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Measurement of the double beta decay half-life of Nd-150 and search for neutrinoless decay modes with the NEMO-3 detector

The half-life for two-neutrino double beta decay of Nd-150 has been measured with data taken by the NEMO 3 experiment at the Modane Underground Laboratory. The limits are also set on the half-life of different neutrinoless double beta decay of this isotope.


Abstract
The half-life for two-neutrino double beta decay of 150 Nd has been measured with data taken by the NEMO 3 experiment at the Modane Underground Laboratory.
This result is the world's most accurate half-life measurement of this isotope. A search for neutrinoless double beta decay of this isotope is presented using the same data taking period. No significant excess of events above the background expectation is observed and the limit on the half-life of this process is set to be T 0ν 1/2 > 1.8 × 10 22 y (90% CL).
This result has significantly improved the previous limit on neutrinoless double beta decay of this isotope. Limits are also set on several other neutrinoless double decay modes.

Declaration
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The Author
The

Chapter 1 Introduction
Neutrinoless double beta decay is a lepton number violating process which provides unique evidence that neutrinos are Majorana particles, i.e. their own anti-particle.
Neutrinoless double beta decay is also sensitive to the neutrino mass scale. The experimental signature of 0νββ decay is the observation of two electrons, for which the total energy sum is equal to the nuclear transition energy.
This thesis presents a measurement of the half-life of neodymium-150 ( 150 Nd) twoneutrino double beta decay (2νββ) and a search for different modes of neutrinoless double beta decay (0νββ). The data used for this thesis were collected by NEMO 3 between 2003 and 2006, corresponding to 924.7 days of data taking.
The 150 Nd isotope has a nuclear transition energy of 3.367 MeV, which is higher than for most of the natural radioactive sources of background, and has a large phase space factor. These properties have made 150 Nd a strong candidate for next generation double beta decay experiments, such as SuperNEMO [1] and SNO+ [2].
In NEMO 3 the backgrounds to 0νββ are divided into 2νββ decay and radioactive backgrounds. The 2νββ background is irreducible as it has the same event topology as 0νββ decay. The precise half-life measurement of 2νββ decay is therefore important for 0νββ searches. It also helps to improve the understanding of nuclear matrix elements (NME), which are the major source of uncertainty in the derivation of the neutrino mass from the 0νββ half-life. The radioactive backgrounds are reduced by applying two-electron event selection criteria. The measurement of their activities is necessary for estimating the number of remaining events due to these backgrounds. This is achieved by studying control channels with final states different from the signal.
The thesis is set out as follows. The theoretical background to the work presented in this thesis and the current status of double beta decay experiments are described in gives details of the different double beta decay theories and modes. Section 2.5.5 introduces the nuclear matrix elemenet of the double beta decay.

The Standard Model
In  Tables 2.1 and Table 2.2, respectively. In the SM fermions and gauge bosons obtain masses through the Higgs mechanism [4] which introduces an additional field with an associated particle, the Higgs boson.

The nature of massive neutrinos
Despite the success of the SM, it is not able to account for massive neutrinos. In the SM, neutrinos are considered to be left-handed and this precludes the possibility that neutrinos have mass, as helicity is conserved only for a massless particle. However, the observation of the oscillation between the different flavours of neutrinos [6,7] has   [5].
shown that the neutrino mass eigenstates (ν 1 , ν 2 and ν 3 ) and the flavour eigenstates particles, which means that they are distinct from their own anti-particles, or they are Majorana particles, meaning they are their own anti-particles.

Dirac neutrino
The Dirac mass term of the neutrino Lagrangian is written as [8] L m D = m D ν 0 where m D is the non-diagonal Dirac mass matrix, and ν 0 R and ν 0 L are the chirally right-handed and left-handed flavour neutrino fields defined as (2. 2) The second term in Equation 2.1 is the Hermitian conjugate (h.c.) of the first term.
As right-handed neutrinos do not couple through the weak interaction, the Dirac model predicts three right-handed sterile neutrinos which do not interact and can not be detected. Dirac mass terms require four independent components (ν 0 L ,ν 0 R , ν 0 L , ν 0 R ) and thus suggest that neutrinos and anti-neutrinos are distinct and lepton number is conserved. The Dirac model leads to small coupling of the neutrino to the Higgs field (in comparison with the coupling of other leptons). The Dirac model can not explain why the neutrino mass is much lower than that of other leptons.

Majorana neutrino
The Majorana mass term may be constructed out of solely the left-handed or righthanded neutrino. However, the left-handed mass is not invariant under the electroweak gauge group. Thus, it is more common to see the mass term in the righthanded construction. The neutrino mass Lagrangian can be written as [8] where m RM is the right-handed symmetric Majorana mass matrix, and (ν 0 R ) c is the charge-conjugate field of ν 0 R , which satisfies the Majorana condition (ν 0 R ) c = Cν 0 T R , where C is the charge conjugation matrix, and T denotes transposition. This field also has the property that (ν 0 R ) c = P L (ν 0 ) c , where P L is the left-handed projection operator, that is (ν 0 R ) c is left-handed. This can be considered in terms of the picture of moving into a reference frame where the helicity of the massive neutrino flips. The Majorana neutrino field is self-conjugate and therefore Majorana neutrinos are their own antiparticles. Among all fermions only neutrinos can be Majorana particles as they do not carry charge. Since the Majorana field has only two independent components (ν 0 R , ν 0 R ) compared to the four in the Dirac field, the Majorana theory is simpler and more natural than the Dirac theory. As there is no reason for the Majorana term to be zero, neutrinos are assumed to be Majorana particles in most theories beyond the SM.

The see-saw mechanism
The see-saw mechanism [9] is a Dirac-Majorana model which is developed in order to justify the smallness of the mass of the neutrinos in comparison with other fermions in the SM. If the neutrino has a Dirac mass as other fermions in the SM do and also a right-handed Majorana mass defined through Equation 2.3, then its total Dirac-Majorana neutrino mass term L mass is constructed as

PMNS matrix
In the full three neutrino mixing framework the weak eigenstates can be expressed as superpositions of three neutrino mass eigenstates (m 1 , m 2 , m 3 ) linked via a unitary This can be rewritten as This unitary mixing matrix is known as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [10,11]. The unitary matrix U can be parametrised in the following form D is the diagonal matrix defined as where s ij , c ij are the sine and cosine of the mixing angle θ ij , δ is the Dirac phase and where m i = ν i . More details about neutrinoless double beta decay are given in Section 2.5.3.

Neutrino mass hierarchy problem
The probability for a neutrino to change from one flavour to another is related to the difference between the masses squared of the mass eigenstates, ∆m 2 ij = m 2 i −m 2 j . Based on the observations from neutrino oscillations, various mass models have been proposed [12]. These can be categorised as the normal hierarchy in which m 1 has the lowest mass among the three, and the inverted hierarchy in which m 3 has the lowest mass. In both scenarios m 1 and m 2 have similar masses. The third model is degenerate, where the three mass eigenstates have similar masses. Figure 2.1 shows a schematic view of the normal and inverted hierarchy models. The approximate mass squared difference between the mass eigenstates found by oscillation experiments is shown [13,14]. Figure 2.2 shows the distribution of m ν with respect to the oscillation parameters for normal (red) and inverted hierarchy (green). The next generation double beta decay experiments will be sensitive to the full degenerate and inverted hierarchy regions. If no neutrinoless double beta decay event is observed by these experiments, then the inverted hierarchy can be excluded.

Absolute mass scale of neutrino
Oscillation experiments are trying to measure the difference between mass eigenstates of the neutrinos, but they are not sensitive to the absolute mass scale of the neutrino. Experimentally there are three possible ways to constrain or measure the absolute mass scale. One way is to search for the neutrino rest mass by exploring the endpoint energy of the electron spectrum in tritium beta decay and thus find the mass scale of the neutrino ( m ν ) directly from the kinematics of the decay. Currently a limit for the electron neutrino mass of less than 2.2 eV has been achieved [15,16]. The advantage of this experimental approach is that it is independent of the nature of the neutrino (Dirac or Majorana); however, the measurement of the mass from the decay spectrum is limited by the energy resolution of the experiments and background considerations. This makes it experimentally difficult to improve the sensitivity of the experiment to an energy scale of meV.
The second way to constrain the neutrino mass is from analysing cosmological data.
The astrophysical experiments are able to set limits on the sum of the neutrino masses ( i m i ). These limits are based on the cosmological models which give the neutrino contribution to the density of the universe. The most recent limits for i m i range from 0.7 to 2 eV at 90% confidence level [17]. This range can constrain the lightest neutrino mass (m 1 or m 3 ). The disfavoured region of lightest neutrino mass is shown by a vertical gray bound in Figure 2.2. As the cosmological models are based on several assumptions, it would be challenging to improve these results.
The third approach to constrain the absolute mass scale of neutrinos is searching for neutrinoless double beta decay (0νββ). As mentioned previously in Section 2.4, neutrinoless double beta decay is sensitive to the effective neutrino mass m ν if the neutrino is a Majorana particle.
2.5 Theory of double beta decay 2.5.1 Beta decay Beta (β) decay is a type of radioactive decay in which the weak interaction converts a down quark of the neutron into an up quark of a proton while emitting an electron and an antineutrino. Thus, this process changes the atomic number, Z, of a nucleus by one unit, while the atomic mass A remains the same: There are also two other possible modes of this process which are known as β + and electron capture (EC) and defined as:

Double beta decay
Double beta decay, 2νββ [18], is a process in which two beta decays occur simultaneously with emission of two electrons and two antineutrinos: Double beta decay can only occur in nuclei with even atomic number and atomic mass (even-even). In these nuclei the single beta decay can be either energetically forbidden or strongly suppressed. The nuclear transition energy, Q ββ , for this process is defined as: where m(A, Z) and m(A, Z + 2) are, respectively, the masses of the initial and final nuclei and m e is the mass of the electron. Figure 2.3 shows a diagram of the energy requirements for double beta decay. This process is the rarest known kind of radioactive decay, and it is predicted in only 36 isotopes. Double beta decay occurs within the SM and involves a second order weak interaction as shown in Figure 2.4a.
The rate of the process is characterised by its very long lifetime (more than 10 18 years).
The half-life (T 2ν 1/2 ) of the double beta decay process is expressed by the equation:  As there are no neutrinos in the final state, the experimental signature of 0νββ is two electrons, with energies summing to the nuclear transition energy (Q ββ ). For 2νββ decay the energy sum of the two electrons forms a continuous spectrum (Figure 2.5).

Neutrinoless double beta decay
The half-life of 0νββ is expressed by: where m νe is the effective Majorana neutrino mass and m e is the electron rest mass. Thus the experimentally measured half-life, or in case of non-observation of the process its lower limit, can provide the effective neutrino mass or an upper limit on it. The phase space integral, G 0ν , is proportional to Q 5 ββ and the atomic number, Z [21]. Thus the neutrinoless double beta decay rate is higher for isotopes with high Q ββ value.

Right-handed current 0νββ
The 0νββ mechanism that involves the helicity flip of the neutrino due to its mass is known as the mass mechanism. This mechanism is not the only possibility that leads to emission of two electrons in the final state. There are theories beyond the Standard Model which predict the existence of pure right-handed W bosons (W R ) or W bosons with mixtures of left-handed and right-handed W bosons (W = W R + W L ) [19]. If W R exists and the neutrino is a Majorana particle, then a righthanded neutrino can interact at the other vertex without need for a helicity flip. This mechanism is known as the right-handed current mechanism of neutrinoless double beta decay. In this thesis a limit is set only on the half-life of pure right-handed neutrinoless double beta decay. The half-life of this process is inversely related to the phase space factor of this process, the NME (M 0ν λ ) and λ, the coupling constant of the right-handed neutrino with W R ,

Double beta decay with emission of Majorons
Several beyond the SM models exist in which global B-L (Baryon-Lepton) symmetry is broken spontaneously. These lead to the prediction of a massless Goldstone boson, called the Majoron [22] which can couple to the neutrino. There are three types of Majoron models: singlet Majoron, doublet Majoron and triplet Majoron [23]. The measurement of the Z boson width at LEP [24] has ruled out the doublet and triplet Majoron models, but singlet and dominantly singlet Majoron models are still possible. These models predict that the two neutrinos in double beta decay couple with a Majoron, χ 0 : Here the term Majoron means light or massless boson with couplings to neutrinos.
In this definition the Majoron is not constrained to be a Goldstone boson.
A model for neutrino masses is proposed in the context of large extra dimensions.
In this model the global B-L symmetry is broken spontaneously by a gauge singlet Higgs field in a bulk [29]. This leads to a bulk singlet Majoron which is observable in neutrinoless double beta decay.
There are in total 10 Majoron models (one bulk Majoron and 9 light or massless (2.22) The index α indicates the Majoron mode, g α is the effective coupling constant of Majoron mode α, G ββ α is the phase space factor and M ββ α is the nuclear matrix element of each Majoron mode.

Nuclear Matrix elements
If the half-life of neutrinoless double beta decay is measured, the effective neutrino mass can be calculated from Equation 2.18. Accurate calculations of the NME are necessary in order to measure the effective neutrino mass with minimum uncertainty.
Two main theoretical methods have been widely used to calculate the nuclear matrix elements for neutrinoless double beta decay: the nuclear shell model (NSM) and the quasi-particle random phase approximation (QRPA). The NSM [30] is useful for calculating single particle states close to the Fermi level and is only reliable for light nuclei such as 48 Ca, 76 Ge and 82 Se. For medium to heavy double beta decay isotopes the shell model calculations are difficult to carry out.
For heavier nuclei the proton-neutron QRPA (pn-QRPA) [31] has been found to be body interaction matrix elements: particle-particle (pp) which is correlated to the proton-proton interaction and particle-hole (ph) which is correlated to the protonneutron interaction. Both matrices contain independent interaction constants g ph and g pp [32].
The ph matrix elements mainly affect the giant Gamow-Teller resonance [33], which is reproduced accurately by the QRPA and as such g ph is generally fixed. The g pp constant, which has a large effect on NME and double beta decay, is a free parameter.
This parameter causes uncertainties in NME calculations and thus much of the work into QRPA has been focused on the g pp problem. One method in the early stage of development is to fix the value of g pp by fitting it to the available 2νββ and β decay data. Another important issue in QRPA is that the nuclei of experimental interest are assumed to have spherical symmetry. As many heavy nuclei are deformed [34]  Chapter 3

Double beta decay experiments
The challenging task in double beta decay experiments is to search for the 0νββ peak in the electron energy spectrum within the continuum of 2νββ and natural radioactive background. This chapter presents the experimental criteria and status of double beta decay experiments.

Experimental criteria
In order to search for 0νββ signal and minimise backgrounds several factors need to be considered in 0νββ experiments [36]: • The double beta decay isotopes and the detector components must have high purity. This goal can be achieved via purification of double beta decay sources and constructing smaller and more granular detector components.
• To suppress cosmic ray backgrounds the detectors are situated underground.
• The isotopes with large Q ββ are favoured as the energy region of interest is above potential backgrounds (more details are given in Section 3.1.1).
• A good energy resolution is required to prevent the tail of the 2νββ spectrum from extending into the 0νββ region of interest. This produces a good signal over background ratio.
• To reject background, event reconstruction and good particle identification is required.
• The NME is understood for some isotopes more than others. The interpretation of limits or, in case of discovery, 0νββ signals requires measurements for a range of isotopes.
• Several 0νββ modes can be studied through energy and angular distributions of the electrons in the final states.
So far, no experiment searching for 0νββ has managed to include all the above criteria for its 0νββ searches. Section 3.2 gives details of the current experimental status of double beta decay detectors.

Neodymium-150
The double beta decay of 150 Nd to the ground state of 150 Sm is accompanied by the simultaneous emission of two electrons. The 2νββ decay to the ground state of 150 Sm has already been observed by several experiments. processes is predicted to be greater than 10 20 y [37]. A precise measurement of the 2νββ decay of this isotope to the ground state is important since it is an irreducible background to 0νββ and also can be used to improve current NME calculations.
The Neodymium-150 ( 150 Nd) isotope has the second highest Q ββ value of all the double beta decay isotopes (after 48 Ca). This, combined with a high atomic number, makes the neutrinoless double beta decay phase space factor G 0ν (Q ββ , Z)) the   process is inversely related to G 0ν (Q ββ , Z), thus the event rate for 150 Nd is expected to be higher than all other isotopes, assuming identical NMEs. This feature makes 150 Nd a strong candidate for next generation double beta decay experiments such as SNO+ [2] and SuperNEMO [1] (Section 3.2). Currently, the only drawback of using 150 Nd for 0νββ searches is the high uncertainty on the NME calculation due to its mass and deformation [38].

Experimental status
The observation of neutrinoless double beta decay can answer important questions in neutrino physics. In the past ten years several experiments have been therefore constructed and many more are currently in the R&D stage. The double beta detectors are divided into two main categories: homogeneous (source equal to detector) experiments in which the double beta decay source is part of the active detector and heterogeneous (source not equal to detector) experiments in which the source is separated from the detector. The homogeneous detection technique gives good energy resolution. The heterogeneous experiments contain tracking detectors which give them a good particle identification.

Experiments following the homogeneous system
Semiconductor germanium (Ge) detectors are among the most popular double beta decay experiments. This is because of the cheap enrichment process for 76 Ge. The NME of this isotope is also relatively well known. There have been two previous semiconductor 76 Ge detectors which produced important 0νββ search results: Heidelberg-Moscow and IGEX. Heidelberg-Moscow was a semi-conductor germanium detector, enriched to 86% in 76 Ge. The final design of the detector consisted of five individual sub-detectors with a total mass of 11.5 kg. The experiment ran between 1990 and 2003. A claim for discovery of 0νββ was made in 2001 by a subgroup of the Heidelberg-Moscow collaboration [40]. The half-life value of 0νββ was obtained to be T 0ν 1/2 = 1.19 ± 2.99 0.50 ×10 25 years, with m ν = 0.2 − 0.6 eV. Figure 3.3 shows the result of this experiment. However, this result has received criticism [41].
It is believed that the background and the systematic uncertainties of this experiment are underestimated. The IGEX experiment used a similar experimental technique to the Heidelberg-Moscow experiment and produced a limit on the half-life of T 0ν 1/2 > 1.57 × 10 25 years at 90% confidence level (CL) [42]. To confirm or disprove the Heidelberg-Moscow results, two experiments are being designed to study the region highlighted by the Heidelberg-Moscow experiment: GERDA [43] and MAJORANA [42]. Both of these detectors will use Ge enriched to 86% 76 Ge. In the first phase of its running, GERDA will utilize the existing One running source equal detector experiment is CUORICINO [44]. This experiment uses bolometers containing double beta decay isotopes, running at extremely low temperatures (order of mK). When a double beta decay event occurs, the electrons of the decay increase the temperature by depositing energy inside a crystal. By measuring the temperature rise the deposited energy is measured. Each bolometer is 5 × 5 × 5 cm 3 and is fixed into a layered tower structure. The total mass of the crystals is 41 kg, corresponding to 11 kg of 130 Te. This detector has set a limit on the half-life of neutrinoless double beta decay of 130 Te to be more than 3.0 × 10 24 years, corresponding to m ν < 0.42 − 0.58 eV [44]. CUORE [45], which is the next generation of the CUORICINO experiment, will have 760 kg of TeO 2 crystals with a total of 206 kg 130 Te. It will consist of 35 towers which have the same size as the CUORICINO tower. It is expected to reach a half-life sensitivity of 2.5 × 10 26 years which corresponds to a mass sensitivity of m ν ∼ 45 −53 meV, depending on NME.
The COBRA [46]  The SuperNEMO detector is a next generation double beta decay experiment which is going to use the same experimental approach as NEMO 3 but will improve the sensitivity to neutrino mass scale by one order of magnitude ( m ν < 45 − 85 meV).
This experiment will comprise 20 individual modules, each of which has a rectangular shape with the source foil in the centre surrounded by the tracker and the calorimeter. This experiment will use 100 kg-200 kg 150 Nd and/or 82 Se isotope.
There are also two other experiments which are planning to use heterogeneous techniques: MOON [51] and DCBA [52]. MOON will use one tonne of 100 Mo and hopes to achieve an effective neutrino mass sensitivity of m ν < 50 meV. DCBA will use natural 150 Nd in its first phase of data taking and then enriched 150 Nd in the second phase. They hope to reach a sensitivity of m ν < 0.12 eV.  The NEMO 3 detector is a heterogeneous system, in which the radioactive sources do not make up part of the active detector. Particle tracking and energy measurements are also performed independently, giving good particle identification and understanding of radioactive backgrounds.

General description of the NEMO 3 detector
NEMO 3 is cylindrical in design and is segmented into 20 sectors. The sectors are numbered from 0 to 19. Each sector contains tracking chambers and calorimeters in order to directly detect decay particles emanating from the radioactive source foils.
The source foils are positioned in the centre of each sector and divide the sectors into two parts: the inner part, which is from the source foil to internal wall of each sector; and the outer part, which is from the source foil to external wall of each sector. The whole detector is surrounded by a solenoid providing a magnetic field of 25 Gauss, external neutron shielding and an anti-radon tent.
The NEMO 3 detector can be defined in both right-handed Cartesian and cylindrical coordinate systems. In the Cartesian system, the x axis starts from the centre of the detector and is along the edge of sector 0. The y axis starts from the centre of the detector and is along the edge of sector 5. The z axis starts from the centre of the detector and points vertically upward. By performing notational conversion, the radius (R = x 2 + y 2 ), polar angle (φ = arccos( x R )) and z of the cylindrical coordinates are formed.

The NEMO 3 sources
Unlike all other currently running double beta decay experiments where calorimeters serve as both double beta decay source and detector, the sources in NEMO 3 are independent from the detector. Inside NEMO 3 there are seven double beta decay isotopes, thus permitting study and comparison of results from different isotopes.
Each sector contains seven foils mounted vertically at a radius of 155 cm from the centre within a metallic support frame. This structure allows multiple isotopes to be mounted within one sector. Each source foil has a height of 2.5 m, a width of 65 mm and an area density of 30-60 mg/cm 2 .
The choice of source nuclei for NEMO 3 was based on several factors: the double beta decay transition energy, Q ββ , the phase space factors, G 0ν and G 2ν , corresponding to neutrinoless decay and two-neutrino decay; the nuclear matrix elements (M 0ν and M 2ν ); the background in the energy region around the Q ββ value; and the natural

The tracking detector
The tracker consists of 6180 octagonal drift cells operating in Geiger mode [56].

Drift cell operation in NEMO 3
The tracking system is immersed in a gas mixture of helium (95%), argon (1%), ethyl alcohol (4%) and water (1500 ppm). The Geiger mode operating voltage of the cells Helium is a low atomic number gas, thus it minimises the energy loss by a charged particle passing through the gas. Ethyl alcohol plays the role of the quencher, which neutralises helium ions and limits the creation of the UV photons and therefore avalanches. A small amount of argon (1%) and water (1500 ppm) were added later during the commissioning of the detector in order to increase the plasma propagation efficiency and to reduce noise and improve the cell stability [57].

The calorimeter
The calorimeter in NEMO 3 measures the energy and the time of flight (TOF) of particles. It consists of 1940 large scintillator blocks coupled to 3" and 5" photomul-tiplier tubes (PMTs) via light guides. The scintillator blocks are produced from low radioactive polystyrene which also provides minimum back scattering. The blocks cover the inner and outer cylindrical walls that surround the tracking volume and the top and bottom of each sector. In order to minimise energy loss, the scintillator blocks are mounted inside the helium-alcohol gas mixture of the tracking volume.

The photomultiplier tubes
The photomultiplier tubes (PMTs) convert optical signals from scintillators to measurable electrical signals. They achieve this via the photoelectric effect and avalanche multiplication. There are two sizes of PMTs used in the calorimeter, 3" and 5". The

The Trigger and readout
The trigger ensures that only events of interest are read out. As NEMO 3 is a low count rate experiment a simple hardware trigger is used to achieve this. The trigger is connected to the calorimeter electronics. When it receives a signal from a PMT which has passed a discriminator threshold of 48 mV, it generates the STOP-PMT signal after 20 ns. This signal stops the charge integration and time measurements performed by an ADC and a TDC, respectively.
In the next stage, the tracking readout system is programmed to search for activation of drift cells in each half-sector (inner or outer). If enough cells are activated (3 out of 9 layers), the STOP-A signal is generated and sent to the drift cell acquisition boards with a programmable delay set at 6.14 µs after the STOP-PMT. The STOP-A signal stops the TDC measurement of the drift cell anodes (TDC A ) and the anode drift times are calculated.
Finally, the STOP-α signal is sent to the drift cell acquisition boards with a fixed delay of 710 µs after the STOP-PMT signal. This signal is to stop the TDC α , which is a TDC independent of TDC A , and is used for calculation of the drift time for delayed hits. This is designed to detect alpha (α) particles from daughter isotopes of radon and have a half-life of 164 µs. The value of 710 µs is more than four times this half-life to have high detection efficiency for α-particles.

Magnetic coil and shielding
A solenoid surrounding the entire detector produces a 25 Gauss magnetic field vertically through the detector. The coil is cylindrical and is 5220 mm in diameter and 2713 mm in height. This causes charged particle tracks to bend as they pass through the detector allowing the identification of electron and positron tracks.

The anti-radon facility
After running the detector for approximately one year, it was discovered that the radon level inside the detector was too high (with a total activity of around 0.7 Bq) and adversely affecting the experiment's sensitivity. This was caused by radon diffusion through the glued joints between sectors. In order to reduce the radon contamination inside the detector an anti-radon facility was installed in September and after (Phase 2) the installation of the anti-radon facility [58].
2004. It consists of an airtight tent and a radon trapping facility.
The detector is fully enclosed in an airtight tent that is made of two layers of polyethylene. It allows only radon-free air to pass through from the radon-trapping facility into the detector and isolates it from laboratory air. The radon trapping system uses activated charcoal that has been treated with oxygen to open up numerous pores inside the material. When air passes through the charcoal, radon is trapped in these pores. The trapping time is greater than the radon decay time.
Therefore radon decays before reaching the tent.
As a result of this system, the radon activity inside the detector has been reduced by a factor of six. Figure 4.8 shows the level of radon inside NEMO 3 before and after the installation of the anti-radon system.

Calibration of the calorimeters
In order to convert the charge output of the PMTs to energy, a calibration procedure is performed. In NEMO 3 there are dedicated calibration runs which take place once per month and last 24 hours. During this time radioactive sources with well known energy spectra are introduced near the double beta decay sources. As NEMO 3 is mainly focused on electron energy measurements, the selected radioactive sources emit electrons. The 207 Bi source decays by a conversion process to two electrons [59] (see Section 6.1.1) with energies of 482 keV and 976 keV, and is suitable for energy calibration up to 1.5 MeV. In order to calibrate the energy up to 3 MeV and higher, 90 Y, which is a pure beta emitter with transition energy of 2.283 MeV, is used.
The timing responses of the PMTs are not identical. The time calibration is performed to make the timing response uniform for all the PMTs in the calorimeter. They are used to check the stability of the laser light intensity which is received by the PMTs in the calorimeter. The laser energy calibration is not applied to the data used in this thesis, as the laser data analysis was not ready for use with the analysis presented. The effect of not using this correction has been studied and is included as systematic uncertainty (Sections 7.3).

The laser time correction
During the analysis of NEMO 3 data in September 2006, it was discovered that the timing measurements of the PMTs were not correct. The time measured by the TDC counter of a given PMT was expected to remain constant for different laser runs; however, a discontinuity was observed in the distribution of the TDC time ( Figure 4.9). This effect was seen in the PMTs of all the 20 sectors. To solve the timing problem a correction was introduced known as laser time correction (LTC) [60]. The measured TDC for each PMT is compared to the reference run as follows: where tdc i j is the time measured by the PMT i in the laser run j, and 1404 is the reference laser run. In this run the maximum number of PMTs were active and they were known to be relatively stable and the timing measurements performed by the TDCs were known to be correct. The value of LTC is added to the TDC values of the PMTs and calculated in each laser run to correct the deviation in timing.   selection that is used in all analysis channels. The statistical methods used to compare simulated events to data and set limits on several 0νββ processes are described in Section 5.7 and Section 5.8, respectively.

Reconstruction of particle tracks
The raw hits in the tracker that are written to data storage tapes are processed by reconstruction software. According to the tracker anode and cathode time values, the triggered drift cell hits are classified as "in time hits", "delayed hits" or as cells fired by neighbouring cells ("re-fired cells" and "noisy cells"). "In time hits" are usually due to the electrons or positrons and "delayed hits" are caused by α particles. Hits are only reconstructed as tracks if they are "in time hits" or "delayed hits". For "in time hits" the anode time values must not be more than a maximum value (currently 1.74 µs found from Geiger calibrations). For "delayed hits", the time measured by the anode is up to 710 µs. The anode drift time and the cathode times provide the transverse and longitudinal position of a hit in a drift cell, respectively.
Tracks are reconstructed if there are "in time hits" in at least three of the nine drift cell layers, with at least two of the hits occurring in neighbouring layers. The pattern recognition is carried out using a cellular automation algorithm [62]. This algorithm defines segments as the lines which connect two hits in neighbouring layers. In order to take into account all inefficiencies, the segments may connect two hits which are not in neighbouring layers if there is no hit in between. The angle between the two segments with a common hit can not be more than 40 degrees. To connect two cells with a segment there are four possibilities, as each cell has two edges, right and left.
The ambiguity is resolved by finding the longest and smoothest pattern of combined segments.
Once the pattern is found, an iterative fit is performed assuming the track follows a helix (due to the magnetic field). The curvature of the track indicates if the particle is an electron or positron. The reconstructed track is extrapolated back to the source foil and projected onto an associated scintillator surface. The coordinates where the track crosses the foil and the scintillator are calculated. Assuming that the tracks originate from the foil, the former gives the vertex of the event. For "delayed hits", the time which is measured by TDC α is used to reconstruct the track in the (x, y) plane.

The vertex resolution
The resolution of the reconstructed vertex of the two electrons in the transverse plane depends on their energy. By using two electrons from 207 Bi source foils (Section 4.8), the vertex resolution is found to be 3 mm at 0.5 MeV and 2 mm at 1 MeV in the

Particle identification
Particle identification is possible by combining the information provided by the tracker and the calorimeter. NEMO 3 is able to identify electrons, positrons, photons and alpha particles.
Due to the magnetic field charged particle trajectories are curved. In NEMO 3 the curvature sign of a trajectory is found with respect to the surface of the source foil. Thus, an electron (positron) originating from the foil is identified as a track of negative (positive) curvature, which is associated with only one scintillator hit. A photon (γ) is identified as a scintillator hit which is not associated with a track. As the energy losses between the scintillator blocks are not known, events are rejected if the photons or electrons are detected by two scintillators. The rate of fake scintillator hits is negligible.
An alpha (α) particle can be identified only if it is accompanied by at least one electron track. Alpha particles are mainly produced by 214 Po decay which is in the radon ( 222 Rn) decay chain. They are defined by drift cell hits delayed by more than 70 µs. As the range of α-particles from 214 Bi decay in the helium gas of the tracking chamber is 25-40 cm [61], the distance between the reconstructed vertex of the electron and the delayed drift cell(s) corresponding to an α-particle is required to be |∆z| < 30 cm in the vertical direction and |∆r| = (∆x) 2 + (∆y) 2 < 25 cm in the x − y plane.

Event simulation in NEMO 3
Monte Carlo (MC) programs are used as event generators that simulate initial particles from isotope decays. The generated events are passed through simulations of the detector response so that the output can be compared to data. Resolution and detector acceptance effects are taken into consideration in the simulation. MC simulations are necessary for analysing experimental data. The MC events are used to measure detector acceptances and to compare the generated physics signal with that seen in the detector. Selection criteria can be optimised by studying the signal to background ratio for specific processes.
The NEMO 3 simulation program [64] uses GENBB as the generator. GENBB simulates the initial kinematics of the particles for two-neutrino double beta decay,   • There was a major problem with the electronics of the detector during the run.
• The run has been taken less than 24 hours after a general electronics shut down, as the PMTs of the calorimeter need time to stabilise.
• They are taken less than 24 hours after 207 Bi calibration runs. During such calibrations PMTs are known to have a high counting rate and require time to stabilise [66].
• No laser run has been performed on the day that data were taken. The laser runs are vital as they give information about the behaviour of the PMTs, and also give the value of LTC (see Section 4.8).
• No appropriate energy or time calibration of the PMTs is available.
After removing these data, the effective data taking time is calculated to be 924.7 days. The data are divided into two sets: Phase 1 and Phase 2. Phase 1 is the data taken before installation of the anti-radon facility in September 2004 and Phase 2 is the data taken after the installation. The Phase 1 and Phase 2 data are combined throughout this thesis unless stated otherwise.

Definition of the 150 Nd source foil boundaries
To define the foil position in the transverse plane, the polar angle, φ, is replaced by a sector number. The sector number is related to the polar angle by: In the construction design of 150 Nd, this value is between 5.74 and 5.87. However, by studying the distribution of event vertex positions, it is observed that data are shifted with respect to the defined MC geometry. Figure 5.1a shows the distribution of the vertex position for two electron events coming from the foil (higher sector number).
The discrepancy between data points and MC simulation can be observed on the right side of the 150 Nd foil. From these plots no conclusion can be drawn for the left side of the foil, since two-electron events from 100 Mo overlap with events from 150 Nd.
To check if the same effect is also observable on the left side, the electron-photon decay channel was analysed. In this channel the event rate for 100 Mo is lower than 150 Nd as this isotope has less contamination than 150 Nd.

Time of flight selection criteria
It is possible that a particle from a source outside the foil deposits energy in one scintillator block, enters the tracking volume, interacts with the source foil, scatters off it and is then detected by another scintillator. These types of events, which are known as external events, can mimic two electron events originating from the foil (internal events). In order to reduce the number of these background events, a time of flight analysis is performed [67]. In this analysis, the time differences between the two calorimeter hits are calculated for two hypotheses. In the first hypothesis the two calorimeter hits and tracks are caused by two electrons from the foil (internal hypothesis). The second hypothesis assumes that the hits are from external sources (external hypothesis). The time differences calculated for both scenarios are then compared with the measured time difference of the calorimeter hits.
For each hypothesis, χ 2 th is defined as: where ∆t mes is the measured time difference between the two scintillator hits; ∆t th is the calculated time difference for the internal and external hypothesis; and σ mes and σ th are the uncertainty on the measured and calculated time differences, respectively.
The value of ∆t th is related to the length of the particle trajectories, l 1 and l 2 , and the energies measured by the calorimeter, E 1 and E 2 . In the case of the internal (external) hypothesis for two-electron events, ∆t th is defined as: where the relativistic factor, β i , is related to the energy measured by each calorimeter block (E i ) and the electron rest mass (m 0 ): The theoretical uncertainty, σ th , is found from differentiation: where t i th = l i β i , σ E is the uncertainty on the energy measurement and σ l is the uncertainty on the track length.
The scenario for internal and external electron-photon events is similar except photons do not leave tracks in the tracker. To measure the length of the photon path, the distance between the scintillator hit and the intersection point of the electron track with the foil (the electron-photon vertex) is found. The length of the photon path, l γ , and the time calculated for the photon to traverse this length, t γ th , are found by: where x ve , y ve and z ve are the coordinates of the vertex position of the events from the detector centre, and x scin , y scin and z scin are the coordinates of the scintillator hit by the photon. As it is not known where in the scintillator the photon hits, these coordinates correspond to the centre of the face. The uncertainty on t γ th is related to the maximum dimensions of the scintillator face and the energy measured by the scintillator. For an internal event, ∆t mes − ∆t int th is expected to be zero ns with some uncertainty due to the time resolution, whereas ∆t mes − ∆t ext th has a value of the order -5 ns. The opposite applies for external events. For both hypotheses, the probabilities, P (χ 2 int ) and P (χ 2 ext ), are calculated. The P (χ 2 ) is defined as: Figure 5.4 shows the P (χ 2 int ) and P (χ 2 ext ) for simulated internal two electron events (2νββ decay of 150 Nd) and simulated crossing electrons events a) However, in the situation where more than one MC components should be estimated, a binned maximum likelihood fit [69] is used. In this technique, to fit m number of MC sources to data, the shapes of the distributions are taken into account. The predicted (normalised) number of MC events in bin i is given by 11) where N D is the total number of events in the data sample, N j is the total number in the MC sample for source j, P j is the proportion of source j in data, and A ij is the expected number of MC simulated events from source j in bin i (which includes the unknown uncertainty in the bin).
The probabilities of observing a particular number of data events, d i , and a particular number of simulated events, a ji , follow Poisson statistics: The total logarithm of likelihoods is then defined by: 14) The proportion of each MC source, P j , is estimated by maximising Equation 5.14 using computing packages [70,71].

Limit setting
The search for new physics in a double beta decay experiment involves dealing with small numbers of expected signal events (in the case of 0νββ) and much larger number of background events than expected signal events (in the case of 0νββ with the emission of Majoron(s)). The exclusion of a signal at a particular confidence level (CL) can be significantly improved relative to a simple counting experiment by using the binned distribution of data, signal and background. In this way each bin is treated as an independent search channel.
A typical new physics search analysis is described by a final variable which is designed to distinguish between two possible hypotheses: the production and detection of the new physics along with the background (S+B hypothesis), or the presence of only the background (B-only hypothesis). The consistency between data and the signal and background models in this final variable is used to set a limit on the maximum number of signal events consistent with the data. The likelihood ratio test statistics [72,73] is an optimal choice for searches with small statistics [74], The limits in this thesis are calculated by using a likelihood-fitter [75] developed at the DØ experiment using a log-likelihood ratio (LLR) test statistic. A description of this method is given in [76]. The LLR is defined as [72]: where g is a random number taken from a Gaussian distribution with a mean of zero and width of one and σ j is the j th fractional uncertainty on the rate of the p 0 .
where P S+B (χ > χ d ) is the probability for the S+B hypothesis to produce an outcome which is less signal-like than that observed in the data and is defined by integrating the expected LLR distribution for S+B hypothesis. Similarly, the calculation of the confidence limit for the background only hypothesis is given by the probability for the B hypothesis to produce an outcome which is less signal-like than that observed in the data and is found by integrating the expected LLR distribution for B-only hypothesis, Downward fluctuations of the background lead to inconsistencies between expected background and observed data and create inaccurate exclusion limits using CL S+B [74].
To deal with this problem, CL S+B is divided by CL B and the value of the signal confidence level (CL s ) is defined as [72]:  Figure 5.6b, it is 0νββ with the emission of a Majoron particle.
In order to calculate the observed limit on the signal rate, this rate is given a new value and the expected LLR distributions are re-generated from another series of pseudo-experiments and the observed LLR is recalculated. This leads to a new value of CL s . This process is repeated through a range of signal rates until CL s reaches the desired confidence level, which in this thesis, by convention, is CL s = 0.1 (1 − CL s = 0.9) or 90% confidence level.

Definition of observed and expected limit
There are two distinct limits, the median expected and observed limit. The values of observed LLR (LLR obs ) and median expected LLR (LLR med ) are found by setting Equation 5.16, respectively. The value of LLR obs relative to LLR med indicates whether data appears to be more background-like or not. The expected limit on the signal rate gives the experiment sensitivity without any bias from actual observation. Thus, different experiments searching for a particular process can be compared using the expected limit results.

The profile likelihood ratio
Systematic uncertainties are often estimated rather than measured and frequently correspond to an upper bound on the true value. These systematic uncertainties widen the expected LLR distributions and decrease the analysis sensitivity for setting limits. This effect of systematics can be reduced by comparing the predicted shape of the final variable to that observed in data, and finding the best fit for systematic uncertainties to data observation [77]: where p(H) ′ i is the systematically varied prediction in bin i for the two hypotheses: (1 + S(H) k ). optimal choice of the test statistic in this case is the profile likelihood ratio [75,78].
This test statistic is the ratio of the χ 2 minimised for each of the two hypothesis. The LLR is defined as: To find the LLR distributions for the S+B and B-only hypotheses, n i in Equation 5.21 is replaced with the MC pseudo-data, and to find LLR obs and LLR med , n i is replaced with the number of observed data and expected background events.
The p(H) ′ i term in Equation 5.21 results in narrower LLR distributions than the ones resulting from Equation 5.16 and therefore increases the analysis sensitivity for setting limits. The Equations 5.18 to 5.20 are used to set limits on the number of observed and expected events at 1 − CL s = 0.9 by varying the signal rate as before.     • A β decay accompanied by an electron from an electron conversion process.

Floating the background normalisation
The latter happens when a photon from an excited daughter nucleus interacts with an electron in one of the inner electron shells, causing the electron to be ejected from the atom.
• Møller scattering of a β-decay electron in the foil which leads to emission of two electrons.
• A β-decay accompanied by a photon. In this case a second electron is produced via a Compton scattering.
• Two-electron conversion processes due to the de-excitation of the daughter  The other elements given in  Background contaminants often decay in multiple channels (and not only the twoelectron channel). It is vital to measure the background activities in channels which are independent of the 2νββ signal. The activity of each background is determined using the decay channel with the highest branching ratio. The activity (A) of each contaminant (in Becquerel) is defined by the number of decays per second, where N data is the number of data events found for the decay channel, N bgr is the number of background events to a specific contaminant, t is the total data taking period in seconds and ǫ is the efficiency of the event selection criteria, which is found by the ratio of simulated events that pass the selections to the total number of generated events.

Electron-photon channel
The isotopes 207 Bi, 152 Eu and 154 Eu decay predominantly to an electron and a photon. 207 Bi can undergo electron capture and decay to the excited state of 207 Pb.
In this case the photon is produced by the de-excitation of the excited 207 Pb and the electron is produced by the internal conversion process [59].  To select electron-photon (eγ) events originating from the foil the following event selection criteria are applied: • Only one electron track associated to a scintillator hit with deposited energy greater than 0.2 MeV.
• The vertex of the eγ event, which is found by extrapolating the electron track to the foil, is reconstructed inside the 150 Nd foil (defined in Section 5.5).
• The track has at least one hit in one of the first two layers of drift cells.
• Only one scintillator block is found which is not associated with a track, with energy greater than 0.2 MeV.
• The internal TOF hypothesis is greater than 4%, and the external TOF hypothesis is less than 1%.
• The energy sum of all other scintillator deposits is less than 0.15 MeV. This requirement rejects events with more than one photon.
• The TOF calculation is less precise for particles with short trajectories. Thus  found to be ten times higher in these hot-spots and therefore this leads to the con-  After applying the photon energy requirement, the only background to 208 Tl which remains in the data sample is the radon from the tracker (described in Section 6.2.1).
The average activity of this background, which was measured by studying the electron-alpha channel [58], is 0.45 ± 0.07 mBq.  normalised to activities found from other decay channels. These plots demonstrate that background MC fits data well in the internal eγ channel.

Electron plus two photon channel
The isotope 208 Tl can undergo a β decay to an excited state of 207 Pb which deexcites to the ground state via two-photon emission. The first photon has an energy of 2.6 MeV and is accompanied by a second photon with an energy of 0.58 MeV or 0.86 MeV. Figure 6.13 shows the decay scheme of 208 Tl. The main internal background to 208 Tl in this channel is 214 Bi which can undergo a β decay and form an excited state of 208 Po which de-excites either by emission of two photons with energies of 0.61 MeV and 1.12 MeV or a photon with energy 1.76 MeV (a decay scheme is shown in Figure 6.14).
To select one electron plus two photons originating from the foil, the following event selection criteria are applied: • Only one electron track associated to a scintillator hit with deposited energy greater than 0.2 MeV.
• The vertex of the eγγ event, which is found by extrapolating the electron track to the foil, is reconstructed inside the 150 Nd foil (defined in Section 5.5).
• The track has a hit in one of the first two layers of the tracker. • The internal TOF hypothesis for each eγ pair is more than 4%, and the external TOF hypothesis is less than 1%.
• The energy sum of all other photon clusters is less than 0.15 MeV.

Single-electron decay channel
The isotopes 234m Pa and 40 K predominantly undergo β decay. Thus the activities of these internal contaminants are measured by studying the single-electron (1e) channel. The following requirements are applied to select 1e events: • Only one electron is found in the event with energy greater than 0.5 MeV.
This energy requirement eliminates a large amount of low energy events which are not relevant for this analysis.
• The electron track must originate from the 150 Nd foil.
• The length of the track is greater than 50 cm.
• The track has at least a hit in one of the first two layers of the tracker. 1.815 < φ < 1.827 and 104 < z < 110 cm, (6.22) which is highly contaminated with 234m Pa (Figure 6.20) and has been subsequently removed from the analysis. Figure 6.21 shows a β decay event display. 210 Bi from the radon decay chain can settle on the tracker wires and undergo a β decay which can mimic 1e events originating from the foil. The average activity of 210 Bi on the surface of the wires is 12.61 mBq [58] (more information about this background is given in Section 6.2.1). Other internal backgrounds are normalised by the activities found in Section 6.1.1. The activities of 40 K and 234m Pa are found by fitting the data simultaneously with the MC contributions of these two contaminants.   1.805 1.81 1.815 1.82 1.825 1.83 1.835 1.84    A( 40 K) = 10.8 ± 0.1 (stat) ± 0.6 (syst) mBq (6.23) A ′ = 213 ± 2 (stat) ± 12 (syst) mBq/kg, (6.24) A( 234m Pa) = 2.42 ± 0.04 (stat) ± 0.14 (syst) mBq (6.25) A ′ = 47.7 ± 0.8 (stat) ± 2.9 (syst) mBq/kg. (6.26) The systematic uncertainties on 40 K and 234m Pa activities are found by varying the energy selections between 0.5 MeV and 1.5 MeV and refitting 234 Pa and 40 K MC distributions to data minus other backgrounds after each energy selection. The systematic uncertainties are found to be 5% for 40 K and 6% for 234m Pa.

The external background
The external background in the NEMO 3 experiment is caused by electrons and photons generated outside the source foil. These particles can interact with the foil and mimic two electron tracks. An external photon's interaction with the foil can produce an electron-positron pair, which can be misidentified as two electrons. A photon can also undergo Compton scattering and produce an electron and another photon. The second electron can be produced by either Møller scattering of the electron or Compton scattering of the second photon. Figure 6.23 shows these three main ways that an external photon can mimic two -electron events originating from the foil.
The second source of external background are crossing electrons which are produced by Compton scattering of the external photons within the detector components.
These electrons can be scattered by the foil and thus detected by two different scintillators. As described in Section 5.6, these events can be reduced by the TOF requirements.
The external background is divided into two main categories based on the origin of Compton followed by Møller scattering (the diagram on the right).
the events: the radon background inside the tracker and the background originating from outside the tracker (i.e. the calorimetry, the shielding and the air surrounding the detector).

The radon background
Radon ( 222 Rn) is out-gassed into the air from the surrounding rocks and can enter the detector either through leaks between the sectors, or through gas piping joins.
Bismuth-214 ( 214 Bi) is one of the descendants of 222 Rn which can settle on the surface of the tracker's wires. The 214 Bi undergoes a β decay to 214 Po. The latter has a halflife of 161 µs and decays to 212 Pb via emission of an α particle. Thus the activity of 214 Bi on the wires was measured by studying electron-alpha (eα) events [58].
Another source of background which is part of the radon decay chain originating from the surface of the wires is 210 Bi. This isotope is a β emitter with Q value of 1.1 MeV. Therefore few 210 Bi events can mimic 2νββ decay. However, since it contributes to the internal single-electron decay channel, it is important to estimate its activity. The activity of 210 Bi was measured by studying single-electron events originating from the surface of the wires [58].   The HPGe measurements [50] show that the glass of the PMTs are contaminated with 214 Bi, 40 K, 208 Tl and 228 Ac. A study of the external eγ and crossing electron events [58] reveals that the iron shield and iron in the sector walls are also contaminated with these isotopes. It is also important to include 60 Co and 234m Pa to the external background model [58] in order to reproduce the crossing electron data.
The magnetic shield of the PMTs and the wall of the sectors are contaminated with these isotopes. In addition, before the installation of the anti-radon facility, a fraction of external 214 Bi and 208 Tl background originated from the air surrounding the detector.

Validation of the external background model
The results presented in Section 6.  Table 6.4: Components of the external background model [58].
for the 150 Nd foil, only particles interacting with this foil are selected. Two decay channels are used to check the external background model: the crossing electron channel and the external eγ channel. In both channels the simulated external backgrounds are normalised to the activities given in Table 6.4. The distribution of the sum of external background is then compared with data.
The one crossing electron events create two tracks in the tracking chamber and two scintillator hits in the calorimeter. The external TOF hypothesis for these events must be more than 4% and the internal TOF hypothesis must be less than 1%.   To select external eγ events, the same selections are applied as described in Section 6.1.1 except that the external TOF hypothesis for an eγ event has to be greater than 4% and the internal TOF hypothesis has to be less than 1%.

Summary and discussion
The eγ, eγγ, 1e and eα [58] channels were studied in order to measure the internal background activities with NEMO 3 data. Table 6.5 summarises the internal backgrounds, the channels studied and the activities measured. The activities of 207 Bi and 40 K measured using NEMO 3 data are inconsistent with the HPGe detector result. The 207 Bi activity is found to be 1.8 times more than HPGe measurement.
As is observed in Figure 6.5, the foil has regions with higher contamination of 207 Bi than others and it is possible that HPGe measured a sample of 150 Nd foil with lower contamination [83]. The activity of 40 K is four times higher than the HPGe measurement. The HPGe detector is a photon detector, whereas 40 K predominantly decays to an electron. The difference can therefore be due to the HPGe detector's inability to detect electrons.
In this thesis a 17% uncertainty on the external background model is used from the validation of the external background model with the eγ channel. As the number of events from external background sources that mimic double beta decay is expected to be low, due to the small surface area of 150 Nd foil (see Table 7.4), this uncertainty will not contribute significantly to the uncertainty on the measurement of 2νββ decay half-life.

Half-life definition
The number of expected events, derived from the radioactive decay law and the event selection efficiency, ǫ, of the considered channel, can be expressed as ) ), (7.1) where N at is the number of atoms in a sample. of T 1/2 for 2νββ decay compared to t, Equation 7.1 can be rewritten as where N exp is the number of expected 2νββ events found from data and N tot bgr is the total number of background events remaining in the data after applying the event selection criteria.

Two-electron event selection
The following selections are applied to the data in order to reduce background to double beta decay: • Two negatively charged particles are required. The tracks must be associated with two scintillator hits.
• Both tracks are required to originate from the 150 Nd foil. • To remove the events from hot-spot regions, events are rejected if they originate from 207 Bi and 234m Pa contaminated regions defined in Equations 6.2 and 6.22.
• The energy of each electron is required to be greater than 0.2 MeV.
• Both electrons originate from a common vertex and have: where ∆R and ∆z are the distances between the two tracks' intersection points with the foil in the horizontal plane, x−y, and in the z coordinate. Figures 7.1a and Figures 7.1b show the distribution of these variables before applying the vertex cuts and after applying all other cuts described above.
• In order to reduce the external background, the internal TOF hypothesis for two-electron events is required to be greater than 4%, and the external TOF hypothesis to be less than 1%. Figure 7.2 shows the distributions of the internal and external TOF hypotheses before applying the TOF hypothesis requirement and after applying all other selections described above. In these figures the number of external background events is normalised to the number of data events for P (χ 2 int < 0.04) and P (χ 2 ext > 0.01).
• The track length for each electron is greater than 30 cm. This cut value is lower than the one applied to channels involving one electron track only, as the TOF measurement for two track events is more precise [50]. • The tracks pass through one of the first two layers of the tracker.      The energy distribution of electrons with the lower energy, the energy sum of the two electrons and the opening angle between them are shown in Figure 7.5. The normalisation of the 2νββ signal is found by scaling the simulated MC events to the same number of data events minus the radioactive background events. The data are in good agreement with the sum of the background and the 2νββ signal distribution.
The efficiency of the 2νββ event selection is 7.2%. After background subtraction

Systematic uncertainties
The breakdown of the systematic uncertainties on the 2νββ half-life is as follows: • The systematic uncertainty on the background estimate, obtained as the quadratic sum of the uncertainties on the estimates for the internal and external background, is 4%, which translates into an uncertainty on the 2νββ half-life of 1.5%.
• An uncertainty of 3% is added due to the uncertainty on the position of the 150 Nd foil in the detector. This is estimated by comparing the half-life results before and after changing the position of the 150 Nd foil (Section 5.5).
• The TOF cut is changed from 4% to 1% in order to estimate the uncertainty on this cut. This yields 1% uncertainty on the half-life results.
• There is a 5% uncertainty on the efficiency calculation due to the inaccuracy of the GEANT simulation and tracking program. This was found by measuring the activity of the calibration sources [54].
• There is 2% uncertainty in the energy calibration coefficient [54] since the laser energy calibration was not used. The uncertainty on the shape of the 2νββ energy sum distribution is studied by smearing the individual electron energies for 2νββ simulated events by 2%. In this case the smeared energy is defined as where E i is the energy of the i th electron and g is a random number taken from a Gaussian distribution with a mean of zero and width of one. This yields a systematic uncertainty of 1.0% on the 2νββ half-life. Figure 7.6 shows the distribution of energy sum of the two electrons for simulated 2νββ events before and after smearing the electron energies by 2%.

Comparison with previous results
The 2νββ decay of 150 Nd was first observed in 1993 by a TPC experiment constructed by the Irvine group in California [87]. The discovery was confirmed by the ITEP Moscow experiment later that year [88]. Both of these groups devel- The Irvine group's TPC experiment was situated in an underground valve house in a canyon wall at the Hoover Dam in the USA [90]. The TPC was a rectangular polycarbonate box with internal dimension of 88 × 88 × 23 cm 3 . The double beta decay source was in the centre and served as the central drift-field cathode. Anode and cathode wires were set perpendicular to each other and located near the walls of the TPC. A 1200 G magnetic field was applied perpendicular to the source plane.
The helix of the particles provided the measurement of the momentum and thus

Signal event selection
The same two-electron event selections as described in Section 7.2 are applied to simulated events from: • 0νββ decay to the ground state of 150 Sm via the mass mechanism (0 + gs m ν ).
• 0νββ decay to the ground state of 150 Sm via right-handed currents (0 + gs RC).
• 0νββ decay with emission of Majoron(s).  • 0νββ decay to the 2 + 1 excited state (2 + 1 RC). The mass mechanism of this decay is highly suppressed due to the nuclear angular momentum and only the right-handed current is enhanced [91].
• 0νββ decay to the 0 + 1 excited state (0 + 1 m ν ). The right-handed current of this decay is highly suppressed due to the nuclear angular momentum and only the mass mechanism is enhanced [92].
The first three decays have the same event topology as 2νββ. The event topologies of the last two decays are two electrons with emission of a photon (2 + 1 RC) or two photons (0 + 1 m ν ). In this thesis it has not been attempted to reconstruct the additional photon(s). Table 8.1 present the event selection efficiency for each of these modes. To set limits on the half-life of these modes, the whole energy sum distributions of the two electrons are investigated and no energy cuts are applied. Figure 8.1 shows the energy sum (E sum ) distribution of different simulated modes, normalised to arbitrary numbers. In these figures the data and the sum of the backgrounds are also shown. it to data minus radioactive background, assuming that there was no new physics in data. Thus, in order to have an unbiased signal search, the 2νββ statistical uncertainty on the 2νββ normalisation, which is found by fitting 2νββ background to data minus radioactive background MC, was "floated" (see Section 5.8.3). This makes the 2νββ background normalisation factor a free parameter.

Systematic considerations
The uncertainty on the energy measurement affects the shape of the energy sum distribution of the signal and is different in each bin. Figure 8.2 shows the energy sum of the two electrons for different simulated neutrinoless double beta decay modes before and after smearing the energy of each electron by 2% (Section 7.3).
To find the shape uncertainty on the energy, the ratio (S-N)/N is plotted, where S is the smeared MC distribution and N is the nominal (unsmeared) MC distribution.
These ratios for different 0νββ modes are shown in Figure 8 histograms (shown in red) are also constructed in order to estimate the negative uncertainty. Figure 8.5 shows the shape uncertainty on the 2νββ background.
The uncertainties that are considered for setting limits on 0νββ decays and 0νββ decays with the emission of Majoron(s) are presented in Table 8 and radioactive backgrounds as considered for the limit setting.

0νββ mass mechanism
In order to search for 0νββ, the E sum distributions of the 0νββ signal, 2νββ and radioactive backgrounds are used as inputs to the limit calculation. The search is performed using the profile likelihood ratio test statistic described in Section 5.8.2 (Equation 5.23). Table 8.3 gives the observed log likelihood ratio (LLR obs ), the median expected log likelihood ratio (LLR med ) and the log likelihood ratio for the background with one and two standard deviations (LLR med ±1σ and LLR med ±2σ). The results in this table show that LLR obs = 4.6 is in agreement with LLR med within about one standard deviation and therefore there is no sign of a 0νββ signal. The CL B value is 0.14 and therefore also consistent with a background only observation.
To set limit on the observed and median expected number of events at 90% CL, the 0νββ MC is re-scaled until CL S+B = 0.014 (CL S = 0.1).
The observed upper limit on the number of events from the 0νββ mass mechanism is found to be 2.7 at 90% CL. With the detector efficiency of 19% and using Equation 7.2, the lower bound on the 0νββ half-life is found to be: T 0ν 1/2 > 1.8 × 10 22 y (90% CL). (8.1) This is consistent with the median expected limit at 90% CL of T 0ν 1/2 > 1.33 × 10 22 y (90% CL). (8.2) This limit on the half-life is converted into a limit on the effective Majorana neutrino mass, m ν , using an NME of 3.14 − 4.04 [93] and G 0ν of 2.69 × 10 −13 y −1 [39]. The experimental lower limit on the half-life of 150 Nd translates into an effective neutrino mass limit of m ν < 1.5 − 2.5 eV. (8.3) The uncertainty on the m ν limit is due to the uncertainty in NME calculations.
Taking into account the nuclear deformation will modify this conclusion. The suppression of the NME for 150 Nd is estimated to be a factor 2.7 [94]. This increases the upper limit to Other backgrounds are found to be negligible in this energy region. A 0νββ distribution with half-life of 1.8 × 10 22 y, corresponding to 2.7 events expected for the observed 90% CL, is shown in blue.   To calculate this limit, the same NME as for the 0νββ mass mechanism and the phase space factor, G ββ = 6.40 × 10 −15 y −1 [39] is used. Considering 150 Nd deformation changes the limit to:  The upper limit on the coupling of neutrino with right-handed currents and Majoron mode two and three are not presented in this thesis as the NME calculations of these modes are out of date [95].

Summary and discussion
The observed limit on the half-life of neutrinoless double beta decay of 150 Nd was found to be T 0ν 1/2 > 1.8 × 10 22 y (90%CL).
This half-life limit has improved the previous limit on this isotope by a factor of ten.
The best previous limit results were measured by the Institute for Nuclear Research of the USSR in 1986 [96] for 50.5 g of enriched 150 Nd. The experiment had a simple set-up using a central foil surrounded by four scintillation counters providing calorimetry and event reconstruction. This experiment gave a lower bound on the 0νββ half-life of, T 0ν 1/2 > 1.7×10 21 y (95% CL). Despite this significant improvement on the half-life limit, the upper bound on effective neutrino mass has a large uncertainty due to the uncertainties on the NME calculations compared to other 0νββ isotopes. Therefore, further progress in the calculation of the NME for 0νββ decay of 150 Nd is required to improve the m ν limit found from this isotope. The NME of other neutrinoless double beta decay modes are also required to be recalculated with improved calculating tools. This is especially true for right-handed current 0νββ searches.

Chapter 9 Conclusion
In this thesis a precise measurement of the 2νββ half-life of 150 Nd and searches for different modes of neutrinoless double beta decay using NEMO 3 data have been presented. These involve measurements of the activities of backgrounds originating from the internal sources and validation of the external background model. The results given in this thesis have been submitted for publication [97].
The internal background activities are measured by studying eγ, eγγ and 1e channels. Hot-spot regions due to 207 Bi and 234m Pa contamination are observed in the eγ and 1e channels, respectively. These regions are subsequently removed from the analysis. The 207 Bi and 40 K activities found in this thesis have invalidated the HPGe results for these isotopes. The 234m Pa, 207 Bi, 40 K, 228 Ac isotopes are found to be the major internal background sources to 2νββ and their activities are measured with uncertainties less than 6%. The 208 Tl isotope is the major background to 0νββ and its activity is measured with an uncertainty of 10%.
The number of two-electron background events originating from the neighbouring double beta decay sources and their contaminants are found to be 22% of the total background events. The external background model used in NEMO 3 was validated by selecting external particles that are interacting with 150 Nd foil. A 17% uncertainty on the model is used from the validation with the external eγ channel. The total external background contribution to 2νββ is found to be 7% of the total background events. After applying the two-electron event selection and subtracting the background events, the half-life of 2νββ is measured to be: T 2ν 1/2 = (9.11 +0.25 −0.22 (stat.) ± 0.62 (syst.)) × 10 18 y.
The dominant systematic uncertainties are due to the uncertainties on the tracking efficiency (5%), 150 Nd foil position (3%) and number of radioactive background events (1.5%).
A search is performed for different 0νββ modes by studying the energy sum distributions of the two electrons in the final state. Limits on 0νββ signals are set using the profile-likelihood ratio technique and by floating the 2νββ background. This limit setting technique has been used for the first time in the NEMO experiment.
The lower limits on the neutrinoless double beta decay half-lives of 150 Nd for the mass mechanism (0νββ) and the Majoron mode one decay (0νββχ) are, The NME calculation of 0νββ must be improved in order to reduce the uncertainty on these limits. Limits are also set for the first time on half-lives of right-handed currents, excited state 0νββ decays and several other models leading to emissions of Majoron(s).