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

Authors: Vaitsa Tsakstara
Journal: Advances in High Energy Physics
 (2015)

---

## Abstract

At first, we evaluate scattering cross sections of low, and intermediate-energy neutrinos scattered off the114Cd 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 ν-nucleus channel addressed here is the dominant reaction channel of the neutral current ν-nucleus scattering. Our ν-nucleus cross sections (calculated with a refinement of the quasiparticle random-phase approximation, QRPA) refer to the g
s
→
g
s transitions for ν-energies ε
ν
≤
100 MeV. Subsequently, simulated ν-signals on 114Cd 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 ν-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.


---

## Body

## 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 throughν-nucleus interactions, the characteristics of the neutrino fluxes emitted by various neutrino sources like the astrophysical ν-sources (supernova, solar, and Earth neutrinos) or the laboratory ν-sources (accelerated β-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 ν-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 ν-nucleus cross sections calculations and realistic descriptions for the ν-beam of the studied neutrino source [25–27], that is, reliable ν-energy distributions.In the present paper, we study extensively the nuclear response to low and intermediate-energy neutrinos of the114Cd 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 ν-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 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 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 ν-nucleus scattering the g
s
→
g
s 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 γ-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 (ν
e) charged-current reactions are important (only a small portion of ν
x neutrinos has energy above the mass of the x-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 areν
~
e 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 0
≤
ε
ν
≤
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-ν fluxes and abundances of natural radioactive elements (K, U, and Th) in the Earth’s interior [40–43].One of the main goals of the present work is to focus on the interpretation of variousν-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 114Cd(
ν
,
ν
′
)
114
C
d
∗ 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 Cd detectors as the CdTe or CdZnTe (the detector medium of COBRA experiment) from low-energy ν sources (geo-, reactor-, and solar-neutrinos) and intermediate-energy neutrinos (supernova and laboratory-neutrinos).The paper is organized as follows. At first (Section2) 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 Coherentν-Nucleus Cross Section

The angle differential cross sectiond
σ
/
d
Ω of the elastic scattering of a neutrino with energy ε
ν on a nucleus (
A
,
Z
) is [34, 44] (1)
d
σ
d
Ω
=
G
F
2
4
π
2
ε
ν
2
1
+
cos
⁡
ϑ
Q
w
2
4
F
q
2
2
,where G
F is the Fermi coupling constant and F
(
q
2
) contains the nuclear dependence of the cross section given by (2)
F
(
q
2
)
=
1
Q
w
N
F
N
q
2
+
1
-
4
sin
2
ϑ
W
Z
F
Z
q
2
.In (1), ϑ represent the scattering angle and Q
w denotes the weak charge of the target nucleus: (3)
Q
w
=
N
+
(
1
-
4
sin
2
ϑ
W
)
Z
,(the weak mixing angle is sin
2
ϑ
W
≈
0.231 [45]).The ground state elastic nuclear form factorsF
Z
(
q
2
) for protons and F
N
(
q
2
) for neutrons are defined by (4)
F
k
(
q
2
)
=
k
4
π
∫
‍
j
0
(
q
r
)
ρ
n
,
p
(
r
)
d
3
r
,
k
=
N
,
Zand are normalized as F
N
,
Z
(
q
2
=
0
)
=
1. In (4), ρ
n
,
p
(
r
) denote the neutron (n) and proton (p) charge density distribution with j
0
(
q
r
)
=
sin
(
q
r
)
/
(
q
r
) being the zero-order spherical Bessel function (we neglect a small correction from the single-nucleon form factors). The proton density ρ
p
(
r
) is often taken from experiment whenever measured charge densities are available [44, 46]. Moreover, some authors assume F
N
≈
F
Z. From (1) we can easily obtain the expression with respect to the scattering angle ϑ as (5)
d
σ
ε
ν
,
ϑ
d
cos
⁡
ϑ
=
G
F
2
8
π
1
+
cos
⁡
ϑ
ε
ν
2
·
f
V
p
Z
F
Z
q
2
+
f
V
n
N
F
N
q
2
2
,where f
V
p and f
V
n stand for the polar-vector coupling for the weak neutral current for proton and neutron, respectively, given by: (6)
f
V
p
=
-
1
-
4
sin
2
ϑ
W
2
,
f
V
n
=
-
1
2
.From the latter two equations one can imply that the consideration ofF
N
(
q
2
) in the cross section is important especially for heavier ν-detection targets. Thus, the coherent cross section depends on the square of the ground state nuclear form factor F
(
q
2
) at momentum transfer q
2 which (in the extreme-relativistic limit) is given by (7)
q
2
=
2
ε
ν
2
(
1
-
cos
⁡
ϑ
)
,or q
=
2
ε
ν
sin
(
ϑ
/
2
).From (2), we see that, since (
1
-
4
sin
2
ϑ
W
≈
0.04
) is small, a neutrino scattered elastically on a spin-zero nucleus couples mostly to the neutron distribution, ρ
n
(
r
). A measurement of the cross section for this process would, at some level, provide a determination of the neutron form factor F
N
(
q
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, 46–48].


### 2.2. Coherent Differential Cross Section with respect to Nuclear Recoil EnergyT
A

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 T
A is also interesting. This is written as [31–33, 49] (8)
d
σ
(
ε
ν
,
T
A
)
d
T
A
=
G
F
2
4
π
Q
W
2
M
1
-
M
T
A
2
ε
ν
2
F
(
2
M
T
A
2
)
,where M is the nuclear mass and F denotes the ground state elastic form factor. More accurate expressions, including higher order terms with respect to T
A, 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 sectionσ
tot
(
ε
ν
) is obtained by integrating numerically Equation (5) over the angle ϑ (ϑ
min
⁡
=
0 to ϑ
max
⁡
=
π) or (8) over T
A between (9)
T
A
min
⁡
=
T
A
2
+
T
A
2
M
A
+
T
A
2to T
A
max
⁡
=
∞ [50]. Some authors [48], however, ignoring the momentum dependence of the nuclear form factors take F
N
(
q
2
)
≈
F
Z
(
q
2
)
≈
1 and in this case the total coherent cross section σ
tot
(
ε
ν
) is approximately written as (10)
σ
tot
(
ε
ν
)
=
G
F
2
8
π
[
Z
(
4
si
n
2
ϑ
W
-
1
)
+
N
]
2
ε
ν
2
.Also, because 4
si
n
2
ϑ
W
-
1
≪
1 many authors take σ
tot
∝
N
2
ε
ν
2 [48].


## 2.1. Angle Differential Coherentν-Nucleus Cross Section

The angle differential cross sectiond
σ
/
d
Ω of the elastic scattering of a neutrino with energy ε
ν on a nucleus (
A
,
Z
) is [34, 44] (1)
d
σ
d
Ω
=
G
F
2
4
π
2
ε
ν
2
1
+
cos
⁡
ϑ
Q
w
2
4
F
q
2
2
,where G
F is the Fermi coupling constant and F
(
q
2
) contains the nuclear dependence of the cross section given by (2)
F
(
q
2
)
=
1
Q
w
N
F
N
q
2
+
1
-
4
sin
2
ϑ
W
Z
F
Z
q
2
.In (1), ϑ represent the scattering angle and Q
w denotes the weak charge of the target nucleus: (3)
Q
w
=
N
+
(
1
-
4
sin
2
ϑ
W
)
Z
,(the weak mixing angle is sin
2
ϑ
W
≈
0.231 [45]).The ground state elastic nuclear form factorsF
Z
(
q
2
) for protons and F
N
(
q
2
) for neutrons are defined by (4)
F
k
(
q
2
)
=
k
4
π
∫
‍
j
0
(
q
r
)
ρ
n
,
p
(
r
)
d
3
r
,
k
=
N
,
Zand are normalized as F
N
,
Z
(
q
2
=
0
)
=
1. In (4), ρ
n
,
p
(
r
) denote the neutron (n) and proton (p) charge density distribution with j
0
(
q
r
)
=
sin
(
q
r
)
/
(
q
r
) being the zero-order spherical Bessel function (we neglect a small correction from the single-nucleon form factors). The proton density ρ
p
(
r
) is often taken from experiment whenever measured charge densities are available [44, 46]. Moreover, some authors assume F
N
≈
F
Z. From (1) we can easily obtain the expression with respect to the scattering angle ϑ as (5)
d
σ
ε
ν
,
ϑ
d
cos
⁡
ϑ
=
G
F
2
8
π
1
+
cos
⁡
ϑ
ε
ν
2
·
f
V
p
Z
F
Z
q
2
+
f
V
n
N
F
N
q
2
2
,where f
V
p and f
V
n stand for the polar-vector coupling for the weak neutral current for proton and neutron, respectively, given by: (6)
f
V
p
=
-
1
-
4
sin
2
ϑ
W
2
,
f
V
n
=
-
1
2
.From the latter two equations one can imply that the consideration ofF
N
(
q
2
) in the cross section is important especially for heavier ν-detection targets. Thus, the coherent cross section depends on the square of the ground state nuclear form factor F
(
q
2
) at momentum transfer q
2 which (in the extreme-relativistic limit) is given by (7)
q
2
=
2
ε
ν
2
(
1
-
cos
⁡
ϑ
)
,or q
=
2
ε
ν
sin
(
ϑ
/
2
).From (2), we see that, since (
1
-
4
sin
2
ϑ
W
≈
0.04
) is small, a neutrino scattered elastically on a spin-zero nucleus couples mostly to the neutron distribution, ρ
n
(
r
). A measurement of the cross section for this process would, at some level, provide a determination of the neutron form factor F
N
(
q
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, 46–48].


## 2.2. Coherent Differential Cross Section with respect to Nuclear Recoil EnergyT
A

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 T
A is also interesting. This is written as [31–33, 49] (8)
d
σ
(
ε
ν
,
T
A
)
d
T
A
=
G
F
2
4
π
Q
W
2
M
1
-
M
T
A
2
ε
ν
2
F
(
2
M
T
A
2
)
,where M is the nuclear mass and F denotes the ground state elastic form factor. More accurate expressions, including higher order terms with respect to T
A, 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 sectionσ
tot
(
ε
ν
) is obtained by integrating numerically Equation (5) over the angle ϑ (ϑ
min
⁡
=
0 to ϑ
max
⁡
=
π) or (8) over T
A between (9)
T
A
min
⁡
=
T
A
2
+
T
A
2
M
A
+
T
A
2to T
A
max
⁡
=
∞ [50]. Some authors [48], however, ignoring the momentum dependence of the nuclear form factors take F
N
(
q
2
)
≈
F
Z
(
q
2
)
≈
1 and in this case the total coherent cross section σ
tot
(
ε
ν
) is approximately written as (10)
σ
tot
(
ε
ν
)
=
G
F
2
8
π
[
Z
(
4
si
n
2
ϑ
W
-
1
)
+
N
]
2
ε
ν
2
.Also, because 4
si
n
2
ϑ
W
-
1
≪
1 many authors take σ
tot
∝
N
2
ε
ν
2 [48].


## 3. Results and Discussion

### 3.1. Original Cross Section Calculations

The neutral-current scattering of low- and intermediate-energy neutrinos,ν
l, and antineutrinos, ν
~
l, 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 (11)
ν
l
(
ν
~
l
)
+
114
Cd
⟶
C
d
∗
114
+
ν
l
′
ν
~
l
′
,(l
=
e
,
μ
,
τ and C
d
∗
114 denotes an excited state of 114Cd isotope). We mention that the above reactions (also the CC reactions for l
=
e) play significant role in astrophysical environment since they affect the electron fraction Y
e 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 ν
l and antineutrinos ν
~
l. However, as we will see in Section 3, the signal on the nuclear detector could be significantly different due to the different ν-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|
J
i
π
i
〉
=
|
0
+
〉 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 (g
pair
p
,
n) and the theoretical energy gaps (Δ
p
,
n
th) for protons (p) and neutrons (n) determined at the BCS level for the 114Cd isotope. These parameters renormalise the pairing force and adjust the gaps Δ
p
,
n
th to the empirical ones Δ
p
,
n
exp
⁡ obtained through the application of the three-point formulae (see [8]). The values of the g
pair
p
,
n 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 (g
pair
p) and neutrons (g
pair
n) and the relevant theoretical values of proton (Δ
p
th) and neutron (Δ
n
th) energy gaps. The latter reproduce well the corresponding (for 114Cd isotope) empirical energy gaps, Δ
p
,
n
exp
⁡, listed also in the table, obtained through the three-point formula [6]. The values of the harmonic oscillator size parameter b, used for 114Cd isotope, are also shown.
Isotope
Z, N
Abundance (%)
b (fm)
g
pair
n
g
pair
p
Δ
p
exp
⁡
Δ
p
th
Δ
n
exp
⁡
Δ
n
th
114Cd
48, 66
28.73
2.214
0.9564
0.9753
1.441
1.44108
1.351
1.35093Afterwards, the proton and neutron nuclear form factors are obtained from the expression(12)
F
k
q
2
=
1
k
∑
j
‍
j
^
n
l
j
j
0
q
r
n
l
j
V
j
k
2
,
k
=
N
,
Z
,where V
j
k is the single particle (
n
l
)
j-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 N
=
3,4
,
5) as well as over the fully occupied core j-levels having V
j
k
=
1.In Figure1, 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 (
F
Z
) and neutrons (
F
N
) obtained with our BCS calculations and Figure 2 shows the momentum dependence of F
N
,
Z
(
q
2
) that enters (1) and (5). In this figure, the results obtained by assuming zero momentum transfer (q
≈
0), that is, F
Z
(
q
2
)
=
F
N
(
q
2
)
=
F
(
q
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
Z
=
F
N
=
F
=
1 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.Figure 1
(a) Form factorF
N
,
Z
(
q
2
) (for neutrons and protons) for 114Cd isotope. (b) The ground state elastic nuclear form factor F
(
q
2
).
(a)
(b)Figure 2
Coherent total cross section for the neutral current reactions114Cd(
ν
l
,
ν
l
′
)
114
C
d
∗,  l
=
e
,
μ
,
τ: (a) as a function of the momentum transfer q and (b) as a function of the incoming neutrino energy ε
ν. The dash dotted curve results by assuming zero momentum transfer; that is, F
Z
(
q
2
)
=
F
N
(
q
2
)
=
F
(
q
2
)
=
1.
(a)
(b)Figure2 illustrates the coherent cross sections of ν-114Cd scattering as a function of (i) the momentum transfer q (Figure 2(a)) and (ii) the incoming neutrino energy ε
ν (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 ν-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-Energyν-Sources

The real neutrino sources, astrophysical (solar, supernova, and geoneutrinos) and laboratory (β-beam, pion-muon stopped neutrino beams, and reactor neutrinos), with few exceptions such as the ν
μ neutrino beam emerging from the π
+ decay at rest (ε
ν = 29.8 MeV), the 7Be solar neutrinos (ε
ν = 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(13)
d
N
ν
(
ε
ν
)
d
ε
ν
≡
η
(
ε
ν
)
,
N
ν denotes the number of neutrinos of the beam. Thus, for example, the ν
e 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, andβ-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 (ν
~
e) generated upon transmutation of neutron-rich β-decay nuclei, accompanied by emission of an electron (e
-) and release of decay energy (Q
β) according to the reaction [40] (14)
(
A
,
Z
)
⟶
(
A
,
Z
+
1
)
+
e
-
+
ν
~
e
+
Q
β
.In the latter reaction A is the mass number and Z the atomic (proton) number of the initial (parent) nucleus. Part of the decay energy, Q
β
=
Q
ν
+
Q
h, is carried away by antineutrinos (Q
ν) while the remainder is available for heating (Q
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 the238U 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.Figure3(a) shows the antineutrino spectra from 40K, 238U series, and 232Th series (τ
1
/
2
=
4.47
×
1
0
9 year, τ
1
/
2
=
14.0
×
1
0
9 year, and τ
1
/
2
=
1.28
×
1
0
9 year, resp.). In this neutrino energy distribution ν
~
e, coming from 82 beta decays in the U series and 70 beta decays in the Th series, are included. Antineutrinos are generated by β-decays of all intermediate radioactive isotopes [40–43].Figure 3
(a) Spectra of the U Series, Th Series, and40K geoneutrinos. Neutrinos from 40K electron capture are also shown in this figure. (b) Normalized reactor neutrino spectra.
(a)
(b)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 intenseν
e sources in many experiments. In the fission of 235U, 239Pu, and 238U, neutron-rich nuclei are produced and ν
~
e antineutrinos are subsequently emitted via β-decay [56, 57]. Experiments using reactor neutrinos are particularly suitable for low Δ
m 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 ν
e flux uncertainty and detector uncertainty are constrained to within a few % at these experiments.Nuclear reactors, as sources ofν
~
e, give fluxes of the order ~1013 ν
~
/
c
m
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 β
- 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 areν
e neutrinos produced through weak, electromagnetic, and strong nuclear processes in the interior of our Sun. They have energy ε
ν
≤
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 Figure4(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].Figure 4
(a) Normalized energy spectrum of8B and hep ν
e solar neutrinos. (b) Energy spectra of ν
e and ν
~
μ neutrino beams, generated from the muon-decay at rest.
(a)
(b)


#### 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),ν
e neutrinos and ν
~
μ antineutrinos are produced from the decay of muons according to the reaction (15)
μ
+
⟶
e
+
+
ν
e
+
ν
~
μ
.The decaying muons result from the decay of slow pions (π
+
→
μ
+
+
ν
μ) and hence, ν
e and ν
~
μ neutrinos have relatively low energies. Their energy spectra are approximately described by normalized distributions of the form [2, 59](16)
η
ν
e
(
ε
ν
)
=
96
ε
ν
2
M
μ
-
4
M
μ
-
2
ε
ν
,
(17)
η
ν
~
μ
(
ε
ν
)
=
16
ε
ν
2
M
μ
-
4
3
M
μ
-
4
ε
ν
,where M
μ
=
105.6 MeV is the muon rest mass (see Figure 4(b)). The maximum energy of ν
e and ν
~
μ in the later equations is ε
ν
max
⁡
=
52.8
MeV
=
M
μ
/
2 [59, 60]. The distribution of these ν
e neutrinos is known as Michel energy spectrum. The pion-muon decay at rest neutrino beams is not completely pure as, for example, the β-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-ν 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 potentialμ. 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 T or the average ν-energy 〈
ε
ν
〉 [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 n
dg shifts the spectrum to higher energies [64, 65]. The Power-Law energy distribution is characterized by the pinching parameter α and average energy 〈
ε
ν
〉. The average energy 〈
ε
ν
〉 reflects the depth of the stars from where the neutrinos are escaping. As 〈
ε
ν
〉 grows, the maximum of the distribution shifts to higher neutrino energy ε
ν [65], while as the width parameter w [8] grows (for the same average energy 〈
ε
ν
〉), both the maximum of the distribution shifts to smaller neutrino energy ε
ν and its peak becomes smaller. For the purposes of the present work, the values of the pinching parameter needed are α =5.1, 3.7, 2.7. The corresponding values of the parameters of 〈
ε
ν
〉 are shown in Table 2.Table 2
Flux averaged coherent cross sections〈
σ
coh
〉 (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〈
σ
coh
〉 (10−42 cm2)
Supernova neutrinos
Neutrinos
Pion-muon stopped
Fermi-Dirac (FD)
Power Law (PL)
Michel spectrum
T
=
3.10
4.14
6.20
〈
ε
ν
〉
=
12
16
24
η
ν
e
(
ε
ν
)
η
ν
~
μ
(
ε
ν
)
〈
σ
〉
=
2648
4457.52
8648.06
〈
σ
〉
=
2653.26
4452.93
8658.47
〈
σ
〉
=
13109.52
15880.76It 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 energyU
ν
=
3
×
1
0
53
erg
N
ν
=
U
ν
/
〈
E
ν
〉. The (time averaged) neutrino flux at a distance D from the source is Φ
=
N
ν
/
(
4
π
D
2
) (for the SN 1987A D
=
10
kpc
=
3.1
×
1
0
22 cm).


#### 3.2.6. Low-Energyβ-Beam Neutrinos

Recently, some acceleratedβ-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 ν
e (for β
+-decaying ions) or the ν
~
e (for β
--decaying ions) to search for standard and nonstandard neutrino physics at low and intermediate energies (ν-nucleus interactions, neutrino properties, neutrino oscillations, etc.) and measure ν-nucleus scattering cross sections [15, 77].For the readers convenience, we summarize here the main features of the low-energyβ-beam neutrinos. Their spectra [8, 75, 76] are characterised by the boost velocities (Lorentz factors or γ-factors). For most applications, we derive normalized synthetic neutrino energy distributions η
b
b
(
ε
ν
) given by linear combinations of the form(18)
η
b
b
(
ε
ν
)
=
∑
j
=
1
N
‍
α
j
η
γ
j
(
ε
ν
)
,where N numbers the different Lorentz (γ) factors included in the synthetic spectrum (expressions for the individual distributions η
γ
j
(
ε
ν
) are given in [7, 8, 74]). Combinations of the type η
b
b
(
ε
ν
) for γ-factors up to 10–12 are used to fit original supernova neutrino spectral distributions, η
SN
(
ε
ν
), reaching terrestrial detectors by adjusting the weight parameters α
j through the minimization procedure described in [8, 75, 76].Many authors in recentβ-beam neutrino simulations employ the energy spectra of the antineutrinos ν
~
e emitted from β
--radioactive He
2
6 ions according to the reaction (19)
He
2
6
⟶
Li
3
6
+
e
-
+
ν
~
eThe Q-value of this reaction is Q
He = 3.5 MeV. Another potential β
--radioactive isotope for ν
~
e beams is the Li
3
8 with Q-value Q
Li = 13.0 MeV. Interesting β
+-radioactive ion source to be accelerated for producing ν
e beams is the Ne
10
18, which decays according to the reaction [15, 77] (20)
Ne
10
18
⟶
F
9
18
+
e
+
+
ν
eThe Q-value of this reaction is Q
Ne = 3.4 MeV. For ν
e beams another promising β
+-radioactive isotope is the B
4
8 (Q
B = 13.9 MeV).From the aforementioned potential targets,He
2
6 and Ne
10
18 are considered to have rather low Q-values, so they are good choices for short baseline neutrino studies while Li
3
8 and B
4
8 have relatively high Q-values and they are the best choices for a large baseline [13].Energy spectra of the reactions (19) and (20) for several integer γ-boost factors (γ
=
3
,
4
,
…
,
15) are discussed in [8, 74].


### 3.3. Simulated Neutrino Signals on Nuclear Detectors

As mentioned in Section1, 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,〈
σ
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〈
σ
coh
〉 is defined as [2] (21)
〈
σ
coh
〉
=
∫
0
∞
‍
σ
coh
(
ε
ν
)
η
(
ε
ν
)
d
ε
ν
.Due to the dominance of the coherent cross section σ
coh
(
ε
ν
) throughout the region of the incoming neutrino energy ε
ν, 〈
σ
coh
〉 is, sometimes, even two or three orders of magnitude larger than the incoherent one, 〈
σ
tot
incoh
〉 [6, 8, 28, 61].The flux averaged cross sections obtained for114Cd with (21) for the neutrino distributions η
(
ε
ν
) 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, ν
e neutrinos, ν
~
e antineutrinos, and ν
x, x
=
ν
μ, ν
τ, ν
~
μ, and ν
~
τ [64, 65, 71]. In the last two columns of Table 2 we tabulate the 〈
σ
coh
〉 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〈
σ
coh
〉 (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〈
σ
coh
〉 (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
9956.57In Table3 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 inν-Detectors

For another connection of the present theoretical results with the neutrino experiments discussed in Section1, and specifically COBRA experiment, we estimate the signals created in the 114Cd detector which is given by the expression [6, 78] (22)
σ
sign
(
ε
ν
)
=
σ
coh
(
ε
ν
)
η
(
ε
ν
)
.By using our theoretical cross sections σ
(
ε
ν
) for 114Cd isotope we may evaluate the neutrino fluxes Φ
ν or the scattering event rates, N
event, for the COBRA detector.Our calculations here are based on a mass 100 Kgr of the COBRA detector with detector materialCdZnTe or CdTe for the typical detection rate of N
event = 1 event h
r
-
1, t. Assuming that N
Cd is the total number of nuclei (atoms) of 114Cd in the detector, we have [78] (23)
d
N
ν
d
t
≡
N
event
=
N
Cd
Φ
ν
(
ε
ν
)
σ
tot
(
ε
ν
)
.The COBRA detector is expected to have a total mass of 114Cd CdZnTe about m
Cd
=
10.6 Kgr which translates to about N
Cd
=
N
Cd
114
=
94.17
N
Avogadro atoms (nuclei). The results from (23) neutrino fluxes for the supernova neutrinos with the mean energies 〈
ε
ν
〉
=
12
MeV (electron neutrinos ν
e), 〈
ε
ν
〉
=
16
MeV (electron anti-neutrinos ν
~
e), and 〈
ε
ν
〉
=
24
MeV (ν
x, ν
~
x, x
=
μ
,
τ) and the 〈
σ
coh
〉 of Table 2 are shown in Table 4 (first three lines).Table 4
Neutrino coherent fluxesΦ
ν
(
ε
ν
) (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 〈
ε
ν
〉 = 12, 16, and 24 MeV. N
0 is the Avogadro’s number.
Neutrino coherent fluxesΦ
ν
Detector medium
Number of atoms
114Cd (Kgr)
〈
ε
ν
〉 (MeV)
Φ
ν (×109 sec−1 cm−2)
CdTe
120.11N
0
13.5
12
1.447
16
0.862
24
0.449
CdZnTe
94.17N
0
10.6
12
1.847
16
1.100
24
0.566Similar calculations will be done assuming that the material of COBRA detector is theCdTe. Again we consider 100 Kgr detector which contains 13.5 Kgr 114Cd or about N
Cd
=
N
Cd
114
=
120.11
N
Avogadro 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 theCd materials to be used in the future as astrophysical neutrino detectors in addition to their main goal of neutrinoless double β-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.


## 3.1. Original Cross Section Calculations

The neutral-current scattering of low- and intermediate-energy neutrinos,ν
l, and antineutrinos, ν
~
l, 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 (11)
ν
l
(
ν
~
l
)
+
114
Cd
⟶
C
d
∗
114
+
ν
l
′
ν
~
l
′
,(l
=
e
,
μ
,
τ and C
d
∗
114 denotes an excited state of 114Cd isotope). We mention that the above reactions (also the CC reactions for l
=
e) play significant role in astrophysical environment since they affect the electron fraction Y
e 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 ν
l and antineutrinos ν
~
l. However, as we will see in Section 3, the signal on the nuclear detector could be significantly different due to the different ν-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|
J
i
π
i
〉
=
|
0
+
〉 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 (g
pair
p
,
n) and the theoretical energy gaps (Δ
p
,
n
th) for protons (p) and neutrons (n) determined at the BCS level for the 114Cd isotope. These parameters renormalise the pairing force and adjust the gaps Δ
p
,
n
th to the empirical ones Δ
p
,
n
exp
⁡ obtained through the application of the three-point formulae (see [8]). The values of the g
pair
p
,
n 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 (g
pair
p) and neutrons (g
pair
n) and the relevant theoretical values of proton (Δ
p
th) and neutron (Δ
n
th) energy gaps. The latter reproduce well the corresponding (for 114Cd isotope) empirical energy gaps, Δ
p
,
n
exp
⁡, listed also in the table, obtained through the three-point formula [6]. The values of the harmonic oscillator size parameter b, used for 114Cd isotope, are also shown.
Isotope
Z, N
Abundance (%)
b (fm)
g
pair
n
g
pair
p
Δ
p
exp
⁡
Δ
p
th
Δ
n
exp
⁡
Δ
n
th
114Cd
48, 66
28.73
2.214
0.9564
0.9753
1.441
1.44108
1.351
1.35093Afterwards, the proton and neutron nuclear form factors are obtained from the expression(12)
F
k
q
2
=
1
k
∑
j
‍
j
^
n
l
j
j
0
q
r
n
l
j
V
j
k
2
,
k
=
N
,
Z
,where V
j
k is the single particle (
n
l
)
j-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 N
=
3,4
,
5) as well as over the fully occupied core j-levels having V
j
k
=
1.In Figure1, 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 (
F
Z
) and neutrons (
F
N
) obtained with our BCS calculations and Figure 2 shows the momentum dependence of F
N
,
Z
(
q
2
) that enters (1) and (5). In this figure, the results obtained by assuming zero momentum transfer (q
≈
0), that is, F
Z
(
q
2
)
=
F
N
(
q
2
)
=
F
(
q
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
Z
=
F
N
=
F
=
1 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.Figure 1
(a) Form factorF
N
,
Z
(
q
2
) (for neutrons and protons) for 114Cd isotope. (b) The ground state elastic nuclear form factor F
(
q
2
).
(a)
(b)Figure 2
Coherent total cross section for the neutral current reactions114Cd(
ν
l
,
ν
l
′
)
114
C
d
∗,  l
=
e
,
μ
,
τ: (a) as a function of the momentum transfer q and (b) as a function of the incoming neutrino energy ε
ν. The dash dotted curve results by assuming zero momentum transfer; that is, F
Z
(
q
2
)
=
F
N
(
q
2
)
=
F
(
q
2
)
=
1.
(a)
(b)Figure2 illustrates the coherent cross sections of ν-114Cd scattering as a function of (i) the momentum transfer q (Figure 2(a)) and (ii) the incoming neutrino energy ε
ν (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 ν-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-Energyν-Sources

The real neutrino sources, astrophysical (solar, supernova, and geoneutrinos) and laboratory (β-beam, pion-muon stopped neutrino beams, and reactor neutrinos), with few exceptions such as the ν
μ neutrino beam emerging from the π
+ decay at rest (ε
ν = 29.8 MeV), the 7Be solar neutrinos (ε
ν = 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(13)
d
N
ν
(
ε
ν
)
d
ε
ν
≡
η
(
ε
ν
)
,
N
ν denotes the number of neutrinos of the beam. Thus, for example, the ν
e 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, andβ-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 (ν
~
e) generated upon transmutation of neutron-rich β-decay nuclei, accompanied by emission of an electron (e
-) and release of decay energy (Q
β) according to the reaction [40] (14)
(
A
,
Z
)
⟶
(
A
,
Z
+
1
)
+
e
-
+
ν
~
e
+
Q
β
.In the latter reaction A is the mass number and Z the atomic (proton) number of the initial (parent) nucleus. Part of the decay energy, Q
β
=
Q
ν
+
Q
h, is carried away by antineutrinos (Q
ν) while the remainder is available for heating (Q
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 the238U 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.Figure3(a) shows the antineutrino spectra from 40K, 238U series, and 232Th series (τ
1
/
2
=
4.47
×
1
0
9 year, τ
1
/
2
=
14.0
×
1
0
9 year, and τ
1
/
2
=
1.28
×
1
0
9 year, resp.). In this neutrino energy distribution ν
~
e, coming from 82 beta decays in the U series and 70 beta decays in the Th series, are included. Antineutrinos are generated by β-decays of all intermediate radioactive isotopes [40–43].Figure 3
(a) Spectra of the U Series, Th Series, and40K geoneutrinos. Neutrinos from 40K electron capture are also shown in this figure. (b) Normalized reactor neutrino spectra.
(a)
(b)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 intenseν
e sources in many experiments. In the fission of 235U, 239Pu, and 238U, neutron-rich nuclei are produced and ν
~
e antineutrinos are subsequently emitted via β-decay [56, 57]. Experiments using reactor neutrinos are particularly suitable for low Δ
m 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 ν
e flux uncertainty and detector uncertainty are constrained to within a few % at these experiments.Nuclear reactors, as sources ofν
~
e, give fluxes of the order ~1013 ν
~
/
c
m
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 β
- 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 areν
e neutrinos produced through weak, electromagnetic, and strong nuclear processes in the interior of our Sun. They have energy ε
ν
≤
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 Figure4(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].Figure 4
(a) Normalized energy spectrum of8B and hep ν
e solar neutrinos. (b) Energy spectra of ν
e and ν
~
μ neutrino beams, generated from the muon-decay at rest.
(a)
(b)


### 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),ν
e neutrinos and ν
~
μ antineutrinos are produced from the decay of muons according to the reaction (15)
μ
+
⟶
e
+
+
ν
e
+
ν
~
μ
.The decaying muons result from the decay of slow pions (π
+
→
μ
+
+
ν
μ) and hence, ν
e and ν
~
μ neutrinos have relatively low energies. Their energy spectra are approximately described by normalized distributions of the form [2, 59](16)
η
ν
e
(
ε
ν
)
=
96
ε
ν
2
M
μ
-
4
M
μ
-
2
ε
ν
,
(17)
η
ν
~
μ
(
ε
ν
)
=
16
ε
ν
2
M
μ
-
4
3
M
μ
-
4
ε
ν
,where M
μ
=
105.6 MeV is the muon rest mass (see Figure 4(b)). The maximum energy of ν
e and ν
~
μ in the later equations is ε
ν
max
⁡
=
52.8
MeV
=
M
μ
/
2 [59, 60]. The distribution of these ν
e neutrinos is known as Michel energy spectrum. The pion-muon decay at rest neutrino beams is not completely pure as, for example, the β-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-ν 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 potentialμ. 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 T or the average ν-energy 〈
ε
ν
〉 [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 n
dg shifts the spectrum to higher energies [64, 65]. The Power-Law energy distribution is characterized by the pinching parameter α and average energy 〈
ε
ν
〉. The average energy 〈
ε
ν
〉 reflects the depth of the stars from where the neutrinos are escaping. As 〈
ε
ν
〉 grows, the maximum of the distribution shifts to higher neutrino energy ε
ν [65], while as the width parameter w [8] grows (for the same average energy 〈
ε
ν
〉), both the maximum of the distribution shifts to smaller neutrino energy ε
ν and its peak becomes smaller. For the purposes of the present work, the values of the pinching parameter needed are α =5.1, 3.7, 2.7. The corresponding values of the parameters of 〈
ε
ν
〉 are shown in Table 2.Table 2
Flux averaged coherent cross sections〈
σ
coh
〉 (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〈
σ
coh
〉 (10−42 cm2)
Supernova neutrinos
Neutrinos
Pion-muon stopped
Fermi-Dirac (FD)
Power Law (PL)
Michel spectrum
T
=
3.10
4.14
6.20
〈
ε
ν
〉
=
12
16
24
η
ν
e
(
ε
ν
)
η
ν
~
μ
(
ε
ν
)
〈
σ
〉
=
2648
4457.52
8648.06
〈
σ
〉
=
2653.26
4452.93
8658.47
〈
σ
〉
=
13109.52
15880.76It 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 energyU
ν
=
3
×
1
0
53
erg
N
ν
=
U
ν
/
〈
E
ν
〉. The (time averaged) neutrino flux at a distance D from the source is Φ
=
N
ν
/
(
4
π
D
2
) (for the SN 1987A D
=
10
kpc
=
3.1
×
1
0
22 cm).


### 3.2.6. Low-Energyβ-Beam Neutrinos

Recently, some acceleratedβ-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 ν
e (for β
+-decaying ions) or the ν
~
e (for β
--decaying ions) to search for standard and nonstandard neutrino physics at low and intermediate energies (ν-nucleus interactions, neutrino properties, neutrino oscillations, etc.) and measure ν-nucleus scattering cross sections [15, 77].For the readers convenience, we summarize here the main features of the low-energyβ-beam neutrinos. Their spectra [8, 75, 76] are characterised by the boost velocities (Lorentz factors or γ-factors). For most applications, we derive normalized synthetic neutrino energy distributions η
b
b
(
ε
ν
) given by linear combinations of the form(18)
η
b
b
(
ε
ν
)
=
∑
j
=
1
N
‍
α
j
η
γ
j
(
ε
ν
)
,where N numbers the different Lorentz (γ) factors included in the synthetic spectrum (expressions for the individual distributions η
γ
j
(
ε
ν
) are given in [7, 8, 74]). Combinations of the type η
b
b
(
ε
ν
) for γ-factors up to 10–12 are used to fit original supernova neutrino spectral distributions, η
SN
(
ε
ν
), reaching terrestrial detectors by adjusting the weight parameters α
j through the minimization procedure described in [8, 75, 76].Many authors in recentβ-beam neutrino simulations employ the energy spectra of the antineutrinos ν
~
e emitted from β
--radioactive He
2
6 ions according to the reaction (19)
He
2
6
⟶
Li
3
6
+
e
-
+
ν
~
eThe Q-value of this reaction is Q
He = 3.5 MeV. Another potential β
--radioactive isotope for ν
~
e beams is the Li
3
8 with Q-value Q
Li = 13.0 MeV. Interesting β
+-radioactive ion source to be accelerated for producing ν
e beams is the Ne
10
18, which decays according to the reaction [15, 77] (20)
Ne
10
18
⟶
F
9
18
+
e
+
+
ν
eThe Q-value of this reaction is Q
Ne = 3.4 MeV. For ν
e beams another promising β
+-radioactive isotope is the B
4
8 (Q
B = 13.9 MeV).From the aforementioned potential targets,He
2
6 and Ne
10
18 are considered to have rather low Q-values, so they are good choices for short baseline neutrino studies while Li
3
8 and B
4
8 have relatively high Q-values and they are the best choices for a large baseline [13].Energy spectra of the reactions (19) and (20) for several integer γ-boost factors (γ
=
3
,
4
,
…
,
15) are discussed in [8, 74].


## 3.2.1. Geoneutrinos

Geoneutrinos (or Earth neutrinos) are mainly electron antineutrinos (ν
~
e) generated upon transmutation of neutron-rich β-decay nuclei, accompanied by emission of an electron (e
-) and release of decay energy (Q
β) according to the reaction [40] (14)
(
A
,
Z
)
⟶
(
A
,
Z
+
1
)
+
e
-
+
ν
~
e
+
Q
β
.In the latter reaction A is the mass number and Z the atomic (proton) number of the initial (parent) nucleus. Part of the decay energy, Q
β
=
Q
ν
+
Q
h, is carried away by antineutrinos (Q
ν) while the remainder is available for heating (Q
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 the238U 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.Figure3(a) shows the antineutrino spectra from 40K, 238U series, and 232Th series (τ
1
/
2
=
4.47
×
1
0
9 year, τ
1
/
2
=
14.0
×
1
0
9 year, and τ
1
/
2
=
1.28
×
1
0
9 year, resp.). In this neutrino energy distribution ν
~
e, coming from 82 beta decays in the U series and 70 beta decays in the Th series, are included. Antineutrinos are generated by β-decays of all intermediate radioactive isotopes [40–43].Figure 3
(a) Spectra of the U Series, Th Series, and40K geoneutrinos. Neutrinos from 40K electron capture are also shown in this figure. (b) Normalized reactor neutrino spectra.
(a)
(b)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 intenseν
e sources in many experiments. In the fission of 235U, 239Pu, and 238U, neutron-rich nuclei are produced and ν
~
e antineutrinos are subsequently emitted via β-decay [56, 57]. Experiments using reactor neutrinos are particularly suitable for low Δ
m 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 ν
e flux uncertainty and detector uncertainty are constrained to within a few % at these experiments.Nuclear reactors, as sources ofν
~
e, give fluxes of the order ~1013 ν
~
/
c
m
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 β
- 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 areν
e neutrinos produced through weak, electromagnetic, and strong nuclear processes in the interior of our Sun. They have energy ε
ν
≤
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 Figure4(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].Figure 4
(a) Normalized energy spectrum of8B and hep ν
e solar neutrinos. (b) Energy spectra of ν
e and ν
~
μ neutrino beams, generated from the muon-decay at rest.
(a)
(b)


## 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),ν
e neutrinos and ν
~
μ antineutrinos are produced from the decay of muons according to the reaction (15)
μ
+
⟶
e
+
+
ν
e
+
ν
~
μ
.The decaying muons result from the decay of slow pions (π
+
→
μ
+
+
ν
μ) and hence, ν
e and ν
~
μ neutrinos have relatively low energies. Their energy spectra are approximately described by normalized distributions of the form [2, 59](16)
η
ν
e
(
ε
ν
)
=
96
ε
ν
2
M
μ
-
4
M
μ
-
2
ε
ν
,
(17)
η
ν
~
μ
(
ε
ν
)
=
16
ε
ν
2
M
μ
-
4
3
M
μ
-
4
ε
ν
,where M
μ
=
105.6 MeV is the muon rest mass (see Figure 4(b)). The maximum energy of ν
e and ν
~
μ in the later equations is ε
ν
max
⁡
=
52.8
MeV
=
M
μ
/
2 [59, 60]. The distribution of these ν
e neutrinos is known as Michel energy spectrum. The pion-muon decay at rest neutrino beams is not completely pure as, for example, the β-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-ν 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 potentialμ. 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 T or the average ν-energy 〈
ε
ν
〉 [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 n
dg shifts the spectrum to higher energies [64, 65]. The Power-Law energy distribution is characterized by the pinching parameter α and average energy 〈
ε
ν
〉. The average energy 〈
ε
ν
〉 reflects the depth of the stars from where the neutrinos are escaping. As 〈
ε
ν
〉 grows, the maximum of the distribution shifts to higher neutrino energy ε
ν [65], while as the width parameter w [8] grows (for the same average energy 〈
ε
ν
〉), both the maximum of the distribution shifts to smaller neutrino energy ε
ν and its peak becomes smaller. For the purposes of the present work, the values of the pinching parameter needed are α =5.1, 3.7, 2.7. The corresponding values of the parameters of 〈
ε
ν
〉 are shown in Table 2.Table 2
Flux averaged coherent cross sections〈
σ
coh
〉 (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〈
σ
coh
〉 (10−42 cm2)
Supernova neutrinos
Neutrinos
Pion-muon stopped
Fermi-Dirac (FD)
Power Law (PL)
Michel spectrum
T
=
3.10
4.14
6.20
〈
ε
ν
〉
=
12
16
24
η
ν
e
(
ε
ν
)
η
ν
~
μ
(
ε
ν
)
〈
σ
〉
=
2648
4457.52
8648.06
〈
σ
〉
=
2653.26
4452.93
8658.47
〈
σ
〉
=
13109.52
15880.76It 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 energyU
ν
=
3
×
1
0
53
erg
N
ν
=
U
ν
/
〈
E
ν
〉. The (time averaged) neutrino flux at a distance D from the source is Φ
=
N
ν
/
(
4
π
D
2
) (for the SN 1987A D
=
10
kpc
=
3.1
×
1
0
22 cm).


## 3.2.6. Low-Energyβ-Beam Neutrinos

Recently, some acceleratedβ-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 ν
e (for β
+-decaying ions) or the ν
~
e (for β
--decaying ions) to search for standard and nonstandard neutrino physics at low and intermediate energies (ν-nucleus interactions, neutrino properties, neutrino oscillations, etc.) and measure ν-nucleus scattering cross sections [15, 77].For the readers convenience, we summarize here the main features of the low-energyβ-beam neutrinos. Their spectra [8, 75, 76] are characterised by the boost velocities (Lorentz factors or γ-factors). For most applications, we derive normalized synthetic neutrino energy distributions η
b
b
(
ε
ν
) given by linear combinations of the form(18)
η
b
b
(
ε
ν
)
=
∑
j
=
1
N
‍
α
j
η
γ
j
(
ε
ν
)
,where N numbers the different Lorentz (γ) factors included in the synthetic spectrum (expressions for the individual distributions η
γ
j
(
ε
ν
) are given in [7, 8, 74]). Combinations of the type η
b
b
(
ε
ν
) for γ-factors up to 10–12 are used to fit original supernova neutrino spectral distributions, η
SN
(
ε
ν
), reaching terrestrial detectors by adjusting the weight parameters α
j through the minimization procedure described in [8, 75, 76].Many authors in recentβ-beam neutrino simulations employ the energy spectra of the antineutrinos ν
~
e emitted from β
--radioactive He
2
6 ions according to the reaction (19)
He
2
6
⟶
Li
3
6
+
e
-
+
ν
~
eThe Q-value of this reaction is Q
He = 3.5 MeV. Another potential β
--radioactive isotope for ν
~
e beams is the Li
3
8 with Q-value Q
Li = 13.0 MeV. Interesting β
+-radioactive ion source to be accelerated for producing ν
e beams is the Ne
10
18, which decays according to the reaction [15, 77] (20)
Ne
10
18
⟶
F
9
18
+
e
+
+
ν
eThe Q-value of this reaction is Q
Ne = 3.4 MeV. For ν
e beams another promising β
+-radioactive isotope is the B
4
8 (Q
B = 13.9 MeV).From the aforementioned potential targets,He
2
6 and Ne
10
18 are considered to have rather low Q-values, so they are good choices for short baseline neutrino studies while Li
3
8 and B
4
8 have relatively high Q-values and they are the best choices for a large baseline [13].Energy spectra of the reactions (19) and (20) for several integer γ-boost factors (γ
=
3
,
4
,
…
,
15) are discussed in [8, 74].


## 3.3. Simulated Neutrino Signals on Nuclear Detectors

As mentioned in Section1, 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,〈
σ
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〈
σ
coh
〉 is defined as [2] (21)
〈
σ
coh
〉
=
∫
0
∞
‍
σ
coh
(
ε
ν
)
η
(
ε
ν
)
d
ε
ν
.Due to the dominance of the coherent cross section σ
coh
(
ε
ν
) throughout the region of the incoming neutrino energy ε
ν, 〈
σ
coh
〉 is, sometimes, even two or three orders of magnitude larger than the incoherent one, 〈
σ
tot
incoh
〉 [6, 8, 28, 61].The flux averaged cross sections obtained for114Cd with (21) for the neutrino distributions η
(
ε
ν
) 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, ν
e neutrinos, ν
~
e antineutrinos, and ν
x, x
=
ν
μ, ν
τ, ν
~
μ, and ν
~
τ [64, 65, 71]. In the last two columns of Table 2 we tabulate the 〈
σ
coh
〉 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〈
σ
coh
〉 (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〈
σ
coh
〉 (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
9956.57In Table3 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 inν-Detectors

For another connection of the present theoretical results with the neutrino experiments discussed in Section1, and specifically COBRA experiment, we estimate the signals created in the 114Cd detector which is given by the expression [6, 78] (22)
σ
sign
(
ε
ν
)
=
σ
coh
(
ε
ν
)
η
(
ε
ν
)
.By using our theoretical cross sections σ
(
ε
ν
) for 114Cd isotope we may evaluate the neutrino fluxes Φ
ν or the scattering event rates, N
event, for the COBRA detector.Our calculations here are based on a mass 100 Kgr of the COBRA detector with detector materialCdZnTe or CdTe for the typical detection rate of N
event = 1 event h
r
-
1, t. Assuming that N
Cd is the total number of nuclei (atoms) of 114Cd in the detector, we have [78] (23)
d
N
ν
d
t
≡
N
event
=
N
Cd
Φ
ν
(
ε
ν
)
σ
tot
(
ε
ν
)
.The COBRA detector is expected to have a total mass of 114Cd CdZnTe about m
Cd
=
10.6 Kgr which translates to about N
Cd
=
N
Cd
114
=
94.17
N
Avogadro atoms (nuclei). The results from (23) neutrino fluxes for the supernova neutrinos with the mean energies 〈
ε
ν
〉
=
12
MeV (electron neutrinos ν
e), 〈
ε
ν
〉
=
16
MeV (electron anti-neutrinos ν
~
e), and 〈
ε
ν
〉
=
24
MeV (ν
x, ν
~
x, x
=
μ
,
τ) and the 〈
σ
coh
〉 of Table 2 are shown in Table 4 (first three lines).Table 4
Neutrino coherent fluxesΦ
ν
(
ε
ν
) (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 〈
ε
ν
〉 = 12, 16, and 24 MeV. N
0 is the Avogadro’s number.
Neutrino coherent fluxesΦ
ν
Detector medium
Number of atoms
114Cd (Kgr)
〈
ε
ν
〉 (MeV)
Φ
ν (×109 sec−1 cm−2)
CdTe
120.11N
0
13.5
12
1.447
16
0.862
24
0.449
CdZnTe
94.17N
0
10.6
12
1.847
16
1.100
24
0.566Similar calculations will be done assuming that the material of COBRA detector is theCdTe. Again we consider 100 Kgr detector which contains 13.5 Kgr 114Cd or about N
Cd
=
N
Cd
114
=
120.11
N
Avogadro 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 theCd materials to be used in the future as astrophysical neutrino detectors in addition to their main goal of neutrinoless double β-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 aboveν-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,〈
σ
coh
〉, reflect to some extent the neutrino signals generated in several selected terrestrial detectors from such ν-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.



---

*Source: 632131-2015-05-12.xml*