# Weighted Sum-Rate Maximization for Movable Antenna-Enhanced Wireless
Networks

Biqian Feng, Yongpeng Wu, _Senior Member, IEEE_ , Xiang-Gen Xia, _Fellow,
IEEE_ ,
and Chengshan Xiao, _Fellow, IEEE_ B. Feng and Y. Wu are with the Department
of Electronic Engineering, Shanghai Jiao Tong University, Minhang 200240,
China (e-mail<EMAIL_ADDRESS>yongpeng.wu@sjtu.edu.cn). X.-G. Xia is
with the Department of Electrical and Computer Engineering, University of
Delaware, Newark, DE 19716, USA. (e-mail: xxia@ee.udel.edu).C. Xiao is with
the Department of Electrical, and Computer Engineering, Lehigh University,
Bethlehem, PA 18015 USA (e-mail: xiaoc@lehigh.edu).Corresponding author:
Yongpeng Wu.

###### Abstract

This letter investigates the weighted sum rate maximization problem in movable
antenna (MA)-enhanced systems. To reduce the computational complexity, we
transform it into a more tractable weighted minimum mean square error (WMMSE)
problem well-suited for MA. We then adopt the WMMSE algorithm and
majorization-minimization algorithm to optimize the beamforming and antenna
positions, respectively. Moreover, we propose a planar movement mode, which
constrains each MA to a specified area, we obtain a low-complexity closed-form
solution. Numerical results demonstrate that the MA-enhanced system
outperforms the conventional system. Besides, the computation time for the
planar movement mode is reduced by approximately 30% at a little performance
expense.

###### Index Terms:

Movable antenna, weighted sum rate maximization, planar movement mode.

##  I Introduction

With the advancement of multiple-input multiple-output (MIMO) technologies,
wireless communication systems have witnessed a substantial increase in
capacity due to the utilization of additional spatial degrees of freedom
(DoFs). However, the conventional MIMO systems equipped with fixed-position
antennas (FPAs) face challenges in further enhancing performance. Since the
last century, the concept of antenna arrays with arbitrarily distributed
elements [1] has gradually surfaced. This concept suggests that fewer, non-
uniformly deployed elements can match the performance as of equally spaced
arrays. Recently, with the gradual maturation of micro-electromechanical
systems (MEMS), and liquid metal technologies, movable antenna (MA) [2], also
known as fluid antenna (FA) [3], flexible antenna [4], and flexible-position
MIMO [5], has been proposed to optimize physical antenna placement adaptively
according to the time-varying dynamic of wireless channels to enhance MIMO
system performance.

Several studies have emerged to analyze and optimize the MA-enhanced MIMO
systems. For instance, in [6, 7], the authors focused on the channel
estimation problem and utilized compressed sensing-based methods to
efficiently estimate the angles of departure, angles of arrival, and complex
coefficients of the multi-path components, which ensures the MA systems to
reap performance gains. Also, based on the estimated channel, [8] analyzed the
maximum channel gain achieved by a single receive MA as compared to its FPA
counterpart, and [9] leveraged the advantages of MA to improve signal-to-noise
ratio (SNR) of multiple destinations for jointly decoding a common message.
Currently, most research focused on rate maximization [10, 11, 12, 13, 14] and
power control [15, 16]. In [10, 11], the transceiver equipped with MAs was
considered to enhance the performance of point-to-point MIMO systems.
Furthermore, in [15, 12, 13, 14], the authors considered a multiuser
transmission scenario where only the BS is equipped with MAs. Specifically,
limited by the realistic electromechanical devices, the motions of the MA
elements were modeled as discrete movements in [15]. In addition, in [16], the
authors considered a multiuser transmission scenario where only users are
equipped with MAs. However, most works employ the gradient descent algorithm
and continuously try to obtain a suitable step size satisfying the
constraints, so the computational complexity is relatively high.

In this letter, we investigate a popular weighted sum-rate (WSR) maximization
problem in an MA-enhanced multiuser MIMO system, where both BS and users are
equipped with MAs. The main contributions are summarized as follows:

  * •

We formulate a WSR maximization problem and transform it into a more tractable
weighted sum mean-square error minimization problem that is compatible with
MAs.

  * •

We employ the block coordinate descent (BCD) method to optimize all variables
alternately and adopt a planar movement mode to achieve a closed-form solution
for optimizing antenna positions, thereby efficiently reducing the
computational complexity.

  * •

Numerical results demonstrate that the MA-enhanced systems ensure superior
performance over the conventional systems. Besides, the planar movement mode
significantly reduces the computational complexity at a little performance
expense.

Notations: We adopt $x$, $\mathbf{x}$, and $\mathbf{X}$ to denote a scalar,
vector, and matrix, respectively. Superscripts $*,T$, and $H$ stand for the
conjugate, transpose, and conjugate transpose, respectively.
$\Pi_{\mathcal{C}}(\mathbf{x})$ denotes the projection of $\mathbf{x}$ onto
the set $\mathcal{C}$. $\odot$ denotes the Hadamard product.

##  II System Model

Figure 1: System model: an MA-enhanced multiuser MIMO system.

As shown in Fig. 1, we consider a multiuser downlink wireless communication
system, where the BS equipped with $M$ MAs serves $K$ single-MA users.
Compared with the conventional communication system, the position of each MA
in the transmit region $\mathcal{C}^{t}$ and the receive region
$\mathcal{C}_{k}^{r},k=1,\cdots,K$, can be adjusted to exploit the spatial
diversity gain via a controller, e.g., stepper motors [2] or liquid metal [3].

Let $\mathbf{t}_{m}=(x_{m},y_{m})^{T}$ and $\mathbf{r}_{k}=(x_{k},y_{k})^{T}$
denote the position of MA $m$ at the BS and the position of the MA at user
$k$, respectively, which are established by the classical Cartesian coordinate
system. By stacking the positions of all antennas together, we denote the
position vectors of the BS and users as
$\mathbf{t}=(\mathbf{t}_{1}^{T},\cdots,\mathbf{t}_{M}^{T})^{T}$ and
$\mathbf{r}=(\mathbf{r}_{1}^{T},\cdots,\mathbf{r}_{K}^{T})^{T}$, respectively.
To avoid potential electrical coupling between adjacent MAs and upgrade the
antenna efficiency in practical implementation, a minimum inter-MA distance
$D$ is required between each pair of MAs at BS [2], i.e.,

$\|\mathbf{t}_{i}-\mathbf{t}_{j}\|\geq D,\forall i\neq j.$ (1)

For the MA-enhanced system, the channel response depends on the position
vector and the channel vector between the BS and user $k$ can be represented
by a function of the position vector, i.e.,
$\mathbf{h}_{k}(\mathbf{t},\mathbf{r}_{k})\in\mathbb{C}^{M\times 1}$. Hence,
the discrete-time signal received at user $k$ is written as

$y_{k}=\mathbf{h}_{k}^{H}(\mathbf{t},\mathbf{r}_{k})\sum_{k=1}^{K}\mathbf{w}_{k}s_{k}+n_{k},$
(2)

where $\mathbf{w}_{k}\in\mathbb{C}^{M\times 1}$ and $s_{k}\in\mathbb{C}$
represent the beamformer and the unit-power complex valued information symbols
intended for user $k$, respectively; $n_{k}\sim\mathcal{CN}(0,\sigma^{2})$
models the additive noise with zero mean and variance $\sigma^{2}$. Based on
(2), the signal-to-interference-plus-noise ratio (SINR) experienced at user
$k$ is obtained as

$\gamma_{k}\triangleq\frac{|\mathbf{h}_{k}^{H}(\mathbf{t},\mathbf{r}_{k})\mathbf{w}_{k}|^{2}}{\sum\limits_{i\neq
k}|\mathbf{h}_{k}^{H}(\mathbf{t},\mathbf{r}_{k})\mathbf{w}_{i}|^{2}+\sigma^{2}}.$
(3)

###  II-A Field-Response-Based Channel Model

Let $L_{k}^{t}$ and $L_{k}^{r}$ denote the total numbers of transmit and
receive channel paths from the BS to user $k$. Then, the signal propagation
phase difference of the $l$-th transmit path for user $k$ between the $m$-th
MA and the reference point $\mathbf{t}_{0}=[0,0]^{T}$ at the BS and the $l$-th
receive path between the MA and reference point $\mathbf{r}_{0}=[0,0]^{T}$ at
user $k$ are, respectively, given by

$\displaystyle\rho_{l,k}^{t}(\mathbf{t}_{m})=\mathbf{t}_{m}^{T}\mathbf{n}_{l,k}^{t}$
$\displaystyle=x_{m}\sin\theta_{l,k}^{t}\cos\phi_{l,k}^{t}+y_{m}\cos\theta_{l,k}^{t},$
(4)
$\displaystyle\rho_{l,k}^{r}(\mathbf{r}_{k})=\mathbf{t}_{m}^{T}\mathbf{n}_{l,k}^{r}$
$\displaystyle=x_{k}\sin\theta_{l,k}^{r}\cos\phi_{l,k}^{r}+y_{k}\cos\theta_{l,k}^{r},$

where
$\mathbf{n}_{l,k}^{t}\triangleq(\sin\theta_{l,k}^{t}\cos\phi_{l,k}^{t},\cos\theta_{l,k}^{t})^{T}$
and
$\mathbf{n}_{l,k}^{r}\triangleq(\sin\theta_{l,k}^{r}\cos\phi_{l,k}^{r},\cos\theta_{l,k}^{r})^{T}$;
$(\theta_{l,k}^{t},\phi_{l,k}^{t})$ and $(\theta_{l,k}^{r},\phi_{l,k}^{r})$
are the elevation and azimuth AoAs for the $l$-th transmit and receive path
between user $k$ and the BS, respectively. Accordingly, the field-response
vector of the receive channel paths between user $k$ and the $m$-th MA at the
BS is given by [8]

$\displaystyle\mathbf{f}_{k}(\mathbf{t}_{m})$
$\displaystyle=\left[e^{j\frac{2\pi}{\lambda}\rho_{1,k}^{t}(\mathbf{t}_{m})},\cdots,e^{j\frac{2\pi}{\lambda}\rho_{L_{k}^{t},k}^{t}(\mathbf{t}_{m})}\right]^{T},$
(5) $\displaystyle\mathbf{g}_{k}(\mathbf{r}_{k})$
$\displaystyle=\left[e^{j\frac{2\pi}{\lambda}\rho_{1,k}^{r}(\mathbf{r}_{k})},\cdots,e^{j\frac{2\pi}{\lambda}\rho_{L_{k}^{r},k}^{r}(\mathbf{r}_{k})}\right]^{T}.$

As a result, the channel vector between user $k$ and the BS is given by

$\mathbf{h}_{k}(\mathbf{t},\mathbf{r}_{k})=\mathbf{F}_{k}^{H}(\mathbf{t})\mathbf{\Sigma}_{k}\mathbf{g}_{k}(\mathbf{r}_{k}),$
(6)

where
$\mathbf{F}_{k}(\mathbf{t})\triangleq(\mathbf{f}_{k}(\mathbf{t}_{1}),\cdots,\mathbf{f}_{k}(\mathbf{t}_{M}))\in\mathbb{C}^{L_{k}\times
M}$ denotes the field-response matrix at the BS, and
$\mathbf{\Sigma}_{k}\in\mathbb{C}^{L_{k}^{t}\times L_{k}^{r}}$ is the path-
response matrix, which represents the multi-path response between all the
transmit and receive channel paths. As can be observed, the movement
introduced by MA causes more significant variations of phases than the
amplitudes [13].

###  II-B Problem Formulation

In this letter, we aim to maximize the weighted sum rate of all the users by
jointly designing the transmit beamformer and the antenna position vectors at
both the BS and the $K$ users, subject to the minimum inter-MA distance (1) at
the BS and the transmit power constraints. Let
$\mathbf{W}\triangleq(\mathbf{w}_{1},\mathbf{w}_{2},\cdots,\mathbf{w}_{K})\in\mathbb{C}^{M\times
K}$. The WSR maximization problem is formulated as

$\displaystyle\underset{\mathbf{W},\mathbf{t},\mathbf{r}}{\text{maximize}}\quad$
$\displaystyle\sum\limits_{k=1}^{K}\alpha_{k}\log(1+\gamma_{k})$ (7a) subject
to $\displaystyle\sum\limits_{k=1}^{K}\|\mathbf{w}_{k}\|^{2}\leq
P_{\mathrm{max}},$ (7b)
$\displaystyle\|\mathbf{t}_{i}-\mathbf{t}_{j}\|_{2}\geq D,\forall i\neq j,$
(7c)
$\displaystyle\mathbf{t}_{i}\in\mathcal{C}^{t},\quad\mathbf{r}_{k}\in\mathcal{C}_{k}^{r},$
(7d)

where the weight $\alpha_{k}\geq 0$ represents the priority of user $k$.
$P_{\mathrm{max}}$ denotes the maximum transmit power. The problem is
challenging mainly due to the nonconvexity of the objective function (7a) and
the constraints (7c), which is an obstacle for the development of an optimal
solution. To reduce the complexity and transform it into a more tractable
problem, we consider the linear receive beamforming strategy so that the
estimated signal at user $k$ is given by

$\hat{s}_{k}=u_{k}^{*}y_{k},$ (8)

where
$\mathbf{u}\triangleq(u_{1},u_{2},\cdots,u_{K})^{T}\in\mathbb{C}^{K\times 1}$
represents the receive beamformer. We assume the independence of $s_{k}$ and
$n_{k}$. Then, the expected mean-square error (MSE) can be written as

$\displaystyle e_{k}\triangleq\mathbb{E}\left[|\hat{s}_{k}-s_{k}|^{2}\right]$
(9)
$\displaystyle=1+|u_{k}|^{2}(\sigma^{2}+\sum_{j=1}^{K}|\mathbf{h}_{k}^{H}(\mathbf{t},\mathbf{r}_{k})\mathbf{w}_{j}|^{2})$
$\displaystyle\quad\quad-2\mathrm{Re}(u_{k}^{*}\mathbf{h}_{k}^{H}(\mathbf{t},\mathbf{r}_{k})\mathbf{w}_{k}).$

where $\mathrm{Re}(\cdot)$ denotes the real part of complex scalar. Then,
inspired by the weighted sum mean-square error minimization (WMMSE) algorithm
proposed for conventional multiuser MIMO systems [17], we introduce an
auxiliary optimization vector variable
$\mathbf{v}=(v_{1},v_{2},\cdots,v_{K})^{T}$ and establish a more tractable MA-
enabled WMMSE problem as follows

$\displaystyle\underset{\mathbf{W},\mathbf{t},\mathbf{r},\mathbf{u},\mathbf{v}}{\text{minimize}}\quad$
$\displaystyle\sum\limits_{k=1}^{K}\alpha_{k}(v_{k}e_{k}-\log(v_{k}))$ (10a)
subject to $\displaystyle\sum\limits_{k=1}^{K}\|\mathbf{w}_{k}\|^{2}\leq
P_{\mathrm{max}},$ (10b)
$\displaystyle\|\mathbf{t}_{i}-\mathbf{t}_{j}\|_{2}\geq D,\forall i\neq j,$
(10c)
$\displaystyle\mathbf{t}_{i}\in\mathcal{C}^{t},\quad\mathbf{r}_{k}\in\mathcal{C}_{k}^{r},$
(10d) $\displaystyle\mathbf{v}\geq\mathbf{0}.$ (10e)

Remark: The equivalence between (7) and (10) can be proved by deriving the
optimal $u_{k},v_{k},\forall k$, in Eq. (10a) and substituting them back into
Eq. (10a), which is similar to [17, Appendix A] for the traditional FPA MIMO
system.

##  III Block Coordinate Descent Method for MA-Enhanced Multiuser MIMO System

BCD method decomposes the optimization variables into multiple blocks and
optimizes different blocks in each iteration with the other blocks fixed.

###  III-A Transceiver Beamformer Design

According to the principle of BCD, we first investigate the optimization of
beamformer $\mathbf{W}$ and related variables $\mathbf{u},\mathbf{v}$ for
fixed MA position vectors $\mathbf{t},\mathbf{r}$. The beamformer design
problem accordingly reduces to

$\displaystyle\underset{\mathbf{W},\mathbf{u},\mathbf{v}}{\text{minimize}}\quad$
$\displaystyle\sum\limits_{k=1}^{K}\alpha_{k}(v_{k}e_{k}-\log(v_{k}))$ (11a)
subject to $\displaystyle\sum\limits_{k=1}^{K}\|\mathbf{w}_{k}\|^{2}\leq
P_{\mathrm{max}},$ (11b) $\displaystyle\mathbf{v}\geq\mathbf{0}.$ (11c)

This problem has been studied extensively in the literature, and one of the
well-known methods to obtain a stationary solution is the WMMSE algorithm with
the following iterative updating rule [17, Table I]:

$\displaystyle u_{k}$
$\displaystyle=\left(\sum_{i=1}^{K}|\mathbf{h}_{k}(\mathbf{t},\mathbf{r}_{k})^{H}\mathbf{w}_{i}|^{2}+\sigma^{2}\right)^{-1}\mathbf{h}_{k}(\mathbf{t},\mathbf{r}_{k})^{H}\mathbf{w}_{k},$
(12a) $\displaystyle v_{k}$
$\displaystyle=\left(1-u_{k}^{*}\mathbf{h}_{k}(\mathbf{t},\mathbf{r}_{k})^{H}\mathbf{w}_{k}\right)^{-1},$
(12b) $\displaystyle\mathbf{w}_{k}$
$\displaystyle=\alpha_{k}u_{k}v_{k}\left(\mu\mathbf{I}_{M}+\sum_{i=1}^{K}\alpha_{i}|u_{i}|^{2}v_{i}\mathbf{h}_{i}(\mathbf{t},\mathbf{r}_{k})\mathbf{h}_{i}(\mathbf{t},\mathbf{r}_{k})^{H}\right)^{-1}$
$\displaystyle\qquad\mathbf{h}_{k}(\mathbf{t},\mathbf{r}_{k}),$ (12c)

where $\mu\geq 0$ is the optimal dual variable for the transmit power
constraint in (11b). To describe the computation of $\mu$, we let
$\mathbf{w}_{k}(\mu)$ denote the right-hand side of (12c). When the matrix
$\sum_{i=1}^{K}\alpha_{i}|u_{i}|^{2}v_{i}\mathbf{h}_{i}(\mathbf{t},\mathbf{r}_{k})\mathbf{h}_{i}(\mathbf{t},\mathbf{r}_{k})^{H}$
is invertible and $\sum_{k=1}^{K}\|\mathbf{w}_{k}(0)\|^{2}\leq
P_{\mathrm{max}}$, then $\mathbf{w}_{k}=\mathbf{w}_{k}(0)$, otherwise the
optimal $\mu$ should satisfy
$\sum_{k=1}^{K}\|\mathbf{w}_{k}(\mu)\|^{2}=P_{\mathrm{max}}$. Note that the
left-hand side is a decreasing function in $\mu$ for $\mu>0$. Hence, it can be
obtained via one dimensional search techniques, e.g., bisection method.

###  III-B MA Position Design at BS

To reduce the computational complexity, we resort to alternative optimization
to design MA positions at the BS. For fixed $\mathbf{W}$,
$\mathbf{t}_{j},\forall j\neq m$, and $\mathbf{r}$, the problem of designing
MA $m$ position at BS accordingly reduces to

$\displaystyle\underset{\mathbf{t}_{m}\in\mathcal{C}^{t}}{\text{minimize}}\,\,\,$
$\displaystyle\sum_{k=1}^{K}\left(\mathbf{f}_{k}^{H}(\mathbf{t}_{m})\mathbf{A}_{k,m}\mathbf{f}_{k}(\mathbf{t}_{m})+\mathrm{Re}(\mathbf{b}_{k,m}^{H}\mathbf{f}_{k}(\mathbf{t}_{m}))\right)$
(13a) subject to $\displaystyle\|\mathbf{t}_{m}-\mathbf{t}_{j}\|_{2}\geq
D,\forall j\neq m,$ (13b)

where

$\displaystyle\mathbf{A}_{k,m}$
$\displaystyle\triangleq\alpha_{k}v_{k}|u_{k}|^{2}\|\mathbf{w}_{m,:}\|^{2}\mathbf{\Sigma}_{k}\mathbf{g}_{k}(\mathbf{r}_{k})\mathbf{g}_{k}(\mathbf{r}_{k})^{H}\mathbf{\Sigma}_{k}^{H},$
(14) $\displaystyle\mathbf{b}_{k,m}$ $\displaystyle\triangleq
2\alpha_{k}v_{k}\bigg{(}|u_{k}|^{2}\sum_{j=1}^{K}\bigg{(}w_{m,j}^{*}\sum_{n\neq
m}w_{n,j}$
$\displaystyle\qquad\mathbf{g}_{k}(\mathbf{r}_{k})^{H}\mathbf{\Sigma}_{k}^{H}\mathbf{f}_{k}(\mathbf{t}_{n})\bigg{)}-u_{k}w_{m,k}^{*}\bigg{)}\mathbf{\Sigma}_{k}\mathbf{g}_{k}(\mathbf{r}_{k}).$

Next, we propose to leverage the minorization maximization (MM) [18] technique
to update antenna position vector. The key to the success of MM lies in
constructing a surrogate function satisfying the upper bound property for both
objective function and constraints. Now, we introduce some Lemmas that
facilitate the implementation of MM.

######  Lemma 1.

The quadratic form $\mathbf{x}^{H}\mathbf{L}\mathbf{x}$, where $\mathbf{L}$ is
a Hermitian matrix, can be upper bounded as [18, Example 13]:

$\displaystyle\mathbf{x}^{H}\mathbf{L}\mathbf{x}\leq\mathbf{x}^{H}\mathbf{M}\mathbf{x}+2\mathrm{Re}(\mathbf{x}^{H}(\mathbf{L}-\mathbf{M})\mathbf{x}_{0})+\mathbf{x}_{0}^{H}(\mathbf{M}-\mathbf{L})\mathbf{x}_{0},$
(15)

where $\mathbf{M}\succeq\mathbf{L}$. Equality is achieved at
$\mathbf{x}=\mathbf{x}_{0}$.

######  Lemma 2.

The linear form
$z_{k}(\mathbf{t}_{m})=\mathrm{Re}(\mathbf{b}^{H}\mathbf{f}_{k}(\mathbf{t}_{m}))$,
where $\mathbf{f}_{k}(\mathbf{t}_{m})$ is the field-response vector defined in
(5), can be lower bouneded and upper bounded as [2, Eq. (26) and Appendix
A-B]:

$\displaystyle z_{k}(\mathbf{t}_{m})$ $\displaystyle\geq
z_{k}(\mathbf{t}_{m,0})+\nabla
z_{k}(\mathbf{t}_{m,0})^{T}(\mathbf{t}_{m}-\mathbf{t}_{m,0})$
$\displaystyle-\frac{4\pi^{2}}{\lambda^{2}}\|\mathbf{b}\|_{1}(\mathbf{t}_{m}-\mathbf{t}_{m,0})^{T}(\mathbf{t}_{m}-\mathbf{t}_{m,0}),$
(16a) $\displaystyle z_{k}(\mathbf{t}_{m})$ $\displaystyle\leq
z_{k}(\mathbf{t}_{m,0})+\nabla
z_{k}(\mathbf{t}_{m,0})^{T}(\mathbf{t}_{m}-\mathbf{t}_{m,0})$
$\displaystyle+\frac{4\pi^{2}}{\lambda^{2}}\|\mathbf{b}\|_{1}(\mathbf{t}_{m}-\mathbf{t}_{m,0})^{T}(\mathbf{t}_{m}-\mathbf{t}_{m,0}),$
(16b)

where

$\displaystyle\nabla z_{k}(\mathbf{t}_{m,0})$
$\displaystyle=\mathrm{Re}(\mathbf{b}^{H}\nabla\mathbf{f}_{k}(\mathbf{t}_{m,0}))^{T},$
(17) $\displaystyle\nabla\mathbf{f}_{k}(\mathbf{t}_{m,0})$
$\displaystyle=j\frac{2\pi}{\lambda}(\mathbf{n}_{1,k},\cdots,\mathbf{n}_{L_{k}^{t},k})^{T}\odot(\mathbf{f}_{k}(\mathbf{t}_{m,0}),\mathbf{f}_{k}(\mathbf{t}_{m,0})).$

Equality is achieved at $\mathbf{t}_{m}=\mathbf{t}_{m,0}$.

Remark: The proof of Eq. (16a) is provided in [2, Eq. (26) and Appendix A-B].
Thus, we can derive Eq. (16b) directly by replacing $\mathbf{b}$ with
$-\mathbf{b}$ in (16a).

From the Cauchy-Schwartz inequality, at iteration $(n)$, a minorizing function
for the constraint is constructed at $\mathbf{t}_{m}=\mathbf{t}_{m}^{(n)}$ as
follows:

$\|\mathbf{t}_{m}-\mathbf{t}_{j}\|_{2}\geq\frac{(\mathbf{t}_{m}^{(n)}-\mathbf{t}_{j})^{T}(\mathbf{t}_{m}-\mathbf{t}_{j})}{\|\mathbf{t}_{m}^{(n)}-\mathbf{t}_{j}\|_{2}}.$
(18)

Based on Lemmas 1-2, at iteration $(n)$, a majorizing function for the
objective function is constructed at $\mathbf{t}_{m}=\mathbf{t}_{m}^{(n)}$ as
follows:

$\displaystyle\sum_{k=1}^{K}\left(\mathbf{f}_{k}^{H}(\mathbf{t}_{m})\mathbf{A}_{k,m}\mathbf{f}_{k}(\mathbf{t}_{m})+\mathrm{Re}(\mathbf{b}_{k,m}^{H}\mathbf{f}_{k}(\mathbf{t}_{m}))\right)$
(19)
$\displaystyle\leq\sum_{k=1}^{K}\mathrm{Re}(\hat{\mathbf{b}}_{k,m}^{H}\mathbf{f}_{k}(\mathbf{t}_{m}))+\mathrm{const.}$
$\displaystyle\leq\sum_{k=1}^{K}\frac{4\pi^{2}}{\lambda^{2}}\|\hat{\mathbf{b}}_{k,m}\|_{1}\|\mathbf{t}_{m}\|^{2}$
$\displaystyle\quad+(\nabla
z_{k}(\mathbf{t}_{m}^{(n)})-\frac{8\pi^{2}}{\lambda^{2}}\|\hat{\mathbf{b}}_{k,m}\|_{1}\mathbf{t}_{m}^{(n)})^{T}\mathbf{t}_{m}+\mathrm{const.}$

where

$\displaystyle\hat{\mathbf{b}}_{k,m}$ $\displaystyle\triangleq
2(\mathbf{A}_{k,m}-\alpha_{k}v_{k}|u_{k}|^{2}\|\mathbf{w}_{m,:}\|^{2}$ (20)
$\displaystyle\qquad\cdot\|\mathbf{\Sigma}_{k}\mathbf{g}_{k}(\mathbf{r}_{k})\|^{2}\mathbf{I})\mathbf{f}_{k}(\mathbf{t}_{m}^{(n)})+\mathbf{b}_{k,m},$
$\displaystyle z_{k}(\mathbf{t}_{m})$
$\displaystyle\triangleq\mathrm{Re}(\hat{\mathbf{b}}_{k,m}^{H}\mathbf{f}_{k}(\mathbf{t}_{m})).$

Therefore, the antenna position vector $\mathbf{t}_{m}$ is updated as

$\displaystyle\mathbf{t}_{m}^{(n+1)}$
$\displaystyle=\underset{\mathbf{t}_{m}\in\mathcal{C}^{t}}{\text{argmin}}\sum_{k=1}^{K}\frac{4\pi^{2}}{\lambda^{2}}\|\hat{\mathbf{b}}_{k,m}\|_{1}\|\mathbf{t}_{m}\|^{2}$
$\displaystyle\qquad\quad+(\nabla
z_{k}(\mathbf{t}_{m}^{(n)})-\frac{8\pi^{2}}{\lambda^{2}}\|\hat{\mathbf{b}}_{k,m}\|_{1}\mathbf{t}_{m}^{(n)})^{T}\mathbf{t}_{m}$
(21a) $\displaystyle\text{ subject to
}\frac{(\mathbf{t}_{m}^{(n)}-\mathbf{t}_{j})^{T}(\mathbf{t}_{m}-\mathbf{t}_{j})}{\|\mathbf{t}_{m}^{(n)}-\mathbf{t}_{j}\|_{2}}\geq
D,\forall j\neq m,$ (21b)

which is a typical convex quadratic programming (QP) problem and can be
efficiently solved by quadprog or CVX.

###  III-C MA Position Design at Users

For fixed $\mathbf{W}$, $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{t}$, each
$\mathbf{r}_{k}$ can be updated in parallel since they are independent in
(10). Specifically, the problem of designing MA position at user $k$
accordingly reduces to

$\displaystyle\underset{\mathbf{r}_{k}\in\mathcal{C}_{k}^{r}}{\text{minimize}}$
$\displaystyle\mathbf{g}_{k}^{H}(\mathbf{r}_{k})\mathbf{C}_{k}\mathbf{g}_{k}(\mathbf{r}_{k})+\mathrm{Re}(\mathbf{d}_{k}^{H}\mathbf{g}_{k}(\mathbf{r}_{k}))$
(22)

where

$\displaystyle\mathbf{C}_{k}$
$\displaystyle\triangleq\sum_{j=1}^{K}|u_{k}|^{2}\mathbf{\Sigma}_{k}^{H}\mathbf{F}_{k}(\mathbf{t})\mathbf{w}_{j}\mathbf{w}_{j}^{H}\mathbf{F}_{k}(\mathbf{t})^{H}\mathbf{\Sigma}_{k},$
(23) $\displaystyle\mathbf{d}_{k}$
$\displaystyle\triangleq-2u_{k}^{*}\mathbf{\Sigma}_{k}^{H}\mathbf{F}_{k}(\mathbf{t})\mathbf{w}_{k}.$

Similar to Lemmas 1-2, at iteration $(n)$, a majorizing function for the
objective function is constructed at $\mathbf{r}_{k}=\mathbf{r}_{k}^{(n)}$ as
follows:

$\displaystyle\mathbf{g}_{k}^{H}(\mathbf{r}_{k})\mathbf{C}_{k}\mathbf{g}_{k}(\mathbf{r}_{k})+\mathrm{Re}(\mathbf{d}_{k}^{H}\mathbf{g}_{k}(\mathbf{r}_{k}))$
(24)
$\displaystyle\leq\frac{4\pi^{2}}{\lambda^{2}}\|\hat{\mathbf{d}}_{k}\|_{1}\|\mathbf{r}_{k}\|^{2}$
$\displaystyle\quad+(\nabla
z_{k}(\mathbf{r}_{k}^{(n)})-\frac{8\pi^{2}}{\lambda^{2}}\|\hat{\mathbf{d}}_{k}\|_{1}\mathbf{r}_{k}^{(n)})^{T}\mathbf{r}_{k}+\mathrm{const.}$

where

$\displaystyle\hat{\mathbf{d}}_{k}$ $\displaystyle\triangleq
2(\mathbf{C}_{k}-\sum_{j=1}^{K}|u_{k}|^{2}\|\mathbf{\Sigma}_{k}^{H}\mathbf{F}_{k}(\mathbf{t})\mathbf{w}_{j}\|^{2}\mathbf{I})\mathbf{g}_{k}(\mathbf{r}_{k}^{(n)})+\mathbf{d}_{k}$
(25) $\displaystyle z_{k}(\mathbf{r}_{k})$
$\displaystyle\triangleq\mathrm{Re}(\hat{\mathbf{d}}_{k}^{H}\mathbf{g}_{k}(\mathbf{r}_{k})).$

Therefore, the MA position at user $k$ is updated as

$\displaystyle\mathbf{r}_{k}^{(n+1)}=\underset{\mathbf{r}_{k}\in\mathcal{C}_{k}^{r}}{\text{argmin}}\frac{4\pi^{2}}{\lambda^{2}}\|\hat{\mathbf{d}}_{k}\|_{1}\|\mathbf{r}_{k}\|^{2}$
(26) $\displaystyle\qquad\qquad\qquad+(\nabla
z_{k}(\mathbf{r}_{k}^{(n)})-\frac{8\pi^{2}}{\lambda^{2}}\|\hat{\mathbf{d}}_{k}\|_{1}\mathbf{r}_{k}^{(n)})^{T}\mathbf{r}_{k}$
$\displaystyle=\Pi_{\mathcal{C}_{k}^{r}}\left(\mathbf{r}_{k}^{(n)}-\frac{\sum_{k=1}^{K}\nabla
z_{k}(\mathbf{r}_{k}^{(n)})}{\sum_{k=1}^{K}\frac{8\pi^{2}}{\lambda^{2}}\|\hat{\mathbf{d}}_{k}\|_{1}}\right).$

##  IV Low-Complexity Design

To reduce the computational complexity, we propose the planar movement mode at
BS, where each MA is only allowed to move in given planar area and the minimum
distance between any two areas is set as $D$ to avoid coupling effect. To
distinguish, the movement mode described in Section III is referred to as the
general movement mode.

###  IV-A Planar Movement Mode at BS

The position optimization problem (13) can be recast as

$\displaystyle\underset{\mathbf{t}_{m}\in\mathcal{C}_{m}^{t}}{\text{minimize}}$
$\displaystyle\sum_{k=1}^{K}\left(\mathbf{f}_{k}^{H}(\mathbf{t}_{m})\mathbf{A}_{k,m}\mathbf{f}_{k}(\mathbf{t}_{m})+\mathrm{Re}(\mathbf{b}_{k,m}^{H}\mathbf{f}_{k}(\mathbf{t}_{m}))\right),$
(27)

where $\mathcal{C}_{m}^{t}$ is the movement area of MA $m$. Therefore, through
constructing the tight upper bound of the objective function as Eq. (19), the
antenna position vector $\mathbf{t}_{m}$ is updated as

$\displaystyle\mathbf{t}_{m}^{(n+1)}=\underset{\mathbf{t}_{m}\in\mathcal{C}_{m}^{t}}{\text{argmin}}\sum_{k=1}^{K}\frac{4\pi^{2}}{\lambda^{2}}\|\hat{\mathbf{b}}_{k,m}\|_{1}\|\mathbf{t}_{m}\|^{2}$
(28) $\displaystyle\qquad\qquad\qquad+(\nabla
z_{k}(\mathbf{t}_{m}^{(n)})-\frac{8\pi^{2}}{\lambda^{2}}\|\hat{\mathbf{b}}_{k,m}\|_{1}\mathbf{t}_{m}^{(n)})^{T}\mathbf{t}_{m}$
$\displaystyle=\Pi_{\mathcal{C}_{m}^{t}}\left(\mathbf{t}_{m}^{(n)}-\frac{\sum_{k=1}^{K}\nabla
z_{k}(\mathbf{t}_{m}^{(n)})}{\sum_{k=1}^{K}\frac{8\pi^{2}}{\lambda^{2}}\|\hat{\mathbf{b}}_{k,m}\|_{1}}\right).$

###  IV-B Convergence and Complexity Analysis

_Convergence analysis:_ Based on the WMMSE algorithm for transmit beamformer
design and the MM algorithm for MA position design, a stationary point is
guaranteed for each subproblem [17, 18]. Therefore, the initial equivalent
function is guaranteed to increase monotonically, and our proposed algorithm
is convergent.

_Complexity analysis:_ The complexity of updating $\mathbf{w}$ at each
iteration of the subproblem is $\mathcal{O}(K^{2}M^{2})$ [17, §IV]. The
complexity of updating $\mathbf{t}$ mainly depends on computing coefficients
$\hat{\mathbf{b}}_{k,m}$ and $\nabla z_{k}(\mathbf{t}_{m}^{(n)})$ and solving
the QP problem. Therefore, the complexity of the general movement mode is
$\mathcal{O}(KM^{2}+KML_{t}L_{r}+KML_{t}^{2}+M^{2.5})$, while for the planar
movement mode, it is $\mathcal{O}(KM^{2}+KML_{t}L_{r}+KML_{t}^{2}+M)$ at each
iteration of subproblem. Similarly, the complexity of updating $\mathbf{r}$ is
$\mathcal{O}(K^{2}M+KML_{t}L_{r}+KML_{r}^{2}+K)$.

##  V Numerical Results

In this section, we numerically evaluate the performance of the proposed
algorithm. The number of MAs at BS is $M=16$. Each element of the path-
response matrix is assumed to be independently and identically distributed
(i.i.d.) Gaussian. The noise power is set as $\sigma^{2}=15$ dBm [11]. The
normalized wavelength is set as $\lambda=1$ m. The minimum distance between
adjacent antennas is set as $D=\lambda/2$. The size of the MA movement areas
at BS is set as $5\lambda\times 5\lambda$.

We compare the proposed design, named TMA-RMA, with the following baselines:
i) Traditional antennas: All antenna positions are fixed; ii) TFPA-RMA: The
antenna positions at the BS are fixed, but the antenna positions at users can
be adjustable in the receive region; iii) TMA-RFPA: The antenna positions at
the users are fixed, but the antenna positions at the BS are adjustable in the
transmit region.

Fig. 4 demonstrates the convergence performance of our proposed algorithms.
Firstly, it can be observed that the differences in the achieved average sum
rate between the two movement modes are relatively small. Secondly, the low-
complexity planar movement mode clearly exhibits a faster convergence speed,
which aligns with our analysis.

Fig. 4 illustrates the average sum rate for various antenna configurations at
both BS and users. Firstly, for different numbers of MAs at BS, the proposed
design consistently outperforms three baselines in terms of the average sum
rate. The performance of the system improves as the number of antennas
increases. Secondly, as the DoF of antenna positions increases, so does the
system gain. It indicates that deploying MAs holds great promise for enhancing
wireless communication systems, especially when compared to the traditional
antennas. Finally, the planar movement mode achieves close performance to the
general movement mode, which motivates us for the further design of low-
complexity systems.

Fig. 4 depicts the average sum rate for different minimum inter-MA distances
at BS. As the minimum distance increases, both the DoF and system gain
decrease accordingly.

Figure 2: Convergence of the proposed algorithms.

Figure 3: Evaluation of weighted sum rate under different number of antennas
$M$.

Figure 4: Evaluation of weighted sum rate under different minimum inter-MA
distance $D$.

##  VI Conclusion

In this letter, we employed the MAs at both BS and users to improve the
weighted sum rate of wireless networks. We adopted BCD algorithm to optimize
all variables, and then confined each antenna to a designated area, i.e.,
planar movement mode, to reduce the computational complexity. Simulation
results confirmed the huge potential of MAs in improving the performance of
future communications systems.

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###### Abstract

Study of the role of the electron screening in the $\alpha$ decay is advanced,
aimed at the experimental test in the storage rings. To this end, systematic
calculation of the effect in heavy ions of the nuclei within the Ra to Po
domain of the alpha emitters is conducted within the adiabatic approach. The
obtained effect is to slow down the decay by an amount within the percent
value. It is of the opposite sign comparing to what is predicted within the
conventional frozen shell model. The reason for this divergence is
demonstrated. Testing this difference experimentally in storage ring
facilities is proposed. It is of great interest for advancing the
astrophysical fundamental research.

Prospects for study of the effect of the electronic screening on the $\alpha$
decay in the storage rings

F. F. Karpeshin1, M. B. Trzhaskovskaya2, and L. F. Vitushkin1
1D. I. Mendeleyev Institute for Metrology, Saint-Petersburg, Russia

2National Research Center “Kurchatov Institute” — Petersburg Nuclear Physics
Institute, Russia

E-mail<EMAIL_ADDRESS>

##  1 Introduction

The $\alpha$ decay plays a significant role both in practical applications and
scientific research, starting from astrophysics (e.g. [1]) and up to the
laboratory research in plasma physics. At the same time, there is a
contradiction between the laboratory study, which considers nuclear reactions
with no respect to a possible role of the electron shell or environment, and
applications, which deal with various electronic shells or environments.
Specifically, the $\alpha$ decay affects the nuclear synthesis in the stars.
The extent of the screening effect in ultra-high-density stellar environments
might become significant and would deserve further investigation. The
beginning was laid, for example, in Refs. [2].

Another astrophysical issue is related with the time reversibility of the
nuclear reactions occurring in the interior of stars, whose study is one of
the most promising areas of modern nuclear physics. Most of these reactions
encounter with the similar problem, as the $\alpha$ decay, aggravated by
extremely small cross-sections at small energies, being still not available
for direct measurements in the laboratory. Consequently, indirect approaches
are developed in order to better know their cross-sections and rates, such as
the asymptotic normalization coefficients, or the Trojan horse method (e.g.
[3]). From this viewpoint, direct information on the rates and role of the
electron screening in the reverse process of $\alpha$ decay becomes important.

Surprisingly, up to paper [4] of 2013, everybody considered this question
within the framework of the frozen electron shell (FS) model (e.g. [5, 6, 7,
8, 9] and Refs. cited therein), although the electrons are four orders of
magnitude lighter than the $\alpha$ particles, and are certainly strongly
affected by the $\alpha$ particle slowly traversing the shell. The calculated
results were even of different signs, in spite of simple arguments, which can
be readily told out based on the physical ground [10]. They suggest that the
electron environment retards the decay. This might promote formation of heavy
elements and actinides in the $r$ process in the stars. The problem is
therefore to involve the interaction of the $\alpha$ particle with the
electronic shell into the consideration. The main difficulty, which however
arises on this way, is in treating the interplay of the two different scales
involved. From one side, one has to take into account a decrease of $\Delta
Q\approx$ 40 keV in the energy of the $\alpha$ particle. This arises and is
being finally formed on the distances of the atomic scale. It is expected to
act as to suppress the decay ([5]). In contrast, from the other side, $\alpha$
decay is predetermined by the strong short-range $\alpha$-nucleus interaction.
Its influence is ruled out on the outer turning point of the Coulomb barrier,
which is essentially inside the electron shell, tens fermi from the nuclear
surface. As a result, an intuitive expectation is that the shell should not
affect the probability significantly. Moreover, in this region the electrons
produce a negative potential for the $\alpha$ particle, which may be assumed
to facilitate the decay. Consecutive account of all these factors can be only
performed within the adiabatic approach, which was proposed in Ref. [4].
Analogy was noted with suppression of prompt fission in muonic atoms of
actinides. That example teaches that it is not the electronic potential so
important but its gradient, determining the force acting on the $\alpha$
particle. Important for experimental research details were elaborated in Refs.
[10, 11]. Furthermore, the adiabatic approach was recognized and extended in
Refs. [12, 13]. Herein, we advance the study of the effect of the electron
screening and discuss a feasibility of experimental test for it, using such
contemporary facilities like storage rings, available at GSI and IMP Lanzhou.
In the next section, we remind the principles of the model. Numerical results
for the isotopes of Po to Ra domain are obtained in Section 3. Prospects of
the experimental study are discussed in Section 4. Resume is derived in the
concluding section 5.

##  2 Remind of the model: physical premises

Within the framework of the Gamow theory, a conventional expression for the
$\alpha$ decay probability is essentially given by the product of two factors:
the cluster preformation and the penetration probabilities. The former is
assumed not be affected by the shell. The latter factor $P$ is determined by
action $S$ as follows:

$P=e^{-2S}\,,$ (1)

with

$S=\int_{R_{1}}^{R_{2}}\sqrt{2m[E-V(R)]}dR\,.$ (2)

Here $V(R)$ is the potential energy of interaction of the emitted $\alpha$
particle with the rest of the system, including interaction with the nucleus
and the electronic shell in the case of atomic system. $m$ is the mass of the
$\alpha$ particle, and $E$ is its kinetic energy in the asymptotic region
where $V(R)$ can be neglected. Furthermore, $R_{1}$, $R_{2}$ are the turning
points. The effect of the electron screening is then expressed as

$Y=P_{a}/P_{n}-1\,,$ (3)

where subscripts ‘n’ and ‘a’ indicate either the case of bare nuclei or atoms,
respectively. The problem of consistent taking into account the aforementioned
factors became a stumbling block for many early calculations. The study of
Patyk et al. [9], who performed the most consistent calculations of alpha
decay for the chain of radon isotopes, put a period to the development of the
FS model.

In more detail, in the case of bare nuclei, the potential energy consists of
the strong short-range interaction $V_{n}(R)$, centrifugal repulsion
$V_{cf}(R)$ and Coulomb attraction $V_{Coul}(R)$. It reads as follows:

$V(R)=V_{n}(R)+V_{cf}(R)+V_{Coul}(R)\equiv V_{N}(R)\,.$ (4)

Nuclear potential $V_{N}(R)$ calculated for 226Ra, with the parameters from
[14], is illustrated in Fig. 1. It looks as a shallow well, formed by the
superposition of the Coulomb repulsion and the strong potential well
$V_{n}(R)$ in the nuclear vicinity, and a flat wide slope formed by
$V_{Coul}(R)$ at more distant $R$. The straight line shows the energy of the
emitted $\alpha$ particle. The centrifugal potential $V_{cf}(R)$ arises in the
case of the angular momentum of the $\alpha$ particle $L\neq 0$, and is not
shown in the Figure. It only slightly modifies the shape of the barrier,
somewhat changing the effect of the electron shell in the third decimal.

Figure 1: Potential energy of the $\alpha$ particle interaction with the
nucleus (thick full curve), resulting from the superposition of the strong
attractive potential well (dashed curve) and the repulsive Coulomb interaction
(dotted line).

In atoms, the adiabatic potential energy of the $\alpha$ interaction with the
shell $U_{e-\alpha}^{\text{ad}}(r)$ is added:

$V(R)\equiv V_{a}(R)=V_{N}(R)+U_{e-\alpha}^{\text{ad}}(R)\,.$ (5)

Furthermore, the $Q$ value is diminished by $\Delta Q$ because of the
rearrangement of the shell in the daughter atoms, with the related change of
the total binding energy $\Delta B\equiv\Delta Q$. The resulting expression
for the action integral becomes as follows:

$S_{a}=\int_{R_{1}^{\prime}}^{R_{2}^{\prime}}\sqrt{2m[E-V_{N}(R)+\Delta
Q-U_{e-\alpha}^{\text{ad}}(R)]}dR\,.$ (6)

In order to better realize the difference between the adiabatic and FS models,
let us dwell in more detail, than it is done in Refs. [4, 10], on the
construction of the $e-\alpha$ prompt electrostatic potential of interaction
of the $\alpha$ particle with the electronic shell
$U_{e-\alpha}^{\text{ad}}(R)$. It is one of the main constituents of the
adiabatic method. Let us put down a general expression as follows:

$U_{e-\alpha}^{\text{ad}}(R)=-\zeta
e^{2}\int\frac{\rho_{e}(r)}{|\mathbf{r}-\mathbf{R}|}\ d^{3}r+\text{const}\,$
(7)

with $\rho_{e}(r)$ being the prompt electron density, and $\zeta$ = 2 — charge
of the $\alpha$ particle. $\rho_{e}(r)$ depends on $R$. It can be considered
as spherically-symmetric at $R<R_{s}$, $R_{s}$ being the point of appearance
of the $\alpha$ particle on the nuclear surface. Its value is usually
calculated by means of the self-consistent method (e.g. [15]):

$\rho_{e}(r)=4\pi\sum_{i}N_{i}[G_{i}^{2}(r)+F_{i}^{2}(r)]\,,$ (8)

where $G_{i}(r)$ and $F_{i}(r)$ are the radial Dirac electron wave functions,
normalized as follows:

$\int_{0}^{\infty}[G_{i}^{2}(r)+F_{i}^{2}(r)]dr=1\,.$ (9)

Summation in (8) is performed over the shells $i$, with $N_{i}$ being the
occupation numbers. We introduced an arbitrary constant in Eq. (7), which we
define later. By definition, $e-\alpha$ interaction potential in the FS model
is obtained with const = 0. Starting from this moment, at $R>R_{s}$,
$G_{i}(r)$ and $F_{i}(r)$ depend on $R$. This is an important element
constituting difference between the FS model and the adiabatic approximation.
Increasing $R$ also increases the r.m.s. radius of the total nuclear charge,
which causes diminishing the binding energy of the atom and rise of the
electronic term [4, 10]. Furthermore, the electronic density gradually becomes
aspherical, following the motion of the $\alpha$ particle. For the
calculation, in the first order of the perturbation theory this effect can be
neglected at small $R$ around the nuclear surface, including a physically
interesting subbarrier region. Then Eq. (7) can be expressed as follows:

$\displaystyle U_{e-\alpha}^{\text{ad}}(R)=4\pi\phi(R)+\text{const}\,,$ (10)
$\displaystyle\phi(R)=u(R)-\frac{\zeta ze^{2}}{R_{e}}\,,$ $\displaystyle
u(R)=-\zeta
e^{2}\sum_{i}N_{i}\int_{0}^{R}[G_{i}(r)^{2}+F_{i}^{2}(r)](\frac{1}{R}-\frac{1}{r})dr\,,$
(11)
$\displaystyle\frac{1}{R_{e}}=\sum_{i}N_{i}\int_{0}^{\infty}[G_{i}^{2}(r)+F_{i}^{2}(r)]\frac{dr}{r}\,.$
(12)

Here we denoted $z$ the number of electrons in the atoms, not to be mixed with
the atomic number $Z$. By definition, function $\phi(R)$ determines the
potential in the FS model. We normalize the constant by the natural boundary
condition:

$U_{e-\alpha}^{\text{ad}}(R)\to 0\quad\text{at}\quad R\to\infty\,.$ (13)

One cannot apply condition (13) to the expression (10) yet for expansion (11)
is not valid at large $R$.

Note that at the starting point, the potential energy of $e-\alpha$
interaction is $U_{e-\alpha}^{\text{ad}}(R_{s})$. Then, according to the
Feynman-Hellman theorem [16], within the framework of the adiabatic approach,
the work $w(R)$ done by the $\alpha$ particles over the electron shell when
moving from the starting point $R_{s}$ to a point $R$ acquires the usual
expression depending on the potential difference at the end and start points:

$w(R)=\int_{R_{s}}^{R}\frac{dU_{e-\alpha}^{\text{ad}}(R^{\prime})}{dR^{\prime}}\
dR^{\prime}=U_{e-\alpha}^{\text{ad}}(R)-U_{e-\alpha}^{\text{ad}}(R_{s})\,.$
(14)

The constant which is not defined yet cancels out in Eq. (14). This equation
with the account of (13) allows one to express the constant in terms of
$\Delta Q$. Going to the limit $R\to\infty$, one arrives at

$\Delta Q=U_{e-\alpha}^{\text{ad}}(R_{s})\,.$ (15)

Now condition (15) defines the constant:

$\text{const}=\Delta Q-\phi(R_{s})\,.$ (16)

Substituting Eq. (16) into (6), we see that the effect of the electron shell
is reduced to the mere addition of a purely electronic effective potential [4]

$\displaystyle W_{eff}^{\text{ad}}(R)=4\pi[\phi(R)-\phi(R_{s})]=$
$\displaystyle-4\pi\zeta
e^{2}\sum_{i}N_{i}\int_{R_{s}}^{R}[G_{i}(r)^{2}+F_{i}^{2}(r)](\frac{1}{R}-\frac{1}{r})dr\,.$
(17)

Eq. (17) differs from Eq. (11) by the low bound of integration. This result is
illustrated in Figure 2.

Figure 2: Thick full curve shows the $Q$ value in the case of a bare nucleus;
dotted line — the adiabatic $e-\alpha$ potential, which should be added in the
case of a neutral atom, after which the alpha energy becomes $Q^{\prime}$. The
variation of the adiabatic electronic potential equals the energy of the
electronic rearrangement $\Delta B$. Dotted line shows that the difference
$\Delta Q=Q-Q^{\prime}$ exactly equals $|\Delta B|$. The action in the case of
neutral atoms does not change if the $Q^{\prime}$ value is shifted
simultaneously with the electronic potential by the same value. The figure
thus illustrates that the action for the atom can be considered as that for
the bare nucleus, with the same $Q$ value, but adding the effective shifted
electronic potential (thin solid line).

Involving the effective electronic potential $W_{eff}^{\text{ad}}(R)$ resolves
the conflict of scales mentioned in the Introduction. It appears as a
perturbative positive definite extra potential, reasonably weak enough, which
slightly affects the barrier penetration probability. Below, we will mean
potential (17) under the alpha-electron interaction. The energy $E$ in formula
(17) remains the same as in Eq. (2) for bare nuclei.

The Feynman—Hellman theorem is not relevant in the FS model, in view of that
it is only valid for stable systems [16], which obviously do not include a FS
model. In this case, one can calculate the $e-\alpha$ interaction potential
directly by means of Eqs. (10) to (12) with const = 0, whilst holding the
$\Delta Q$ value in Eq. (6). For the purpose of comparison of the models, let
us introduce the effective potential for the FS model $W_{eff}^{\text{FS}}(R)$
by mere inclusion of $\Delta Q$ value into $\phi(R)$:

$W_{eff}^{\text{FS}}=4\pi\phi(R)-\Delta Q\,.$ (18)

Then in the both models, the expression for the action integral reads as
follows:

$S_{a}=\int_{R_{1}^{\prime}}^{R_{2}^{\prime}}\sqrt{2m[E-V_{n}(R)-V_{Coul}(R)-W_{eff}(R)}dR\,,$
(19)

with $W_{eff}(R)$ given by either (17) or (18).

In Figs. 3 the resulting interaction potential of the $\alpha$ particle with
the electron shell is presented in more detail in the case of neutral
${}^{226}_{88}$Ra atoms $\alpha$ decay. The calculations are performed by
means of Eqs. (10) to (12), using the RAINE package of the computer codes
[17]. Fig. 3a shows general view of the calculated potentials. At small $R$,
the adiabatic potential starts from $U_{e-\alpha}^{\text{ad}}(R_{s})=\Delta Q$
at $R=R_{s}$. The $\Delta Q$ value is calculated as the difference of the
total binding energies of the mother and daughter atoms. At large $R$,
qualitative view of the curve is shown. It vanishes at large $R$ as
$-\zeta(z-\zeta)e^{2}/R$ in the main term. At small distances, calculation
shows that the adiabatic potential turns out to be confined between the FS
potentials of the atoms $Z$ and $Z-2$. This is reasonable in view of that the
real electronic density evolves from the parent to the daughter atoms.

Fig. 3b shows the effective potentials in the subbarrier region. The adiabatic
potential starts from zero value. The effective FS potentials start from the
negative and positive values in the initial and final atoms, respectively.
Numerically, in the case of 226Ra, $U_{e-\alpha}^{\text{FS}}(0)$ = $-$39.572
keV was

Figure 3: a (left) — schematic $e-\alpha$ potentials in the case of decay of
neutral 226Ra atoms: the adiabatic approach (full curve), in comparison with
the potentials of the frozen shell in the initial (dashed curve) and final
(dash-dotted curve) atoms, respectively; b (right) — the effective calculated
$e-\alpha$ potentials in the subbarrier region.

obtained, and $\Delta B$= $-$38.805 keV. Thus, the FS curve starts from
$-$0.767 keV at $R$ = 0. This is in a close agreement with Ref. [9], where the
value of about 750 eV can be concluded (cf. Fig 1 of Ref. [9]). According to
Ref. [18], $\Delta B$ = 640927 $-$ 679616 = $-$38789 eV, that is by only 16 eV
less than our value. The left and right barrier turning points are $R_{1}$ = 9
fm, $R_{2}$ = 51 fm. The three conclusions drawn below clearly follow fig. 3:

1) The effect of the shell on the $\alpha$ decay rate is negative (as the
adiabatic effective potential is represented by a positive definite curve).

2) Contrary, within the framework of the FS model, the effect is positive (as
the corresponding lowest curve really lies below zero in the subbarrier area).

3) In the subbarrier region, the absolute value of the adiabatic potential is
considerably less than that of the FS model. This explains why the effect
within the consecutive adiabatic approach turns out not only to be of the
opposite sign, but also considerably less in absolute value.

Moreover, the adiabatic approach allows one to consistently calculate the
contribution of each electron shell to the overall effect. This is performed
directly by means of Eq. (17). Results are presented in Fig. 4. It follows
from Fig. 4 that about 80% of the total screening potential is expected to the
contribution from the $K$-shell electrons. This gives an opportunity of
performing an accurate and beautiful experiment, which we will discuss in the
next section.

Figure 4: Contributions of separate shells into the $e-\alpha$ adiabatic
potential in the case of neutral Ra atoms. Left: $K$ shell — full curve, $L$
shell — dotted curve, $M$ shell — dash-dotted curve, and $N$ shell — dotted
curve. Right: the $e-\alpha$ adiabatic potential of the $He$-like ions (dashed
curve), Ne-like ions (dash-dotted curve), and neutral atoms (full curve).

##  3 Results

Representative calculations were performed in Refs. [4, 10, 11] for various
$\alpha$ emitters throughout the periodic table, and with different decay
energies. They show that the effect of screening strongly decreases with
increasing the $Q$ value and decreasing the lifetime. Within the adiabatic
approximation, it is evident that the inner electrons produce more effect, as
they are more sensitive to the motion of the $\alpha$ particle in the
subbarrier area near the nucleus. According to Fig. (4), more than 80% of the
effect are produced by the $K$ electrons. This suggests an elegant and basic
way of experimental check e.g. through measurement of the difference in the
decay rate between the He-like and bare ions of the same nuclei — alpha
emitters. Monochromaticity parameters of the storage ring beam are good
enough, in order to detect the recoil nuclei by the Schottky method [19].

Consider dependence of the decay rate on stripping the electron shell in more
detail. Results of calculation for 226Ra and 212Rn atoms are presented in
Table 1.

Table 1: The effect of the electron screening $Y$ as calculated for different electronic configurations of the 226Ra atom Nuclide |  $Q$, MeV | Neutral |  $K$-shell |  $K$\- and $L$-shells | He-like ion | Ne-like ion
---|---|---|---|---|---|---
226Ra | 4.87063 |  $-$0.246 |  $-$0.200 |  $-$0.235 |  $-$0.201 |  $-$0.236
212Rn | 6.385 |  $-$0.086 |  $-$0.070 |  $-$0.081 |  $-$0.071 |  $-$0.082

The $Q$ values (column 2) are cited according to Ref. [20]. In column 3, the
$Y$ values (3), calculated within the adiabatic approach, are presented for
the neutral atoms. The partial contributions from the $K$ shells and both the
$K$ and $L$ shells, respectively, are listed in columns 4 and 5. In the sixth
and seventh columns the results are listed for the He-like and Ne-like ions,
respectively. It follows from these results that 82% of the effect are due to
the contribution of the $K$-shell electrons. 96% — nearly full effect — is
achieved in the case of Ne-like atoms. Note that in the case of He-like or Ne-
like ions the effect is a little greater than if calculated for neutral atoms
with allowance for only the $K$\- or both $K$\- and $L$-shells, respectively.
This fact has a simple explanation on the physical ground. In the ions, the
electronic orbitals are more compact around the nuclei, and therefore the
wavefunctions are greater in the area under the Coulomb barrier. This
consequently causes bigger values of the integrand in Eq. (17) and the related
increase of the effect.

The experiment can be realized in a similar way to which was applied in search
for the time modulation in beta decay of Pm ions [21]. The daughter product
from the $\alpha$ decay will stay in the ring and should be seen by Schottky
analysis. If one starts from 212Rn84+, one should have most of the time
208Po82+, it will be good to detect both. If one uses bunches of ions then the
Schottky needs to be calibrated and one has to find out how well this can be
done for the four cases of decay from 212Rn84+ and 212Rn86+. Concerning the
“background” arising by ionization and shake-off from Rn84+, and there could
also be electron capture by Rn86+, these could be made quite weak.

On one hand, the $\alpha$ lifetimes of seconds to minutes seem feasible from
the viewpoint of experiment using storage rings. On the other hand, the
calculated values of $Y$ are mainly determined by the $Q$ values. They weakly
depend on the atomic and mass numbers $Z$, $A$ of the nuclides within the Ra –
Po domain. In Fig. 5 we present the $Y$ values as calculated for the isotopes
of Ra and Rn against their $Q$ values in neutral atoms. For comparison, the
results, obtained within the FS model, are also shown in the same figure.

Figure 5: Calculated relative change of the $\alpha$ decay rate $Y$ in neutral
atoms as compared to that of bare nuclei in the isotopes of Ra (full line) and
Rn (dotted line) $Y$ against the released energy $Q$. Dashed curve — the same
for Ra isotopes as calculated in the FS model.

In order to have a big effect, it is desirable to use the isotopes with small
$Q_{\alpha}$. But small $Q_{\alpha}$ correlate with long halflives which may
comprise days and years, which is inconvenient for measuring in the storage
rings. In Table 2 we present the resulting $Y$ values for radium isotopes with
halflives within seconds to minutes and the $\alpha$ branching ratios 100% or
close. Most abundant isotope 226Ra and the next 222Ra are characterized with a
feasible effect $Y$ = $-$0.25 and $-$0.14 percent, respectively, with
comparatively long halflives. Among the lighter isotopes, there are two groups
around $A\approx$ 222 and 213. They are characterized with the $Q_{\alpha}$
values of 6–7 MeV, and the effect $|Y|\lesssim$ 0.1%. Results for other such
suitable isotopes of Fr, Rn, Ac, At and Po are listed in Table 3.

Table 2: Relative change of the decay rate $Y$ between the decay in bare nuclei and He-like ions in the case of Ra isotopes  $A$ | $Q_{\alpha}$ | $T_{1/2}$ |  $Y$, %
---|---|---|---
226 | 4.87063 | 1600 y |  $-$0.246
224 | 5.78887 | 3.66 d |  $-$0.14
222 | 6.681 | 38.0 s |  $-$0.091
221 | 6.884 | 28 s |  $-$0.083
220 | 7.595 | 18 ms |  $-$0.060
214 | 7.273 | 2.46 s |  $-$0.069
213 | 6.861 | 2.74 m |  $-$0.084
212 | 7.0319 | 13.0 s |  $-$0.071
Table 3: Relative change of the decay rate $Y$ as in Table 2 calculated for other elements  Nuclide | $Q_{\alpha}$ | $T_{1/2}$ |  $Y$, %
---|---|---|---
221Fr | 6.4579 | 4.9 m |  $-$0.10
220Fr | 6.8007 | 27.4 s |  $-$0.086
213Fr | 6.9051 | 34.6 s |  $-$0.082
220Rn | 6.40467 | 55.6 s |  $-$0.10
219Rn | 6.9461 | 3.96 s |  $-$0.080
212Rn | 6.385 | 23.9 m |  $-$0.11
223Ac | 6.7831 | 2.10 m |  $-$0.087
222Ac | 7.1374 | 5 s |  $-$0.074
202At | 6.3537 | 184 s |  $-$0.11
201At | 6.4733 | 89 s |  $-$0.10
200At | 6.5964 | 43 s |  $-$0.095
199At | 6.780 | 7.2 s |  $-$0.087
198At | 6.893 | 4.2 s |  $-$0.083
197At | 7100 | 0.35 s |  $-$0.075
196At | 7200 | 0.3 s |  $-$0.072
211Po | 7.5945 | 0.516 s |  $-$0.061

##  4 Discussion

Although the predicted magnitude of the effect is as small as about $10^{-3}$,
there is a circumstance favorable for experimental studies: about 80% of the
effect is due to the contribution of the $K$-shell electrons. In order to
observe the effect, it is therefore sufficient to compare the decay halflives
for bare nuclei with the halflives for respective one- and (or) two-electron
atoms, without involving neutral atoms — their accumulation in accelerator
rings is impossible. And generally, it is more reliable to observe small
effects via difference measurements at the same facility. Thus, comparative
measurements can be performed in the same storage rings with bare nuclei and
respective ions of different multiplicity — for example, helium-like, or neon-
like ions. An experiment on comparison of the alpha decay half-lives in
neutral atoms and their ions was already tested at GSI a few years ago [22].
H-like 213Fr ions were produced and injected in the storage ring. The
decreasing number of parent ions has been observed by the Schottky technics
after each injection. However, because of a large statistical uncertainty, the
measured half-life, $T_{1/2}$ = 34(6) s at rest, was compatible with the
neutral half-life. Analyzing that experiment in the light of the present
consideration, the following conclusions are worthy of noting.

First, the main idea of that experiment sounds striking: comparison of the
lifetimes of neutral atoms with those in H-like ions. Why not just bare
nuclei? One $1s$ electron already produces close to half the effect.

Second, they compared the lifetimes in the H-like ions with those in neutral
atoms, as measured by traditional techniques in stoppers. In the case of 213Fr
two independent neutral half-life measurements have been carried out, one at
GSI and the other at the LNS-INFN. Even though they agree, a discrepancy has
been found with the value present in the literature. Our present proposal
excludes the development of such an unforeseen situation. It assumes that both
lifetimes — of the bare nuclei and He-like ions — will be measured in the same
channel of the storage ring by the Schottky method. This way will considerably
compensate systematic uncertainties, which will be merely the same, whereas in
the previous experiment they could be of different origin and therefore, not
correlated with the measurements of the ionic half-life.

Third, a general idea was told out in Ref. [22] that such an experiment is
feasible with the resulting uncertainty within 0.1%, which is a typical value
of the effect calculated herein. Therefore, one can think this estimation
opens green light to realization of the experiment as discussed above.
Moreover, one can speak about experimental test for correctness of either the
FS model or adiabatic approach. This is even more feasible, as one should
choose between the estimates of 10-2 and $-10^{-3}$, resulting from the FS and
adiabatic models, respectively.

##  5 Conclusion

We consecutively considered the effect of the electron screening on the
$\alpha$ decay rate from the viewpoint of its experimental investigation. The
results presented previously demonstrate that within the adiabatic approach
the effect of the electron screening on the $\alpha$ decay rate is certainly
negative, in spite of the attractive potential provided by the shell. The
physics arguments in favor of this conclusion can be formulated as follows.

1) Raising electron terms create resisting force acting from the shell on the
$\alpha$ particle.

2) Experimentally observed drastic suppression of prompt fission in muonic
atoms of actinide elements.
The main error of the FS model is in the identification of the inner
electrostatic Coulomb potential times 2$e$ with the potential energy of the
$\alpha$ particle in its motion through the electronic shell [2, 9]. Non-
identity of these concepts is demonstrated in the best way by addition of the
constant into Eq. (7). Quint essence of the adiabatic approach is expressed by
Eq. (15). This condition results in the negative effect in the adiabatic
approach and shows the reason for the positive effect within the FS model.
Flight of $\alpha$ particle through the electron shell rearranges the latter.
In turn, this weakens the effect of the shell on the particle. Using the
classic analogy, one can say that if the electrons are not fixed firmly on
their places, then they take part of the momentum of the $\alpha$ particle and
are partially carried by the particle. It is figuratively said in Ref. [10]
that the alpha particle is accompanied with a tail of the atomic electrons
when it crosses the shell. Such a loosed dynamics leads to softening of the
electric field strength around the $\alpha$ particle in comparison with what
is expected in the FS model.

General effect of retardation of the decay provided by the adiabatic approach
is at the level of 10-3. FS model results in the effect of the opposite sign
and percent value. Its detection could be a difficult task. However,
contemporary technique makes the observation of the effect, and all the more
testing for the difference between the models quite feasible. Moreover,
application of the storage ring facilities makes unnecessary use of any
detectors. The Schottky analysis allows one to conduct such an experiment. It
needs no counters or other special technics apart from the beams of the target
nuclei or ions. Minutes or even seconds of the beam time is enough for one
experimental run. Such an impressive progress is provided by development of
the experimental capabilities. Inferring the experimental results will
certainly put a milestone on the way of astrophysical research of the
processes occurring in the stellar plasma, including the Sun.

## Acknowledgements

The authors are grateful to X. Ma, M. Steck and R. Schuch for many fruitful
discussions of the questions considered herein. They want to thank T. Stölker
and Yu. Litvinov for helpful remarks.

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