text:
\section{Introduction}

  Pairing correlation between nucleons is a key ingredient
to describe the structure of neutron stars and finite nuclei.
There are two distinct ways of description of such {\it finite-density}
nuclear many-body systems; the non-relativistic and the relativistic ones.
The latter incorporates the mesons
explicitly in addition to the nucleons in terms of a field theory.
Both describe the basic properties such as the saturation with a similar
quality in different manners.
Irrespective of whether non-relativistic or relativistic, however,
various theoretical approaches can be classified into two types:
One is realistic studies adopting phenomenological interactions
constructed for finite-density systems from the beginning (hereafter
we call this the P-type, indicating ``phenomenological"), as often
done in the studies of heavy nuclei. And the other is microscopic
studies based on bare nucleon-nucleon interactions in free space
(the B-type, indicating ``bare").
In relativistic studies, typical examples of these two types as for the
particle-hole (p-h) channel are the relativistic mean field
(RMF) model and the Dirac-Brueckner-Hartree-Fock (DBHF) method,
respectively. As for the particle-particle (p-p) channel, that is,
pairing correlation, a bare interaction was used as the lowest-order
contribution in the gap equation~\cite{mig} in a study of the
B-type~\cite{rel3}. This is thought to be a good approximation at least
for the $^1S_0$ channel (Refs.\cite{baldo,gar}, for example).
The first relativistic study of the P-type of pairing correlation
in nuclear matter was done by Kucharek and Ring~\cite{kr}.
They adopted a one-boson exchange (OBE) interaction with the coupling
constants of the RMF model, which we call the RMF interaction hereafter,
aiming at a fully selfconsistent Hartree-Bogoliubov calculation,
which we call the P1-type, in the sense that both the p-h and the p-p
channel interactions are derived from a common Lagrangian.
But the resulting pairing gaps were about three times larger than
those accepted in the non-relativistic studies (see Fig.\ref{f6}).
The reason can be ascribed to the fact that the coupling constants of the
RMF model were determined by physics involving only low momenta
($k\le k_{\rm F}^0$, $\frac{2}{3\pi^2}(k_{\rm F}^0)^3=\rho_0$
denoting the saturation density of symmetric nuclear matter),
and therefore the adopted OBE interaction is not reliable at
high momenta. After a five-year blank, some attempts to improve this were
done~\cite{rel1,rel2,mm}. But their results were insufficient.

  An alternative way is to adopt another interaction in the p-p channel
while the single-particle states are still given by the RMF model.
We call this the P2-type. There are some variations of this.
The first one, which we call the P2a-type, adopts another phenomenological
interaction for the p-p channel. Actually, the non-relativistic Gogny
force~\cite{gogny} was used combined with the single-particle states
of the RMF model for finite nuclei in Ref.\cite{rhfb} and subsequent works,
and gave excellent results. The second variation, which we call the P2b-type,
is to adopt a bare interaction that describes the high-momentum part
realistically, the Bonn potential, again combined with
the single-particle states of the RMF model~\cite{rr} (see also Ref.\cite{mt}).
If one assumes that the RMF model simulates roughly the DBHF calculation,
this P2b-type can be regarded as simulating the B-type calculation~\cite{rel3}
mentioned above. The results
of these P2a- and P2b-types are very similar at densities $\rho<\rho_0$.
This supports a statement that the Gogny force resembles a realistic free
interaction in the low-density limit~\cite{bert} (see also Ref.\cite{gar}).
But a clear difference
can be seen at $\rho\sim\rho_0$. This difference can also be seen in
fully non-relativistic calculations; compare the results in
Refs.\cite{nu1} and~\cite{nu2}, for example. The precise origin of this
difference has not been understood well. A comparison after taking
the polarization effects which have been known to be important at
finite densities~\cite{chen,pol1,pol2} into account may be necessary
(see also Ref.\cite{mt}).
The third variation, which we call the P2b$^\prime$-type, is to parameterize
the p-p channel interaction in terms of a few scattering
parameters. This was actually examined by being combined with the DBHF
calculation~\cite{shift}, which we call the B$^\prime$-type (see also
Ref.\cite{papen}).
This method is free from model-dependent ambiguities and meets the
viewpoint of modern effective field theories~\cite{wein2,sw2} but is
applicable only to dilute systems. These classifications are summarized
in Table 1.

  Since we are interested in a wide density range where the $^1S_0$
gap exists and would like to respect the selfconsistency in the sense
that both the p-h and the p-p channel interactions are derived from a
common Lagrangian, here we construct a phenomenological relativistic
nucleon-nucleon interaction based on the RMF interaction (the P1-type)
by adjusting to the pairing properties given by the RMF+Bonn calculation
(the P2b-type). In other words, we aim at constructing an interaction
similar to the Gogny force in the sense that it reproduces the pairing
properties given by the bare interactions in spite of the fact that
it was constructed for the finite-density system from the beginning.

  The contents of this paper are as follows: In sect. 2 we present our
method of constructing a phenomenological p-p channel
interaction for the superfluid gap equation. This procedure is applied
both to the Hartree and to the Hartree-Fock model. In sect. 3, we discuss
the pairing properties obtained in the Hartree approximation. Note here
that we adopt the no-sea approximation throughout this paper and therefore
the Relativistic Hartree model means the so-called Relativistic Mean
Field model. In sect. 4, the Relativistic Hartree-Fock-Bogoliubov
calculations with or without the modulation of the high-momentum interaction
are presented. Conclusions are given in sect. 5.

\section{Construction of a relativistic particle-particle channel interaction}

  We start from the ordinary $\sigma$-$\omega$ model Lagrangian density,
\begin{eqnarray}
{\mathcal L}&=&\bar\psi(i\gamma_\mu\partial^\mu-M)\psi
\nonumber\\
 &+&\frac{1}{2}(\partial_\mu\sigma)(\partial^\mu\sigma)
  -\frac{1}{2}m_\sigma^2\sigma^2
  -\frac{1}{4}\Omega_{\mu\nu}\Omega^{\mu\nu}
  +\frac{1}{2}m_\omega^2\omega_\mu\omega^\mu
\nonumber\\
 &+&g_\sigma\bar\psi\sigma\psi-g_\omega\bar\psi\gamma_\mu\omega^\mu\psi ,
\nonumber\\
&\Omega_{\mu\nu}&=\partial_\mu\omega_\nu-\partial_\nu\omega_\mu .
\end{eqnarray}
The antisymmetrized matrix element of the RMF interaction $V$ derived from
this Lagrangian is defined by,
\begin{equation}
\bar v({\bf p},{\bf k})
=\langle{\bf p}s',\tilde{{\bf p}s'}\vert V\vert
           {\bf k}s,\tilde{{\bf k}s}\rangle
   -\langle{\bf p}s',\tilde{{\bf p}s'}\vert V\vert
           \tilde{{\bf k}s},{\bf k}s\rangle ,
\end{equation}
under an instantaneous approximation, with tildes denoting time reversal.
After a spin average and an angle integration are performed to project out
the $S$-wave component, its concrete form is given by
\begin{eqnarray}
&&\bar v(p,k)
\nonumber \\
&&=-\frac{g_\sigma^2}{2E^\ast_pE^\ast_k}
          \Biggl\{1+\frac{2(E^\ast_pE^\ast_k+M^{\ast\,2})-(p^2+k^2+m_\sigma^2)}{4pk}
                    \ln{\left(\frac{(p+k)^2+m_\sigma^2}{(p-k)^2+m_\sigma^2}\right)}
          \Biggr\}
\nonumber \\
&&\,+\frac{g_\omega^2}{2E^\ast_pE^\ast_kpk}\left(2E^\ast_pE^\ast_k-M^{\ast\,2}\right)
              \ln{\left(\frac{(p+k)^2+m_\omega^2}{(p-k)^2+m_\omega^2}\right)}\, ,
\nonumber
\end{eqnarray}
where
\begin{equation}
E^\ast_k=\sqrt{{\bf k}^2+M^{\ast\,2}}\, .
\end{equation}
  Our policy of constructing a phenomenological interaction proposed above is to
introduce a {\it density-independent} parameter $\Lambda$ so as not to change
the Hartree part with the momentum transfer ${\bf q}=0$ which determines
the single-particle energies,
respecting that the original parameters of the RMF are density-independent.
In the Hartree-Fock model, the ${\bf q}\neq0$ part also contributes to the
mean field. This will be discussed in sect.4 in detail.
Since the high-momentum part of the RMF interaction does not have
a firm experimental basis as mentioned above, we suppose there is room
to modify that part. Needless to say, such a modification should be checked
by studying independent phenomena, for example, medium-energy
heavy-ion collisions.
Some adjustments of the density-dependent $\delta$ force to the ones that
give realistic pairing properties have already been examined in the
non-relativistic studies \cite{bert,gar}. Among them, a fit to a bare
interaction in the $T=0$ channel was done \cite{gar}. We aim at a similar
procedure in the relativistic model for the $T=1$ channel.
In the preceding letter~\cite{tm}, the upper bounds of the momentum
integration in the gap equation,
\begin{equation}
\Delta(p)
=-\frac{1}{8\pi^2}
  \int_0^\infty \bar v(p,k)
         \frac{\Delta(k)}{\sqrt{(E_k-E_{k_{\rm F}})^2+\Delta^2(k)}}k^2dk\, ,
\label{gapeq}
\end{equation}
where
\begin{equation}
E_k=E^\ast_k+g_\omega\langle\omega^0\rangle \, ,
\label{spe}
\end{equation}
and the nucleon effective mass equation,
\begin{equation}
  M^\ast=M-\frac{g_\sigma^2}{m_\sigma^2}\frac{\gamma}{2\pi^2}
  \int_0^\infty
  \frac{M^\ast}{\sqrt{k^2+M^{\ast\,2}}}v_k^2k^2dk\, ,
\label{effmass}
\end{equation}
where the spin-isospin factor $\gamma$ = 4 and 2 indicate
symmetric nuclear matter and pure neutron matter, respectively, were cut
at a finite value $\Lambda$, as usually done in condensed-matter physics,
since the gap increases monotonically until reaching Kucharek and
Ring's value when the model space is enlarged as shown in Fig.3 of
Ref.\cite{mm}, while $\bar v(p,k)$ is left unchanged. We call this method the
sudden cutoff hereafter. This was done first in Ref.\cite{kr} by inspection.
We proposed a quantitative method to determine $\Lambda$, which is described
below, and obtained 3.60 fm$^{-1}$ for the linear $\sigma$-$\omega$
parameter set in Ref.\cite{tm}. This value almost coincides with their value,
about 3.65 fm$^{-1}$, for the NL2 parameter set.

  In the present paper, we examine smooth cutoffs that weaken the
high-momentum part; a form factor $f({\bf q}^2)$, ${\bf q}={\bf p}-{\bf k}$,
is applied to each nucleon-meson vertex in
$\bar v({\bf p},{\bf k})$ while the upper bounds of
the integrals are conceptually infinity. They are replaced numerically by
a finite number, 20 fm$^{-1}$ which has been proved to be large enough in
Ref.\cite{mm}. Since there is no decisive reasoning to choose
a specific form, we examine four types,
\begin{eqnarray}
\mbox{monopole: }\,\,f({\bf q}^2)&=&\frac{\Lambda^2}{\Lambda^2+{\bf q}^2}\, ,
\nonumber \\
\mbox{dipole: }\,\,f({\bf q}^2)&=&\left(\frac{\Lambda^2}
                                             {\Lambda^2+{\bf q}^2}\right)^2\, ,
\nonumber \\
\mbox{strong(a): }\,\,f({\bf q}^2)&=&\frac{\Lambda^2-{\bf q}^2}
                                       {\Lambda^2+{\bf q}^2}\, .
\nonumber \\
\mbox{strong(b): }\,\,f({\bf q}^2)&=&\frac{\Lambda^2-m_i^2}
                                       {\Lambda^2+{\bf q}^2}\, , \quad\quad
(i=\sigma, \omega) .
\end{eqnarray}
Note that the sudden cutoff above was applied to ${\bf k}$, not to ${\bf q}$.

  The parameter $\Lambda$ is determined so as to minimize the difference
in the pairing properties from the results of the P2b-type RMF+Bonn
calculation. Assuming the P2b-type roughly simulates the B-type as mentioned
in sect.1, conceptually we aim at fitting to the pairing properties given
by the fully microscopic B-type calculation.
Here we adopt the Bonn-B potential because this has a moderate property
among the available (charge-independent) versions A, B, and C~\cite{bonn}.
The pair wave function,
\begin{eqnarray}
&&\phi(k)=\frac{1}{2}\frac{\Delta(k)}{E_{\rm qp}(k)}\, ,
\nonumber \\
&\,&E_{\rm qp}(k)=\sqrt{(E_k-E_{k_{\rm F}})^2+\Delta^2(k)}\, ,
\label{pairwf}
\end{eqnarray}
is related to the gap at the Fermi surface,
\begin{equation}
  \Delta(k_{\rm F})=-\frac{1}{4\pi^2}
  \int_0^\infty \bar v(k_{\rm F},k)\phi(k)k^2dk\, ,
\end{equation}
and its derivative determines the coherence length~\cite{coh},
\begin{equation}
\xi=\left(\frac{\int_0^\infty\vert\frac{d\phi}{dk}\vert^2k^2dk}
               {\int_0^\infty\vert\phi\vert^2k^2dk}\right)
      ^\frac{1}{2}\, ,
\end{equation}
which measures the spatial size of the Cooper pairs. These expressions
indicate that $\Delta(k_{\rm F})$ and $\xi$ carry independent information,
$\phi$ and $\frac{d\phi}{dk}$, respectively, in strongly-coupled systems,
whereas they are intimately related to each other in weakly-coupled ones.
Therefore we search for $\Lambda$ which minimizes
\begin{equation}
\chi^2
=\frac{1}{2N}\sum_{k_{\rm F}}
\left\{\left(\frac{\Delta(k_{\rm F})_{\rm RMF}-\Delta(k_{\rm F})_{\rm Bonn}}
                   {\Delta(k_{\rm F})_{\rm Bonn}}\right)^2
+\left(\frac{\xi_{\rm RMF}-\xi_{\rm Bonn}}
            {\xi_{\rm Bonn}}\right)^2
\right\}\, ,
\end{equation}

where the subscripts ``RMF" and ``Bonn" denote the RMF interaction
including $\Lambda$ and the Bonn-B potential, respectively, while the
single-particle states are determined by the original RMF model in both
cases.

  The actual numerical task is to solve the gap equation (\ref{gapeq}) and
the effective mass equation for the nucleon (\ref{effmass}).
They couple to each other through Eq.(\ref{spe}) and
\begin{equation}
v_k^2=\frac{1}{2}\left(1-
   \frac{E_k-E_{k_{\rm F}}}{\sqrt{(E_k-E_{k_{\rm F}})^2+\Delta^2(k)}}\right)\, .
\label{v2}
\end{equation}
The parameters of the standard $\sigma$-$\omega$ model that we adopt are
$g_\sigma^2=$ 91.64, $g_\omega^2=$ 136.2, $m_\sigma=$ 550 MeV,
$m_\omega=$ 783 MeV, and $M=$ 939 MeV~\cite{sw}. $N$ in $\chi^2$ is taken
to be 11; $k_{\rm F}=$ 0.2, 0.3,$\ldots$, 1.2 fm$^{-1}$. In the following,
the results
for symmetric nuclear matter are presented. Those for pure neutron matter
are very similar except that $\Delta(k_{\rm F})$ is a little larger due to
a larger effective mass $M^\ast$ as shown in Fig.~1\,(b) of Ref.\cite{tm}.
Minimizations of $\chi^2$ give the optimal values, $\Lambda=$ 7.26, 10.66,
and 10.98 fm$^{-1}$ for the first
three types of form factor, respectively, as shown in Fig.\ref{cutdep}(a).
We do not choose an optimal $\Lambda$ for the strong(b) type because of its
pathological $\Lambda$-dependence shown in Fig.\ref{cutdep}(b).
Figure~\ref{cutdep}(b) shows that the $\Lambda$-dependence of these smooth
cutoff cases is very mild, except for the strong(b) type, in comparison with
the sudden cutoff case. The very steep $\Lambda$-dependence in the strong(b)
case is due to the consecutive depression of the attraction and the
repulsion at around $\Lambda\sim m_\sigma$ and $\sim m_\omega$, respectively.
In the strong(a) case, although another smaller $\Lambda$ around 3 fm$^{-1}$
can give similar $\Delta(k_{\rm F})$, $\bar v(k_{\rm F},k)$ and consequently
$\Delta(k)$ exhibit an unphysical staggering at around $k\sim m_\sigma$.
Therefore we discard this. In addition, although the strong(a) case with
$\Lambda=$ 10.98 fm$^{-1}$ gives practically the same $k_{\rm F}$-dependence
of $\Delta(k_{\rm F})$ and $\xi$ (in Fig.\ref{kfdep} shown later),
$\bar v(p,k)=0$ at $\Lambda=\vert{\bf q}\vert=\vert{\bf p}-{\bf k}\vert$
brings about an unphysical staggering in $\Delta(k)$ and $\phi(k)$; this leads
to an oscillatory structure in $r$-space with a period $\sim \pi/\Lambda$.
Therefore we discard this, too.
Form factors with similar $\Lambda$ are also suggested in a study of medium-energy
heavy-ion collisions~\cite{hi2}. This indicates that the present results
have a physical meaning.

\section{Pairing properties obtained in the Relativistic Hartree-Bogoliubov
calculation by using the constructed interaction}

  Figure~\ref{kfdep} presents the results for $\Delta(k_{\rm F})$ and
$\xi$ as functions of the Fermi momentum $k_{\rm F}$, obtained by using
the cutoff parameters so determined. Both the monopole and the dipole
cases reproduce the results
from the Bonn-B potential very well, as the sudden cutoff case studied in
Ref.\cite{tm}, in a wide and physically relevant density
range, in the sense that pairing in finite nuclei occurs near the nuclear
surface where density is lower than the saturation point~\cite{surf1,surf2,surf3}
and that the calculated range of $k_{\rm F}$ almost corresponds to that of
the inner crust of neutron stars~\cite{nrel2}.
This is our first conclusion. In the present method with a
{\it density-independent} cutoff parameter, some small deviations remain:
The overall slight
peak shift to higher $k_{\rm F}$ in $\Delta(k_{\rm F})$ and the deviation
in $\xi$ at the highest $k_{\rm F}$ are brought about by the systematic
deviation in the critical density where the gap closes, between the
calculations adopting bare interactions and those adopting phenomenological
ones as mentioned above.
The deviation at the lowest $k_{\rm F}$ is due to the feature that the
present model is based on the mean-field picture for the finite-density
system; this is a different point from the three-parameter fitting in
Refs.\cite{bert,gar}. Actually, in such an extremely dilute system, the
effective-range approximation for free scattering holds well~\cite{shift}.

  Next we look into the momentum dependence at $k_{\rm F}=$ 0.9 fm$^{-1}$,
where $\Delta(k_{\rm F})$ becomes almost maximum. Figure~\ref{kdep}\,(a)
shows $\phi(k)$.
It is evident that the monopole form factor gives the result identical to
the Bonn-potential case as in the sudden cutoff case. This demonstrates
clearly the effectiveness of the interaction constructed here.
Since we confirmed that the results of the monopole and the dipole cases
coincide within the width of the line, hereafter only the monopole case
will be shown as a representative. This quantity peaks at $k=k_{\rm F}$
as seen from Eq.(\ref{pairwf}). The width of the peak represents the
reciprocal of the coherence length. Equation (\ref{pairwf}) shows that $\phi(k)$
is composed of $\Delta(k)$ and the quasiparticle energy $E_{\rm qp}(k)$.
Figure~\ref{kdep}\,(b) graphs the former. The gaps of both the sudden cutoff
and the monopole form factor cases are almost identical up to
$k\sim 2k_{\rm F}$, and deviations
are seen only at larger momenta where $E_{\rm qp}(k)$ are large and
accordingly pairing is not important. This is not a trivial result
since the bare interaction is more repulsive than the phenomenological ones
constructed here even at the momentum region where $\Delta(k)$ are
almost identical as shown in Fig.~\ref{kdep}\,(c).
The reason why we compare the constructed {\it p-p channel} interaction with the
bare interaction is as follows: Although evidently the RMF interaction
corresponds to a medium-renormalized one, not to a bare one, here we aim at
constructing an interaction
similar to the Gogny force in the sense that it reproduces the pairing
properties given by the bare interactions in spite of the fact that
it was constructed for the finite-density system from the beginning.
Accordingly we compare them to see the difference between an effective
and a bare interactions which give similar pairing properties.
This figure indicates that the difference is roughly $k$-independent.

  Here we turn to the dependence on $r$, the distance between the two
nucleons that form a Cooper pair, in order to look into the physical contents
further. The gap equation, before the angle integration that results in
Eq. (\ref{gapeq}), can be Fourier-transformed to
the local form, $\Delta({\bf r})=-\bar v({\bf r})\phi({\bf r})$
in $r$-space in the non-relativistic limit~\cite{cor1}.
One can see from this expression that, assuming
$\Delta({\bf r})$ is finite, $\phi({\bf r})$ is pushed outwards when
$\bar v({\bf r})$ has a repulsive core, as the Brueckner wave
functions~\cite{baldo,nrel4}. This is related to an observation that the
gap equation reduces to a Schr\" odinger equation for the relative motion
of the two particle that form a Cooper pair in the limit of
$v_k^2\rightarrow 0$, i.e., at high $k$\cite{shr1,shr2,gar}.
This, on the other hand, masks practically the differences in the repulsive
interactions at short range, in other words, widely spread in $k$-space.
The $r$-space pair wave functions,
\begin{equation}
\phi(r)=\frac{1}{2\pi^2}\int_0^\infty\phi(k)j_0(kr)k^2dk\, ,
\end{equation}
where $j_0(kr)$ is a spherical Bessel function,
at $k_{\rm F}=$ 0.9 fm$^{-1}$ are shown in Fig.~\ref{rdep}\,(a).
Appreciable differences are seen only in the core region as mentioned above.
The coherence
length, that is a typical spatial scale of pairing correlation, is about
6 fm at this $k_{\rm F}$ as shown in Fig.~\ref{kfdep}\,(b);
this is almost one order of magnitude
larger than the size of the core region. Therefore, practically we can
safely use the p-p channel interaction, including the sudden cutoff one,
constructed here for the gap equation.
Figure~\ref{rdep}\,(b) shows the corresponding $\Delta(r)$.
The gaps are positive at the outside of the core and negative inside in
all cases. Note here that the gap equation is invariant with respect
to the overall sign inversion; we defined as $\Delta(k_{\rm F})>0$.
Their depths at the inside region reflect the heights of the repulsive
core. In the sudden cutoff case, $\Delta(r)$ behaves somewhat
differently from others due to the lack of high-momentum components.

\section{Relativistic Hartree-Fock-Bogoliubov calculation with a cutoff}

  In the preceding sections, we have shown that we can construct
phenomenologically relativistic p-p channel interactions
which give realistic pairing properties by introducing a {\it density-independent}
momentum-cutoff parameter to the (no-sea) Relativistic Hartree model.
To do this, we have fully utilized the property that only the ${\bf q}=0$
part of the interaction contributes to the Hartree mean field.
To see the further applicability of the method presented above, the Relativistic
Hartree-Fock (RHF) model, in which the ${\bf q}\neq0$ part of the interaction
also contributes to the p-h channel, should be examined.
In the following, we investigate the Relativistic Hartree-Fock-Bogoliubov (RHFB)
model with a momentum-cutoff form factor, by comparison with the
corresponding Hartree-Bogoliubov calculation.
An RHFB calculation with a sudden cutoff was previously done by Guimar\~aes
et al.\cite{rel1}. They obtained very large pairing gaps in the no-sea
approximation (Fig.~4 of Ref.\cite{rel1}).
Before studying the effects of the form factor to modulate smoothly the
high-momentum interaction, we examine the sudden cutoff --- this is a
straightforward extension of our previous calculation in Ref.\cite{tm} to the
RHFB.

  The RHF model was described in detail in Refs.\cite{sw,jam,bou},
for example. The difference from the Hartree model is the
second term in Fig.~\ref{f5}. This introduces the space component of the
vector selfenergy, $\Sigma^v$, and all the Lorentz components of $\Sigma$
--- the scalar $\Sigma^s$, the vector (time) $\Sigma^0$, and the vector (space)
$\Sigma^v$ --- become momentum dependent:
\begin{equation}
\Sigma(p)=\Sigma^s(p)-\gamma^0\Sigma^0(p)+{\bm \gamma}\cdot{\bf p}\Sigma^v(p)\, .
\end{equation}
In contrast, in the Hartree model, $\Sigma^v$ is not present, $\Sigma^s$ and
$\Sigma^0$ are momentum independent, and $\Sigma^0$ is fully determined by the
input density. Their concrete forms are
\begin{eqnarray}
\Sigma^s(p)&&=-\frac{\gamma}{(2\pi)^3}\frac{g_\sigma^2}{m_\sigma^2}
\int_0^{k_{\rm F}}d^3k\frac{M^\ast(k)}{E^\ast(k)}
\nonumber \\
&&+\frac{1}{4\pi^2k}\int_0^{k_{\rm F}}dkk\frac{M^\ast(k)}{E^\ast(k)}
\Bigl[\frac{1}{4}g_\sigma^2\Theta_\sigma(p,k)-g_\omega^2\Theta_\omega(p,k)\Bigr]\, ,
\nonumber \\
\Sigma^0(p)&&=-\frac{\gamma}{(2\pi)^3}\frac{g_\omega^2}{m_\omega^2}
\int_0^{k_{\rm F}}d^3k
\nonumber \\
&&-\frac{1}{4\pi^2k}\int_0^{k_{\rm F}}dkk
\Bigl[\frac{1}{4}g_\sigma^2\Theta_\sigma(p,k)
     +\frac{1}{2}g_\omega^2\Theta_\omega(p,k)\Bigr]\, ,
\nonumber \\
\Sigma^v(p)&&=-\frac{1}{4\pi^2k^2}\int_0^{k_{\rm F}}dkk\frac{k^\ast}{E^\ast(k)}
\Bigl[\frac{1}{2}g_\sigma^2\Phi_\sigma(p,k)+g_\omega^2\Phi_\omega(p,k)\Bigr]\, ,
\end{eqnarray}
with
\begin{eqnarray}
&&\Theta_i(p,k)=\ln\left|\frac{A_i(p,k)+2pk}{A_i(p,k)-2pk}\right|\, ,
\nonumber \\
&&\Phi_i(p,k)=\frac{1}{4pk}A_i(p,k)\Theta_i(p,k)-1\, ,
\nonumber \\
&&A_i(p,k)={\bf p}^2+{\bf k}^2+m_i^2\, , \quad\quad\quad\quad(i=\sigma, \omega)\, ,
\end{eqnarray}
as Eqs. (5.76) - (5.79) in Ref.\cite{sw}. The retardation effect is neglected
here. Note that the momentum dependence of
$M^{\ast}$ is stemming from that of $\Sigma^s$: $M^{\ast}=M+\Sigma^s$
(see Eq.(\ref{effmass})).
When pairing is introduced, $\int_0^{k_{\rm F}}$ is replaced by
$\int_0^\infty v_k^2$, and therefore $\Sigma^s(p)$, $\Sigma^0(p)$, $\Sigma^v(p)$,
and $\Delta(p)$ (Eq.(\ref{gapeq})) couple through Eqs.(\ref{v2}) and (\ref{spe}).
When the momentum space is discretized to $n$ meshes, these quantities form
a set of $4n$-dimensional coupled non-linear equations; this is a contrast to the
$n+1$-dimensional ones --- $\Delta(p)$ and a momentum-independent $M^{\ast}$ or
$\Sigma^s$ --- in the Hartree case.

  In order to reproduce the saturation, the coupling constants are adjusted to
$g_\sigma^2=$ 83.11, $g_\omega^2=$ 108.05 with the masses being unchanged
\cite{sw}. We adopt this parameter set for the time being. Other sets will be
examined later. Note that the pairing contribution to the energy density
around the saturation point is negligible. First we compare the gap at the
Fermi surface calculated without a cutoff in RHFB and RHB,
in Fig.~\ref{f6}. We call them the full calculations hereafter.
The calculated gaps in the former are larger than those in the latter, by about
4 MeV at the maximum, for example. This difference is brought about by that in
the coupling constants rather than the Fock effect itself \cite{carl}.
Actually we confirmed that a Hartree calculation with the coupling constants
of the Hartree-Fock gave almost the same gaps as those given by the Hartree-Fock
calculation although such a calculation destroys the saturation completely.
This result reflects the property that the gap is not sensitive to the detail
of the single-particle states. If the coupling constants of Ref.\cite{rel1},
$g_\sigma^2=$ 96.392, $g_\omega^2=$ 129.260 are adopted, calculated gaps become
even larger; about 15 MeV at the maximum, for example. But it is evident that
this is still smaller than their calculated value in their Fig.~4. The reason
for this difference is not clear.

  Now let us introduce a sudden cutoff in the upper bound of the momentum
integrations in $\Sigma(p)$'s and $\Delta(p)$ as in the previous Hartree case.
The result is graphed in Fig.~\ref{f7} by the long-dashed curve.
Although a plateau appears around $\Lambda=$ 2 fm$^{-1}$ as in the
corresponding case in Fig.~\ref{cutdep}\,(b), the gap value of the present
plateau is about 4.5 MeV, which is larger than the physical value given by the
bare interaction, about 2.8 MeV at this density.  Applying the same procedure
as in the previous Hartree case, the obtained optimal cutoff is 1.26 fm$^{-1}$.
This indicates that the attractive interaction alone accounts for the physical
magnitude of the gap since the $\Lambda$ of the plateau corresponds to the $k$
at which $\bar v(k_{\rm F},k)$ turns from attractive to repulsive~\cite{tm}.
Note that evidently the RHFB calculation with this sudden cutoff can not be
applied to the case with $k_{\rm F}>\Lambda$.

  Next we proceed to the smooth cutoff --- the form factor in the nucleon-meson
vertices. The monopole and the dipole types are examined here. Their
$\Lambda$-dependence is included in Fig.~\ref{f7}. The optimal cutoffs are
$\Lambda=$ 3.40 fm$^{-1}$ and 5.02 fm$^{-1}$ for the monopole and the dipole
types, respectively. These values are smaller than the corresponding ones in the
Hartree cases since the original gap values given by the full calculation
are larger. Interestingly, however, their ratio,
$\Lambda({\rm monopole})/\Lambda({\rm dipole})$, almost
coincides with that of the Hartree case. The p-p channel
interactions including the optimal monopole form factor are compared in
Fig.~\ref{f8}. The high-momentum repulsive part, in particular at lower
densities, is strongly suppressed in the RHFB case. This testifies the discussion
about Fig.~\ref{f7} above that the optimal cutoffs are smaller.
In Fig.~\ref{f9} we present $\Delta(k_{\rm F})$ and $\xi$ calculated by
adopting the optimal cutoffs. They reproduce the values given by the Bonn-B
potential to an extent similar to the Hartree case or a little better except
$\Delta(k_{\rm F})$ of the sudden cutoff case. Again the results of the monopole
and the dipole form factors coincide with each other within the width of
the line.

  Although the pairing properties of the RHFB with the optimal form factor
presented in Fig.~\ref{f9} are essentially the same as those of the
corresponding Hartree calculation, an essential feature of the Hartree-Fock
model is that the form factor can affect the saturation property.
Therefore we have to check this before concluding the applicability of the
present procedure. The energy density is graphed as a function of the
Fermi momentum in Fig.~\ref{f10}. This figure can be compared with Fig.~40
in Ref.\cite{sw}. Here the pairing correlation energy (both cases) and the
cutoff contribution (Hartree-Fock case only) are additionally included.
Although these two curves almost coincide with each other at low densities,
the cutoff effect destroys the saturation in the RHFB case. This is because
suppressing the
high-momentum repulsion breaks the balance between the attraction and the
repulsion, and the latter contribute more at higher densities (see
Fig.~\ref{f8}\,(a)). To see this more closely, the cutoff dependence of the
energy density without the pairing contribution at the saturation density,
$k_{\rm F}^0=$ 1.42 fm$^{-1}$, is shown in Fig.~\ref{f11}. In the
large-$\Lambda$ limit, the energy densities approaches to about -15.69 MeV,
which is 0.38 \% less bound than the original value in Ref.\cite{sw},
-15.75 MeV, because of the instantaneous approximation. This figure shows that
\begin{equation}
\Lambda_{\rm p-p}<\Lambda_{\rm p-h}\, ,
\end{equation}
if we call the cutoff which reproduces the pairing properties and which does
the saturation properties $\Lambda_{\rm p-p}$ and $\Lambda_{\rm p-h}$,
respectively.

  Up to now, we used a parameter set of Serot and Walecka. To complete the
discussion, other sets with different characteristics are also examined here.
One is a set used by Bouyssy et al., $g_\sigma^2=$ 69.62, $g_\omega^2=$ 153.81,
$m_\sigma=$ 440 MeV, $m_\omega=$ 783 MeV, and $M=$ 938.9 MeV, which gives
saturation at $k_{\rm F}^0=$ 1.30 fm$^{-1}$\cite{bou}. A distinct feature of
this set is that the plateau appears at very small pairing gap (see
Fig.~\ref{f7}). This indicates that the repulsive part also contributes to the
gap as in the Hartree case, and accordingly, $\Lambda_{\rm p-p}$ will be
larger than in the case of Serot and Walecka's set. Actually, the optimal
monopole cutoff is 5.22 fm$^{-1}$, which is
larger than 3.40 fm$^{-1}$ in the previous case. But Fig.~\ref{f11} shows that
still $\Lambda_{\rm p-p}<\Lambda_{\rm p-h}$. Another is a set used by
Jaminon et al.\cite{jam}. An interesting feature of this set is that the
saturation (at $k_{\rm F}^0=$ 1.36 fm$^{-1}$) is given by introducing a ``weak"
form factor~\cite{bro},
\begin{equation}
f(q^2)=\sqrt{\frac{\Lambda^2}{\Lambda^2-q^2}}\, ,
\end{equation}
from the beginning.
Note that here $q^2$ is the square of the 4-momentum transfer and therefore
this contains a retardation effect. The parameters are $g_\sigma^2=$ 93.87,
$g_\omega^2=$ 127.55, $m_\sigma=$ 550 MeV, $m_\omega=$ 782.8 MeV, and
$\Lambda=$ 7.754 fm$^{-1}$. Since $M$ is not shown explicitly, 939 MeV
is adopted. The RHFB calculation with this form factor under the instantaneous
approximation gives pairing gaps which is larger than the physical values;
$\Delta(k_{\rm F})\simeq$ 8.5 MeV at the maximum, for example.
This means that still $\Lambda_{\rm p-p}<\Lambda_{\rm p-h}$ is necessary to
reproduce the physical pairing gap.

  In all the calculations above, the cutoff in the form factors is common to
$\sigma$ and $\omega$. Referring to the Bonn potential and Ref.\cite{hi2},
for example, $\Lambda_\sigma\neq\Lambda_\omega$ is another option.
Of course large deviations from $\Lambda_\sigma=\Lambda_\omega$ destroy
the saturation at least when the coupling constants are kept unchanged.
We examined some cases in a limited range,
$\frac{\Lambda_\omega}{\Lambda_\sigma}=$
0.70 $(\simeq\frac{m_\sigma}{m_\omega})$ - 1.2, but the results were negative.

\section{Conclusions}

  We have constructed phenomenologically a particle-particle channel
interaction which suits the gap equation for nuclear matter.
This was done by introducing a density-independent
momentum-cutoff parameter to the one-boson exchange interaction derived
from the Lagrangian of the RMF model and adjusting it to the pairing properties
obtained by the Bonn-B potential. The model pairing properties were
calculated by using the RMF model in the p-h channel and the Bonn-B potential
in the p-p channel utilizing the properties that the RMF model simulates
crudely\footnote
{
The coupling constants must be density-dependent in order to simulate
the $G$-matrices quantitatively.
}
the $G$-matrices in the DBHF calculation and that the pairing properties
are not sensitive to the detail of the single-particle states.
By this procedure we aimed at constructing an interaction like the Gogny
force in the sense that it reproduces the pairing
properties given by the bare interactions in spite of the fact that
it was constructed for the finite-density system from the beginning.
The actual determination of the optimal cutoff parameter was done by the
method proposed in our preceding letter for the Hartree mean field plus
a sudden cutoff~\cite{tm}.

  In the present paper, first we applied this method to the (no-sea)
Relativistic Hartree model plus various types of form factor which
modulates the high-momentum repulsion that spoils the pairing properties.
Among the four types examined, the monopole and the dipole form factors
exhibit desired properties. Close analyses in momentum space and coordinate
space have clarified that the gap equation involves a mechanism to mask
the difference in the short-range repulsion between the bare and the
in-medium effective interactions and that, in the typical spatial scale
of pairing phenomena determined by the coherence length, practically there are
no difference in the pairing properties.

  Second we performed RHFB calculations with and without a momentum cutoff
--- a sudden cutoff and two types of form factor. The reason why we examined
the RHFB separately is that the form factor which modulates the
${\bf q}\neq0$ part of the interaction also affects the p-h channel.
The same procedure was applied; the resulting optimal cutoffs are smaller
than those for the Hartree model because the original gap values given by the
full calculation without a cutoff are larger. Interestingly, the ratio
$\Lambda({\rm monopole})/\Lambda({\rm dipole})$ almost coincides with
that for the Hartree model. Having confirmed that these optimal form factors
reproduce $\Delta(k_{\rm F})$ and $\xi$ with a quality similar to that in the
Hartree calculation, we looked at the saturation curve of the energy density.
Although the results of the RHB and the RHFB each with the optimal form factor
coincide with each other up to $k_{\rm F}\sim$ 1 fm$^{-1}$, the result for
the latter starts to deviate at larger $k_{\rm F}$. This indicates that
$\Lambda_{\rm p-p}<\Lambda_{\rm p-h}$ is necessary in order also to reproduce
the saturation simultaneously using the RMF interaction both in the p-h and
the p-p channels. This holds also for the other parameter sets with
characteristic features mentioned in the previous section.

  These results support the observation that the high-momentum components
of the original RMF interaction should be refined.
The modulated interaction obtained here by the proposed one-parameter fitting
can be successfully used for the Relativistic Hartree-Bogoliubov calculation.
But its applicability to the Relativistic Hartree-Fock-Bogoliubov
calculation is limited to low densities.

redpajama_set_name: RedPajamaArXiv


text:
La provincia eclesiástica de Zaragoza está formada por la archidiócesis de Zaragoza y las diócesis sufragáneas de Tarazona, Huesca, Barbastro-Monzón y Teruel y Albarracín.

La diócesis de Jaca, aunque pertenece a la comunidad autónoma de Aragón y fue la cuna del Reino de Aragón, pertenece a la Provincia Eclesiástica de Pamplona - Tudela.

Historia
La diócesis de Zaragoza fue restaurada en 1118 durante la Reconquista. El 18 de julio de 1318, el obispo Pedro López de Luna, consiguió que el papa Juan XXII elevara la sede episcopal al rango de archidiócesis, desmembrándola de la provincia eclesiástica de Tarragona. Bajo su provincia eclesiástica quedaron las diócesis sufragáneas de Huesca, Tarazona y Segorbe-Albarracín, Pamplona y Calahorra, pertenecientes a los reinos de Aragón, Valencia, Navarra y Castilla.

En 1574 la diócesis de Burgos fue elevada a archidiócesis, pasando a su provincia eclesiástica las diócesis de Calahorra y Pamplona. Con el paso del tiempo, Zaragoza pasó a ser únicamente sede metropolitana del reino de Aragón. En 1571 se erigieron las diócesis de Barbastro y Jaca y en 1577 la de Teruel. Segorbe-Albarracín se segregó en dos diócesis: Albarracín (que más tarde se uniría con Teruel) y Segorbe (que pasó a ser sufragánea de la provincia eclesiástica de Valencia).

En el , tras la firma del Concordato de 1851 se modificaron varias circunscripciones de las diócesis de España. Pamplona fue reincorporada y se suprimieron las diócesis de Albarracín y Barbastro. Desde 1858 los obispos de Tarazona ejercieron como administradores apostólicos de Tudela.

En 1955 Pamplona fue elevada a archidiócesis y las diócesis de Jaca y Tudela quedaron bajo su provincia eclesiástica. El 11 de agosto de 1984 mediante la bula Cor Nostrum, se unió aeque principaliter las diócesis de Teruel y Albarracín. Los últimos cambios se realizaron entre 1995 y 1998 con la incorporación a Barbastro de los pueblos oscenses que dependían de Lérida.

Organización territorial

Datos de la provincia eclesiástica
El arzobispo de Zaragoza es el metropolitano de la provincia y tiene autoridad limitada sobre las diócesis sufragáneas.

Actualmente, la provincia tiene alrededor de 1.136 parroquias, abarca unos 42.739 km² en donde habitan aproximadamente 1.288.067 de personas de las cuales el 96,92% son católicos.

Episcopologio
 Ver Lista de Obispos y Arzobispos de Zaragoza
 Ver Lista de Obispos de Tarazona
 Ver Lista de Obispos de Huesca
 Ver Lista de Obispos de Barbastro-Monzón
 Ver Lista de Obispos de Teruel y Albarracín

Referencias

Provincia eclesiástica de Zaragoza

redpajama_set_name: RedPajamaWikipedia


A new study says Walmart is the place to find love. Here's how to score a date at the megastore.
If you feel like your soulmate keeps passing you by, try shopping at Walmart; a surprising new study suggests the love of your life might be there!
According to the study of "missed connection" posts on Craigslist, Walmart was found to be the most popular place for people to find love. The findings stretched across 15 states, as people thought they may have met their future spouse at a Walmart more than anywhere else.
We're thinking that the next time you rush in to pick up a few things at the megastore, slow down and take note of the potential dates shopping alongside to you — and you may even want to give that cute cashier a second glance too.
We've uncovered ten different ways categorized by department to help you to meet your match at Wal-Mart. And if you do find love in the aisles of Wal-Mart, whether it's shopping for patio furniture or rushing through checkout … don't worry, you can thank us with an invite to your retail-themed wedding.
In a mad five-minute dash at the front of the store, you've grabbed more off the shelves than you probably could handle. Now, you're wandering around with your arms full of groceries and a helpless look on your face, looking for a cart or a basket. You could ask an associate standing guard at the door to help you … but let's be real here: a pair of strong, manly arms belonging to a good-looking guy would be extra helpful and probably preferable.
Next time you're in need of a haircut, stop by SmartStyle Family Hair Salon in Wal-Mart and while you're getting a touch-up to your luscious locks, ask your hairdresser if she's seen any cute, single guys stop in for a trim lately. Who knows? She might be able to play matchmaker for the two of you.
Say, you caught the smoldering gaze of a hot stranger while looking for a cheap, but cute anniversary card for your parents in the card section … how do you know if they're single or (sadly) taken? Slide up beside them and ask him, "Do you have a girlfriend? Would she think this is a cheesy card?" Hopefully, the answer is no. Otherwise, it may be time to head to the Home department.


Discussion in 'The Edge of the Forum' started by Song of storms, Oct 20, 2018.
This is my thread. It's just mine. I will post here every once in a while about stuff.
A Magikarp enters in a cafe. But it has no effect!
Quantumcat and Subtle Demise like this.
I took over your thread.
Subtle Demise and Lilith Valentine like this.
I will buy raspberries today. Or shouldn't I?
I feel like today is more of a strawberry day. Get strawberries instead.
Especially during days ending in "y".
BORTZ and Brigand like this.
Subtle Demise, smileyhead, ry755 and 4 others like this.
I'm having a cheeseburger with a large Coke.
Update of the previous post: I cleaned the dishes and now I'm watching The Simpsons.
I can't see the rest of your signature just the first quote.
RedBlueGreen and Ricken like this.
@Song of storms Hi it me again why dont you tell us a Story?


text:
Q: Can I make the webview take only a part of the screen? I am trying to make a webview to open on the bottom of the application to display little information but it keeps taking the whole screen and covering other widgets with information.
Is there a way to make it have a certain size and place on the application?

A: Oh, you should provide some of your layout xml so we can see what your are wrapping the webview inside. Is your webview the root element of your app layout - than it will always use the full screen. But try wrap it inside a framelayout or something else...
<FrameLayout
android:layout_height="fill_parent"
android:layout_width="fill_parent"
android:gravity="bottom">
  <WebView
  android:layout_height="100dp"
  android:layout_width="fill_parent"/>
</FrameLayout>

redpajama_set_name: RedPajamaStackExchange


PetsSA - cats and kitters classified ads in South Africa - Stunning English Bulldog babies!
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Puppies will be Micro - Chipped, Vaccined, Kusa Registered and they have lines of Mervander in them which is the best in SA.
We based in Elarduspark, Attie Pelzer Street.
We can fly a puppy anywhere in SA as well.


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text:
Reducing Stress: What Scientists Learned From the Children Who Survived a Famine During the Deadliest War in History | James Clear
Reducing Stress: What Scientists Learned From the Children Who Survived a Famine During the Deadliest War in History
by James Clear — Get free updates of new posts here
Source: National Institute for War Documentation, Amsterdam
There was nothing left to eat.
The butter had disappeared in October. By November, adult food rations had been cut to 1000 calories per day. A few months later, in the dead of winter, rations dropped to 500 calories per day. Food stocks throughout the country were empty. If you were lucky enough to have food ration coupons, you could get 100 grams of cheese every two weeks. Meat was a fantasy. By April of 1945, each person was limited to 1 loaf of bread and 5 potatoes — for the entire week. [1]
It was the middle of terrible famine known as the Dutch Hunger Winter. World War II was nearing an end and Allied forces were able to push the German army out of the southern half of the Netherlands. As the Nazi's retreated, however, they destroyed docks and bridges, flooded the farm lands, and set up blockades in the northern half of the country to cut off shipments of food and fuel. What little food had been stockpiled and saved was nearly impossible to transport. Starving and without options, many people ate tulip bulbs and sugar beets.
Among those struggling to survive was a 9-year-old boy from Amsterdam named Henkie Holvast. During the worst period of the famine, Henkie was one of the many children who would carry spoons with them wherever they went "just in case." Photographer Martinus Meijboom captured this iconic image of Henkie during the Dutch Hunger Winter. Two of Henkie's younger siblings died during the famine. Somehow, he managed to survive.
To make matter worse, winter had come early that year. Canals and waterways had frozen, further restricting food transport. Gas and electricity were either unavailable or inoperable because of the war. The Holvast family, like many others throughout the Netherlands, had begun burning their furniture to stay warm. By April 1945, the situation was desperate. Approximately 20,000 Dutch had died from malnutrition.
In April 1945, the Royal Air Force flew from Great Britain and coordinated a series of air drops known as Operation Manna. In total, they dropped more than 6,600 tons of food in German-occupied territory. The Dutch responded with a simple message of "MANY THANKS" written in tulips on the countryside. [2]
The famine mercifully ended the next month, May of 1945, when Allied forces regained control of the Netherlands.
The most surprising part of the famine, however, was just beginning.
The Impact of Stress
As far as famines go, the Dutch Hunger Winter was remarkably unique. Most famines occur in areas that suffer from overpopulation, severe crop failure, or repeated periods of political instability. The Netherlands experienced none of these influences. Once the war ended and Allied troops arrived, the Dutch quickly recovered to a normal diet.
From a scientific perspective the Dutch survivors were perfect for study. The population consisted of a well-defined group of people who experienced one period of malnutrition at exactly the same time.
In the 1990s, Dr. Tessa Roseboom, a medical faculty member from the University of Amsterdam, began diving into the data about the children conceived and born during the Dutch Hunger Winter. Thanks to meticulous record keeping by the Dutch, Roseboom was able to track thousands of the children throughout their lives. What she discovered was remarkable. [3]
According to Roseboom's research, children who were conceived during the Dutch Hunger Winter have:
Higher risk of cardiovascular disease as an adult (up to 2x greater risk)
Higher rates of obesity throughout life
Increased risk of high blood pressure as an adult
Higher rates of hospitalization as an adult (i.e. increased illness)
Lower likelihood of being employed [4]
In other words, the children who were still in their mother's womb during that brutal winter have poorer health six decades later. These studies were groundbreaking because they revealed just how deeply stress can burrow into our lives. Not only do the effects of stress and malnutrition impact us at the time they occur, they can have lingering effects on ourselves and our children for decades to come.
Stress In Our Lives
The studies on the Dutch Hunger Winter offer a clear and dramatic look at how stress changes our bodies and stays with us throughout our lives. While we don't have to live in such extreme situations (hopefully), we do deal with stress on a day-to-day basis.
Because this is something we deal with everyday, our best defense against the effects of stress is to build daily habits that counteract those effects.
In other words, reducing stress isn't something that only those in dire circumstances need to consider. It is something we all need to handle. And the research above makes it clear: reducing stress is something you need to do not only for yourself, but also for your children and grandchildren as well.
Now for the million dollar question: What can we do to reduce stress in our lives?
7 Scientifically Proven Ways to Reduce Stress
Here are 7 scientifically proven ways to reduce stress in your life.
Exercise. I can't tell you how many times exercise has saved my sanity. If I didn't lift weights consistently, I wouldn't have a business. The stress of entrepreneurship would have run me into the ground by now. There are many studies linking exercise to reduced stress levels. My method of choice is strength training or sprinting, but all types of exercise are useful. (Yoga, for example.)
Meditation / Deep breathing. Yes, meditation can reduce stress. [5-7] That's probably not a surprise. If you're like me, then you know meditation is good for you, but you just never find a way to fit it in. Here's a tip I recently got from a monk: start your meditation habit by meditating for 1 minute. Do that for a month. Then increase to 2 minutes and do that for a month. And so on, until you get to the level you desire. Talk about slow gains. I love it!
Music. Listening to music can actually trigger the release of stress-reducing chemicals in the body, which is pretty awesome. [8,9] (Want more? I wrote a previous article on the health benefits of music.)
Sleep. If you are feeling stressed, a nap or a solid 8 or 9 hours of sleep can really help. In some cases, sleep is not only the solution, but actually the problem. Sleep deprivationcan be brutal on your health. Most people aren't getting enough sleep each night and sleep debt is a cumulative problem. The stress of too little sleep can add up and the only real solution is to give yourself the chance to rest. Make time to rest and rejuvenate now or make time to be sick and injured later.
Laughter. Everything is better when you laugh, including your stress levels. [10-12]
Stand up straight. Surprisingly, research from Harvard has revealed that your body language can impact the amount of testosterone and cortisol in your bloodstream. I wrote about the research here, but this TED Talk is a fantastic summary as well.
Art. I have written about the health benefits of art previously and one of them is stress reduction. Don't confuse creating art with being artistic. What we are really talking about here is creating something rather than sitting around and passively consuming. Worrying about all of the things on your to-do list is passive and naturally provides a feeling of being out of control. Creating something—whether that means writing in a journal, taking a photo, crafting a ceramic pot, and flashing your scrapbooking skills—naturally makes you feel in control of something and gives you a healthy outlet for your energy.
Where to Go From Here
Thankfully, most of us will never have to live through a period of intense stress like the Dutch Hunger Winter. That said, stress is still part of our daily lives and it is perhaps the greatest burden to our long-term health. Stress can decrease your heart health. It can increase the rate at which you age. It can disrupt your immune system.
The best path forward is to build stress-reducing habits into our lives (like the ones listed above), so that we can curtail the long-term impact that it has on us and our loved ones.
Technically, rations were measured exactly as 400 grams of bread and 1 kilogram of potatoes. This is approximately 1 loaf of bread and 5 large potatoes.
After much searching, I can't find the original source for the "Many Thanks" photo. If you know who took it, please share and I will happily cite them.
Effects of Prenatal Exposure to the Dutch Famine on Adult Disease in Later Life: An Overview by Tessa J. Roseboom, Jan H.P. van der Meulen, Anita C.J. Ravelli, Clive Osmond, David J.P. Barker, Otto P. Bleker.
Long-Run Effects on Gestation During the Dutch Hunger Winter Famine on Labor Market and Hospitalization Outcomes by Robert S. Scholte, Gerard J. van den Berg, and Maarten Lindeboom
Mindfulness-based stress reduction by Marchand
A randomized, controlled trial of meditation for work stress, anxiety and depressed mood in full-time workers by Manocha, Black, Sarris, and Stough
Effects of mental relaxation and slow breathing in essential hypertension by Kaushik, Kaushik, Mahajan, and Rajesh
From music-beat to heart-beat: a journey in the complex interactions between music, brain and heart by Cervellin and Lippi
Emotional foundations of music as a non-pharmacological pain management tool in modern medicine by Bernatzky, Presch, Anderson, and Panksepp
Effects of laughter therapy on postpartum fatigue and stress responses of postpartum women by Shin, Ryu, and Song
A case of laughter therapy that helped improve advanced gastric cancer by Noji and Takayanagi
Laughter and depression: hypothesis of pathogenic and therapeutic correlation by Fonzi, Matteucci, and Bersani
James Clear writes about science-based ideas for living a better life and building habits that stick. If you enjoyed this article, then join his free newsletter.
Dr. Tessa Roseboom Dutch Hunger Winter James Clear Stress The Netherlands
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redpajama_set_name: RedPajamaCommonCrawl


text:
The University of Law in Liverpool
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Supportive staff and personal training sessions to help you achieve your fitness and wellbeing goals.
Energetic, beautiful, creative… you will soon realise there is nowhere quite like Liverpool. Whether you're shopping in Liverpool One, or visiting the many art galleries, restaurants, bars and museums there's always something to see or do.
* 97% of our full-time 2016/17 and accelerated 2017/17 LPC students graduating in summer 2017 secured employment, a training contract, or further study within nine months of successfully completing their course.
How to find the Liverpool campus
The University of Law
The Law and Social Justice Building-2nd floor
Chatham Street
L69 7ZR
Conveniently located in the north west of England, The University of Law's campus at Liverpool has excellent transport connections. The campus is based on the South campus of the University of Liverpool which can be easily reached by road, rail or air
From the M62: at the end of the motorway continue straight ahead onto Edge Lane (A5080 then A5047) and follow signs for Liverpool City Centre and the University.
Postcode for sat nav or online directions: L3 5TR
The National Express coach station is in Liverpool One Bus Station, Canning Place, Liverpool, L1 8JX, an 18 minute walk from the Campus. From the bus station walk along Hanover Street, pass Liverpool Central Train Station. Cross Renshaw Street onto Brownlow Hill and head towards the Metropolitan Catholic Cathedral.
National Express Coaches
Website: www.nationalexpress.com
The campus is a ten minute walk from the nearest mainline station at Lime Street.
Take the main exit and turn left into Lime Street. Then turn left again at the Britannia Adelphi Hotel and continue up Brownlow Hill towards the Metropolitan Catholic Cathedral and Red Brick Building with its clock tower.
Use the Merseyrail website or National Rail Enquries to plan your journey.
Travelling by air
Liverpool is well connected to both London Heathrow and London Gatwick airports.
Liverpool John Lennon Airport is eight miles from the city and is best reached by taxi or the express bus, directly outside the entrance to the airport terminal. You can also get buses 500, 86, 80A and 82A to Liverpool city centre.
Website: www.liverpoolairport.com/from-airport/by-bus.html
A train service runs between Manchester Airport and Liverpool Lime Street station. There is also a direct coach service which takes about an hour.
Website: www.manchesterairport.co.uk
Attend an open day


text:
Harry Potter and the Half-Blood Prince - JK Rowling new softcover book
Book 6. The sixth book in the amazingly popular Harry Potter series. New softcover book
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HARRY POTTER AND THE HALF BLOOD PRINCE
by JK ROWLING
See other Harry Potter books click here
New softcover book. 607 pages. First published 2005, this Celebratory Edition published 2009.
Harry has yet again spent the summer holidays at the Dursleys'. He has had plenty to think about, though - from the death of his beloved godfather Sirius Black, to the terrifying chase through the Ministry of Magic by the Death Eaters, to the fierce duel he witnessed between Professor Dumbledore and Lord Voldemort.
It is the middle of the summer, but there is an unseasonal mist pressing against the windowpanes. Harry is waiting nervously for a visit from Professor Dumbledore himself. He can't quite believe that Professor Dumbledore will actually appear at the Dursleys' of all places. Why is the Professor coming to visit him now? What is it that cannot wait until Harry returns to Hogwarts in a few weeks' time?
Harry's sixth year at Hogwarts has already got off to an unusual start, as the worlds of Muggle and magic start to intertwine
J.K. Rowling charts Harry Potter's adventures in his sixth year at Hogwarts with a mix of detail and humour that is unsurpassed, pace that is breathless and above all a flair that is magical. In her darkest and most breathtaking adventure yet, JK Rowling skilfully begins to unravel the complex web she has woven, as we discover more of the truth about Harry, Dumbledore, Snape and, of course, He Who Must Not Be Named ...
Harry Potter and the Half-Blood Prince was released on 16 July 2005. It broke all sales records, selling nine million copies in its first 24 hours of release. While writing, she told a fan online, "Book six has been planned for years, but before I started writing seriously I spend two months re-visiting the plan and making absolutely sure I knew what I was doing." She noted on her website that the opening chapter of book six, which features a conversation between the Minister of Magic and the British Prime Minister, had been intended as the first chapter first for Philosopher's Stone, then Chamber of Secrets then Prisoner of Azkaban.
In 2006, Half-Blood Prince received the Book of the Year prize at the British Book Awards.
Joanne "Jo" Murray OBE, née Rowling, (born 31 July 1965), who writes under the pen name, J. K. Rowling, is a British author, best known as the creator of the Harry Potter fantasy series, the idea for which was conceived whilst on a train trip from Manchester to London in 1990. The Potter books have gained worldwide attention, won multiple awards, and sold more than 400 million copies.
Although she writes under the pen name "J. K. Rowling", her name when her first Harry Potter book was published was simply "Joanne Rowling". Before publishing her first book, her publisher Bloomsbury feared that the target audience of young boys might be reluctant to buy books written by a female author. It requested that Rowling use two initials, rather than reveal her first name. As she had no middle name, she chose K. for Kathleen as the second initial of her pseudonym, from her paternal grandmother. The name Kathleen has never been part of her real name. Following her marriage, her legal name is Joanne Murray, which she uses when conducting her personal business. She calls herself "Jo" and says, "No one ever called me 'Joanne' when I was young, unless they were angry."
Aside from writing the 7 Harry Potter novels, Rowling is perhaps equally famous for her "rags to riches" life story, in which she progressed from living on welfare to multi-millionaire status within five years. The 2008 Sunday Times Rich List estimated Rowling's fortune at £560 million (8 million), ranking her as the twelfth richest woman in Britain. Forbes ranked Rowling as the forty-eighth most powerful celebrity of 2007, and Time magazine named her as a runner-up for its 2007 Person of the Year, noting the social, moral, and political inspiration she has given her fandom. She has become a notable philanthropist, supporting such charities as Comic Relief, One Parent Families, Multiple Sclerosis Society of Great Britain, and the Children's High Level Group.
As a child, Rowling enjoyed writing fantasy stories, which she often read to her sister. "I can still remember me telling her a story in which she fell down a rabbit hole and was fed strawberries by the rabbit family inside it," she recalls, "Certainly the first story I ever wrote down (when I was five or six) was about a rabbit called Rabbit. He got the measles and was visited by his friends, including a giant bee called Miss Bee."
In 1990, while she was on a four-hour-delayed train trip from Manchester to London, the idea for a story of a young boy attending a school of wizardry "came fully formed" into her mind. "I really don't know where the idea came from", she told The Boston Globe, "It started with Harry, then all these characters and situations came flooding into my head." When she had reached her Clapham Junction flat, she began to write immediately.
In June 1997, Bloomsbury published Philosopher's Stone with an initial print-run of 1000 copies, five hundred of which were distributed to libraries. Today, such copies are valued between £16,000 and £25,000. Five months later, the book won its first award, a Nestlé Smarties Book Prize. In February, the novel won the prestigious British Book Award for Children's Book of the Year, and later, the Children's Book Award. Its sequel, Harry Potter and the Chamber of Secrets, was published in July, 1998. In October 1998, Scholastic published Philosopher's Stone in the US under the title of Harry Potter and the Sorcerer's Stone: a change Rowling claims she now regrets and would have fought if she had been in a better position at the time.
Harry Potter is now a global brand worth an estimated £7 billion, and the last four Harry Potter books have consecutively set records as the fastest-selling books in history. The series, totalling 4,195 pages, has been translated, in whole or in part, into 65 languages.
The Harry Potter books have also gained recognition for sparking an interest in reading among the young at a time when children were thought to be abandoning books for computers and television, although the series' overall impact on children's reading habits has been questioned.
You may also be interested in the Harry Potter audiobook range, available from The House of Oojah
See other childrens fiction click here
Harry Potter and the Half-Blood Prince by JK Rowling
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Harry Potter and the Deathly Hallows - New hardcover book
Book 7. The final book in the amazingly popular Harry Potter series. New hardcover book with dustjacket, choice of Adult or Children's edition. How will it end?
Harry Potter's School Books - new boxed set
New boxed set of 2 hardcover books - Quidditch Through the Ages by Kennilworthy Whisp & Fantastic Beasts and Where to Find Them by Newt Scamander
Wizard - Harry Potter's Brand Magic - Stephen Brown new softcover book
The marketing story behind the greatest storyteller of our time and one of the most successful brands of recent times.
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Women's Ring Set features ONE of 6mm x 6mm (Approx. 0.7ct equivalent) round-cut stone in the center and TEN of 1.5mm x 1.5mm (1.0ct) round-cut stones on the side. The main stone sits 7mm above finger. There is also a matching band with ELEVEN of 1.5mm x 1.5mm (1.1ct) round-cut stones. The setting is Gold Ion Plated over 316 stainless steel. This ring will not tarnish. Men's Gold IP stainless steel band is 5mm wide. Total 3 pieces in this item.


text:
Angry Faces
A new study suggests that not only do we cue into angry faces, we're more likely to think they're looking right at us even if they're not.
http://podcast.scienceupdate.com/100113_sciup_face.mp3
BOB HIRSHON (host):
Recognizing anger…I'm Bob Hirshon and this is Science Update.
Angry people often seem like they're looking directly at us even if they aren't. This according to Cambridge University psychologists Andrew Calder and Michael Ewbank. They showed volunteers a series of photographs of angry, fearful or neutral facial expressions. Ewbank says the volunteers tended to perceive the angry faces as looking directly at them even if the eyes were gazing slightly to the right or to the left.
MICHAEL EWBANK (Cambridge University):
When you see an angry face and the gaze is quite ambiguous, you're more likely to interpret that it's looking at you than it's not, because in a sense it's better to be safe than to be sorry.
HIRSHON:
He says this makes sense from an evolutionary perspective, because being able to tell whether an angry human or predator was staring at them helped our ancestors survive.
So in a sense it's a kind of adaptive mechanism to be prepared for danger.
I'm Bob Hirshon, for AAAS, the science society.
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\section{Introduction}
In a two-dimensional bilayer system of charged particles where the
separation of the layers are large enough to prevent tunnelling,
an external current applied in one (active) layer can drag the
charges in the adjacent (passive) via the inter-layer Coulomb
interaction, and thus, induce a current. This phenomenon, known as
the Coulomb Drag (CD) effect, which was proposed
theoretically~\cite{Price83:750}, and investigated
experimentally~\cite{Gramila91:1216}. The Coulomb drag is of
fundamental interest, since it can be utilized as a probing tool
to investigate the screened interaction in mesoscopic transport
systems. A quantitative measure of the Coulomb drag is the
transresistivity $\rho_{21} = E_2 / J_1$ where $J_1$ is the
applied current in the active layer and $E_2$ is the induced
electric field in the passive layer.

The presence of perpendicularly applied magnetic field enriches
the physics of the CD effect by introducing Landau levels. The
electrostatic and transport properties of the bilayer system were
investigated first within the independent electron
picture~\cite{Jauho93:4420} and recently within a self-consistent
screening scheme~\cite{siddikikraus:05,Bilayersiddiki06:}. The
main outcomes of the self-consistent scheme are (i) the formation
of incompressible strips in the active layer at integer Landau
filling factor values, which affects the density and potential
profile in the passive layer, and (ii) the tilting of the total
potential within the active layer in the out-of-linear response
(OLR) regime, induced by a large external
current~\cite{Guven03:115327}.

In this work, we combine these effects to formulate the interaction between
the active and passive layers within the OLR regime, in
order to provide a self-consistent description of the CD effect.
We investigate the effect of the current intensity, temperature
and density mismatch between the layers on the inter-layer Coulomb
interaction in the OLR regime.
\section{Model, Results and Discussion}
The electron bilayer system in the present work consists of
two $\rm{GaAs}$ quantum wells (having a width of $2d\sim3$ $\mu$m) that are
sandwiched between two silicon doped
thick $\rm{(AlGa)As}$ layers. This structure is grown on top of a $\rm{GaAs}$
substrate. The layers are translationally invariant.
The electron density of the top layer is manipulated by
a top gate. The electron layers are symmetric (with respect to the
growth direction) and ($h\sim$) $10-30$ nm apart separated by an
undoped $\rm{(AlGa)As}$ layer. Such a bilayer system is known to
be electronically decoupled and can be represented by two
different electrochemical potentials. We note that the
electron tunnelling between the layers is not possible. Both of
the electron channels are formed in the interval $-b<x<b$, where
$|d-b|$ sets the depletion length.

The total electrostatic potential of an electron on the line
$(x,z)$ due to a line-charge at $(x_{0},z_{0})$ is given
by~\cite{Bilayersiddiki06:} \begin{eqnarray} V(x,z)= V_{bg}(x,z) + V_H(x,z),
\label{eq:Vvon_x}\end{eqnarray} and \begin{eqnarray} V_H(x,z)= \frac{2e^2}{\bar{\kappa}}
\int_{x_{l}}^{x_{r}} dx_{0} K(x,x_{0},z,z_{0}) n_{\rm
el}(x_{0},z_{0}), \label{eq:V_Hartree} \end{eqnarray} where $-e$ is the
charge of an electron, $\bar{\kappa}$ an average background
dielectric constant, and the kernel $K(x,x_{0},z,z_{0})$ solves
Poisson's equation under the relevant boundary conditions given by
\begin{eqnarray} \label{perp-bilayer} K(x,x_{0},z,z_{0})=-\ln\!
\left(\frac{\cos ^2 \frac{\pi}{4d}(x+x_{0}) + \gamma ^2}{ \sin ^2
\frac{\pi}{4d} (x-x_{0}) + \gamma ^2}\right) \end{eqnarray} where the $z-$
dependence is given by $\gamma=\sinh (\pi |z-z_{0}|/4d)$. The
confining (background) potential is obtained by inserting a
constant number density ($n_{0}$) of the background charges into
Eq.~(\ref{eq:V_Hartree}). The gate can be
described by an induced charge distribution $n_{g}(x)$, that is residing
on the surface. In order to obtain a flat (gate) potential profile
in the bulk we choose the induced charge distribution as
$n_{g}(x)=n_{g}^{0}(1+\alpha(x/d)^2)$ where $n_{g}^{0}$ determines
the strength of the gate potential whereas $\alpha (=0.7)$ gives
the slope of the induced charge distribution. With such treatment
of the gate, the average electron densities can be modified while keeping
the depletion length constant. The Hartree potential
can be calculated from Eq.(\ref{eq:V_Hartree}) for the top (bottom) layer
yielding a total potential  \begin{eqnarray} V(x,z)=
-V^{T}_{bg}(x,z)-V^{B}_{bg}(x,z)-V_{g}(x,z)\nonumber \end{eqnarray} \begin{eqnarray}
+V_{H}^{T}(x,z)+V_{H}^{B}(x,z) .\label{int1d}\end{eqnarray}
The electron densities are calculated within the TFA: \begin{eqnarray}
\label{eq:TFA}n^{T,B}_{el}(x)=\int
dED(E)f([E+V(x,z_{T,B})-\mu^{*}_{T,B}]), \end{eqnarray} with $D(E)$ is the
Landau density of states (DOS), $f(E)=1/[\exp(E/k_BT)+1]$ is the
Fermi function, $\mu^{*}_{T,B}$ is the chemical potential (being
constant in the equilibrium state) and $z_{T,B}$ is the position of
top (bottom) layer, respectively. We employ the Gaussian
broadened Landau DOS~\cite{Guven03:115327}. The self-consistent numerical
calculations start with obtaining the confining potential created by its own donors for each
layer and the electron densities and the intralayer and interlayer Hartree
potentials~\cite{Siddiki03:125315} at zero temperature and zero magnetic field.
The energies are scaled by the average pinch-off energy $E_0=2\pi e^2 n_0 d/\bar{\kappa}$. The
lengths are scaled by the screening length $a_{0}$
($=a^{*}_{B}/2$) expressed in terms of effective Bohr radius,
$a^{*}_{B}=\bar{\kappa}\hbar^{2}/(me^{2})$. The electron density
and the electrostatic potential for finite temperature and magnetic field can be calculated
self-consistently by the above scheme within the TFA.

The crucial and distinguishing feature of the approach we
implement is that we neither rely on a one-electron description
nor use any phenomenological parameters to account for electron
interactions, both for inter- and intra-layer. In the next step,
we drive a relatively high current from the active (here, top)
layer and calculate the effect of this on the passive layer. We
use a local version of the Ohm's law to describe current densities
employing a Gaussian broadened DOS defining the conductivity
model~\cite{Guven03:115327,siddiki2004}. We assume a local thermal
equilibrium, where local quantities vary slowly on the
quantum mechanical length scales. In the presence of current, it is
well known that the Landau levels are tilted due to the
Hall potential developed~\cite{Guven03:115327}, therefore the
electrochemical potential $\mu^{*}_{T,B}$, becomes position
dependent and thus must be included within the self-consistent scheme.
This brings two other self-consistent loops:
(i) the re-calculation of the $\mu^{*}_{T,B}(x)$ in each layer;
(ii) the response of the other layer to this modification. In this
work, we also include this tilting to investigate its effect on
the potential (see Fig~\ref{fig:fig1}) and the density
profile of the passive layer.
\begin{figure}
{\centering
\includegraphics[width=1.0\linewidth]{fig1.eps}
\caption{ \label{fig:fig1}The calculated potential profiles as a
function of lateral coordinate, for zero and finite current applied
to the top layer. The gate potential is chosen such that the
resulting density mismatch is at the order of few percents, namely
$V_g/E_0=0.05$. Calculations are done at $k_BT/E_0=5.5\times10^{-3}$ for
$n_0=4\times10^{11}$ cm$^{-2}$ and interlayer spacing of $h=12$
nm. The $B$ field is selected such that
$\hbar\omega_c/E_0=5.5\times10^{-4}$ assuming a fixed depletion
length $b/d=0.9$}}
\end{figure}

Here, we aim to show the effect of the combination of the inter-
and the intra-layer Coulomb interactions together with the effect
of external current driven in at least one of the layers. If one
drives a current perpendicular to the applied magnetic field a
Hall potential develops in the transverse direction to the
current, i.e. Hall potential, resulting from the Lorentz force. If
the system is all compressible, the Hall potential is linear in
$x$, as one would expect to observe in a normal metal. Here we
observe a similar behavior for the top layer, depicted in
Fig.\ref{fig:fig1}. On the other hand the response of the second
layer is as expected in the first glance, since, if the top layer
has a tilted electrochemical potential in one direction the other
should be the opposite. In Fig.~\ref{fig:fig1}, we show the
self-consistently calculated potential profiles of both layers.
For zero bias the top layer (broken black line) and the bottom
layer (dotted cyan line) are symmetric in $x$. However, for a
large current intensity ($U_H$, see Ref.~\cite{Guven03:115327}) at
the top layer to induce an electrochemical potential of this layer
(red short dashed-dotted lines) a tilt is induced at the bottom
layer with an opposite slope (dark blue dotted lines). More
interestingly, if a finite current is driven from the bottom layer
with same current direction of the top layer, the tilting observed
is slightly suppressed (brown dashed-dotted line).

In summary, by exploiting the smooth variation of the external
potential we have explicitly calculated the Coulomb interaction in
and between the bilayer quantum Hall samples in the presence of a
strong current driven in one layer. Our preliminary results
indicate that, a strong Coulomb coupling induces modifications in
the potential landscape of both layers, which clearly points that
a self-consistent microscopic model is necessary to grasp the
underlying physics of the Coulomb drag effect observed in the
bilayer systems.

The authors acknowledge the support of the Marmaris Institute of
Theoretical and Applied Physics (ITAP), TUBITAK grant 105T110,
SFB631 and DIP.