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 144Ce-144Pr source used in theCeLAND 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. Theoreticaluncertainties to the model are estimated. The possibility to predict neutrino spectrum shape isof primary importance for CeLAND preparation, as the experiment aims at measuring neutrinooscillations 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 . . . . . . . . . . . . . . . . . . . . . . . 31.2 Experimental overview of neutrino knowledge . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 The solar and atmospheric neutrino anomalies . . . . . . . . . . . . . . . . . 41.2.2 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 . . . . 82.1.1 Reactor antineutrino anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Other anomalies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 CeLAND: search for a sterile neutrino . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Antineutrino source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Choice of the source material . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 144Ce-144Pr source characteristics . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Elements of beta decay theory 20

3.1 Extended Fermi theory of beta decays . . . . . . . . . . . . . . . . . . . . . . . . . . 203.1.1 Fermi theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.1.2 Beta decay classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.1.3 Fermi function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Atomic and nuclear small eects on the spectrum shape . . . . . . . . . . . . . . . . 243.2.1 Finite size corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2.2 Screening correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.3 Radiative correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2.4 Finite masses and recoil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.5 Weak magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4 Implementation for the modeling and spectra predictions 45

4.1 Program structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 Result presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.1 Predicted spectra and comparison of corrective terms . . . . . . . . . . . . . 484.2.2 Application to CeLAND experiment . . . . . . . . . . . . . . . . . . . . . . . 504.2.3 Comparison with BESTIOLE modeling . . . . . . . . . . . . . . . . . . . . . 59

A Numerical tables 69

B Screening correction derivation 71

1

Introduction

Neutrinos are a center of concern in modern Physics. These ghost particles have already shownthat 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 beyondthe Standard Model. Some neutrino experiments have already observed surprising results, whichwait now to be enlightened.

Among these results, the short baseline neutrino detection experiments have revealed a lackof 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 oscillationbetween the three known avors of neutrinos (electron, muon and tau neutrinos). An hypothesis toexplain these anomalies is to introduce a new oscillation mode with a heavier neutrino. This fourthneutrino must have a square mass dierence ∆m2 ≈ 1 eV2 with the known neutrinos. It must alsobe a sterile neutrino, which is not coupled to any interaction of the Standard Model. The CeLANDexperiment aims at testing this hypothesis with an articial antineutrino source and a very lowbackground 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 sourceantineutrino 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 inthe 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. Finallythe 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.

2

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 makeup matter and their interactions. It is a relativistic quantum eld theory with symmetry groupSU(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, weakand electromagnetic interactions. The strong interaction can be considered separately, but theother two can be described within a unied framework: the electroweak theory. The resultingparticles are classied in twelve gauge bosons with integer spin, which are interactions carriers, andtwenty 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 onegeneration to the other but dierent masses. The minimal formalism of a quantum eld theorydoes not include any mass term. The supplementary Higgs eld introduces a symmetry breakingfor the electroweak interaction and has for consequences to generate mass terms for all particlesexcept neutrinos, gluons and photons in the nal Lagrangian of the model. Tables 1.1 and 1.2show 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 withthe General Relativity is a major problem for the modern physics. Gravity is supposed to aectall massive particles and in the General Relativity own frame, all particles are impacted by thespace-time curvature.

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

sensitive to 1st generation 2nd generation 3rd generation

all interactions quarksup (u) charm (c) top (t)

down (d) strange (s) bottom (b)electroweakinteraction leptons

electron (e) muon (µ) tau (τ)

weak interaction electron neutrino (νe) muon neutrino (νµ) tau neutrino (ντ )

Table 1.1: Fermions of the Standard Model

3

interaction gauge bosons scalar bosonstrong 8 gluons (g)

Higgs boson (H0)electromagnetic photon (γ)weak Z0, W+, W−

Table 1.2: Bosons of the Standard Model

0

10

20

30

86 88 90 92 94

Ecm

[GeV]

σh

ad [

nb

]

3ν

2ν

4ν

average measurements,error bars increased by factor 10

ALEPHDELPHIL3OPAL

Figure 1.1: Measurements of the hadron production cross-section around the Z resonance. Thecontribution of Z0 decay to neutrinos has a strong inuence on the cross-section peak. Predictedcross-section for two, three and four neutrino species are shown. These neutrinos are massless andcoupled to the Standard Model via weak interaction. Figure taken from [44].

checked from the decay width of the Z0 boson by an group of experiments at the Large Electron-Positron Collider [44]. The cross-section of the hadronic decay of Z0 is linked to the possible Z0

decay via neutrinos. It constrains the number of neutrinos sensitive to the weak interaction. Asit can be seen on gure 1.1, the high precision measurements conrm with very strong condencelevel 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 twentiethcentury. The Sun and the Earth atmosphere (through impact of cosmic rays) are two neutrinosources carefully studied in many occasion. But predicted neutrino detection rates were stronglyoverestimated when compared to the eective results. This apparent lack of events resulted inthe solar neutrino anomaly and the atmospheric neutrino anomaly. These anomalies have beenexplained at the end of the 90s respectively by the Sudbury Neutrino Observatory (SNO) experiment[9] and the Super-Kamiokande experiment [24].

4

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

p+ p→ D + e+ + νe (1.1)

p+ e− + p→ D + νe

3He + p→ 4He + e+ + νe

e− + 7Be→ 7Li + νe

p+ 8B→ 8Be∗+ e+ + νe

The neutrino detection rate from experiments focusing on electron neutrino was about one thirdof the theoretical expected rate. The possibility for neutrinos to oscillate between their threeknown avors was an hypothesis to explain the anomaly. The missing part of the detected neutrinowould have been the part which has oscillated between the emission and the detection. The SNOexperiment could detect neutrinos using two reactions. On the one hand

νe +D → p+ p+ e− (1.2)

concerns only electron neutrino. On the other hand

να +D → n+ p+ να (1.3)

is sensitive to all neutrino avors. The electron neutrino detection process conrmed the anomalybut 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 butpart 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 theincoming particle direction [24].

1.2.2 Neutrino experimental state of the art

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

Pα→β =∑i=(1,2,3) |Uαi|

2 |Uβi|2 + 2R∑

i<j<3 UαiU∗βiU

∗αjUβj exp (iφosc)

(1.4)

where i = (1, 2, 3) associated with Hamiltonian mass eigenstates |νi〉 and (α, β) = (e, µ, τ) representsstates|να〉 and|νβ〉 from the avor eigenstates (eigenstates for the weak interaction charged current).The phase φosc between avors determines the oscillation frequency, which depends on the squaredmass dierence ∆m2 between two mass eigenstates. The unitary matrix U allows to switch fromone base to the other. The mixing matrix U is dened as νe

νµντ

= U

ν1

ν2

ν3

=

Ue1 Ue2 Ue3Uµ1 Uµ2 Uµ3

Uτ1 Uτ2 Uτ3

ν1

ν2

ν3

,

5

and is called PMNS matrix (from its authors Pontecorvo-Maki-Nakagawa-Sakata). This unitarymatrix can be split up into a product of three rotations associated with the three mixing anglesand one phase. With cij = cos θij and sij = sin θij we have:

U =

1 0 00 c23 s23

0 −s23 c23

c13 0 s13e−iδ

0 1 0−s13e

iδ 0 c13

c12 s12 0−s12 c12 0

0 0 1

(1.5)

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

As it will be seen in section 1.3, neutrino oscillation experiments are generally sensitive to onlyone oscillation frequency among the three possible ones, which correspond to the three possible∆m2 parameters. Solar neutrino experiments are associated with the oscillation between |ν1〉 and|ν2〉. They have measured ∆m2

21 = 7.58+0.23−0.20 × 10−5 eV 2 [17].

∣∣∆m232

∣∣ = 2.35+0.12−0.09 × 10−3 eV 2 is

known from atmospheric neutrinos experiments. Its sign is not known yet, because the oscillationprobability depends on ∆m2. The interaction of neutrino with matter in the Sun allows to get rido this problem for ∆m2

21 but no equivalent method is usable with atmospheric neutrino at themoment. The problem of neutrino mass hierarchy arises from this uncertainty. One is unable todiscriminate between m1 < m2 < m3 and m3 < m1 < m2. The two order of magnitude dierencebetween ∆m2

21 and ∆m232 made ∆m2

31 degenerated and indistinguishable from ∆m232. The absolute

values of the neutrino masses have not been determined yet. An upper limit of 2 eV on mνe is thebest available at the moment. Experiments on high precision energy measurement of β-decays (forinstance on Tritium) are in progress to improve this constraint.

The corresponding sin2 θ12 = 0.306+0.018−0.015 and sin2 θ23 = 0.42+0.08

−0.03 are known from the sameexperiments. Finally, the last generation of accelerator and reactor experiments have gained accessto sin2 θ13 = 0.0251± 0.0034 [17].

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

Finally, new anomalies from short baseline experiments have been identied recently. A fourthneutrino, 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 thishypothesis, will be described in chapter 2.

1.3 Phenomenology of two-avor neutrino oscillations

Due to ∆m2 inuence on the phase term in equation 1.4, a usual case for neutrino experiment isto be sensitive to only one of the possible mass dierences as a rst approximation [25]. The twoavor oscillation probability for plane waves then writes:

Pα→β = Uα1U∗β1U

∗α1Uβ1 + Uα2U

∗β2U

∗α2Uβ2 + Uα1U

∗β1U

∗α2Uβ2e

iφosc + Uα2U∗β2U

∗α1Uβ1e

−iφosc . (1.6)

6

Neutrino source solar and reactor atmospheric accelerator∆m2 1 MeV 10 to 1000 MeV 1 GeV

8× 10−5 eV 2 12 500 m 12.5× 104 to 12.5× 106 m 12 500 km2× 10−3 eV 2 500 m 5× 103 to 5× 105 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 ∆m2 andsource.

Since the U matrix can be written as a rotation matrix

(cos θij sin θij− sin θij cos θij

), Pα→β can be ex-

pressed as

Pα→β = sin2(2θij) sin2

(c3

~∆m2

ijL

4E

)(1.7)

The angle θij determines the oscillation amplitude and the phase c3

~∆m2

ijL

4E leads to the oscillationfrequency. This expression shows clearly that the oscillations take place only if the neutrino havenon-zero mass dierences (and so at least two of the three neutrino are massive). Phase dependencyon L

E is the crucial parameter to determine to which ∆m2 an experiment is sensitive to. Using moreeasily understandable units, equation 1.7 is rewritten:

Pα→β = sin(2θij) sin2

(1.27

∆m2ij(eV2)L(m)

E(MeV)

)(1.8)

Thus, the condition to achieve the best sensitivity to a given oscillation process is then 1.27∆m2

ij(eV2)L(m)

E(MeV) =π2 , leading to

∆m2ij(eV2)L(m)

E(MeV)≈ 1

Table 1.3 summarizes the optimal parameters as a function of the studied ∆m2 and the energyrange of the used neutrino source. These estimates are only orders of magnitude.

7

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 isproportional to the ratio between the source-detector distance and the neutrino energy (equation1.7). Depending on the baseline, neutrino detection experiments constrain the existence of suchoscillation processes, whose frequency depends on ∆m2 and amplitude on sin2 2θosc.

2.1.1 Reactor antineutrino anomaly

In 2011, new evaluations of the nuclear reactor neutrino ux, based on the new experimental value ofthe 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 baselinereactor 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 uponthe possibility of a brand new phenomenon. Figure 2.1 from [16] shows a summary of the resultsof short baseline reactor antineutrino experiments. The dotted line is the prediction from the threeneutrino mixing model, and the solid line includes a sterile neutrino with ∆m2 ≈ 1 eV 2. This gureshows that a new oscillation mode, leading to the existence of a fourth neutrino species, can explainthe anomaly.

2.1.2 Other anomalies

Equally another anomaly has been observed in gallium solar neutrino detectors [32, 31](GALLEXand Sage, see [10, 37] and [3, 4] for further informations). Those detectors were calibrated withintense neutrinos sources (37Ar and 51Cr), for which the detected neutrino event rate was found tobe lower than expected. The Gallium to Germanium reaction νe +71 Ga→71 Ge + e− cross-sectioncomes with major uncertainties, lowering the statistical signicance of the neutrino decit. Howeverthe newest interpretation still supports a Gallium anomaly [31].

A third anomaly comes from cosmological observations, which constrain the number of possibleneutrinos as the Universe expansion rate is linked to the energy density in relativistic particlesduring 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

8

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 (dashedline), on the other hand the possibility to have a 3+1 neutrinos system scenario (solid line), with∣∣∆m2

∣∣ ≈ 1 eV 2. The anomaly is visible for 10 to 100 m baseline. At larger baselines, the datapoints correspond to experiments associated with solar and atmospheric neutrino oscillations.

depends on an eective number of neutrino species equal to Neff = 3.046 with the parameters of theStandard Model. The existence of an additional neutrino species would increase Neff . Althoughnot signicant enough, cosmological data do not exclude a non-standard Neff > 3.046 hypothesis.Recent Planck results, which found Neff = 3.30± 0.27 [6], are a good example. A sterile neutrino ismarginally compatible with cosmological data.

Finally, the accelerator experiments LSND and MiniBooNE, with short baselines and neutrinosin 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 (∆m2new, sin

2 2θnew) parameter space have been published in [31],using data from reactor, gallium and accelerator experiments. The gure 2.2 summarizes thepreferred parameter space. Two region are preferred around ∆m2

new = 7.6 eV2, sin2(2θnew) = 0.12and ∆m2

new ' 2 eV2, sin2(2θnew) ' 0.1. Tritium beta decay and double beta decay experimentshave also constrained the (∆m2

new, sin2 2θnew) parameter space, with upper limits for ∆m2

new lyingin the 102-104 eV2 range at the 95 % condence level. Finally, cosmological data have also beenused to set a 95% C.L. upper limit on the mass of a new sterile neutrino [33], which is somehow intension with the mixing parameters preferred by the terrestrial experiment data:

ms < 0.45 eV. (2.1)

9

sin22ϑee

∆m

412

[e

V2]

+

10−2

10−1

1

10−2

10−1

1

10

102

++

+

95% CL

GalliumReactorsνeC

SunCombined

95% CL

GalliumReactorsνeC

SunCombined

Figure 2.2: Allowed 95% condence level regions in the sin2(2θnew)−∆m2new plane obtained from

separate ts of Gallium, reactor, solar and νeC scattering from KARMEN [11] and LSND data andfrom 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 ∆m2 ≈ eV

2

.To avoid averaging eects that could erase any oscillation pattern seen in the detector and thereforedegrade the sensitivity to oscillations, an experiment looking for such oscillation should be a shortbaseline one (as seen in section 1.3), with a compact neutrino source. It will allow to observethe specic pattern as a function of the baseline and the spectrum distortion from the oscillationsin the event rate. The optimal distance between source and detector is at the meter scale, for∆m2

new ≈ 1 eV2 and MeV neutrinos (as can be achieved with a radioactive source or nuclear

10

reactors). The activity of the source will be chosen to obtain a statistically signicant detectionrate 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, thesedetectors 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 bigenough to both have the necessary sensitivity to the weakly interacting neutrinos to clearly pickout 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 moredetails). Situated in the Kamioka mine in Japan, KamLAND was built to observe the neutrino uxfrom nuclear power plants with a very long baseline (about 200 km). It is composed of a 1 ktonliquid-scintillator inner detector in a spherical plastic balloon with 6.5 m radius, surrounded bynon-scintillating oil, in a 9 m steel sphere, and an outer water-Cerenkov detector lling the gapbetween the sphere and the cylindrical cavity in the rock. The outer detector is used as a vetoagainst cosmic rays, while the inner part is the proper neutrino detector. The scintillation lightis detected with photomultiplier tubes which are mounted inside the steel sphere. The gure 2.3presents a schematic view of the detector. For CeLAND, an articial antineutrino source will beplaced in the outer detector. See section 2.3 for details about the selected 144Ce-144Pr source. Theantineutrino will be detected via inverse beta decay (described in section 2.3.1). The KamLANDliquid-scintillator is characterized by a proton density of ρP = 6.62× 1028 protons per m−3.

Simultaneously to CeLAND, the KamLAND-Zen experiment aimed at the 2β0ν detection with136Xe will run [27]. Both experiments focus on the same energy range, typically 2 to 3 MeV forinverse beta decay and for the 136Xe double decay with or without neutrino. Background eectshave 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 wewill see in the following, the 144Ce−144Pr 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 compactnessof the system, particularly for transportation, tungsten was selected because it is a high densitymaterial 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. Thiswill allow a relatively correct solid angle coverage, the use of the water from outer detector andbuer oil as an additional neutron and gamma shieldings. Furthermore it will ease the source heatdissipation, as it emits about 600 W, by using the water circulation system of the detector. Thesource 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 distanceand energy. The L

E neutrino oscillation frequency dependency (see section 1.3) leads both to adistortion in the neutrino energy spectrum and neutrino rate variation as a function of length. Itallows also to observe a neutrino detection rate with a source to detector length variation. Thegure 2.5 illustrates this two-dimensional oscillation pattern. A precise activity measurement is alsonecessary 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 accommodateup to 100 kCi. Both β and γ spectroscopy will be performed on source samples in order to control thesource purity and characterize the source antineutrino energy spectrum shape. The beta spectrum

11

18m

φ

Buffe

r Oil

Liq

uid S

cinti

llato

r

PMTs(inner detector)

Stainless steelSpherical tank

PMTs(outer detector)

Acrylic sphere

Water pool

Plastic balloon 16.5

m φ

13m

φ

Figure 2.3: KamLAND detector schematic from [16]

12

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.

13

0.70.9

1.21.4

1.61.8

2.12.3

2.5

02

46

810

1214

16

0

10

20

30

40

50

60

Evis

(MeV)

2−D reconstructed spectrum for Ue4

= 0.20 and ∆m2

41 = 2.0 ev

2

Lrec

(m)

N in

[10

cm

, 1

00

keV

] b

in

Figure 2.5: Expected spectrum with oscillation into a fourth neutrino (∆m2new = 2 eV2 and

sin2 2θnew ≈ 0.15) as a two-dimensional function of the visible energy and the distance betweensource and detector. The visible energy in the detector is the positron energy, which is linked tothe νe 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 51Cr or 37Ar, which decay by electron capture and secondlyelectron antineutrino emitters with a continuous β-spectrum such as 144Ce, 90Sr, 42Ar or 106Ru.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 variousreasons. In such a case the detection is performed by inverse beta decay νe + p → e+ + n. The

14

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

σIBD(Ee) ∼ 0.96× 10−43 × peEe cm2, (2.2)

where pe and Ee are the momentum and energy (in MeV) of the detected e+, neglecting hereneutron recoil, weak magnetism eects, and radiative second order corrections. It is one to twoorder of magnitude higher than the electron scattering νe + e− → νe + e− cross-section σES at theMeV 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 neutronobservation, the rst being promptly annihilated with an electron, the second being captured by afree proton. As the average capture time of the neutron is a few µs, the positron annihilation isthe prompt event and the neutron capture the delayed event. The time coincidence between themallows ecient suppression of background.

In order to obtain statistically signicant results in a reasonable time of about two years, thenumber of expected events can be linked to detector parameters to compute activity requirementsfor the source. With N the number of emitted neutrinos, V the detector volume, ρP the protondensity and introducing a 1

4πd2 geometrical factor to account for the coverage of the source by thedetector (d being the source-detector distance), we have:

NIBD ≈ N1

4πd2V ρPσIBD (2.3)

NIBD ≈ N × 6.6× 10−20

To obtain NIBD ≈ 10000, we need N ≈ 7.6× 1023. If we consider 3.2× 107 seconds for a year andthat 1 Ci = 3.7× 1010 decays per second, we need an activity of roughly 65 kCi. To compare with aneutrino source, the e− scattering cross-section is two order of magnitude lower, but the density ofelectrons is about ten times the density of protons, so at rst glance 10 times higher activity shouldbe required. However, the electron scattering detection process lacks of the time coincidence of theIBD and suers from larger backgrounds (solar neutrinos mainly). The necessary activity shouldthen rather be at the MCi level.

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

144Ce-144Pr pair

The inverse beta decay reaction has a 1.8 MeV energy threshold, so the source isotope must havehigh enough end-point energy Qβ for a considerable part of emitted antineutrino spectrum to haveenough energy to be detected. A high Qβ means a short lifetime for the nucleus under consideration.It implies that the source production, transportation, and generally the experiment handling will bedicult. The chosen solution is to rely on a pair of nuclei, which are both beta emitters, and wherethe father nucleus is a long lifetime isotope and the daughter a short lifetime one with high Qβ .Four pairs 144Ce−144 Pr, 106Ru−106 Rh, 90Sr−90 Y and 42Ar−42 K matching these requirementshave been identied. The rst three are common ssion products of nuclear spent fuel, when 42Arneeds a double neutron capture from stable 40Ar with a low cross-section and a highly unstableintermediate 41Ar to be produced. Reprocessing of nuclear spent fuel seems far more accessible.Then, considering the cumulative ssion yield1 of the three other mother nucleus presented in table2.1, the 106Ru 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 thereactor is at equilibrium.

15

Cumulative ssion yield (%)14458 Ce 106

44 Ru 9038Sr

235U 5.50 (4) 0.401 (6) 5.78 (6)239Pu 3.74 (3) 4.35 (9) 2.10 (4)

Table 2.1: Cumulative ssion yield for a thermal ssion of 235U and 239Pu for 144Ce, 106Ru and90Sr [1] in commercial nuclear reactors.

Branching ratio Quantum numbers Qβ(keV) type V-A ratio

76.5 % 0− 318.7 First non uniqueforbidden

A 100 %

3.9 % 1− 238.6 First non uniqueforbidden

unknown

19.6 % 1− 185.2 First non uniqueforbidden

unknown

Table 2.2: Summary of the 14458 Ce86 beta decay features from ground state 0+.

the high energy threshold of inverse beta decay, the more we have neutrinos above it the better, sothe selected couple should be 144Ce−144 Pr because of the highest Qβ in comparison to 90Sr−90 Y(2997.5 keV versus 2280.1 keV).

2.3.2 144Ce-144Pr source characteristics

Cerium and Praseodymium have respectively an atomic number of 58 and 59. In order to producethe source, Cerium will be extracted from nuclear spent fuel by chemical rare-earth separationtechniques [16]. It does not allow to separate between isotopes for a given element. The 144Ceisotope is the most stable Cerium radioisotope with a 284.91 days half-life. Most of the Ceriumisotopes have a short lifetime with respect to the 144Ce, and have a low cumulative ssion yield.We can then safely neglect any background radiation from other Cerium isotopes. The long lifetimeof 144Ce makes the source handling possible without insurmountable problems. It will drive theactivity of the source in front of 144Pr which has 17.28 minutes half-life.

144Ce always decays to144Pr, which then decays to 144Nd. The 144Nd is an alpha emitter withroughly a 1015 years half-life. The source β-spectrum will be a combination of the beta emissionfrom Cerium and Praseodymium. The relevant decay schemes are given on gures 2.6, 2.7 and 2.8from the National Nuclear Data Center [1]. We note that the 144Pr can decay from an excited 3−

state. The following β-transitions will have a quite small branching ratio that we try to estimatein the following. From 3− level, no transition has more than 0.033 % branching ratio, and it isnecessary 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 Qβ), which hasa 19.6 % branching ratio. From the resulting 1− state of Praseodymium, only one of the threeγ transitions leads to the 3− state, with intensity of 0.2 %, followed by another transition withintensity 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 to3.33× 10−6 % at most, so we can actually neglect all transitions from 144Pr excited 3− state.

The β-transitions are classied depending on the actual process involved and the quantum statechange for the nucleus. The relevant characteristics are summarized in tables 2.2 and 2.3. Detailsabout the β-decay classication are given in section 3.1.2.

16

Figure 2.6: 144Ce decay scheme from its ground state. Data from the National Nuclear Data Center[1]

Branchingratio

Quantumnumbers

Qβ(keV) type V-A Ratio

97.9 % 0+ 2997.5 First non uniqueforbidden

A 100 %

1.04 % 2+ 2301.0 First uniqueforbidden

A 100 %

1.05 % 1− 811.8 Allowed A 100 %

6.70 · 10−3% 0+ 913 First non uniqueforbidden

A 100 %

1.4 · 10−3% 2+ 1436.5 First uniqueforbidden

A 100 %

8.7 · 10−4% 0+ 322.2 First non uniqueforbidden

A 100 %

6.2 · 10−4% 2+ 924.7 First uniqueforbidden

A 100 %

3.0 · 10−4% 0+ 254.6 First non uniqueforbidden

A 100 %

1.5 · 10−4% 1+ 342.6 First non uniqueforbidden

unknown

Table 2.3: Summary of the 14458 Pr85 beta decay features from ground state 0-.

17

Figure 2.7: 144Pr 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 shapedistortion and variation in the νe rate as a function of distance. Spectrum predictions are then animportant tool to foresee the experiment performances and check on which energy domains andwith which precision we can expect any oscillation pattern detection to be signicant, dependingon the setup and operational mode variations. The next chapters describe the construction of anadapted model for the spectrum of the CeLAND neutrino source. The possibility to predict thespectrum shape will be useful for checking the β-spectroscopic measurements. Such measurementswill be done on material samples and are scheduled by the end of the year (2013).

18

Figure 2.8: 144Pr decay scheme from excited states. Data from the National Nuclear Data Center[1]

19

Chapter 3

Elements of beta decay theory

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

AZX → A

Z+1X′ + e− + νe

where the conversion from the father nucleus X with mass MX to the daughter nucleus X' withmass MX′ is done at the nucleon level by n→ p+ e− + ν. The Qβ value of a transition is denedas the available kinetic energy for the electron and the antineutrino

Qβ = (MX −MX′ −me −mν) c2 (3.1)

Both involved nuclei are possibly in excited states. This must be taken into account in the energeticsof 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

Te = Electron kinetic energy

W =Temec²

+ 1

W0 =Qβmec2

+ 1

α =e²

~c

γ =√

1− (αZ)2

Moreover, all explicitly computed uncertainties are supposed to be at 68 % condence levelexcept if dierently stated.

20

3.1.1 Fermi theory

Fermi's description of beta decay was a coherent framework, giving accurate predictions for theobserved 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 WolfgangPauli in 1930 as a way to solve incoherence observed in beta decay particle energy spectrum whenone considers only one emitted particle. The electron neutrino was undetected and its inclusionin Fermi's theory gave support to this then hypothetic particle. Fermi's computation is based onquantum mechanics with a relativistic treatment for leptons. He used a perturbation method wherethe interaction term of the Hamiltonian is a perturbation term in comparison to the Hamiltonianof the four involved particles. [38]

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

P =2π

~|Vfi|2 ρ(Ef ). (3.2)

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

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

Vfi = 〈ψf |V |ψi〉 =

ˆψ∗fV ψidv. (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 supposedparity conservation with a vector operator (V). Later, this theory has been successfully extended toaxial currents (A) allowing parity violation by George Gamow and Edward Teller [26]. The correctform 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∗Nfφ∗eφ∗ν

where the φ∗ represent respectively nucleus state, electron state and neutrino state. In the initialstate, we have simply ψi = φNi. Using free particle wave functions normalized on volume v forneutrino and electron, at the zeroth order in momentum dependency, we have the simplied formfor the matrix elementVfi = g

v

´φ∗NfV φNidv. g is the interaction strength constant. We dene

then the nuclear matrix element Mfi =´φ∗NfV φNidv, which does not depend on leptons energies.

This approximation is named allowed approximation. β-transitions for which Mfi 6= 0 can beunderstood under the allowed approximation are logically called allowed transitions. If under thisapproximationMfi = 0, the wave function must be extended to rst order in momentum to computeVfi and so on. Nevertheless Vfi will be referred by extension as the nuclear matrix element in thefollowing, 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:

dne =4πp2v

h3dp, (3.4)

with pν the neutrino momentum dnν =4πp2

νvh3 dpν . So

ρ(Ef ) =d2n

dEf=

(4π)2p2p2

νv2

h6

dpdpνdEf

(3.5)

21

Kinetic Energy (keV)

Ele

ctr

on

s p

er

ke

V

0 500 1000 1500 2000 2500 30000

1

2

3

4

5

6x 10

−4

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

Kinetic Energy (keV)

Ele

ctr

on

s p

er

ke

V

0 500 1000 1500 2000 2500 30000

1

2

3

4

5

6x 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 thePraseodymium 144 showing the inuence of the Fermi function.

Finally the energy Ef is Ee + Eν , 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 isbetween 10 keV and 10 MeV) dpdpν

dEf= dp

c and we have the dierential transition probability:

dP =2π

~cg2 |Mfi|2

(4π)2p2p2

ν

h6dp. (3.6)

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

N(p)dp = Kp2p2νF (Z, p)dp (3.7)

and pν = Qβ −Te. Using our notations dened at the beginning of section 3.1, and using W as thedierential variable, equation 3.7 writes:

N(W )dW = KpW (W −W0)2F (Z,W )dW (3.8)

In order to compare dierent nuclei, we should equally be interested in the total decay rate:

P =1

2π3~7c3g2 |Mfi|2

ˆ pmax

0

F (Z, p)p2(Qβ − Te)2dp. (3.9)

We can use the dimensionless Fermi integral

f(Z,E0) =1

(mec)3(mec2)2

ˆ pmax

0

F (Z, p)p2(E0 − Ee)2dp (3.10)

with E0 the maximum electron total energy and Ee the total electron energy for momentum p toreplace the integral in equation 3.9. With P = 0.693

decay half−life , it is possible to dene the so-called

22

comparative half-life, the ft-value:

ft = 0.6932π3~7

g2m5ec

4∣∣M

fi

∣∣2 (3.11)

This quantity depends only on the nuclear matrix element, and is then useful when comparingdierent nuclei, or dierent β transitions with various Qβ 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. Yeta 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 selectionrules for β-decay. The allowed approximation implies that neither the electron nor the neutrino cancarry 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 decaysand 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 forbiddendecays. We must expand the wave-functions in the computation to the rst order in momentumdependency where the matrix element does not vanish. This order corresponds to the orbitalmomentum value carried by the emitted particles. For a l-th forbidden decay, emitted particlescarry 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 summarizedin table 3.1.

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

For forbidden decays, the nuclear matrix elements then introduce a so-called form factor, amomentum dependent term in the spectrum expression. It depends on the order of forbiddennessof the transition. As we have seen in section 2.3 we will be interested mainly in rst-forbiddenunique decays, for which this term can be approximated by FF = p2 + p2

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. Onlyone nuclear matrix element is involved in unique transitions [13, section 14.1]. It is directly linkedto the possibility of excluding the vector current we have just mentioned before. The commonlyaccepted rule is to use the empirical approximation that a l-th non-unique transition uses the factorfrom the (L− 1)-th unique transition, so the rst-forbidden non-unique decays will use the allowedfactor FF = 1[15]. The form factor nuclear matrix elements will be called forbidden phase spacefactor in the following.

3.1.3 Fermi function

The Fermi function account for the nuclear Coulomb eld in a simplied manner with a point-likenucleus [50, 35]. It is expressed by:

23

∆π∆I No Yes

0 Allowed, if 0→ 0 vector: superallowed 1st non-unique, if 0→ 0 axial1 Allowed axial 1st non-unique2 2nd non-unique 1st unique3 2nd unique 3rd non-unique4 4th non-unique 3rd unique5 4th unique

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

F (W,Z) = 4(2pR)2(γ−1)eπαZWp

|Γ(γ + iαZWp |2Γ(2γ + 1)2

, (3.12)

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

13 + 2.426A−

13 − 6.614A−1 with radius given in fm.

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

R1 = Rme

~c= 0.0029A

13 + 0.0063A−

13 − 0.017A−1. (3.13)

The Fermi function eect is qualitatively visible for both the electron spectrum, which will neverbe zero even with a zero kinetic energy, and the neutrino spectrum, which will present a discontinuityfor a kinetic energy equal to the endpoint energy for the transition [41]. An illustrative comparisonfor 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 appliedin order to match the model with experimental measurements of spectra, some of them beinghighly controversial, others being well understood. These corrective terms are generally acceptedas symmetrical between neutrino and electron spectra with one exception. We provide here a shortlist of the eects that we have gathered from the literature. They will be shortly described inthe 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 thatthe processing for corrective terms can be slightly dierent for Fermi and Gamow-Teller transitiontypes.

1. Finite size of the nucleus correction for electromagnetic interaction. The charge distributionis not point-like. This term will be called L0 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 aboveeect for weak hypercharge. It follows the convolution between the leptonic and nucleonicwave 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 ) anddiscussed in 3.2.2

24

4. Radiative QED corrections from the emission of real and virtual photons by the chargedparticles. This correction is not symmetrical for electrons and neutrinos. Noted Gβ(Z,W )and Gν(Z,Wν) 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) anddiscussed in 3.2.4

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

8. Moving Coulomb eld source due to recoil. It is linked with the recoil eect. NotedRF (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 ourcase (as for example the electron capture eect, which concerns positive β-decay only). For themost controversial ones, we could not obtain a precise enough description to include them in themodel. The possibility to take them into account is open for modeling improvements, particularlyfor the so-called electron-exchange term. This term describes the scenario of a decay where thebeta particle is in a bound state with the nucleus at the end of the process and one of the electronbounded to the father nucleus is ejected instead.

3.2.1 Finite size corrections

Electromagnetic eect L0

The nite nuclear size eect is computed through the solution of the Dirac equation in the eld ofthe daughter nucleus charge distribution. In order to manage the calculation, the uniformly chargedsphere is used as an approximation of the real charge distribution. From [51], the correction term isthen 2

1+γL0(Z,W ) where L0 is the result of a numerical integration of the Dirac equation. In thisstudy, a γ dependency is integrated in the Fermi function by switching the constant 4 in equation3.12 with a term F1 = 2(γ+1) and L0 is a generalized polynomial form, followed by empirical termsparametrized by coecients tabulated in table 1 for electron case, reproduced here in appendix A.1

L0 = 1 +13

60(αZ)2 −WRαZ

41− 26γ

15(2γ − 1)− αZRγ 17− 2γ

30W (2γ − 1)+ a−1

R

W(3.14)

+

5∑n=0

an(WR)n + 0.41(R− 0.0164)(αZ)4.5

where an =∑6x=1 bx(αZ)x with a dierent set of bx for each a.

For our purpose, we will retain this L0 expression as the electromagnetic correction and absorbthe γ dependency in F0 with F1 = 2(γ + 1) 2

1+γ = 4, keeping in mind this term combination forthe Fermi function and nite size corrections, in order to obtain an easily understandable termfor the nite-size eect, with the (1 + δ) form, comparable to the other corrective terms. Thisslightly dierent formulation can be found in [13, p. 105]. The correction is illustrated on gure3.2, which presents the relative deviation between corrected and uncorrected 144Pr spectra. Itsenergy dependency is mainly linear and it favors high-energy neutrinos. With a total amplitude ofmore than 4.5% it is the stronger of the corrections included in the model.

The error on L0 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 L0, so we cansafely neglect it.

25

Kinetic Energy (keV)

rela

tive c

orr

ecti

on

in

%

0 500 1000 1500 2000 2500 3000−3

−2

−1

0

1

2

3

8 branches

Figure 3.2: 144Pr neutrino relative nite-size electromagnetic term.

26

Weak-interaction eect

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

C(Z,W ) = 1 + C0 + C1W + C2W2 +

C−1

W, (3.15)

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

CA0 = −233

630(αZ)2 − (W0R)2

5+

2

35W0RαZ

CA1 = −21

35RαZ +

4

9W0R

2

CA2 = −4

9R2

CA−1 = 0

CV0 = −233

630(αZ)2 − (W0R)2

5− 6

35W0RαZ

CV1 = −13

35RαZ +

4

15W0R

2

CV2 = − 4

15R2

CV−1 =

(2

15γW0R

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 section2.3), we could check for both transitions if it is dominated either by vector decay or axial decay anduse the corresponding formula, where the ratio allows an evaluation of the error. Nevertheless, it isnoticeable that neither 144Ce nor 144Pr are in the particular case called mirror decay. These decayscorrespond to nuclei AZXN which have Z protons and A = 2Z + 1 nucleons so N = Z + 1 neutrons(for negative β-decays). They decay to A

Z′YN ′ which have Z ′ = Z + 1 and N ′ = Z. The symmetrybetween the initial and nal states imply that the initial and nal wave-functions for nucleons arevery close to each other [38, p. 290]. So the nuclear matrix elements can be computed and the ratioof 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 onlyhope that a measurement of it is available. But the benet is that for nuclei having asymmetricaldecays, the Fermi transition is strongly suppressed, so we can expect an accurate modeling usingthe Gamow-Teller expressions for all our branches.

27

Kinetic Energy (keV)

rela

tive c

orr

ecti

on

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 144Pr neutrino spectrum.Once again, this one has a rst order linear energy dependency, with same characteristic as theelectromagnetic term and a smaller amplitude.

28

The nite-size term errors are estimated in [51] for C to be 1 to 4 % of the correction itself forZ = 60 nuclei. The C term is itself averaged for a correction in the 144Pr and 144Ce cases. Asthe correction amplitude is already less than 3 % of the spectrum, the error should not have aninuence in the nal result, i.e less than 1 part per thousand.

3.2.2 Screening correction

The electrons in bound states in the atom produce screening of the nuclear charge for the emittedbeta particle. This change in electromagnetic eld modies the beta spectrum. An accurate com-putation of the spectrum is made by numerical integration of the Dirac equation of the emittedparticle in the atomic eld. The result strongly depends on the atomic model.

According to [28] and [13, p. 145], the computation of the screening correction is based onan energy shift on the Fermi function with a screening potential V0. The nite-size eects shouldbe taken into account. The potential V0 is generally represented as a function of α, Z and me.Omitting the electron mass in our units system,

V0 = α2 × (Z − 1)43 ×N(Z − 1), (3.16)

where N is a function of Z depending on the model. We dene:

W = W − V0

p =√W 2 − 1

Energy shift application

In order to obtain a similar expression to the other corrective terms, the screening correction termis expressed as the ratio between shifted and non-shifted spectrum with form KpWp2

νF (Z,W )dW .

S(Z,W ) =

(p

p

)W

W

F0(Z, W )

F0(Z,W )

L0(Z, W )

L0(Z,W )(3.17)

The last ratio is neglected as the energy dependence of L0 is weak giving a ratio of almost 1. Theexplicit expression below is the one kept in our modeling and is plotted on gure 3.4 [35].

S(Z,W ) =W

W

(p

p

)2γ−1

eπ(αZWp −αZWp )|Γ(γ + iαZWp )|2

|Γ(γ + iαZWp )|2 (3.18)

For minus beta decay case arises the problem of the momentum denition with energy shift. AsW is not always larger than 1, p is not dened for small W . The screening correction is then set toone on its undened domain, following the prescription of [35]. Further investigation are needed tounderstand what happens for this small part of the spectrum (V0 is a few tens of eV with a pessimisticevaluation). Finally, a recent work on 138La with high precision measurement on low energy betaparticles shows a deviation with theoretical predictions [43]. This deviation is present at energiesless than 75 keV, where the screening eect is the strongest for electron spectrum. The deviationraises when energy decreases. At its greatest extent, the theoretical spectrum underestimatesexperimental data by 24.5 %. A possible interpretation is a wrong estimation of the screeningeect.

29

Kinetic Energy (keV)

rela

tive c

orr

ecti

on

in

%

0 500 1000 1500 2000 2500 3000−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

8 branches

Figure 3.4: Eect of the screening term. This term has not a linear eect as the majority of theother corrections. It is the stronger for low-energy electron and high-energy neutrinos.

30

Screening potential

The screening potential has been evaluated by integrating the Dirac equations for the electron inthe atomic eld with dierent methods. With more details, they match dierent models for theatom [28]. The simplest uses a Thomas-Fermi statistical approach (TF method in the following),

leading to a V0 potential proportional to Z43 . From then, this has stayed the classical way to express

the screening potential, allowing simple comparison between the dierent evaluations, even if moreprecise methods lead to new dependencies. The TF approach assumes that the electron densitydistribution around the atom is a uniform statistical distribution function. The electrons feel apotential eld from the nucleus and their own distribution. It aims at computing electron densityand kinetic energy as a functional of this density [42]. Thomas-Fermi statistic result is N = 1.795,giving V0 = 22.1 eV for 144Pr.

A better evaluation comes from the resolution of the Hartree-Fock equations for the atomic sys-tem. The Hartree-Fock wave-function is computed with a coupled system of Schrödinger equations,where each particle sees the mean eld originating from the other atomic electrons. The correctmean eld V0 is obtained by solving iteratively the Hartree-Fock equations until self-consistencyis achieved. We will use in the following tabulated values for N from [28] and [13]. The rst setof tabulated values corresponds to Hartree-Fock equations resolution in the Slater approximation(HFS method), where the many-particle wave-function is set down as a Slater determinant [47].The second set of tabulated values comes from the relativistic Hartree-Fock equations are solvedinstead (RHF method), which are expected to be more accurate for heavy atoms. The tables arereproduced in table A.2 . A comparison of the N coecient as a function of Z is given on gure3.5. Since no exact values corresponding to the Z of Cerium and Praseodymium are available, alinear interpolation has been used to estimate N . The RHF method is used by default in the modeland the other ones are available.

Figure 3.5: Screening potential coecients comparison.

Screening uncertainty

The screening correction is a well understood process, but suers from doubts on the atomic models.As V0 has dierent values depending on the computation method, we retain the screening as a

31

Kinetic Energy (keV)

Ne

utr

ino

pa

r k

eV

2300 2400 2500 2600 2700 2800 2900 3000 3100

1.8

2

2.2

2.4

2.6

2.8

3

3.2

3.4

x 10−4

Figure 3.6: Detail of the 144Pr neutrino spectrum for high energies where the screening error is thelargest. As we can see, the nal spectrum uncertainty is quite small. See also gure 3.7

source of theoretical error. In order to estimate the error on this term, we consider than the RHFtreatment will lead to the best result in our cases, so for Z ≈ 60, and use it as the central value.The deviation is then roughly assessed by using the half-width between the extremal estimatesfrom HFS method and the TF method, which is expected to be conservative enough. The erroris computed by a Taylor expansion of our analytic formula 3.18, where the biggest diculty arisesfrom the analytical dierentiation of S(Z,W, V0) at xed W and Z. The computation of ∂S

∂V0and

∂2S∂V 2

0is detailed in appendix B. Despite the rather large variation on N (between 1.5 and 1.8),

leading for instance in the 144Pr case to V0 = 20.7 ± 1.4 eV, the propagated error seen on gures3.6 and 3.7 shows that we expect a decent precision on the screening correction.

The second order of the Tailor expansion has been checked. It indeed is negligible because itamounts to 0.1% of the rst order term at worst. It is illustrated on gure 3.8. We have also useda Toy Monte-Carlo simulation in order to check the error propagation and the possibility of highorder resonance in the derivation. The results support the analytical propagation and conrm thathigher orders are negligible. A comparison of both nal errors on the spectrum can be seen ongure 3.9, which is the relative deviation between Monte-Carlo propagated error and analyticallypropagated error. The analytical error includes only the rst order error. We can see that the peakof the rst order error at high energy dominates strongly the error term. The overall agreement

32

Kinetic Energy (keV)

Rela

tive e

rro

r o

n t

he s

pectr

um

0 500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5x 10

−3

Figure 3.7: Ratio between the error from screening term and the corresponding dierential neutrinospectrum for 144Pr. As we can see, we have less than 0.25 % error from this term, but the error isthe highest close to the undened part, where we cannot compute the correction.

stays good with no more than 0.9 % error on the error propagation.

3.2.3 Radiative correction

Radiative corrections come from the emission of virtual and real photons in the decay process(e.g. photon loops in the Feynman diagram, see gure 3.10). An extensive treatment for thesecorrections leads to a number of problems, some impossible to solve analytically, some dependenton the nucleus model, and is far beyond the scope of this study. Moreover, they are not symmetricalbetween electron and neutrino spectra, forcing a separate treatment, because the neutrino cannot becoupled with a photon as a neutral particle. In the case of real photon emission, energy conservationimplies that we should take into account the photon energy in the Qβ of the transition. As thephoton spectrum varies in 1

Eγ, the main contribution is from low energy photons and we will then

neglect their contribution in the energy conservation equation [41].

Electron case

We will use the results from [45] concerning the electron spectrum, where the corrections are spito into two parts with strongly dierent behaviors. The radiative corrections can be separated intwo types [13, p. 430 and following]:

the inner radiative corrections depend on the form of the strong and weak interactions,and on the nucleus structure. They do not depend on the energy of the electron. So innerradiative corrections term only aect the renormalization of the coupling constant and notthe spectrum shape. As we are primarily interested in the spectrum shape, we can ignore

33

Kinetic Energy (keV)

rela

tiv

e s

ec

on

d o

rde

r u

nc

ert

ain

ty f

rom

sc

ree

nin

g

0 500 1000 1500 2000 2500 30000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10

−6

Figure 3.8: Ratio between the second order error from screening term and the dierential neutrinospectrum for 144Pr. As we can see, we have less than 0.00016 % error from the second order, so itis clearly negligible in comparison to the rst order error term, which is shown on gure 3.7.

34

Kinetic Energy (keV)

Re

lati

ve

de

via

tio

n b

etw

ee

n e

rro

rs i

n %

0 500 1000 1500 2000 2500 30000.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

Figure 3.9: Relative deviation between Monte-Carlo propagated and analytically propagated screen-

ing error for the main β-branch of 144Pr neutrino spectrum |errorMC−errorTH |errorTH

.

35

p e− ν

n

(a)

p e− ν

n

(b)

p e− ν

n

(c)

p e− ν

n

(d)

p e− ν

n

(e)

Figure 3.10: Feynman diagrams for radiative corrections from [45]

36

them and hence the diculties they bring along themselves, particularly the dependency onthe nuclear model.

the outer radiative corrections contain energy dependent terms and modify the spectrumshape, but they can be exactly computed as they are independent of the details of the inter-actions, the vector or axial nature of the transition and the nuclear structure. It allows usinga well understood bare nucleon model to compute them.

Radiative outer corrections to the electron beta spectrum are expressed by [45]:

Gβ(Z,W ) = 1 +α

2πgβ(W ) (3.19)

gβ(W ) = 3lnMN −3

4+ 4

(argtanhβ

β− 1

)×(W0 −W

3W− 3

2+ ln(2(W0 −W ))

)+

4

βL

(2β

1 + β

)+

1

βargtanhβ

(2(1 + β2) +

(W0 −W )2

6W 2− 4argtanhβ

)(3.20)

where β = pW , L is the Spence function (or dilogarithm) L(x) =

´ x0

ln(1−t)t dt andMN is the nucleon

mass in units of electron mass. We have retained this expression in our model. The radiativecorrections inuence is shown on gure 3.11. The energy dependency is not linear. For electrons,it is the strongest corrective term with more than 6 % relative amplitude.

This idyllic description must be tempered nevertheless. It only accounts for the lowest orderouter corrections (so correction of order α). The superior orders Zα2 and Z2α3 are not so wellknown, in particular for Gamow-Teller transitions and the polynomial form αm(αZ)n is demon-strated only for superallowed Fermi transitions [50],[13, p. 432]. Problems arise because even if theneglected terms are one order of magnitude lower than α

2π gβ , the rst order radiative correctionis large enough so that higher orders could have contribution to the nal spectrum. But a roughevaluation of the second order term shows that the energy dependent part is small and so that theinuence on spectrum shape is at the 0.1% level for electrons. We then neglect it. A particularlyin depth study for outer radiative terms is available in [55, 54, 53], where higher order terms andtheir validity are at length discussed.

Another particularity for radiative term is that the electron results are not applicable to neutrinoterms computation.

Neutrino case

As for the electron radiative QED correction, we can use an equivalent analytic expression at theorder α1Z0 derived by Sirlin in [46]. However, the status of superior order terms is even less known.Fortunately, when the high Z of our nuclei should force the consideration of the most detailed modelbecause of the importance of the radiative correction for electrons, the neutrino terms are at leastabout an order of magnitude weaker. We can then expect a better prediction even with our sparseknowledge. For practical purposes, this term is expressed with the dierence between the end pointenergy and the total neutrino energy W = W0 −Wν , and so on for p and β. This notation allowsus to take into account the eventuality of a non negligible energy used in the nucleus recoil. Assoon as this condition is fullled, we have simply equivalence between electronic quantities and thehat notation.

Gν(Z, W ) = 1 +α

2πh(W ), (3.21)

37

Kinetic Energy (keV)

rela

tiv

e c

orr

ec

tio

n i

n %

0 500 1000 1500 2000 2500 3000−6

−5

−4

−3

−2

−1

0

1

2

8 branches

Figure 3.11: Relative eect of the radiative term for 144Pr electron spectrum.

38

Kinetic Energy (keV)

rela

tiv

e c

orr

ec

tio

n i

n %

0 500 1000 1500 2000 2500 3000−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

8 branches

Figure 3.12: Relative eect of the radiative term for 144Pr neutrino spectrum.

h(W ) = 3lnMN +23

4− 8

βL

(2β

1 + β

)(3.22)

+8

(argtanhβ

β− 1

)ln(2W β) + 4

argtanhβ

β

(7 + 3β2

8− 2argtanhβ

)

This expression is once again obtained by considering only the energy dependent terms, when abroader and possibly more accurate expression is given in [12]. According to [35], the dierencebetween both expressions for the relevant terms in our case is within 5%, which seems prettynegligible for a corrective term of the order of 1% on the spectrum like the radiative QED correctionsfor neutrinos. This term is illustrated on gure 3.12, which can be compared to gure 3.11 with thesame non-linear energy dependency. It promotes high-energy neutrinos. As opposed to the electroncase, it has the weakest amplitude of the non-negligible terms.

3.2.4 Finite masses and recoil

Both nucleus and neutrino masses are neglected in Fermi's theory. First the neutrino mass isexpected to have a small inuence with the old 30 eV limit [51]. As the contemporary constraintsgive a neutrino mass smaller than 2 eV, we can indeed consider that it will be negligible.

With a nite nuclear mass, the phase space factor is modied for a three-body decay [51]. Thisterm depends on the axial or vector nature of the transition. With Wilkinson's nomenclature, we

39

have the nuclear mass M = 1837×A (A being the nucleon number) and

R(W,M) = 1 + r0 +r1

W+ r2W + r3W

2, (3.23)

with respectively

rV0 =W 2

0

2M2− 11

6M2and rA0 = −2W0

3M− W 2

0

6M2− 77

18M2,

rV1 =W0

3M2and rA1 = − 2

3M+

7W0

9M2,

rV2 =2

M− 4W0

3M2and rA2 =

10

3M− 28W0

9M2,

rV3 =16

3M2and rA3 =

88

9M2.

As already stated in section 3.2.1, in our case all transitions come down to pure axial case. Itallows us to use the axial form directly expecting then a correct precision of the model.

The nucleus recoil leads to a corrective term for the Fermi function. The source of the Coulombeld from the nucleus is now moving so the electromagnetic eect on the electron is modied. From[49, 51], we have the leading term of this eect in a non-relativistic approximation (considering therelative speed of the electron and the nucleus).

RF (Z,W,M) = 1− παZ 1 + ΩW0−W3W

Mp(3.24)

where ΩA = − 13 and ΩV = 1. The Ω quantity actually depends on the ratio of the coupling

constants∣∣∣ gAgV ∣∣∣.

In our case, the recoil eect is extremely small, as it can be seen on the gure 3.13. It has anegligible eect in comparison to other corrections and particularly considering the precision weexpect from other terms. We can note that the energy dependency is not simply linear but a peakoccurs at transitions end-point. In all cases it will not have any noticeable contribution to the nalspectrum and as a consequence we do not consider the eventuality of an uncertainty associated withthis term.

3.2.5 Weak magnetism

This eect is a consequence of the nucleus recoil, which can be roughly described as a formallyanalogue of the magnetic eect from the non static Coulomb eld, but for the weak interaction[35, 50]. For the tritium nucleus, a combined recoil term including both eects from section 3.2.4and this one is available [52], but the general case is not known with accuracy. Following [35], webegin by writing B(W ) = 1 + δWMW , with

δWM =4

3

b

MNCAλme (3.25)

The electron mass occurs because of our natural units. λ =∣∣∣ gAgV ∣∣∣ ≈ 1.23 is linked to the ratio

of the Gamow-Teller and Fermi coupling strengths [34, 29]. CA is here the Gamow-Teller nuclearmatrix element, and b is linked with the magnetic transition moment µ by b =

√2µ. The magnetic

transition moment is dened as the change in nuclear magnetic moment occurring to the nucleusduring the β-transition.

40

Kinetic Energy (keV)

rela

tive c

orr

ecti

on

in

%

0 500 1000 1500 2000 2500 3000−5

−4

−3

−2

−1

0

1

2

3

4x 10

−3

8 branches

Figure 3.13: Relative eect of the combined recoil terms R and RF for 144Pr neutrino spectrum.

41

In order to evaluate the weak magnetism term δWM , a rst method is to use the impulseapproximation. It considers the nucleus wave-function to be a linear superposition of independentnucleons wave-functions. Each one has the behavior of a free nucleon in the interaction withleptons. The impulse approximation also neglects any angular momentum dependencies. Underthese assumptions, the δWM coecient has no dependence on nuclear structure. It is equivalent touse the neutron decay case for any Gamow-Teller decay and so [48]

b

MNCAλ≈

√2µ

MN

√2λ≈ µp − µn

MN

∣∣∣∣gVgA∣∣∣∣ .

Then we havedB

dW=

4

3

µp − µnMN

∣∣∣∣gVgA∣∣∣∣me = 0.5% MeV−1 (3.26)

A second method uses the conserved vector current hypothesis to obtain the weak magnetismterm from the measured isotope gamma decay parameters. This hypothesis states that a similarbehavior is expected from electromagnetic interactions and vector terms in the weak interactions.Particularly, following the electromagnetic current conservation which implies the universality of thecoupling strength, the coupling strength for vector part of the weak interaction is also a universalcoupling constant. A major consequence is that an electromagnetic interaction process in a nucleusis analog to a beta decay process with the same change in nucleus quantum numbers and hasproportional nuclear matrix elements [29]. The weak magnetism is then analogous to the classicalmagnetism, and the parameter b is equivalent to the electromagnetic M1 decay form factor bγ =√

6ΓM1M2

αE3γ

, where ΓM1 is the transition radiative width (M1 describes here a magnetic dipole decay)

[19].Unfortunately, the necessary gamma decay data are available only for 13 nuclei [35, 36], because

it is essential to have for a given nucleus a beta decay and a pure gamma M1 decay with equivalentquantum number change. It is dicult to estimate how representative this sample is. All nuclei arequite light when compared to 144Pr or 144Ce. Using the mean value from the complete set as thecentral value for dB

dE and the standard deviation as the error leads to

dB

dE= (4.78± 10.5)% MeV−1. (3.27)

We can also try to search for a possible link between dBdE and transition parameters. A possible

trend has been noticed with ft-values [35], because large ft-values imply small matrix element CAas it can be seen on equation 3.11. We can see two domains on a graphical representation of dBdE asa function of log ft, but the variation range in ft value makes logarithmic scales attractive and thesame kind of separation is less obvious with such scales. Moreover, 144Pr and 144Ce branches havelog ft between 6.5 and 9.2 for the main transitions we are interested in. They could then be placedin the long-lifetime category, for which we have only three isotopes, and a large deviation betweenthem. The correction evaluation based on these three isotopes would give

dB

dE= (18.47± 17.04)% MeV−1 (3.28)

but we cannot consider that sample as representative enough to estimate δWM .So we will primarily use the analytical approximation from equation 3.26, which linear eect is

shown on gure 3.14, and apply an uncertainty of 100 % of the correction term on it. The tableresults will be used as a secondary computation method. As we lack of a better description, theweak magnetism is a weak point of the model. With its importance and the associated uncertainty,

42

Kinetic Energy (keV)

rela

tiv

e c

orr

ec

tio

n i

n %

0 500 1000 1500 2000 2500 3000−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

8 branches

Figure 3.14: Relative eect of the weak magnetism term for 144Pr neutrino spectrum. The lin-ear dependency in energy corresponds to the analytical calculation B(W ) = 1 + δWMW . As anexception, it promotes low-energy neutrinos.

the weak magnetism correction is expected to be the main source of errors. The error is propagatedusing once again Taylor series. With our linear model for weak magnetism, the rst order is theonly one which can contribute to the nal result, as we have the straightforward dB

( dNdE )= 1 +meW

and d2B

( dNdE )2 = 0. The gure 3.15 shows the comparison between screening and weak magnetism

driven uncertainties on the spectrum.

43

Kinetic Energy (keV)

Ne

utr

ino

s p

er

ke

V

0 500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5x 10

−6

Screening error

Weak mag error

Figure 3.15: Screening and weak magnetism parts of the nal uncertainty for the nal neutrinospectrum of 144Pr. The optimistic evaluation of weak magnetism is used and is clearly the mainerror source. Yet, for the more entertaining part of the spectrum beyond 2.5 MeV, both errors aresimilar in amplitude.

44

Chapter 4

Implementation for the modeling and

spectra predictions

This chapter describes the computation of β-spectra, following the considerations from chapter 3,specically for CeLAND source case. The implementation has been done using the MatLab softwaretools. The goal is to obtain both e− and νe theoretical spectra for 144Pr and 144Ce isotopes. Theβ-branches can either be combined or separately considered. The choice of applying any of thecorrections previously discussed is left at the user's discretion. The nal results essentially consistin dierential spectra, which are adapted from equation 3.8 with the corrective terms, so:

N(W )dW = KFF pW (W −W0)2F (Z,W )L0(Z,W )C(Z,W )S(Z,W )G(Z,W )R(M,W )B(W )dW.(4.1)

The computation of integrated spectra over variable energy bins is also possible.

4.1 Program structure

The les can roughly be divided into two groups with dierent purposes and some useful scriptedmodeling processes. The scheme available on gure 4.1 shows all the essential les in these cat-egories and the relation between each other within a group. For additional information, MatLabconventional documentation is included in the code.

Interface group

The central part is organized around Isotope.m and is in charge of the management of availableoptions in the computation. An specic isotope is represented by an adapted subclass which includesthe necessary data and initializes the program with corresponding parameters. The choice betweenneutrino and electron spectra and of the energy scale used in any output are necessary at the callof one of these subclasses. Moreover, one can select any combination of the β-transitions, which areimplemented for a particular nucleus for the nal results. The default choice is to use all possibletransitions for the computation of a total spectrum.

The provided methods at this level act generally as top-level controllers and use the second levelgroup to provide the spectra. The main method allows to compute a combined and normalizedspectrum with the possibility to select which corrective terms must be used and which of theavailable computation methods must be used for terms which have this feature. The errors fromscreening and weak magnetism terms are combined and propagated to the nal spectrum. A specic

45

Main.m

Test.m

ElectronSpectrum.m

NeutrinoSpectrum.m

Spectrum.m

gammac.m

psic.mpsin.m

Transition.m

Cerium.m

Huberium.m

Praseodymium.m

Isotope.m

PrettyFigureFormat.m

errorbar_tick.m

SubclassUsed by

BestImport.mIntegrated

Spectrum.m

ClassScriptFunction

a.ma.m

a.m

Spectrum calculation at the beta branch level.Various terms involved

are separately computed.

Management for termchoice, computationmethods and beta

branches combination

Scripts for tests and results, including plotsand Bestiole spectrum

handling.

Figure 4.1: Structural overview of the computer program, with a three-category organization.46

method with the same features but using Monte-Carlo error propagation is present. It can drawrandom numbers following a gaussian distribution to account for uncertainties in a given correction.The nal value for spectrum and error are respectively the mean value and standard deviation ofthe resulting distribution at a specic energy point. Direct access to the complete default spectrumis possible. It should be noted that a method is included to compute the inverse beta decay crosssection with the energy scale from the class instance The cross-section is taken from [22]. It includesneutron recoil correction, weak magnetism eects, and radiative second order corrections. It allowsto simply apply the cross-section to the spectra for applications.

Finally, display methods allow to obtain simply graphical results as classical spectrum plots orFermi-Kurie plots. See also section 4.2.

Computation group

With data and model parameters provided by the previous level, Spectrum.m along with its sub-classes are in charge of the practical spectrum computation. Each corrective term in equation 4.1 isassociated to a specic method or method group implementing the theoretical expressions detailedin chapter 3. Vector transitions, which need peculiar expressions, have been implemented in previ-sion to future changes but are inactivated. The screening and weak magnetism corrections errorsare propagated to the interface group methods.

For screening term and weak magnetism term, optional computation methods are available asstated respectively in section 3.2.2 and 3.2.5.

Moreover, the specic methods used in the Monte-Carlo error propagation draw random num-bers according to a gaussian distribution for a parameter θ with uncertainty δθ The correction isexpressed as a function of θ, C(θ). The values of θ and δθ depend on the nucleus parameters.Applying the function C to each of those events allows us to obtain a distribution with mean valueC and standard deviation δC.

Numerical tables are used for nite-size electromagnetic, screening and weak magnetism correc-tions. They are respectively Finite_size_parametrization.mat (Implementing table A.1 repro-duced from [51]), screeningpotential.mat (Table A.2 from [28] and [13]) and Huber_weak_mag.mat(Implementing the relevant parts of table I from [36]).

Error propagation

A particular care has been focused on the error propagation from the computation of a correction tothe nal spectrum. Errors are computed for the relevant terms by the method which computes thecorrection. We consider that uncertainties from dierent corrections are uncorrelated and combinethem by a quadratic sum, with S a spectrum point and i = (screening,weak magnetism),

∆S

S=

√∑(∆SiSi

)2

. (4.2)

The error on the integrated spectrum is then computed by a direct integration of the error on thedierential spectrum, under the same condition as the spectrum itself, as we consider on the contrarya perfect correlation between the spectrum points. Finally, the uncertainty on the cross-section isan output from the cross-section method. It is combined as an uncorrelated error with the globalspectrum error, should this function be activated in the suitable script IntegratedSpectrum.m.

New nuclei

So far we have only considered 144Pr and 144Ce as these isotopes will be the antineutrino emitters ofthe CeLAND experiment. The program could be adapted to nuclei with resembling decay properties

47

with minimal work by creating new subclasses for Isotope.m with the correct parameters. Asafeguard is coded at the nucleus selection level that must be overcome if one wants to study anucleus which is not yet implemented in the code. Any other nucleus could be considered givingminor supplementary changes in the main classes Isotope.m and Spectrum.m to take into accountits specicity. For instance, the forbidden phase space factor is only implemented up to secondforbidden transitions and new cases will be necessary for uncommon isotopes with third and fourthforbidden transitions. The main work would be to take into account β-transitions with leadingvector part, as can be a superallowed transition. The specic expressions are currently disabled sothe process to select the right formulas for this kind of nuclei must be implemented.

4.2 Result presentation

As we are interested in the shape of the spectrum i.e the relative abundance between particles ofvarious energy, we consider that the integral of the dierential spectrum is 1, whatever terms couldbe included. It allows to easily compare the inuence of the corrections. All spectra represent thena dierential number of particles with respect to the energy. The tools to obtain a spectrum as anhistogram of number of particles in an energy bin are available in IntegratedSpectrum.m. Theinverse beta decay cross-section and its uncertainty due to neutron lifetime can be added by thescript.

A specic display method converts a spectrum to a Fermi-Kurie plot. These plots are a tradi-tional way to present results for β-decay. They will be used in further development to compare themodel with already published experimental data. A rewriting of the equation 3.8 leads to√

N(W )

pWF (Z,W )K= (W −W0) (4.3)

Then plotting the energy versus the left member, we expect a straight line crossing the x-axis forW = W0. In term of classical notations, W = Te

mec²+1 andW0 =

Qβmec²

+1, so the Fermi-Kurie plots

show generally√

N(W )pWF (Z,W ) versus Te and allows a direct reading for Qβ . But as it can be seen, we

consider the case of an allowed decay without any corrective terms. The linearity of the plot can

be restored for the general case by using√

N(W )pWF (Z,W )SF (Z,W ) , with SF (Z,W ) the forbidden phase

space factor, taking into account the order of forbiddenness of the transition [38].

4.2.1 Predicted spectra and comparison of corrective terms

With our default parameters, the nal 144Pr and 144Ce neutrino spectra are shown on gures 4.3and 4.8. The main transition end-points lead to clear discontinuities for neutrino spectra.

Table 4.1 summarizes the corrections with the associated relative total amplitude, computedfrom the amplitude of the relative deviation between corrected spectrum and Fermi spectrum foreach correction. These deviation are shown on gure 4.2. It allows us to observe the relative im-portance of nite-size corrections, and to clearly dismiss the recoil correction. The weak magnetismcorrection is clearly the dominant source of error. An evaluation of the uncertainties is also shownin table 4.1.

The result with the weak magnetism optimistic hypothesis are illustrated on gures 4.3 and4.4. The uncertainty is quite small. If the pessimistic hypothesis which makes use of the gammatransition parameters is used (see section 3.2.5), the amplitude and uncertainty are both greatlyimpacted. The weak magnetism is the dominant term from both point of view as it can be seen ongures 4.5, 4.6 and 4.7. The error reaches more than 25 % of the total spectrum value. Nevertheless

48

Kinetic Energy (keV)

rela

tiv

e c

orr

ec

tio

ns

in

%

0 500 1000 1500 2000 2500 3000−3

−2

−1

0

1

2

3

Screening

Finite−size

Hypercharge convolution

Radiative

Recoil

Weak mag

Figure 4.2: Combination of the gures from chapter 3 which show the relative corrective terms forall considered corrections with all beta branches included for 144Pr neutrino spectrum. We can seethat the recoil eect is much smaller than the other terms for our heavy nuclei and so we expectno inuence from it. All other corrections are present for the high energy neutrino spectrum. Wecan also note that compensation eects between corrections can occur.

49

144Pr corrections relative amplitude of thecorrection in comparison to

spectrum (%)

estimate of uncertaintyrelatively to spectrum (%)

screening 2.94 0.25electromagnetic nite-size 4.68 < 0.00005weak interaction nite-size 2.53 < 0.1

radiative 1.56 ≈0.07nite mass and recoil eects 0.013 -

impulse approx. weak magnetism 1.34 1.34weak magnetism from gamma decay 11.93 26.21

Table 4.1: Relative importance of the considered corrective terms for neutrinos.

we can hope to at least constrain the weak magnetism from β-spectrum measurement on sourcesamples.

4.2.2 Application to CeLAND experiment

As the Praseodymium's lifetime is only 17.28 minutes when Cerium's is 284.91 days, we considerthat 144Ce will limit the source activity and that the 144Ce-144Pr pair is at secular equilibrium.With this hypothesis, we can combine both isotopes and obtain the full source spectrum shown ongure 4.9. As the same neutrinos number is expected from both nuclei and the energy range is alot smaller for Cerium, we can see the strong domination from it at low-energy levels.

Number of inverse beta decay interactions in KamLAND

We will then use our model to predict what could be expected during the CeLAND experiment.The antineutrinos will be detected by inverse beta decay (IBD, see section 2.3.1), so the observablespectrum is obtained by multiplying the IBD cross-section term to the computation. This spectrumis shown on gure 4.10. The threshold of IBD is high enough to negate all inuence of Ceriumas its transitions end-points are below 1.8 MeV. The cross-section is computed using prescriptionsfrom [22]. Then we can obtain the total number of events expected during the experiment.

Nν = NCeρpFgeo

ˆ ∞0

σIBD (Eν)S (Eν) dEν (4.4)

We take into account only NCe the number of expected Cerium decays because of the secularequilibrium between isotopes, as already mentioned. ρp is the density of protons in the detector andFgeo is a geometric factor (with dimension of a length) including the detector and source geometry.The source is a 13.5 cm diameter and 13.5 cm high cylinder. For a spherical detector with 6.5 mradius such as KamLAND with center at 9.6 m from the source, Fgeo = 1.10m. Considering thepossible backgrounds (possible gamma rays and neutrons escaping the source), we can be forced touse a ducial cut on the detector. Then we will be reduced to a 6 m radius, giving Fgeo = 0.85m.The number of Cerium decays corresponding to a data taking period T is:

NCe =

ˆ T

0

A0e−ln(2)tT1/2 dt (4.5)

= A0

T1/2

ln(2)

[1− e

−ln(2)TT1/2

],

50

Kinetic Energy (keV)

ne

utr

ino

s p

er

ke

V

−500 0 500 1000 1500 2000 2500 3000 35000

1

2

3

4

5

6x 10

−4

Figure 4.3: Complete neutrino spectrum for eight rst β-branches of 144Pr with optimistic weakmagnetism correction.

51

Kinetic Energy (keV)

Err

or

ba

r s

ize

in

%

0 500 1000 1500 2000 2500 30000.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Figure 4.4: Error bars for the 144Pr spectrum presented on gure 4.3.

52

Kinetic Energy (keV)

rela

tiv

e c

orr

ec

tio

n i

n %

0 500 1000 1500 2000 2500 3000−6

−4

−2

0

2

4

6

8

Figure 4.5: Relative deviation for pessimistic weak magnetism for 144Pr neutrino spectrum. Therelative amplitude is then 11.93 %. this has to be compared with gure 4.2.

53

Kinetic Energy (keV)

ne

utr

ino

s p

er

ke

V

−500 0 500 1000 1500 2000 2500 3000 35000

1

2

3

4

5

6

7

8x 10

−4

Figure 4.6: 144Pr neutrino spectrum with uncertainty for pessimistic weak magnetism. The expectedaccuracy of the model is strongly degraded. This must be compared to gure 4.3.

54

Err

or

ba

r s

ize

in

%

Kinetic Energy (keV)

0 500 1000 1500 2000 2500 30005

10

15

20

25

30

Figure 4.7: Error bar for the pessimistic weak magnetism hypothesis for 144Pr.

55

Kinetic Energy (keV)

ne

utr

ino

s p

er

ke

V

−50 0 50 100 150 200 250 300 3500

1

2

3

4

5

6x 10

−3

Figure 4.8: Complete neutrino spectrum for the three β-branches of 144Ce.

56

Kinetic Energy (keV)

ne

utr

ino

s p

er

ke

V

0 500 1000 1500 2000 2500 30000

0.5

1

1.5

2

2.5

3

3.5x 10

−3

Figure 4.9: Combined spectrum of the CeLAND source (without error bars).

57

Kinetic Energy (keV)

dif

fere

nti

al

nu

mb

er

of

ne

utr

ino

s c

m2 p

er

ke

V

1600 1800 2000 2200 2400 2600 2800 3000 32000

1

2

3

4

5

x 10−47

Figure 4.10: Dierential observable neutrino spectrum from 144Pr including the IBD cross-section.

58

where A0 is the initial source activity. The liquid scintillator from KamLAND is characterized bya density of protons ρp = 6.62× 1022 cm−3. So with a 75 kCi source and 18 months of data taking,we could expect respectively about 24165 ± 189 or 18673 ± 146 detected neutrinos depending onconsidered detector radius, where the uncertainties come from the β-spectrum modeling, assumingoptimistic weak magnetism, and from the IBD cross-section.

Comparison with expected spectrum distortion from a 4th neutrino oscillation process.

The CeLAND experiment will use the oscillation process dependency on LE , with L the distance

between source and detector and E the antineutrino energy. The link with energy leads to aspectrum distortion by the oscillation process. Our modeling does not take into account any neutrinooscillation. But the precision of the expected spectrum for CeLAND experiment is critical, becausethe more uncertainty we have on the expected antineutrino spectrum, the less the experiment issensitive to spectrum distortion. The gures 4.11, 4.12 and 4.13 show the comparison between ourpredicted detected spectrum and a model where oscillation inuence has been added for a fourthneutrino candidate. The oscillation pattern is visible. The rst gure takes only the optimistictheoretical uncertainty into account and the deviation with the oscillated model is much largerthan the spectrum uncertainty. In such a case, it would not be dicult to obtain a signicantresult. A more realistic plot is shown on the second gure where statistical uncertainties havebeen added to the error bars. Finally, the third case shows articially expanded error bars, but isrepresentative of what could be a pessimistic hypothesis (and even quite underestimated). If theoscillation pattern is included in the error bars, it will be much more dicult to conclude.

4.2.3 Comparison with BESTIOLE modeling

BESTIOLE stands for Beta Energy Spectrum Tool for an Improved Optimal List of Elements. Itaims at predicting electron and neutrino spectra originating from nuclear reactors [41, 40]. It canalso be used to predict single-isotope spectra. BESTIOLE modeling is based on the same approachthan ours, with a corrected Fermi theory. Some dierences exist between the models, for instancescreening terms are not included in BESTIOLE.

Our spectrum modeling for 144Pr agrees quite well with BESTIOLE, conrming the robustnessof our implementation. As BESTIOLE model has already been confronted to experiment [41, 40],particularly with data from electron spectra from nuclear fuel, it gives us condence for our ownresults. It is a good omen for the future confrontation with source sample beta spectroscopy.

59

Theoretical L/E spectra expected in the KamLAND detector

L/E (m MeV−1

)

Co

un

ts p

er

0.1

m M

eV

−1 b

ins

1 2 3 4 5 6 7 8 9

200

400

600

800

1000

1200

Oscillated model with Ue4

= 0.20 and ∆ m2 = 2.00 eV

2

Model with no oscillations

Figure 4.11: kComparison of expected detected spectra with or without oscillation with optimistictheoretical uncertainties. The candidate fourth neutrino has ∆m2

new = 2 eV2 and sin2 2θnew ≈ 0.15.

60

Theoretical L/E spectra expected in the KamLAND detector

L/E (m MeV−1

)

Co

un

ts p

er

0.1

m M

eV

−1 b

ins

2 3 4 5 6 7 8 90

200

400

600

800

1000

1200

Oscillated model with Ue4

= 0.20 and ∆ m2 = 2.00 eV

2

Model with no oscillations

Figure 4.12: Comparison of expected detected spectra with or without oscillation with theoreticaluncertainties and statistical uncertainties. The candidate fourth neutrino has ∆m2

new = 2 eV2 andsin2 2θnew ≈ 0.15.

61

Theoretical L/E spectra expected in the KamLAND detector

L/E (m MeV−1

)

Co

un

ts p

er

0.1

m M

eV

−1 b

ins

1 2 3 4 5 6 7 8 90

200

400

600

800

1000

1200

1400

Oscillated model with Ue4

= 0.20 and ∆ m2 = 2.00 eV

2

Model with no oscillations

Figure 4.13: Comparison of expected detected spectra with or without oscillation with statisticaluncertainties and articially expanded theoretical uncertainties (10 times). The candidate fourthneutrino has ∆m2

new = 2 eV2 and sin2 2θnew ≈ 0.15.

62

Kinetic Energy (keV)

Neu

trin

os p

er

100 k

eV

0 500 1000 1500 2000 2500 30000

0.01

0.02

0.03

0.04

0.05

0.06

new model spectrum

BESTIOLE spectrum

Figure 4.14: Comparison between both programs on 144Pr antineutrino spectrum. The agreementbetween the two models is good.

63

Conclusion

The developed beta spectrum modeling is now ready to be used to determine the inuence of thespectrum shape uncertainty on the oscillation detection process. On the predicted spectra, theuncertainties are shown quite small except for weak magnetism. The other uncertainties achievea 1 % precision on the spectrum shape and the agreement with BESTIOLE model conrms thenew model as a useful basis. The weak magnetism is noticeably the less well-known among thecorrections that have been included. Its uncertainty dominates strongly any other error source andthe two available approaches give dierent results. Little model improvements need to be donebefore applying it to spectroscopy. The inuence of the uncertainties on the nuclear data, such asthe experimentally measured branching ratio and Qβ , will be quantied.

This model will be compared to already existing beta spectroscopic data and it will be used tointerpret the data from source sample beta spectroscopy, which will be performed next year. Perhapsthe combination between experimental and theoretical inputs will make possible to constrain theweak magnetism eect, which will be of great help for the next steps of the CeLAND experiment.

64

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68

Appendix A

Numerical tables

Table A.1: Coecients of the parametrization of L0 for electrons

b1 b2 b3 b4 b5 b6a−1 0.115 -1.8123 8.2498 -11.223 -14.854 32.086a0 -0.00062 0.007165 0.01841 -0.53736 1.2691 -1.5467a1 0.02482 -0.5975 4.84199 -15.3374 23.9774 -12.6534a2 -0.14038 3.64953 -38.8143 172.137 -346.708 288.787a3 0.008152 -1.15664 49.9663 -273.711 657.629 -603.703a4 1.2145 -23.9931 149.972 -471.299 662.191 -305.68a5 -1.5632 33.4192 -255.133 938.53 -1641.28 1095.358

69

Table A.2: Screening potential coecient N(Z) from Hartree-Fock-Slater and relativistic Hartree-Fock methods.

Z N from HFS N from RHF

1 17 1.3998 1.42 1.429 1.44410 1.47111 1.47412 1.47613 1.48414 1.48115 1.48416 1.488 1.49717 1.494 1.50218 1.49620 1.49623 1.504 1.5225 1.51327 1.518 1.54429 1.56130 1.5432 1.55635 1.5536 1.55138 1.55239 1.55345 1.56149 1.566 1.63752 1.56753 1.56854 1.56855 1.56760 1.57264 1.57766 1.57968 1.58670 1.5974 1.59376 1.59580 1.59984 1.83886 1.692 1.601 1.90794 1.603

70

Appendix B

Screening correction derivation

We have the analytic expression of the screening correction as a function of W and V0

S(V0,W ) =W

W

(p

p

)2γ−1

eπ(αZWp −αZWp )|Γ(γ + iαZWp )|2

|Γ(γ + iαZWp )|2 (B.1)

In order to compute the error on the corrective term as a function of our incertitude on the com-putation of V0, we rewrite it as

S(V0,W ) =e−

παZWp

Wp2γ−1|Γ(γ + iαZWp )|2 ∗ W ∗ p2γ−1eπ

αZWp |Γ(γ + i

αZW

p)|2 (B.2)

so S(V0,W ) = KWF1F2F3 where K gathers the terms which are independent on V0 and the Fterms are arranged to simplify dierentiation.

K =e−

παZWp

Wp2γ−1|Γ(γ + iαZWp )|2

W = W − V0

F1 = p2γ−1 =

(√(W − V0)

2 − 1

)2γ−1

F2 = eπαZWp = e

παZ(W−V0)√(W−V0)2−1

F3 = |Γ(γ + iαZW

p)|2 = |Γ(γ + i

αZ(W − V0)√(W − V0)

2 − 1)|2

∂S

∂V0= K

(∂W

∂V0F1F2F3 + W

∂F1

∂V0F2F3 + WF1

∂F2

∂V0F3 + WF1F2

∂F3

∂V0

)(B.3)

∂W

∂V0= −1 (B.4)

∂F1

∂V0= (2γ − 1)

V0 −W√(W − V0)

2 − 1

((W − V0)

2 − 1)γ−1

= (1− 2γ)W p2γ−3 = (1− 2γ)W

p2F1 (B.5)

71

∂F2

∂V0= παZ

1

(W − V0)2 − 1

(W − V0)2√(W − V0)

2 − 1−√

(W − V0)2 − 1

eπαZ(W−V0)√(W−V0)2−1 =

παZ

p3eπαZWp =

παZ

p3F2

(B.6)Now we want to derive the gamma function term. The logarithmic derivative of the gamma function

dened as Γ(z) =´∞

0tz−1e−tdt is known as the dilogarithm function ψ(z) =

´∞0

(e−t

t − e−zt

1−e−t

)dt.

Then we have Γ′(z) =´∞

0tz−1lnt e−tdt = ψ(z)Γ(z).

In order to derive |Γ(γ + i αZ(W−V0)√(W−V0)2−1

)|2 we use the property Γ(z) = Γ(z), so as we consider

only αZWp real by restricting ourselves on W values, we can simply write

∂F3

∂V0=

∂Γ

(γ + i αZ(W−V0)√

(W−V0)2−1

)∂V0

Γ

γ − i αZ(W − V0)√(W − V0)

2 − 1

+Γ

γ + iαZ(W − V0)√(W − V0)

2 − 1

∂Γ

(γ − i αZ(W−V0)√

(W−V0)2−1

)∂V0

∂F3

∂V0=

∣∣∣∣Γ(γ + iαZW

p

)∣∣∣∣2 iαZp3

[ψ

(γ + i

αZW

p

)− ψ

(γ − iαZW

p

)]∂F3

∂V0= F3

iαZ

p3

[ψ

(γ + i

αZW

p

)− ψ

(γ − iαZW

p

)](B.7)

As we can express the complex dilogarithm function as series related to harmonic numbers ψ(z) =

−γ +∑∞n=0

(1

n+1 − 1n+z

), where γ is the Euler-Mascheroni constant, we obtain the same relation

as for the gamma function ψ(z) = −γ +∑∞n=0

(1

n+1 − 1n+z

)= ψ(z).

∂F3

∂V0= F3

iαZ

p3∗ 2iIm

[ψ

(γ + i

αZW

p

)]= −2F3

αZ

p3Im

[ψ

(γ + i

αZW

p

)](B.8)

∂S

∂V0= KF1F2F3

(−1 + (1− 2γ)

(W

p

)2

+ WπαZ

p3− 2W

αZ

p3Im

[ψ

(γ + i

αZW

p

)])(B.9)

Due to some diculties with the analytical form of the derivative, we have also computed a

numerical one for ∂|Γ(γ+ix)|2∂x , which can be used to check the analytical result. The included gures

show that our analytic computation is consistent with the numerical gradient for |Γ(γ + ix)|2 for

0 < x < 1, where we will physically use the domain 0 < x < αZ as x stands for αZWp .

In order to check the second order, we will need ∂2S∂V 2

0. With

F4 =

(−1 + (1− 2γ)

(W

p

)2

+ αZπW

p3− 2W

αZ

p3Im

[ψ

(γ + i

αZW

p

)])

∂2S

∂V 20

= K

(∂F1

∂V0F2F3F4 + F1

∂F2

∂V0F3F4 + F1F2

∂F3

∂V0F4 + F1F2F3

∂F4

∂V0

)(B.10)

∂(Wp

)2

∂V0= 2(W − V0)

(W − V0)2 −

((W − V0)

2 − 1)

((W − V0)

2 − 1)2 =

2(W − V0)((W − V0)

2 − 1)2 =

2W

p4

72

∂ Wp3

∂V0=

1 + 2 (W − V0)2√

(W − V0)2 − 1

5 =1 + 2W 2

p5

∂Im[ψ(γ + iαZWp

)]∂V0

=1

2i

∂ψ(γ + iαZWp

)− ψ

(γ + iαZWp

)∂V0

=1

2i

∂ψ(γ + iαZWp

)∂V0

−∂ψ(γ − iαZWp

)∂V0

=

αZ

2p3

(ψ1

(γ + i

αZW

p

)+ ψ1

(γ + i

αZW

p

))

=αZ

p3Re

(ψ1

(γ + i

αZW

p

))

∂F4

∂V0=

((1− 2γ)

2W

p4+ αZπ

1 + 2W 2

p5− 2αZ

(1 + 2W 2

p5Im

[ψ

(γ + i

αZW

p

)]+ W

αZ

p6Re

[ψ1

(γ + i

αZW

p

)]))(B.11)

∂2S

∂V 20

=

((F4 + 1)

∂S

∂V0+KF1F2F3

∂F4

∂V0

)(B.12)

73