text:
\section{Introduction} \label{sec.introduction}

According to Gamow theory of alpha decay \cite{1928Gamow,1928Condon}, this fundamental radioactive decay   can be considered as a two-step process \cite{1998Lovas,2003Hodgson,2010Delion}. In the first step,  an alpha  cluster is formed inside the parent nucleus.
The resulting alpha particle  resides  in a metastable state of an average potential  of the daughter system.  In the second step, the particle tunnels  through the potential barrier. Each step requires different theoretical treatment. To compute the preformation factor that  describes the alpha formation probability, one needs to evaluate the overlap integral involving wave functions of the parent and daughter nuclei, and that of the alpha particle. The estimate of the penetration probability requires a careful treatment of
the  resonance state.

The commonly used  formulation of the alpha-decay problem employs the R-matrix expression \cite{1954Thomas,1957Mang}
\begin{equation}\label{Grmatrix}
\Gamma_L = 2 P_L \gamma_L^2
\end{equation}
for  the absolute width. In this formalism, the first stage (formation of alpha particle with angular momentum $L$) is given by the reduced width $\gamma_L^2$,  while the second stage (decay) is expressed by means of the penetrability $P_L$.
Alternatively, the absolute width can be obtained from  the general reaction-theory expression \cite{1972Arima,1974Arima,1976Fliessbach-jpg,1977Fliessbach}
\begin{equation}\label{Goverlap}
\Gamma_L = S_L \Gamma_L^\text{sp},
\end{equation}
where $S_L$ is the alpha-spectroscopic factor and  $\Gamma_L^\text{sp}$ is the
single-particle (s.p.)  decay width.

Historically, expression (\ref{Grmatrix}) was derived in 1954 by Thomas \cite{1954Thomas} using  the time-independent R-matrix theory of nuclear reactions.
In 1957, Mang \cite{1957Mang} developed the alpha-decay formalism based on the time-dependent perturbation theory. He made the connection with the shell model  and succeeded in expressing the alpha-decay formation amplitude in a basis of s.p. states. As shown in Refs.~\cite{1963Zeh,1964Mang}  formulations of Thomas and Mang are formally equivalent; there are, however, many differences when it comes to practical implementations.

The reduced width
calculated in a  shell-model configuration expressed in the harmonic oscillator (h.o.) basis
is too  small. This can be partly cured by  means of configuration mixing involving extended shell-model spaces \cite{1961Harada,1979Tonozuka} as each admixed configuration contributes coherently to $\gamma_L^2$.  To improve asymptotic properties of s.p. wave functions, the particle continuum was taken into account \cite{1983Janouch} by considering h.o. expansion \cite{1988Dodig-Crnkovic} or within a Woods-Saxon (WS) basis consisting of bound and outgoing single-particle resonant (Gamow) states \cite{1993Lenzi,2000Delion}.  The configuration mixing calculations of  Refs.~\cite{1979Tonozuka,1993Lenzi} in the valence space of $^{212}$Po assumed the seniority-zero (pairing vibrational) wave functions
obtained by considering the monopole pairing interaction between like nucleons.
However, all these improvements were not sufficient to reproduce the experimental alpha decay in $^{212}$Po. It is only after the valence proton-neutron interaction had been considered together with  a generalized wave function expressed as a combination of cluster and shell model components \cite{1989Okabe} that theoretical  and  experimental widths could be reconciled \cite{1992Varga}.

The R-matrix expression for the width (\ref{Grmatrix}) depends on the channel radius $R$. This radius should be chosen large enough so that the alpha-daughter interaction  in the external region   is given by the   Coulomb force alone \cite{2010Descouvemont}. The infinite range of the Coulomb force implies, however,  that the asymptotic behavior of the R-matrix expression is  reached only at  large values of $R$, at which  the  asymptotic behavior of the shell-model s.p. basis (h.o. basis in most applications) used to calculate $\gamma_L^2(R)$,  does matter. Due to the mismatch between the internal part of the s.p. wave function (well described in the h.o. basis) and the asymptotic part (poorly or not described in the h.o. basis), rather small changes in $R$ may  produce appreciable variations in penetrability. Physically, the reason for this sensitivity is the fact that the alpha cluster is formed in the surface region  of the nucleus in which the coupling to the  alpha continuum that impacts the radial behavior of the formation amplitude  is important \cite{1983Janouch}. Consequently, the absolute R-matrix width depends in general on the channel radius \cite{1991Insolia,2010Delion}, and this is an obvious drawback of the method \cite{2010Descouvemont}.

Our renewed interest in the alpha-decay problem is stimulated by the recent experimental data above the doubly-magic $^{100}$Sn \cite{2005Janas,2006Liddick} that demonstrate the presence of  very fast alpha decays. Indeed, the observed enhancement of the reduced widths of $^{105,106}$Te relative to
$^{213,212}$Po is two-to-three, thus confirming earlier expectations \cite{1965Macfarlane} of ``superallowed" alpha decays in this region due to the large  overlaps of valence s.p. shell model proton and neutron wave functions.
Our  long-term goal  is to estimate  alpha preformation factors in nuclei above $^{208}$Pb  and  $^{100}$Sn  by using  large valence s.p. spaces, including positive-energy Gamow states of a finite-depth WS potential \cite{1993Lenzi,2000Delion}. In this study, we focus on
$^{212}$Po and $^{104}$Te  nuclei having two valence protons and two valence neutrons outside  doubly-magic cores.

Our paper is organized as follows. Section~\ref{sec.formalism} briefly describes the alpha-decay formalism used in this work, with  special emphasis on approximations used to describe wave functions of parent and daughter nuclei. Section \ref{sec.model} deals with the approximations employed and parameters used. In particular, we discuss the sensitivity of the calculated spectroscopic factor to the parameters defining the shifted Gaussian basis that is used to compute the normalization of the channel function. In Sec.~\ref{sec.applications.ho} we study the sensitivity of the reduced alpha width in $^{212}$Po  on the choice of s.p. basis used. In Section \ref{sec.aw} we discuss the absolute alpha-decay width  of $^{212}$Po and in Sec.~\ref{sec.applications.compare} we compare it with the absolute width of the superallowed alpha emitter  $^{104}$Te. Finally, the main conclusions of this work are summarized in Sec.~\ref{sec.conclusions}.

\section{Formalism}\label{sec.formalism}
In this section, we discuss the  R-matrix (\ref{Grmatrix}) and
spectroscopic factor (\ref{Goverlap})
expressions for the decay width.
The connection between the two formulations is given in Ref.~\cite{1972Arima}. We also discuss the so-called delta-approximation for the formation amplitude.

\subsection{R-matrix expression for the decay width}\label{subsec.formalism.r}

Within the R-matrix theory \cite{1954Thomas,1957Mang,1963Zeh},  the absolute width is given by Eq.~(\ref{Grmatrix})
with  $P_L(R)$ being the barrier penetrability  and $\gamma_L(R)$  -- the reduced width amplitude  \cite{1960Lane}. While both quantities strongly depend on the value of the channel radius  $R$, the absolute width should be $R$-independent.

For $P_L(R)$ we  use the standard expression \cite{1954Thomas}:
\begin{equation}
 P_L(R) = \frac{kR}{|H_L^+(\eta, k R)|^2},
\end{equation}
where $k$ is given by the  alpha energy
 $E_\alpha=\frac{ \hbar^2 \; k^2}{2 \; \mu}$, obtained
from the experimental $Q_\alpha$ value
by correcting for  electron screening;
$\mu=\frac{m_d m_\alpha}{m_d + m_\alpha}$ is the reduced mass of alpha particle
with $m_d$ being the mass of the daughter nucleus;
$H_L^+(\eta, kR)$ is the outgoing spherical Coulomb-Hankel function; and  $\eta=\frac{2 Z_d \mu e^2 }{\hbar^2 k}$ is the Sommerfeld Coulomb parameter.

The reduced width amplitude  $\gamma_L(R)$ may be written in terms of the formation amplitude $g_L(R)$ \cite{1961Harada,2010Delion}:
\begin{equation} \label{eq.gamma}
 \gamma_L = \sqrt{\frac{\hbar^2 R}{2 \mu}} g_L(R),
\end{equation}
with
\begin{eqnarray} \label{eq.flfirst}
 g_L(R)&=& \int d\Omega_R \int d\xi_\alpha \int d\xi_D \; \nonumber \\
	 && \Phi^P_{J M}\;
 	 \mathcal{A} \left[  \phi_\alpha(\xi_\alpha) \; \Psi^D_j(\xi_D)\; Y_L(\hat{R}) \right]^*_{J M},
\end{eqnarray}
where $\phi_\alpha$ is the normalized wave function of the alpha particle with zero angular momentum, $Y_{L M_L}$ is the angular part of the center-of-mass  (c.o.m.) motion of the alpha particle, $\Psi^D_{j m_j}$ is the wave function of the daughter nucleus, and $\Phi^P_{J M}$ is the wave function of the parent nucleus.
The coordinates $\xi_\alpha$ and $\xi_D$ are the intrinsic coordinates of the  alpha particle and  daughter nucleus, respectively.
All wave functions are normalized in terms of the internal and c.o.m. coordinates \cite{1960Lane}.
By construction, the parent and daughter wave functions   are antisymmetric.
The antisymmetrization with respect to inter-fragment nucleons
is done by means of the  operator $\mathcal{A}$. Its action can be approximated by means of a factor $\left[ \binom{N_v}{2}\binom{Z_v}{2} \right]^{1/2}$ \cite{1960Lane,1960Mang,1961Harada}, with $N_v$ and $Z_v$ being, respectively,  the numbers of valence neutrons and protons in the parent nucleus.

For the internal alpha-particle wave function we take the standard Gaussian ansatz \cite{1990Bayman,1993Lenzi}:
\begin{eqnarray}
 \phi_\alpha(\rho_1 \rho_2 \rho_3, \sigma_1 \sigma_2 \sigma_3 \sigma_4) &=&
       \phi(\rho_1 \rho_2 \rho_3) \chi_{00}(\sigma_1 \sigma_2) \chi_{00}(\sigma_3 \sigma_4),  \nonumber \\
  \chi_{00}(\sigma_1 \sigma_2) &=& [\chi_{1/2}(\sigma_1) \chi_{1/2}(\sigma_2)]_{00},  \nonumber \\
  \phi(\rho_1 \rho_2 \rho_3) &=& \left( \frac{8 \beta}{\pi} \right)^{9/4} e^{-4 \beta (\rho_1^2 + \rho_2^2 + \rho_3^2)}.
\end{eqnarray}
The parameter $\beta=\frac{9}{64 r_\alpha^2}$=0.057\,fm$^{-2}$ depends on  the  root-mean-square alpha radius $r_\alpha$=1.57\,fm \cite{1990Bayman}.

The transformation between the intrinsic $\xi_\alpha=\{ \bm{\rho}_1, \bm{\rho}_2, \bm{\rho}_3 \}$ and nucleonic  $\{\gras{r_i}\}$ ($i$=1,2,3,4) coordinates reads:
\begin{eqnarray} \label{eq.rel}
 \gras{\rho}_1&=& \frac{\gras{r}_1 - \gras{r}_2}{\sqrt{2}}, \nonumber \\
 \gras{\rho}_2&=& \frac{\gras{r}_3 - \gras{r}_4}{\sqrt{2}}, \\
 \gras{\rho}_3&=& \frac{(\gras{r}_1 + \gras{r}_2) - (\gras{r}_3 + \gras{r}_4)}{2}\nonumber,
\end{eqnarray}
and
\begin{equation}\label{com}
\gras{R}= \frac{\gras{r}_1+\gras{r}_2+\gras{r}_3+\gras{r}_4}{4}
\end{equation}
is the c.o.m. coordinate of alpha particle.
Let us denote the spherical components of intrinsic coordinates by $\gras{\rho}_i=(\rho_i,\tilde{\theta}_i,\tilde{\varphi}_i)$. Assuming $\theta_R=\varphi_R=0$, the nucleonic coordinates  can be written as:
\begin{eqnarray}\label{eq.rhoandr}
 4 r^2_{1,2} &=& 4R^2 + \rho^2_3 + 2 \rho^2_1 \pm 2\sqrt{2}  \rho_3 \rho_1  \cos \tilde{\theta}_{31} \nonumber \\
        &&      + 4  R \left( \rho_3 \cos \tilde{\theta}_3 \pm \sqrt{2} \rho_1
        \cos \tilde{\theta}_1 \right),   \nonumber \\
4 r^2_{3,4} &=& 4R^2 + \rho^2_3 + 2 \rho^2_2 \mp 2\sqrt{2}\rho_3\rho_2 \cos \tilde{\theta}_{32} \nonumber \\
         &&     - 4 R \left( \rho_3 \cos \tilde{\theta}_3 \pm \sqrt{2} \rho_2 \cos \tilde{\theta}_2 \right),
\end{eqnarray}
where  $\tilde{\theta}_{ij} = \tilde{\theta}_j - \tilde{\theta}_i$, and
\begin{eqnarray}\label{eq.cos}
  \cos \theta_{1,2} &=& \frac{2R  + \rho_3 \cos \tilde{\theta}_3 \pm \sqrt{2} \rho_1 \cos \tilde{\theta}_1}{2r_{1,2}}, \nonumber \\
  \cos \theta_{3,4} &=& \frac{2R  - \rho_3 \cos \tilde{\theta}_3 \pm \sqrt{2} \rho_2 \cos \tilde{\theta}_2}{2r_{3,4}}.
\end{eqnarray}

This paper deals with  g.s.$\rightarrow$g.s. alpha decays to the magic  daughter nucleus. Assuming the  seniority-zero wave function, the corresponding formation amplitude is \cite{1961Harada,1964Mang}
\begin{equation}\label{eq.f0}
 F_0(R) =  \frac{\sqrt{8}}{16 \pi^{3/2}}  \sum_{\nu_n,\nu_p}  (-)^{l_n+l_p} b_{\nu_n,\nu_p} \hat{j}_n  \hat{j}_p I_{v_n,\nu_p}(R),
\end{equation}
where
\begin{eqnarray}\label{eq.inp}
    I_{\nu_n,\nu_p}(R) &=&   \int d\gras{\rho}_1 d\gras{\rho}_2 d\gras{\rho}_3
 	      \phi(\rho_1 \rho_2 \rho_3)   \nonumber \\
    &\times&    \frac{u_{\nu_n}(r_1)}{r_1} \frac{u_{\nu_n}(r_2)}{r_2} P_{l_n}(\cos\theta_{12})  \\
    &\times&    \frac{u_{\nu_p}(r_3)}{r_3}  \frac{u_{\nu_p}(r_4)}{r_4}
    P_{l_p}(\cos\theta_{34}), \nonumber
\end{eqnarray}
with $\theta_{ij}=\theta_j - \theta_i$, $\nu= \{n,l,j\}$, and $u_{\nu}(r)$ being s.p.  radial wave functions.
The factor $\sqrt{8}$ comes from the Jacobian of the transformation between
the nucleonic coordinates $\{\gras{r_i}\}$  and the
internal and  c.o.m.   coordinates \cite{1965Eichler,2010Delion}.
In Eq.~(\ref{eq.inp}) and in the following, the s.p. indices $1,2$ refer to neutrons while $3,4$ refer to protons.
The coefficients $b_{\nu_n,\nu_p}$ are the shell-model four-particle wave function amplitudes.

\subsection{Delta-function approximation}\label{subsec.formalism.delta}

In the calculation of alpha-decay rates based on h.o. wave functions, it was noticed \cite{1960Mang} that the relative rates   change little with the oscillator length $b_\text{h.o.}$ of the basis. Using this argument,  Mang proposed to take  $\beta \gg 1/b^2_\text{h.o.}$. In this limit, the expression for the formation amplitude  can be simplified (see also  Ref.~\cite{1958Brussaard}).
In the literature, this is known as delta-function approximation
\cite{1963Rasmussen}.

In practice, one  assumes that the alpha particle wave function
is  constant inside a small volume of radius $s_\alpha =2.34$\,fm \cite{1963Rasmussen} and zero outside.
Within this approximation  $\gras{\rho}_i=0$; hence, it immediately follows from Eqs.~(\ref{eq.rhoandr})  that $\gras{r}_1=\gras{r}_2=\gras{r}_3=\gras{r}_4=\gras{R}$ \cite{1957Devons,1963Rasmussen},  and the formation amplitude reduces to
\begin{equation}
 F^\delta_0(R) = \frac{\sqrt{8}}{16 \pi^{3/2}}  \left( \frac{4\pi s^ 3_\alpha}{3} \right)^{3/2}
	\left( \sum_{\nu_n} I^n_{\nu_n} \right) \left( \sum_{\nu_p} I^p_{\nu_p} \right),
\end{equation}
with
\begin{equation}\label{Bcorr2}
   I^\tau_\nu = (-)^{l_\nu} b^\tau_\nu  \hat{j}_\nu   B_\nu \frac{u^2_{\nu\tau}(R)}{R^2},
\end{equation}
where $\tau=n, p$. The correction factor  $B_\nu$ depends on the relative angular momentum \cite{1963Rasmussen}:
\begin{equation}\label{Bcorr}
B_\nu=1-0.013 l_\nu(l_\nu+1).
\end{equation}

\subsection{Four-particle amplitudes} \label{sec.fpa}
For the g.s. alpha decay of $^{212}$Po and $^{104}$Te, we are going
to assume that the four valence nucleons move around the rigid, doubly-magic core.
The parent-nucleus  wave function is approximated by a product of two-neutron and two-proton  seniority-zero  states:
\begin{equation}
  |\Phi^P_{J=0,M=0}\rangle= |\Psi_{2n,{00}} \rangle \otimes |\Psi_{2p,{00}} \rangle,
\end{equation}
where
\begin{equation}\label{twop}
 |\Psi_{2\tau,0} \rangle = \sum_{\nu} X_{\nu}^{\tau} |\nu \nu,00 \rangle ,
\end{equation}
$|\nu \nu,00 \rangle=\frac{ [a_{\nu}^\dagger a_{\bar\nu}^\dagger]_{00} }{ \sqrt{2} } |0_\tau \rangle$, and $|0\rangle=|0_n \rangle\otimes |0_p \rangle$ is the shell-model vacuum representing the  $^{208}$Pb or $^{100}$Sn g.s. wave function. The four-particle amplitudes $b_{\nu_n,\nu_p}$ in (\ref{eq.f0})  can thus be written in a separable form:
\begin{equation}
 b_{\nu_n,\nu_p} = X_{\nu_n}^n X_{\nu_p}^p.
\end{equation}

\subsection{Alpha decay spectroscopic factor}\label{subsec.formalism.s}
Based on the general theoretical arguments \cite{1972Arima,1974Arima,1976Fliessbach-jpg,1977Fliessbach}, the absolute width can be expressed as a product of the alpha-particle spectroscopic factor and the single particle width, see Eq.~(\ref{Goverlap}). The spectroscopic factor  $S_L$  contains information about the probability of forming an alpha cluster in the parent system. Since the alpha particle, when formed,  occupies the resonant state, the s.p. width can be obtained from the so-called current expression \cite{1961Humblet,2000Barmore,2010Delion}:
\begin{equation} \label{eq.widthsp1}
 \Gamma_L^\text{sp} = i\frac{\hbar^2 }{2 \mu} \frac{u'^*_L(R)\; u_L(R) - u'_L(R)\; u^*_L(R)}{\int \; |u_L(R)|^2 \; dR},
\end{equation}
where the Gamow function $u_L(R)$ is obtained as a solution of the Schr{\"o}dinger equation with  outgoing boundary condition.
When the imaginary part of the complex energy eigenvalue $\mathcal{E}_\alpha=\frac{ \hbar^2 \; k^2}{2 \; \mu}$ is small, which is always the case for the considered g.s. alpha emitters, one can approximate (\ref{eq.widthsp1}) with \cite{2004Kruppa}:
\begin{equation} \label{eq.gammasp}
 \Gamma_L^{sp} = \frac{\hbar^2 \Re(k)}{\mu} \frac{|u_L(R)|^2}{|H^{+}_L(\eta,k R)|^2 }.
\end{equation}
The s.p. width obtained in this way should be identical to the value
$-2$Im($\mathcal{E}_\alpha$) given by the imaginary part of the Gamow resonance energy, if the latter is computed with a sufficient precision.

The conventional alpha spectroscopic factor as introduced in Ref.~\cite{1972Arima} is defined by
\begin{equation} \label{eq.sbaket}
  S_L= | \langle \mathcal{A} \left[ \phi_\alpha(\xi_\alpha) \; \Psi^D_j(\xi_D)\; \psi_L(\gras{R}) \right]_{J M} | \Phi^P_{J M} \rangle |^2,
\end{equation}
where $\psi_{LM}(\gras{R})= \frac{u_L(R)}{R} Y_{LM}(\hat{R})$ represents  the relative motion alpha particle with respect to the daughter.
In terms of the formation amplitude, $S_L$ reads \cite{1998Lovas,1976FliessbachMang,2010Delion}:
\begin{equation} \label{eq.sint}
 S_L = \int_0^\infty g_L^2(R) R^2  dR.
\end{equation}

\subsection{Modified spectroscopic factor} \label{app.norm}

Since the formation amplitude Eq. (\ref{eq.flfirst}) represents the overlap of the parent wave function with the daughter-alpha product state,  one would be tempted to associate it with the probability amplitude that in the parent wave function $\Phi^P_{JM}$ an alpha particle $\phi_\alpha$ and a daughter nucleus $\Psi^D_{j m_j}$ are  at a distance $R$. The value of $S_L$  would then be associated with the total probability of formation of an alpha particle. However, the fundamental problem with this interpretation is that the channel function
$\mathcal{A} \left[ \phi_\alpha(\xi_\alpha) \; \Psi^D_j(\xi_D)\; \psi_L(\gras{R}) \right]_{J M}$ is not properly normalized \cite{1975Fliessbach,1976Fliessbach-jpg,1977Fliessbach,1978Fliessbach,1980Watt,1987Blendowske,1998Lovas}.

The properly defined spectroscopic factor (sometimes referred to as ``the amount of clustering") \cite{1975Fliessbach,1976Fliessbach,1976Fliessbach1,1987Beck,1992VargaLovas,1992Varga} is given by
\begin{equation}\label{eq.s2}
 \mathcal{S}_L= \int_0^\infty G_L^2(R) R^2  dR,
\end{equation}
where
\begin{equation}\label{eq.gbig}
 G_L(R)= \int  \mathcal{N}_L^{-1/2}(R,R') \; g_L(R')R'^2  \;dR'
\end{equation}
is the modified formation amplitude. The norm kernel $\mathcal{N}_L$ appearing in Eq.~(\ref{eq.gbig}) is \cite{1976Fliessbach1}
\begin{eqnarray} \label{eq.nk}
&& \mathcal{N}_L(R,R') =  \\
&&     \langle \mathcal{A}
                           \frac{\delta(R_\alpha - R)}{R^2} \phi_\alpha
                           \left[ Y_L \Psi_j^D \right]_J
                          |
             \mathcal{A}
                           \frac{\delta(R_\alpha - R')}{R'^2} \phi_\alpha
                           \left[ Y_L \Psi_j^D \right]_J
       \rangle. \nonumber
\end{eqnarray}
The presence of the norm kernel $\mathcal{N}$ effectively enhances the spectroscopic factor by one-to-two orders of magnitude \cite{1979Conze,1980Watt,1987Blendowske,1992VargaLovas,1992Varga}.

To compute  $\mathcal{N}_L^{-1/2}(R,R')$, we expand the eigenfunctions of the norm kernel in an orthonormalized shifted Gaussian basis (SGB) \cite{1976Fliessbach1},
\begin{equation} \label{eq.tildeF}
 \tilde{F}_L(R,R_k) = \sum_{k'}  \left( N^{-1/2}_F \right)_{k k'} \; F_L(R,R_{k'}),
\end{equation}
with $R_k$ equidistant mesh points in the interval $(0,R_{\textrm{max}})$ and $k=1,\dots,M$, where $M$ is the dimension of the basis. The SGB is given by
\begin{equation}\label{eq.sgb}
 F_L(R,R_k) = 4 \pi \left( \frac{8 \beta'}{\pi} \right)^{3/4} e^{-4 \beta' (R^2 + R_k^2)}  i^L j_L(-i8 \beta' R R_k),
\end{equation}
while the SGB overlap $(N_F)_{k k'}$ is given by
\begin{eqnarray} \label{eq.nf}
 (N_F)_{k k'} &=& \int  F^*_L(R,R_k)  F_L(R,R_{k'}) R^2 dR  \\
            &=& 4 \pi e^{-2 \beta' (R_k^2 + R^2_{k'})}   i^L j_L(-i 4 \beta' R_k R_{k'}). \nonumber
\end{eqnarray}
Using the SGB  overlaps, the eigenvalue equation for the norm matrix can be expressed in the form:
\begin{equation}\label{fnc}
 \sum_{k'}^M \mathcal{N}^{\tilde{F}}_{kk'} \; c^\nu_{k'} = n_\nu \;  c^\nu_k,
\end{equation}
where
\begin{equation}\label{NtF}
 \mathcal{N}^{\tilde{F}}_{kk'} = \sum_{nn'}^M \; \left( N^{-1/2}_F \right)_{k n} \; \mathcal{N}^F_{nn'} \; \left( N^{-1/2}_F \right)_{n' k'}
\end{equation}

For $\beta'=4 \beta$, the core-projected norm $\mathcal{N}^F$ in Eq.~(\ref{NtF}) reduces to a simple expression \cite{1992Varga,1976Fliessbach1,1966Brink}:
\begin{equation} \label{eq.nkkp}
  \mathcal{N}^F_{kk'} =  \left( \langle \psi^{(\nu),L}_k | \psi^{(\nu),L}_{k'} \rangle \right)^2 \;
                         \left( \langle \psi^{(\pi),L}_k | \psi^{(\pi),L}_{k'} \rangle \right)^2
\end{equation}
where
\begin{eqnarray}\label{eq.nkkp2}
\langle \psi^{(\mu),L}_k | \psi^{(\mu),L}_{k'} \rangle & = &
                \langle \phi^L_k | \phi^L_{k'} \rangle \nonumber \\
&-& \sum_{nlj\tau \in {\rm core}} \delta_{lL} \langle \phi^l_k | R_{nlj} \rangle \langle R_{nlj} | \phi^l_{k'} \rangle
\end{eqnarray}
with $\phi^L_k(R) =  F_L(R,R_k) (\beta'\rightarrow \beta)$
and $R_{nlj}(R)=u_{nlj}/R$ are the radial s.p. wave functions of the core.

In terms of eigenstates $c^\nu_k$ of (\ref{fnc}), the spectral representation of the norm kernel can be written as:
\begin{equation}
 \mathcal{N}_L^{-1/2}(R,R') = \sum_{\nu \atop (n_\nu > n_{\rm min})}  \; n_\nu^{-1/2} \; u^{L*}_\nu(R) \; u^L_\nu(R'),
\end{equation}
where the  eigenfunctions $u^L_\nu(R)$ of the norm kernel are
\begin{equation} \label{eq.ur}
 u^L_\nu(R) = \sum_k^M \; c^\nu_k \; \tilde{F}_L(R,R_k),
\end{equation}
and  $n_{\rm min}$ represents the usual cutoff on the eigenvalue of the norm kernel.  The final expression for the  modified formation amplitude in the normalized SGB becomes \cite{1976Fliessbach1}:
\begin{equation} \label{eq.gbig2}
 G_L(R) =\sum_{\nu \atop (n_\nu > n_{\rm min})} \; n_\nu^{-1/2} \; u^L_\nu(R) \; g^L_\nu
\end{equation}
with
\begin{equation} \label{eq.gnu}
 g^L_\nu = \int  u^L_\nu(R) \; g_L(R) \;  R^2 dR.
\end{equation}

\section{The model} \label{sec.model}

\subsection{Single-particle space} \label{subsec.sp}

The s.p. space is spanned on resonant states of a WS+Coulomb average potential. The parameters of the s.p. Hamiltonian, namely
the WS potential depth $V_0$, spin-orbit potential depth   $V_\text{so}$, diffuseness $a$ ($=a_\text{so}$), radius $r_0$ ($=r_{0,\text{so}}$), and the radius of the uniform charge distribution $r_c$ defining the Coulomb potential  are listed in Table~\ref{table.ws}.
 \begin{table}[ht]
 \caption{\label{table.ws} Parameters of the average WS Hamiltonian used in this work to compute s.p. neutron and proton states  of  $^{208}$Pb and $^{100}$Sn cores.}
 \begin{ruledtabular}
 \begin{tabular}{ccccccc}
 Core   & $\tau$     & \mbox{$V_0$} & \mbox{$V_\text{so}$} & \mbox{$a$} & \mbox{$r_0$}  & \mbox{$r_c$} \\
     &                  & \mbox{(MeV)} &  \mbox{(MeV)}     &  \mbox{(fm)} &  \mbox{(fm)}   & \mbox{(fm)} \\
 \hline
\multirow{2}{*}{$^{208}$Pb}  & $n$   & 44.40          & 16.5                & 0.70      & 1.27      & \\
 & $p$  & 66.04           & 19.0                & 0.75      & 1.19        & 1.27              \\[2pt]
\multirow{2}{*}{$^{100}$Sn} & $n$   & 51.60          & 11.3                & 0.70      & 1.27       &\\
  & $p$  & 52.20          & 10.5                & 0.70      & 1.27          & 1.27
 \end{tabular}
 \end{ruledtabular}
 \end{table}
The resulting neutron and proton s.p. energies for $^{208}$Pb and $^{100}$Sn are given in Tables~\ref{table.spPb} and \ref{table.spSn}, respectively.
The nucleus $^{101}$Sb is proton-unbound; the values in Table~\ref{table.spSn} are generally consistent with systematics \cite{2002Isakov}. In particular, we predict
a very small splitting between the $0g_{7/2}$ and $1d_{5/2}$ neutron shells outside $^{100}$Sn,  and a  $0g_{7/2}$ g.s. in $^{101}$Sn as suggested by recent experiment \cite{2010Darby}.

 \begin{table}[htb]
 \caption{\label{table.spPb} The eigenstates (in MeV) of the s.p. Hamiltonian of Table~\ref{table.ws} for $^{208}$Pb calculated
with the Gamow solver {\sc anti} \cite{1995Ixaru}. The positive-energy eigenvalues represent Gamow resonances; their imaginary energies reflect nonzero particle width.}
 \begin{ruledtabular}
 \begin{tabular}{cc|cc}
   Orbit          & Neutrons & Orbit         & Protons \\
 \hline
  $1g_{9/2}$  & $-3.926$                  & $0h_{9/2}$  & $-3.784$ \\
  $0i_{11/2}$ & $-2.797$                   & $1f_{7/2}$  & $-3.542$  \\
  $2d_{5/2}$  & $-2.072$                   & $0i_{13/2}$ & $-1.844$ \\
  $0j_{15/2}$ & $-1.883$                   & $2p_{3/2}$  & $-0.690$ \\
  $3s_{1/2}$  & $-1.438$                   & $1f_{5/2}$  & $-0.518$ \\
  $2d_{3/2}$  & $-0.781$                   &  $2p_{1/2}$  &  $0.491-i0.200 \times 10^{-11}$ \\
  $1g_{7/2}$  & $-0.768$                   &  $1g_{9/2}$  &  $4.028-i
  0.130 \times 10^{-7}$ \\
  $1h_{11/2}$ &  $2.251 - i0.026$ & $0i_{11/2}$ &  $5.434 - i0.992 \times 10^{-8}$  \\
  $0j_{13/2}$ &  $5.411 - i0.009$ &  $0j_{15/2}$ &  $5.960 - i0.115\times 10^{-7}$ \\
                    &                            & $2d_{5/2}$  &  $6.748 - i0.184\times 10^{-2}$  \\
                    &                            &$3s_{1/2}$  &  $7.843 - i0.367\times 10^{-1}$   \\
                    &                            & $1g_{7/2}$  &  $8.087 - i0.898\times 10^{-3}$  \\
                    &                            & $2d_{3/2}$  &  $8.530 - i0.284\times 10^{-1}$  \\
                    &                            &$1h_{11/2}$  &  $11.390- i0.215\times 10^{-1}$   \\
                    &                            &$0j_{13/2}$  &  $15.086 - i0.493\times 10^{-2}$   \\
                    &                            & $1h_{9/2}$  &  $15.964 - i0.393$
 \end{tabular}
 \end{ruledtabular}
 \end{table}

\begin{table}[htb]
\caption{\label{table.spSn} Similar as in Table~\ref{table.spPb} except for $^{100}$Sn.}
 \begin{ruledtabular}
\begin{tabular}{ccc}
  Orbit          & Neutrons &  Protons \\
\hline
 $0g_{7/2}$  & $-10.830$  & $2.669 - i0.207\times 10^{-7}$\\
 $1d_{5/2}$  & $-10.674$  & $2.869 - i0.963\times 10^{-5}$\\
 $2s_{1/2}$  & $-9.074$   &  $4.150 - i0595\times 10^{-2}$\\
 $1d_{3/2}$  & $-8.927$   &  $4.393 - i0.166\times 10^{-2}$\\
 $0h_{11/2}$ & $-5.793$  &  $7.280 - i0.110\times 10^{-2}$ \\
 $1f_{7/2}$  & $-2.346$                       & $9.649 -i 0.452$   \\
 $2p_{3/2}$  & $-1.531$                       & \\
 $2p_{1/2}$  & $-0.912$                       & \\
 $0h_{9/2}$  & $-0.641$                       & $12.012 -i 0.0736$ \\
 $1f_{5/2}$  & $-0.171$                       & \\
 $0i_{13/2}$ & $3.254 -i 0.132\times 10^{-2}$ & $15.572 -i 0.185$
\end{tabular}
 \end{ruledtabular}
\end{table}

\subsection{Two-particle interaction} \label{subsec.force}

The correlated two particle wave functions $|\Psi_{2\tau,0} \rangle$ (\ref{twop}) have been obtained using a separable two-body  $T=1$ pairing interaction \cite{1971Bes}:
\begin{equation}\label{vres}
 \langle \nu \nu, 00 | V | \nu' \nu',00 \rangle =
          -G_{\tau} f(\nu,\tau) f(\nu',\tau),
\end{equation}
where
\begin{equation}\label{fntau}
 f(\nu,\tau)= \frac{(-)^{l_\nu}}{\sqrt{2}} \langle j_\nu|| Y_0 || j_\nu \rangle I(\nu,\tau).
\end{equation}
In Eq.~(\ref{fntau}) we used the Condon-Shortley phase convention for $\langle j_\nu|| Y_0 || j_\nu \rangle$ and
\begin{equation}
 I(\nu,\tau)=\int u_{\nu \tau}^2(r) f_\tau (r) dr.
\end{equation}
For the radial form factor $f_\tau(r)$ we took  the derivative of the WS potential
multiplied by $r$:
\begin{equation}\label{ff}
 f_\tau (r)=\frac{r}{a_{v\tau}}
    \frac{e^{\frac{r-R_{v\tau}}{a_{v\tau}}}}{\left( 1+ e^{\frac{r-R_{v\tau}}{a_{v\tau}}} \right)^2}.
\end{equation}

In the case of $^{212}$Po and $^{104}$Te  the two-particle  amplitudes of Eq.~(\ref{twop}) were obtained exactly in the  Tamm-Dancoff approximation \cite{1964Lane,2004Ring}:
\begin{equation} \label{eq.x}
 X_{\nu}^{\tau}=N_0 \frac{f(\nu,\tau)}{2 \epsilon^\tau_\nu - E^\tau_0},
\end{equation}
where $\epsilon^\tau_\nu$  are  s.p. energies,  $E^\tau_0$ is the correlated two-particle energy and $N_0$ is the normalization constant fixed by the condition $\sum_{\nu} \left( X_{\nu}^{\tau} \right)^2=1$.

The parameters $R_{v\tau}$ and $a_{v\tau}$ defining  the radial form factor (\ref{ff}) for $^{210}$Pb and $^{210}$Po were chosen to reproduce  the wave functions used by Harada \cite{1961Harada}. Since such data are not available for $^{102}$Sn and $^{102}$Te, in this case we adopted  the values of the WS potential for $^{100}$Sn shown in  Table~\ref{table.ws}.
The pairing strength $G_{\tau}$ was adjusted to fit the experimental two-nucleon separation energies $S_{2\tau}$ through the dispersion relation
\begin{equation}
  \frac{1}{G_\tau} = \sum_\nu \frac{ f^2(\nu,\tau)}{2 \epsilon_\nu^\tau - E^\tau_0}.
\end{equation}
Since the proton-unbound nucleus  $^{102}$Te is not known experimentally, for this system we adopted the value of $S_{2p}=-2.14$\,MeV obtained by extrapolating down from the heavier Te isotopes  \cite{nndc}. This value is in  reasonable agreement with recent phenomenological estimates \cite{2002Isakov}.
Table~\ref{table.2sp} lists the parameters of the residual interaction used in our study.
\begin{table}[htb]
\caption{\label{table.2sp} Parameters $R_{v\tau}$ and $a_{v\tau}$ of the residual interaction (\ref{vres}). The last column lists the value of $S_{2\tau}$ that has been used to constrain the pairing strength $G_{\tau}$ for various configuration spaces considered.}
 \begin{ruledtabular}
\begin{tabular}{cccc}
  nucleus           & $R_v$ (fm) & $a_v$ (fm) & $S_{2\tau}$ (MeV) \\
\hline
 $^{210}$Pb & 7.525 & 0.70 & 9.123  \\
 $^{210}$Po & 5.451 & 0.75 & 8.783 \\[2pt]
 $^{102}$Sn & 5.895 & 0.70 & 24.3 \\
 $^{102}$Te  & 5.895 & 0.70 &  $-$2.14 \\
\end{tabular}
 \end{ruledtabular}
\end{table}

\subsection{Configuration space}
To study the dependence of the  formation amplitude on the size of valence space, and to compare with previous work, we considered  several model spaces. Those used in the description of  the alpha decay of $^{212}$Po are given in
Table \ref{table.msPb}.
The model space M0 contains only one valence shell. The space M1 contains one  major shell, including  the unusual-parity intruder orbit. The model space M2 is that  used by Harada \cite{1961Harada}. The model space M3 is that of Glendenning and Harada \cite{1965Glendenning}. Finally, M4 is  the extended shell model space employed by Tonozuka and Arima.
\begin{table}[ht]
\caption{\label{table.msPb} Model spaces used in this work to describe  $^{212}$Po alpha decay.}
\begin{ruledtabular}
\begin{tabular}{|c|c|c|}
   Model       & Neutron States & Proton States \\
\hline
   M0   &  $1g_{9/2}$     &  $0h_{9/2}$ \\
\hline
   M1        &  $1g_{9/2},0i_{11/2},2d_{5/2},0j_{15/2}$     &  $0h_{9/2},1f_{7/2},0i_{13/2},2p_{3/2}$ \\
                          &  $3s_{1/2},2d_{3/2},1g_{7/2}$                     &  $1f_{5/2},2p_{1/2}$ \\
\hline
   M2         & $1g_{9/2},0i_{11/2},2d_{5/2}$                     &  $0h_{9/2},1f_{7/2},0i_{13/2}$ \\
\hline
   M3        & $1g_{9/2},0i_{11/2},2d_{5/2},0j_{15/2}$     &  $0h_{9/2},1f_{7/2},0i_{13/2}$ \\
\hline
  M4       & $1g_{9/2},0i_{11/2},2d_{5/2},0j_{15/2}$     & $0h_{9/2},1f_{7/2},0i_{13/2},2p_{3/2}$ \\
           & $3s_{1/2},2d_{3/2},1g_{7/2},1h_{11/2}$      &  $1f_{5/2},2p_{1/2},1g_{9/2},0i_{11/2}$ \\
           & $0j_{13/2}$                                 &  $0j_{15/2},2d_{5/2},3s_{1/2},1g_{7/2}$ \\
           &                                             &  $2d_{3/2},1h_{11/2},0j_{13/2},1h_{9/2}$ \\
\end{tabular}
\end{ruledtabular}
\end{table}
The model spaces used to describe $^{104}$Te alpha decay are shown in Table~\ref{table.msSn}; M1 consists of  one  major shell, including  the unusual-parity intruder orbit, while M4 consists of states with width less than $1$\;MeV.
\begin{table}[ht]
\caption{\label{table.msSn} Model spaces used in this work to describe $^{104}$Te alpha decay.}
\begin{ruledtabular}
\begin{tabular}{|c|c|c|}
  Model              & Neutron States & Proton States \\
\hline
   M1                   & $0g_{7/2},1d_{5/2},2s_{1/2},1d_{3/2}$     &  $0g_{7/2},1d_{5/2},2s_{1/2},1d_{3/2}$ \\
                      &  $0h_{11/2}$                                             &  $0h_{11/2}$ \\
\hline
   M4                   & $0g_{7/2},1d_{5/2},2s_{1/2},1d_{3/2}$     &  $0g_{7/2},1d_{5/2},2s_{1/2},1d_{3/2}$ \\
                      &   $0h_{11/2},1f_{7/2},2p_{3/2},2p_{1/2}$    &  $0h_{11/2},1f_{7/2},0h_{9/2},0i_{13/2}$ \\
                      &   $0h_{9/2},1f_{5/2},0i_{13/2}$             & \\
\end{tabular}
\end{ruledtabular}
\end{table}

\subsection{Wave functions} \label{sec.bcswf}
For the alpha formation amplitude in  $^{212}$Po  discussed in Sec.~\ref{sec.applications.ho} we considered the model spaces M2, M3 and M4. The wave function amplitudes in  M2 were taken from Refs.~\cite{1960Banerjee,1961Harada}.
For calculations in M3, we took the $T$=1 seniority-zero amplitudes  of Ref.~\cite{1965Glendenning} and renormalized them accordingly.
For calculations in the extended space M4, we used the renormalized amplitudes of Ref.~\cite{1979Tonozuka}; here we retained only  configurations having  width smaller than 1\,MeV.
The comparison between $^{212}$Po and $^{104}$Te discussed in Sec.~\ref{sec.applications.compare} was carried out in the model spaces M1 and M4. The corresponding  wave functions were calculated in the two-particle approximation described in Sec.~\ref{subsec.force}, except for $^{212}$Po in the M4 model space, where Ref.~\cite{1965Glendenning} was used instead.

\subsection{Penetration factor}\label{sec.gammasp}

The s.p. alpha width $\Gamma_0^{sp}$ has been  obtained from the current expression (\ref{eq.gammasp}). The alpha-core potential was assumed to be of a WS+Coulomb form with the parameters of Ref.~\cite{1976DeVries}: $r_0=R_c=1.315$\,fm, $a=0.65$\,fm.
The strength of the WS potential has been adjusted to reproduce the
measured $Q_\alpha$ value corrected by the electron screening term \cite{1954Thomas,Toth1960,1964Mang,1974Rasmussen,Hatsukawa1990}:
\begin{equation}
Q_\alpha = E_\alpha \frac{A_P}{A_D} + \Delta E_{\rm sc}
\end{equation}
where
\begin{equation}\label{escr}
\Delta E_{\rm sc} = 65.3 Z_P^{1.4}-80 Z_P^{0.4}~(\rm eV).
\end{equation}

For $^{212}$Po, $E_\alpha = 8.785$\,MeV \cite{nndc} and $\Delta E_{\rm sc}=31.8$\,keV; hence, $Q_\alpha= 8.986$\,MeV. The g.s. alpha decay of  $^{104}$Te has not been observed. For that reason, we took the value  $Q^{\rm BE}_\alpha$=5.135\,MeV extrapolated down from the binding energy differences in   $^{108}$Te (3.445\,MeV) and $^{106}$Te (4.290\,MeV) \cite{RykGrz11}. By adding the screening correction $\Delta E_{\rm sc}=16.1$\,keV, we arrived at
$Q_\alpha = 5.151$\,MeV.
The resulting WS
potential strength is  $V_0=143.49$\,MeV for $^{212}$Po and $149.64$\,MeV for
 $^{104}$Te.

The Gamow wave functions were obtained by means of the code  {\sc anti} \cite{1995Ixaru}. The   complex energy of the metastable alpha state is $\mathcal{E}_\alpha=(8.986-i0.632 \times 10^{-13})$\,MeV for $^{212}$Po and
$\mathcal{E}_\alpha=(5.151-i0.814 \times 10^{-13})$\,MeV for $^{104}$Te. The outgoing spherical Coulomb-Hankel function $H^+$ was calculated using the code \cite{1985Thompson}.

\subsection{Calculation of the spectroscopic factor}

The radial integration in the expressions for the spectroscopic factor (\ref{eq.s2}) and the formation amplitude in the normalized SGB (\ref{eq.gnu}) have been carried out  using $200$ Gauss-Legendre mesh-points with the maximum radius of  $20$ fm.

The s.p. core wave functions  entering Eq. (\ref{eq.nkkp2}) are those of the  s.p. Hamiltonian of Table~\ref{table.ws}.
The radial mesh $R_k$ defining the normalized SGB (\ref{eq.tildeF}) was taken at equidistant points $R_k=k\, \Delta R$. In order to determine the step $\Delta R$ we expanded the s.p. core states $u(r)$ in the normalized SGB: $\tilde{u}(r) \equiv \sum_{k=1}^M \; a_k\; [r \tilde{F}_0(r,R_k)]$. Under the condition that $u_{{\rm diff}}(r)=|u(r)-\tilde{u}(r)|| < 0.005$\,fm$^{-1/2}$ we found that $0.44\,\rm{fm} \lesssim \Delta R \lesssim 0.57$\,fm  and $R_{\textrm{max}} \gtrsim 14$\,fm. For this range of $\Delta R$  and $R_{\textrm{max}}$ the normalized SGB is orthonormal with an accuracy better than $10^{-9}$. To illustrate the quality of the resulting  expansion, Fig. \ref{fig.udiff} shows $u_{{\rm diff}}(r)$ for the neutron core states in $^{208}$Pb.
\begin{figure}[htb]
 \includegraphics[width=0.45\textwidth]{01.figdiffuexpminucore-PoNeutron.eps}
 \caption{ \label{fig.udiff} $u_{{\rm diff}}(r)=|u(r)-\tilde{u}(r)|$ for  the  neutron core states in $^{208}$Pb for $\Delta R=0.5$\,fm, $M=30$, and  $R_{\textrm{max}}=15$\,fm.}
\end{figure}

To calculate the modified formation amplitude $G(R)$, one needs to determine the  eigenvalue cutoff $n_{\rm min}$. To this end, we show in Figs.~\ref{fig.nPo} and \ref{fig.nTe} typical distribution of the eigenvalues $n_\nu$ of the norm kernel (\ref{eq.nk}) for  $^{212}$Po  and $^{104}$Te, respectively,  for different values of $\Delta R$. One may observe that a significant fraction of them accumulate at zero \cite{1990Bonche,2004Ring}. To eliminate these spurious eigenvectors, we  define the cutoff at the value where the eigenvalue distribution changes slope. For $^{212}$Po and $^{104}$Te this happens at $n_\nu$ around $10^{-3}$. Consequently, in our calculations, we adopt the cutoff value of $n_{\rm min}=0.001$.
\begin{figure}[htb]
 \includegraphics[width=0.45\textwidth]{02.figeigenvalues-PoRmax13.eps}
 \caption{ \label{fig.nPo} Eigenvalues of the norm kernel (\ref{eq.nk}) for $^{212}$Po for $R_{\textrm{max}}=13$ fm for different values of $\Delta R$.}
\end{figure}
\begin{figure}[htb]
 \includegraphics[width=0.45\textwidth]{03.figeigenvalues-TeRmax10.eps}
 \caption{ \label{fig.nTe} Similar as in Fig.~\ref{fig.nPo} except for  $^{104}$Te and  $R_{\textrm{max}}=10$ fm.}
\end{figure}

The eigenfunctions $u^L_\nu(R)$ of the norm kernel (\ref{eq.ur})  are orthonormal with an accuracy of  $10^{-10}$ for all eigenvalues. The eigenfunctions with  $n_\nu < n_{\rm min}$ oscillate inside the nuclear volume  and vanish outside the surface region.  To further check the quality  of  $u^L_\nu(R)$ we compute expression  (\ref{eq.gbig2}) by assuming  $n_{\rm min}=0$ and $n_\nu=1$ for all $\nu$. In this case, Eq.~(\ref{eq.gbig2}) formally reduces  to $g(R)$. Figure \ref{fig.gexp} shows
$g(R)$ for $^{212}$Po calculated in this way. The agreement with the original formation amplitude is excellent, except for a small deviation close to $R=0$ and a  small oscillation around and beyond the nuclear surface, which is not visible in the scale of  Fig.~\ref{fig.gexp}.
\begin{figure}[htb]
 \includegraphics[width=0.45\textwidth]{04.figcomparegsmallgexpnormbasis.eps}
 \caption{ \label{fig.gexp} Formation amplitude $g(R)$ for $^{212}$Po in the M4 model space expanded in eigenfunctions of the norm kernel for $\Delta R=0.500$ fm and  $R_{\textrm{max}}=13$ fm.}
\end{figure}

Next we study the sensitivity of $\mathcal{S}$ to the choice of $R_{\textrm{max}}$, $\Delta R$, and $n_{\rm min}$. For this analysis we relax the condition for $u_{{\rm diff}}(r)$ in order to access a wider range of $\Delta R$. First, we study the sensitivity of $\mathcal{S}$ as a function of $R_{\textrm{max}}$ for various values of  $\Delta R$. Figure \ref{fig.svsrmax} shows the result for $0.53 \rm{fm} \leq \Delta R \leq 0.59$ fm for $^{212}$Po in the model space M4 and $n_{\rm min}=0.001$. Except for a small value of $\Delta R=0.53$\,fm, which does not produce stable results, a plateau in  $R_{\textrm{max}}$ is reached around 14\,fm.
\begin{figure}[htb]
 \includegraphics[width=0.45\textwidth]{05.figSvsRmax_parameinDR.eps}
 \caption{\label{fig.svsrmax} Convergence of $\mathcal{S}$ for $^{212}$Po (model space M4) as a function of $R_{\textrm{max}}$ of the normalized SGB for different values of  $\Delta R$ (with $n_{\rm min}=0.001$.)}
\end{figure}

The dependence of $\mathcal{S}$ on $\Delta R$ displayed in Fig.~\ref{fig.svsrmax} reflects the fact that for too small values of the step
the basis functions become numerically linearly dependent, while for too large $\Delta R$'s the basis cannot capture  high Fourier components \cite{1976Fliessbach1,1990Bonche,2004Ring}.
Figure~\ref{fig.svsDR} shows $\mathcal{S}$ for $^{212}$Po in the model space M4 and $n_{\rm min}=0.001$ as a function of $\Delta R$. In general, appreciable oscillations of $\mathcal{S}$ can be seen except for the ``safe" region $0.54\,\rm{fm} \leq \Delta R \leq 0.59$ fm, where results weakly depend on $R_{\textrm{max}}$.
\begin{figure}[htb]
 \includegraphics[width=0.45\textwidth]{06.figSvsDR_parameinRmax.eps}
 \caption{\label{fig.svsDR} Similar as in Fig.~\ref{fig.svsrmax} but as a function of the step size $\Delta R$  for different values of $R_{\textrm{max}}$.}
\end{figure}

Finally, Fig.~\ref{fig.svsnmin} shows the behavior of $\mathcal{S}$ as a function of the eigenvalue cutoff $n_{\rm min}$ for $\Delta R=0.57$\,fm. The cutoff used in Figs. \ref{fig.svsrmax} and \ref{fig.svsDR} corresponds to $n^{-1/2}_{\rm min}=(0.001)^{-1/2} \approx 31.5$.
\begin{figure}[htb]
 \includegraphics[width=0.45\textwidth]{07.figlogSvssqrtnmin.eps}
 \caption{\label{fig.svsnmin} Similar as in Fig.~\ref{fig.svsrmax} but as a function of  $n_{\rm min}$ for  different values of $R_{\textrm{max}}$ and $\Delta R=0.57$.}
\end{figure}

\subsection{Integral over intrinsic coordinates}
The multidimensional integral (\ref{eq.inp}) depends on the nucleonic coordinates, which are parametrized in terms of intrinsic  variables through Eqs.~(\ref{eq.rhoandr}) and (\ref{eq.cos}). The integration over $\tilde{\varphi}_i$ can easily be done analytically. Since the coordinates of particles 1 and 2  depend only on the relative coordinates 1 and 3, and the particle coordinates 3 and 4  depend only on the relative coordinates 2 and 3,  one can greatly simplify the remaining six-dimensional integral by making first the integration over the relative coordinates 1 and 2 and then the integration over the coordinate 3:
\begin{equation}
    \int d\gras{\rho}_3 \left[ \dots \left(\int \dots d\gras{\rho}_1\right)  \left( \int \dots d\gras{\rho}_2 \right) \right].
\end{equation}
The integration has been  carried out using the Gauss-Legendre quadrature
using 10 points for the radial integrals and 8 points for the
the angular coordinates. This guarantees the convergence up to the fourth significant digit.

\section{Reduced width for $^{212}$P\lowercase{o}} \label{sec.applications.ho}

\subsection{Single-$j$ configuration}

Following Rasmussen \cite{1963Rasmussen}, it is instructive to compute
relative reduced widths
assuming a pure single-$j_n$ shell model orbital assignment for the neutron
pair, while  the proton pair fills the  $0h_{9/2}$ shell.
For
simplicity,  the results are expressed relative to the $^{210}$Po reference (a neutron pair in  $2p_{1/2}$).

In the delta-function approximation of Sec.~\ref{subsec.formalism.delta}, the ratio $r_\delta$ of the reduced widths  is given by a simple expression \cite{1963Rasmussen}:
\begin{equation} \label{eq.ratiodelta}
  r_{\delta} = \frac{\gamma^2_{j_n}}{\gamma^2_{2p_{1/2}}}=\frac{2j_n+1}{2} \; \left( \frac{u_{j_n}(R)}{u_{2p_{1/2}}(R)} \right)^4.
\end{equation}
In a more general case expressed by Eq.~(\ref{eq.f0}), the ratio $r$ depends on the proton wave function:
\begin{equation}  \label{eq.ratio}
  r = \frac{\gamma^2_{j_n,0h_{9/2}}}{\gamma^2_{2p_{1/2},0h_{9/2}}} =  \frac{2j_n+1}{2} \; \left( \frac{I_{j_n,0h_{9/2}}(R)}{I_{2p_{1/2},0h_{9/2}}(R)} \right)^2.
\end{equation}
Table~\ref{table.ratio} compares the ratio $r_\delta$ given by Eq.~(\ref{eq.ratiodelta}) using the WS wave functions with that of Table~I of Rasmussen  \cite{1963Rasmussen} based on the rounded square well potential of Blomqvist and Wahlborn \cite{1960Blomqvist} for several neutron configurations at $R=9.5$\,fm. We find excellent agreement between these two calculations, and we checked that
this agreement also holds for $R=9.0$\,fm. This is not surprising as both calculations employ finite-depth potentials.
\begin{table}
\caption{ \label{table.ratio} Single-$j$ alpha reduced width ratios  at $R=9.5$\, fm. Shown are: $r_\delta$ of Ref.~\cite{1963Rasmussen}, $r_\delta$ of Eq.~(\ref{eq.ratiodelta}), $r$ of Eq.~(\ref{eq.ratio}), and $r$ of Ref.~\cite{1962Zeh}.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
 Orbital $j_n$    &  $r_\delta$ \cite{1963Rasmussen} & $r_\delta$  & $r$  &  $r$ \cite{1962Zeh}  \\
\hline
 $0i_{13/2}$ & 0.44							  &  0.46    &    0.20 & 0.10\\
 $0i_{11/2}$ & 0.32							  &  0.34  &    0.21  &  0.08\\
 $1g_{9/2}$  & 7.50							 &  7.50     &  6.50  & 3.73     \\
 $1f_{5/2}$   & 0.73							 &  0.74  &  0.58  &  0.55   \\
 $2p_{3/2}$  & 1.89							 &  1.89     &   1.73  &  1.89
\end{tabular}
\end{ruledtabular}
\end{table}
The fourth column of Table \ref{table.ratio} displays  the ratio $r$ given by
Eq.~(\ref{eq.ratio}) using the WS wave functions; they are compared with the h.o. values of Ref.~\cite{1962Zeh} (last column). It is seen that h.o. calculations
underestimate  WS values for high-$j$ orbits by
a factor two-to-three.

It has been early  recognized \cite{1960Mang,1963Rasmussen} that the delta-function  approximation  overestimates the contributions of high-$j$ orbitals. One can see it clearly by comparing the values of $r_\delta$  of Eq.~(\ref{eq.ratiodelta}) with those of $r$ (\ref{eq.ratio}), i.e., the third and fourth columns of Table \ref{table.ratio}. To cure this deficiency,
a correction factor $B_\nu$  (\ref{Bcorr})  was introduced \cite{1963Rasmussen} in Eq.~(\ref{Bcorr2}) that depends on the relative angular momentum.

\subsection{Enhancement due to configuration mixing}\label{enhan}

As   was first shown by Harada \cite{1961Harada}, the reduced width  at the surface region is strongly enhanced by the configuration mixing  because
contributions from various shell model orbits add  coherently.
To assess the effect of collective enhancement due to the configuration mixing,
we carried out calculations in the M2 space. For $R=8$\,fm,
our WS calculations yield the enhancement factor of $\zeta=8.5$ with respect to the valence-shell configuration M0. This is to be compared with $\zeta=11$ obtained
in the delta-function  approximation;  $\zeta=10$ obtained by Rasmussen \cite{1963Rasmussen}; and $\zeta=5.5$ of  Harada \cite{1961Harada} using h.o. wave functions.

For the model space M3 of Glendenning and Harada \cite{1965Glendenning},
obtained by adding the intruder neutron state $0j_{15/2}$ to M2, we obtain $\zeta=21$. This should be  compared with $\zeta=24$ obtained
in the delta-function  approximation and  $\zeta=30$ obtained in Ref.~\cite{1965Glendenning} (also within the delta-function  approximation) using a fairly rich wave function that also includes proton-neutron correlations and $J>0$ two-particle couplings. It is worth noting  that our enhancement is around $80\%$ of that by Glendenning and Harada, and that  the seniority-zero component   in  their wave function is also 80\%.

\subsection{Extended shell model space}

Due to the strong collective enhancement of the reduced width due to configuration mixing, it is important to consider extended shell-model space by taking into account higher-lying  orbitals~\cite{1979Tonozuka}. For finite-depth
shell-model potentials, such as the WS potential used in this study,  this necessitates a proper treatment of the particle continuum. An appropriate representation to deal with the continuum space is the complex Berggren ensemble representing bound and unbound s.p. states \cite{1968Berggren,Michelrev}.

Here we consider the large configuration space M4 of Tonozuka and Arima \cite{1979Tonozuka}, i.e.,
all s.p.  orbits up to $N = 7$ harmonic oscillator shell except for broad resonances  with widths greater than 1 MeV. The shell-model amplitudes were taken from  Ref.~\cite{1979Tonozuka} and renormalized to the reduced model space. For the sake of comparison with Ref.~\cite{1979Tonozuka}, we consider
the relative reduced width
\begin{equation}\label{thetaw}
\theta^2(R)=\frac{\gamma^2(R)}{\gamma^2_W(R)},
\end{equation}
where  $\gamma^2_W(R)=\frac{3 \hbar^2}{2 \mu R^2}$ is the  Wigner limit  \cite{1958Lane}.

Table \ref{table.tonozuka} compares our WS results for $\theta^2(R)$ with those   of Ref.~\cite{1979Tonozuka} obtained in the h.o. basis for several values of $R$.
\begin{table}[htb]
\caption{\label{table.tonozuka} The relative reduced width $\theta^2$  (\ref{thetaw})
obtained in Ref.~\cite{1979Tonozuka} and this work.}
\begin{ruledtabular}
\begin{tabular}{cccc}
 Model space &  $R$ (fm)  & Ref.~\cite{1979Tonozuka}  & This work \\
 \hline
 M0          &  8.4        &    $6.3 \times 10^{-6}$    &    $0.60 \times 10^{-6}$     \\
 M3          &  8.5        &    $4.4 \times 10^{-5}$    &    $0.48 \times 10^{-5}$     \\
 M4          &  9.0        &    $2.9 \times 10^{-4}$    &    $0.41 \times 10^{-4}$
\end{tabular}
\end{ruledtabular}
\end{table}
Generally,  the reduced width obtained in the WS model is about one order of magnitude smaller than that in the h.o. basis. This is because the h.o. basis knows nothing about the particle thresholds, and the radial behavior at large distances is solely determined by the oscillator length. For that reason, calculations based on the h.o. wave functions show large sensitivity to this parameter \cite{1962Harada}.

\begin{figure}[htb]
 \includegraphics[width=0.45\textwidth]{08.figgtimesgPoM0toM4.eps}
 \caption{ \label{fig.fl} Formation amplitude $g(R)$ for $^{212}$Po obtained in this work in the model spaces M0 to M4 as defined in table \ref{table.msPb}. The imaginary part of the formation amplitude in M4 (dotted line) is also shown.}
\end{figure}
The formation amplitude obtained in this work is shown in
Fig. \ref{fig.fl} for the configuration spaces M0, M1, M2, M3, and M4.
Compared with the formation amplitudes of Ref.~\cite{1979Tonozuka},  the maximum
of the formation amplitudes obtained in
the WS model are significantly larger, and appear at lower values of $R$,  than in the h.o. model. Also
the overall shape of the formation amplitude is very different in the two cases. A  characteristic two-humped shape of $g(R)$ calculated in M4 resembles the formation amplitude $G(R)$ obtained in Refs.~\cite{1976Fliessbach,1979Tonozuka}.
A similar result was also obtained in Refs.~\cite{1989Okabe,1992Varga}.
It is indeed interesting to see that a two-humped behavior of  the formation amplitude for $^{212}$Po has been obtained by considering large configuration space and the Berggren ensemble of the WS potential.

Figure~\ref{fig.fl} also shows that the formation amplitude  in the M4 model space has a small imaginary part. This is because our calculations are carried out in the pole approximation that ignores the non-resonant continuum \cite{1993Beggren,2003IdBetan,2003Michel,Michelrev}. This  spurious component of $g(R)$ results in a very small imaginary contribution to the reduced width, which can be safely neglected considering the expected accuracy of our model.

\section{Absolute alpha-decay width of $^{212}$P\lowercase{o}} \label{sec.aw}

The  g.s. alpha-decay  width of  $^{212}$Po has been determined in the  seniority-zero approximation using three different models spaces listed in  Table \ref{table.msPb}: M0, M3, and M4. The corresponding four-particle shell-model wave function contains one configuration in the M0 space, 12 configurations in M3, and 144 seniority zero configurations in M4.

The absolute width from Eq.~(\ref{Grmatrix}) should not depend on the channel radius  $R$. However, in  R-matrix studies involving approximations, such as the  one-channel R-matrix treatment, this  condition cannot be met
\cite{2010Descouvemont}. Therefore, in practical calculations, in which
the dependence of $\Gamma_L$ on $R$ around the nuclear surface is small relative to the appreciable $R$-dependence of the formation amplitude, one is trying to meet the plateau condition for $\Gamma_L(R)$ in which the absolute width varies weakly  around the nuclear surface \cite{1988Insolia}.  Figure~\ref{fig.gamma} shows  the dependence of the R-matrix width (\ref{Grmatrix}) on the channel radius.
It is seen that the plateau condition is met only in the case of the extended configuration space M4 involving particle continuum. Here, we find  a fairly weak variation of $\Gamma(R)$  between 7\,fm and 11\,fm.
\begin{figure}[htb]
 \includegraphics[width=0.45\textwidth]{09.widthfromthomas.eps}
 \caption{\label{fig.gamma} Dependence of the absolute  alpha-decay width
 (\ref{Grmatrix}) of  $^{212}$Po  on the R-matrix channel radius $R$
for  three different model spaces M0, M3 and M4.}
\end{figure}

As seen in Fig.~\ref{fig.gamma}, and discussed in  Sec.~\ref{enhan} and Refs.~\cite{1962Harada,1979Tonozuka,2010Qi}, the width strongly increases with the size of the shell-model space. Indeed, in the surface region,
$\Gamma(R)$ obtained in M3 shows an enhancement $\sim$15 with respect to M0, and in the extended space M4 the enhancement is $\sim$260. Compared to experimental value, however, the width obtained in M4 is still 600 times smaller than the experimental value  $\Gamma_{\rm exp}=0.153 \times 10^{-14}$\, MeV \cite{nndc}.

A further enhancement in the reduced width is due to the antisymmetrization and normalization of the channel decay \cite{1975Fliessbach,1976Fliessbach}. This is achieved by replacing the standard formation amplitude $g(R)$ with the modified formation amplitude $G(R)$ of Eq.~(\ref{eq.gbig}). Figure~\ref{fig.gbig} shows $G(R)$ calculated in the M4 model space with $\Delta R=0.56$ fm, $R_{\textrm{max}}=11.76$ fm ($M=21$) and $n_{\rm min}=0.001$. A small oscillation at the tail of $G(R)$  can be seen. The amplitude of this oscillation, around the asymptotic behavior given by  $H_0^+(\eta,kR)$, varies very little with  $R_{\textrm{max}}$ for this value of $\Delta R$.  As discussed in, e.g., \cite{1976Fliessbach,1992VargaLovas,1998Lovas}, the behavior  of $g(R)$ and $G(R)$ is generally very different. This can be seen by comparing Figs.~\ref{fig.fl} and \ref{fig.gbig}.
\begin{figure}[htb]
\includegraphics[width=0.45\textwidth]{10.figcompareasymptoticGtimesRandHplus.eps}
 \caption{ \label{fig.gbig} Modified formation amplitude $G(R)$ of Eq.~(\ref{eq.gbig}) in the extended  model space M4 with $n_{{\rm min}}=0.001$, $\Delta R=0.56$ fm and $R_{\textrm{max}}=11.76$. Unlike $g(R)$, $G(R)$ properly accounts for the normalization and antisymmetrization of the decay channel. The asymptotic behavior of $G(R)$ is given by  the Coulomb-Hankel function at the alpha-decay energy $Q_\alpha=8.986$ MeV (dashed line).}
\end{figure}

The  absolute alpha-decay width obtained by using the R-matrix expression (\ref{Grmatrix}) with the formation amplitude $G(R)$ of Fig.~\ref{fig.gbig}  is shown in Fig.~\ref{fig.gamma2}. There appears a small plateau in the region of nuclear surface that corresponds to $\Gamma \approx 0.0042 \times 10^{-14}$ MeV. This value  is $\sim 36$ times smaller than $\Gamma_{\rm exp}$.
At larger distances $R> 9$\,fm, the result is affected by spurious oscillations
of $G(R)$ around $H_0^+(\eta,kR)$, i.e., it is  quite unreliable.
\begin{figure}[htb]
 \includegraphics[width=0.45\textwidth]{11.figwidthvsr.eps}
 \caption{ \label{fig.gamma2} Absolute width from R-matrix expression (\ref{Grmatrix}) calculated in the M4 model space using the modified formation amplitude $G(R)$ of Fig.~\ref{fig.gbig}. At $R>9$\,fm, the result obtained by assuming $G(R) \propto H_0^+(\eta,kR)$  is marked by a dotted line.}
\end{figure}

The absolute width  can also be obtained from  expression (\ref{Goverlap}), which involves the alpha-particle spectroscopic factor $\mathcal{S}$  and the s.p. decay width.  Figure~\ref{fig.gammasp} shows the result of  the current expression (\ref{eq.gammasp}) for $\Gamma^{sp}$ as a function of the channel radius.
As discussed in Ref.~\cite{2004Kruppa},  $\Gamma^{sp}$ calculated this way
should be independent of $R$   if $R$  is
large enough. This is precisely what is seen in Fig.~\ref{fig.gammasp}:
the s.p. width converges beyond the range of the WS potential to $\Gamma^{sp}=0.1247 \times 10^{-12}$ MeV, which is indeed very close to the value of $-2$Im$({\cal E}_\alpha)=0.1265  \times 10^{-12}$ MeV given by the imaginary part of the Gamow resonance.
\begin{figure}
 \includegraphics[width=0.45\textwidth]{12.widthsp.eps}
 \caption{ \label{fig.gammasp} Single particle  width of $^{212}$Po from current expression (\ref{eq.gammasp}).}
\end{figure}

Using the modified formation amplitude $G(R)$ of Fig.~\ref{fig.gamma2}, we  compute the spectroscopic factor $\mathcal{S} = 0.011$, which -- combined with the value of $\Gamma^{sp}$ above --  yields $\Gamma= 0.14 \times 10^{-14}$ \,MeV. Using $\Delta R=0.55$ fm we obtain $\mathcal{S} = 0.0080$ and  $\Gamma= 0.10 \times 10^{-14}$ \,MeV. Both these values are close to $\Gamma_{{\rm exp}}=0.153 \times 10^{-14}$\,MeV.

\section{Comparison between ground-state alpha decay of $^{212}$P\lowercase{o} and $^{104}$T\lowercase{e}} \label{sec.applications.compare}

To compare  absolute widths of $^{212}$Po and $^{104}$Te in a consistent way, we consider similar M1 and M4 model spaces for both nuclei.  The norm kernel eigenvalues $n_\nu$ do not depend on the model space in which $g(R)$ is calculated, so we  take the cutoff $n_{{\rm min}}=0.001$.

Let us begin with $^{212}$Po  by making a convergence analysis of $\mathcal{S}$ in the M1 model space as a function of $\Delta R$ and $R_{\rm{max}}$ (as in Fig. \ref{fig.svsrmax}). For $\Delta R=0.53,\, 0.54,\, 0.55$, and $0.56$ fm, we found $\mathcal{S} = 0.0041,\, 0.0011,\, 0.00030,\, 0.00032$, respectively. The resulting converged value $\mathcal{S} = 0.0003$ is  too small, as expected from Fig. \ref{fig.fl}. This deficiency is related to the poor quality of the interaction used to describe  $^{212}$Po in M1. To better understand this fact, let us take a look of the spectroscopic factor in terms of the spectral representation of the norm kernel,
\begin{equation} \label{eq.s3}
 \mathcal{S} =\sum_\nu \frac{g^2_\nu}{n_\nu},
\end{equation}
where the sum is truncated by the condition $n_\nu > n_{{\rm min}}$.
The summation range and eigenvalues $n_\nu$ are the same for  M1 and M4; the only difference comes from $g_\nu$. Because of the rapid oscillation of the eigenfunctions inside the nucleus, only the eigenfunctions which are peaked at and beyond the nuclear surface will contribute significantly to the sum. But -- because $g(R)$ in M1 is small in the surface region -- the overlap with those eigenfunctions is small, and this gives rise to a very reduced value of $\mathcal{S}$.

By making a similar analysis for $^{104}$Te in M1, we found $\mathcal{S} = 0.067,\, 0.024,\, 0.0066$, and $0.00046$ for $\Delta R=0.53,\, 0.54,\, 0.55$, and 0.56 fm,  respectively. In the model space M4 we found $\mathcal{S} = 0.21$, 0.088, 0.032, and 0.0051 for the same  values of $\Delta R$.
Clearly, the convergence in $\mathcal{S}$ has not been achieved for $^{104}$Te. We would like to attribute this to the impact of the proton continuum on $g_\nu$, which results in increased oscillations of $G(R)$ in the surface area.
Table \ref{table.s} compares the  values of  $\mathcal{S}$ and the corresponding  absolute widths for $^{212}$Po and $^{104}$Te at $\Delta R=0.56$\,fm. (The single particle width for Te is $\Gamma^{sp}=0.162 \times 10^{-12}$ MeV.)
\begin{table}[htb]
 \caption{\label{table.s} Alpha decay spectroscopic factor and absolute width for $^{212}$Po and $^{104}$Te computed in the configuration spaces M1 and M4, with $n_{{\rm min}}=0.001$, $\Delta R=0.56$ and $R_{\textrm{max}}=11.76\,\rm{fm}\, (M=21)$.}
\begin{ruledtabular}
\begin{tabular}{c|cc|cc}
 Model Space  &
 \multicolumn{2}{c|}{$\mathcal{S}$} &
  \multicolumn{2}{c}{$\Gamma\times 10^{14}$ MeV} \\
  & $^{212}$Po    & $^{104}$Te    & $^{212}$Po              & $^{104}$Te \\
\hline
 M1          & $0.00032$     & $0.00046$          & $0.0040$               & $0.0075$  \\
 M4          & $0.011$       & $0.0051$           & $0.14$                 & $0.083$
\end{tabular}
\end{ruledtabular}
\end{table}

It is interesting to compare our current results for $^{104}$Te with the estimates of phenomenological alpha-decay models based on semi-classical approximation \cite{Xu2006,Ren2006,Mohr07}. The assumed large value of $Q_\alpha=6.12$\,MeV in Ref.~\cite{Xu2006} results in a very short half-life of $7\times 10^{-11}$\,sec. The alpha-decay energies of 5.05\,MeV \cite{Ren2006} and 5.42$\pm$0.07\,MeV \cite{Mohr07} result in   $T_{1/2}\sim10^{-7}$\,sec and $\sim5\times 10^{-9}$\,sec, respectively, and these estimates are not inconsistent with our value (M4 model space) $T_{1/2}=5.5 \times 10^{-7}$\,sec ($Q_\alpha=5.151$\,MeV).
As the value of  $Q_\alpha$  in  $^{104}$Te is very uncertain, we show in Fig.~\ref{fig.gammate} the absolute width and half-life  $T_{1/2}$ as a function of $Q_\alpha$ for the model space M4.
\begin{figure}[htb]
\includegraphics[width=0.45\textwidth]{13.gammaandhalflive_vs_q-Te.eps}
 \caption{\label{fig.gammate} Ground-state alpha-decay width (left scale)  and half-life (right scale) in $^{104}$Te as functions of the decay energy.}
\end{figure}

Our predicted spectroscopic factors in M4 for $^{104}$Te  and $^{212}$Po are about 0.5\% and 1\%, respectively. As mentioned above,  a fairly small  value of $\mathcal{S}$ in $^{104}$Te could be a consequence of the proximity of the proton  continuum. Indeed, all the valence proton shells are resonances.
The small value of $\mathcal{S}$   in $^{104}$Te could also be attributed to the poor quality of the valence interaction assumed, and the neglect of the $T=0$ force. The effect of the proton-neutron interaction was examined in, e.g., Refs.~\cite{1961Harada,Sasaki1970} for $^{212}$Po and was found to be minor due to the fact that   neutrons and protons in $^{212}$Po occupy different shells. This is no longer true in the $N=Z$ nucleus $^{104}$Te, in which the major enhancement of $\mathcal{S}$ is expected due to $T=0$
correlations. Therefore, our predictions for $\mathcal{S}$ and $\Gamma$ in  $^{104}$Te  given in Table~\ref{table.s} should be considered as a very conservative lower limit.

\section{Conclusions} \label{sec.conclusions}

The g.s. alpha decay of $^{212}$Po has been studied within the complex-energy shell model framework with the Berggren ensemble of the average Woods-Saxon potential. We applied the pole approximation by considering   s.p. resonant states
only. The overlap integral involving alpha-cluster nucleons was computed exactly, without resorting to the delta-function approximation. We considered the large valence space of Tonozuka and Arima that is necessary to produce the collective enhancement of the formation amplitude.

The absolute alpha-decay width was computed using the reduced width obtained in the framework of the R-matrix theory and also from the alpha spectroscopic factor. The latter approach yielded  results consistent with experimental value, but only after considering
the antisymmetrization and normalization of the decay channel wave function.
The R-matrix estimate underestimates the experimental width by a factor of
$\sim 36$.
The R-matrix expression depends on the asymptotic value of the formation amplitude that is very sensitive to the size of the configuration space. On the other hand, the reaction-theory expression (\ref{Goverlap}) involves the spectroscopic factor -- an integral quantity that depends less on the size of the basis used.
It is very encouraging to see that a reasonable agreement with the experimental width of $^{212}$Po has been obtained without  explicitly considering the alpha-cluster component in the wave function of the parent nucleus. In this context, we believe that the improved treatment of the particle continuum has been essential.

We have  also provided an estimate of the alpha-decay rate in $^{104}$Te.
Unfortunately, due to the fact that the valence proton shells in this nucleus lie in the continuum, no fully convergent result has been achieved. We hope to improve the situation in the future by inclusion of the non-resonant continuum space that will remove some of the undesired oscillations in $G(R)$ at large distances.
In addition, since the residual interaction employed in our work neglects the proton-neutron components, and the wave function has a seniority-zero character based on $T=1$
nucleonic pairs, the predicted alpha width in this $N=Z$ nucleus should be viewed as a conservative low limit. Indeed, the inclusion of $T=0$ correlations is expected to increase the value of $\Gamma$ significantly.

The  calculations presented in this study should be considered as an important step   towards an improved  microscopic understanding of the alpha-decay process. Still, as this work demonstrates, further improvements are  needed.
The neglect of the non-resonant continuum, i.e., complex-energy scattering states in the Berggren ensemble, slightly violates the completeness relation at a one-body level. This results in small imaginary contributions to spectroscopic factors and reduced widths, and -- most importantly -- can  affect the behavior of formation amplitudes at very large distances. The second crucial development will be the use of large-scale shell model calculations, including realistic
$T=0$ and $T=1$ interactions, to compute wave function amplitudes. This will enable us to provide a more meaningful estimate of $^{104}$Te alpha decay rate.
The work in both directions is underway.

\begin{acknowledgments}
Useful discussions with Doru Delion, Torsten Fliessbach, Robert Grzywacz, Kiyoshi Kato, and Krzysztof Rykaczewski  are gratefully acknowledged. Special thanks to  Rezso Lovas for his insights on the norm kernel and his patience in explaining the underlying physics. This work was supported  by the Office of
Nuclear Physics,  U.S. Department of Energy under Contract No.
DE-FG02-96ER40963 and  by the National Council of Research PIP-77 (CONICET, Argentina).
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


text:
71 dead after plane carrying Brazilian football team crashes in Colombia
Published time: 29 Nov, 2016 05:10 Edited time: 11 Apr, 2018 10:07
Wreckage from a plane that crashed into Colombian jungle with Brazilian soccer team Chapecoense is seen near Medellin, Colombia, November 29, 2016. © Fredy Builes © Reuters
A plane carrying Brazil's Chapecoense football team has crashed in Colombia en route to the biggest match in the team's history, killing 71 of the 77 people on board. Three days of national mourning have been declared.
Earlier, the number of people on board was reported as 81.
"This wasn't just a group of people who respected each other professionally. It was a family, a group of friends," club President Plínio David de Nes Filho, who was supposed to travel with the team before changing his plans, told an interviewer on Brazilian television.
"Everybody laughed so much, even in defeat... There was a great atmosphere, great joy. Yesterday morning, when I said goodbye to them, they said that they were going off to make our dreams come true. We shared this dream with all our emotion. And in the early hours of this morning, that dream came to an end," he said.
Rescue crew work in the wreckage from a plane that crashed into Colombian jungle with Brazilian soccer team Chapecoense, seen near Medellin, Colombia, November 29, 2016. © Fredy Builes © Reuters
The team had been traveling from its hometown of Chapeco, in the southern state of Santa Catarina, to Medellin in Colombia. It was due to play the first leg of the Copa Sudamericana continental tournament against local side Atletico Nacional on Wednesday.
READ MORE: Brazilian football team took doomed flight after being denied charter request
Airport authorities state 72 passengers from Chapecoense football team and 9 crew onboard #LMI2933. https://t.co/ayIDZCJVZspic.twitter.com/HwgVWg5e9w
— Flightradar24 (@flightradar24) 29 November 2016
After a layover in Bolivia, the chartered 1999-built British Aerospace 146 aircraft set off for Colombia, but reported electrical problems at about 10pm local time on Monday night as it was nearing its destination. Authorities are also investigating reports that the plane – operated by LaMia Airlines, a Venezuelan-Bolivian company – had run out of fuel before reaching the airport. There have also been local reports that the pilot may have dumped fuel on purpose seconds before the crash, preventing a massive explosion.
A rescue worker checks a body from a plane that crashed into Colombian jungle with Brazilian soccer team Chapecoense, near Medellin, Colombia, November 29, 2016. © Fredy Builes © Reuters
The plane crashed on a hillside outside Medellin. Photos from the site show that most of the tail and body of the plane were completely destroyed as the aircraft split in two. Two flight recorders have been retrieved from the wreckage.
A downpour and low visibility complicated the rescue operation, as helicopters were dispatched to the inaccessible location, but then had to be grounded. Survivors had to be carried by foot on stretchers, then loaded on to trucks that made their way through the slippery mud to the ambulances, as temperatures remained at 5 degrees Centigrade.
At least seven passengers were found alive at the crash site, though one has since died in hospital.
"The possibility that other people will be found alive has not been ruled out," said a statement from Colombia's disaster risk management agency.
Aerocivil publica el listado de pasajeros sobrevivientes del vuelo de Lamia CP 2933 accidentando en Antioquia https://t.co/kHymbYbYaipic.twitter.com/7xmV4t1s36
— AERONÁUTICA CIVIL (@AerocivilCol) 29 November 2016
Four footballers were among those pulled out of the wreckage, along with two crewmembers and a journalist, but Marcos Danilo Padilha, the team's starting goalkeeper, died from his injuries.
"The government will do everything possible to alleviate the pain," said a tweet from Brazilian President Michel Temer.
This pic. Punch in stomach:
club's locker room: players of #Chapecoense who have not travelled,after being told news pic.twitter.com/EZthaTu2yb
— Tancredi Palmeri (@tancredipalmeri) 29 November 2016
Chapeco, a city of only 200,000 that is most famous for its food processing industry, will be scarred by the sudden demise of its Cinderella team, which rose from the fourth to the top division in less than a decade, thanks to savvy management.
Fans of Chapecoense soccer team react in front of the Arena Conda stadium in Chapeco, Brazil, November 29, 2016. © Paulo Whitaker © Reuters
Hundreds of supporters gathered for an early-morning vigil outside the club's stadium, Arena Conda, singing together to commemorate the team.
La afición de Chapecoense reza y canta para homenajear a sus héroes. Qué imágenes... pic.twitter.com/50xqOUBgUe
— Invictos (@InvictosSomos) 29 November 2016
Reaching the final of the continent's second most-prestigious club tournament was by far the biggest achievement in the club's history, and many players had tweeted photos of their journey to the game in anticipation.
#BreakingNow Insta Vid of player Filipe Machado, rec on plane just hrs before the crash in Colombia. #Chapecoensepic.twitter.com/uwckTgit8F
— Breaking Now! (@BreakingNowThis) 29 November 2016
"We're starting the journey to Colombia! We're starting the journey to Colombia!" Filipe Machado says to the camera, as his teammates cheer and mug for the camera, in one of the last videos of the team together. Machado was not on the survivors' list.
COMNEBOL, South America's ruling football body, has now suspended the Copa Sudamericana and other activities until further notice. Atletico Nacional has conceded the title to Chapecoense, whose players will now likely be honored as champions posthumously.
Trends:Brazil newsColombia news

redpajama_set_name: RedPajamaCommonCrawl


text:
Checklist for Drafting Children and Armed Conflict Provisions in Ceasefire and Peace Agreements
See full resource on watchlist.org
This checklist for mediators and those involved in negotiating peace terms will assist in ensuring that child protection is represented in the language of peace agreements. Acknowledging the significance of incorporating children early on in the peace process, this source provides guidance on the effective release and reintegration of child soldiers and integrating the six grave violations against children during armed conflict into transitional justice mechanisms to end impunity and ensure children's access to justice. It also outlines the role of former child soldiers in the transition to peace as primarily victims who must be reintegrated into society with long-term support rather than other armed group members who may be integrated into the regular armed forces.
Child Soldiers: A Handbook for Security Sector Actors
Child Soldiers Initiative
See full resource on childsoldiers.org
The Roméo Dallaire Initiative's landmark publication, Child Soldiers: A Handbook for Security Sector Actors aims to highlight and reinforce the various roles that security sector actors (i.e. military, police, prison personnel and peacekeepers) can play in protecting all children affected by war. Pages 68-76 cover security sector actors' roles in supporting monitoring and reporting serious violations against children in armed conflict. This Handbook is available in French and English.
Children and Truth Commissions
UNICEF Innocenti Research Centre
Download full PDF from unicef-irc.org
This resource is intended to inform the work of child protection advocates and organizations, legal experts and other professionals in their efforts to protect the rights of children involved in truth and reconciliation processes. It includes emerging good practices and recommended procedures for children's participation in truth commissions. The annex also provides model checklists for taking statements from children and model memorandum-of-understanding (MOU) agreements between truth commissions and child protection agencies.
What is Transitional Justice?
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This two-page fact sheet describes transitional justice; where it came from and how it is used as a response to systematic or widespread violations of human rights. The aims of transitional justice will vary depending on the context but these features are constant: the recognition of the dignity of individuals; the redress and acknowledgment of violations; and the aim to prevent them happening again. Transitional justice is often approached in the following forms: criminal prosecutions, truth seeking/fact finding, reparations, reforms. More information is also available on the International Center for Transitional Justice website: https://www.ictj.org/about/transitional-justice
Rule of Law tools for Post-Conflict States: Reparations Programmes
This resource provides a practical guide for implementing effective reparations programs. It focuses on establishing out-of-court reparations programs to redress gross and serious violations of human rights in conflict settings. It also examines kinds of benefits, financing reparations, linking with other justice measures, and making programs gender-sensitive.
Child and Youth Participation Resource Guide
This resource compiles materials on child and youth participation, including children's participation in emergencies, conflict situations and peace building. Part four focuses on children's involvement in political decision making, such as their role in policy analysis in National Plans of Action, Committee on the Rights of the Child, and other organizations and conferences.


text:
Q: How can I find out the file causing complaints during boot time? (persistent live stick) I have made a persistent live stick with Ubuntustudio 20.04 from ubuntustudio-20.04.2.0-dvd-amd64.iso.
I applied some changes to it (e.g. install Chromium and uninstall Firefox, because it caused memory freeze on a Lenovo T430 laptop in Jitsi meetings with camera enabled). Yeah: it took an awful lot of time installing and uninstalling software (I had to let it run over night) on a thumbdrive with persistence, although I used an USB3-device.
Now I have almost reached my target. But I noticed a message booting from that pendrive (it was not there in the precedent bootings from it)

./boot/grub/x86_64-efi/div.mod: OK Checking ./boot/grub/loopback.cfg
./boot/grub/loopback.cfg: OK
Check finished: errors found in 1 files! You might encounter errors.

The system booted despite that message.
Unfortunately the time consumption for booting has grown to almost 4 minutes. Initially it was less than 2 minutes!
There has always been a file md5sums.txt at the root of my thumbdrive. From another system, I went to the directory at the root of the thumbdrive and then applied md5sum -c md5sums.txt. According to the output of that, they are all ok.
How can I find out for which file the complaint at boot time was?
BTW: When shutting down the computer booted from the persistent live drive, lots of file IO happens in the partition casper-rw. Unfortunately after clicking on the button to really shut down the computer, the screen immediately blanks and only a tiny cursor is in the left upper corner of the screen which might easily get overlooked. It looks like this is when casper-rw is updated. Depending on what one has done before, this "dark phase" before the Ubuntustudio logo with that rotating indicator can take up to one minute! It would be nice to display a warning against pulling out the thumbdrive all the time, when casper-rw is updated! Otherwise the persistent stick is easily ruined.

A: Full Install vs Persistent install
Ubuntu can be installed to a USB in different ways. A Live install does not save between sessions. A Persistent install extracts the OS from a compressed file and saves data to an overlay file or partition each session, and a Full install installs the complete OS to the USB just like an install to internal disk.
Comparison between Persistent and Full install USB
Advantages of a persistent install:

*

*You can use the persistent pendrive to install Ubuntu to another computer.

*A persistent install takes up less space on the pendrive.

*You can reset the pendrive by overwriting the old casper-rw file with a new one.

*The install to pendrive takes less time.

*Slightly less wear on the drive.
Advantages of a Full install:

*

*You can update and upgrade.

*If you have problems or wish to modify, the solution is the same as with an internal install, (You can ask for help in the forums).

*No ugly startup / install screen.

*Better security, you can use full encryption

*You can use proprietary drivers.

*Swapfiles and partitions work and Hibernation can be enabled.

*Many persistent installs are limited to a 4GB casper-rw and a 4GB home-rw persistence file, to get more persistence requires persistence partitions. Once casper-rw is full, the drive will not boot.

*More efficient usage of disk space. Does not require reserved space for persistence.

*Faster boot, no automatic disk checking or Try Ubuntu/Install Ubuntu screen.

*You can run VBox and use virtual machines.

*Generally faster boot than Live or Persistent USB's.

*More stable, better for day to day use. I have run Ubuntu off a flash drive for 5 years making only LTS upgrades.
Note that once booted, both methods run at about the same speed. If the computer has lots of RAM Ubuntu should run mainly in RAM and there will not be a big difference between running off internal HDD and USB3 flash drive f.
Full Install Method
A method for creating a Full install USB from scratch can be found here: How to Create a Full Install of Ubuntu 20.04 to USB Device Step by Step

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Keep the kitchen clean and eat out!
Do you live in or plan to be in Florida on Thanksgiving? Then perhaps the idea of dining out is more appealing than peeling potatoes and washing dishes. The tradition of families gathering around a large table for their Thanksgiving Day meal is still alive, but that table is less likely to be filled with homemade fare than with takeout turkey and trimmings. And, not too surprising, is that for at least one in 10 Americans, that table may be located in a restaurant instead of the family home, according to a 2016 National Restaurant Association study.
The study also found that one in 20 Americans will get a full takeout meal to celebrate the holiday.
While there are many reasons for families to dine out for Thanksgiving—convenience and travel come to mind—people say that going to restaurants with family and friends is a better use of leisure time than cooking and cleaning up.
So, if you're going to be dining out for Thanksgiving in Florida, reservations are a must to ensure a quality meal in a desirable location. Although, many hotel restaurants offer Thanksgiving buffets, but space is limited and restaurants in popular locations, such as Disney World in Orlando, are often booked months in advance. And, did you know that you don't have to be a hotel guest to dine at their restaurants?
Here are just a few suggestions on where to dine out for Thanksgiving in Florida (prices, dishes, and availability are subject to change).
Omni Amelia Island Plantation: Enjoy a Thanksgiving grand buffet at the Amelia Plantation's Sunrise Cafe from 10:30 to 7:30 p.m. Adults $62, children $26. Call 904-261-6161. The Verandah, also located at the Omni Amelia Island Plantation, offers a Thanksgiving dinner served from 4:30 to 8:30 p.m. Call 904-261-6161 for reservations. Other restaurants at the Omni that are open on Thanksgiving are Oceanside, Marche Burette, Natural Slice, and Falcon's Nest.
The Ritz-Carlton: Enjoy a special four-course meal of roasted turkey in the elegant oceanfront surrounding of Salt at the Ritz-Carlton. Reservations are recommended; call 904-277-1100.
Swan and Dolphin Resort: The Garden Grove will serve a traditional Thanksgiving buffet from 1 to 10 p.m. Favorite Disney characters will make a special appearance. The Fresh Mediterranean Market will also serve a special Thanksgiving buffet from 12:30 to 6 p.m. Both Todd English's Bluezoo and Il Mulino restaurants will serve a full menu and Thanksgiving special from 4 to 10 p.m. All four Swan and Dolphin Resort restaurants can be reached at 407-934-1609. View the Thanksgiving menus.
If you'll be at Disney World in November, you can also get traditional Thanksgiving Day fare at these other restaurants within Disney World resorts.
Rusty Pelican: Enjoy a Thanksgiving Day champagne brunch buffet with outstanding views of Biscayne Bay served from 10 a.m. to 4 p.m. Adults $85, children $35. Children 4 and younger eat free. Reservations recommended; call 305-361-3818. If you'd like to eat later in the day, Rusty Pelican also offers a Thanksgiving dinner from 6 to 8:30 p.m. that features turkey, stuffing, scallops, and pork shank. $70 per person. Reservations recommended; call 305-361-3818.
Byblos Miami: This Miami Beach hot spot puts a Mediterranean twist on their Thanksgiving Menu. Expect dishes like, turkey porchetta with saffron brioche stuffing and duck kibbeh. The meal starts at $65 per person with an additional $30 to include wine pairings.
The Ivy House Restaurant: Feast on a Southern-style Thanksgiving menu that's every bit as good as your mother's. The Ivy House has repeatedly been named by Florida Trend magazine as one of the top 400 places to eat in the state. Call 352-652-5550 for reservations.
Caribe Royale: Located on World Center Drive in Orlando, Caribe Royale will serve a traditional Thanksgiving Buffet from 1 to 7 p.m. in its Tropicále Restaurant, featuring live jazz music for entertainment. Adults $55, children (ages 3 to 9) $18. Children younger than 3 eat free. Florida residents receive a 20-percent discount. Reservations recommended; call 407-238-8020.
A Land Remembered: Located at Rosen Shingle Creek Resort, A Land Remembered, specializing in steak, has put together a special nontraditional Thanksgiving Day menu. Dine from 5:30 to 10 p.m. Adults $49.99, children $19.99. Call 407-996-3663 for reservations.
Marina Jack: Enjoy a panoramic view of the marina, downtown, Sarasota Bay, Big Pass, and the Gulf of Mexico as you partake in a traditional Thanksgiving meal of roasted turkey, mashed potatoes and gravy, green bean almandine, and cranberry sauce at The Dining Room. All areas open at 11 a.m. Reservations recommended; call 941-365-4232.
Columbia Restaurants: With six locations in Florida, including St. Augustine, Columbia Restaurant will serve a traditional Thanksgiving feast of roast turkey, Columbia stuffing, fresh vegetables, whipped potatoes, sweet potato pecan casserole, cranberry sauce, and giblet gravy beginning at 11 a.m. Dinner also includes a choice of Columbia soups or Caesar salad. Adults $21.95, children (10 and younger) $5.95. Reservations required. Call the St. Augustine location at 904-824-3341 for reservations.
400 Beach Seafood & Tap House: This dining establishment in downtown St. Petersburg has put together a traditional Thanksgiving menu that will be served from 11 a.m. to 7 p.m. Adults $36, children (12 and younger) $15. Reservations recommended; call 727-896-2400.
Parkshore Grill: Located in downtown St. Petersburg, Parkshore Grill will offer a three-course Thanksgiving menu in addition to its regular menu from noon to 8 p.m. Adults $32, children $17. Reservations recommended; call 727-896-9463.
The Rusty Pelican: Enjoy a Thanksgiving Day champagne buffet with outstanding views of Tampa Bay. The buffet is open from 10:30 a.m. to 4:30 p.m. Adults $65, children (ages 5 to 12) $25. For reservations, call 813-281-1943.
Sheraton Sand Key Resort: Enjoy the elegant atmosphere of Rusty's and its mouthwatering Thanksgiving buffet featuring traditional and nontraditional selections served from 1 to 9 p.m. Adults $39, children (ages 6 to 12) $15. Children 5 and younger eat free. Reservations recommended; call 727-595-1611, extension 7001.
Yacht StarShip: Dine aboard a yacht while cruising Tampa Bay on Thanksgiving Day. Enjoy a traditional Thanksgiving buffet, cash bar, entertainment, and dancing. Adults $59.95, children $29.95. Departures at 1:30 and 5:30 p.m. Reservations required; call 813-223-7999.


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text:
#pragma once

#include <atomic>
#include <cassert>
#include <memory>
#include <utility>
#include <glog/logging.h>
#include <folly/experimental/channels/detail/AtomicQueue.h>
#include <folly/lang/Assume.h>

namespace apache {
namespace thrift {
namespace detail {
namespace twowaybridge_detail {

template <typename T>
using Queue = folly::channels::detail::Queue<T>;

template <typename T>
class QueueWithTailPtr : public Queue<T> {
 public:
  QueueWithTailPtr() = default;
  template <typename F>
  QueueWithTailPtr(Queue<T>&& queue, F&& visitor) : Queue<T>(std::move(queue)) {
    for (auto* node = Queue<T>::head_; node; node = node->next) {
      visitor(node->value);
      tail_ = node;
    }
  }

  void append(QueueWithTailPtr&& other) {
    if (!other.head_) {
      return;
    }
    if (!Queue<T>::head_) {
      Queue<T>::head_ = std::exchange(other.head_, nullptr);
    } else {
      tail_->next = std::exchange(other.head_, nullptr);
    }
    tail_ = other.tail_;
  }

 private:
  // holds invalid pointer if head_ is null
  typename Queue<T>::Node* tail_;
};

template <typename Consumer, typename Message>
using AtomicQueue = folly::channels::detail::AtomicQueue<Consumer, Message>;

// queue with no consumers
template <typename Message, typename Value>
class AtomicQueueOrPtr {
 public:
  using MessageQueue = Queue<Message>;

  AtomicQueueOrPtr() {}
  ~AtomicQueueOrPtr() {
    auto storage = storage_.load(std::memory_order_relaxed);
    auto type = static_cast<Type>(storage & kTypeMask);
    auto ptr = storage & kPointerMask;
    switch (type) {
      case Type::EMPTY:
      case Type::CLOSED:
        return;
      case Type::TAIL:
        MessageQueue::fromReversed(
            reinterpret_cast<typename MessageQueue::Node*>(ptr));
        return;
      default:
        folly::assume_unreachable();
    };
  }
  AtomicQueueOrPtr(const AtomicQueueOrPtr&) = delete;
  AtomicQueueOrPtr& operator=(const AtomicQueueOrPtr&) = delete;

  // returns closed payload and does not move from message on failure
  Value* pushOrGetClosedPayload(Message&& message) {
    auto storage = storage_.load(std::memory_order_acquire);
    if (static_cast<Type>(storage & kTypeMask) == Type::CLOSED) {
      return closedPayload_;
    }

    std::unique_ptr<typename MessageQueue::Node> node(
        new typename MessageQueue::Node(std::move(message)));
    assert(!(reinterpret_cast<intptr_t>(node.get()) & kTypeMask));

    while (true) {
      auto type = static_cast<Type>(storage & kTypeMask);
      auto ptr = storage & kPointerMask;
      switch (type) {
        case Type::EMPTY:
        case Type::TAIL:
          node->next = reinterpret_cast<typename MessageQueue::Node*>(ptr);
          if (storage_.compare_exchange_weak(
                  storage,
                  reinterpret_cast<intptr_t>(node.get()) |
                      static_cast<intptr_t>(Type::TAIL),
                  std::memory_order_release,
                  std::memory_order_acquire)) {
            node.release();
            return nullptr;
          }
          break;
        case Type::CLOSED:
          message = std::move(node->value);
          return closedPayload_;
        default:
          folly::assume_unreachable();
      }
    }
  }

  MessageQueue closeOrGetMessages(Value* payload) {
    assert(payload); // nullptr is used as a sentinel
    // this is only read if the compare_exchange succeeds
    closedPayload_ = payload;
    while (true) {
      auto storage = storage_.exchange(
          static_cast<intptr_t>(Type::EMPTY), std::memory_order_acquire);
      auto type = static_cast<Type>(storage & kTypeMask);
      auto ptr = storage & kPointerMask;
      switch (type) {
        case Type::TAIL:
          return MessageQueue::fromReversed(
              reinterpret_cast<typename MessageQueue::Node*>(ptr));
        case Type::EMPTY:
          if (storage_.compare_exchange_weak(
                  storage,
                  static_cast<intptr_t>(Type::CLOSED),
                  std::memory_order_release,
                  std::memory_order_relaxed)) {
            return MessageQueue();
          }
          break;
        case Type::CLOSED:
        default:
          folly::assume_unreachable();
      }
    }
  }

  bool isClosed() const {
    return static_cast<Type>(storage_ & kTypeMask) == Type::CLOSED;
  }

 private:
  enum class Type : intptr_t { EMPTY = 0, TAIL = 1, CLOSED = 2 };

  static constexpr intptr_t kTypeMask = 3;
  static constexpr intptr_t kPointerMask = ~kTypeMask;

  // These can be combined if the platform requires Value to be 8-byte aligned.
  // Most platforms don't require that for functions.
  // A workaround is to make that function a member of an aligned struct
  // and pass in the address of the struct, but that is not necessarily a win
  // because of the runtime indirection cost.
  std::atomic<intptr_t> storage_{0};
  Value* closedPayload_{nullptr};
};
} // namespace twowaybridge_detail

template <
    typename ClientConsumer,
    typename ClientMessage,
    typename ServerConsumer,
    typename ServerMessage,
    typename Derived>
class TwoWayBridge {
  using ClientAtomicQueue =
      twowaybridge_detail::AtomicQueue<ClientConsumer, ClientMessage>;
  using ServerAtomicQueue =
      twowaybridge_detail::AtomicQueue<ServerConsumer, ServerMessage>;

 public:
  using ClientQueue = twowaybridge_detail::Queue<ClientMessage>;
  using ServerQueue = twowaybridge_detail::Queue<ServerMessage>;
  using ClientQueueWithTailPtr =
      twowaybridge_detail::QueueWithTailPtr<ClientMessage>;

  struct Deleter {
    void operator()(Derived* ptr) { ptr->decref(); }
  };
  using Ptr = std::unique_ptr<Derived, Deleter>;

  Ptr copy() {
    auto refCount = refCount_.fetch_add(1, std::memory_order_relaxed);
    DCHECK(refCount > 0);
    return Ptr(derived());
  }

 protected:
  TwoWayBridge() = default;

  // These should only be called from the client thread

  void clientPush(ServerMessage&& value) {
    serverQueue_.push(std::move(value));
  }

  bool clientWait(ClientConsumer* consumer) {
    return clientQueue_.wait(consumer);
  }

  ClientConsumer* cancelClientWait() { return clientQueue_.cancelCallback(); }

  ClientQueue clientGetMessages() { return clientQueue_.getMessages(); }

  void clientClose() { clientQueue_.close(); }

  bool isClientClosed() { return clientQueue_.isClosed(); }

  // These should only be called from the server thread

  void serverPush(ClientMessage&& value) {
    clientQueue_.push(std::move(value));
  }

  bool serverWait(ServerConsumer* consumer) {
    return serverQueue_.wait(consumer);
  }

  ServerConsumer* cancelServerWait() { return serverQueue_.cancelCallback(); }

  ServerQueue serverGetMessages() { return serverQueue_.getMessages(); }

  void serverClose() { serverQueue_.close(); }

  bool isServerClosed() { return serverQueue_.isClosed(); }

 private:
  void decref() {
    if (refCount_.fetch_sub(1, std::memory_order_acq_rel) == 1) {
      delete derived();
    }
  }

  Derived* derived() { return static_cast<Derived*>(this); }

  ClientAtomicQueue clientQueue_;
  ServerAtomicQueue serverQueue_;
  std::atomic<int8_t> refCount_{1};
};
} // namespace detail
} // namespace thrift
} // namespace apache

redpajama_set_name: RedPajamaGithub


text:
India looks for intelligence-gathering aircraft
By Vivek Raghuvanshi
Kiran jets of the Indian Air Force's Suryakiran aerobatics team fly past the Indian national flag as they perform during the inaugural function of the Aero India 2005 air show at the Yelahanka Air Force Station on the outskirts of Bangalore on Feb. 9, 2005. (Indranil Mukherjee/AFP via Getty Images)
Electronic intel capabilities would help Air Force 'get her act together'
NEW DELHI — Aimed at countering a perceived threat from China, India is seeking to acquire seven aircraft with signals intelligence and communication-jamming capabilities with a budget of about $570 million.
"The requirement is for aircraft to carry out a unique list of missions centered around a two-front threat from China and Pakistan," said Daljit Sigh, a defense analyst and retired Indian Air Force air marshal.
India's global hunt is for twin-engine jets, and preference will be given to overseas companies able to provide custom, Indian-built electronic warfare equipment, a senior Indian Air Force official said.
"We need more airborne electronic intelligence (ELINT) assets, particularly to meet the Chinese threat as they are likely to extensively use electronic warfare and spoofing during any hostilities, and the IAF needs to get her act together," said Vijainder K. Thakur, a defense analyst and retired Air force squadron leader.
The seven sigint aircraft will be operated by the Indian Air Force.
A request for information was sent to various foreign original equipment manufacturers seeking details of the aircraft and the signals-intelligence and commutations-jamming systems available, according to an official at India's Ministry of Defence.
After responses to the RFI, a formal request for proposal will likely follow in mid-2018, with the expectation for a tentative product delivery in 24 months from the date a contract is signed, the MoD official explained.
The main competitors for the program are likely to be Israel Aerospace Industries; Northrop Grumman, Boeing, Raytheon, Gulfstream, Textron of the United States; Dornier of Germany; Bombardier of Canada; Dassault Aviation of France; and Embraer of Brazil.
"The main mission of these aircraft would be to map and update the electronic order of battle of the adversary at regular intervals during peacetime and in addition, degrade or disrupt the communication network of the adversary during hostilities. These aircraft are required to be networked with ground stations and other airborne platform for secure exchange of information," Singh said.
The electronic warfare database fed from a ground station with the help of airborne an ELINT system is considered a key capability for the Air Force.
"Airborne ELINT has the advantage of being able to pick up radio frequency emissions from deep within adversary territory. Also, airborne ELINT can eavesdrop over mountain ranges such as along the Line of Actual Control on the Chinese border," Thakur noted.
"Typically, if your database is good, the EW equipment on a fighter aircraft like Su-30MKI multirole aircraft used by IAF would be able to not only passively detect an adversary fighter but also identify the fighter type and thereby more effectively counter the threat. It's even possible to identify the missile that has been fired at you," he added.
The latest RFI would be the third of its kind. The first was issued in 2006; and the second in 2012 for nine aircraft. The latest RFI, however, involves additional requirements, according to the MoD official.
About Vivek Raghuvanshi
Vivek Raghuvanshi is the India correspondent for Defense News.
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