Advances in High Energy Physics
Volume 2015 (2015), Article ID 632131, 11 pages
http://dx.doi.org/10.1155/2015/632131

Research Article

Convoluted -Signals on ¹¹⁴Cd Isotope from Astrophysical and Laboratory Neutrino Sources

Vaitsa Tsakstara1,2

¹Technological Education Institute of Western Macedonia, 51100 Grevena, Greece
²Division of Theoretical Physics, University of Ioannina, 45110 Ioannina, Greece

Received 9 July 2014; Revised 28 February 2015; Accepted 2 March 2015

Academic Editor: Theocharis Kosmas

Copyright © 2015 Vaitsa Tsakstara. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

Abstract.

At first, we evaluate scattering cross sections of low, and intermediate-energy neutrinos scattered off the 114 Cd isotope, the most abundant Cd isotope present also in the COBRA detector (CdTe and CdZnTe materials) which aims to search for double beta decay events and neutrino observations at Gran Sasso laboratory (LNGS). The coherent -nucleus channel addressed here is the dominant reaction channel of the neutral current -nucleus scattering. Our -nucleus cross sections (calculated with a refinement of the quasiparticle random-phase approximation, QRPA) refer to the transitions for -energies  MeV. Subsequently, simulated -signals on 114 Cd isotope are derived. Towards this purpose, the required folded cross section comes out of simulation techniques by employing several low, and intermediate-energy neutrino distributions of the astrophysical -sources, like the solar, supernova, and Earth neutrinos, as well as the laboratory neutrinos, the reactor neutrinos, the pion-muon stopped neutrinos, and the β-beam neutrinos.

1. Introduction

In the past few decades neutrinos attracted a great part of the effort of the authors working in Nuclear, Astroparticle Physics and Cosmology [1–8]. Many neutrino experiments like the KamLAND [9], Borexino [10, 11], SNO+ [12], LENA [13], and other experiments, with high precision detectors aim to measure different types of neutrinos as well as neutrino-nucleus cross sections. Measurements in the beta-beam neutrino experiments [14–16]   or in the expected to be built near spallation neutron sources (ORLAND experiment, Spallation Source) [17–20] may help answer key questions about the neutrino properties, the evolution of distant massive stars, the structure of the interior of our planet [21–23], and the fundamental electroweak interactions.

In recent terrestrial experiments aiming at neutrino studies through -nucleus interactions, the characteristics of the neutrino fluxes emitted by various neutrino sources like the astrophysical -sources (supernova, solar, and Earth neutrinos) or the laboratory -sources (accelerated -decay ions in storage rings [15, 16], pion-muon decay at rest, e.g., at Fermilab [24], reactor neutrinos, etc.) are encoded on the nuclear response of the detector materials. On the theoretical side, the nuclear responses of -detectors to the energy spectra of the observed neutrino flux could be simulated by convoluted (folded) cross sections. The latter are obtained by using original -nucleus cross sections calculations and realistic descriptions for the -beam of the studied neutrino source [25–27], that is, reliable -energy distributions.

In the present paper, we study extensively the nuclear response to low and intermediate-energy neutrinos of the ¹¹⁴Cd isotope focusing on the computation of flux averaged cross sections of this nucleus for various neutrino energy spectra. The isotopes constitute significant materials of the semiconductors CdZnTe and CdTe detectors of the COBRA experiment which is aiming to search for double beta decay events and neutrino observations at Gran Sasso laboratory (LNGS) [4, 5, 12]. Calculation of the induced -signal in such nuclear detectors involves, first, evaluation of the flux of the neutrinos arriving at the detector and, second, folding of the calculated cross section of the neutrino interaction with the nuclear detector [6, 8]. The present work is an extension of our previous studies which address inelastic cross sections of neutrinos scattered off the and isotopes contents of the COBRA detector [6, 8, 28]. We pay special attention to the calculations of the elastic (coherent) neutral current (NC) neutrino-nucleus reaction cross sections for the ¹¹⁴Cd nucleus (the most abundant isotope in the natural Cd). For similar calculations on other Cd isotopes the reader is referred to [29]. As it is well known, in NC -nucleus scattering the transitions represent the dominant reaction channel [6, 30]. In terrestrial experiments the detection of this channel may be achieved through a measurement of the nuclear recoil signal [31–33] which is a rather different signature compared to that of the incoherent channel [1, 3, 6, 8, 34]. Inelastic scattering of neutrinos on nuclei creates different signal and could be studied through the outgoing charged-lepton and extracted nucleons and/or -ray emission. These channels have been studied in detail elsewhere (see, e.g., [6, 8, 30]).

For detectors of supernova neutrinos, all neutrino and antineutrino flavours take part in NC scattering on nuclear detectors (specifically heavy flavour neutrinos can be detected too) [4, 5, 35] while only electron-neutrino () charged-current reactions are important (only a small portion of neutrinos has energy above the mass of the -lepton which constitutes the main detection signal).

The solar neutrinos, which represent a powerful tool to investigate the Sun’s deep interior [36, 37], have provided precious information to understand new physics and the interpretation of neutrino oscillations. Future experiments, like the SNO+ [12], are expected to contribute crucially in order to explore the CNO-cycle neutrino flux and clarify the open issue of the Sun’s core metalicity.

Recently, geoneutrinos which are antineutrinos produced from decays of natural radioactive elements (predominantly ⁴⁰K and nuclides in the ²³⁸U and ²³²Th chains) mainly in the crust and mantle of the Earth have been successfully detected by the liquid-scintillator experiments of the KamLAND [38] and Borexino Collaborations [39], but the geological information contained in these measurements is still limited [40, 41]. Geoneutrinos are direct messengers of the abundance and matter distribution of radioactive elements deep within our planet, information that provides strong constraints on several phenomena occurring inside the Earth [42, 43]. Concerning the energy distribution of geoneutrinos, up to now little is known due to the fact that this is a new research field, but in the near future experiments like LENA, SNO+, and others, having in their objectives to explore geoneutrinos, are expected to provide us with new data. From the currently known information we imply that their energy range is  MeV.

It is expected [28] that the response of the isotopes in the particle-bound excitation region, which coincides with the energy range of geoneutrinos, is rather rich and this motivates our present calculations. The next-generation detectors (LENA, Borexino, SNO) are expected to give useful answers to several questions of geological importance regarding the precise geo- fluxes and abundances of natural radioactive elements (, , and ) in the Earth’s interior [40–43].

One of the main goals of the present work is to focus on the interpretation of various -signals generated in nuclear detectors of terrestrial experiments through the investigation of the nuclear response of detector medium to the corresponding neutrino energy spectra. We emphasize on signals coming from geo-, supernova-, solar-, reactor-, pion-muon stopped-neutrinos by using the following basic theoretical ingredients: (i) the calculated coherent total cross sections of the neutral-current reaction ¹¹⁴Cd computed with a refinement of the quasiparticle random-phase approximation (QRPA), (ii) reliable descriptions of the shapes of neutrino energy distributions provided mostly from numerical simulations of distributions of low energy neutrinos, and (iii) computational tools required for the folding procedure in order to simulate the signal expected to be recorded on detectors as the CdTe or CdZnTe (the detector medium of COBRA experiment) from low-energy sources (geo-, reactor-, and solar-neutrinos) and intermediate-energy neutrinos (supernova and laboratory-neutrinos).

The paper is organized as follows. At first (Section 2) the main formalism is described and our cross section calculations are presented. Then, (Section 3), a description of the main characteristics of the low- and intermediate-energy neutrino sources addressed here is briefly summarized and folded cross sections as well as event rates for neutral current neutrino scattering off the ¹¹⁴Cd isotope are discussed. Finally (Section 4), the main conclusions of the present work are extracted.

2. Brief Description of the Formalism

2.1. Angle Differential Coherent -Nucleus Cross Section

The angle differential cross section of the elastic scattering of a neutrino with energy on a nucleus is [34, 44]

where is the Fermi coupling constant and contains the nuclear dependence of the cross section given by

In (1), represent the scattering angle and denotes the weak charge of the target nucleus:

(the weak mixing angle is [45]).

The ground state elastic nuclear form factors for protons and for neutrons are defined by

and are normalized as . In (4), denote the neutron () and proton () charge density distribution with being the zero-order spherical Bessel function (we neglect a small correction from the single-nucleon form factors). The proton density is often taken from experiment whenever measured charge densities are available [44, 46]. Moreover, some authors assume . From (1) we can easily obtain the expression with respect to the scattering angle as

where and stand for the polar-vector coupling for the weak neutral current for proton and neutron, respectively, given by:

From the latter two equations one can imply that the consideration of in the cross section is important especially for heavier -detection targets. Thus, the coherent cross section depends on the square of the ground state nuclear form factor at momentum transfer which (in the extreme-relativistic limit) is given by

or .

From (2), we see that, since is small, a neutrino scattered elastically on a spin-zero nucleus couples mostly to the neutron distribution, . A measurement of the cross section for this process would, at some level, provide a determination of the neutron form factor . Some authors consider that this would be complementary to parity violating experiments because it would provide additional data, obtained at different energy ranges and with different nuclei that could be used to calibrate nuclear structure calculations [34, 44, 46–48].

2.2. Coherent Differential Cross Section with respect to Nuclear Recoil Energy

From an experimental point of view and particularly for the neutrino facilities near spallation sources [18, 33] the expression of the coherent differential cross section with respect to the nuclear recoil energy is also interesting. This is written as [31–33, 49]

where is the nuclear mass and denotes the ground state elastic form factor. More accurate expressions, including higher order terms with respect to , can be found in [31–33, 50]. It should be noted that the signal on the coherent neutrino-nucleus scattering experiments is significantly different to that of the incoherent scattering where the signal could be an outgoing particle.

2.3. Total Coherent Cross Section

The total coherent cross section is obtained by integrating numerically Equation (5) over the angle ( to ) or (8) over between

to [50]. Some authors [48], however, ignoring the momentum dependence of the nuclear form factors take and in this case the total coherent cross section is approximately written as

Also, because many authors take [48].

3. Results and Discussion

3.1. Original Cross Section Calculations

The neutral-current scattering of low- and intermediate-energy neutrinos, , and antineutrinos, , off the most abundant Cd isotope of the COBRA detectors, that is, the ¹¹⁴Cd isotope (with abundance 28.8%), the main subject of our present work, is represented by the reactions

( and denotes an excited state of ¹¹⁴Cd isotope). We mention that the above reactions (also the CC reactions for ) play significant role in astrophysical environment since they affect the electron fraction of the stellar matter [21–23].

In the first stage of the present work, we evaluate original cross sections for the reactions of (11). These calculations refer to the coherent channel (ground state to ground state transitions) of the target nucleus ¹¹⁴Cd [1, 6, 8, 51–53]. As can be seen from (5) the original cross section identically comes for scattering of neutrinos and antineutrinos . However, as we will see in Section 3, the signal on the nuclear detector could be significantly different due to the different -energy distributions.

In this work, we use theoretical densities obtained from simple mean-field calculations using the successful Woods-Saxon effective interaction plus the monopole (pairing) Bonn C-D interaction.

The ground state of the studied (even-even) ¹¹⁴Cd isotope is computed by solving iteratively the BCS equations. In Table 1, we list the values of the pairing parameters () and the theoretical energy gaps () for protons () and neutrons () determined at the BCS level for the ¹¹⁴Cd isotope. These parameters renormalise the pairing force and adjust the gaps to the empirical ones obtained through the application of the three-point formulae (see [8]). The values of the that adjust the energy gaps in both cases are reliable (see Table 1).

Table 1: Parameters determining the BCS ground state pairing interactions for protons () and neutrons () and the relevant theoretical values of proton () and neutron () energy gaps. The latter reproduce well the corresponding (for ¹¹⁴Cd isotope) empirical energy gaps, , listed also in the table, obtained through the three-point formula [6]. The values of the harmonic oscillator size parameter , used for ¹¹⁴Cd isotope, are also shown.

Isotope
,
Abundance (%)
(fm)

¹¹⁴Cd
48, 66
28.73
2.214
0.9564
0.9753
1.441
1.44108
1.351
1.35093

Afterwards, the proton and neutron nuclear form factors are obtained from the expression

where is the single particle -level occupation probability amplitude for protons or neutrons (the summation runs over the 15 active levels of the model space chosen which consists of the major harmonic oscillator shell with quantum number ) as well as over the fully occupied core -levels having .

In Figure 1, we show the ingredients required for (1) and (5) in order to calculate the differential and integrated coherent cross section for the neutrino reactions of (11). Figure 1(a) shows the theoretical form factors for protons and neutrons obtained with our BCS calculations and Figure 2 shows the momentum dependence of that enters (1) and (5). In this figure, the results obtained by assuming zero momentum transfer (), that is, , are also presented (dash dotted curve). We see that, for energies above about 40 MeV, the two methods give significantly different results; the approximation is good only for small momentum transfer (e.g., solar and low-energy SN-neutrinos). This shows also the sensitivity of the total cross sections of the total cross sections on the nuclear form factor.

(a)

(b)

Figure 1: (a) Form factor (for neutrons and protons) for ¹¹⁴Cd isotope. (b) The ground state elastic nuclear form factor .

(a)

(b)

Figure 2: Coherent total cross section for the neutral current reactions ¹¹⁴Cd,  : (a) as a function of the momentum transfer and (b) as a function of the incoming neutrino energy . The dash dotted curve results by assuming zero momentum transfer; that is, .

Figure 2 illustrates the coherent cross sections of -¹¹⁴Cd scattering as a function of (i) the momentum transfer (Figure 2(a)) and (ii) the incoming neutrino energy (Figure 2(b)). The original cross sections will be used below for evaluations of flux averaged folded cross sections for various neutrino sources. Towards this purpose, the -energy distributions of each source are required. We mention that here we have neglected the threshold energy of the detector (calculations where the threshold energy is considered have been performed in [54]).

In the next subsection, we summarize the main features of the neutrino energy distributions adopted in this work.

3.2. Energy Spectra of Low- and Intermediate-Energy -Sources

The real neutrino sources, astrophysical (solar, supernova, and geoneutrinos) and laboratory (-beam, pion-muon stopped neutrino beams, and reactor neutrinos), with few exceptions such as the neutrino beam emerging from the decay at rest ( = 29.8 MeV), the ⁷Be solar neutrinos ( = 0.862 MeV [10, 11]), and so forth, produce neutrinos that present a spectral distribution, characteristic of the source itself (i.e., on the reactions producing the considered neutrinos), and are defined by

denotes the number of neutrinos of the beam. Thus, for example, the neutrinos originating from pion-muon decay at rest have energy spectra approximately described by the well-known Michel distribution, while the supernova neutrinos are commonly interpreted by using for their energy spectra a two-parameter Fermi-Dirac or power law distributions [6, 8, 28] (see below).

In this section, we summarize briefly the basic features of the currently interesting low-energy astrophysical and laboratory neutrino sources: solar, supernova, geoneutrinos, reactor neutrinos, pion-muon stopped neutrinos, and -beam neutrinos. We focus on their energy distributions which drop in the neutrino energy range of our original cross sections. These neutrino spectra will be used in the folding procedure in the next section, in order to simulate the nuclear detector response of the ¹¹⁴Cd nucleus and calculate event rates.

3.2.1. Geoneutrinos

Geoneutrinos (or Earth neutrinos) are mainly electron antineutrinos () generated upon transmutation of neutron-rich -decay nuclei, accompanied by emission of an electron () and release of decay energy () according to the reaction [40]

In the latter reaction is the mass number and the atomic (proton) number of the initial (parent) nucleus. Part of the decay energy, , is carried away by antineutrinos () while the remainder is available for heating (). KamLAND is the first detector to conduct an investigation on geoneutrinos [40]. As it is well known, cosmochemical analysis expects significant amount of radioactive isotopes contained in the Earth and radiogenic heat generation of which totals up to about 20 TW [41].

The abundant radioactive isotopes that are in the present Earth are classified into three groups: (i) isotopes in the ²³⁸U decay series, (ii) isotopes in ²³²Th decay series, and ⁴⁰K isotope. These isotopes are the geologically important isotopes that heat the Earth’s interior (they finally decay into stable nuclei). Radiogenic heat is produced by decays of isotopes, in which electron-type (anti-) neutrinos are emitted.

Figure 3(a) shows the antineutrino spectra from ⁴⁰K, ²³⁸U series, and ²³²Th series ( year, year, and year, resp.). In this neutrino energy distribution , coming from 82 beta decays in the U series and 70 beta decays in the Th series, are included. Antineutrinos are generated by -decays of all intermediate radioactive isotopes [40–43].

(a)

(b)

Figure 3: (a) Spectra of the U Series, Th Series, and ⁴⁰K geoneutrinos. Neutrinos from ⁴⁰K electron capture are also shown in this figure. (b) Normalized reactor neutrino spectra.

The most recent measurements from KamLAND [38] and Borexino [39] are reaching the precision where they can start to constrain Earth models. However, these detectors are not sensitive to the neutrino direction. The amount of heat-producing elements in the Earth’s mantle is of great interest and hence a detector located away from neutrinos produced in continental crust or on the ocean would be ideal [55]. The next-generation liquid-scintillator neutrino observatory, LENA [13], thanks to its large volume, would be a real breakthrough in geoneutrino detection and geologically significant results could be obtained. LENA could measure the total geoneutrino flux at the level of few percent, by far more precise than other current experiments (e.g., Borexino or KamLAND) could reach. The event and background rates expected for LENA (both in Pyhäsalmi and Fríejus), and projects the precision at which the total geoneutrino flux as well as the U/Th ratio could be measured [13]. The large number of events expected for geoneutrinos will give valuable information on the abundances of Uranium and Thorium and their relative ratio in the Earths crust and mantle.

3.2.2. Reactor Neutrinos

Nuclear reactors have been used as intense sources in many experiments. In the fission of ²³⁵U, ²³⁹Pu, and ²³⁸U, neutron-rich nuclei are produced and antineutrinos are subsequently emitted via -decay [56, 57]. Experiments using reactor neutrinos are particularly suitable for low measurement because the mean energy of the reactor neutrinos is a few MeV (much smaller than that of accelerator neutrinos). The distance from the reactor core in many reactor experiments is ranged from several tens of meters to 1 km while the overall systematic errors including the flux uncertainty and detector uncertainty are constrained to within a few % at these experiments.

Nuclear reactors, as sources of , give fluxes of the order ~10¹³  at distances ~10 m from the reactor core. These antineutrinos have an energy spectrum peaked at very low energies (~0.3 MeV) and extending up to ~10 MeV, characteristic of the decay of the fission products. Figure 3(b) illustrates the reactor neutrino spectra normalized so that the sum over all data-points is equal to one. The fuel composition was adopted to be 62% ²³⁵U, 30% ²³⁹Pu, and 8% ²³⁸U [56, 58].

3.2.3. Solar Neutrinos

Solar neutrinos are neutrinos produced through weak, electromagnetic, and strong nuclear processes in the interior of our Sun. They have energy  MeV and are created either via the well-studied pp-chain reactions or via the CNO-cycle processes [36]. Their energy depends not only on the pertinent nuclear processes, but also on the densities and temperatures in the Sun’s environment. The detection of solar neutrinos by terrestrial experiments provides unique information about the interior of the Sun and constitutes excellent probes for astrophysics, nuclear physics, and particle physics searches.

In Figure 4(a), we show the energy spectra of the important ⁸B and hep neutrino sources predicted by the standard solar model. Each of these spectra has a characteristic shape which is independent of the conditions in the solar interior. The ⁸B spectrum, on the other hand, is more nearly symmetric, with a peak at 6.4 MeV and a somewhat extended tail. The hep spectrum is rather symmetric and peaks at 9.6 MeV. We mention that measurements of the spectrum of neutrinos that reaches us from the sun provided a decisive test of whether the solar neutrino puzzle is due to our lack of understanding of the solar interior or due to new physics [36, 37].

(a)

(b)

Figure 4: (a) Normalized energy spectrum of ⁸B and hep solar neutrinos. (b) Energy spectra of and neutrino beams, generated from the muon-decay at rest.

3.2.4. Pion-Muon Decay at Rest Neutrino Energy Distributions

In the operating pion-muon decay at rest neutrino sources (Fermilab at USA, J-PARC at Japan) and the expected to operate neutrino facilities at the Neutron Spallation Sources (ORNL at USA, and Lund in Sweden), neutrinos and antineutrinos are produced from the decay of muons according to the reaction

The decaying muons result from the decay of slow pions () and hence, and neutrinos have relatively low energies. Their energy spectra are approximately described by normalized distributions of the form [2, 59]

where  MeV is the muon rest mass (see Figure 4(b)). The maximum energy of and in the later equations is [59, 60]. The distribution of these neutrinos is known as Michel energy spectrum. The pion-muon decay at rest neutrino beams is not completely pure as, for example, the -beam neutrinos.

Obviously, the analytic expressions of (16) and (17) are convenient for the required numerical integration in the folding procedure [6, 8, 28, 61–63]. Their energy range and shape roughly resemble those of supernova neutrinos and give us a unique opportunity to study neutrino interactions in this important energy range. This will improve our understanding of SN dynamics and help us to design and calibrate the response of supernova neutrino detectors. We should mention, however, that the pion-muon decay at rest neutrino distributions is closed for high energies while the SN neutrino spectra are open at their high energy tail (see below).

3.2.5. Supernova Neutrinos

According to predictions of recent numerical simulations [64, 65], the creation of the supernova neutrino fluxes is a very complicated process. The shape of SN-neutrino energy distributions is determined by the conditions under which the neutrinos are emitted from the star causing the cooling of the protoneutron star formed at the center of the collapsing star [21, 66–69]. In earlier studies, a thermal spectrum was employed to describe the SN- energy distribution [70]. Recent stellar evolution simulations, however, have shown that several effects modify the spectral shape from a purely thermal one [64].

In stellar modelling, authors use analytic expressions that include various modulation effects by inserting a chemical potential . Such an expression is as the well-known two-parameter Fermi-Dirac (FD) distribution [64]. A similar expression is the analytically simpler two-parameter Power-Law (PL) energy distribution [64, 65, 71]. Both parametrizations FD and PL yield similar distributions characterized by the temperature or the average -energy [8, 28, 72–74]. It is important to note that the flavour-dependent fluxes and spectra emitted by supernovae at any distance from the source can be different from those originally produced, which is mainly due to neutrino oscillations but also due to other phenomena [70].

In Figure 1 of [75, 76] some specific cases of Fermi-Dirac (FD) and Power-law (PL) distributions have been presented for various values of their parameters [64, 65]. Concerning the behaviour of FD and PL distributions throughout their energy range we mention the following features. For the Fermi-Dirac spectral distribution, as the temperature grows both the maximum of the PL distribution shifts to greater neutrino energy and its peak becomes smaller. Also, increasing the degeneracy parameter shifts the spectrum to higher energies [64, 65]. The Power-Law energy distribution is characterized by the pinching parameter and average energy . The average energy reflects the depth of the stars from where the neutrinos are escaping. As grows, the maximum of the distribution shifts to higher neutrino energy [65], while as the width parameter [8] grows (for the same average energy ), both the maximum of the distribution shifts to smaller neutrino energy and its peak becomes smaller. For the purposes of the present work, the values of the pinching parameter needed are =5.1, 3.7, 2.7. The corresponding values of the parameters of are shown in Table 2.

Table 2: Flux averaged coherent cross sections (in units 10⁻⁴² cm²) for ¹¹⁴Cd isotope obtained in the case of neutrino energy spectra coming from two neutrino sources: (i) supernova neutrinos and (ii) laboratory neutrinos.

Flux averaged cross sections (10⁻⁴² cm²)

Supernova neutrinos

Neutrinos

Pion-muon stopped

Fermi-Dirac (FD)
Power Law (PL)
Michel spectrum

3.10
4.14
6.20

12
16
24

2648
4457.52
8648.06

2653.26
4452.93
8658.47

13109.52
15880.76

It is important to note that the flavour-dependent fluxes and spectra emitted by an SN at any distance from the source can be different from those originally produced mainly due to neutrino oscillations in propagation and also due to other phenomena [70]. The high statistics of neutrino signal from a future galactic SN may allow us to unravel the relevant SN-neutrino scenarios.

The number of the emitted neutrinos can be obtained from the total emitted energy . The (time averaged) neutrino flux at a distance from the source is (for the SN 1987A  cm).

3.2.6. Low-Energy -Beam Neutrinos

Recently, some accelerated -radioactive nuclei have been proposed as sources of neutrino beams (beta-beam neutrinos) [14, 15, 77]. Such facilities may produce pure beam neutrinos in which the possible flavors are either the (for -decaying ions) or the (for -decaying ions) to search for standard and nonstandard neutrino physics at low and intermediate energies (-nucleus interactions, neutrino properties, neutrino oscillations, etc.) and measure -nucleus scattering cross sections [15, 77].

For the readers convenience, we summarize here the main features of the low-energy -beam neutrinos. Their spectra [8, 75, 76] are characterised by the boost velocities (Lorentz factors or -factors). For most applications, we derive normalized synthetic neutrino energy distributions given by linear combinations of the form

where numbers the different Lorentz () factors included in the synthetic spectrum (expressions for the individual distributions are given in [7, 8, 74]). Combinations of the type for -factors up to 10–12 are used to fit original supernova neutrino spectral distributions, , reaching terrestrial detectors by adjusting the weight parameters through the minimization procedure described in [8, 75, 76].

Many authors in recent -beam neutrino simulations employ the energy spectra of the antineutrinos emitted from -radioactive ions according to the reaction

The -value of this reaction is = 3.5 MeV. Another potential -radioactive isotope for beams is the with -value = 13.0 MeV. Interesting -radioactive ion source to be accelerated for producing beams is the , which decays according to the reaction [15, 77]

The -value of this reaction is = 3.4 MeV. For beams another promising -radioactive isotope is the ( = 13.9 MeV).

From the aforementioned potential targets, and are considered to have rather low -values, so they are good choices for short baseline neutrino studies while and have relatively high -values and they are the best choices for a large baseline [13].

Energy spectra of the reactions (19) and (20) for several integer -boost factors () are discussed in [8, 74].

3.3. Simulated Neutrino Signals on Nuclear Detectors

As mentioned in Section 1, the characteristics of the arriving at a nuclear detector neutrino flux are concealed in the nuclear response of the detector medium, that is, in the material CdTe or CdZnTe for the case of the COBRA detectors. Theoretically, these features could be simulated by convoluted cross sections calculations carried out as discussed in [64, 65, 71].

In the present work, the convolution (folding) method was performed with the original cross sections obtained as discussed before, in order to compute the flux averaged total cross sections, , for the low-energy neutrino spectra of the previous section.

For the coherent channel, which is possible only in neutral current neutrino-nucleus reactions studied in the present work, the flux averaged cross section is defined as [2]

Due to the dominance of the coherent cross section throughout the region of the incoming neutrino energy , is, sometimes, even two or three orders of magnitude larger than the incoherent one, [6, 8, 28, 61].

The flux averaged cross sections obtained for ¹¹⁴Cd with (21) for the neutrino distributions of Section 3 are listed in Tables 2 and 3. In Table 2 the flux averaged cross sections refer to various supernova neutrino scenarios described by the Fermi-Dirac (F-D) and Power-Low (P-L) distributions corresponding to the parameters given in this Table, that is, neutrinos, antineutrinos, and , , , , and [64, 65, 71]. In the last two columns of Table 2 we tabulate the calculated for the distributions of (16) and (17). Here the flux averaged cross sections have been calculated as in [30].

Table 3: Flux averaged coherent cross sections (in units 10⁻⁴⁰ cm²) for ¹¹⁴Cd isotope obtained in the case of neutrino energy spectra coming from two neutrino sources: (i) geoneutrinos, (ii) reactor neutrinos, and (iii) solar neutrinos.

Flux averaged cross sections (10⁻⁴² cm²)

Detector
Geoneutrinos
Reactor neutrinos
Solar neutrinos

¹¹⁴Cd
⁴⁰K
²³⁸U
²³²Th
²³⁵U
²³⁸U
²³⁹Pu
⁸B
hep

151.38
1504.40
972.56
192.20
508.90
9604.36
8503.99
9956.57

In Table 3 we list the flux averaged cross sections evaluated by adopting the neutrino distributions of the geoneutrinos (see Figure 3(a)), of the reactor neutrinos (see Figure 3(b)) and solar neutrinos (see Figure 4(a)) for the ⁸B-neutrinos and Figure 4(b) for the hep neutrinos).

3.4. Number of Events in -Detectors

For another connection of the present theoretical results with the neutrino experiments discussed in Section 1, and specifically COBRA experiment, we estimate the signals created in the ¹¹⁴Cd detector which is given by the expression [6, 78]

By using our theoretical cross sections for ¹¹⁴Cd isotope we may evaluate the neutrino fluxes or the scattering event rates, , for the COBRA detector.

Our calculations here are based on a mass 100 Kgr of the COBRA detector with detector material or for the typical detection rate of = 1 event , t. Assuming that is the total number of nuclei (atoms) of ¹¹⁴Cd in the detector, we have [78]

The COBRA detector is expected to have a total mass of ¹¹⁴Cd CdZnTe about  Kgr which translates to about atoms (nuclei). The results from (23) neutrino fluxes for the supernova neutrinos with the mean energies (electron neutrinos ), (electron anti-neutrinos ), and (, , ) and the of Table 2 are shown in Table 4 (first three lines).

Table 4: Neutrino coherent fluxes (in units 10⁹ sec⁻¹ cm²) for ¹¹⁴Cd isotope for the two materials (CdTe and CdZnTe) of the COBRA experiment obtained in the case of supernova neutrinos with mean energies = 12, 16, and 24 MeV. is the Avogadro’s number.

Neutrino coherent fluxes

Detector medium
Number of atoms
¹¹⁴Cd (Kgr)
(MeV)
(×10⁹ sec⁻¹ cm⁻²)

CdTe
120.11
13.5
12
1.447

16
0.862

24
0.449

CdZnTe
94.17
10.6
12
1.847

16
1.100

24
0.566

Similar calculations will be done assuming that the material of COBRA detector is the . Again we consider 100 Kgr detector which contains 13.5 Kgr ¹¹⁴Cd or about atoms (nuclei). The results from (23) neutrino flux for the supernova neutrino scenarios adopted above are shown in Table 4 (last three lines).

These results are encouraging for the materials to be used in the future as astrophysical neutrino detectors in addition to their main goal of neutrinoless double -decay search. We stress, however, that even though the above neutrino fluxes are of the same order with those expected at the Spallation Neutron Source at ORLaND, Oak Ridge [17–19], in choosing a neutrino cross section measurement target other experimental criteria usually lead to more popular choices (Xe, Cs, etc.). For such targets similar calculations to those we performed here for ¹¹⁴Cd could be also done.

4. Conclusions

Astrophysical neutrinos (solar, supernova, and Earth neutrinos) are key particles in investigating the structure and evolution of stars, the astronuclear reactions, and also in deepening our knowledge on the fundamental interactions and the nuclear weak responses. In this work we applied the convolution procedure to calculate flux averaged cross sections and event rates for the above -sources based on neutrino-nucleus cross sections obtained with realistic nuclear structure calculations (QRPA method). For the computed folded cross sections we employed specific spectral distributions describing neutrino-energy spectra of supernova and solar neutrinos, geoneutrinos, reactor neutrinos, laboratory neutrinos, and neutrinos produced from the decay of pions and muons at rest.

The flux-averaged total coherent cross sections, , reflect to some extent the neutrino signals generated in several selected terrestrial detectors from such -sources. In this work, we estimated coherent neutrino fluxes for ¹¹⁴Cd which is content of the CdTe and CdZnTe materials of the COBRA detector at LNGS. The goal of this experiment is to search for double beta decay events and neutrino observation.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to Professor T. S. Kosmas for fruitful and stimulating discussions and to the Division of Theoretical Physics of the University of Ioannina for technical support and computational assistance to perform the present calculations.

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