 𝑁,𝑚 provides an estimate of the response matrix i.e. how each energy bin of the true histogram is modified for the given absorber configuration, and we may write (see Proposition 8.3): 1

𝑹𝑁 = 𝑺 𝑁,𝑚 + 𝑶𝑁×𝑁 (𝑚− 2 ), where 𝑶𝑵×𝑵 (𝑚

− 21

(10)

) denotes an 𝑁 ×𝑁-dimensional error matrix in which 1

each component converges as 𝑂(𝑚− 2 ), as only statistical errors are considered due to assumption (A1) above. Inserting the estimate into to the statistical discrete unfolding problem, it follows that [ ] − 21 𝒉𝑚𝑒𝑎𝑠 ) 𝒉𝑁,𝑛 . (11) 𝑁,𝑛 = 𝑹 𝑁 𝒉𝑁,𝑛 = 𝑺 𝑁,𝑚 + 𝑶𝑵×𝑵 (𝑚 ( )−1 3.4. Calculating the backward operator 𝑺 𝑁,𝑚 and the unfolded histogram

(5) When the algorithm is applied, an estimate 𝒉𝑎𝑙𝑔𝑜 is calculated for 𝑁,𝑛 the true histogram 𝒉𝑁,𝑛 by formally inverting the total forward matrix to get

where the single-energy bin forward matrix 𝑺 𝑗𝑁,𝑚 is defined for each 𝑗 as 𝑗,1←𝑗 𝑆𝑁,𝑚 ⋮ 𝑗,𝑁←𝑗 𝑆𝑁,𝑚

0 ⋮ 0

The corresponding estimate of the total forward matrix is given by the sum of the single-energy bin forward matrices over all energy bins:

Next, the single-energy bin forward unfolding problem is considered:

0 ⋮ 0

⋯ ⋱ ⋯

3.3. Calculating the total forward matrix

1 . = 𝛥𝐸𝑁

3.2. Calculating the single-energy bin forward matrix operators

⋯ ⋱ ⋯

0 ⋮ 0

𝑗,𝑘←𝑗 where the matrix entries 𝑆𝑁,𝑚 are easily solved for since the input and output vectors are known as a priori inputs and as simulation result outputs, respectively.

It is noted that in 4𝜋 absorbers, for small 𝑗 (small input energy) the vector above has all components equal to zero, except for the 𝑗th component. This is because bremsstrahlung is very improbable at low energies. However, for large 𝑗 (high input energy), bremsstrahlung generation and hence, energy escape, is quite likely and the components will be non-zero for some components with indices less than or equal to 𝑗. In general, probability mass will only ‘‘migrate to the left’’, by conservation of energy. Thus, given that the simulated input particles have energies exclusively in the 𝑗th energy bin, the only migration that may occur is to a lower or within the identically indexed bin, see Fig. 2.

⎛ 0 ⎜ ∶= ⎜ ⋮ ⎜ 0 ⎝

𝑗,1←𝑗 𝑆𝑁,𝑚 ⋮ 𝑗,𝑁←𝑗 𝑆𝑁,𝑚

i.e.

The 𝑚 simulated beta particles interact with the absorber material and, depending on their energy, cause bremsstrahlung, scattering or other radiation that may escape from the absorber. The total energy deposited by each of the m simulated decays in the absorber via primary and secondary particles is recorded in the simulation. The simulation output for each input bin vector 𝑗 is given by the corresponding normalized simulated output histogram heights, i.e. an 𝑁 × 1 vector: ( )𝑇 𝒉𝑠𝑖𝑚_𝑜𝑢𝑡𝑝𝑢𝑡,𝑗 = ℎ𝑠𝑖𝑚_𝑜𝑢𝑡𝑝𝑢𝑡,𝑗 , … , ℎ𝑠𝑖𝑚_𝑜𝑢𝑡𝑝𝑢𝑡,𝑗 (4) 𝑁,𝑚 𝑁,𝑚,𝑁 𝑁,𝑚,1

𝑺 𝑗𝑁,𝑚

0 ⋮ 0

⎛ ℎ𝑠𝑖𝑚_𝑜𝑢𝑡𝑝𝑢𝑡,𝑗 ⎜ 𝑁,𝑚,1 =⎜ ⋮ ⎜ ℎ𝑠𝑖𝑚_𝑜𝑢𝑡𝑝𝑢𝑡,𝑗 ⎝ 𝑁,𝑚,𝑁

(3)

𝑠𝑖𝑚_𝑖𝑛𝑝𝑢𝑡,𝑗 𝑺 𝑗𝑁,𝑚 𝒉𝑁 = 𝒉𝑠𝑖𝑚_𝑜𝑢𝑡𝑝𝑢𝑡,𝑗 , 𝑁

⋯ ⋱ ⋯

0 ⋮ 0

⋯ ⋱ ⋯

0 ⋮ 0

⎞ ⎟ ⎟, ⎟ ⎠

( )−1 𝑚𝑒𝑎𝑠 𝒉𝑁,𝑛 = 𝒉𝑎𝑙𝑔𝑜 = 𝑺 𝑁,𝑚 𝒉𝑁,𝑛 𝑁,𝑚,𝑛

(6)

[𝑎𝑙𝑔𝑜𝑟𝑖𝑡ℎ𝑚 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒 𝑜𝑓 𝑡𝑟𝑢𝑒 ℎ𝑖𝑠𝑡𝑜𝑔𝑟𝑎𝑚]

(12) ( )−1 where the inverse 𝑺 𝑚,𝑁 exists with probability 1 as shown in Proposition 8.2, see Fig. 3. Various numerical techniques, such as least square methods, have been developed to solve the equation without inverting the matrix with e.g. Gauss–Jordan-elimination, which may be numerically costly and imprecise, see e.g. [36].

[𝑠𝑖𝑛𝑔𝑙𝑒 − 𝑒𝑛𝑒𝑟𝑔𝑦 𝑏𝑖𝑛 𝑓 𝑜𝑟𝑤𝑎𝑟𝑑 𝑚𝑎𝑡𝑟𝑖𝑥] 𝑗,𝑘←𝑗 and 𝑆𝑁,𝑚 ∈ [0, 1] denotes the transformation coefficients for the probability mass migration from energy bin 𝑗 to energy bin 𝑘 for m 𝑗,𝑘←𝑗 samples in the simulations. The properties of 𝑆𝑁,𝑚 are derived in Lemma 8.1. Please note that all elements of the forward matrix are zero except for in the 𝑗th column, as only the 𝑗th histogram energy

3

Please cite this article as: M. Paulsen, K. Kossert and J. Beyer, An unfolding algorithm for high resolution microcalorimetric beta spectrometry, Nuclear Inst. and Methods in Physics Research, A (2019) 163128, https://doi.org/10.1016/j.nima.2019.163128.

M. Paulsen, K. Kossert and J. Beyer

Nuclear Inst. and Methods in Physics Research, A xxx (xxxx) xxx

Fig. 2. Illustration of 𝒉𝑠𝑖𝑚_𝑖𝑛𝑝𝑢𝑡,𝑗 (left), the probability mass is confined to the 𝑗th energy bin. This is generally not the case for 𝒉𝑠𝑖𝑚_𝑜𝑢𝑡𝑝𝑢𝑡,𝑗 (right), where an isotropic point source 𝑚,𝑁 𝑚,𝑁 of monoenergetic electrons having an energy of 1 MeV, embedded at the center of a Au cube of side length 0.6 mm (4𝜋 absorber) was simulated for 𝑚 = 106 particles.

Fig. 3. Heat map of a forward matrix (left) and its inverse i.e. the backward matrix (right). The gray area corresponds to zero-entries. The forward matrix was generated by simulating within 710 energy bins of width 1 keV for an isotropic point source embedded in a 4𝜋 Au absorber (0.6 × 0.6 × 0.6 mm3 ) and in each energy bin, m = 106 particles were simulated. Column-wise, it contains the energy loss information of Fig. 2. The diagonal elements are close to 1 in both matrices, the off-diagonal elements x 𝑓 𝑜𝑟𝑤 ∈ (0, 0.0055) and x 𝐛𝐚𝐜𝐤 ∈ (−0.0062, 10−6 ).

the following, the implementation procedure is described along with results and a discussion of calculated examples. While the mentioned procedures are software specific, the algorithm could be implemented using many types of Monte Carlo software e.g. GEANT4; PENELOPE or EGSnrc [33–35].

3.5. Error of the complete algorithm when used on measurement data It should be noted that the measured histogram typically contains an 𝑁 ×1 error vector since it is the result of a noisy measurement combined with a filtering procedure as noted in the introduction: ( 1) 𝑚𝑒𝑎𝑠 −2 , (13) 𝒉𝑚𝑒𝑎𝑠 𝑁,𝑛 = 𝒉𝑁,𝑛 + 𝑶𝑵 𝑛

4.1. Procedure for algorithm implementation

𝑚𝑒𝑎𝑠 𝒉𝑁,𝑛

where denotes an idealized, noise-free measured histogram. The error component of each bin is assumed to be normally distributed and small compared to the measured bin height such that the propagation of uncertainty, when the matrix inverse (which has small elements as depicted in Fig. 3) is applied, does not lead to erratic behavior. The magnitude of the algorithm estimate error is formally given by ‖ 𝑎𝑙𝑔𝑜 ‖ ‖𝒉𝑁,𝑛 − 𝒉𝑁,𝑚,𝑛 ‖ ‖ ‖ [ ( 1 )]−1 ‖ ‖ ‖ ‖ = ‖𝒉𝑁,𝑛 − 𝑺 𝑁,𝑚 + 𝑶𝑵×𝑵 𝑚− 2 𝒉𝑚𝑒𝑎𝑠 ‖ 𝑁,𝑛 ‖ ‖ ‖ ‖ [ ( 1 )]−1 ( 𝑚𝑒𝑎𝑠 ( 1 ))‖ ‖ ‖ ‖ −2 = ‖𝒉𝑁,𝑛 − 𝑺 𝑁,𝑚 + 𝑶𝑵×𝑵 𝑚 𝒉𝑁,𝑛 + 𝑶𝑵 𝑛− 2 ‖ (14) ‖ ‖ ‖ ‖ and depends on the numerical method used to solve for the algorithm estimate via the matrix, the number of measured decays n, the number of samples m used to construct the total forward matrix, the number of bins N and of course on the choice of norm ‖⋅‖.

The unfolding algorithm was implemented in Python and the Monte Carlo simulations were carried out using the software EGSnrc [35]. This consisted of the following steps: (1) Simulation of the true histogram: A 36 Cl (𝐸max = 709.5 keV) beta spectrum provided by EGSnrc, based on the nuclear decay scheme [37], was used for the true spectrum. From this the true histogram 𝒉𝑁,𝑛 heights were generated by sampling n∼106 energies in a point-like source geometry. The energies, positions and trajectories were saved in the IAEA phase space (phsp) file format [38] using the code EGS Brachy [39]. (2) Simulation of the measured histogram: The measured 36 Cl histogram heights 𝒉𝑚𝑒𝑎𝑠 𝑁,𝑛 were simulated in absorber geometries having typical MMC dimensions, see Table 1 using the very energies of the true histogram 𝒉𝑁,𝑛 found in the phsp file. (3) Construction of the total forward matrix: The total forward matrix 𝑺 𝑁,𝑚 was constructed via single-energy bin simulations in the absorber geometry used for the measured histogram heights. These simulations covered the energy range from 0 to 710 keV. (4) Calculation of the unfolded histogram heights: The unfolded histogram heights i.e. the algorithm estimate 𝒉𝑎𝑙𝑔𝑜 = 𝑁,𝑚,𝑛 ( )−1 𝑚𝑒𝑎𝑠 𝑺 𝑁,𝑚 𝒉𝑁,𝑛 were calculated in Python using the forward matrix and the simulated measured histogram heights via the built-in least squares method numpy.linalg.solve [40]. (5) Calculation of the algorithm estimate error:

4. Implementation and consistency results of the algorithm In this section, the algorithm is applied on simulated measured histograms. A practical advantage of using simulated data is that the true histogram is known first hand and thus, the error of the algorithm estimate can be calculated in a straight-forward manner. In 4

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Table 1 Overview of the simulation results for the algorithm implementation.

Fig. 4. Plots of the simulated measured histogram (top left) and the unfolded histogram (top right) in a 4𝜋 absorber along with the true histogram. There are some bremsstrahlung losses at higher energies, which shifts the simulated measured histogram skew slightly to the left. The skew becomes evident in the bottom residual plot which also shows the correction provided by the unfolding algorithm. Here, the number of events for each single-energy bin simulation was 𝑚 = 2.5 ⋅ 105 and the number of energy bins is 𝑁 = 71 i.e. the bin width is 10 keV.

In order to calculate the algorithm error, the true histogram heights 𝒉𝑁,𝑛 were retrieved from the phsp file via simulation of the same point-like source geometry embedded in a large Au block (dimensions: 1 × 1 × 1 m3 ) to ensure complete energy deposition of primary and secondary particles, in order to exclude any bremsstrahlung effects or information loss in the true histogram. In step (2), (3) and (5) the pulse height distribution code tutor7pp [41] was used. To quantify the consistency and correction capacity of the unfolding 𝑎𝑙𝑔𝑜 algorithm, the residual vectors 𝒓𝑚𝑒𝑎𝑠 ∶ = 𝒉𝑁,𝑛 − 𝒉𝑚𝑒𝑎𝑠 𝑁,𝑛 and 𝒓𝑁,𝑚,𝑛 ∶ = 𝑁,𝑛

and analogously for the measured histogram. This corresponds to summing the absolute values of the corresponding residual vector elements.

4.2. Consistency results

Two different absorber geometries and various measurement relevant values of the algorithm parameters were considered. The steps (1)–(5) described in the previous section were carried out for 100 random 36 Cl histograms, by randomly sampling the initial seeds of the ranmar random number generator [42] in step (1) according to a uniform distribution. An overview of the results is found in Table 1 along with plots for selected parameter values in Figs. 4–6.

𝒉𝑁,𝑛 − 𝒉𝑎𝑙𝑔𝑜 , tracking the bin-wise height differences, were defined. 𝑁,𝑚,𝑛 The algorithm error was calculated in the 𝓁 1 -norm: ∑ ‖ 𝑎𝑙𝑔𝑜 ‖ |ℎ𝑁,𝑛,𝑗 − ℎ𝑎𝑙𝑔𝑜 | ‖𝒉𝑁,𝑛 − 𝒉𝑁,𝑚,𝑛 ‖ 1 = 𝑁,𝑚,𝑛,𝑗 ‖ ‖𝓁 𝑁

(15)

𝑗=1

5

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Fig. 5. The simulated measured spectrum for a 2𝜋 absorber geometry are plotted on the top left. To the top right, the true spectrum and the algorithm unfolded spectrum are depicted. At the bottom, the residual plot shows the drastic effect of the unfolding correction Here, the number of events for each single-energy bin simulation was 𝑚 = 5 ⋅ 105 and the number of energy bins is 𝑁 = 142 i.e. the bin width is 5 keV.

Fig. 6. For smaller energy bin width (1 keV, 𝑁 = 710 energy bins), the variance in the simulation becomes more visible. A histogram representation of the simulated measured spectrum (top left) and the unfolded spectrum (top right) in a 4𝜋 absorber is depicted along with the simulated true spectrum. The bremsstrahlung skew is still visible in the bottom residual plot and the correction provided by the unfolding algorithm is substantial. The number of events for each single-energy bin simulation was 𝑚 = 106 .

are lost in the measurement process. In practice, a rather precise knowledge of the source geometry is needed for 2𝜋 absorbers, as variations of the source distance above the absorber showed. A variation of 10 μm approximately doubled the 𝓁 1 -error, nonetheless the unfolded histogram would still be quite useful as this corresponds to a moderately distorted spectrum. As expected, an increased number of samples for the matrix generation results in a corresponding variance reduction of the algorithm error in the outermost right column. The results could be improved by optimizing the code e.g. by reducing

4.3. Discussion of implementation and consistency results The implementation shows that the algorithm can be used to unfold measured beta spectra in a consistent manner, especially when 4𝜋 absorbers are used. Concerning the rather subtle bremsstrahlung effect, the algorithm gives notable corrections. Table 1 indicates that the unfolding algorithm reduces the simulated measured 𝓁 1 -error by at least 82% at an energy bin width of 1 keV. This is also the case for the 2𝜋 absorbers although approximately half of the information/statistics 6

Please cite this article as: M. Paulsen, K. Kossert and J. Beyer, An unfolding algorithm for high resolution microcalorimetric beta spectrometry, Nuclear Inst. and Methods in Physics Research, A (2019) 163128, https://doi.org/10.1016/j.nima.2019.163128.

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Fig. 7. Approx. 𝑛 = 107 beta particles of 36 Cl (𝐸𝑚𝑎𝑥 = 709.5 keV) were simulated in PENELOPE and plotted over 142 bins (red dots). The 4𝜋 geometry consisted of a Au cylinder (height = 0.6 mm, radius = 0.3 mm) and EGSnrc was used to calculate the response matrix (𝑁 = 142 energy bins, 𝑚 = 106 events per bin), which in turn was applied to unfold the simulated measured histogram and the unfolded histogram was plotted (green crosses).

Fig. 8. Approx. 𝑛 = 107 of 36 Cl (𝐸𝑚𝑎𝑥 = 709.5 keV) were simulated in PENELOPE and plotted over 142 bins (red dots). The 2𝜋 geometry consisted of a Au cylinder (height = 0.3 mm, radius = 0.3 mm) and EGSnrc was used to calculate the response matrix (𝑁 = 142 energy bins, 𝑚 = 106 events per bin) for an idealized point source on top of the absorber, which in turn was applied to unfold the simulated measured histogram (plotted as green crosses).

and 2𝜋) were simulated for a high number of beta particles to make any systematic deviations visible. The results are depicted in Figs. 7–8. The 𝓁 1 -error is reduced by 79% using the unfolding algorithm for the PENELOPE data in the 4𝜋 geometry. The remaining variation seems to be statistical. The 𝓁 1 -error is reduced by 98% using the unfolding algorithm for the PENELOPE data in a 2𝜋 geometry, with a deviation below ∼15 keV that is sensitive to the cut-off energy (below this energy, the considered particles are no longer simulated) and scattering parameters set in the respective software. In both software codes, a very high sensitivity was chosen and cut-off energies of 0.5 keV and 1 keV were used, respectively.

the so-called cut off energies of the simulated particles (allowing for simulation at lower energies) or using variance reduction methods. Calculation times may also need to be considered. For instance, in the 4𝜋 absorber example (𝑁 = 71 energy bins, 𝑚 = 2.5 ⋅ 105 events per bin) the calculation time was approximately three hours on an Intel i7 laptop PC using a single core to construct the forward matrix. For smaller energy bin widths and more samples (𝑁 = 710 energy bins, 𝑚 = 1 ⋅ 106 events per bin) the calculation time scaled by a factor of 40 to around 120 h. The Monte Carlo simulations consisted of more than 99% of the calculation time. These may easily be parallelized as the single-energy bin simulations can be run independently. Furthermore, most of the EGSnrc code allows for built-in parallelization. Thus, a large part of the simulations was carried out on a multicore computer cluster.

5.2. The unfolding algorithm applied to measurement data 5. Accuracy of the proposed unfolding algorithm Concerning the unfolding of experimental data, two conditions need to be fulfilled: (i) the detailed geometry of the microcalorimeter setup needs to be known, (ii) the measurement needs to be accurate and ideally be free of systematic errors that are not due to bremsstrahlung such as high energy thresholds. Unfortunately, this is only partially true regarding the data that is available in the literature. Typically, the setups using Si(Li) detectors are not able to measure beta spectra at energies below 35 keV, see e.g. Singh et al. [44] or Behrens and Szybisz [45]. In this section, we consider a 36 Cl spectrum that was measured by Willett and Spejewski [27] using a lithium drifted silicon crystal absorber having a 4𝜋 geometry as described in Spejewski [26]. After the measurement data had been resolution corrected, using the methods in Wortman and Cramer Jr. [21], the shape-factor function 𝐶𝑊 𝑆 (𝑊 ) = 1 − 1.0 ⋅ 𝑊 − 0.24 ⋅ 𝑊 −1 + 0.42 ⋅ 𝑊 2 was calculated [27], where W denotes the energy of the beta particle.

In order to test the accuracy of the proposed method, the unfolding algorithm was applied to data that was independent from the software EGSnrc, which was used for the above consistency checks. Firstly, we simulated measurement data in the Monte Carlo software PENELOPE [34] for parameters that are comparable with those that are found in state-of-the-art MMC measurements concerning statistics, energy threshold and energy resolution, see e.g. Loidl et al. 2019 [43]. Secondly, the 36 Cl spectrum measured by Willett and Spejewski [27] using Si(Li) detectors was unfolded. 5.1. Unfolding simulated histograms from another Monte Carlo software 36 Cl

histograms were simulated using the pencyl code in PENELOPE [34] and unfolded with the proposed algorithm implemented in EGSnrc as described above. Two cylindrical absorber geometries (4𝜋 7

Please cite this article as: M. Paulsen, K. Kossert and J. Beyer, An unfolding algorithm for high resolution microcalorimetric beta spectrometry, Nuclear Inst. and Methods in Physics Research, A (2019) 163128, https://doi.org/10.1016/j.nima.2019.163128.

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Fig. 9. Unfolding of a measured 36 Cl spectrum; a spectrum plot (left) and a residual plot (right) of the measured spectrum and the unfolded spectrum are depicted along with the reference spectrum. The simulations (energy bin width = 1.4 keV, 𝑁 = 500 energy bins, 𝑚 = 106 events per bin) required for the unfolding algorithm were done for an isotropic point source embedded in a 4𝜋 Si(Li) absorber (50 × 16 × 16 mm3 ).

Declaration of competing interest

To facilitate the application of the algorithm, a measured histogram having 500 energy bins was generated by discretizing the continuous spectrum function, based on the shape-factor function 𝐶WS . It should be noted that this corresponds to a resolution corrected measured histogram, where the statistical fluctuation has been greatly reduced. To determine the unfolding effect of the algorithm interpolation, plots of the generated measured histogram and the unfolded histogram were compared with a reference spectrum. The reference spectrum was calculated using a shape-factor function 𝐶𝑟𝑒𝑓 (𝑊 ) = 1 − 1.326 ⋅ 𝑊 + 0.6328 ⋅ 𝑊 2 which was obtained by Kossert et al. [28] when fitting experimental MMC data from Rotzinger et al. [12]. The results are depicted in Fig. 9. While there is a large systematic deviation below 100 keV, probably due to a relatively high energy threshold (∼60 keV) in the measurements of [27], the bremsstrahlung skew is rather small (Silicon has a Z value of only 14, simulation yields a total energy loss in the absorber of approximately 0.2%) but becomes visible above 100 keV. The relative 𝓁 1 -error correction of the unfolding algorithm is approximately 24% over the entire energy range and 39% above 100 keV, which is consistent with assumption (A3) of Section 3 not being fulfilled at lower energies for this measurement. Despite this fact, the correction provides a reasonable improvement.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by the European Metrology Programme for Innovation and Research (EMPIR), Germany [grant number 15SIB10 MetroBeta, [25]]. The EMPIR initiative is co-funded by the European Union’s Horizon 2020 research and innovation program and the EMPIR Participating States, Germany. We thank the project partners at the CEA and PTB for their input during interesting discussions. In connection with running the egsNRC code on the Linux-Compute Cluster at the PTB Berlin Rolf Behrens, Gert Lindner and Hayo Zutz provided generous support. Appendix A. Supplementary data Supplementary material related to this article can be found online at https://doi.org/10.1016/j.nima.2019.163128.

6. Conclusion and discussion An unfolding algorithm for high resolution microcalorimetric beta spectrometers was presented. Provided that one utilizes a reliable Monte Carlo code, has sufficient knowledge of the considered sourceabsorber geometry and reasonable computing power, it is well applicable for the unfolding of measured beta spectra, in particular for partial energy escape corrections in microcalorimetric detection using 4𝜋 absorbers. Furthermore, the method can be used for other detectors and geometries (<4𝜋) as demonstrated by the calculated 2𝜋 examples. In cases where it is not possible to completely embed a radionuclide in an ideal absorber, the presented unfolding algorithm may thus be combined with simpler beta spectrometers to allow for novel measurements. Of course, the quality of the outcome from the proposed unfolding algorithm depends on the accuracy of the code to simulate the particle tracks. This, in turn, is strongly related to the (limited) accuracy of some interaction probabilities, e.g. the cross section for bremsstrahlung production. Moreover, a prerequisite, when applying the algorithm to experimental data is a meticulous determination of the source-absorber geometry, which then needs to be implemented into the Monte Carlo simulations. However, a study of such effects goes far beyond the scope of this work. Furthermore, a very low energy threshold and a wide energy range in the measured data is needed for the proposed unfolding method to reach its full potential. The algorithm will be applied on spectra of beta emitters that will be measured in a recently developed setup that utilizes state-of-the-art MMCs, see Paulsen et al. 2019 [46].

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