# User Localization with HRIS and Backscatter Modulation for Next-Generation
Networks

Mattia Piana and Stefano Tomasin
Dept. of Information Engineering, University of Padova, Italy
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>This work was
partially supported by the European Union under the Italian National Recovery
and Resilience Plan (NRRP) of NextGenerationEU, partnership on
“Telecommunications of the Future” (PE0000001 - program “RESTART”).

###### Abstract

Hybrid reflective intelligent surfaces (HRISs) can support localization in
sixth-generation (6G) networks thanks to their ability to generate narrow
beams and at the same time receive and process locally the impinging signals.
In this paper, we propose a novel protocol for user localization in a network
with an HRIS. The protocol includes two steps. In the first step, the HRIS
operates in full absorption mode and the user equipment (UE) transmits a
signal that is locally processed at the HRIS to estimate the angle of arrival
(AoA). In the second step, the base station transmits a downlink reference
signal to the UE, and the HRIS superimposes a message by a backscatter
modulation. The message contains information on the previously estimated AoA.
Lastly, the UE, knowing the position of the HRIS, estimates the time of flight
(ToF) from the signal of the second step and demodulates the information on
the AoA to obtain an estimate of its location. Numerical results confirm the
effectiveness of the proposed solution, also in comparison with the Cramér Rao
lower bound on the estimated quantities.

##  I Introduction

Accurate user equipment (UE) localization is a key target for next-generation
cellular networks [1] to provide new services to the UE. This target can be
facilitated by new technologies that will be deployed in sixth-generation (6G)
networks. In particular, reconfigurable intelligent surfaces (RISs) are seen
as a transformative technology that can control the physical propagation
environment in which they are embedded by passively reflecting radio waves in
preferred directions [2]. UE localization and sensing are also among the
applications that can benefit from RIS deployments. In particular, integrated
sensing and communications (ISAC) systems can benefit from RISs as shaping the
wireless environment can increase the target’s illumination and facilitate its
detection by the UEs in radar applications [3].

In this paper, we focus on the localization of the UE at the UE itself.
Localization at the UE in RIS-aided scenarios was studied for base stations
(BSs) and UEs with single antennas in [4], for BSs with multiple antennas and
UEs with single antennas in [5], and for both BS and UE with multiple antennas
in [6]. The localization of UE by the BS in a no line of sight (nLOS) scenario
is particularly challenging due to the interaction of the RIS with the channel
and the limited observation (channel matrix size) of the cascade channel
through the RIS [7].

While RISs are equipped with a large number of elements and can create narrow
electromagnetic beams, the channel seen by the RISs (and the entailed
information on UE location) remains largely unknown to both the BS and the UE,
as they are equipped with much fewer antennas. To overcome this issue, a new
technological solution is provided by hybrid RISs (HRISs) that, beyond
reflecting, also capture part of the impinging signal for local processing
[8]. Early works have started exploring the use of HRISs for localization. In
[9] the UE and HRIS positions are jointly estimated by the UE while in [10]
both BS-HRIS and HRIS-UE channels are estimated by the BS, as well as the
angle of arrival at the HRIS. In both works, the HRIS actively shares with the
BS its locally gathered information to further process it, via a control link.
It can be argued that while sharing the local information with the BS is easy,
as an infrastructure connecting the HRIS to the BS is expected to be deployed
(see for example [9]), sharing the local information with the UE is not
straightforward.

In this paper, we propose a novel localization technique by the UE, which
exploits the sensing capability of an HRIS. In our scheme, the UE transmits
signals to the HRIS that operates in full absorption mode, i.e., the received
signal is not reflected but fully processed locally to estimate the angle of
arrival (AoA) at the HRIS. Then, after a preset time, the BS transmits a
downlink signal through the HRIS that enables the UE to estimate the time of
flight (ToF) from the HRIS. Moreover, in this second phase, the HRIS transmits
to the UE the information on the AoA estimated in the first phase by a
backscattered modulation superimposed on the signal transmitted by the BS. The
UE, knowing the location of the HRIS, can then estimate its position. We
denote the resulting localization protocol as localization with HRIS and
backscattered signal (LHBS). LHBS exploits the control channel between the
HRIS and the BS only to trigger the transmission of the downlink reference
signal in the second phase. Lastly, the assumption that the UE knows the
position of the HRIS is also reasonable since the HRIS is fixed and is part of
the network infrastructure, thus being equivalent to that of knowing the
position of BSs.

The rest of this work is organized as follows: Section II introduces the
System model. The proposed LHBS protocol is described in Section III. The
performance of the proposed solution is investigated in Section IV, where we
also provide a comparison with the Cramér-Rao lower bound on the AoA, the
range, and the position estimators. Lastly, in Section V we draw the main
conclusions.

##  II System Model

Figure 1: Reference scenario: an UE at position $\bm{p}$ localizes itself
using an HRIS at position $\bm{q}$ and a BS in a nLOS scenario. The AoA and
AoD to and from the HRIS are $\theta_{\rm BH}$ and $\phi_{\rm HU}$,
respectively. The AoD from the BS is $\phi_{\rm BH}$.

We consider the scenario of Fig. 1 in which a UE estimates its position in a
cellular network assisted by a single BS. The estimation is assisted by an
HRIS controlled by the BS.

The BS is located at the origin of our reference system. The HRIS is located
at position $\bm{q}=[q_{x},q_{y}]$, which is known to the UE. The UE aims at
estimating its position $\bm{p}=[p_{x},p_{y}]$.

The BS is equipped with a uniform linear array (ULA) with $M$ antennas, while
the UE is equipped with a single antenna. The HRIS comprises $R$ evenly spaced
elements organized along a line, each receiving and reflecting signals as an
omnidirectional antenna while introducing a configurable phase shift. The HRIS
is hybrid because it can also process impinging signals locally. The
coefficient $\rho\in[0,1]$ represents the fraction of the signal power that is
reflected, while a fraction $(1-\rho)$ of the signal power is absorbed for
local processing purposes by each HRIS element. We assume that the HRIS has a
direct/control link (e.g., via optical fiber) with the BS but it is not
equipped with a transmitter. Consequently, the HRIS cannot directly
communicate with the UE but it can only process locally the impinging signals
and change its element phase shifts. We define two extreme operative modes of
the HRIS, namely:

  * •

_Full Absorption Mode_ : in this case $\rho=0$, meaning that the whole
incoming signal is absorbed by the HRIS for local processing.

  * •

_Fully Reflective Mode_ : in this case $\rho=1$, so the whole incoming signal
is reflected by the HRIS.

We consider a scenario without strict synchronization between the BS, the
HRIS, and the UE. In the following, time is referred to the system of the UE,
without loss of generality. We denote with $\epsilon$ the normalized (to
$\lambda/(2\pi c)$) time offset between the UE and the HRIS or BS clocks,
where $\lambda$ is the wavelength, and $c$ is the speed of light in air.

###  II-A Channel Model

#### Geometrical Properties of Signal Propagation

Considering a transmission at mmWaves, we assume that each channel is
characterized by a single path. The AoA and angle of departure (AoD) to and
from the HRIS are denoted as $\theta_{\rm BH}$ and $\phi_{\rm HU}$,
respectively. The AoD from the BS is $\phi_{\rm BH}$. We define the ToF
between the HRIS and the UE as $\tau_{\rm HU}$, consequently the distance
between UE and HRIS is

$r_{\rm HU}=c\tau_{\rm HU}.$ (1)

Similarly, defining the ToF between the HRIS and the BS as $\tau_{\rm HB}$,
the distance between BS and HRIS is $r_{\rm HB}=c\tau_{\rm HB}$. Let $d_{\rm
tot}=r_{\rm HU}+r_{\rm HB}$ be the total distance of the downlink path, i.e.,
the distance that the signal sent from the BS and reflected from the HRIS
travels to reach the UE. We denote as $\tau=d_{\rm tot}/c$ the ToF between the
BS and the UE.

#### Baseband Channel Model

The free-space path loss between the UE and the HRIS is
$\gamma=\left(\frac{\lambda}{4\pi r_{\rm HU}}\right)^{2}$. We define the
channel coefficient from the UE to the HRIS as $\xi_{\rm
HU}=\sqrt{\gamma}e^{-j\left(\epsilon+\frac{2\pi r_{\rm HU}}{\lambda}\right)}$,
including the effect of path loss, group delay between the UE and the HRIS,
and clock differences. Similarly, we define $\xi_{\rm BH}$ as the channel
coefficient from the BS and the HRIS similarly defined (using $r_{\rm hb}$).
Lastly, we define the steering vector operator at the receiver as

$\bm{a}_{A}(\theta)=[1,e^{-j\kappa\sin{\theta}},\ldots,e^{-j\kappa\sin{\theta}(A-1)}]^{T},$
(2)

where $\kappa=2\pi d/\lambda$, $d$ is the spacing between two adjacent
antennas (or HRIS elements), $A\in\\{M,R\\}$ is the number of antennas (or
elements), and $\theta$ represents the AoA. The baseband single-path channel
matrix from the BS to the HRIS is therefore

$\bm{H}_{\rm BH}=\frac{1}{\sqrt{M}}\xi_{\rm BH}\,\bm{a}_{R}(\theta_{\rm
BH})\bm{a}_{M}^{H}(\phi_{\rm BH})\in\mathbb{C}^{R\times M}.$ (3)

The column-vector channel from the UE to the HRIS is

$\bm{h}_{\rm UH}=\xi_{\rm UH}\,\bm{a}_{R}(\phi_{\rm HU})\in\mathbb{C}^{R\times
1}.$ (4)

The BS-HRIS channel $\bm{H}_{\rm BH}$ is assumed to be known both at the BS
and at the HRIS as it is fully determined by their positions. Let $\bm{b}_{\rm
BS}\in\mathbb{C}^{M}$ be the beamforming vector at the BS.

#### HRIS Model

We characterize the effect of the HRIS by the diagonal matrix
$\bm{\Omega}\in\mathbb{C}^{R\times R}$, where each element introduces the
following gain and phase shift

$\bm{\Omega}_{ii}=\sqrt{\rho}e^{j\alpha_{i}},\quad\alpha_{i}\in[0,2\pi),\;i=0,\ldots,R-1.$
(5)

We assume that the line of sight (LOS) between the UE and the BS is blocked,
so the communication is assisted by an HRIS.

##  III Localization With HRIS
and Backscattered Signal

In this section, we describe the localization with HRIS and backscattered
signal (LHBS) at the UE. Our approach is based on the estimation of two
quantities: the AoD at the HRIS $\phi_{\rm HU}$ and the ToF $\tau_{\rm HU}$.
From this information, the UE will estimate its position. Our algorithm is
divided into three phases, which are first briefly outlined and then explained
in detail:

  * •

Phase 1- _AoA estimation at the HRIS_ : the single-antenna UE transmits a
pilot signal. The HRIS, in full absorption mode, detects the signal coming
from the UE and estimates its AoA. When the HRIS detects the UE pilot signal,
it notifies the BS of the reception via the control link.

  * •

Phase 2- _Acknowledgment_ : After a pre-agreed time, the BS sequentially
transmits two pilot signals that are fully reflected through the HRIS to the
UE. In this phase, the HRIS sets its phases according to the estimated AoA in
the first phase to steer the BS signal towards the UE. Moreover, the reflected
signals are modulated by the HRIS (in backscattering) to convey the
information regarding $\phi_{\rm HU}$.

  * •

Phase 3- _UE position estimation_ : when the UE receives the reflected
signals, it demodulates the AoD $\phi_{\rm HU}$ and estimates the ToF
$\tau_{\rm HU}$. With this information, it finally estimates its position.

The timing scheme of the LHBS protocol is summarized in Fig. 2. Note that the
processing times are ignored but their inclusion in our model is
straightforward.

Figure 2: Timing scheme of the LHBS protocol: $\tau_{c}$ is the control-link
propagation time, $\tau_{\rm HU}$ is the ToF between UE and HRIS, and
$\tau_{\rm BH}$ is the ToF between BS and HRIS. Processing times are ignored.

###  III-A AoA Estimation at the HRIS

The LHBS procedure starts (at time $t=0$, UE time) with the UE transmitting
the reference sequence. This sequence is used at the HRIS to detect the
impinging signal by correlating it with a local replica and to estimate the
AoA from the UE. We assume here that the HRIS operates in full absorption
mode. When the HRIS detects the signal, it notifies the BS of the reception
via the control link, the latter will use this information in the next phase.

In detail, the UE transmits a reference signal of $N$ symbols
$\bm{x}=[x(0),\ldots,x(N-1)]^{T}$, using a unit-power square root raised
cosine (SRRC) pulse $q(t)$, with symbol period $T_{c}$. Without loss of
generality, we assume $N$ to be even. We use a constant-amplitude zero-
autocorrelation (CAZAC) sequence [11], where

$x_{n}=e^{j\pi\frac{n(n+1)}{N}},\quad n=0,\ldots,N-1.$ (6)

This specific sequence, having good autocorrelation properties, allows for
precisely detecting the time of arrival of the signal at the HRIS.

At the HRIS, the received signal is filtered with a filter with impulse
response $q(-t)$ and sampled at period $T_{c}$ (with a sample falling in
correspondence with the peak of the convolution between $q(t)$ and $q(-t)$),
providing

$\bm{y}_{\rm H}(n+k^{\prime})=\bm{h}_{\rm UH}x_{n}+\bm{w}_{\rm
H}(n+k^{\prime})\in\mathbb{C}^{R\times 1},$ (7)

where entries of noise vector $\bm{w}_{\rm H}(n)$ are independent and
identically distributed as $\mathcal{CN}(0,\,\sigma^{2})$, and $k^{\prime}$ is
a suitable delay due to both propagation and sampling delays. We notice that
(7) is a typical signal model for the multiple signal classification (MUSIC)
algorithm [12, Sec. 8.6.3] that provides the estimate $\hat{\phi}_{\rm HU}$ of
the AoA. Hence, the HRIS estimates the AoA.

###  III-B Acknowledgment

In the second phase of LHBS, the HRIS modulates the information on the
estimated AoA by backscattering on a pilot signal transmitted by the BS. The
modulated signal also enables the UE to estimate the ToF from the HRIS to the
UE. In detail, the backscatter transmission is obtained by the modulation of
the incoming signal with two pilot sequences by the HRIS, which will enable
the demodulation of the backscattered information by differential
demodulation. This solution avoids that the UE needs to know in advance the
channel from the BS to the UE.

In our protocol, also the BS will transmit a stream of symbols
$x_{n}\in\mathbb{C}$ using an SRRC pulse $q(t)$ with symbol period $T_{c}$. In
this phase, the HRIS works in fully reflective mode. The BS aligns its
beamformers to the HRIS direction by setting

$\bm{b}_{\rm BS}=\frac{1}{\sqrt{M}}\bm{a}_{M}(\phi_{\rm BH}).$ (8)

#### First Transmission

In transmitting the first acknowledgment signal, the HRIS will convey the
information on the AoA by a backscatter modulation. In particular, in this
phase, the HRIS will use this information to set its phase shifters as

$\bm{\Omega}_{ii}=e^{jk[\sin{\hat{\phi}_{\rm HU}}-\sin{\theta_{\rm
BH}}]i}e^{j\hat{\phi}_{\rm HU}}.$ (9)

The timing of the first transmission is such the HRIS receives the first
sample at time $t_{1}=\kappa_{1}T_{c}$, so time $T$ is elapsed from the
reception at the HRIS of the UE signal of phase 1. Using (8) and (9) we obtain
(see Fig. 2)

$\begin{split}r_{\rm UE}(t)=&R\xi_{\rm BH}\xi_{\rm UH}e^{j\hat{\phi}_{\rm
HU}}\times\\\ &\times\sum_{n=0}^{N-1}x_{n}q(t-nT_{c}-2\tau_{\rm HU}-T)+\\\
&+w^{\prime}(t),\end{split}$ (10)

where $w^{\prime}(t)$ is the AWGN and both AoA and ToF appear. This model will
be used in Section III-D to assess the AoA and range estimator performance.
Then the UE filters the received signal by $q(-t)$ and upsamples the filter
output with over-sampling period $T_{s}=\frac{T_{c}}{u_{\rm f}}$, where
$u_{\rm f}$ is the oversampling factor. Let us define

$s_{i}=R\xi_{\rm BH}\xi_{\rm UH}e^{j\hat{\phi}_{\rm
HU}}\sum_{n=0}^{N-1}x_{n}\tilde{q}(iT_{s}-nT_{c}-2\tau_{\rm HU}-T-\tau_{1}),$
(11)

for $i=0,\ldots,Nu_{\rm f}-1$, where $\tilde{q}(t)$ is the convolution between
$q(t)$ and $q(-t)$, and $\tau_{1}$ is a suitable delay so that the maximum of
the convolution is one of the samples of $s_{i}$ and all non-zero terms fall
in the considered window of $i$. The oversampled signal model is

$\bm{r}=\bm{s}+\bm{w},$ (12)

where entries of $\bm{r}$ are $r_{i}=s_{i}+w(iT_{s})$ and
$w(iT_{s})\sim\mathcal{CN}(0,\,\sigma^{2}_{\rm u})$ for $i=0,...,Nu_{\rm
f}-1$. Note that, $\sigma^{2}_{\rm u}=u_{\rm f}\sigma^{2}$ as up sampling the
received signal also increases the noise variance. Recalling (1), we notice
that $\bm{r}\sim\mathcal{CN}(\bm{s}(\hat{\phi}_{\rm HU},r_{\rm
HU}),\,\sigma^{2}_{\rm u}\bm{I}_{Nu_{\rm f}})$, i.e., the range and AoA
information is contained in the mean of a Gaussian vector.

Note that we could obtain the same goal via digital modulation of the HRIS
elements. As an example, in [13] a passive (non-hybrid) RIS is exploited and
the state of its elements (on/off) is used to convey its local information via
a spatial modulation scheme.

The estimation of the AoA at the HRIS in phase 1 is crucial, as it will affect
the modulation that HRIS will apply in phase 2. Since the path loss of the
signal received by the HRIS when estimating the UE AoA is much lower than the
one that affects the BS-HRIS-UE path, we can take the estimation at the HRIS
as perfect. In other words, due to the large number of HRIS elements and the
mmWave frequencies, if we want our algorithm to work then the estimation error
at the HRIS can be made much smaller with respect to the end-to-end one.
Hence, the bottleneck of the system is the estimation at the UE of the AoA
with the signal coming from the BS and reflected by the HRIS.

#### Second Transmission

In the transmission of the second acknowledgment signal, the HRIS optimally
aligns the channel between BS and UE and creates a virtual LOS between the
two. To do that, the HRIS sets its phase shifters as

$\bm{\Omega}_{ii}=e^{jk[\sin{\hat{\phi}_{\rm HU}}-\sin{\theta_{\rm BH}}]i}.$
(13)

In this case, no additional modulation is added by the HRIS. The second
transmission starts immediately after the first and the received discrete-time
signal at the UE (after filtering and sampling) can be written similarly to
(12) as

$\bm{r}^{\prime}=\bm{s}^{\prime}+\bm{w}^{\prime},$ (14)

where $\bm{w}^{\prime}$ is the AWGN vector and entries of $\bm{s}^{\prime}$
are

$s_{i}^{\prime}=R\xi_{\rm BH}\xi_{\rm
UH}\sum_{n=0}^{N-1}x_{n}\tilde{q}(iT_{s}-nT_{c}-2\tau_{\rm
HU}-T-\tau_{1}-NT_{c}),$ (15)

for $i=0,\ldots,Nu_{\rm f}-1$.

The UE demodulates $\hat{\phi}_{\rm HU}$ by aligning in time the two signals,
and then combining them as follows

$\begin{split}z&=\frac{1}{Nu_{\rm f}}\sum_{n=0}^{Nu_{\rm
f}-1}r_{n}^{*}r^{\prime}_{n}\\\ &\simeq||\xi_{\rm BH}\xi_{\rm
UH}||^{2}R^{2}e^{-j\hat{\phi}_{\rm HU}}.\end{split}$ (16)

The approximation in (16) arises from the large number theorem as we
substituted the average with the expectation operator when $N$ is sufficiently
large. The UE can now obtain the analog demodulated AoA at the HRIS as

$\dot{\phi}_{\rm HU}=-\angle{z}.$ (17)

Note that in both transmissions we use a CAZAC sequence of length $N$. In
principle, we could use a general pilot sequence not necessarily known at the
UE to estimate the AoA, as (16) holds when we use the same sequence in both
transmissions. Still, we will use these two sequences also in Section III-C
for the range estimation.

###  III-C UE Position Estimation

In the third phase of LHBS, the UE computes its position. In particular, the
UE knows the time of its transmission (time zero) and also the fixed time $T$.
Consequently, the UE can extract the ToF $\tau_{\rm HU}$ from both the
received signals of the first and second transmission by the BS, and then take
the average. Note the estimate of the dalay cannot be done on the two
sequences together, since they have an unknown phase shift, due to backscatter
modulation.

In detail, the UE correlates the over-sampled signal with a local replica of
the over-sampled transmitted sequence. The relation between the transmitting
and arrival times at the UE is (e.g., for the first sequence)

$t_{\rm RX}=2\tau_{\rm HU}+T,$ (18)

and so a first estimate of the ToF at the UE is

$\tilde{\tau}_{\rm HU}=\frac{t_{\rm RX}-T}{2}.$ (19)

Similarly, we obtain the second estimation $\tilde{\tau}^{\prime}_{\rm HU}$
from the second transmission. Lastly, the estimation of the range at the UE is
obtained by computing

$\dot{r}_{\rm HU}=c\left(\frac{\tilde{\tau}_{\rm
HU}+\tilde{\tau}^{\prime}_{\rm HU}}{2}\right).$ (20)

Note that estimating the range in this way relies only on the absolute times
at the UE and is not affected by clock misalignment between UE and BS.

With the AoA and range estimations $\dot{\phi}_{\rm HU}$ and $\dot{r}_{\rm
HU}$, and the known HRIS position $\bm{q}$, the UE estimates its position
$\dot{\bm{p}}$. In particular, defining $\bm{\theta}=[\phi_{\rm HU},r_{\rm
HU}]^{T}$, we have

$\left[\begin{matrix}\dot{p}_{x}\\\
\dot{p}_{y}\end{matrix}\right]=\bm{g}(\dot{\bm{\theta}})=\bm{q}+\left[\begin{matrix}\,\,\,\,\dot{r}_{\rm
HU}\cos{\dot{\phi}_{\rm HU}}\\\ -\dot{r}_{\rm HU}\sin{\dot{\phi}_{\rm
HU}}\end{matrix}\right].$ (21)

###  III-D Cramér-Rao Lower Bounds

Figure 3: RMSE and $\sqrt{CRLB}$ of the ranging estimate vs the SNR for a
sequence length $N=100$ or 150. Figure 4: RMSE and $\sqrt{CRLB}$ of the AoA
estimate vs the SNR for a sequence length $N=100$ or 150.

To assess the performance of the LHBS algorithm, we compared the MSE of the
range $r_{\rm HU}$ and the demodulated AoA $\dot{\phi}_{\rm HU}$ with their
Cramér Rao lower bounds (CRLB) that give a lower bound on the error variance
of all the possible unbiased estimators of the parameters. Due to space
limitations, the algebraic operations to compute the quantities in the
following can be found in the Appendix.

#### CRLB of the Received AoA

We assume that the estimate of the AoA at the HRIS is without error, while we
take into account the error introduced in the transmission from the BS to the
UE through the HRIS. From [14, Sec. 15.7] we compute

$\begin{split}\frac{\partial^{2}\,\ln\,p(\bm{r}|\hat{\phi}_{\rm HU},r_{\rm
HU})}{\partial^{2}\hat{\phi}_{\rm HU}}&=\frac{2}{\sigma^{2}_{\rm
u}}\frac{\partial}{\partial\hat{\phi}_{\rm
HU}}\Re\left[(\bm{r}-\bm{s}(\hat{\phi}_{\rm HU},r_{\rm
HU}))^{H}\times\right.\\\ &\times\left.\frac{\partial\,\bm{s}(\hat{\phi}_{\rm
HU},r_{\rm HU})}{\partial\hat{\phi}_{\rm HU}}\right]=\frac{2}{\sigma^{2}_{\rm
u}}\Re[-\bm{r}^{H}\bm{s}].\end{split}$ (22)

The CRLB on the AoA is then

$\begin{split}CRLB_{\phi_{\rm
HU}}&=-\left[\mathbb{E}\left(\frac{\partial^{2}\,\ln\,p(\bm{r}|\hat{\phi}_{\rm
HU},r_{\rm HU})}{\partial^{2}\hat{\phi}_{\rm
HU}}\right)\right]^{-1}=\frac{\sigma^{2}_{\rm
u}}{2||\bm{s}||^{2}}.\end{split}$ (23)

#### CRLB of the Ranging Estimation at the UE

We also compute the CRLB of the estimate of the ranging using the results of
[15, Sec. 2.4.6] that provide

$CRLB_{r_{\rm HU}}=\frac{c^{2}}{8\cdot 2B_{2}^{2}\frac{E_{s}}{N_{0}}},$ (24)

where $E_{s}$ is the total energy used for the correlation, $B_{2}$ is the
root mean square bandwidth, and $N_{0}=\sigma^{2}_{\rm u}T_{s}$. The
coefficient 8 at the denominator of (24) is due to our use of two sequences to
perform the estimation of the ToF and both of them estimate $2\tau_{\rm HU}$.

#### CRLB of the Position Estimation

To obtain the CRLB of the position estimation we first compute the Fisher
information matrix (FIM) on the AoA and range. In detail, the FIM is

$\bm{I}=\begin{bmatrix}CRLB_{\phi_{\rm HU}}^{-1}&\alpha\\\ \alpha&CRLB_{r_{\rm
HU}}^{-1}\end{bmatrix}.$ (25)

where

$\begin{split}\alpha&=-\mathbb{E}\left(\frac{\partial^{2}\,\ln\,p(\bm{r}|\hat{\phi}_{\rm
HU},r_{\rm HU})}{\partial\hat{\phi}_{\rm HU}\partial r_{\rm HU}}\right)\\\
&=-\frac{2}{\sigma^{2}_{\rm
u}}\Im\left[\bm{s}^{H}\frac{\partial\bm{s}}{\partial r_{\rm
HU}}\right].\end{split}$ (26)

From [14, Sec. 3.8] we obtain the CRLB on the position estimation at the UE as

$CRLB_{\rm pos}={\rm
tr}\left[\frac{\partial\bm{g}(\bm{\theta})}{\partial\bm{\theta}}\bm{I}^{-1}\frac{\partial\bm{g}(\bm{\theta})^{T}}{\partial\bm{\theta}}\right].$
(27)

Figure 5: RMSE or $\sqrt{CRLB}$ of the position estimation vs SNR for a
sequence length of $N=100$ or 150.

##  IV Numerical Results

In this section, we assess the performance of the proposed LHBS protocol and
compare it with the obtained CRLBs. The transmit power is
$\mathbb{E}(||x_{n}||^{2})=1$, the carrier frequency is $f_{c}=25$ GHz, the
distance HRIS-UE is $r_{\rm HU}=100$ m, the distance BS-HRIS is $r_{\rm
BH}=100$ m, the sequence length is $N=100$ or $150$, the signal bandwidth is
$B=20$ MHz, the roll-off factor is $\beta=0.8$, and the up sampling factor is
$u_{\rm f}=10$.

We define the signal-to-noise ratio (SNR) on the received signal as

$\mathrm{SNR}=\frac{||\xi_{\rm BH}\xi_{\rm UH}||^{2}R^{2}}{\sigma^{2}}.$ (28)

Fig. 3 shows the root mean square error (RMSE) of the ranging $r_{\rm HU}$ as
a function of the SNR. We also plot the corresponding square root of CRLB
(24). We observe that the RMSE of the ranging error is in the range of tens of
centimeters and for high SNR values the performance of the estimator gets
close to the CRLB. Also, we observe that increasing the sequence length ($N$)
provides a performance improvement. We also observe that the improvement
saturates at high SNR: this is due to the time quantization. Upsampling even
more the received signal would lower the saturation level.

Fig. 4 shows the RMSE of the AoA estimated at the HRIS as a function of the
SNR, again in a comparison with the square root of the CRLB (23). We note that
in this case, the estimator achieves a performance that is closer to the CRLB.

Lastly, Fig. 5 shows the RMSE of the estimated UE position as a function of
the SNR. We observe that in the considered scenario we can achieve a
positioning precision in the range of meters.

##  V Conclusions

In this paper, we presented a novel protocol for HRIS-aided user localization.
We exploited the sensing property available at the HRIS to locally process the
incoming signal from the UE and estimate the AoA of the signal. The UE then
computes its position from two signals transmitted from the BS and reflected
by the HRIS. They encode by a differential backscatter modulation the AoA
information. We proved the effectiveness of our approach via simulation and
compared the performances with the Cramér Rao lower bounds.

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##  VI Appendix

###  VI-A CRLB of the Received AoA

In this section, the CRLB of the Received AoA will be derived.

$\begin{split}\frac{\partial^{2}\,\ln\,p(\bm{r}|\hat{\phi}_{\rm HU},r_{\rm
HU})}{\partial^{2}\hat{\phi}_{\rm HU}}&=\frac{2}{\sigma^{2}_{\rm
u}}\frac{\partial}{\partial\hat{\phi}_{\rm
HU}}\Re\left[(\bm{r}-\bm{s}(\hat{\phi}_{\rm HU},r_{\rm
HU}))^{H}\frac{\partial\,\bm{s}(\hat{\phi}_{\rm HU},r_{\rm
HU})}{\partial\hat{\phi}_{\rm HU}}\right]\\\ &=\frac{2}{\sigma^{2}_{\rm
u}}\frac{\partial}{\partial\hat{\phi}_{\rm
HU}}\Re\left[(\bm{r}^{H}-\bm{s}^{H}(\hat{\phi}_{\rm HU},r_{\rm
HU}))j\bm{s}(\hat{\phi}_{\rm HU},r_{\rm HU})\right]\\\
&=\frac{2}{\sigma^{2}_{\rm u}}\frac{\partial}{\partial\hat{\phi}_{\rm
HU}}\Re\left[j\,\bm{r}^{H}\bm{s}(\hat{\phi}_{\rm HU},r_{\rm HU})\right]\\\
&=\frac{2}{\sigma^{2}_{\rm u}}\Re[-\bm{r}^{H}\bm{s}(\hat{\phi}_{\rm HU},r_{\rm
HU})].\end{split}$ (29)

If we take the expectation of (29) we simply substitute $\bm{r}$ with
$\bm{s}(\hat{\phi}_{\rm HU},r_{\rm HU})$, and from that the CRLB immediately
follows.

###  VI-B Off-diagonal coefficient

In this section, the off-diagonal coefficient

$\alpha=-\mathbb{E}\left(\frac{\partial^{2}\,\ln\,p(\bm{r}|\hat{\phi}_{\rm
HU},r_{\rm HU})}{\partial\hat{\phi}_{\rm HU}\partial r_{\rm HU}}\right)$ (30)

is derived. We start by computing

$\frac{\partial^{2}\,\ln\,p(\bm{r}|\hat{\phi}_{\rm HU},r_{\rm
HU})}{\partial\hat{\phi}_{\rm HU}\partial r_{\rm HU}}=\frac{2}{\sigma^{2}_{\rm
u}}\frac{\partial}{\partial\hat{\phi}_{\rm
HU}}\Re\left[(\bm{r}-\bm{s}(\hat{\phi}_{\rm HU},r_{\rm
HU}))^{H}\frac{\partial\,\bm{s}(\hat{\phi}_{\rm HU},r_{\rm HU})}{\partial
r_{\rm HU}}\right].$ (31)

Knowing that $\partial r_{\rm HU}=c\,\partial\tau_{\rm HU}$ , we rewrite (31)
as

$\frac{\partial^{2}\,\ln\,p(\bm{r}|\hat{\phi}_{\rm HU},r_{\rm
HU})}{\partial\hat{\phi}_{\rm HU}\partial r_{\rm HU}}=\frac{2}{\sigma^{2}_{\rm
u}}\frac{\partial}{\partial\hat{\phi}_{\rm
HU}}\Re\left[(\bm{r}-\bm{s}(\hat{\phi}_{\rm HU},r_{\rm
HU}))^{H}\frac{\partial\,\bm{s}(\hat{\phi}_{\rm HU},r_{\rm
HU})}{c\,\partial\tau_{\rm HU}}\right].$ (32)

We compute $\check{\bm{s}}$ with entries

$\check{s}_{i}=\left[\frac{\partial\,\bm{s}(\hat{\phi}_{\rm HU},r_{\rm
HU})}{\,\partial\tau_{\rm HU}}\right]_{i}=-2R\xi_{\rm BH}\xi_{\rm
UH}e^{j\hat{\phi}_{\rm
HU}}\sum_{n=0}^{N-1}x_{n}\tilde{q}^{\prime}(iT_{s}-nT_{c}-2\tau_{\rm
HU}-T-\tau_{1}),$ (33)

where $\tilde{q}^{\prime}(\cdot)$ is the raised cosine derivative. We notice
that $\bm{s}(\hat{\phi}_{\rm HU},r_{\rm
HU})^{H}\frac{\partial\,\bm{s}(\hat{\phi}_{\rm HU},r_{\rm
HU})}{\,\partial\tau_{\rm HU}}$ does not depend on $\hat{\phi}_{\rm HU}$,
consequently its derivative with respect to $\hat{\phi}_{\rm HU}$ is zero.
Thus, we rewrite (32) as

$\begin{split}\frac{\partial^{2}\,\ln\,p(\bm{r}|\hat{\phi}_{\rm HU},r_{\rm
HU})}{\partial\hat{\phi}_{\rm HU}\partial r_{\rm
HU}}&=\frac{2}{\sigma^{2}_{\rm u}}\frac{\partial}{\partial\hat{\phi}_{\rm
HU}}\Re\left[\bm{r}^{H}\frac{\partial\,\bm{s}(\hat{\phi}_{\rm HU},r_{\rm
HU})}{c\,\partial\tau_{\rm HU}}\right]\\\ &=\frac{2}{\sigma^{2}_{\rm
u}}\Re\left[j\,\bm{r}^{H}\frac{\partial\,\bm{s}(\hat{\phi}_{\rm HU},r_{\rm
HU})}{c\,\partial\tau_{\rm HU}}\right].\end{split}$ (34)

Finally, we find

$\begin{split}\alpha&=-\mathbb{E}\left(\frac{\partial^{2}\,\ln\,p(\bm{r}|\hat{\phi}_{\rm
HU},r_{\rm HU})}{\partial\hat{\phi}_{\rm HU}\partial r_{\rm HU}}\right)\\\
&=-\frac{2}{\sigma^{2}_{\rm
u}}\Im\left[\bm{s}^{H}\check{\bm{s}}\right]=-\frac{2}{\sigma^{2}_{\rm
u}}\Im\left[\bm{s}^{H}\frac{\partial\bm{s}(\hat{\phi}_{\rm HU},r_{\rm
HU})}{\partial r_{\rm HU}}\right].\end{split}$ (35)


$\displaystyle\sup_{g\in\mathcal{G}}\frac{1}{kT}\cdot\sum_{t=1}^{T}\sum_{j=1}^{k}\left|g(Z_{t,j})-\pi\circ
g(Z_{t,j})\right|$
$\displaystyle\leq\varepsilon+2\cdot\frac{\mathfrak{R}_{kT}(\mathcal{G})}{kT}+2\cdot\sqrt{\frac{\log\left(\frac{1}{\delta}\right)}{kT}}$
(177) $\displaystyle\leq
2\varepsilon+2\cdot\sqrt{\frac{\log\left(\mathcal{N}(\mathcal{G},\varepsilon)\right)+\log\left(\frac{1}{\delta}\right)}{kT}},$
(178)

where the second inequality follows by standard bounds on Rademacher
complexity by covering numbers (e.g. Van Handel [2014, Corollary 5.25]). Thus,
as long as

$\displaystyle
v>6k\varepsilon+6\cdot\sqrt{\frac{k}{T}\left(\log\left(\frac{\mathcal{N}(\mathcal{G},\varepsilon)}{\delta}\right)\right)},$
(179)

the third term is bounded by $\delta$. The result follows. ∎

Finally, we prove a result akin to contraction, showing that if
$\mathcal{G}_{\widetilde{\delta}}$ is as in (152), then the covering number of
$\mathcal{G}_{\widetilde{\delta}}$ is upper bounded by the covering number of
$\mathcal{F}$.

######  Lemma 13.

Let $\mathcal{F}:\mathcal{X}\to[-1,1]$ be a function class with
$\left\|\cdot\right\|_{\nu}$ and $\left\|\cdot\right\|_{kT}$ as in Definition
7. For $\widetilde{\delta}>0$, let $\mathcal{G}_{\widetilde{\delta}}$ be as in
(152). Let $\mathcal{N}(\mathcal{F},\varepsilon)$ denote the covering number
of $\mathcal{F}$ at scale $\varepsilon$ with respect to
$\left\|\cdot\right\|_{\mu}$. Then for any $1\geq c,\widetilde{\delta}>0$, it
holds that

$\displaystyle\mathcal{N}\left(\mathcal{G}_{\widetilde{\delta}},\varepsilon\right)\leq\mathcal{N}\left(\mathcal{F},\frac{c\sigma\widetilde{\delta}^{2}}{8}\cdot\varepsilon\right).$
(180)

###### Proof.

We compute for any $X\in\mathcal{X}$, and any $f,f^{\prime}\in\mathcal{F}$,

$\displaystyle\left|\frac{f(X)}{\left\|f\right\|_{\nu}}-\frac{f^{\prime}(X)}{\left\|f^{\prime}\right\|_{\nu}}\right|$
$\displaystyle\leq\frac{\left|f(X)-f^{\prime}(X)\right|}{\left\|f\right\|_{\nu}}+\left|f^{\prime}(X)\right|\cdot\left|\frac{1}{\left\|f\right\|_{\nu}}-\frac{1}{\left\|f^{\prime}\right\|_{\nu}}\right|$
(181)
$\displaystyle\leq\frac{\left|f(X)-f^{\prime}(X)\right|}{\left\|f\right\|_{\nu}}+\frac{\left\|f-f^{\prime}\right\|_{\nu}}{\left\|f\right\|_{\nu}\cdot\left\|f^{\prime}\right\|_{\nu}},$
(182)

where the second inequality follows by the boundedness of $\mathcal{F}$ and
the triangle inequality. If $\left\|f\right\|_{\nu}\geq\widetilde{\delta}$,
then, we have that

$\displaystyle\left|\frac{f(X)}{\left\|f\right\|_{\nu}}-\frac{f^{\prime}(X)}{\left\|f^{\prime}\right\|_{\nu}}\right|\leq\frac{1}{\widetilde{\delta}}\cdot\left|f(X)-f^{\prime}(X)\right|+\frac{1}{\widetilde{\delta}^{2}}\cdot\left\|f-f^{\prime}\right\|_{\nu}.$
(183)

Noting that $\phi_{c}$ is $\frac{1}{c}$-Lipschitz, we see that

$\displaystyle\left\|\phi_{c}\left(\frac{f}{\left\|f\right\|_{\nu}}\right)-\phi_{c}\left(\frac{f^{\prime}}{\left\|f^{\prime}\right\|_{\nu}}\right)\right\|_{\mu}$
$\displaystyle\leq\frac{1}{c}\cdot\left(\frac{1}{\widetilde{\delta}}\cdot\left\|f-f^{\prime}\right\|_{\mu}+\frac{1}{\widetilde{\delta}^{2}}\cdot\left\|f-f^{\prime}\right\|_{\nu}\right)$
(184)
$\displaystyle\leq\frac{2}{c\sigma\widetilde{\delta}^{2}}\cdot\left\|f-f^{\prime}\right\|_{\mu},$
(185)

where we used the fact that $\widetilde{\delta}\leq 1$. The result follows
immediately. ∎


#  $\alpha$ \+ 92Zr cluster structure in 96Mo

S. Ohkubo1 and Y. Hirabayashi2 1 Research Center for Nuclear Physics, Osaka
University, Ibaraki, Osaka 567-0047, Japan  2Information Initiative Center,
Hokkaido University, Sapporo 060-0811, Japan

###### Abstract

In the evaluation of the half-life of the neutrinoless double-$\beta$ decay
($0\nu\beta\beta$) of a doubly closed-subshell nucleus 96Zr, the structure of
the nucleus 96Mo is essentially important. The $\alpha$-clustering aspects of
96Mo are investigated for the first time. By studying the nuclear rainbows in
$\alpha$ scattering from 92Zr at high energies and the characteristic
structure of the excitation functions at the extreme backward angle at the
low-energy region, the interaction potential between the $\alpha$ particle and
the 92Zr nucleus is determined well in the double folding model. The validity
of the double folding model was reinforced by studying $\alpha$ scattering
from neighboring nuclei 90Zr, 91Zr, and 94Zr. The double-folding-model
calculations reproduced well all the observed angular distributions over a
wide range of incident energies and the characteristic excitation functions.
By using the obtained potential the $\alpha$ +92Zr cluster structure of 96Mo
is investigated in the spirit of a unified description of scattering and
structure. The existence of the second-higher nodal band states with the
$\alpha$\+ 92Zr cluster structure, in which two more nodes are excited in the
relative motion compared with the ground band, is demonstrated. The
calculation reproduces well the ground-band states of 96Mo in agreement with
experiment. The experimental $B(E2)$ value of the transition in the ground
band is also reproduced well. The effect of $\alpha$ clustering in 96Mo on the
the half-life of the $0\nu\beta\beta$ double-$\beta$ decay of 96Zr is
discussed.

##  I INTRODUCTION

The observation of neutrinoless double-$\beta$ decay, $0\nu\beta\beta$, which
violates lepton number conservation, is expected to serve to shed light on the
fundamental questions beyond the standard model, such as determining the
nature of neutrino, Dirac, or Majorana particles. Since supersymmetric
particles have not been observed in Large Hadron Collider experiments, much
more attention than ever has been paid to study of $0\nu\beta\beta$ Ejiri2005
; Avignone2008 ; Vergados2012 . The inverse half-life of $0\nu\beta\beta$ is
given by
$[T^{0\nu}_{1/2}]^{-1}=G_{0\nu}\left|{<m_{\beta\beta}>/}{m_{e}}\right|^{2}|M^{0\nu}|^{2}$,
where $<m_{\beta\beta}>$ is the effective Majorana neutrino mass, ${m_{e}}$ is
the electron mass, and $G_{0\nu}\sim 10^{-14}$ yr-1 is a phase-space factor.
For the evaluation of the nuclear matrix element (NME) of the transition
$M^{0\nu}$ Suhonen1998 ; Faessler1998 ; Suhonen2012 ; Engel2017 ;
Jokiniemi2018 ; Dolinski2019 , it is essential to know the ground-state wave
functions of the initial- and final- state nuclei.

Up to now theoretical $0\nu\beta\beta$ decay study has been done based on the
mean-field theory such as the shell-model Iwata2016 ; Coraggio2020 , ab initio
calculations Yao2020 ; Belley2021 , quasiparticle random phase approximation
(QRPA) Simkovic2013 ; Hyvarinen2015 ; Jokiniemi2021 , the projected Hartree-
Fock Bogoliubov model (PHFB) Chaturvedi2008 ; Rath2010 ; Nautiyal2022 , the
generator coordinate method (GCM) Rodriguez2010 ; Hinohara2014 ; Song2017 ;
Jiao2019 , the energy density functional (EDF) Vaquero2013 ; Yao2015  and the
interacting boson model (IBM) Barea2009 ; Barea2013 . No attention has been
paid to the $\alpha$ cluster structure viewpoint until the study of 48Ca decay
to 48Ti Ohkubo2021 . This is probably because it has been believed intuitively
that strong spin-orbit force would break $\alpha$ clustering and partly
because experimental data of $\alpha$-transfer reactions such as (6Li,d),
(d,6Li), and $(p,p\alpha)$ are scarce.

$\alpha$ cluster structure has been established in the light mass region
Suppl1972 ; Suppl1980  and medium-weight mass region around 44TiMichel1998 ;
Yamaya1998 ; Sakuda1998 ; Ohkubo1999  and recently extended to the 52Ti region
Fukada2009 ; Bailey2019 ; Ohkubo2020 ; Ohkubo2021 ; Bailey2021 . In a previous
paper Ohkubo2021 , paying attention to the $0\nu\beta\beta$ decay of 48Ca to
48Ti, one of the present authors (S.O.) has shown that the ground $0^{+}$
state of 48Ti has $\alpha$-clustering aspects, which significantly quenches
the half-life than the conventional shell model calculations in which
excitations to the higher several major shells are not considered.

In the $0\nu\beta\beta$ of the parent nucleus 96Zr Fukuda2020 , the structure
of the ground state of the daughter nucleus 96Mo, whose $\alpha$ threshold
energy 2.76 MeV is small, is crucial in evaluating the NME of $0\nu\beta\beta$
decay transitions. The persistency of $\alpha$ clustering in the heavier mass
region around $A=90$ has been explored for the typical nucleus 94Mo with two
protons and two neutrons outside the closed shell core 90Zr in Refs.
Ohkubo1995 ; Buck1995 ; Michel2000 . Later $\alpha$ cluster model study
Ohkubo2009 ; Souza2015 ; Ni2011  also supports $\alpha$ clustering in the 94Mo
region. Recent observations of $\alpha$ particles in the pick-up reactions
$(p,p\alpha)$ in the Sn isotopes Tanaka2021  seem to reinforce the importance
of $\alpha$ clustering in the heavy mass region.

The ground state of 96Zr is spherical being a doubly closed-subshell nucleus
and is analog to the doubly closed shell 16O in light nuclei Molnar1986 . The
first excited $0^{+}$ state is considered to be a four-particle four-hole
excited-state analog to the mysterious $0^{+}$ state at 6.05 MeV in 16O.
Recent large-scale shell-model calculations for the Zr isotopes Togashi2016
have confirmed that the ground state of 96Zr is spherical and that shape
transition to deformed occurs at 100Zr as the number of the excess neutrons
increases. As for the structure of 96Mo, studies including $2\nu\beta\beta$
decay using QRPA Stoica1995 , phase transition from spherical 92Mo to deformed
toward 104Mo Lesher2007 , octupole collective motion Gregor2017  and shell-
model structure Coraggio2018  have been reported. $0\nu\beta\beta$ of 96Zr has
been investigated using many models, which includes the QRPA, PHFB, EDF, IBM,
and GCM Nautiyal2022 . However, no study of $0\nu\beta\beta$ of 96Zr from the
viewpoint of $\alpha$ cluster of 96Mo has been challenged.

The purpose of this paper is to show that $\alpha$ clustering persists in the
ground state of 96Mo by studying bound states and scattering for the
$\alpha$+92Zr system in a unified way and that the half-life of
$0\nu\beta\beta$ of 96Zr is quenched significantly. For this, by using a
double folding model the interaction potential between $\alpha$ particle and
92Zr is determined by analyzing angular distributions of nuclear rainbows in
$\alpha$+92Zr scattering at high energies, backward angle anomaly (BAA) or
anomalous large angle scattering (ALAS) at lower energies. The potential
reproduces well the excitation functions with a characteristic dip at the
extreme backward angles near 180∘ in the lower-energy region systematically
not only for $\alpha$+92Zr scattering but also for $\alpha$+90,91,94Zr
scattering. The existence of the second-higher nodal band states with the
$\alpha$+92Zr cluster structure, which is responsible for the emergence of the
characteristic dip in the back-angle excitation function, is shown for the
first time. The ground band of 96Mo is understood well in the $\alpha$-cluster
model study using the obtained double folding potential. $\alpha$ clustering
of 96Mo gives significant effect to quench the $0\nu\beta\beta$ decay of 96Zr.

The paper is organized as follows. In Sec. II the double folding model is
presented. Section III is devoted to the analysis of $\alpha$+92Zr scattering
over a wide range of incident energies by using a double folding model. To
confirm the validity of the obtained interaction potential for $\alpha$+92Zr,
$\alpha$ scattering from neighboring nuclei 90,91,94Zr is also investigated.
In Sec. IV the origin of the characteristic dip in the back-angle excitation
function in $\alpha$+92Zr scattering is investigated from the viewpoint of
persistent existence of the $\alpha$ cluster structure at the highly excited
energies in 96Mo. In Sec. V, $\alpha$+92Zr clustering of 96Mo is studied and
discussions of $\alpha$ clustering on the $0\nu\beta\beta$ decay of 96Zr is
given. A summary is given in Sec. VI.

##  II  DOUBLE FOLDING MODEL

We study $\alpha$ scattering from 92Zr and neighboring nuclei 90,91,94Zr with
a double folding model using a density-dependent nucleon-nucleon force. The
double folding potential is calculated as follows:

$\displaystyle V({\bf r})=\int\rho_{00}^{\rm(^{4}He)}({\bf
r}_{1})\;\rho_{00}^{\rm(Zr)}({\bf r}_{2})$ (1) $\displaystyle\times v_{\it
NN}(E,\rho,{\bf r}_{1}+{\bf r}-{\bf r}_{2})\;{\it d}{\bf r}_{1}{\it d}{\bf
r}_{2},$

where $\rho_{00}^{\rm(^{4}He)}({\bf r_{1}})$ and $\rho_{00}^{\rm(Zr)}({\bf
r_{2}})$ represent the nucleon density of the ground states of 4He and Zr,
respectively, which are obtained by the convolution of the proton size from
the charge density distribution taken from Ref.DeVries1987 . For the effective
interaction $v_{\rm NN}$ we use the density($\rho$)-dependent M3Y interaction
Kobos1982 . In the calculations we introduce the normalization factor $N_{R}$
for the real double folding potential Satchler1979 ; Brandan1997 . The Coulomb
folding potential is calculated similarly by the folding prescription in Eq.
(1). An imaginary potential with a Woods-Saxon volume-type form factor
(nondeformed) is introduced phenomenologically to take into account the effect
of absorption due to other channels.

##  III ANALYSIS OF ALPHA SCATTERING FROM 92Zr and 90,91,94Zr

In exploring the $\alpha$ cluster structure in the medium-weight mass region
where the level density is high, a unified description of $\alpha$ scattering
including rainbow scattering, prerainbows and BAA (ALAS), and the $\alpha$
cluster structure in the bound and quasibound energy region has been very
powerful Michel1986 ; Michel1988 ; Ohkubo1999 ; Ohkubo2020 ; Ohkubo2021 .

The angular distributions in $\alpha$ scattering from 92Zr have been measured
systematically at $E_{\alpha}$=40, 65, 90 and 120 MeV in Ref. Put1977  and
35.4 MeV in Ref. Singh1986 . The interaction potential can be uniquely
determined from the analysis of the angular distributions in the rainbow
energy region, which show the Airy minimum in the lit side of the nuclear
rainbow followed by the falloff of the cross sections corresponding to the
darkside of the nuclear rainbow.

We started to analyze the angular distribution at the highest energy
$E_{\alpha}$= 120 MeV to fit to the experimental angular distribution by
introducing $N_{R}$=1.26 and a phenomenological imaginary potential with a
strength parameter $W$=18.5 MeV, a radius parameter $R_{W}=$7.1 fm and a
diffuseness parameter $a_{W}=$0.6 fm. Then by keeping the fixed $N_{R}$ =1.26
for 90 and 65 MeV and with a slightly reduced value $N_{R}$ =1.22 for 40 and
35.4 MeV, all the angular distributions are easily reproduced by the
calculations with a small adjustment to reduce the strength and/or diffuseness
parameters of the imaginary potential with decreasing incident energies. The
calculated angular distributions are in good agreement with the experimental
data as displayed in Fig. 1. In Table I the values of the volume integral per
nucleon pair $J_{V}$ and the rms radius $\sqrt{<r_{V}^{2}>}$ of the double
folding potential, and the parameters of the imaginary potential together with
the volume integral per nucleon pair $J_{W}$ and the rms radius
$\sqrt{<r_{W}^{2}>}$ are listed. The energy dependence of the volume integrals
$J_{V}$ is reasonable, which is consistent with the previous calculations for
$\alpha$+92Zr scattering in Refs. Ohkubo1995 ; Put1977 .

Figure 1: (Color online) The angular distributions in $\alpha$+92Zr scattering
at $E_{\alpha}$=35.4, 40, 65, 90 and 120 MeV calculated with the optical
potential model with the double folding potential (solid lines) are compared
with the experimental data (filled circles) Put1977 ; Singh1986 . The
calculated farside (dotted lines) and nearside (dashed lines) contributions
are also indicated.

In order to see the contributions of the refractive farside scattering, the
calculated angular distributions are decomposed into the farside and nearside
components Fuller1975 . In Fig. 1, we see that the falloff of the cross
sections in the angular distributions in the intermediate angular region above
$E_{\alpha}$=65 MeV, which is peculiar to nuclear rainbow scattering, are all
due to farside scattering. A clear first-order Airy minimum $A1$ of the
nuclear rainbow is seen at $\theta=50^{\circ}$ at $E_{\alpha}$=120 MeV, which
shifts backward as the incident energy decreases, at around
$\theta=70^{\circ}$ for $E_{\alpha}$=90 MeV and at $\theta=125^{\circ}$ for
$E_{\alpha}$=65 MeV. At $E_{\alpha}$=40 MeV no Airy minimum is observed. The
appearance of the oscillations in the backward angular distributions shows
that the nearside contributions are involved since the oscillations are the
consequence of interference of the two amplitudes of farside and nearside
scattering. This backward rise of the cross sections with the oscillations at
$E_{\alpha}$=40 MeV is the indication of BAA under incomplete absorption,
which is typically observed and explained in $\alpha$+16O Ohkubo1977 ;
Michel1983  and $\alpha$+40Ca scattering Delbar1978  in the energy region
$E_{\alpha}$=20-30 MeV.

In the energy region below $E_{\alpha}$=40 MeV the concept of farside and
nearside scattering is no more powerful in understanding the characteristic
features of the angular distributions. It is useful to understand the
characteristics of the angular distributions of BAA in terms of the concept of
internal waves and barrier waves Brink1985 . The scattering amplitude
$f(\theta)$ can be decomposed into $f^{I}(\theta)$, which is due to the
internal waves penetrating the barrier deep into the internal region of the
potential and $f^{B}(\theta)$, which is due to the barrier waves reflected at
the barrier of the potential in the surface region,
$f(\theta)=f^{I}(\theta)+f^{B}(\theta)$. In the case of incomplete absorption
the internal waves, $f^{I}(\theta)$, carry the information of the internal
region of the potential. Unfortunately at the lower energies below 30 MeV no
angular distributions have been measured for $\alpha$+92Zr scattering where
the effect of the internal waves is clearly seen Ohkubo1977 ; Michel1983 ;
Delbar1978 ; Brink1985 .

Table 1:  The normalization factor $N_{R}$, volume integral per nucleon pair $J_{V}$, rms radius $\sqrt{<r_{V}^{2}>}$ of the double folding potentials, and the strength $W$, radius $R_{W}$, diffuseness $a_{W}$, volume integral per nucleon pair $J_{W}$ and rms radius $\sqrt{<r_{W}^{2}>}$ of the imaginary potentials used in $\alpha$+92Zr scattering in Fig. 1. Energies are in MeV, volume integrals in MeVfm3 and radii in fm.  $E_{\alpha}$ | $N_{R}$ | $J_{V}$ | $\sqrt{<r_{V}^{2}>}$ | $W$ | $R_{W}$ | $a_{W}$ | $J_{W}$ | $\sqrt{<r_{W}^{2}>}$
---|---|---|---|---|---|---|---|---
35.4 | 1.22 | 318.9 | 5.00 | 14.0 | 7.1 | 0.55 | 60.4 | 5.87
40 | 1.22 | 315.7 | 5.00 | 14.0 | 7.1 | 0.50 | 59.8 | 5.81
65 | 1.26 | 313.4 | 5.01 | 17.0 | 7.1 | 0.60 | 74.1 | 5.94
90 | 1.26 | 298.2 | 5.01 | 18.5 | 7.1 | 0.60 | 80.7 | 5.94
120 | 1.26 | 280.5 | 5.02 | 18.5 | 7.1 | 0.60 | 80.7 | 5.94
Figure 2:  (Color online) The calculated angular distributions (solid lines)
in (a) $\alpha$+90Zr scattering at $E_{\alpha}$= 21, 23.4 and 25 MeV and (b)
$\alpha$+91Zr scattering at $E_{\alpha}$= 21, 23, and 25 MeV are compared with
the experimental data (filled circles) Wit1975 .

However, we note that the angular distributions in $\alpha$ scattering from
neighboring nuclei 90Zr and 91Zr have been measured up to the backward angles
at the lower energies $E_{\alpha}$=23-25 MeV. In Fig. 2 the angular
distributions show a BAA rising toward the extreme backward angles at 21 and
25 MeV. Note that the angular distributions for both 90Zr and 91Zr decrease
sharply toward 180∘ at $E_{\alpha}$=23 MeV in the BAA energy region, which is
not seen in the typical $\alpha$+16O Ohkubo1977 ; Michel1983  and
$\alpha$+40Ca scattering Delbar1978 . This characteristic decrease is
intriguing because angular distributions at other energies generally increase
toward $\theta=180^{\circ}$, see Fig. 1, as expected from the behavior of the
Legendre polynomials whose moduli increases toward $\theta=180^{\circ}$ at the
extreme back angles. In Fig. 2 the angular distributions in $\alpha$\+ 90Zr
and $\alpha$+91Zr scattering calculated using the double folding potential
derived from Eq. (1) are compared with the experimental data Wit1975 . The
potential parameters used are listed in Table II. The calculations reproduce
the experimental angular distributions well. Note that the particular behavior
at 23 MeV that decreases sharply toward 180∘ is reproduced excellently. This
shows that the calculated double folding potentials for $\alpha$\+ 90Zr and
$\alpha$+91Zr work very well in this low-energy region, which reinforces the
validity of the double folding potential in the $E_{\alpha}$=23- MeV to
$E_{\alpha}$=25- MeV region.

In Fig. 3 the excitation functions at the extreme backward angle
$\theta$=176.2∘ ($\theta_{Lab}$=176∘) in $\alpha$ scattering from 90Zr, 91Zr,
92Zr and 94Zr calculated using the potentials at $E_{\alpha}$= 23 MeV in Table
II are displayed in comparison with the experimental data. All the calculated
excitation functions show a dip and its position shifts to lower energy from
90Zr to 94Zr. The position of the dips in the calculated excitation functions
for $\alpha$+90Zr, $\alpha$+91Zr and $\alpha$+94Zr agrees with the
experimental data excellently. The energy of the observed dips for 90Zr, 91Zr
and 94Zr deceases linearly with the mass number $A$ of the target nucleus,
$E_{\alpha}$=54.5 - 0.346$A$, which predicts a dip at $E_{\alpha}$=22.7 MeV
for 92Zr. As seen in Fig. 3, the double folding model calculation locates a
dip at $E_{\alpha}$= 22.7 MeV for $\alpha$+92Zr, which is in good agreement
with the above-predicted energy, 22.7 MeV.

Figure 3: (Color online) The calculated excitation functions in $\alpha$ scattering from 90Zr, 91Zr, 92Zr, and 94Zr at the extreme backward angle $\theta$=176.2∘ ($\theta_{Lab}$=176∘) (solid lines) are compared with the experimental data (filled circles) Wit1975 .  Table 2:  The volume integral per nucleon pair $J_{V}$, rms radius $\sqrt{<r_{V}^{2}>}$ of the double folding potentials, and the strength $W$, radius $R_{W}$, diffuseness $a_{W}$, volume integral per nucleon pair $J_{W}$, and rms radius $\sqrt{<r_{W}^{2}>}$ of the imaginary potentials used in $\alpha$+90,91,92,94Zr scattering in Fig. 2 and Fig. 3. Energies are in MeV, volume integrals in MeVfm3, and radii in fm. $N_{R}$=1.22 is used for all target nuclei and incident energies.  | $E_{\alpha}$ | $J_{V}$ | $\sqrt{<r_{V}^{2}>}$ | $W$ | $R_{W}$ | $a_{W}$ | $J_{W}$ | $\sqrt{<r_{W}^{2}>}$
---|---|---|---|---|---|---|---|---
90Zr | 21 | 331.5 | 4.96 | 10.0 | 7.55 | 0.40 | 51.5 | 6.03
| 23.4 | 329.5 | 4.96 | 10.0 | 7.55 | 0.43 | 51.7 | 6.06
| 25 | 327.5 | 4.96 | 10.0 | 7.55 | 0.48 | 52.1 | 6.11
91Zr | 21 | 333.1 | 5.00 | 10.6 | 7.60 | 0.37 | 54.8 | 6.05
| 23 | 331.7 | 5.00 | 10.2 | 7.60 | 0.41 | 53.0 | 6.08
| 25 | 330.3 | 5.01 | 10.2 | 7.60 | 0.45 | 53.3 | 6.12
92Zr | 23 | 329.6 | 4.99 | 10.7 | 7.63 | 0.43 | 55.8 | 6.12
94Zr | 23 | 330.0 | 5.02 | 11.8 | 7.70 | 0.48 | 62.3 | 6.23

The mechanism explaining why the dip emerges in the excitation function at the
extreme backward angle near $\theta$=180∘, namely why the angular distribution
decreases sharply toward $\theta$=180∘ at a particular energy, has been
investigated in detail for the typical $\alpha$+90Zr system by one of the
present authors (S.O.) and his collaborators, see Ref. Michel2000 . The
mechanism is understood as follows. The dip appears at the energy where the
scattering amplitude $f(\theta)$ becomes vanishingly small. When
$f^{I}(\theta)$$\approx$$-f^{B}(\theta)$, the cancellation of the two
amplitude occurs, i.e., in the case when $|f^{I}(\theta)|$ $\approx$
$|f^{B}(\theta)|$ and arg($|f^{I}(\theta))$\- arg($f^{B}(\theta))$ $\approx$
$k\pi$ where $k$ is an odd integer. At near $\theta$=$180^{\circ}$ this
condition is satisfied at the energy $E_{\alpha}$=22-24 MeV under moderate
absorption not only for $\alpha$+90Zr but also for $\alpha$+91Zr,
$\alpha$+92Zr and $\alpha$+94Zr since both the real potential and the
imaginary potential change little from that of $\alpha$+90Zr as seen in Table
II. The good agreement of the calculated excitation functions, especially the
energy position and width of the dip for $\alpha$+91Zr, and $\alpha$+94Zr with
the experimental data, is the natural consequence that their potentials
resemble that for $\alpha$+90Zr. Although no experimental data are available
for $\alpha$+92Zr, the emergence of the dip at the predicted energy in the
excitation function would be confirmed in the future experiment. Since the
internal waves, which are responsible for the emergence of the dip, are
sensitive to the internal region of the real potential, the present good
agreement in Fig. 3 shows that the obtained double folding potential is
reliable sufficiently in this low-energy region above the Coulomb barrier.

##  IV MECHANISM OF THE CHARACTERISTIC DIP IN THE BACK-ANGLE EXCITATION
FUNCTION IN $\alpha$+92Zr SCATTERING

In this section, paying attention to the highly lying excited $\alpha$ cluster
structure in 96Mo, we investigate how the anomalous dip in the back-angle
excitation in $\alpha$+92Zr scattering in Fig. 3 is created.

For this purpose, in Fig. 4 (a) back-angle excitation functions calculated by
reducing gradually the strength of the imaginary potential, $W$=3$W_{0}$/4,
$W_{0}$/2, $W_{0}$/4, $W_{0}$/8, and 0 MeV, are compared with the original one
with $W_{0}=10.7$ in Table II. For $W=0$ the peaks in the excitation function
at $E_{\alpha}$=20.5 and 23 MeV are due to the resonances with the
$\alpha$+92Zr structure. That the $\alpha$ cluster structure at the highly
excitation energies can be seen in the excitation function at the extreme
backward angles near $180^{\circ}$ has been already shown for the
$\alpha$+40Ca cluster structure in 44Ti Michel1986B ; Ohkubo1987 .

Figure 4: (Color online) (a)The excitation functions at the extreme backward
angle $\theta$=176.2∘ in $\alpha$+92Zr scattering calculated with the reduced
strengths of the imaginary potential $W$=0 (dotted lines), $W_{0}$/8 (dashed
lines), $W_{0}$/4 (long dashed lines), $W_{0}$/2 (dash-dotted lines), and
3$W_{0}$/4 ( dashed-and-double-dotted lines) are compared with the original
one with $W=$$W_{0}$=10.7 MeV (solid lines). (b) The calculated partial wave
cross sections of elastic scattering under $W=$0 for even $L$ and for (c) odd
$L$. Figure 5: (Color online) Phase shifts in $\alpha$ \+ 92Zr scattering
calculated with the double folding potential with $N_{R}$=1.22, (a) even $L$
partial waves and (b) odd $L$ partial waves, are displayed for $L\leq$15\. The
lower abscissa shows cener-of-mass energy $E$ and the upper abscissa shows
laboratory incident energy $E_{\alpha}$.

In Fig. 4 (b) and (c), the partial wave cross sections of elastic scattering
are displayed. Fig. 4 (b) shows that the peaks at $E_{\alpha}$=20.5 and 23 MeV
are caused by the even $L$ partial waves and Fig. 4 (c) shows that the odd $L$
partial waves do not contribute to create the peaks. Thus we find that the
peaks at $E_{\alpha}$=20.5 and $E_{\alpha}$=23 MeV in the excitation function
with $W=0$ are caused by the resonant waves $L=10$ and $L=12$, respectively.

To see the details of the resonances responsible for the peaks, in Fig. 5 the
phase shifts in $\alpha$+92Zr scattering calculated by switching off the
imaginary potential are displayed. We see that the phase shifts for the even
parity and odd parity partial waves show different behavior in the relevant
energies, $E_{\alpha}$=18 - 26 MeV (center-of-mass energy $E$=17.3-24.9 MeV).
Although the the phase shifts of the even-parity partial waves, $L=$10 and 12,
pass through $\delta_{L}$=270∘ slowly at the resonance energies, those of the
odd-parity partial waves, $L=$11 - 15, cross $\delta_{L}$=90∘ sharply at the
resonance energies. The narrow odd-parity resonances hardly contribute to the
peaks in the excitation functions as seen in Fig. 4 (a) and (c). This is why
even-parity waves are dominantly responsible for the peaks and the dip in Fig.
4. The broad resonant nature of the even- parity waves is the consequence that
they are the high-lying second-higher nodal $\alpha$-cluster resonance states,
in which the relative motion is two more excited compared with the lowest
Pauli-allowed ground-band states in 96Mo. The nature of the resonant
$\alpha$+92Zr cluster structure in 96Mo is discussed in detail in the next
section.

##  V  ALPHA CLUSTER STRUCTURE IN 96Mo AND NEUTRINOLESS DOUBLE $\beta$ DECAY
of 96Zr

In order to reveal the cluster structure of 96Mo underlying in the excitation
function with the characteristic dip at the extreme backward angle, the
resonant states and the bound and quasibound energy levels calculated in the
double folding potential with $N_{R}$=1.22 by switching off the imaginary
potential of the optical potential used in Fig. 3 are displayed in Fig. 6 (a).
The resonance energies are given at the energies where the phase shifts
steeply pass through $\delta_{L}$=90${}^{\circ}(270^{\circ})$ in Fig. 5. By
investigating the resonant wave function for $L=12$ at $E=22.04$ MeV, we find
that the wave function has four nodes in the relative motion, see Fig. 7. The
resonances with $L$=10, 12, and 14 in the range of $E$=19-25 MeV are found to
belong to the band with $N=2n+L=20$, where $n$ is the number of the nodes in
the relative wave function between $\alpha$ and 92Zr. The $N=20$ band states
energies are on the $J(J+1)$ plot with the bandhead $J^{\pi}$=$0^{+}$ state at
$E$=14.4 MeV and the rotational constant $k$=$\hbar^{2}/2\cal{J}$=0.0492 MeV,
where $\cal{J}$ is the moment of inertia of the band. The band has a well-
developed $\alpha$+92Zr cluster structure. The large separation distance
between $\alpha$ and 92Zr can be seen in the the wave functions of the
$10^{+}$ and $12^{+}$ states in Fig. 7. The outermost peak, which is located
at around $R$=7-8 fm, is much larger than the sum of the experimental radius
Angeli2013  of $\alpha$ and 92Zr, 6.0 fm. Although the phase shifts for the
lower $L=0-6$ of the $N=20$ band show rising toward $\delta_{L}$=270∘, they do
not cross $\delta_{L}$=270∘ sufficiently. However, since the number of the
nodes $n$ of their wave functions satisfy the condition $N=2n+L=20$, they are
considered to belong to the persistent member of the rotational band with
$N=20$. From the $J(J+1)$ plot they are extrapolated to exist persistently at
the energies indicated by the dotted lines in Fig. 6 (a). The resonance
energies and widths of these broad resonances can be calculated in the complex
scaling method Aguilar1971 ; Ohkubo2002 . The presence of the $12^{+}$ state
of the $N=20$ band, which manifests itself in the emergence of the
characteristic dip in the back-angle excitation function, demonstrates for the
first time the existence of a second-higher nodal band member state with the
$\alpha$ \+ 92Zr cluster structure in 96Mo, in which two more nodes are
excited in the relative motion compared with the $N=$16 ground band.

Figure 6: (Color online) (a) Energy levels of 96Mo calculated in the
$\alpha$\+ 92Zr cluster model with the double folding potential with
$N_{R}$=1.22. The calculated excitation energy of the $N$=16 ground-band
states, $0^{+}$, $2^{+}$, $4^{+}$, $6^{+}$ and $8^{+}$, which look compressed,
increases as the spin increases. (b) The $N=16$ band energy levels calculated
using the double folding model with $L$ dependence. (c) Experimental energy
levels of the ground band. The horizontal dashed lines (blue) correspond to
$E_{\alpha}$=18 MeV (center-of-mass energy $E$=17.3) and $E_{\alpha}$=26 MeV
($E$=24.9 MeV), between which the characteristic dip in the excitation
function appear.

The wave functions of the resonances with odd $L$ in Fig. 5 have $N=19$ and
form a negative-parity rotational band with the $\alpha$+92Zr cluster
structure. The band states are well located on the $J(J+1)$ plot with its
bandhead $1^{-}$ state at $E=11.1$ MeV and $k$=0.0383 MeV. The $N=19$ band is
a higher nodal band with developed $\alpha$ clustering, in which the relative
motion is one more excited compared with the lower-lying $N=17$ band states.
The calculation locates the $N=18$ rotational band with its bandhead $0^{+}$
at $E$=7.19 MeV and $k$=0.0236 MeV, which is a higher nodal band with one more
node in the wave functions compared with those of the Pauli-allowed lowest
$N=16$ band. The $N=17$ rotational band states are well located on the
$J(J+1)$ plot with its bandhead $1^{-}$ state at $E$=1.33 MeV. The calculation
locates the band states with $N=16$ below the $\alpha$ threshold. It is
surprising that the Pauli-allowed lowest $N=16$ band states satisfying the
Wildermuth condition falls in good correspondence with the ground band of
96Mo. The calculated $0^{+}$ state of the $N=16$ band with $E$=-5.56 MeV is
slightly overbound by 2.8 MeV compared with the experimental energy of the
ground state with $E=$-2.76 MeV from the $\alpha$ threshold. This is because
the potential determined at the highly excited energy region, $E_{\alpha}$=23
MeV, is straightforwardly applied to the calculations in the bound-state
energy region. The energy levels for $N=$18, 17 and 16 in Fig. 6, most of
which are located below the Coulomb barrier, are the ones calculated in the
bound-state approximation.

Figure 7: (Color online) The calculated $u(R)$ of the relative wave function
$u(R)$/$R$ of the $10^{+}$ and $12^{+}$ states of the $N=$20 band with the
$\alpha$+92Zr cluster structure in 96Mo. The wave functions are calculated as
scattering states and normalized to unity for $R\leq$10 fm.

According to the dispersion relation Mahaux1986 , the energy dependence of the
volume integral of the real potential shows the threshold anomaly. Namely the
volume integral $J_{V}$ increases as the incident energy decreases from the
rainbow energy region to the lower-energy region of BAA and reaches a maximum
followed by a decrease toward $E_{\alpha}$=0. In fact, in Table I and II we
see that $J_{V}$ increases from 280.5 MeVfm3 at the rainbow energy
$E_{\alpha}$=120 MeV to 318.9 MeVfm3 at $E_{\alpha}$=35.4 MeV and 329.6 MeVfm3
at $E_{\alpha}$=23 MeV. The dispersion relation tells that a potential with a
reduced $J_{V}$ value should be used in the bound and quasibound energy region
below and near the threshold energy $E$=0. The overbinding of the ground-state
energy in Fig. 6 (a) is simply ascribed to that it is calculated using the
potential with $N_{R}$=1.22 at $E_{\alpha}$=23 MeV with the large $J_{V}$
value without taking into account the energy dependence of the real potential
due to the dispersion relation Mahaux1986 . By using the double folding
potential with a slightly reduced strength, $N_{R}$=1.182 with $J_{V}$=319.3
MeVfm3, the calculated ground-state $0^{+}$ energy agrees with the
experimental value as seen in Fig. 6 (b). A similar situation where $J_{V}$
must be reduced in the bound and quasibound energy region compared with that
used in the higher scattering energy region has been reported in the recent
unified description of bound and scattering states for the $\alpha$+48Ca
cluster structure in 52Ti Ohkubo2020  and the $\alpha$+44Ca cluster structure
in 48Ti Ohkubo2021 .

Table 3:  The excitation energies $E_{x}$, intercluster rms radii $\sqrt{<R^{2}>}$ and $B(E2)$ values for the $J\rightarrow(J-2)$ transitions of the ground band in 96Mo calculated in the $\alpha$+92Zr cluster model with the double folding potential are compared with the experimental data Abriola2008  and the large-scale shell model calculation Coraggio2018 .  $J^{\pi}$ |  $E_{x}$ (MeV) |  $\sqrt{<R^{2}>}$ (fm) |  $B(E2)$ (W.u.)
---|---|---|---
| exp | cal | cal | exp Abriola2008  | cal | Ref.Coraggio2018
$0^{+}$ | 0.00 | 0.000 | 5.20 |  |  |
$2^{+}$ | 0.778 | 0.770 | 5.21 | 20.7$\pm$0.4 | 20.7 | 18.7
$4^{+}$ | 1.628 | 1.611 | 5.19 | 41$\pm$7 | 28.7 | -
$6^{+}$ | 2.441 | 2.543 | 5.14 | - | 29.1 | -
$8^{+}$ | 2.978 | 3.583 | 5.05 | - | 26.5 | -

In Fig. 6 (a) the calculated $N$=16 ground-band states are very compressed,
which is also the case for the $N=16$ states calculated with $N_{R}$=1.182.
Although the conventional Wood-Saxon potential gives an inverted energy level
spectrum in this heavy mass region, namely the excitation energy of the
ground- band states decreases as the spin increases in disagreement with
experiment, the present double folding model potential gives the energy
spectrum consistent with experimental ground band. In fact, the excitation
energy of the calculated energy levels of the $N$=16 band, which looks almost
degenerate in Fig. 6 (a), increases as the spin increases from $0^{+}$ to
$8^{+}$. This compression is because the angular momentum dependence of the
local potential has not been taken into account. In order to discuss the
spectroscopic properties in the low-energy region, it is necessary to take
into account the $L$ dependence of the potential. The nucleus-nucleus
potential, which is originally non-local due to the the Pauli principle, has
$L$ dependence when represented as a local potential. The $L$ dependence is
usually not important and often neglected in the scattering energy region.
However, this $L$ dependence is important when we study the cluster structure
in the bound and quasibound energy region. The necessity of the $L$ dependence
of the intercluster potential due to the Pauli principle has been
theoretically founded in the microscopic studies of interactions between
composite particles Tohsaki1980 ; Aoki1982 . In fact, it has been shown that
this $L$ dependence is indispensable in the $\alpha$ cluster structure using a
local potential, for example, in 20Ne Michel1989 , 44Ti Michel1986 ;
Michel1988 , 94Mo Ohkubo1995 ; Souza2015 , 212Po Ohkubo1995 ; Ni2011 , and
46,50Cr Mohr2017 . Following the double folding potential model study of the
$\alpha$ cluster structure in 94Mo in Ref. Ohkubo1995  where linear $L$
dependence in the double folding potential is first discussed, we use
$N_{R}^{(L)}$=$N_{R}^{(L=0)}$ \- $c$ $L$ with $N_{R}^{(L=0)}$=1.182 and
$c$=5.00$\times 10^{-3}$ for 96Mo. The calculated energy levels of the $N=16$
ground band are displayed in Fig. 6 (b). In Table III the calculated $B(E2)$
values as well as the excitation energies, intercluster rms radii of the
ground band of 96Mo are listed in comparison with the experimental data. The
excitation energy of the ground band is reproduced well by the double folding
potential model with small $L$ dependence. The experimental $B(E2)$ values
Abriola2008  are also reproduced well by introducing an additional small
effective charge $\Delta e=0.3e$ for protons and neutrons. We note that in the
large-scale shell-model calculations in Ref. Coraggio2018  rather large
additional effective charges $\Delta e=0.5e$ for protons and $\Delta
e=0.5-0.8e$ for neutrons are introduced. The rms charge radius
$<r^{2}>_{{}^{96}{\rm Mo}}^{1/2}$=4.36 fm of the ground state calculated using
the experimental values $<r^{2}>_{{}^{4}{\rm He}}^{1/2}$=1.676 fm and
$<r^{2}>_{\rm{}^{92}Zr}^{1/2}$=4.306 fm Angeli2013  is in good agreement with
the experimental value 4.38 fm Angeli2013 . The calculated intercluster
distance of the ground state is about 87% of the sum of the experimental rms
charge radii of the two clusters, which is compared to 87% for the ground
state of 44Ti Michel1988 .

Note that the value of the parameter $c$=5.00$\times 10^{-3}$ lies in the
expected range of $\alpha$-cluster states $c$$\approx$(2.5-5)$\times 10^{-3}$,
as observed for many $\alpha$-cluster states in a wide range of nuclei
Mohr2007  including the mass region near $A=100$ such as 94Mo Ohkubo1995 ;
Michel1998 , 93Nb Ohkubo2009 ; Kiss2009 , and 104Te Mohr2007 ; light- and
medium-weight mass regions such as 20Ne Abele1993  and 44Ti Atzrott1996 , and
heavy mass region such as 212Po Hoyler1994 ; Ohkubo1995 . The $L$-dependent
potential calculation locates the negative-parity $N=17$ band with the head
$1^{-}$ state at $E_{x}$=6.83 MeV well below the Coulomb barrier. Although
$1^{-}$ states have been observed at $E_{x}$=3.600 MeV and 3.895 MeV
Abriola2008 , experimental spectroscopic properties about $\alpha$ clustering
of the excited states near and above $E_{x}$$\approx$4 MeV are not clear.

Figure 8: (Color online) The calculated $u(R)$ of the relative wave function
$u(R)$/$R$ of the ground state of 96Mo calculated in the $\alpha$\+ 92Zr
cluster model (solid line) is compared with the harmonic oscillator wave
function with $N_{\rm HO}$ =16 (dashed line).

The potential embeds the eight deeply bound unphysical Pauli-forbidden $0^{+}$
states. The overlaps of the eight $0^{+}$ wave functions with the harmonic
oscillator wave functions with $\nu$=0.22 fm-2 ($\nu=m\omega/\hbar$ and
$\hbar\omega$=9.12 MeV) are 0.96, 0.93, 0.93, 0.95, 0.97 and 0.98, 0.99, and
0.97 for the oscillator quanta $N_{\rm HO}$=$0$, $2$, $\cdots$, and $14$,
respectively. This means that the Pauli-forbidden states with $N_{\rm HO}<16$
in the resonating group method are highly excluded from the obtained ground-
state wave function, thus mimicking Saito’s orthogonality condition model
Saito1968 .

As seen in Fig. 8, the ground-state wave function resembles the shell-model
function with $N_{\rm HO}=16$ in the internal region. However, the outermost
peak at around 6 fm is slightly shifted outward compared with that of the
harmonic oscillator wave function causing significant enhancement of the
amplitude at the outer surface region due to $\alpha$ clustering. This
enhancement means that the obtained wave function contains significant amount
of components in the shells higher than $N_{\rm HO}$ =16.

In Fig. 9 the occupation probability of the quanta $N_{\rm HO}\geq$16 in the
ground state wave function is displayed. The dominant occupation probability
in the lowest Pauli-allowed shell-model like $N_{\rm HO}=16$ is 78%. The
significant amount of higher $N_{\rm HO}\geq 18$ components, 22% is due to the
$\alpha$ clustering of the ground state. The $2^{+}$ and $4^{+}$ states have
the similar character. This $\alpha$ clustering is responsible for the
enhancement of the $B(E2)$ values in 96Mo and should be also taken into
account in the evaluation of NME of $0\nu\beta\beta$ decay of 96Zr to 96Mo.

Figure 9: (Color online) The occupation probability of the harmonic oscillator
quanta $N_{\rm HO}$ in the ground-state wave function of 96Mo.

We discuss the effect of $\alpha$ clustering of 96Mo on the NME of
$0\nu\beta\beta$ decay of 96Zr, which was the motivation of the present
$\alpha$ cluster structure study of 96Mo as introduced in Sec. I. The NME
values of $0\nu\beta\beta$ decay of 96Zr evaluated by using various nuclear
models are summarized in Ref. Engel2017  and most recently in Ref.
Nautiyal2022 . The QRPA calculations give NME values 2.72 with the Argonne V18
nucleon-nucleon potential and 2.96 with the CD-Bonn nucleon-nucleon potential
with the axial vector coupling constant $g_{A}$=1.27 in Ref. Simkovic2013  and
3.14 with $g_{A}$=1.26 in Ref. Hyvarinen2015 . The IBM calculation by Barea et
al. gives 2.53 in Ref. Barea2013 . The latest PHFB calculations give 2.5
Nautiyal2022 . On the other hand, the EDF calculations give considerably large
NME values, about twice as large as the above values. The nonrelativistic EDF
calculations by Vaquero et al. Vaquero2013  give 5.65 with $g_{A}$=1.26 when
evaluated with shape fluctuation and 6.50 when evaluated with both the shape
and pairing fluctuations. The relativistic EDF calculation by Yao et al.
Yao2015  gives almost the same large result. Yao et al. claim Yao2015  that
the EDF calculations are unable to reproduce the properties of 96Zr giving
too-low excitation energy of E$(2_{1}^{+})$ and too-large $B(E2:0_{\rm
g.s.}\rightarrow 2_{1}^{+})$ value, which is one order of magnitude large
compared with the experimental data. Yao et al. Yao2015  ascribe this to the
overestimation of the collectivity in 96Zr due to the “common problem of most
EDF-based GCM or collective Hamiltonian calculations.” Moreover, the GCM
calculation in the frame of covariant density functional theory Song2017
gives the largest value of 6.37 among the nuclear model calculations. The
overestimation of the collectivity of the doubly closed-subshell nucleus 96Zr
increases the overlap of the wave functions of 96Zr and 96Mo, which leads to
the large NME values. Although the present cluster model is unable to
calculate NME values because nucleon degree of freedom is not involved, it can
qualitatively tell whether the NME is enhanced or reduced by $\alpha$
clustering of 96Mo compared with the shell-model calculations in which the
excitations to the higher major shells are not included. Taking into account
that the excitation energy 1.58 MeV of the first excited state $0^{+}$ of 96Zr
is rather high in this mass region resembling the mysterious $0^{+}$ of the
double magic nucleus 16O Molnar1986 , and that there is no evidence that 96Zr
has $\alpha$+92Sr clustering Molnar1986 , the ground-state wave function can
be well considered to have a doubly closed-subshell shell-model structure.
Thus the $\alpha$ clustering of 96Mo reduces considerably the overlap of the
ground-state wave function of 96Zr with that of 96Mo in the evaluation of the
NME. That is, the $0\nu\beta\beta$ decay of 96Zr to 96Mo would be
significantly quenched, thus have a longer half-life, due to the $\alpha$
clustering than that in the shell-model calculations which do not take into
account the four particle excitations. Unfortunately, NME values in the shell
model have not been reported. The shell-model calculations with configuration
mixing including $N_{\rm HO}=$ 18, 20 and 22 major shells are presently
formidably difficult even with modern computers. We note that both the QRAP
and IBM calculations do not include $\alpha$-like four-particle four-hole
excitations and $\alpha$-like correlations.

Finally, we briefly mention about the large $B(E2)$ value 51.7 W.u. of the
transition from $0_{2}^{+}$ ($E_{x}$=1.148 MeV) to $2^{+}_{1}$ ($E_{x}$=0.778
MeV) in 96Mo Abriola2008 , which may suggest that the $0_{2}^{+}$ state has
$\alpha$ clustering in which the core is excited. If the $0_{2}^{+}$ state has
significant amount of $[\alpha_{(L=2)}$+92Zr(2${}^{+}_{1})]_{J=0}$ clustering
component, then the $B(E2)$ value can be enhanced because in addition to the
$E2$ transition matrix element due to the intercluster relative motion,
<2${}^{+}_{1}(\alpha_{L=0})|\hat{O}_{E2}({\bf r})|0_{2}^{+}(\alpha_{L=2}$)>,
the internal transition of the core 92Zr, <92Zr(g.s.)
$|\hat{O}_{E2}(\xi)|^{92}$Zr($2^{+}_{1}$)>, contributes to the total $E2$
transition where $\xi$ is the internal coordinate of 92Zr. Coupled- channels
calculations with excitations of 92Zr would be a future challenge to
understand the origin of the large $B(E2)$ value of the $0_{2}^{+}$ state of
96Mo and the effective charge.

##  VI SUMMARY

In the evaluation of nuclear matrix element of neutrinoless double $\beta$
decay $0\nu\beta\beta$ of the doubly closed-subshell nucleus 96Zr to 96Mo, it
is important to take into account the collectivity due to $\alpha$ clustering
in the structure of 96Mo, which has extra two neutrons on the 94Mo nucleus,
which is analog of 20Ne and 44Ti and has been considered to have $\alpha$
cluster structure. We have studied for the first time $\alpha$ clustering
aspects of 96Mo by using a double folding potential determined from the
analysis of nuclear rainbows at high energies and the characteristic structure
of the angular distributions at low energies in $\alpha$ particle scattering
from 92Zr. The validity of the double folding potential used is also confirmed
by studying $\alpha$ scattering from 90,91.94Zr in the low-energy region where
a characteristic dip appears in the excitation functions at the extreme
backward angle near $180^{\circ}$. The double folding model calculations
reproduced well all the observed angular distributions over a wide range of
incident energies and the excitation functions with a characteristic dip at
the extreme backward angle. By studying the $\alpha$ cluster structure with
the obtained double folding potential, the existence of the second-higher
nodal $N=20$ band states with the $\alpha$\+ 92Zr cluster structure, in which
two more nodes are excited in the relative motion compared with the $N=16$
ground band in 96Mo, is demonstrated for the first time at the highly excited
energy region. The $\alpha$-cluster model using this potential locates the
ground state in agreement with experiment and reproduces the observed $B(E2)$
value of 96Mo. The effect of $\alpha$ clustering in 96Mo on the the half-life
of the $0\nu\beta\beta$ double-$\beta$ decay of 96Zr is discussed.

###### Acknowledgements.

One of the authors (S.O.) thanks the Yukawa Institute for Theoretical Physics,
Kyoto University where part of the work was done during a stay in 2022.

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