### Beta spectrum modeling for the CeLAND experiment antineutrino source

### Mathieu Durero

### Master of Science Thesis Kungliga Tekniska Högskolan,

### Stockholm, Sweden

### Project performed at: CEA Saclay, IRFU, Service de Physique des Particules Gif-sur-Yvette, France

### Supervisors: Matthieu Vivier and Thierry Lasserre

### October 18, 2013

### Abstract

### This Master Thesis dissertation reports on the development of a model for β-decay spectra com-

### putation. It is specically adapted to spectra predictions for the

^{144}

### Ce-

^{144}

### Pr source used in the

### CeLAND experiment. Spectra are modeled using Fermi Theory and taking into account supplemen-

### tary small eects, namely the nucleus nite-size eects, screening eect by the atomic electrons,

### nucleus recoil, outer radiative corrections and the so-called weak magnetism eect. Theoretical

### uncertainties to the model are estimated. The possibility to predict neutrino spectrum shape is

### of primary importance for CeLAND preparation, as the experiment aims at measuring neutrino

### oscillations using notably neutrino spectrum distortion among other things.

## Contents

### 1 Neutrinos physics and sterile neutrino hypothesis 3

### 1.1 The Standard Model and massless neutrinos . . . . 3

### 1.2 Experimental overview of neutrino knowledge . . . . 4

### 1.2.1 The solar and atmospheric neutrino anomalies . . . . 4

### 1.2.2 Neutrino experimental state of the art . . . . 5

### 1.3 Phenomenology of two-avor neutrino oscillations . . . . 6

### 2 The CeLAND project 8 2.1 The reactor antineutrino anomaly and other hints for a new neutrino species . . . . 8

### 2.1.1 Reactor antineutrino anomaly . . . . 8

### 2.1.2 Other anomalies . . . . 8

### 2.2 CeLAND: search for a sterile neutrino . . . 11

### 2.3 Antineutrino source . . . 14

### 2.3.1 Choice of the source material . . . 14

### 2.3.2

^{144}

### Ce-

^{144}

### Pr source characteristics . . . 16

### 3 Elements of beta decay theory 20 3.1 Extended Fermi theory of beta decays . . . 20

### 3.1.1 Fermi theory . . . 21

### 3.1.2 Beta decay classication . . . 23

### 3.1.3 Fermi function . . . 23

### 3.2 Atomic and nuclear small eects on the spectrum shape . . . 24

### 3.2.1 Finite size corrections . . . 25

### 3.2.2 Screening correction . . . 29

### 3.2.3 Radiative correction . . . 33

### 3.2.4 Finite masses and recoil . . . 39

### 3.2.5 Weak magnetism . . . 40

### 4 Implementation for the modeling and spectra predictions 45 4.1 Program structure . . . 45

### 4.2 Result presentation . . . 48

### 4.2.1 Predicted spectra and comparison of corrective terms . . . 48

### 4.2.2 Application to CeLAND experiment . . . 50

### 4.2.3 Comparison with BESTIOLE modeling . . . 59

### A Numerical tables 69

### B Screening correction derivation 71

## Introduction

### Neutrinos are a center of concern in modern Physics. These ghost particles have already shown that they do not behave as predicted by the Standard Model of particle physics predicted for them.

### They are then an active research eld to look for for new phenomena, leading to physics beyond the Standard Model. Some neutrino experiments have already observed surprising results, which wait now to be enlightened.

### Among these results, the short baseline neutrino detection experiments have revealed a lack of detected neutrinos in comparison to what is predicted by nuclear and particle physics models.

### These anomalies remind the older solar neutrino anomaly, which is now understood as an oscillation between the three known avors of neutrinos (electron, muon and tau neutrinos). An hypothesis to explain these anomalies is to introduce a new oscillation mode with a heavier neutrino. This fourth neutrino must have a square mass dierence ∆m

^{2}

### ≈ 1 eV

^{2}

### with the known neutrinos. It must also be a sterile neutrino, which is not coupled to any interaction of the Standard Model. The CeLAND experiment aims at testing this hypothesis with an articial antineutrino source and a very low background detector.

### As concerns the CeLAND source, the antineutrinos are emitted by beta decays of radioisotopes.

### In order to reduce the systematic errors linked to the source antineutrino spectra and hence im- prove the experiment sensitivity to the detection of oscillations, it is necessary to model the source antineutrino spectrum. The development of this model is the subject of the present Master Thesis.

### An introduction to the neutrino physics and a phenomenological overview of the neutrino oscil- lations are given in chapter 1. The experimental hints for a sterile fourth neutrino are exposed in the same section. The chapter 2 describes the CeLAND project and the antineutrino source char- acteristics. The theory of beta decay, which is used in the model, is detailed in chapter 3. Finally the practical implementation and the results obtained from this model are discussed in chapter 4.

### Appendix contains some of the computations, which were too long to be included in chapter 3.

### Chapter 1

## Neutrinos physics and sterile neutrino hypothesis

### 1.1 The Standard Model and massless neutrinos

### The Standard Model of particle physics intends to describe all elementary particles which make up matter and their interactions. It is a relativistic quantum eld theory with symmetry group SU (3)

C### × SU(2)

^{L}

### × U(1)

^{Y}

### , where C, L and Y are respectively color, weak isospin and hypercharge.

### The three symmetry groups are each associated with an interaction, respectively the strong, weak and electromagnetic interactions. The strong interaction can be considered separately, but the other two can be described within a unied framework: the electroweak theory. The resulting particles are classied in twelve gauge bosons with integer spin, which are interactions carriers, and twenty four fermions and antifermions with half-integer spin, which are the matter components.

### Fermions are also divided in three generations, the particles having the same behavior from one generation to the other but dierent masses. The minimal formalism of a quantum eld theory does not include any mass term. The supplementary Higgs eld introduces a symmetry breaking for the electroweak interaction and has for consequences to generate mass terms for all particles except neutrinos, gluons and photons in the nal Lagrangian of the model. Tables 1.1 and 1.2 show the particles of the model as they are sorted by their behavior relatively to the interactions.

### The gravity is not currently included and the impossibility to make the Standard Model agree with the General Relativity is a major problem for the modern physics. Gravity is supposed to aect all massive particles and in the General Relativity own frame, all particles are impacted by the space-time curvature.

### In the Standard Model, three active neutrinos are included, each being associated to a charged lepton. They are supposed massless. The possible existence of more active neutrino species has been

### sensitive to 1st generation 2nd generation 3rd generation

### all interactions quarks up (u) charm (c) top (t)

### down (d) strange (s) bottom (b)

### electroweak

### interaction leptons electron (e) muon (µ) tau (τ)

### weak interaction electron neutrino (ν

e### ) muon neutrino (ν

µ### ) tau neutrino (ν

τ### )

### Table 1.1: Fermions of the Standard Model

### interaction gauge bosons scalar boson strong 8 gluons (g)

### Higgs boson (H

^{0}

### ) electromagnetic photon (γ)

### weak Z

^{0}

### , W

^{+}

### , W

^{−}

### Table 1.2: Bosons of the Standard Model

**0** **10** **20** **30**

**86** **88** **90** **92** **94**

**E** _{cm} ** [GeV]**

_{cm}

### σ **had** ** [** **nb** ]

**3ν** **2ν**

**4ν**

**average measurements,**
**error bars increased**
** by factor 10**

**ALEPH** **DELPHI** **L3**

**OPAL**

### Figure 1.1: Measurements of the hadron production cross-section around the Z resonance. The contribution of Z

^{0}

### decay to neutrinos has a strong inuence on the cross-section peak. Predicted cross-section for two, three and four neutrino species are shown. These neutrinos are massless and coupled to the Standard Model via weak interaction. Figure taken from [44].

### checked from the decay width of the Z

^{0}

### boson by an group of experiments at the Large Electron- Positron Collider [44]. The cross-section of the hadronic decay of Z

^{0}

### is linked to the possible Z

^{0}

### decay via neutrinos. It constrains the number of neutrinos sensitive to the weak interaction. As it can be seen on gure 1.1, the high precision measurements conrm with very strong condence level that exactly three neutrino species are coupled to the Standard Model.

### 1.2 Experimental overview of neutrino knowledge

### 1.2.1 The solar and atmospheric neutrino anomalies

### Neutrino detection experiments have been realized during all the second half of the twentieth

### century. The Sun and the Earth atmosphere (through impact of cosmic rays) are two neutrino

### sources carefully studied in many occasion. But predicted neutrino detection rates were strongly

### overestimated when compared to the eective results. This apparent lack of events resulted in

### the solar neutrino anomaly and the atmospheric neutrino anomaly. These anomalies have been

### explained at the end of the 90s respectively by the Sudbury Neutrino Observatory (SNO) experiment

### [9] and the Super-Kamiokande experiment [24].

### Model of the Sun predicts that the nuclear fusion reactions in Sun's core produce electron neutrinos as a byproduct, following the processes:

### p + p → D + e

^{+}

### + ν

e### (1.1)

### p + e

^{−}

### + p → D + ν

^{e}

3

### He + p →

^{4}

### He + e

^{+}

### + ν

e### e

^{−}

### +

^{7}

### Be →

^{7}

### Li + ν

_{e}

### p +

^{8}

### B →

^{8}

### Be ∗ + e

^{+}

### + ν

e### The neutrino detection rate from experiments focusing on electron neutrino was about one third of the theoretical expected rate. The possibility for neutrinos to oscillate between their three known avors was an hypothesis to explain the anomaly. The missing part of the detected neutrino would have been the part which has oscillated between the emission and the detection. The SNO experiment could detect neutrinos using two reactions. On the one hand

### ν

_{e}

### + D → p + p + e

^{−}

### (1.2)

### concerns only electron neutrino. On the other hand

### ν

_{α}

### + D → n + p + ν

^{α}

### (1.3)

### is sensitive to all neutrino avors. The electron neutrino detection process conrmed the anomaly but the second detection process obtained results in harmony with the theoretical expected rate.

### SNO then conrmed that the Sun modeling leads to a correct electron neutrino emission rate but part of these neutrinos switches to muon neutrino or tau neutrino before reaching the Earth [9].

### In the atmospheric neutrino sector, Super-Kamiokande equally conrmed the neutrino oscilla- tions by measuring the variation in the atmospheric neutrino detection rate as a function of the incoming particle direction [24].

### 1.2.2 Neutrino experimental state of the art

### Neutrino oscillations between avors prove that neutrinos are massive because the oscillation process implies that the avor eigenstates are not equivalent to the mass eigenstates. It is a typical quantum mechanics process [30]. With three neutrino avors, the oscillation probability is given by:

### P

_{α→β}

### = P

i=(1,2,3)

### |U

^{αi}

### |

^{2}

### |U

^{βi}

### |

^{2}

### + 2 R n P

i<j<3

### U

_{αi}

### U

_{βi}

^{∗}

### U

_{αj}

^{∗}

### U

_{βj}

### exp (iφ

_{osc}

### ) o

### (1.4) where i = (1, 2, 3) associated with Hamiltonian mass eigenstates |ν

^{i}

### i and (α, β) = (e, µ, τ) represents states|ν

^{α}

### i and|ν

^{β}

### i from the avor eigenstates (eigenstates for the weak interaction charged current).

### The phase φ

osc### between avors determines the oscillation frequency, which depends on the squared mass dierence ∆m

^{2}

### between two mass eigenstates. The unitary matrix U allows to switch from one base to the other. The mixing matrix U is dened as

###

### ν

_{e}

### ν

_{µ}

### ν

_{τ}

###

### = U

###

### ν

_{1}

### ν

_{2}

### ν

_{3}

###

### =

###

###

### U

_{e1}

### U

_{e2}

### U

_{e3}

### U

_{µ1}

### U

_{µ2}

### U

_{µ3}

### U

_{τ 1}

### U

_{τ 2}

### U

_{τ 3}

###

###

###

### ν

_{1}

### ν

_{2}

### ν

_{3}

###

### ,

### and is called PMNS matrix (from its authors Pontecorvo-Maki-Nakagawa-Sakata). This unitary matrix can be split up into a product of three rotations associated with the three mixing angles and one phase. With c

ij### = cos θ

ij### and s

ij### = sin θ

ij### we have:

### U =

###

###

### 1 0 0

### 0 c

_{23}

### s

_{23}

### 0 −s

^{23}

### c

_{23}

###

###

###

###

### c

_{13}

### 0 s

_{13}

### e

^{−iδ}

### 0 1 0

### −s

^{13}

### e

^{iδ}

### 0 c

_{13}

###

###

###

###

### c

_{12}

### s

_{12}

### 0

### −s

^{12}

### c

_{12}

### 0

### 0 0 1

###

### (1.5)

### Neutrinos are not the best known particles. They are associated to various problems of the modern Physics and are a promising research eld. The neutrino oscillation experiments have already measured the three mixing angles but they cannot access to the phase δ. δ is linked to the CP symmetry violation in the neutrino sector. This parameter could explain an asymmetry in the behavior of neutrino in comparison to antineutrino. It is then intimately linked to one of the biggest problem of the modern Physics, which is to account for asymmetry between matter and antimatter at the Universe scale [2].

### As it will be seen in section 1.3, neutrino oscillation experiments are generally sensitive to only one oscillation frequency among the three possible ones, which correspond to the three possible

### ∆m

^{2}

### parameters. Solar neutrino experiments are associated with the oscillation between |ν

^{1}

### i and

### |ν

^{2}

### i. They have measured ∆m

^{2}21

### = 7.58

^{+0.23}

_{−0.20}

### × 10

^{−5}

### eV

^{2}

### [17]. ∆m

^{2}

_{32}

### = 2.35

^{+0.12}

_{−0.09}

### × 10

^{−3}

### eV

^{2}

### is known from atmospheric neutrinos experiments. Its sign is not known yet, because the oscillation probability depends on ∆m

^{2}

### . The interaction of neutrino with matter in the Sun allows to get rid o this problem for ∆m

^{2}21

### but no equivalent method is usable with atmospheric neutrino at the moment. The problem of neutrino mass hierarchy arises from this uncertainty. One is unable to discriminate between m

1### < m

_{2}

### < m

_{3}

### and m

3### < m

_{1}

### < m

_{2}

### . The two order of magnitude dierence between ∆m

^{2}21

### and ∆m

^{2}32

### made ∆m

^{2}31

### degenerated and indistinguishable from ∆m

^{2}32

### . The absolute values of the neutrino masses have not been determined yet. An upper limit of 2 eV on m

νe### is the best available at the moment. Experiments on high precision energy measurement of β-decays (for instance on Tritium) are in progress to improve this constraint.

### The corresponding sin

^{2}

### θ

12### = 0.306

^{+0.018}

_{−0.015}

### and sin

^{2}

### θ

23### = 0.42

^{+0.08}

_{−0.03}

### are known from the same experiments. Finally, the last generation of accelerator and reactor experiments have gained access to sin

^{2}

### θ

13### = 0.0251 ± 0.0034 [17].

### Moreover a neutral particle is the only candidate fermion having the possibility to be its own antiparticle (called a Majorana particle). This hypothesis makes possible for instance the double beta decay without neutrino emission, which is actively researched today [27].

### Finally, new anomalies from short baseline experiments have been identied recently. A fourth neutrino, which must be sterile because of the constraints from ALEPH evoked in section 1.1, could be an explanation for these results. The CeLAND experiment, which aims at testing this hypothesis, will be described in chapter 2.

### 1.3 Phenomenology of two-avor neutrino oscillations

### Due to ∆m

^{2}

### inuence on the phase term in equation 1.4, a usual case for neutrino experiment is to be sensitive to only one of the possible mass dierences as a rst approximation [25]. The two

### avor oscillation probability for plane waves then writes:

### P

_{α→β}

### = U

_{α1}

### U

_{β1}

^{∗}

### U

_{α1}

^{∗}

### U

_{β1}

### + U

_{α2}

### U

_{β2}

^{∗}

### U

_{α2}

^{∗}

### U

_{β2}

### + U

_{α1}

### U

_{β1}

^{∗}

### U

_{α2}

^{∗}

### U

_{β2}

### e

^{iφ}

^{osc}

### + U

_{α2}

### U

_{β2}

^{∗}

### U

_{α1}

^{∗}

### U

_{β1}

### e

^{−iφ}

^{osc}

### . (1.6)

### Neutrino source solar and reactor atmospheric accelerator

### ∆m

^{2}

### 1 MeV 10 to 1000 MeV 1 GeV

### 8 × 10

^{−5}

### eV

^{2}

### 12 500 m 12.5 × 10

^{4}

### to 12.5 × 10

^{6}

### m 12 500 km 2 × 10

^{−3}

### eV

^{2}

### 500 m 5 × 10

^{3}

### to 5 × 10

^{5}

### m 500 km

### 1 eV

^{2}

### 1 m 10 to 1000 m 1 km

### Table 1.3: Optimal experimental baseline to highlight oscillation process for a given ∆m

^{2}

### and source.

### Since the U matrix can be written as a rotation matrix

### cos θ

_{ij}

### sin θ

_{ij}

### − sin θ

^{ij}

### cos θ

_{ij}

### , P

α→β### can be ex- pressed as

### P

α→β### = sin

^{2}

### (2θ

ij### ) sin

^{2}

### c

^{3}

### ~

### ∆m

^{2}

_{ij}

### L 4E

### !

### (1.7)

### The angle θ

ij### determines the oscillation amplitude and the phase

^{c}

_{~}

^{3}

^{∆m}

_{4E}

^{2}

^{ij}

^{L}

### leads to the oscillation frequency. This expression shows clearly that the oscillations take place only if the neutrino have non-zero mass dierences (and so at least two of the three neutrino are massive). Phase dependency on

^{L}

_{E}

### is the crucial parameter to determine to which ∆m

^{2}

### an experiment is sensitive to. Using more easily understandable units, equation 1.7 is rewritten:

### P

α→β### = sin(2θ

ij### ) sin

^{2}

### 1.27 ∆m

^{2}

_{ij}

### (eV

^{2}

### )L(m) E(MeV)

### !

### (1.8)

### Thus, the condition to achieve the best sensitivity to a given oscillation process is then 1.27

^{∆m}

^{2}

^{ij}

_{E(MeV)}

^{(eV}

^{2}

^{)L(m)}

### =

π

2

### , leading to

### ∆m

^{2}

_{ij}

### (eV

^{2}

### )L(m)

### E(MeV) ≈ 1

### Table 1.3 summarizes the optimal parameters as a function of the studied ∆m

^{2}

### and the energy

### range of the used neutrino source. These estimates are only orders of magnitude.

### Chapter 2

## The CeLAND project

### 2.1 The reactor antineutrino anomaly and other hints for a new neutrino species

### As seen in section 1.3, an oscillation comes out as an event detection rate variation, which is proportional to the ratio between the source-detector distance and the neutrino energy (equation 1.7). Depending on the baseline, neutrino detection experiments constrain the existence of such oscillation processes, whose frequency depends on ∆m

^{2}

### and amplitude on sin

^{2}

### 2θ

osc### .

### 2.1.1 Reactor antineutrino anomaly

### In 2011, new evaluations of the nuclear reactor neutrino ux, based on the new experimental value of the neutron lifetime, better spectrum predictions and better modeling of nuclear fuel components, lead to a new anomaly in the antineutrino sector. The expected neutrino rate of short baseline reactor antineutrino experiments is now higher than the original detection rate at the 3σ level [39].

### However, the deviation is not signicant enough to immediately conclude, yet it casts doubt upon the possibility of a brand new phenomenon. Figure 2.1 from [16] shows a summary of the results of short baseline reactor antineutrino experiments. The dotted line is the prediction from the three neutrino mixing model, and the solid line includes a sterile neutrino with ∆m

^{2}

### ≈ 1 eV

^{2}

### . This gure shows that a new oscillation mode, leading to the existence of a fourth neutrino species, can explain the anomaly.

### 2.1.2 Other anomalies

### Equally another anomaly has been observed in gallium solar neutrino detectors [32, 31](GALLEX and Sage, see [10, 37] and [3, 4] for further informations). Those detectors were calibrated with intense neutrinos sources (

^{37}

### Ar and

^{51}

### Cr), for which the detected neutrino event rate was found to be lower than expected. The Gallium to Germanium reaction ν

e### +

^{71}

### Ga →

^{71}

### Ge + e

^{−}

### cross-section comes with major uncertainties, lowering the statistical signicance of the neutrino decit. However the newest interpretation still supports a Gallium anomaly [31].

### A third anomaly comes from cosmological observations, which constrain the number of possible neutrinos as the Universe expansion rate is linked to the energy density in relativistic particles during the radiation domination era [2]. The photons and neutrinos are the only particles involved, and photon energy density is determined from the cosmic microwave background measurements.

### The expansion rate of the early Universe is then used to constrain the neutrino energy density. It

### Figure 2.1: Illustration of the short baseline reactor antineutrino anomaly taken from from [16].

### Two scenarios are represented. On one hand, a three known neutrinos mixing scenario (dashed line), on the other hand the possibility to have a 3+1 neutrinos system scenario (solid line), with ∆m

^{2}

### ≈ 1 eV

^{2}

### . The anomaly is visible for 10 to 100 m baseline. At larger baselines, the data points correspond to experiments associated with solar and atmospheric neutrino oscillations.

### depends on an eective number of neutrino species equal to N

ef f### = 3.046 with the parameters of the Standard Model. The existence of an additional neutrino species would increase N

ef f### . Although not signicant enough, cosmological data do not exclude a non-standard N

eff### > 3.046 hypothesis.

### Recent Planck results, which found N

eff### = 3.30 ± 0.27 [6], are a good example. A sterile neutrino is marginally compatible with cosmological data.

### Finally, the accelerator experiments LSND and MiniBooNE, with short baselines and neutrinos in the GeV range, have also recorded dierent neutrino rates in comparison to the predictions [7, 8].

### A detailed description of the anomalies is given in [2].

### Recent constraints in the (∆m

^{2}new

### , sin

^{2}

### 2θ

new### ) parameter space have been published in [31], using data from reactor, gallium and accelerator experiments. The gure 2.2 summarizes the preferred parameter space. Two region are preferred around ∆m

^{2}new

### = 7.6 eV

^{2}

### , sin

^{2}

### (2θ

new### ) = 0.12 and ∆m

^{2}new

### ' 2 eV

^{2}

### , sin

^{2}

### (2θ

new### ) ' 0.1. Tritium beta decay and double beta decay experiments have also constrained the (∆m

^{2}new

### , sin

^{2}

### 2θ

new### ) parameter space, with upper limits for ∆m

^{2}new

### lying in the 10

^{2}

### -10

^{4}

### eV

^{2}

### range at the 95 % condence level. Finally, cosmological data have also been used to set a 95% C.L. upper limit on the mass of a new sterile neutrino [33], which is somehow in tension with the mixing parameters preferred by the terrestrial experiment data:

### m

_{s}

### < 0.45 eV. (2.1)

*s i n* ^{2} 2ϑ _{ee}

### ∆ *m* 41 2 [eV 2 ]

## +

### 10 ^{−} ^{2} 10 ^{−} ^{1} 1

### 10 ^{−} ^{2} 10 ^{−} ^{1} 1 10 10 ^{2}

## + +

## +

### 95% CL Gallium Reactors ν _{e} C Sun

### Combined 95% CL

### Gallium Reactors ν _{e} C Sun

### Combined

### Figure 2.2: Allowed 95% condence level regions in the sin

^{2}

### (2θ

_{new}

### ) − ∆m

^{2}new

### plane obtained from separate ts of Gallium, reactor, solar and ν

e### C scattering from KARMEN [11] and LSND data and from the combined ts of all data. Crosses indicate the best-t points. Figure taken from [31].

### All the above mentioned anomalies point toward a new oscillation frequency with ∆m

^{2}

### ≈ eV

^{2}

### . To avoid averaging eects that could erase any oscillation pattern seen in the detector and therefore degrade the sensitivity to oscillations, an experiment looking for such oscillation should be a short baseline one (as seen in section 1.3), with a compact neutrino source. It will allow to observe the specic pattern as a function of the baseline and the spectrum distortion from the oscillations in the event rate. The optimal distance between source and detector is at the meter scale, for

### ∆m

^{2}

### ≈ 1 eV

^{2}

### and MeV neutrinos (as can be achieved with a radioactive source or nuclear

### reactors). The activity of the source will be chosen to obtain a statistically signicant detection rate with a very low background detector. A large liquid-scintillator detector such as KamLAND, Borexino or SNO+ seems perfectly tted for a light sterile neutrino experiment. Moreover, these detectors are situated underground and then guarantee a good protection against cosmic rays.

### 2.2 CeLAND: search for a sterile neutrino

### The principle of the experiment is to place an antineutrino source close to a detector which is big enough to both have the necessary sensitivity to the weakly interacting neutrinos to clearly pick out the source's signal and to give access to the oscillation pattern at the meter scale.

### The CeLAND experiment will run at the KamLAND detector (see for example [5, 20]for more details). Situated in the Kamioka mine in Japan, KamLAND was built to observe the neutrino ux from nuclear power plants with a very long baseline (about 200 km). It is composed of a 1 kton liquid-scintillator inner detector in a spherical plastic balloon with 6.5 m radius, surrounded by non-scintillating oil, in a 9 m steel sphere, and an outer water-Cerenkov detector lling the gap between the sphere and the cylindrical cavity in the rock. The outer detector is used as a veto against cosmic rays, while the inner part is the proper neutrino detector. The scintillation light is detected with photomultiplier tubes which are mounted inside the steel sphere. The gure 2.3 presents a schematic view of the detector. For CeLAND, an articial antineutrino source will be placed in the outer detector. See section 2.3 for details about the selected

^{144}

### Ce-

^{144}

### Pr source. The antineutrino will be detected via inverse beta decay (described in section 2.3.1). The KamLAND liquid-scintillator is characterized by a proton density of ρ

P### = 6.62 × 10

^{28}

### protons per m

^{−3}

### .

### Simultaneously to CeLAND, the KamLAND-Zen experiment aimed at the 2β0ν detection with

136

### Xe will run [27]. Both experiments focus on the same energy range, typically 2 to 3 MeV for inverse beta decay and for the

^{136}

### Xe double decay with or without neutrino. Background eects have been studied and both experiments should be able to run together without interfering.

### The source of Cerium will be placed close to the detector, inside a tungsten-alloy shielding. As we will see in the following, the

^{144}

### Ce −

^{144}

### Pr pair emits also gamma rays with energy in the MeV range.

### The primary role of the tungsten-alloy shielding is to protect humans against gamma radiations, though the shielding has been designed to achieve further suppression of the escaping gamma rays, in order to limit any background radiation entering the detector. In order to achieve compactness of the system, particularly for transportation, tungsten was selected because it is a high density material and it easily resists to corrosion and heat.

### The current deployment scheme is to insert the source in a steel structure in the outer detector, close to the steel sphere delimiting both parts of KamLAND. It is illustrated on gure 2.4. This will allow a relatively correct solid angle coverage, the use of the water from outer detector and buer oil as an additional neutron and gamma shieldings. Furthermore it will ease the source heat dissipation, as it emits about 600 W, by using the water circulation system of the detector. The source will be about 9.6 m away from the detector center and will stay here for 18 months.

### The experiment is based on a double analysis of both the event rate as a function of distance and energy. The

^{L}

_{E}

### neutrino oscillation frequency dependency (see section 1.3) leads both to a distortion in the neutrino energy spectrum and neutrino rate variation as a function of length. It allows also to observe a neutrino detection rate with a source to detector length variation. The

### gure 2.5 illustrates this two-dimensional oscillation pattern. A precise activity measurement is also

### necessary to improve the analysis sensitivity. This will be done through a calorimetric measurement

### [16]. The source should be above 75 kCi and the source container has been designed to accommodate

### up to 100 kCi. Both β and γ spectroscopy will be performed on source samples in order to control the

### source purity and characterize the source antineutrino energy spectrum shape. The beta spectrum

### 18m φ

### Buffer Oil

### Liquid Scintillator

### PMTs

### (inner detector)

### Stainless steel Spherical tank PMTs

### (outer detector)

### Acrylic sphere Water pool

### Plastic balloon

### 16.5m φ 13m φ

### Figure 2.3: KamLAND detector schematic from [16]

### Figure 2.4: CeLAND source location in the water outer detector, 9.6 m from the detector center,

### 0.6 m from the stainless steel sphere (1 cm thick), 2.8 m from the liquid scintillator active volume.

**0.7** **0.9** **1.2** **1.4** **1.6** **1.8** **2.1** **2.3** **2.5** **2** **0**

**6** **4** **10** **8**

**14** **12** **16** **0**

**10** **20** **30** **40** **50** **60**

**E**

**vis**

** (MeV)** 2−D reconstructed spectrum for U

e4

### = 0.20 and ∆m

^{2}

_{41}

### = 2.0 ev

^{2}

**L**

**rec**

** (m)**

### N in [10cm, 100 keV] bin

### Figure 2.5: Expected spectrum with oscillation into a fourth neutrino (∆m

^{2}new

### = 2 eV

^{2}

### and sin

^{2}

### 2θ

new### ≈ 0.15) as a two-dimensional function of the visible energy and the distance between source and detector. The visible energy in the detector is the positron energy, which is linked to the ¯ ν

e### energy in the IBD detection process.

### modeling detailed in this report will be used to interpret the β-spectroscopic measurements.

### 2.3 Antineutrino source

### 2.3.1 Choice of the source material

### For a neutrino or antineutrino source, two suitable categories can be considered. First monochro- matic electron neutrino emitters like

^{51}

### Cr or

^{37}

### Ar, which decay by electron capture and secondly electron antineutrino emitters with a continuous β-spectrum such as

^{144}

### Ce,

^{90}

### Sr,

^{42}

### Ar or

^{106}

### Ru.

### All the nuclear data are taken from [1].

### An antineutrino source

### The choice has been made to use an antineutrino source, which seems the most adapted for various

### reasons. In such a case the detection is performed by inverse beta decay ¯ν

e### + p → e

^{+}

### + n . The

### inverse beta decay (IBD) cross-section is expressed as

### σ

IBD### (E

e### ) ∼ 0.96 × 10

^{−43}

### × p

^{e}

### E

e### cm

^{2}

### , (2.2) where p

e### and E

e### are the momentum and energy (in MeV) of the detected e

^{+}

### , neglecting here neutron recoil, weak magnetism eects, and radiative second order corrections. It is one to two order of magnitude higher than the electron scattering ν

e### + e

^{−}

### → ν

^{e}

### + e

^{−}

### cross-section σ

ES### at the MeV scale, which would be the main detection process with a neutrino source.

### The IBD signature in a detector is the time coincidence of the positron detection and the neutron observation, the rst being promptly annihilated with an electron, the second being captured by a free proton. As the average capture time of the neutron is a few µs, the positron annihilation is the prompt event and the neutron capture the delayed event. The time coincidence between them allows ecient suppression of background.

### In order to obtain statistically signicant results in a reasonable time of about two years, the number of expected events can be linked to detector parameters to compute activity requirements for the source. With N the number of emitted neutrinos, V the detector volume, ρ

P### the proton density and introducing a

_{4πd}

^{1}2

### geometrical factor to account for the coverage of the source by the detector (d being the source-detector distance), we have:

### N

IBD### ≈ N 1

### 4πd

^{2}

### V ρ

P### σ

IBD### (2.3)

### N

IBD### ≈ N × 6.6 × 10

^{−20}

### To obtain N

IBD### ≈ 10000, we need N ≈ 7.6 × 10

^{23}

### . If we consider 3.2 × 10

^{7}

### seconds for a year and that 1 Ci = 3.7 × 10

^{10}

### decays per second, we need an activity of roughly 65 kCi. To compare with a neutrino source, the e

^{−}

### scattering cross-section is two order of magnitude lower, but the density of electrons is about ten times the density of protons, so at rst glance 10 times higher activity should be required. However, the electron scattering detection process lacks of the time coincidence of the IBD and suers from larger backgrounds (solar neutrinos mainly). The necessary activity should then rather be at the MCi level.

### The monochromatic electron neutrino emission solution has been more seriously considered by some of the competitors to CeLAND (particularly the SOX experiment, for further information, [18] can be read).

144

### Ce-

^{144}

### Pr pair

### The inverse beta decay reaction has a 1.8 MeV energy threshold, so the source isotope must have high enough end-point energy Q

β### for a considerable part of emitted antineutrino spectrum to have enough energy to be detected. A high Q

β### means a short lifetime for the nucleus under consideration.

### It implies that the source production, transportation, and generally the experiment handling will be dicult. The chosen solution is to rely on a pair of nuclei, which are both beta emitters, and where the father nucleus is a long lifetime isotope and the daughter a short lifetime one with high Q

β### . Four pairs

^{144}

### Ce −

^{144}

### Pr,

^{106}

### Ru −

^{106}

### Rh,

^{90}

### Sr −

^{90}

### Y and

^{42}

### Ar −

^{42}

### K matching these requirements have been identied. The rst three are common ssion products of nuclear spent fuel, when

^{42}

### Ar needs a double neutron capture from stable

^{40}

### Ar with a low cross-section and a highly unstable intermediate

^{41}

### Ar to be produced. Reprocessing of nuclear spent fuel seems far more accessible.

### Then, considering the cumulative ssion yield

^{1}

### of the three other mother nucleus presented in table 2.1, the

^{106}

### Ru will be excluded due to its low abundance in Uranium ssion products. Considering

1The cumulative ssion yield of an isotope is the number of nuclei of the considered isotope per ssion when the reactor is at equilibrium.

### Cumulative ssion yield (%)

144

58

### Ce

^{106}44

### Ru

^{90}38

### Sr

235

### U 5.50 (4) 0.401 (6) 5.78 (6)

239

### Pu 3.74 (3) 4.35 (9) 2.10 (4)

### Table 2.1: Cumulative ssion yield for a thermal ssion of

^{235}

### U and

^{239}

### Pu for

^{144}

### Ce,

^{106}

### Ru and

90

### Sr [1] in commercial nuclear reactors.

### Branching ratio Quantum numbers Q

β### (keV) type V-A ratio

### 76.5 % 0

^{−}

### 318.7 First non unique

### forbidden A 100 %

### 3.9 % 1

^{−}

### 238.6 First non unique

### forbidden unknown

### 19.6 % 1

^{−}

### 185.2 First non unique

### forbidden unknown Table 2.2: Summary of the

^{144}58

### Ce

86### beta decay features from ground state 0

^{+}

### .

### the high energy threshold of inverse beta decay, the more we have neutrinos above it the better, so the selected couple should be

^{144}

### Ce −

^{144}

### Pr because of the highest Q

β### in comparison to

^{90}

### Sr −

^{90}

### Y (2997.5 keV versus 2280.1 keV).

### 2.3.2

^{144}

### Ce-

^{144}

### Pr source characteristics

### Cerium and Praseodymium have respectively an atomic number of 58 and 59. In order to produce the source, Cerium will be extracted from nuclear spent fuel by chemical rare-earth separation techniques [16]. It does not allow to separate between isotopes for a given element. The

^{144}

### Ce isotope is the most stable Cerium radioisotope with a 284.91 days half-life. Most of the Cerium isotopes have a short lifetime with respect to the

^{144}

### Ce, and have a low cumulative ssion yield.

### We can then safely neglect any background radiation from other Cerium isotopes. The long lifetime of

^{144}

### Ce makes the source handling possible without insurmountable problems. It will drive the activity of the source in front of

^{144}

### Pr which has 17.28 minutes half-life.

144

### Ce always decays to

^{144}

### Pr, which then decays to

^{144}

### Nd. The

^{144}

### Nd is an alpha emitter with roughly a 10

^{15}

### years half-life. The source β-spectrum will be a combination of the beta emission from Cerium and Praseodymium. The relevant decay schemes are given on gures 2.6, 2.7 and 2.8 from the National Nuclear Data Center [1]. We note that the

^{144}

### Pr can decay from an excited 3

^{−}

### state. The following β-transitions will have a quite small branching ratio that we try to estimate in the following. From 3

^{−}

### level, no transition has more than 0.033 % branching ratio, and it is necessary to compute a global ratio including the gamma decays to obtain the 3

^{−}

### state. First, this state is only accessible after the less energetic of the Cerium decays (smallest Q

β### ), which has a 19.6 % branching ratio. From the resulting 1

^{−}

### state of Praseodymium, only one of the three γ transitions leads to the 3

^{−}

### state, with intensity of 0.2 %, followed by another transition with intensity of 0.257 %. So nally, we have a maximal proportion of 1.01 × 10

^{−4}

### % of Cerium atoms, which end in the 3

^{−}

### Praseodymium state. Finally, we can estimate the total branching ratio to 3.33 × 10

^{−6}

### % at most, so we can actually neglect all transitions from

^{144}

### Pr excited 3

^{−}

### state.

### The β-transitions are classied depending on the actual process involved and the quantum state

### change for the nucleus. The relevant characteristics are summarized in tables 2.2 and 2.3. Details

### about the β-decay classication are given in section 3.1.2.

### Figure 2.6:

^{144}

### Ce decay scheme from its ground state. Data from the National Nuclear Data Center [1]

### Branching

### ratio Quantum

### numbers

### Q

_{β}

### (keV) type V-A Ratio

### 97.9 % 0

^{+}

### 2997.5 First non unique

### forbidden A 100 %

### 1.04 % 2

^{+}

### 2301.0 First unique

### forbidden A 100 %

### 1.05 % 1

^{−}

### 811.8 Allowed A 100 %

### 6.70 · 10

^{−3}

### % 0

^{+}

### 913 First non unique

### forbidden A 100 % 1.4 · 10

^{−3}

### % 2

^{+}

### 1436.5 First unique

### forbidden A 100 % 8.7 · 10

^{−4}

### % 0

^{+}

### 322.2 First non unique

### forbidden A 100 % 6.2 · 10

^{−4}

### % 2

^{+}

### 924.7 First unique

### forbidden A 100 % 3.0 · 10

^{−4}

### % 0

^{+}

### 254.6 First non unique

### forbidden A 100 % 1.5 · 10

^{−4}

### % 1

^{+}

### 342.6 First non unique

### forbidden unknown

### Table 2.3: Summary of the

^{144}58

### P r

85### beta decay features from ground state 0

^{-}

### .

### Figure 2.7:

^{144}

### Pr decay scheme from its ground state. Data from the National Nuclear Data Center [1]

### CeLAND aims at highlighting short baseline neutrino oscillations using the spectrum shape

### distortion and variation in the ¯ ν

e### rate as a function of distance. Spectrum predictions are then an

### important tool to foresee the experiment performances and check on which energy domains and

### with which precision we can expect any oscillation pattern detection to be signicant, depending

### on the setup and operational mode variations. The next chapters describe the construction of an

### adapted model for the spectrum of the CeLAND neutrino source. The possibility to predict the

### spectrum shape will be useful for checking the β-spectroscopic measurements. Such measurements

### will be done on material samples and are scheduled by the end of the year (2013).

### Figure 2.8:

^{144}

### Pr decay scheme from excited states. Data from the National Nuclear Data Center

### [1]

### Chapter 3

## Elements of beta decay theory

### Our antineutrino will thus be emitted by negative beta decay described as

A

Z

### X →

^{A}Z+1

### X

^{0}

### + e

^{−}

### + ¯ ν

_{e}

### where the conversion from the father nucleus X with mass M

X### to the daughter nucleus X' with mass M

X^{0}

### is done at the nucleon level by n → p + e

^{−}

### + ¯ ν . The Q

β### value of a transition is dened as the available kinetic energy for the electron and the antineutrino

### Q

β### = (M

X### − M

^{X}

^{0}

### − m

^{e}

### − m

^{ν}

^{¯}

### ) c

^{2}

### (3.1) Both involved nuclei are possibly in excited states. This must be taken into account in the energetics of the transition.

### A description of the beta decays was rst presented by Enrico Fermi in 1934 [23].

### 3.1 Extended Fermi theory of beta decays

### Nomenclature

### In the following, we will use a derivative of Wilkinson's nomenclature [50, 51].

### E = Electron total energy T

_{e}

### = Electron kinetic energy

### W = T

e### m

_{e}

### c ² + 1 W

0### = Q

β### m

e### c

^{2}

### + 1 α = e ²

### ~c γ = p

### 1 − (αZ)

^{2}

### Moreover, all explicitly computed uncertainties are supposed to be at 68 % condence level

### except if dierently stated.

### 3.1.1 Fermi theory

### Fermi's description of beta decay was a coherent framework, giving accurate predictions for the observed decays at the time, written from an analogy with photons emission by excited atoms.

### Particularly the involved leptons were considered created (and annihilated in the inverse process) as the photons were in the radiation theory. It uses the neutrino, which was proposed by Wolfgang Pauli in 1930 as a way to solve incoherence observed in beta decay particle energy spectrum when one considers only one emitted particle. The electron neutrino was undetected and its inclusion in Fermi's theory gave support to this then hypothetic particle. Fermi's computation is based on quantum mechanics with a relativistic treatment for leptons. He used a perturbation method where the interaction term of the Hamiltonian is a perturbation term in comparison to the Hamiltonian of the four involved particles. [38]

### In β-transition, the beta particle shows a continuous distribution of energy. The transition probability is given by Fermi's Golden Rule:

### P = 2π

### ~ |V

^{f i}

### |

^{2}

### ρ(E

f### ). (3.2)

### ρ is the density of nal states. With dn the number of nal states in the energy interval dE, ρ(E) =

_{dE}

^{dn}

### . V

f i### is known as the matrix element of the transition operator V associated with initial state i and nal state f, and can be written in the volume integral form

### V

f i### = hψ

^{f}

### | V |ψ

^{i}

### i = ˆ

### ψ

^{∗}

_{f}

### V ψ

i### dv. (3.3)

### In a relativistic context, the transition operator can either be vector, axial vector, scalar, pseu- doscalar or tensor operator, or a linear combination of them depending on its transformation prop- erties. The right form of the operator took decades to be determined. The rst hypothesis supposed parity conservation with a vector operator (V). Later, this theory has been successfully extended to axial currents (A) allowing parity violation by George Gamow and Edward Teller [26]. The correct form is now understood as a linear combination V-A.

### ψ

_{f}

^{∗}

### represents the nal state of the whole system and as such is broken down in φ

^{∗}N f

### φ

^{∗}

_{e}

### φ

^{∗}

_{ν}

### where the φ

^{∗}

### represent respectively nucleus state, electron state and neutrino state. In the initial state, we have simply ψ

i### = φ

N i### . Using free particle wave functions normalized on volume v for neutrino and electron, at the zeroth order in momentum dependency, we have the simplied form for the matrix elementV

f i### =

^{g}

_{v}

### ´

### φ

^{∗}

_{N f}

### V φ

N i### dv . g is the interaction strength constant. We dene then the nuclear matrix element M

f i### = ´

### φ

^{∗}

_{N f}

### V φ

_{N i}

### dv , which does not depend on leptons energies.

### This approximation is named allowed approximation. β-transitions for which M

f i### 6= 0 can be understood under the allowed approximation are logically called allowed transitions. If under this approximation M

f i### = 0 , the wave function must be extended to rst order in momentum to compute V

f i### and so on. Nevertheless V

f i### will be referred by extension as the nuclear matrix element in the following, even when considering such transitions.

### The density of nal states for an e

^{−}

### with momentum between p and p + dp in a volume v is:

### dn

e### = 4πp

^{2}

### v

### h

^{3}

### dp, (3.4)

### with p

ν### the neutrino momentum dn

ν### =

^{4πp}

_{h}

_{3}

^{2}

^{ν}

^{v}

### dp

_{ν}

### . So

### ρ(E

_{f}

### ) = d

^{2}

### n

### dE

_{f}

### = (4π)

^{2}

### p

^{2}

### p

^{2}

_{ν}

### v

^{2}

### h

^{6}

### dpdp

ν### dE

_{f}

### (3.5)

**Kinetic Energy (keV)**

**Electrons per keV**

**0** **500** **1000** **1500** **2000** **2500** **3000**

**0**
**1**
**2**
**3**
**4**
**5**
**6****x 10**^{−4}

(a) Electron spectrum without F(Z,p).

**Kinetic Energy (keV)**

**Electrons per keV**

**0** **500** **1000** **1500** **2000** **2500** **3000**

**0**
**1**
**2**
**3**
**4**
**5**
**6****x 10**^{−4}

(b) Electron Fermi spectrum. It includes F(Z,p).

### Figure 3.1: Comparison of the electron spectrum in the case of the main branch of the Praseodymium 144 showing the inuence of the Fermi function.

### Finally the energy E

f### is E

e### + E

ν### , so at xed electron energy with a relativistic massless neutrino (or neglecting the neutrino mass, which is appropriate where the energy scale of a beta decay is between 10 keV and 10 MeV)

^{dpdp}

_{dE}

_{f}

^{ν}

### =

^{dp}

_{c}

### and we have the dierential transition probability:

### dP = 2π

### ~c

### g

^{2}

### |M

^{f i}

### |

^{2}

### (4π)

^{2}

### p

^{2}

### p

^{2}

_{ν}

### h

^{6}

### dp. (3.6)

### The Fermi Theory main point is to add to this expression the so-called Fermi function term F (Z, p) which is linked with the number of protons of the daughter nucleus and accounts for the inuence of the nuclear Coulomb eld, see also section 3.1.3. It uses the nuclear Coulomb potential in the determination of the electron wave-function instead of the free particle approach we have detailed. The inuence of the Fermi function is shown on gure 3.1. It has a clear shift eect favoring the low energy electrons. Then, grouping all non momentum dependent terms in a constant K gives the shape of the electron spectrum:

### N (p)dp = Kp

^{2}

### p

^{2}

_{ν}

### F (Z, p)dp (3.7)

### and p

ν### = Q

β### − T

^{e}

### . Using our notations dened at the beginning of section 3.1, and using W as the dierential variable, equation 3.7 writes:

### N (W )dW = KpW (W − W

^{0}

### )

^{2}

### F (Z, W )dW (3.8) In order to compare dierent nuclei, we should equally be interested in the total decay rate:

### P = 1

### 2π

^{3}

### ~

^{7}

### c

^{3}

### g

^{2}

### |M

^{f i}

### |

^{2}

### ˆ

p_{max}

0

### F (Z, p)p

^{2}

### (Q

β### − T

^{e}

### )

^{2}

### dp. (3.9) We can use the dimensionless Fermi integral

### f (Z, E

0### ) = 1 (m

e### c)

^{3}

### (m

e### c

^{2}

### )

^{2}

### ˆ

p_{max}0

### F (Z, p)p

^{2}

### (E

0### − E

^{e}

### )

^{2}

### dp (3.10)

### with E

0### the maximum electron total energy and E

e### the total electron energy for momentum p to

### replace the integral in equation 3.9. With P =

^{0.693}

### , it is possible to dene the so-called

### comparative half-life, the ft-value:

### f t = 0.693 2π

^{3}

### ~

^{7}

### g

^{2}

### m

^{5}

_{e}

### c

^{4}

### M

_{f i}

2

### (3.11)

### This quantity depends only on the nuclear matrix element, and is then useful when comparing dierent nuclei, or dierent β transitions with various Q

β### and number of protons Z.

### 3.1.2 Beta decay classication

### The beta transitions are classied according to the initial and nal nuclear state quantum numbers [38]. First we must dened Fermi (spins of the neutrino and electron are antiparallel) and Gamow- Teller (spins are parallel) transitions, which correspond to vector and axial currents respectively. Yet a given transition might not be pure and have a Fermi component and a Gamow-Teller component, as the quantum number changes are compatible with both.

### A stricter classication is given by the allowed approximation, from which follow the selection rules for β-decay. The allowed approximation implies that neither the electron nor the neutrino can carry orbital angular momentum. So, there is no possible change for parity of the nucleus. Then, the possible change of nuclear spin comes only from particle spins, which can be 0 for Fermi decays and 0 or 1 for Gamow-Teller decays. The allowed selection rules are then a nuclear spin change

### ∆I = 0 or 1 without parity change.

### Should the nuclear matrix elements vanish in the allowed approximation, we can have forbidden decays. We must expand the wave-functions in the computation to the rst order in momentum dependency where the matrix element does not vanish. This order corresponds to the orbital momentum value carried by the emitted particles. For a l-th forbidden decay, emitted particles carry a total angular momentum of l. For odd orders, parity change is possible because π = (−1)

^{l}

### . For each order the possible changes in nuclear spin get higher. The selections rules are summarized in table 3.1.

### We must note that in some cases we can exclude the vector current because the quantum number changes are not compatible with antiparallel spins for electron and neutrino. It is useful to simplify the computations.

### For forbidden decays, the nuclear matrix elements then introduce a so-called form factor, a momentum dependent term in the spectrum expression. It depends on the order of forbiddenness of the transition. As we have seen in section 2.3 we will be interested mainly in rst-forbidden unique decays, for which this term can be approximated by F

F### = p

^{2}

### + p

^{2}

_{ν}

### .

### A third classication discriminates unique and non-unique decays. It comes from the expression of form factors computed from nuclear matrix elements and electron radial wave-functions. Only one nuclear matrix element is involved in unique transitions [13, section 14.1]. It is directly linked to the possibility of excluding the vector current we have just mentioned before. The commonly accepted rule is to use the empirical approximation that a l-th non-unique transition uses the factor from the (L − 1)-th unique transition, so the rst-forbidden non-unique decays will use the allowed factor F

F### = 1 [15]. The form factor nuclear matrix elements will be called forbidden phase space factor in the following.

### 3.1.3 Fermi function

### The Fermi function account for the nuclear Coulomb eld in a simplied manner with a point-like

### nucleus [50, 35]. It is expressed by:

### ∆π

### ∆I No Yes

### 0 Allowed, if 0 → 0 vector: superallowed 1st non-unique, if 0 → 0 axial

### 1 Allowed axial 1st non-unique

### 2 2nd non-unique 1st unique

### 3 2nd unique 3rd non-unique

### 4 4th non-unique 3rd unique

### 5 4th unique

### Table 3.1: β-decays classication, the order and unicity are indicated as a function of quantum number changes. If applicable, the vector or axial nature is mentioned and unique transitions are moreover known purely axial. Otherwise the V-A ratio should be computed through the nuclear matrix elements.

### F (W, Z) = 4(2pR)

^{2(γ−1)}

### e

^{παZW}

^{p}

### |Γ(γ + i

^{αZW}p

### |

^{2}

### Γ(2γ + 1)

^{2}

### , (3.12)

### where Γ is the gamma function and R is the nuclear radius. In the following, this radius will be evaluated using Elton's formula [21] R = 1.121A

^{1}

^{3}

### + 2.426A

^{−}

^{1}

^{3}

### − 6.614A

^{−1}

### with radius given in fm.

### With our unit system, using electron mass in eV, we have the dimensionless expression for R:

### R

_{1}

### = R m

_{e}

### ~c

### = 0.0029A

^{1}

^{3}

### + 0.0063A

^{−}

^{1}

^{3}

### − 0.017A

^{−1}

### . (3.13) The Fermi function eect is qualitatively visible for both the electron spectrum, which will never be zero even with a zero kinetic energy, and the neutrino spectrum, which will present a discontinuity for a kinetic energy equal to the endpoint energy for the transition [41]. An illustrative comparison for electron spectrum is presented on gure 3.1.

### 3.2 Atomic and nuclear small eects on the spectrum shape

### Fermi's theory is a simplied description of beta decay. Various corrections need to be applied in order to match the model with experimental measurements of spectra, some of them being highly controversial, others being well understood. These corrective terms are generally accepted as symmetrical between neutrino and electron spectra with one exception. We provide here a short list of the eects that we have gathered from the literature. They will be shortly described in the following. The fth term is supposed to be the major source of uncertainty on the model, as it suers from lack of a unique and detailed theoretical model. It is important to notice that the processing for corrective terms can be slightly dierent for Fermi and Gamow-Teller transition types.

### 1. Finite size of the nucleus correction for electromagnetic interaction. The charge distribution is not point-like. This term will be called L

0### in the following and is discussed in section 3.2.1 2. Finite size of the nucleus correction for weak interaction. It is the analogous to the above eect for weak hypercharge. It follows the convolution between the leptonic and nucleonic wave functions. Noted C(Z, W ) and discussed in 3.2.1

### 3. Screening corrections of the nuclear charge by the atom's electrons. Noted S(Z, W ) and

### discussed in 3.2.2

### 4. Radiative QED corrections from the emission of real and virtual photons by the charged particles. This correction is not symmetrical for electrons and neutrinos. Noted G

β### (Z, W ) and G

ν### (Z, W

ν### ) and discussed in 3.2.3

### 5. Weak magnetism contribution. Noted B and discussed in 3.2.5

### 6. Finite mass for daughter nucleus and recoil eect on the phase space. Noted R(W, M) and discussed in 3.2.4

### 7. Finite neutrino mass. This eect is negligible, considering the actual limit on the neutrino mass. Also discussed in 3.2.4

### 8. Moving Coulomb eld source due to recoil. It is linked with the recoil eect. Noted R

F### (Z, W, M ) and discussed again in 3.2.4

### Other corrections have been mentioned in the literature. Some of them are clearly irrelevant to our case (as for example the electron capture eect, which concerns positive β-decay only). For the most controversial ones, we could not obtain a precise enough description to include them in the model. The possibility to take them into account is open for modeling improvements, particularly for the so-called electron-exchange term. This term describes the scenario of a decay where the beta particle is in a bound state with the nucleus at the end of the process and one of the electron bounded to the father nucleus is ejected instead.

### 3.2.1 Finite size corrections

### Electromagnetic eect L

0### The nite nuclear size eect is computed through the solution of the Dirac equation in the eld of the daughter nucleus charge distribution. In order to manage the calculation, the uniformly charged sphere is used as an approximation of the real charge distribution. From [51], the correction term is then

_{1+γ}

^{2}

### L

_{0}

### (Z, W ) where L

0### is the result of a numerical integration of the Dirac equation. In this study, a γ dependency is integrated in the Fermi function by switching the constant 4 in equation 3.12 with a term F

1### = 2(γ + 1) and L

0### is a generalized polynomial form, followed by empirical terms parametrized by coecients tabulated in table 1 for electron case, reproduced here in appendix A.1

### L

0### = 1 + 13

### 60 (αZ)

^{2}

### − W RαZ 41 − 26γ

### 15(2γ − 1) − αZRγ 17 − 2γ

### 30W (2γ − 1) + a

_{−1}

### R

### W (3.14)

### +

5

### X

n=0

### a

n### (W R)

^{n}

### + 0.41(R − 0.0164)(αZ)

^{4.5}

### where a

n### = P

6x=1

### b

x### (αZ)

^{x}

### with a dierent set of b

x### for each a.

### For our purpose, we will retain this L

0### expression as the electromagnetic correction and absorb the γ dependency in F

0### with F

1### = 2(γ + 1)

_{1+γ}

^{2}

### = 4 , keeping in mind this term combination for the Fermi function and nite size corrections, in order to obtain an easily understandable term for the nite-size eect, with the (1 + δ) form, comparable to the other corrective terms. This slightly dierent formulation can be found in [13, p. 105]. The correction is illustrated on gure 3.2, which presents the relative deviation between corrected and uncorrected

^{144}

### Pr spectra. Its energy dependency is mainly linear and it favors high-energy neutrinos. With a total amplitude of more than 4.5% it is the stronger of the corrections included in the model.

### The error on L

0### has been checked with numerical results from [14] versus the formula 3.14.

### According to [35] this error is smaller than 10

^{−5}

### relatively to the corrective term L

0### , so we can

### safely neglect it.

**Kinetic Energy (keV)**

**relative correction in %**

**0** **500** **1000** **1500** **2000** **2500** **3000**

**−3**

**−2**

**−1** **0** **1** **2** **3**

**8 branches**

### Figure 3.2:

^{144}

### Pr neutrino relative nite-size electromagnetic term.

### Weak-interaction eect

### The weak-interaction nite-size eect is analogous to the previous electromagnetic eect from the weak interaction point of view. It takes into account the eect of a nite-size nucleus. When solving the Dirac equations, the leptonic wave-functions interact with the nucleonic wave-functions for the nucleus components. Here, the nucleonic functions are rectangular single-nucleon functions with width R [51]. The analytical expression is not well dened and its choice can lead to dierent results for the spectrum. A generic model, such as the one we are building up, cannot expect the precision that follows from the use of a more realistic nuclear model than a uniformly charged sphere. We will follow the latest recommended forms. As a summary of the results from [13, p. 462], Wilkinson's presents the generic form

### C(Z, W ) = 1 + C

0### + C

1### W + C

2### W

^{2}

### + C

_{−1}

### W , (3.15)

### where the coecients are dierent for Fermi (vector) and Gamow-Teller (axial) decays.

### C

_{0}

^{A}

### = − 233

### 630 (αZ)

^{2}

### − (W

_{0}

### R)

^{2}

### 5 + 2

### 35 W

0### RαZ C

_{1}

^{A}

### = − 21

### 35 RαZ + 4 9 W

0### R

^{2}

### C

_{2}

^{A}

### = − 4

### 9 R

^{2}

### C

_{−1}

^{A}

### = 0 C

_{0}

^{V}

### = − 233

### 630 (αZ)

^{2}

### − (W

_{0}

### R)

^{2}

### 5 − 6

### 35 W

_{0}

### RαZ C

_{1}

^{V}

### = − 13

### 35 RαZ + 4 15 W

0### R

^{2}

### C

_{2}

^{V}

### = − 4

### 15 R

^{2}

### C

_{−1}

^{V}

### = 2

### 15 γW

0### R

^{2}

### + γRαZ 70

### Most of our considered beta transitions are pure axial decays, so the pure Gamow-Teller form can be safely used without modications. Concerning the two minor decays of the Cerium (see section 2.3), we could check for both transitions if it is dominated either by vector decay or axial decay and use the corresponding formula, where the ratio allows an evaluation of the error. Nevertheless, it is noticeable that neither

^{144}

### Ce nor

^{144}

### Pr are in the particular case called mirror decay. These decays correspond to nuclei

^{A}Z

### X

_{N}

### which have Z protons and A = 2Z + 1 nucleons so N = Z + 1 neutrons (for negative β-decays). They decay to

^{A}Z

^{0}

### Y

_{N}

^{0}

### which have Z

^{0}

### = Z + 1 and N

^{0}

### = Z . The symmetry between the initial and nal states imply that the initial and nal wave-functions for nucleons are very close to each other [38, p. 290]. So the nuclear matrix elements can be computed and the ratio of Fermi to Gamow-Teller components in the transition follows more easily from this computation.

### The drawback of not being in this case is that we cannot access to this ratio by calculation, and only

### hope that a measurement of it is available. But the benet is that for nuclei having asymmetrical

### decays, the Fermi transition is strongly suppressed, so we can expect an accurate modeling using

### the Gamow-Teller expressions for all our branches.

**Kinetic Energy (keV)**

**relative correction in %**

**0** **500** **1000** **1500** **2000** **2500** **3000**

**−1.5**

**−1**

**−0.5** **0** **0.5** **1** **1.5**

**8 branches**

### Figure 3.3: Relative eect of the weak-interaction nite-size term for

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