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    {\small\rm\zihao{6}\songti\thepage\hfill
    \quad\quad ACTA ASTRONOMICA SINICA \hfill
    \hspace{2mm} Vol.44\vspace{2mm}}}} %%%%%%%
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    {\small\rm \zihao{6}\songti suppl.\hfill B. Zhang: Radio Pulsar Death  %%（作者：题目名称）
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    \zihao{6}\songti
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    Vol.44~ Suppl.
    \hfill\hspace{8mm} %%% if ``October/January'' then \hspace{8/10 mm} July
    \small ACTA ASTRONOMICA SINICA
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      Feb.,~2003 \\[-1.5 true mm] %%January
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\vs\vs\vs \bc {\Large\bf
Radio Pulsar Death    %%（文题）
%\\[1ex]
%$\!\!^*$
}

\vs\vs{\normalsize\zihao{4}\fangsong Bing Zhang}  %%（本文作者）

\vs
({\footnotesize {\it The Pennsylvania State University, USA}}) \\%%（本文作者单位）
\vs\vs \ec

\normalsize
\begin{quotation}

 \no{\bf A{\scriptsize\bf BSTRACT }}~~~\small
 Pulsar radio emission is believed to be originated from the
electron-positron pairs streaming out from the polar cap region.
Pair formation, an essential condition for pulsar radio emission,
is believed to be sustained in active pulsars via one photon
process from either the curvature radiation (CR) or the inverse
Compton scattering (ICS) seed photons, or sometimes via two photon
process. In pulsars with super-critical magnetic fields, some more
exotic processes, such as magnetic photon splitting and bound pair
formation, will also play noticeable roles. All these effects
should be synthesized to discuss radio pulsar death both in the
conventional long-period regime due to the turnoff of the active
gap, and in the high magnetic field regime due to the possible
suppression of the free pair formation. Here I briefly review some
recent progress in
understanding radio pulsar death.

 %\vn
 %\no{\bf Key words}~~  pulsar-mode  change-spectrum
\end{quotation}

%%%% 正文
 \para{\large\bf{1}~~ Why Pair Production Essential?}

\vs\vs A pulsar drags its magnetosphere to co-rotate. In the
observer's rest frame, the local charge density required for
co-rotation is the ``Goldreich-Julian'' (1969) density, which when
$r \ll r_{lc}$ ($r_{lc}$ is the light cylinder radius) is
approximately $\rho_{_{\rm GJ}} \simeq -({\bm \Omega \cdot B})/({2
\pi c})$, where ${\bm\Omega}$ is the angular velocity of the star,
and ${\bm B}$ is the local magnetic field vector. In a simplest GJ
magnetosphere, the positive and negative charges are separated
from each other in space, and pair production is not required.

Pair production comes in for two reasons. First, it is obligatory.
The most prominent feature of pulsar radio emission is its very
high brightness temperature (typical value $T_B \sim
10^{25}-10^{30}$ K). Due to self-absorption, the maximum
brightness temperature for any incoherent emission is limited by
the kinetic energy of the emitting electrons, i.e. $T_{\rm
incoh,\max} = \gamma_e m_e c^2/k \sim 6\times 10^{12}
\gamma_{e,3}$\,K. The pulsar radio emission mechanism therefore
must be {\em extremely} coherent. Guided by our understanding of
other coherent sources in the universe, the emission source is
very likely a plasma in which various instabilities can be
developed to achieve coherent emission. A pair plasma is therefore
required in an otherwise charge-separated magnetosphere.

Second, it is inevitable. A rotating magnetic pulsar is a unipolar
generator, with a potential drop across the open field line region
\begin{equation}
\Phi=(B_p R^3 \Omega^2/2c^2) = 6.6\times 10^{12}~{\rm V}~B_{p,12}
P^{-2} R_6^3,
\end{equation}
where $B_p$ is the surface magnetic field at the pole, $P$ is the
rotation period, $R$ is the neutron star radius, and the
convention $Q=10^n Q_n$ has been adopted. Such a huge potential is
likely dropped along the open field lines (see \S2 for the
reasons), which accelerates a test particle up to an energy of
$\gamma_p m_e c^2 = e \Phi \sim 6.6$ TeV, or $\gamma_p
\sim 10^7$ ($\gamma_p$ is the Lorentz factor of the ``primary''
particles to be distinguished with the secondary pairs). These
particles emit curvature radiation (CR) or inverse Compton scattering
(ICS) photons (eqs.[2-4]) during
acceleration. These primary $\gamma$-rays inevitably materialize in
a strong magnetic field or a hot thermal photon bath near the surface
(see \S2 for detailed discussions).

\vspace*{1mm}
\para{\large\bf{2}~~Pulsar Inner Gaps and Pair
Production Mechanisms}

\vs\vs In a force-free magnetosphere where $\rho=\rho_{\rm GJ}$,
the electric field parallel to the local magnetic field,
$E_\parallel$, is equal to zero everywhere. In order to have
particles accelerated in a pulsar magnetosphere, there must be a
deficit of $\rho$ with respect to $\rho_{_{\rm GJ}}$ so that an
$E_\parallel$ can be developed. There are two preferred
charge-depleted regions, namely gaps, in a pulsar magnetosphere:
the polar cap region above the neutron star surface (inner gap),
and the region near or beyond the ${\bm \Omega \cdot B}=0$ ``null
charge surface'' (outer gap). Since outer gaps can only be
developed in young and millisecond pulsars while a much larger
population of pulsars have radio emission, it is generally
believed that radio emission may not be closely related to the
existence of the outer gap. Rather, it should be closely related
to an active inner gap which provides copious pairs from the polar
cap cascade.

There are two ways to classify the inner gaps. The first way is
according to the boundary condition at the pulsar surface, and
inner gaps can be divided into two subtypes: a vacuum gap
($E_\parallel \neq 0$ at the surface, Ruderman \& Sutherland 1975)
and a space-charge-limited flow ($E_\parallel=0$ at the surface,
Arons \& Scharlemann 1979). The first type (hereafter VG) requires
that the charges (usually positive ions) are tightly bound in the
surface, so that a large vacuum gap naturally develops as charges
in the polar regions move away from the surface due to centrifugal
forces. Such a suggestion, however, is questioned later since
calculations show that ion binding energies are usually much
smaller than the one required for binding (e.g. Usov \& Melrose
1995 for a review). The picture may be still pertained by
conjecturing either that some pulsars are more exotic objects with
exposed strange quark surface (Xu, Qiao \& Zhang 1999; Xu, Zhang
\& Qiao 2001), or that much stronger (compared with the dipolar
field inferred from the $P$ and $\dot P$ data) sun-spot like local
magnetic fields anchor in the polar caps (Gil \& Mitra 2001; Gil
\& Melikidze 2002). The second type of inner gaps (hereafter SCLF)
is the natural outcome of a neutron star (or a strange star with a
normal matter crust) with none or weakly distorted dipolar
magnetic field configuration. In such a case, charges (either
electrons for ${\bm \Omega \cdot B} >0$, or ions for ${\bm \Omega
\cdot B} <0$) will be freely pulled out from the surface. Taking
$\rho =\rho_{_{\rm GJ}}$ at the surface, it is natural that a deficit
of $\rho$ with respect to $\rho_{_{\rm GJ}}$ will grow as the current
flow outwards in the open field line region, since both $\rho$ and
$\rho_{_{\rm GJ}}$ follow different $r-$dependences. In the early
years of pulsar theories when a general relativistic effect is not
noticed, such $\rho$-deficit growth was believed to be due to the
curvature of the pulsar dipolar field lines (Arons \& Scharlemann
1979; Arons 1983). In an oblique pulsar, for those field lines
curving towards the rotational axis (the so-called favorably
curved lines), an decreasing angle between ${\bm \Omega}$ and
${\bm B}$ result in a net gain in $\rho_{_{\rm GJ}} \propto {\bm
\Omega\cdot B}$ with respect to the local $\rho \propto |{\bm B}|
\propto r^{-3}$ ($r$ is the radial coordinate of the point of
interest) value defined by the magnetic flux conservation. The
small $E_\parallel$ gradient generated from a small
$(\rho-\rho_{_{\rm GJ}})$ deviation results in an elongated gap
compared with VGs. The growth of $E_\parallel$ at the edges of the
gap is even smaller, resulting in a ``slot gap'' (Arons 1983).
Later Muslimov \& Tsygan (1992) first realized that by taking into
account the general relativistic effect, another more prominent
acceleration mechanism in the SCLF picture is available. For a
Kerr metric, an observer at infinity views that local inertia
frames are dragged by the rotating body. This modified the
requires ``co-rotating'' magnetosphere charge density to a
relatively smaller value: $\rho_{_{\rm GJ}}({\rm GR}) \simeq
-[({\bm \Omega-\Omega_{\rm LIF}) \cdot B}] / 2\pi c \alpha \simeq
-({\bm \Omega \cdot B}/2\pi c\alpha) [1-\kappa_g (R/r)^3]$, where
$\alpha \sim 0.78$ and $\kappa_g \sim 0.15-0.27$ are some
constants depending on the equation of state of the neutron star
(Muslimov \& Harding 1997; Harding \& Muslimov 1998, hereafter
HM98). Clearly, besides the ``flaring'' term due to field line
curvature (the $[{\bf \Omega \cdot B}]$ factor), an additional
$r$-dependence of $\rho_{_{\rm GJ}}$ is introduced (the $[1-\kappa_g
(R/r)^3]$ factor). It turns out that this component results in a
larger $(\rho-\rho_{_{\rm GJ}})$ deviation, and a much faster growth
of $E_\parallel$, as long as the inclination angle is not very
close to 90\dg. Such an effect significantly influences the
properties of the SCLF gaps (HM98). In the VG scenario, however,
since $\rho = 0$ in the gap, the frame-dragging modification is
only minor, i.e., by a factor of $(1-\kappa_g) \sim 0.85$ (Zhang,
Harding \& Muslimov 2000, hereafter ZHM00).

The second way to classify inner gaps is according to the source of
$\gamma$-ray photons that trigger the pair cascade. For a long time
since the early pulsar theories, curvature radiation (CR) is regarded
as the only source of the primary $\gamma$-rays (Sturrock 1971;
Ruderman \& Sutherland 1975; Arons \& Scharlemann 1979). The typical
CR photon energy is (${\cal R}$ is the field line curvature)
\begin{equation}
\epsilon_{_{\rm CR}}=(3/2) (\hbar c/{\cal R}) \gamma_p^3 \simeq 76
~{\rm GeV}~ {{\cal R}_6}^{-1} \gamma_{p,7}^3\,.
\end{equation}
The importance of inverse Compton scattering (ICS) off the thermal
X-rays near the neutron star surface was gradually appreciated later
(e.g. Xia et al. 1985; Daugherty \& Harding 1989; Dermer 1990; Sturner
et al. 1995). Zhang \& Qiao (1996) and Luo (1996), within the
frameworks of VG model and the
(non-relativistic) SCLF model, respectively, first clearly suggested
that the ICS photons may be more energetic and therefore have shorter
mean free path to generate pairs than the CR photons. Zhang et al.
(1997) further pointed out that there are two preferred ICS photon
energies (though K-N limit ignored), so that by combining the CR
typical energy, there are altogether three inner gap ``modes''.
The understanding of these three modes were greatly advanced by HM98,
Hibschman \& Arons (2001a,
b, hereafter HA01a, HA01b) and Harding \& Muslimov (2001, 2002,
hereafter HM01, HM02) within the framework of the relativistic SCLF
models. The first ICS mode is defined by the
characteristic photon energy due to ``resonant'' scatterings (which
are generally related to transition of
electrons between the first Landau state and the ground state, and
have much larger cross section than the Thomson cross section), i.e.,
($B'=B/B_q$ and $B_q \equiv m_e^2 c^3/\hbar e \simeq 4.414 \times
10^{13}$ G is the critical field)
\begin{equation}
\epsilon_{_{\rm ICS,R}}=2 \gamma_p B' (m_e c^2) \simeq 23 ~{\rm
GeV}~ \gamma_{p,6} B_{12}\,.
\end{equation}
Scatterings above the resonance (non-resonant scatterings) also
contribute to a significant higher energy component in the final IC
spectrum. The typical energy of these scatterings is defined by the
minimum of the IC-boosted thermal peak energy and the Klein-Nishina
limit (i.e. the electron's kinetic energy)
\begin{equation}
\epsilon_{_{\rm ICS,NR}}={\min} (\gamma_p^2 k T, \gamma_p m_ec^2)
={\min} (8.6 \gamma_{p,4}^2 T_6, 5.1\gamma_{p,4})~{\rm GeV}\,.
\end{equation}
The K-N limit takes over as long as $\gamma_p > 5.9\times 10^3
T_6^{-1}$. Resonant scattering also reaches the K-N regime when $B'>1$.

Gamma-rays produced via CR or ICS can materialize essentially in
two ways, i.e., one photon production (1p: $\gamma \rightarrow
e^{+}e^{-}$) in strong magnetic fields and sometimes two photon
production (2p: $\gamma\gamma \rightarrow e^{+}e^{-}$). The 1p is
believed to be the dominant source of pairs (Sturrock 1971). The
pair generation rate exponentially grows as the factor
$(1/2)(\epsilon_\gamma/m_ec^2) B'\sin\theta_{\gamma B}$ reaches a
critical value of $\sim 1/10$, or in high magnetic fields
($B'>0.1$), as the threshold condition
$(1/2)(\epsilon_\gamma/m_ec^2)\sin\theta_{\gamma B}>1$ is reached
(Daugherty \& Harding 1983), where $\theta_{\gamma B}$ is the
incident angle between the gamma-ray and the local magnetic field.
These define the {\em ``mean free path'' (i.e. attenuation length)
of the photons}, $l_{ph}$, which is shorter for a larger
$\epsilon_\gamma$, for a larger $\theta_{\gamma B}$ (or a shorter
pulsar period which gives a larger field line curvature), and for a
higher $B$ when the near-threshold effect is not important. In
most pulsars, usually more than one generation of pairs are
produced. The first generation pairs lose their perpendicular
energies through synchrotron radiation (SR), and the resultant
secondary $\gamma$-rays also usually meet one photon pair
production condition, so that a photon-pair cascade develops
(Daugherty \& Harding 1982, 1996). The remaining parallel energy
of the pairs will also be converted to radiation through resonant
ICS, and the resulting photons may sometimes (although the
condition is more stringent) be converted to further pairs,
leading to a full polar cap cascade (Zhang \& Harding 2000a).

The 2p process is in principle expected, since the typical energy of
primary gamma-rays, $\epsilon_\gamma$, and the typical energy of the
thermal X-rays, $\epsilon_x$, usually satisfy the kinetic condition
$\epsilon_\gamma \epsilon_x (1-\cos \theta_{\gamma x}) \geq (m_e
c^2)^2$, where $\theta_{\gamma x}$ is the incident angle between the
gamma-ray and the X-ray (Zhang \& Qiao 1998). However, the mean free
path of a typical gamma-ray for 1p is usually smaller than that of 2p,
so that a gamma-ray usually has materialized in the strong magnetic
field before interacting with the thermal X-rays. To have 2p competitive
against 1p, one has to either raise the thermal luminosity and
$\theta_{\gamma x}$ or considerably lower the magnetic field. The hot
magnetar environment (Zhang 2001) and the low-$B$ millisecond pulsars
(Harding, Muslimov \& Zhang 2002, hereafter HMZ02) are therefore
possible sites where the 2p process may become noticeable.

\para{\large\bf{3}~~Energy Budget Deathlines and Death Valley}

We have shown above that pair production is generally expected in
normal pulsars, and in \S1 we have made the argument that pair
production is almost obligatory in the current radio emission models.
{\em Production of free pairs is the essential condition for pulsar
radio emission. The condition that free pair production is prohibited
or suppressed therefore defines radio pulsar death} (recent
discussions on this topic include, e.g. Arons 2000; ZHM00; HA01a,b;
HM02; HMZ02). One important caveat is that free pair production may
not be {\em the sufficient condition} for pulsar radio emission, which
is model-dependent and
poorly known. There exists a missing link between the pulsar high
energy emission theories from which the cascade pair properties are
derived, and the pulsar radio emission theories in which various
pair properties are required (but see Arendt \& Eilek 2002 for a
recent attempt to connect the missing link).

When a radio pulsar ages, it slows down so that the unipolar potential
drops. The accelerated particles achieve less energies, and their
subsequent CR/ICS photons are less energetic. Eventually, these
photons no longer pair produce, either via 1p or 2p. No further pair
plasma is ejected into the plasma, and the pulsar stops shining in
the radio band. A radio pulsar dies. The death of this type is due to
inadequate rotational energy budget. The condition defines the so-called
energy budget ``deathline'' (e.g. Ruderman \& Sutherland 1975 and
many literatures thereafter) in the $P-B_p$ space or $P-\dot P$ space.
For a star-centered dipole, the surface magnetic field at the pole
reads ($I$ and $R$ are moment of inertia and radius of the neutron
star, respectively)
\begin{equation}
B_p=6.4\times 10^{19}~{\rm G}~(P \dot P)^{1/2} I_{45}^{1/2}
R_6^{-3}\,.
\end{equation}

In some work where detailed theories are not needed, the pulsar
deathline is conventionally taken as a line of constant $\Phi$
(eq.[1]), with a slope 3 in the $P-\dot P$ diagram. We'd like to
caution here, however, that the real death lines could considerably
differ from such a line, since the final potential drop across the gap
at the pulsar death is defined by the condition of pair production,
which gives a lower $\Phi$ with a different slope than 3.
In fact, it is not easy to draw a single line to
define radio pulsar death, since many factors will affect the position
of the deathline.
(This is why I did not present any analytical expression of the
deathline in this chapter.)
What is more relevant should be a ``death valley'', a term first
invented by Chen \& Ruderman (1993).

Before moving into detailed discussions about the energy budget
deathlines, two more length parameters need to be introduced. One
is {\em the acceleration length}, $l_{acc}$, which is the typical
distance an electron has to travel for acceleration to achieve a
typical Lorentz factor $\gamma_{p,c}$, with which the electron's
CR/ICR emission photons can pair produce. This is
acceleration-model-dependent (VG vs. SCLF). Another
one is {\em the mean free path of the electrons}, $l_e$, which is the
distance of the electron travels before emitting one CR or ICS
photon, i.e., $l_e \simeq (\epsilon_\gamma / \dot \gamma m_e c^2)
c$. This is radiation-mechanism-dependent (CR
vs. ICS). These two length parameters, together with {\em the mean
free path of the photon}, $l_{ph}$, as discussed in \S2, are essential
for the following discussions. In all the models, a test particle has
to be accelerated through a length scale of $l_{acc}$ to reach a
Lorentz factor $\gamma_c$. This particle emits a test photon (via CR
or ICS) with a probability (depends on $l_e$), and the photon
attenuation length is $l_{ph}$. By minimizing the length $l_{acc}+
l_{ph}$, one gets a typical height of the gap, and hence, the
potential drop across the gap ($\Delta V$). By demanding $\Delta V$
equal to the maximum achievable potential (some fraction
of $\Phi$ in eq.[1]), one gets a deathline (ZHM00).

Lack of a well-defined deathline is due to many uncertainties involved:

(1) Criterion: In the literatures, there exist essentially two
criteria to define the deathlines. ZHM00 adopted the criterion of
$l_e \leq l_{acc}+l_{ph}$ (see also Zhang et al. 1997), which
ensures that the pair ``multiplicity'' is at least 1. HA01a
adopted a (usually) less demanding criterion for pair production
to reach a minimum multiplicity adequate to screen the
$E_\parallel$. These could be regarded as the ``strong
criterion''. On the other hand, HM02 proposed a ``weak
criterion''. They did not introduce any demand on $l_e$, but just
evaluated the condition that pair production happens at all. In
this criterion, $E_\parallel$ is not necessarily screened, and the
pair number density at the deathline may be very low, i.e., could
be well below the GJ density. Two comments need to be addressed. First,
neither criteria are necessarily the criterion for radio emission.
The latter merely defines a pair deathline. Second, the
discrepancies between both criteria only step out in the
ICS-controlled gaps. For CR-gaps, the difference between these
treatments disappear, essentially because $l_e \ll l_{ph}$ in the
CR case (ZHM00).

(2) Model-dependence:  Clearly different models (VG or SCLF; CR-,
RICS-, or NRICS-controlled) result in different $\Delta V$'s of
the gap, and hence, different deathlines within that particular
model (e.g. ZHM00). An adequate study needs to address the
parameter regimes for the different modes to dominate so as to
achieve a synthesized deathline in the whole $P-\dot P$ space
(e.g. HA01a, HM01, HM02, for SCLF models). The following essential
features are noticeable. In VGs, usually $l_{acc}$ is small and
negligible; in SCLFs, $l_{acc}$ is comparable to (although smaller
than) $l_{ph}$ for the minimized case; in CR gaps, $l_e$ is small
and negligible; while in ICS gaps, $l_e$ is comparable to
$l_{ph}+l_{acc}$.

(3) Equation of state (EOS):  Since 1p process is the dominant
pair formation process in normal pulsars, the magnetic fields in
the pair formation region is a crucial parameter to define the
deathline in the $P-\dot P$ space (although not in the $P-B_p$
space). For the pure dipolar field configuration, equation (5)
indicates that the neutron star EOS influences the estimated polar
cap surface magnetic fields through influencing $I_{45}$ and
$R_6$. If pair production region is right above the surface (or a
fixed altitude above the surface), then different EOSs result in
different deathlines. In SCLF models, different EOSs also modify
the frame-dragging terms. The combined effects (HMZ02) generally
makes a larger difference in short period (e.g. millisecond)
regimes than in the long period regime. Softer neutron star EOSs
or strange star EOSs tend to facilitate pair production and hence,
lower the deathlines.

(4) Unknown surface field configuration:  A potentially even more
important factor that may influence the surface magnetic field
configuration and hence the deathlines is the unknown near-surface
magnetic field configuration. It has been long recognized that a
CR-controlled gap in a star-centered dipolar field can not sustain
pair production in all known pulsars, so that near-surface
multipole magnetic fields have long been speculated (Ruderman \&
Sutherland 1975; Arons \& Scharlemann 1979). Several possible
distorted magnetic configurations have been discussed in Chen \&
Ruderman (1993) within the VG models, and these are also valid in
discussing other models. The magnetic configuration is the key
factor that influences the deathlines although it is sparsely
modeled due to many uncertainties involved. Recently, Gil et al.
(Gil \& Mitra 2001; Gil \& Melikidze 2002; Gil et al. 2002)
investigated some possible consequences when the polar cap fields
are extremely curved and strengthened.

(5) Refined geometry and physics:  Most deathline discussions have
been analytical. With the ``weak criterion'' and the assumption of
a star-centered dipole, HM01 and HM02 performed detailed numerical
simulations to study pulsar deathlines for the CR- and
ICS-controlled SCLF gaps, respectively. Refined geometry (e.g.
curved spacetime) and physics (e.g. pair formation details) are
included. Their results suggest that many of the previous
analytical treatments may not be reliable. One important finding
is that at the deathlines, the maximum usable potential is only a
small fraction of the value expected analytically (cf. ZHM00), and
is model-dependent (CR vs. ICS). This cautions us that numerical
calculations may be also essential to discuss deathlines in other
models under other criteria and/or assumptions.

\para{\large\bf{4}~~A Death Valley in the High-$B$ Regime?}

Two other QED processes can potentially suppress free pair formation
in the strong magnetic field regime. These are magnetic photon
splitting (sp: $\gamma \rightarrow \gamma \gamma$) and bound pair
(positronium) formation.

Baring \& Harding (1998, 2001, hereafter BH98, BH01)
suggested a possible radio pulsar deathline in the high magnetic
field regime, due to possible suppression of 1p pair production by
magnetic photon splitting. (Strictly speaking, pulsars do not
evolve across this line, so the ``deathline'' is essentially a
radio quiescence line.) Such a line, by definition, is defined by
$l_{\gamma \rightarrow \gamma\gamma} < l_{\gamma\rightarrow
e^{+}e^{-}}$, where $l$'s are the $\gamma$-ray attenuation lengths
of the relevant processes (sp or 1p), which are dependent on both
the field strength and the $\gamma$-ray energy, $\epsilon_\gamma$.
By specifying a characteristic $\epsilon_\gamma$, a deathline in
the $P-B_p$ or $P-\dot P$ space can be plotted. BH98's photon
splitting deathline ($\dot P \simeq 7.9\times 10^{-13}
P^{-11/15}$) is defined by specifying $\epsilon_\gamma =
\epsilon_{\rm esc}$, where $\epsilon_{\rm esc}$ is critical energy
of the $\gamma$-ray that can just evade both 1p pair production
and photon splitting.

There are several caveats concerning the BH98 photon splitting deathline:

(1) How many modes split? The BH98 high-$B$ deathline is
contingent upon the assumption that photons for both the $\|$ and
$\perp$ modes (defined by the electric vector with respect to the
magnetic field plain) split in superstrong magnetic fields. Within
the linear vacuum polarization treatment, only the $\perp$-mode
photons are allowed to split at least for fields below $B_q$ (e.g.
Usov 2002). Whether the $\|$-mode photons split in superstrong
fields due to non-linear vacuum polarizations is fundamental for
all the discussions in this section.

(2) High altitude pair formation? Even if both photon modes split,
whether 1p pair production is completely suppressed still depends
on some further conditions. If particles can be accelerated at
higher altitudes and emit photons, these $\gamma$-rays can still
produce pairs since the local field has degraded with respect to
the high value near the surface (Zhang \& Harding 2000b, hereafter
ZH00b). For a VG, on the other hand, since downwards photons can
not pair produce and there is no free source of the primary
electrons, essentially few pairs could be generated. The
magnetosphere would be dead, so strictly speaking, the photon
splitting deathline is valid only for such a case (ZH00b).

(3) Model-dependence? Both $l_{\gamma \rightarrow \gamma\gamma}$
and $l_{\gamma\rightarrow e^{+}e^{-}}$ depend on $\epsilon_\gamma$
in different ways. It is more relevant to adopt the characteristic
$\epsilon_\gamma$ of a certain type of the VG gap, rather than
$\epsilon_{esc}$ to define the photon splitting deathlines (Zhang
\& Harding 2002; Zhang 2001). The deathlines drawn with such a
criterion tends to be tilted up with respect to the BH98 deathline
in the short period regime, since fast pulsars have larger
potentials to accelerate particles to higher energies and the more
energetic photons are more facilitated to 1p pair production
(Zhang \& Harding 2002). Distorted magnetic configurations will
also modify the competition between the sp and the 1p processes
(e.g. Gil et al. 2002).

(4) 2p pairs in magnetars? Finally, in magnetars, even if 1p pair
production could be completely suppressed by photon splitting, 2p
pair production typically has a shorter attenuation length in the
hot magnetar environment (Zhang 2001). So pair production might
not be completely suppressed in magnetars. A photon splitting
death valley, if exist, is therefore only valid for the high
magnetic field pulsars without substantial magnetic decay but with
strong binding at the surface.

Bound pair formation (e.g. Usov \& Melrose 1995; Usov 2002)
essentially delay the free pair formation front, but in principle does
not suppress free pair formation. Its role is similar to the two-mode
photon splitting processes in a SCLF gap, i.e., to increase the gap
height and the gap potential, which is helpful to interpret pulsar
high energy emission within the polar cap models.

\para{\large\bf{5}~~Concluding Remarks}

The following statements may be pertinent:

(1) We now have a clear framework about the particle acceleration
and photon-pair cascade in the pulsar polar cap region. Pair
production from the polar cap is believed to be an essential
condition for pulsar radio emission. The sufficient condition for
radio emission, however, is unknown. Personally, I think that
radio emission condition should be more stringent than the pair
production condition. Thus, (moderate) non-dipolar fields may
indeed exist at least in some pulsars. An important advance in the
pulsar study in the recent years is the realization that ICS plays
a crucial role in stead of CR in determining gap properties at
least in some pulsars.

(2) Theories are generally successful to define the conventional
radio pulsar death in the long period regime, although many
uncertainties prevent us from achieving a solid deathline. A death
valley is more pertinent. Without introducing distortions from a
star-centered-dipolar configuration, one can not include all
pulsars above the deathline. The 8.5 second pulsar (Young et al.
1999) remains a challenge for any pure-dipole model after detailed
numerical simulations. A best guess is that an ICS-controlled gap
anchors in this pulsar with a moderate non-dipolar near-surface
field configuration.

(3) In the high magnetic field regime, pulsar death is not
unambiguously defined. There is no strong reason against the
possible radio emission from high magnetic field pulsars and
magnetars. Possible reasons of apparent radio quiescence of
magnetars include (a) the beaming effect; (b) that the coherent
condition is destroyed; (c) that the main energy band of the coherent
emission is not in radio; or
(d) that the soft gamma-ray repeaters and the anomalous X-ray
pulsars are not magnetars at all (but might be accretion-powered
systems).

\vs\vn{\bf ACKNOWLEDGEMENTS}~~ I am grateful to Alice Harding, G.
J. Qiao, Alex Muslimov, J. L. Han, R. X. Xu, and W. P. Lin for
stimulating collaborations, to Jon Arons, Matthew Baring, Janusz Gil
for discussions concerning various topics covered in this talk, and to
R. X. Xu for presenting the materials at the meeting on behalf of me.

\bc {\bf References} \ec

\vspace{-4mm} \no \fns

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