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# The Decay Q Value of Neutrinoless Double Beta Decay Revisited

D.-M. Mei  Corresponding author.
Email<EMAIL_ADDRESS>Department of Physics, The University of South
Dakota, Vermillion, SD 57069, USA  W.-Z. Wei  Department of Physics, The
University of South Dakota, Vermillion, SD 57069, USA

(September 3, 2024)

###### Abstract

An earlier publication ”The implication of the atomic effects in neutrinoless
double beta (0$\nu\beta\beta$) decay” written by Mei and Wei has motivated us
to compare the decay Q value ($Q_{\beta\beta}$) derived from the decay of the
parent nucleus to the daughter nucleus with the two ejected beta particles in
the final state to the $Q_{\beta\beta}$ directly derived from the decay of the
initial neutral atom to the final state of double-ionized daughter ion with
the two ejected beta particles in the final state. We show that the results
are the same, which is the mass-energy difference ($\Delta Mc^{2}$) subtracted
by the total difference of the atomic electron binding energy ($\Delta E_{b}$)
between the ground states of initial and final neutral atoms. We demonstrate
that $\Delta Mc^{2}$ is the sum of $Q_{\beta\beta}$ and the atomic relaxation
energy ($\Delta E_{b}$) of the atomic structure after the decay. Depending on
the atomic relaxation time, the release of the atomic binding energy may not
come together with the energy deposition of the two ejected beta particles.

Suggested keywords

††preprint: APS/123-QED

Neutrinos take a key role in understanding the universe aat . The discovery of
neutrino oscillation indicates that neutrinos have mass superk . This has
motivated physicists to postulate new properties of neutrinos moh , which have
created a possible connection between the observed asymmetry of matter over
antimatter in our universe apa . If neutrinos are Majorana particles majorana
, this means that neutrinos are their own anti-particles. This Majorana nature
of neutrinos, if confirmed, might offer an explanation of the prevalence of
matter over anti-matter bar . The only experimentally feasible way to answer
whether neutrinos are Majorana particles is to search for neutrinoless double-
beta (0$\nu\beta\beta$) decay fra ; agi , a proposed form of rare nuclear
decay. The experimental signature is the decay Q value from this decay process
as explored by many experiments gerda ; mjd ; exo ; cuore .

In an earlier paper meiwei , Mei and Wei derived the decay Q value from the
decay of the parent nucleus to the daughter nucleus with the two ejected beta
particles in the final state. In this case, a 0$\nu\beta\beta$ decay process
can be described as below fra :

$^{A}_{Z}X_{N}\longrightarrow^{A}_{Z+2}X_{N-2}^{{}^{\prime}}+e^{-}+e^{-},$ (1)

where ${}^{A}_{Z}X_{N}$ is the mother nucleus in the initial state while
${}^{A}_{Z+2}X_{N-2}^{{}^{\prime}}$ is the daughter nucleus in the final
state. The two electrons in the final state are the two ejected beta particles
as the result of this nuclear decay process. The decay Q value, which is
defined as the available kinetic energy for the two ejected beta particles,
for this nuclear decay process can be calculated using the mass difference in
the rest frame between the initial and final products meiwei :

$Q_{\beta\beta}=[m_{N}(^{A}_{Z}X)-m_{N-2}(^{A}_{Z+2}X^{{}^{\prime}})-2m_{e}]c^{2},$
(2)

where $m_{N}(^{A}_{Z}X)$ stands for the mass of the mother nucleus in the rest
frame while $m_{N-2}(^{A}_{Z+2}X^{{}^{\prime}})$ represents the mass of the
daughter nucleus in the rest frame. $m_{e}$ is the mass of electron in the
rest frame. $c$ is the speed of light. The mass of a nucleus is related to the
corresponding mass of the neutral atom, which is denoted as $M(^{A}_{Z}X)$,
through the following relation:

$M(^{A}_{Z}X)c^{2}=m_{N}(^{A}X)c^{2}+Zm_{e}c^{2}-\sum_{i=1}^{Z}B_{i},$ (3)

where $B_{i}$ stands for the atomic binding energy of the $i$th electron. Note
that the sign of the binding energies in Eq. 3 must be taken as positive
values. Therefore, the atomic binding energies are all positive values in this
work and the previous work by Mei and Wei meiwei . Replacing the masses of the
mother and daughter nuclei in Eq. 2 using the corresponding atomic masses in
the neutral form expressed in Eq. 3, by rearranging the terms,
$Q_{\beta\beta}$ is found to be:

$Q_{\beta\beta}=[M(^{A}_{Z}X)-M(^{A}_{Z+2}X^{{}^{\prime}})]c^{2}-[\sum_{i=1}^{Z+2}B_{i}-\sum_{i=1}^{Z}B_{i}].$
(4)

As can be seen in Eq. 4, $Q_{\beta\beta}$ is given by two terms. The first
term,

$\Delta Mc^{2}=[M(^{A}_{Z}X)-M(^{A}_{Z+2}X^{{}^{\prime}})]c^{2},$ (5)

is the mass-energy difference between the atomic masses of the mother and
daughter atoms in the neutral form, and the second term,

$\Delta E_{b}=[\sum_{i=1}^{Z+2}B_{i}-\sum_{i=1}^{Z}B_{i}]$ (6)

represents the difference of the total atomic binding energy of the mother and
daughter atoms in the neutral form. Therefore, $Q_{\beta\beta}$ can be written
as:

$Q_{\beta\beta}=\Delta Mc^{2}-\Delta E_{b}.$ (7)

It is important to point out that this is a standard definition for the decay
Q value, which is similar to a normal beta decay process where the decay Q
value is expressed as gas ; kra :

$Q_{\beta}=[M(^{A}_{Z}X)-M(^{A}_{Z+1}X^{{}^{\prime}})]c^{2}-[\sum_{i=1}^{Z+1}B_{i}-\sum_{i=1}^{Z}B_{i}],$
(8)

Considering that 0$\nu\beta\beta$ decay in a nucleus occurs very fast in a
level of picoseconds, the atomic structure may have no time to respond to the
change of the nuclear charge right after the decay. The daughter nucleus would
be surrounded by the atomic electrons from the mother atom. In this case, the
daughter atom is a double-ionized ion. The decay Q value can be directly
derived from the decay of the ground state of the mother atom to the double-
ionized daughter ion with the two ejected beta particles in the final state.
This process can be expressed as:

$^{A}_{Z}X\longrightarrow^{A}_{Z}X^{{}^{\prime\prime}++}+e^{-}+e^{-},$ (9)

where ${}^{A}_{Z}X$ is the parent atom in the neutral form and
${}^{A}_{Z}X^{"++}$ is the double-ionized daughter ion in which the daughter
nucleus is surrounded by the parent atomic electrons right after the decay.
Therefore, the atomic member, $Z$, is the same as the parent atom when the
nuclear charge is altered by two units in the final state of the nucleus. The
decay Q value for this decay is calculated as:

$Q_{\beta\beta}=M(^{A}_{Z}X)c^{2}-M(^{A}_{Z}X^{{}^{\prime\prime}++})c^{2}-2m_{e}c^{2},$
(10)

where $M(^{A}_{Z}X)$ is the mass of the parent atom in the neutral form and
$M(^{A}_{Z}X^{{}^{\prime\prime}++})$ is the mass of the double-ionized
daughter ion. Note that $M(^{A}_{Z}X^{{}^{\prime\prime}++})$ is related to the
mass of the daughter nucleus through the following relation:

$M(^{A}_{Z}X^{{}^{\prime\prime}++})c^{2}=m_{N-2}(^{A}_{Z+2}X^{{}^{\prime}})c^{2}+Zm_{e}c^{2}-\sum_{i=1}^{Z}B_{i},$
(11)

where $m_{N-2}(^{A}_{Z+2}X^{{}^{\prime}})c^{2}$ is the mass of the daughter
nucleus, $Zm_{e}$ represents the total mass of the orbital electrons from the
parent atom and $\sum_{i=1}^{Z}B_{i}$ stands for the total atomic binding
energy from the parent atom. Since the mass of the daughter nucleus can be
calculated using the mass of the daughter atom in the neutral form using Eq.
3, thus, $m_{N-2}(^{A}_{Z+2}X^{{}^{\prime}})c^{2}$ can be expressed as:

$m_{N-2}(^{A}_{Z+2}X^{{}^{\prime}})c^{2}=M(^{A}_{Z+2}X^{{}^{\prime}})c^{2}-(Z+2)m_{e}c^{2}+\sum_{i=1}^{Z+2}B_{i},$
(12)

where $M(^{A}_{Z+2}X^{{}^{\prime}})c^{2}$ is the mass of the daughter atom in
the neutral form, $(Z+2)m_{e}c^{2}$ is the total mass of the orbital electrons
surrounding the daughter nucleus, and $\sum_{i=1}^{Z+2}B_{i}$ represents the
total atomic binding energy of the daughter atom in the neutral form. Putting
Eq. 12 into Eq. 11 and rearranging the terms,
$M(^{A}_{Z}X^{{}^{\prime\prime}++})c^{2}$ can be expressed as:

$M(^{A}_{Z}X^{{}^{\prime\prime}++})c^{2}=M(^{A}_{Z+2}X^{{}^{\prime}})c^{2}-2m_{e}c^{2}+\sum_{i=1}^{Z+2}B_{i}-\sum_{i=1}^{Z}B_{i}.$
(13)

Replacing $M(^{A}_{Z}X^{{}^{\prime\prime}++})c^{2}$ in Eq. 10 using Eq. 13,
one obtains the following:

$Q_{\beta\beta}=M(^{A}_{Z}X)c^{2}-M(^{A}_{Z+2}X^{{}^{\prime}})c^{2}-[\sum_{i=1}^{Z+2}B_{i}-\sum_{i=1}^{Z}B_{i}].$
(14)

Eq. 14 is the decay Q value directly derived from the decay of the ground
state of the mother atom in the neutral form to the double-ionized daughter
ion as described in Eq. 9. Note that Eq. 14 is exactly the same as Eq. 4,
which is the decay Q value derived from the decay of the parent nucleus to the
daughter nucleus with the two ejected beta particles in the final state as
described in Eq. 1. Since the decay Q value is defined as the mass difference
between the initial state and the final state products, the decay Q value is
expected to be the same regardless of how it is derived.

From Eq. 7, it is apparent that the decay Q value for 0$\nu\beta\beta$ is
equal to the mass difference of the neutral atoms in the initial and final
states subtracted by the difference of the total atomic electron binding
energy between the mother and daughter atoms. This indicates that the
calculation of $Q_{\beta\beta}$ must take into account the difference of the
total atomic binding energy between the mother and daughter neutral atoms. As
long as $Q_{\beta\beta}$ is calculated by using the mass-energy difference
between the ground state of mother and daughter neutral atoms, it is required
by the energy conservation as shown in Eq. 3 that $Q_{\beta\beta}$ = $\Delta
Mc^{2}$ \- $\Delta E_{b}$.

To calculate $\Delta E_{b}$, one must know the total electron binding energy
for a given atomic system. Since this quantity cannot be easily measured, a
good approximation is given by Lunney, Perrson, and Thibault lpt :

$B_{e}(Z)=14.4381Z^{2.39}+1.55468\times 10^{-6}Z^{5.35}eV,$ (15)

where $B_{e}(Z)$ represents the total electron binding energy for a given
atomic system. Therefore, $\Delta E_{b}$ can be expressed as:

$\Delta E_{b}=B_{e}(Z+2)-B_{e}(Z),$ (16)

for any given 0$\nu\beta\beta$ decay process. Figures 1, 2, and 3 display the
calculated electron binding energy as a function of $Z$ according to Eq. 15
for nine 0$\nu\beta\beta$ decay candidates and their corresponding decay
daughters. The difference in the total electron binding energy, $\Delta
E_{b}$, is labelled respectively for the nine 0$\nu\beta\beta$ decay
processes.

Figure 1:  The electron binding energy as a function of atomic number $Z$ and
the labelled electron binding energy difference for
${}^{48}_{20}Ca\rightarrow^{48}_{22}Ti$,
${}^{76}_{32}Ge\rightarrow^{76}_{34}Se$, and
${}^{82}_{34}Se\rightarrow^{82}_{36}Kr$.  Figure 2:  The electron binding
energy as a function of atomic number $Z$ and the labelled electron binding
energy difference for ${}^{96}_{40}Zr\rightarrow^{96}_{42}Mo$,
${}^{100}_{42}Mo\rightarrow^{100}_{44}Ru$, and
${}^{116}_{48}Cd\rightarrow^{116}_{50}Sn$.  Figure 3:  The electron binding
energy as a function of atomic number $Z$ and the labelled electron binding
energy difference for ${}^{130}_{52}Te\rightarrow^{130}_{54}Xe$,
${}^{136}_{54}Xe\rightarrow^{136}_{56}Ba$, and
${}^{150}_{60}Nd\rightarrow^{150}_{62}Sm$.

As can be seen in Figures 1, 2, and 3, the difference in the total electron
binding energy, $\Delta E_{b}$, ranges from a minimum of 4.76 keV for
${}^{48}_{20}Ca\rightarrow^{48}_{22}Ti$ to a maximum of 21.89 keV for
${}^{150}_{60}Nd\rightarrow^{150}_{62}Sm$. This will result in a significant
contribution to the calculation of $Q_{\beta\beta}$ for a given
0$\nu\beta\beta$ decay process using Eq. 7. Therefore, $\Delta E_{b}$ cannot
be ignored.

The decay Q value used by the current experiments gerda ; mjd ; exo ; cuore
is just taken as the mass-energy difference of the initial and the final state
atoms in the neutral form. This refers to a decay of the following form:

$^{A}_{Z}X\longrightarrow^{A}_{Z+2}X^{{}^{\prime}},$ (17)

where ${}^{A}_{Z}X$ and ${}^{A}_{Z+2}X^{{}^{\prime}}$ are the neutral atoms
from the initial and the final states. This does not represent
0$\nu\beta\beta$ decay, which is expected to have two beta particles present
in the final state. Therefore, the decay Q value derived from this decay form,
$Q_{\beta\beta}$ = $M(^{A}_{Z}X)c^{2}$ \- $M(^{A}_{Z+2}X^{{}^{\prime}})c^{2}$,
is not the decay Q value for 0$\nu\beta\beta$ decay. From Eq. 7, one obtains
the following:

$\Delta Mc^{2}=Q_{\beta\beta}+\Delta E_{b}.$ (18)

$\Delta Mc^{2}$ is the sum of the decay Q value for 0$\nu\beta\beta$ decay
($Q_{\beta\beta}$) and the relaxation energy ($\Delta E_{b}$) of the initial
atomic structure after the decay. It is critical to distinguish the decay Q
value, which is the available energy for the two ejected beta particles in the
final state for 0$\nu\beta\beta$ decay, and the relaxation energy of the
initial atomic structure. The former is the experimental signature for
0$\nu\beta\beta$ decay while the latter is the release of the atomic binding
energy difference after the decay. The release of the atomic binding energy
difference depends on the relaxation time of the atomic structure after the
decay. This relaxation time is not well understood for 0$\nu\beta\beta$ decay
since it has not been observed yet. The atomic relaxation time in Rydberg
blockaded-$\Lambda$ atoms through spontaneous decay qiao  may shed light on
the atomic relaxation time of 0$\nu\beta\beta$ decay. In the recent paper
written by Qiao et al. qiao , the atomic relaxation time is calculated using
the atomic number and the decay rate as below:

$t_{r}=\frac{2Z}{\gamma_{rd}}\sqrt[2Z]{2\pi Z}(1-\frac{2}{\sqrt{Z}}),$ (19)

where $t_{r}$ is the atomic relaxation time, $\gamma_{rd}$ is the decay rate.
If one uses $Z$ (the atomic number) and $\gamma_{rd}$ (the decay rate) for
0$\nu\beta\beta$ decay, the relaxation time of the atomic orbits can be very
long. In particular, the lifetime of the double-ionized daughter ion in the
final state can be very long. This is because the thermalized two ejected beta
particles cannot be easily captured by the double-ionized daughter ion due to
the repulsive Coulomb forces from large numbers of electrons surrounding the
daughter nucleus. These repulsive forces provide a stabilizing energy balance
between the two thermalized electrons and the positively charged daughter
nucleus. An indicator of long-lived ions from $\beta$ decay is a good example
of the above theory. The 42K ions, which are from the $\beta$ decay of 42Ar,
produced uniformly throughout the Ar volume, are transported near the
detectors by convective flow in the Gerda detector gerda , where ion
collection onto the detector surfaces is aided by the stray electric fields
from the biased detector surface and any unshielded high voltage components.
The observed enhancement of $\gamma$ rays from the $\beta$ decay of 42K ion in
the Gerda experiment indicates that the lifetime of 42K ions can be very long.
Thus, it is reasonable to assume that the release of the atomic binding energy
after the 0$\nu\beta\beta$ decay may not come within the detection time window
for the energy deposition from the two ejected beta particles.

Even if the release of the atomic binding after the 0$\nu\beta\beta$ decay is
within the detection time window of the two ejected beta particles, the full
relaxation of the atomic orbital structure is a multistep process, which
results in a cascade of atomic radiation. Depending on the atomic number, the
electron orbital configuration involved, and electron shells, the energy of
emitted X-rays and/or Auger electrons is typically in the range from a few eV
to a couple of keV. Such low energy X-rays or Auger electrons have a short
range (nm to $\mu$m) to lose all energies. Therefore, the linear energy
transfer (LET) is very high. A high LET means that the majority of energy loss
is converted into the production of phonons. Depending on the detection
technology, there is a chance that the majority of the atomic energy released
by atomic radiation may not be detected.

In any case, for calculating $Q_{\beta\beta}$ for 0$\nu\beta\beta$ decay, the
difference in the total atomic binding energy between the mother and daughter
neutral atoms must be taken into account because it is in the level of a few
keV to more than 10 keV as also calculated in our earlier publication meiwei .
This difference in the total atomic binding energy is larger than the
detection energy window, which is so called the energy region of interest
(ROI), used for several 0$\nu\beta\beta$ decay experiments such as Gerda gerda
, Majorana Demonstrator mjd , LEGEND legend , CUORE cuore  and CUPID cupid . A
narrow energy window can be used by these experiments because they possess
excellent energy resolution. A narrow energy window allows these experiments
to keep background events especially the 2$\nu\beta\beta$ decay events out of
the ROI and thus have the potential to discover the 0$\nu\beta\beta$ decay
process. However, based on the discussions in this work and the previous
publication meiwei , the size of a narrow energy window, which can be used to
search for 0$\nu\beta\beta$ decay, should be evaluated using the atomic
binding energy difference between the mother and daughter atoms in the neutral
form. Otherwise, 0$\nu\beta\beta$ decay experiments could miss the decay
signature because the real decay Q value is out of the ROI, which is
determined using $\Delta Mc^{2}$ used by the current experiments. This
indicates that the study of the cascade of the atomic binding energy and its
relaxation time becomes important for 0$\nu\beta\beta$ decay.

The currently used decay Q value is the atomic mass difference of the neutral
atoms from the initial and the final states. The PENNING-trap technique has
been used to measure directly the atomic mass differences between the neutral
atoms eron  in the initial and the final states. These measured values are
4267.98 keV for 48Ca $\rightarrow^{48}$Ti kwia , 2039.04 keV for 76Ge
$\rightarrow^{76}$Se raha , 2997.73 keV for 82Se $\rightarrow^{82}$Kr linc ,
3355.85 keV for 96Zr $\rightarrow^{96}$Mo guly , 3034.40 keV for 100Mo
$\rightarrow^{100}$Ru raha , 2813.50 keV for 116Cd $\rightarrow^{116}$Sn raha1
, 2527.01 keV for 130Te $\rightarrow^{130}$Xe raha1 , 2457.83 keV for 136Xe
$\rightarrow^{136}$Ba reds , and 3367.72 keV for 150Nd $\rightarrow^{150}$Sm
kolh , respectively. These values should be corrected for the total difference
in the atomic binding energy between the initial and the final neutral atoms
as shown in Figures 1, 2, and 3.

Note that the decay Q-value can be measured using 2$\nu\beta\beta$ decay
spectra as conducted by the NEMO collaboration for 100Mo in R. Arnold et al.
arno . However, the spectra measured using the calorimeter technique should
correct for the energy loss of the emitted electrons in order to determine the
end-point energy for the 2$\nu\beta\beta$ decay process. Thus, this end-point
is the decay Q-value for 0$\nu\beta\beta$.

In conclusion, we derive the decay Q value for 0$\nu\beta\beta$ decay in the
forms of the parent nucleus decay into the daughter nucleus with the two
ejected beta particles in the final state and the parent neutral atom decay
into the double-ionized daughter ion with the two ejected beta particles in
the final state, respectively. We show that the decay Q value is the same in
both cases. If one uses the atomic masses of the mother and daughter atoms in
the neutral form to calculate the decay Q value, $Q_{\beta\beta}$ = $\Delta
Mc^{2}$ \- $\Delta E_{b}$, which is different from the current value ($\Delta
Mc^{2}$) used in the experiments. For example, this difference ($\Delta
E_{b}$) can be as large as 8.97 keV for 0$\nu\beta\beta$ decay in 76Ge, 17.73
keV for 0$\nu\beta\beta$ in 130Te, and 18.74 keV for 0$\nu\beta\beta$ in
136Xe, respectively, as shown in Figures 1, 2, and 3. We emphasize that
$\Delta Mc^{2}$ is not the decay Q value for 0$\nu\beta\beta$ decay and is the
sum of the decay Q value and the relaxation energy of the total atomic binding
energy difference between the mother and daughter atoms in the neutral form.
This atomic relaxation after the decay may not come within the detection time
window depending on the atomic relaxation time, which is not well understood
for 0$\nu\beta\beta$ decay since it has not been observed yet. Therefore, it
is worth pointing out that the current and planned 0$\nu\beta\beta$ decay
experiments may need to re-evaluate the ROI, which should be larger than the
current ROI, as we suggested in the previous publication meiwei .

This work was supported in part by NSF OISE 1743790, DE-SC0004768, and a
governor’s research center supported by the State of South Dakota. The authors
would like to thank Dr. Christina Keller for a careful reading of the
manuscript.

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