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

At first, we evaluate scattering cross sections of low, and intermediate-energy neutrinos scattered off the 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 (calculatedwith a refinement of the quasiparticle random-phase approximation, QRPA) refer to thegs → gs transitions for ]-energies ε] ≤ 100MeV. Subsequently, simulated ]-signals on 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.

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][26][27], that is, reliable ]-energy distributions.
In the present paper, we study extensively the nuclear response to low and intermediate-energy neutrinos of the 114 Cd isotope focusing on the computation of flux averaged cross sections of this nucleus for various neutrino energy spectra.The Cd 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 2 Advances in High Energy Physics work is an extension of our previous studies which address inelastic cross sections of neutrinos scattered off the Te and Zn 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 114 Cd 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][32][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 40 K and nuclides in the 238 U and 232 Th 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 0 ≤  ] ≤ 10 MeV.
It is expected [28] that the response of the Cd 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 (K, U, and Th) in the Earth's interior [40][41][42][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 Cd 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 114 Cd(], ]  ) 114 Cd * computed with a refinement of the quasiparticle randomphase 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 Cd detectors as the CdTe or CdZnTe (the detector medium of COBRA experiment) from lowenergy ] sources (geo-, reactor-, and solar-neutrinos) and intermediate-energy neutrinos (supernova and laboratoryneutrinos).
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 114 Cd isotope are discussed.Finally (Section 4), the main conclusions of the present work are extracted.

Angle Differential Coherent
]-Nucleus Cross Section.The angle differential cross section /Ω of the elastic scattering of a neutrino with energy  ] on a nucleus (, ) is [34,44] where   is the Fermi coupling constant and F( 2 ) 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 sin 2   ≈ 0.231 [45]).The ground state elastic nuclear form factors   ( 2 ) for protons and   ( 2 ) for neutrons are defined by and are normalized as  , ( 2 = 0) = 1.In (4),  , () denote the neutron () and proton () charge density distribution with  0 () = sin()/() 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   ( 2 ) 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 F( 2 ) at momentum transfer  2 which (in the extreme-relativistic limit) is given by or  = 2 ] sin(/2).From (2), we see that, since (1 − 4sin 2   ≈ 0.04) 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   ( 2 ).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][47][48].

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][32][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][32][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.

Total Coherent Cross
Section.The total coherent cross section  tot ( ] ) is obtained by integrating numerically Equation (5) over the angle  ( min = 0 to  max = ) or ( 8) over   between to  max  = ∞ [50].Some authors [48], however, ignoring the momentum dependence of the nuclear form factors take   ( 2 ) ≈   ( 2 ) ≈ 1 and in this case the total coherent cross section  tot ( ] ) is approximately written as Also, because 4 sin 2   − 1 ≪ 1 many authors take

Results and Discussion
(ℓ = , , and 114 Cd * denotes an excited state of 114 Cd isotope).We mention that the above reactions (also the CC reactions for ℓ = ) play significant role in astrophysical environment since they affect the electron fraction   of the stellar matter [21][22][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 114 Cd [1,6,8,[51][52][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 |    ⟩ = |0 + ⟩ of the studied (eveneven) 114 Cd isotope is computed by solving iteratively the BCS equations.In Table 1, we list the values of the pairing parameters ( , pair ) and the theoretical energy gaps (Δ th , ) for protons () and neutrons () determined at the BCS level for the 114 Cd isotope.These parameters renormalise the pairing force and adjust the gaps Δ th , to the empirical ones Δ exp , obtained through the application of the three-point formulae (see [8]).The values of the  , pair that adjust the energy gaps in both cases are reliable (see Table 1).Afterwards, the proton and neutron nuclear form factors are obtained from the expression where    is the single particle (ℓ)-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  = 3, 4, 5) as well as over the fully occupied core -levels having    = 1.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  , ( 2 ) that enters (1) and (5).In this figure, the results obtained by assuming zero momentum transfer ( ≈ 0), that is,   ( 2 ) =   ( 2 ) = F( 2 ) = 1, are also presented (dash dotted curve).We see that, for energies above about 40 MeV, the two methods give significantly different results; the approximation   =   = F = 1 is good only for small momentum transfer (e.g., solar and low-energy SNneutrinos).This shows also the sensitivity of the total cross sections of the total cross sections on the nuclear form factor.
Figure 2 illustrates the coherent cross sections of ]-114 Cd 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.

Energy Spectra of Low-and Intermediate-Energy ]-
Sources.The real neutrino sources, astrophysical (solar, supernova, and geoneutrinos) and laboratory (-beam, pionmuon stopped neutrino beams, and reactor neutrinos), with few exceptions such as the ]  neutrino beam emerging from the  + decay at rest ( ] = 29.8MeV), the 7 Be 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 114 Cd nucleus and calculate event rates.
In the latter reaction  is the mass number and  the atomic (proton) number of the initial (parent) nucleus.Part of the decay energy,   =  ] +  ℎ , is carried away by antineutrinos ( ] ) while the remainder is available for heating ( ℎ ).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 238 U decay series, (ii) isotopes in 232 Th decay series, and 40 K 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.
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.

Reactor Neutrinos. Nuclear reactors have been used
as intense ]  sources in many experiments.In the fission of 235 U, 239 Pu, and 238 U, neutron-rich nuclei are produced and ] antineutrinos are subsequently emitted via -decay [56,57].Experiments using reactor neutrinos are particularly suitable for low Δ 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 ∼10 13 ]/cm 2 sec 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% 235 U, 30% 239 Pu, and 8% 238 U [56,58].

Solar Neutrinos. Solar neutrinos are ]
neutrinos produced through weak, electromagnetic, and strong nuclear processes in the interior of our Sun.They have energy  ] ≤ 18 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 8 B 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 8 B 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].

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   = 105.6MeV is the muon rest mass (see Figure 4(b)).The maximum energy of ]  and ] in the later equations is  max ] = 52.8MeV =   /2 [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][62][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).
Advances in High Energy Physics

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][67][68][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 twoparameter 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][73][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  dg 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.
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 SNneutrino scenarios.

Low-Energy 𝛽-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,  SN ( ] ), 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 6  2 He ions according to the reaction The -value of this reaction is  He = 3.5 MeV.Another potential  − -radioactive isotope for ] beams is the 8  3 Li with -value  Li = 13.0MeV.Interesting  + -radioactive ion source to be accelerated for producing ]  beams is the 18  10 Ne, which decays according to the reaction [15,77] 18 10 Ne → 18  9 The -value of this reaction is  Ne = 3.4 MeV.For ]  beams another promising  + -radioactive isotope is the 8 4 B ( B = 13.9MeV).
From the aforementioned potential targets, 6  2 He and 18  10 Ne are considered to have rather low -values, so they are good choices for short baseline neutrino studies while 8  3 Li and 8 4 B have relatively high -values and they are the best choices for a large baseline [13].

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, ⟨ tot ⟩, 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 ⟨ coh ⟩ is defined as [2] ⟨ Due to the dominance of the coherent cross section  coh ( ] ) throughout the region of the incoming neutrino energy  ] , ⟨ coh ⟩ is, sometimes, even two or three orders of magnitude larger than the incoherent one, ⟨ incoh tot ⟩ [6,8,28,61].The flux averaged cross sections obtained for 114 Cd 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 ⟨ coh ⟩ calculated for the distributions of ( 16) and (17).Here the flux averaged cross sections have been calculated as in [30].
In Table 3 we list the flux averaged cross sections evaluated by adopting the neutrino distributions of the geoneutrinos (see Figure 3 By using our theoretical cross sections ( ] ) for 114 Cd isotope we may evaluate the neutrino fluxes Φ ] or the scattering event rates,  event , for the COBRA detector.
Our calculations here are based on a mass 100 Kgr of the COBRA detector with detector material CdZnTe or CdTe for the typical detection rate of  event = 1 event hr −1 , t.Assuming that  Cd is the total number of nuclei (atoms) of 114 Cd in the detector, we have [78] 4 (first three lines).Similar calculations will be done assuming that the material of COBRA detector is the CdTe.Again we consider 100 Kgr detector which contains 13.5 Kgr 114 Cd or about  Cd = 114 Cd = 120.11Avogadro 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 Cd 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][18][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 114 Cd could be also done.

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, ⟨ coh ⟩, 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 114 Cd 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.

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

Figure 4 :
Figure 4: (a) Normalized energy spectrum of 8 B and hep ]  solar neutrinos.(b) Energy spectra of ]  and ] neutrino beams, generated from the muon-decay at rest.
(a)), of the reactor neutrinos (see Figure 3(b)) and solar neutrinos (see Figure 4(a)) for the 8 Bneutrinos 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 114 Cd detector which is given by the expression[6,78]

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

Table 2 :
Flux averaged coherent cross sections ⟨ coh ⟩ (in units 10 −42 cm 2 ) for 114 Cd 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 ⟨ coh ⟩ (10 −42 cm 2 )