SINS (Single Isotope Neutrino Spectrum generator)

Table of Contents

1. Purpose of Package

2. Installation Instructions

3. Use Instructions

   a. CSV File Construction

   b. Test Cases

4. Methodology

5. References

1. Purpose of Package

This package is designed to create neutrino spectra for the radioactive decay of any isotope. To do this it takes a CSV file of the different decay paths of said isotope, and for those that are beta decay or electron capture, it generates the resulting spectrum for neutrinos per keV per decay.

2. Installation Instructions

You can simply use the package manager pip to install SING.

pip install sins

The other Python packages required to run this package successfully are sys, csv, matplotlib, numpy, scipy, sklearn, and ast. To install them you can simply run:

pip install sys
pip install csv
pip install matplotlib
pip install numpy
pip install scipy
pip install skleanr
pip install ast

3. Use Instructions

Once you have made the CSV file for your isotopes decay paths, running this program is quite simple.

import sins
sins('your_csv_file.csv')

From here it should return the plot of your neutrino spectrum as well as a csv file with the spectrum data. If any of the decay paths were via beta decay, you will also get a plot of the beta spectrum and the associated csv file.

a. CSV File Construction

Writing the proper csv file is essential to the function of this package. The file should contain 8 columns: Decay Type, Isotope, Z, A, Spin, Parity, Q, and Branching Ratio. For this information I would recommend using the KAERI Table of Nuclides [1] and the IAEA Table of Nuclides[2]. Here is an example csv file for Ir-192, which has both beta and electron capture paths:

Decay,Isotope,Z,A,Spin,Parity,Q,Branching Ratio
Start,Ir-192,77,192,4.0,1,0,0
Beta,Pt-192,78,192,3.0,1,48.1,0.000059
Beta,Pt-192,78,192,5.0,-1,70.5,0.000039
Beta,Pt-192,78,192,3.0,-1,76.5,0.001026
Beta,Pt-192,78,192,4.0,1,240.0,0.0560
Beta,Pt-192,78,192,3.0,1,535.0,0.4142
Beta,Pt-192,78,192,4.0,1,672.0,0.4798
EC,Os-192,76,192,0.0,1,136.7,0.00094
EC,Os-192,76,192,0.0,1,355.9,0.0393
EC,Os-192,76,192,0.0,1,466.0,0.00670

Some important notes when writing these csv files:

• Use Beta and EC as the decay path type labels.
• Make sure your Z, A, and parity can be turned into integers.
• Make sure your spin, Q, and branching ratio can be turned into floats.

b. Test Cases

In the test case folder there are four different example cases: Cs-137, Cd-109, Co-57, and Ir-192. Feel free to run these to make sure everything is working before you begin using the package.

4. Methodology [4][5]

For electron capture, calculating the neutrino energy is simple, as it is equal to the Q value. For beta decay, it is a bit more complicated. The methodology, as well as the accuracy of the method used, are elaborated upon here.

To create the antineutrino spectrum I used Fermi's Theory of Beta Decay [3] and followed the approximations given in [4]. While corrective terms have since been added to Fermi's original theory, they have yet to be added to this package.

The allowed beta spectrum is given by,

$$N_\beta(W_e)=Kp_eW_ep_\nu W_\nu F(Z,W_e)C(W_e),$$

where K is the normalization constant, $p_eW_ep_vW_v$ is the phase space factor, $F(Z,W_e)$ is the Fermi function, and $C(W_e)$ is the theoretical shape factor. All of these factors end up being unitless, and $N_\beta(W_e)$ is defined as the probability of a beta being created at normalized energy $W_e$. As stated in [4], the Fermi function is given by,

$$F(Z,W)=2(\gamma+1)(2pR)^{2(\gamma-1)}e^{\pi\alpha ZW/p}\frac{|\Gamma(\gamma+i\alpha ZW/p)|^2}{\Gamma(2\gamma+1)^2},$$

where $\gamma=\sqrt{1-(\alpha Z)^2}$. For all other variable definitions, see Table 2.2 from [5] (shown below). The momentum and normalized energy definitions are in the $\beta$'s reference frame. Their derivation can be found in Appendix B of [5].

Symbol	Meaning	Definition
$E_e$	Electron Kinetic Energy	Varies
$p_e$	Normalized Electron Momentum	$\sqrt{\frac{E^2}{m_e^2c^4}+\frac{2E}{m_ec^2}}$
$p_\nu$	Normalized Neutrino Momentum	$Q/m_ec^2-E/m_ec^2 $
$W_e$	Normalized Electron Energy	$E/m_ec^2+1$
$W_\nu$	Normalized Neutrino Energy	$Q/m_ec^2-E/m_ec^2 $
$W_0$	Normalized Endpoint Energy	$Q/m_ec^2+1$
$Q$	Maximum Energy	Isotope Dependent
$\alpha$	Fine Structure Constant	1/137.035999206
$m_e$	Electron Mass as Energy	510.9989461 keV
$R$	Nuclear Radius	$0.42587\alpha A^{1/3}$ [6]
$Z$	Proton Number	Isotope Dependent
$A$	Nucleon Number	Isotope Dependent

While $C(W)=1$ for allowed decay paths, it requires more calculation for forbidden decay paths. Allowed decay paths are where the angular momentum of the beta and the antineutrino are zero ($\Delta l=0$). For forbidden decay paths, the change in angular momentum of these particles is greater than 0. For the conservation of angular momentum to still hold, this requires both a spin and parity change. This was originally thought to be forbidden in the laws of physics, but it is now known to be possible, just heavily suppressed due to the parity and spin changes required. There are two types of forbidden decay: unique and non-unique. To determine what type (unique or non-unique) and degree of forbiddenness ($l$) each path is we need to consider change in angular momentum ($I$) and parity ($\Pi$). From parity, we can determine whether or not $l$ is even or odd using the following equation,

$$\Pi_P=\Pi_D*(-1)^l$$

From this equation, it is evident that if there is a change in parity the forbiddenness is odd whereas if there is no change in parity the forbiddenness is even. From there we know $\Delta I=l+1$ if the decay is unique forbidden and $Delta I=l,l-1$ if the decay is non-unique forbidden.

To find the antineutrino spectrum from the beta spectrum, we have to perform a change of variables to get to the neutrino's reference frame. This affects the momentum and energies of both the electron and the neutrino. The definition of these variables in the neutrino's reference frame are given in Table 2.3 from [5] (shown below). Their derivation can be found in Appendix B of [5].

Symbol	Meaning	Definition
$E_\nu$	Neutrino Kinetic Energy	Varies
$p_e$	Normalized Electron Momentum	$\sqrt{(Q/m_ec^2-E_\nu/m_ec^2+1)^2-1}$
$p_\nu$	Normalized Neutrino Momentum	$E_\nu/m_ec^2 $
$W_e$	Normalized Electron Energy	$Q/m_ec^2-E_\nu/m_ec^2+1$
$W_\nu$	Normalized Neutrino Energy	$E_\nu/m_ec^2$

A statement on the accuracy of this method can be found in Chapter 2.1.1 of [5].

5. References

1. Y.-S. Cho, Korea atomic energy research institute table of nuclides, Accessed Jan-May 2024, 2000. url: https://atom.kaeri.re.kr/nuchart/.
2. International Atomic Energy Agency Nuclear Data Section, Iaea table of nuclides, Ac- cessed Jan-May 2024, 2009- 2024. url: https://www-nds.iaea.org/relnsd/vcharthtml/ VChartHTML.html.
3. F. L. Wilson, “Fermi’s Theory of Beta Decay,” Am. J. Phys., vol. 36, no. 12, pp. 1150– 1160, 1968. doi: 10.1119/1.1974382.
4. P. Huber, “Determination of antineutrino spectra from nuclear reactors,” Phys. Rev. C, vol. 84, p. 024 617, 2 Aug. 2011. doi: 10.1103/PhysRevC.84.024617.
5. B. Ryan, Cevns in natural zinc superconductors and its applications for nuclear non-proliferation, chapter 2.1 (2024), https://dspace.mit.edu/handle/1721.1/155639.
6. H. Behrens and J. Jänecke, “Numerical tables for beta-decay and electron capture: Z = 1 - 14,” Group I Elementary Particles, Nuclei and Atoms, vol. 4, H. Schopper, Ed., Springer-Verlag Berlin Heidelberg. doi: 10.1007/10201072_11.