text:
\section{Introduction}
Many properties of metals depend crucially on the type and
concentration of defects that perturb the ideal crystal structure. Of
these, the simplest are point defects, such as self-interstitials and
vacancies.  Since their formation energies are of order electron
volts, their equilibrium concentrations tend to be very low. They do
form in abundance, however, in radiation environments due to the
collisions between the irradiating species (electrons, heavy ions or
neutrons) and the atoms of the host crystal \cite{johnson}. When the
energy of the impinging particle is close to, but above, the
displacement threshold, the collision typically produces a single
Frenkel pair. Particles with higher kinetic energy, for instance
neutrons produced in fusion reactions, create collision cascades which
can produce not only Frenkel pairs, but an ensemble of mobile and
immobile self-interstitial and vacancy clusters of different
sizes. The ensuing evolution of the point defect distributions due to
diffusion \cite{flynn} determines the long time scale degradation of
the mechanical properties of the material, in addition to volume
swelling at large doses.

For body centered cubic (bcc) metals, such as $\alpha$-Fe, V and Mo,
molecular dynamics simulations have revealed a detailed microscopic
picture of the diffusive motion of individual defects and
vacancies\cite{delarubia97}. While the migration barrier $\Delta E_v$
for vacancy migration is rather high $(\sim0.5$ eV), both
self-interstitial atoms and small self-interstitial cluster are highly
mobile and easily diffuse along particular crystallographic directions
($\langle 111\rangle$-directions). In V, the lowest energy
self-interstitial configuration is a $\langle 111\rangle$-oriented
dumbell, which migrates easily with a crowdion transition state
\cite{han1}. However, this easy migration, with barriers as low as
$\Delta E_i\sim 0.02$ eV \cite{zepeda04}, leads to long
one-dimensional self-interstitial diffusional trajectories. The
self-interstitial dumbbell can change direction by rotating into other
$\langle111\rangle$-directions. The barrier associated with such
rotations $\Delta E_r$ is of the same order as $\Delta E_v$ (i.e.,
$\Delta E_r\gg\Delta E_i$).

In $\alpha$-Fe, by contrast, the ground state of the dumbbell
self-interstitial is the $\langle 110\rangle$ configuration, and
accessing the easy-glide $\langle 111\rangle$ configuration requires
overcoming a qualitatively similar rotational barrier that separates
the $\langle 110\rangle$ from the $\langle 111\rangle$ configurations
\cite{wirth97}.  Although the migration barriers for the crowdion
mechanism in $\alpha$-Fe are also believed to be very small $(<0.04$
eV), the observed effective migration barrier (including rotation into
the $\langle 111\rangle$ orientation and migration along the $\langle
111\rangle$-direction) are higher than in V
\cite{marian01,soneda01}. Similar arguments apply to Mo and Ta. Both
diffusion mechanisms imply that interstitial transport in bcc metals
takes place in the form of long one-dimensional trajectories with
occasional directional changes that become more frequent as the
temperature increases. In fcc metals such as Cu, for instance,
simulations showed that self-interstitial diffusion occurs through
much more conventional, isotropic diffusion mechanisms
\cite{osetsky01}. In most metals, however, large separations in time
scale between self-interstitial and vacancy motion exist.

The intent of the present study is to illuminate the consequences of
the intriguing microscopic diffusion mechanisms in bcc metals on the
evolution of the point defect population using simple, but readily
generalizable models. Our models shall minimally include local
self-interstitial/vacancy mutual annihilation and absorption into
(unsaturable) sinks which, in a real system, are in the forms of
dislocations or grain boundaries. Together with the continuous
production of point defects, these processes form the fundamental
events in metals under irradiation conditions.

On the continuum level, point defect dynamics can be described by a
set of kinetic master or rate equations that treat the point defects
as a continuous density whose temporal evolution is governed by
various gain and loss processes. These equations do not take into
account spatial correlations and are only analytically tractable in
the simplest situations. Although we focus the discussion specifically
on bcc metals, the continuum theory makes no reference to the
underlying crystal structure and could be readily applied to fcc
metals as well. In many cases, single interstitials and
vacancies combine with defects of the same type to form stable
clusters. The evolution of the cluster size distribution and other
microscopic variables is more conveniently studied in a particle based
model that explicitly represents defects and their diffusion
mechanisms (translation, rotation) on a lattice.  The competition
of events occuring with different rates can then be followed using a
kinetic Monte Carlo (kMC) scheme.

Both master equation and Monte Carlo approaches have frequently been
employed to illuminate the physics of damage evolution in irradiated
materials. Starting from the elementary processes mentioned above
\cite{brailsford72}, steady-state rate equations have been used to
study the effects of preferential absorption of self-interstitial
atoms at sinks (absorption bias) \cite{woo96,singh97}, the formation
of interstitial cluster during the cascade phase (production bias)
\cite{woo96,singh97,golubov00,trinkaus00}, and the one-dimensional
motion of these clusters in bcc metals
\cite{singh97,golubov00,trinkaus00}. kMC models were first used to
evolve the primary damage state obtained from ns-long MD simulations
to macroscopic time scales, but have increasingly been used to study
the evolution of defect structure during continuous irradiation in
copper \cite{heinisch96,heinisch97,heinisch99}, vanadium \cite{alonso}
and iron \cite{soneda03}.

Although many treatments have included a high level of atomistic
detail from the start, we begin by analyzing the simplest situation
that includes only production of Frenkel pairs, point defect
recombination and absorption at sinks in Section
\ref{simple-sec}. Even though this simplified case has been studied
many times before \cite{brailsford72}, we shall see that the inclusion
of specific features of bcc metals leads to a surprisingly rich
evolutionary picture. This approach allows us to gradually increase
the level of complexity of the model in a verifiable, controlled
manner. We introduce first interactions between vacancies that lead to
vacancy cluster formation in Section \ref{vaccluster-sec} and then
interactions between interstitials in Section \ref{intclust-sec}. In
particular, we shall discuss how interstitial cluster mobility affects
vacancy cluster formation. While the discussion in the text is
applicable to bcc metals broadly, we apply our findings to the
specific cases of V and $\alpha$-Fe in Section \ref{real-sec}.

\section{Point defect dynamics in simple situations}
\label{simple-sec}
\subsection{Atomistic details and kMC model}
\label{model-sec}

The physics of radiation damage evolution is governed by the
production of defects due to irradiation and their subsequent
elimination from the population through diffusional processes. Typical
values for the defect production rate $\sigma F$ range between
$10^{-3}$ dpa/s, in ion irradiation experiments \cite{iwai96}, and
$10^{-10}$ dpa/s, for neutron irradiation \cite{heinisch99}, where $F$
is the irradiation flux and $\sigma$ is the cross-section. A
``displacement per atom'' (dpa) refers to the production of one
Frenkel defect pair per lattice site.

Self-interstitial transport is composed of two parts, a
one-dimensional diffusive motion along one of the 4 distinct $\langle
111 \rangle$ directions and dumbbell rotations from one $\langle 111
\rangle$ direction to another. The temperature dependence of the
(one-dimensional) diffusivity $D_i$ is usually described through the
Arrhenius form in most cases except for vanadium, where the unusually
low activation barrier can lead to more complicated behavior at higher
$T$ (see Ref.~\onlinecite{zepeda04}).  For $T<600K$, however, the
interstitial diffusivity is well described by the Arrhenius expression
$D_i/a^2=f\nu_i\exp{[-\Delta E_i/k_BT]}$, where $\Delta E_i$ is the
activation energy barrier, $\nu_i$ an attempt frequency and
correlations are expressed through the correlation factor $f$ ($f=1$
when the diffusion is uncorrelated). The temperature dependence of the
rotation rate $\gamma_r$, by contrast, is always of the Arrhenius
form, $\gamma_r=\nu_r\exp{[-\Delta E_r/k_BT]}$, where $\Delta E_r$ the
characteristic rotation barrier and $\nu_r$ is an attempt
frequency. Values for both $\nu_i$ and $\nu_r$ range between
$10^{12}s^{-1}$ and $10^{13}s^{-1}$. Finally, vacancies diffuse
three-dimensionally with rate $D_v/a^2=\nu_v\exp{[-\Delta E_v/k_BT]}$,
where $\Delta E_v$ and $\nu_r$ are the activation barriers and attempt
frequencies, respectively. The defects can undergo two basic
reactions: recombination once an interstitial and vacancy defect are
within a certain ``recombination volume'', or absorption at sinks.
Typical dislocation densities in metals are of the order $10^{12} -
10^{14}\,{\rm m}^{-2}$, which translates into dislocation sink
densities (per lattice site) of $n_s \sim 10 ^{-4}-10 ^{-6}$.

Since the relevant time scales (ns-sec) for the evolution of the point
defect distributions far exceed those of molecular dynamics
simulations, we employ a coarse grained description, in which we
represent vacancies, dumbbell interstitial configurations and sinks as
pointlike objects that occupy ideal lattice sites of a bcc lattice
with lattice parameter $a$.  In order to mimic constant irradiation
conditions, new pairs of defects (self-interstitial and vacancy) are
introduced randomly onto defect-free lattice sites with rate $\sigma
F$. The microscopic diffusion process is replaced by instantaneous
hops of point defects to vacant neighboring lattice sites with rates
$D_i/a^2$ and $D_v/a^2$, respectively.  A self-interstitial is
constrained to forward-backward hops along one of the four $\langle
111 \rangle$ directions, but can also rotate to another $\langle 111
\rangle$ direction with rate $\gamma_r$. Vacancies diffuse
isotropically.  A self-interstitial or vacancy recombines if it finds
itself next to a vacancy or self-interstitial, respectively, or is
absorbed if one of its 8 neighboring sites contains a sink.  At each
time step, an event is chosen according to its probability and then
executed. Time is advanced according to the usual continuous time
algorithm \cite{binder92}, where the time increment is chosen from an
exponential distribution.

\subsection{Basic rate equations}
The elementary processes described in the kMC model can,
alternatively, be described within a rate equation formalism. Before
the advent of large scale computer simulations, this method
represented the only viable theoretical approach for simulating long
time radiation damage evolution.  The rate equations can be solved by
direct numerical integration or direct analysis in limiting cases.  In
the minimal model discussed above, the time evolution of the number
densities of the interstitals $n_i(t)$ and vacancies $n_v(t)$ is given
by the coupled nonlinear equations \cite{sizmann68,brailsford72,sizmann74}
\begin{eqnarray}\nonumber
\frac{dn_i}{dt}&=&\sigma
F-\kappa_v\omega_{iv} n_i-\kappa_i\omega_{vi} n_v-\kappa_{is}\omega_{is} n_i\\
\frac{dn_v}{dt}&=&\sigma
F-\kappa_v\omega_{iv}n_i-\kappa_i\omega_{vi} n_v-\kappa_{vs}\omega_{vs} n_v
\label{rate-eq}
\end{eqnarray}
Defect pairs are added to the population at a rate proportional to the
particle flux $F$ and a cross-section $\sigma$. Loss can occur through
a diffusing interstitial recombining with a vacancy with rate
$\omega_{iv}$ and a diffusing vacancy recombining with an interstitial
with rate $\omega_{vi}$. The recombination rates are all proportional
to the diffusion constant of the moving defect, but depend on the
dimensionality of the diffusion process. $\kappa_i$ and $\kappa_v$ are
dimensionless capture numbers that represent the spatial extent of the
defects and their effective (possibly long ranged) interactions.
Losses can also occur through absorption at sinks with rates
$\omega_{is}$ and $\omega_{vs}$ and corresponding capture numbers
$\kappa_{is}$ and $\kappa_{iv}$ for interstitials and vacancies,
respectively.  An alternative representation of Eqs.~(\ref{rate-eq})
is to define sink strengths $k_x^2$ via the relation
$k_{x}^2D_{x}=\kappa_{x}\omega_x$, where the subscript $x$ refers to any
of the combination of indices used above.

The encounter rates of the defects are given by the number of distinct
sites visited by a random walker per unit time. Since the mean squared
displacement $\langle R^2 \rangle=l^2N$ for a random walk with step
length $l$, the number of sites visited is $s=[\langle R^2
\rangle/l^2]^{1/2}\sim N^{1/2}$ in one dimension. By contrast, a
detailed analysis of random walks on three-dimensional cubic lattices
shows that a random walker visits $\mathcal{O}(N)$ distinct sites
after $N$ hops.  For a given density of target sites $n$, the typical
collision time $\tau_c$ is given by the condition $D\tau_c/a^2=1/n$
(3D) and $(D\tau_c/a^2)^{1/2}=1/n$ (1D), from which we deduce the
encounter rates
\begin{equation}
\omega_{3D}\sim nD/a^2 \qquad {\rm and} \qquad \omega_{1D}\sim n^2D/a^2.
\end{equation}
In a similar manner, the capture numbers $\kappa$ are also affected by
the dimensionality of the random walk. Assuming ideal spherical
defects of linear dimension $r$, $\kappa \sim r/a$ for a 3D random
walk in a mean-field approximation \cite{brailsford81}, but in
general, capture numbers can also depend on spatial fluctuations and
dose. For a 1D random walk, the scaling with defect size becomes much
stronger \cite{barashev01}, $\kappa \sim (r/a)^4$.

These expressions apply to strictly 1D or 3D random walks. As
discussed in the Introduction, we encounter an intermediate case in
bcc metals, where rotations interrupt 1D random walks and lead to 3D
trajectories. The encounter rate of this random walk must, therefore,
be larger than the purely 1D case. For a rotation rate $\gamma_r$, the
random walker performs on average $D/\gamma_r a^2$ hops along a
particular direction before rotating into a new direction.  There are
$\gamma_r\tau$ of these segments during time $\tau$. The mixed 1D/3D
collision time thus follows from the condition $(D/\gamma_r
a^2)^{1/2}\gamma_r \tau_c=1/n$, which implies
\begin{equation}
\omega_{1D/3D}=\omega_{3D}\sqrt{\gamma_ra^2/D}
\label{om-mixed-eq}
\end{equation}
The mixed encounter rate $\omega_{1D/3D}$ scales like the 3D encounter
rate $\omega_{3D}$, but is reduced by the square root of the ratio of
the number of rotations to hops. This reaction rate has also been
derived in ref.~\onlinecite{barashev01}. This work views the kinetics
in the intermediate 1D/3D regime as an enhanced 1D reaction rate, but
the resulting expressions agree up to numerical prefactors. These
authors also showed that the size dependence of the mixed capture
number $\kappa \sim (r/a)^2$, and ref.~\onlinecite{trinkaus02}
provides an interpolation formula between the limiting cases using a
continuum description. While the encounter rates and reaction kinetics
of random walkers decrease when the dimensionality of the random walk
changes from three to one, the diffusivity does not, since the mean
squared displacement of a random walk of $N$ steps of length $l$ is
$\langle {\bf R}(N)^2\rangle=l^2N$ independent of dimensionality.

Note that these encounter rates are derived under the assumption of
collisions with a stationary target. The case of several colliding 1D
random walkers becomes equivalent to a 3D random walk because, from
the rest frame of a given walker, the other walkers appear to be
executing a 3D random walk. This case would be relevant for describing
the collisions of interstitials with each other, but not with the
vacancies or sinks.

Inserting these encounter rates into Eqs.~(\ref{rate-eq}), assuming
mixed 1D/3D encounter for diffusing interstitials, yields rate
eqations of the form
\begin{eqnarray}\nonumber
\frac{dn_i}{dt}&=&\sigma
F-n_i n_v(\kappa_v \sqrt{\beta}D_i/a^2-\kappa_i D_v/a^2)\\&-&\kappa_{is} n_s n_i \sqrt{\beta}D_i/a^2\\
\frac{dn_v}{dt}&=&\sigma \nonumber
F-n_i n_v(\kappa_v \sqrt{\beta}D_i/a^2-\kappa_i D_v/a^2)\\&-&\kappa_{vs} n_s n_v D_v/a^2,
\label{rate2-eq}
\end{eqnarray}
where $\beta=\gamma_ra^2/D_i$ is a dimensionless ratio that describes
the relative frequency of dumbbell rotations and diffusional hops.

\subsection{Simple scaling analysis}
We now specialize these results to the common case of bcc metals,
where $\gamma_ra^2/D_i\ll 1$ and $D_i/D_v\gg 1$. Before performing the
kMC simulations and solving the full rate equations numerically, it is
instructive to analyse the limiting behaviors of this system
\cite{sizmann68}. Inititally, there are no defects in the metal, and
only the first term in the rate equations is important. In this regime
(regime I), defect densities increase linearly with time,
\begin{equation}
n_i^{\rm I}=n_v^{\rm I}=\sigma F t.
\end{equation}

Once a sufficient density of defects has been produced such that the
encounter times between defects becomes smaller than the time between
creation events, loss through recombination ($2^{nd}$ and $3^{rd}$
terms) becomes important \cite{sizmann64}. As we shall see below,
typical parameter ranges lead to a situation in which defect
recombination becomes important before sink loss.  The balance between
creation and recombination makes a steady state possible:
$dn_i/dt=dn_v/dt=0$. Ignoring the sink terms, one readily finds for
this Regime II
\begin{equation}
\label{regII-eq}
n_i^{\rm II}=n_v^{\rm
II}=\left(\frac{\kappa_v\sqrt{\beta}D_i+\kappa_iD_v}{\sigma
Fa^2}\right)^{-1/2}\sim\Gamma^{-1/2},
\end{equation}
which only depends on the dimensionless ratio
$\Gamma=\frac{(\sqrt{\beta}D_i+D_v)}{\sigma Fa^2}$. The crossover from
Regime I to Regime II occurs at time $t_{\text{I/II}}=F\sigma
t_{\text{I/II}} \simeq \Gamma^{-1/2}$, i.e., $t_{\text{I/II}}\sim
n_i^{-1}(D_i/a^2)^{-1}$. Note that this scaling regime is bound from
above by the following condition: if $n_v$ becomes so large that the
time between interstitial-vacancy encounters is on average shorter
than the time between two subsequent rotations (i.e., if
$\Gamma<\beta^{-1}$), the scaling of the encounter time becomes that
of a 1D random walk. In this case, the scaling of the steady state
defect density with $\Gamma$ changes from $n_i^{\rm II}=n_v^{\rm
II}\sim\Gamma^{-1/2}$ to $n_i^{\rm II}=n_v^{\rm II}\sim
\Gamma^{-1/3}$.  This regime is unlikely to be of experimental
relevance.

Eventually, loss through sinks becomes important and the steady state
Regime II ends. Clearly, there exists a terminal steady state (see
below) in which all loss terms in Eqs.~(\ref{rate-eq}) balance the
creation of defects. If $D_i\gg D_v$ as in the bcc metal case,
however, the system will initially loose mostly interstitials and very
few vacancies. This breaks the symmetry between interstitials and
vacancies and $n_v\gg n_i$.  The second steady state is, therefore,
reached through a transient Regime III with distinct
scaling. Subtracting the two rate equations (and neglecting vacancy
loss at sinks),
\begin{equation}
\frac{d(n_v -n_i)}{dt}\simeq\frac{dn_v}{dt}=\kappa_{is}n_s\sqrt{\beta}\frac{D_i}{a^2}n_i,
\end{equation}
where $n_s$ denotes the sink density. This equation allows us to
eliminate $n_i$ in the rate equation for $n_v$, so that
\begin{equation}
\frac{dn_v}{dt}\simeq\sigma F -\frac{\kappa_vn_v}{\kappa_{is}n_s}\frac{dn_v}{dt}.
\label{regIII-eq}
\end{equation}
Integrating Eq.~(\ref{regIII-eq}), we
obtain (to leading order) power-law growth of $n_v$ with time,
\begin{equation}
n_v^{\rm III}\simeq (\kappa_sn_{is}\sigma F t/\kappa_v)^{1/2}.
\end{equation}
In the case of extremely large sink densities $n_s$, the encounter
rates $\omega_{is}$ between interstitials and sinks would not be 3D as
assumed above, but 1D-like. In this case, $n_s$ would have to be
replaced by $n_s^2$, but again, such high densities are unrealistic.
The crossover time $t_{\text{II/III}}$ between Regimes II and III can
be obtained from the condition $\Gamma\simeq \kappa_{is}n_s\sigma F
t_{\text{II/III}}/\kappa_v$, which implies $t_{\text{II/III}}\sim
(\kappa_v/\kappa_{is}n_s)^{1/2}/(D_i/a^2)$. The crossover occurs at
constant time, independent of the defect densities.

If there are only sinks for one species of defect and no sinks for the
other, Regime III will simply continue. In all other cases, a final
steady state will occur when all loss terms are important. Setting
again $\frac{dn_i}{dt}=\frac{dn_v}{dt}=0$ as in
Ref.~\onlinecite{sizmann74}, we obtain a condition for steady state,
\begin{equation}
\label{regIV-1}
\frac{n_v^{\rm IV}}{n_i^{\rm IV}}=\frac{\sqrt{\beta}D_i}{D_v}=\frac{\kappa_{is}}{\kappa_{vs}}\alpha\sqrt{\beta},
\end{equation}
where we have introduced a third dimensionless ratio $\alpha=D_i/D_v$.
The interstitial and vacancy populations will, in general, be
different.  The steady state values are the roots of the quadratic
equation
\begin{eqnarray}
\label{regIV-2}
(n_v^{\rm IV})^2 \frac{\kappa_{vs}\Gamma}{\kappa_{is}\alpha\sqrt{\beta}}-n_v^{\rm IV}\frac{\kappa_{vs}n_sD_v}{\sigma Fa^2}=1.
\end{eqnarray}
Since typically $n_s\ll 1$, the linear term can be neglected relative
to the quadratic term, and we obtain the scaling of the vacancy
density in Regime IV,
\begin{equation}
n_v^{\rm IV}\sim \left(\frac{\Gamma}{\alpha\sqrt{\beta}}\right)^{-1/2}.
\end{equation}
The crossover time $t_{\text{III/IV}}$ follows again from the
condition $\kappa_{is}n_s\sigma F t_{\text{III/IV}}/\kappa_v\simeq
(\Gamma/\alpha\sqrt{\beta})^{-1}$, i.e.,~$t_{\text{III/IV}}\sim
(\alpha\sqrt{\beta}\kappa_v/\kappa_{is}n_s)^{1/2}/(D_i/a^2)$ is again
independent of $n_i$ or $n_v$.

The volume swelling is proportional to the excess vacancy population
in this regime, $S=n_v-n_i$. Note that the imbalance arises here due
to the very different diffusivities of the two types of defects. Upon
termination of the irradiation, both the $n_i$- and $n_v$-populations
would relax exponentially to zero in this simple model.

\subsection{Numerical integration and kMC}
A full solution of the rate equations (\ref{rate-eq}) is only possible
numerically. In the following, we show a series of such numerical and
kinetic Monte Carlo simulation results for a representative choice of
parameters $D_i=1000 D_v$, $\gamma_r=0.01 D_i/a^2$, so that
$\alpha=1000$ and $\beta=0.01$ (see Section \ref{real-sec} for typical
experimental regimes). In addition, we set the density of sinks to
$n_s=10^{-4}$. Figure~\ref{time-fig} shows the evolution of the
average interstitial and vacancy density (symbols) based on the above
parameters obtained from the kMC simulation and numerical integration
of the corresponding rate equations (solid lines) using the expression
Eq.~(\ref{om-mixed-eq}) for the mixed encounter rates. The
dimensionless parameter $\Gamma$ was varied by changing the defect
creation rate $F\sigma$. As predicted by the scaling analysis, four
distinct regimes appear with increasing time. After first rising
linearly with time (Regime I), $n_i$ and $n_v$ reach the steady state
plateau of Regime II. Once loss through sinks becomes important
(Regime III), $n_v$ increases as $t^{1/2}$ while $n_i$
decreases. Finally, all curves reach the steady-state Regime IV, where
$n_i$ and $n_v$ are given by Eqs.~(\ref{regIV-1}) and
(\ref{regIV-2}). At the highest defect densities, the kMC simulations
where not carried out into the final steady state regimes, because the
computational effort becomes very large.

\begin{figure}[t]
\includegraphics[width=8cm]{fig1finsm}
\caption{\label{time-fig}Self-interstitial ($\blacksquare$), $n_i$,
and vacancy ($\lozenge$), $n_v$, densities as a function of time for
$\Gamma=10^2, 10^3, 10^4, 10^5,$ and $10^6$ ($\Gamma$ increases from
the top to bottom of the figure). Time is measured in units of the
inverse interstitial hopping rate $a^2/D_i$. The solid lines show the
result of direct numerical integration of Eqs.~(\ref{rate-eq}) with
$\kappa_v=\kappa_i=\kappa_{is}=\kappa_{vs}=21$, and the symbols correspond to the
results of the kMC simulations in a periodic simulation box. The
straight solid lines have slope $1$ and $1/2$.  }
\end{figure}

Excellent agreement between continuum theory and kMC model for the 3
largest values of $\Gamma$ is achieved by adjusting all capture
numbers to a single numerical constant. In the simple situation
examined here, kMC and rate theory are completely equivalent. Since we
have used $\omega_{\rm 1D/3D}$ in the rate equations, this agreement
also validates the scaling argument for the mixed 1D/3D encounter
rate.  For the two smallest $\Gamma$-values, we observe increasing
discrepancies between rate equations and kMC. As discussed above, we
expect a transition from the mixed 1D/3D encounter rates $\omega_{iv}$
to 1D dominated encounter kinetics when the interstitial density
becomes so high that they typically collide with vacancies or sinks
before rotating ($\Gamma<100$). This transition is not faithfully
reproduced by the rate theory which assumes only the limiting cases of
either 1D or mixed 1D/3D encounter rate scaling, but is properly
captured by the kMC which includes explicit 1D/3D trajectories.

Note also that with increasing $\Gamma$, Regime II begins to shrink
and Regime I crosses over directly into Regime III.  This happens
because as the defect densities decrease below the sink density, point
defect loss at sinks will become dominant before recombination plays a
significant role. Since the crossover time $t_{\text{I/III}}\simeq
\kappa_sn_s/\kappa_v\sigma F$ only depends on the point defect
production rate, a further decrease of the production rate will
eventually lead to a direct crossover from Regime I to Regime IV.  A
smaller sink density would push the appearance of Regimes II and III
to higher values of $\Gamma$.

\section{Vacancy cluster formation}
\label{vaccluster-sec}
\subsection{Irreversible aggregation}
To this point, neglected interactions between vacancies. Experimental
vacancy-vacancy binding energies $E_b$ can be of order the single
vacancy migration energy.  This suggests that it is not unreasonable
to expect vacancies to form stable clusters or microvoids once they
meet. A relatively simple approximation for this situation is to
consider the limiting case of ``irreversible aggregation'', which
neglects any dissociation of a vacancy from a cluster and thus sets
$E_b=\infty$. The realistic situation, which includes finite binding
energies, can be viewed as an intermediate case between irreversible
aggregation and the case of $E_b=0$ of the previous section.

In the ``irreversible aggregation'' model, vacancies bind to form
stable divacancies, leaving the population of free vacancies, while
interstitials recombining with a divacancy recreate a single mobile
vacancy. Stable cluster grow or shrink under the influence of arriving
vacancies and interstitials. Denoting the number density of clusters
of size $m$ as $n_c(m)$, we can write the following set of rate
equations\cite{ghoniem89}:
\begin{eqnarray}\nonumber
\frac{dn_i}{dt}&=&\sigma
F-\kappa_i\omega_{vi} n_v\\\nonumber
&-&(\kappa_v\omega_{iv}+\kappa_{is}\omega_{is}+\sum_m\kappa_i^m\omega_{ic(m)})n_i\\\nonumber
\frac{dn_v}{dt}&=&\sigma
F-(\kappa_i\omega_{vi}+\kappa_{vs}\omega_{vs}+\sum_m\kappa_v^m\omega_{vc(m)}\\\nonumber
&-&2 \kappa_v \omega_{vv})n_v-(\kappa_v\omega_{iv}-\kappa_i^2\omega_{is(2)})n_i\\\nonumber
\frac{dn_c(m)}{dt}&=&(\kappa_v^{m-1}\omega_{vc(m-1)}-\kappa_v^{m}\omega_{vc(m)})n_v\\
&+&(\kappa_i^{m+1}\omega_{ic(m+1)}-\kappa_i^{m}\omega_{ic(m)})n_i ,
\label{rate-irr-eq}
\end{eqnarray}
where $n_v\equiv n_c(1)$ is now understood to refer to the free
vacancy density (or cluster of size 1). The additional terms in the
equations for the evolution of $n_i$ and $n_v$ account for
interstitial/vacancy-cluster encounters, vacancy/vacancy-cluster
encounters, vacancy-cluster nucleation and divacancy
decomposition. $\omega_{ic(m)}= n_c(m)\sqrt{\beta}D_i/a^2$,
$\omega_{vc(m)}= n_c(m)D_v/a^2$ and $\omega_{vv}=n_vD_v/a^2$ denote
the corresponding encounter rates. The last expression in
(\ref{rate-irr-eq}) describes the evolution of stable, immobile
vacancy clusters of size $m>1$ due to the diffusive arrival of
interstitials and vacancies. The hierarchy of rate equations
(\ref{rate-irr-eq}) is, in principle, amenable to a semi-analytical
treatment via numerical integration, if all constants $\kappa^m_{i/v}$ are
specified. However, such a solution requires termination of the set of
equations at a finite cluster size $m_{max}$, i.e., one imposes a
boundary condition $n_c(m_{max})=0$. The corresponding kMC model does
not require any ad hoc assumptions and takes all these processes
naturally into account.

Before examining this model in kMC, we can draw some premliminary
conclusions. In Regime I, only nucleation of divacancies will be
important. Since here the vacancy density grows linearly with time, we
predict the scaling regime
\begin{equation}
n_c(2)^I\sim(\sigma Ft)^3.
\end{equation}
The divacancy density grows cubically with time. In the steady-state
Regime II, the vacancy cluster density is slaved to the free vacancy
density and, therefore, is constant as well. For $\alpha\gg1$, we
expect the total cluster density $n_c=\sum_m n_c(m)$ to be much
smaller than $n_v$ and $n_i$, so that the presence of vacancy
clustering does not yet strongly alter the interstitial and vacancy
population. Using Eq.~(\ref{regII-eq}), we find
\begin{equation}
n_c^{II}\simeq \Gamma^{-1/2}/(\alpha\sqrt{\beta}).
\end{equation}
Once past Regime II, the cluster density will begin to rise again as
$n_v$ increases. $n_v$ will then increasingly fall below the total
number of vacancies in the system $n_{vtot}$, because increasing
numbers of vacancies $\sum_{m>1} mn_c(m)$ become immobilized in
clusters. These clusters act as additional sinks for the diffusing
vacancies, but do not remove them from the system. The higher the
cluster density, the smaller the loss of vacancies through sinks and
interstitial recombination. We therefore expect that Regime III will
extend to larger times and the presence of the immobile vacancy
cluster delays the onset of the final steady-state Regime IV.

For the total number of vacancies $n_{vtot}$, we can write the rate
equations as
\begin{eqnarray}\label{nvtot-scaling-eq}
\frac{dn_{vtot}}{dt}&=&\sigma
F-\kappa_i\omega_{vi}n_v-\sum_{m=1}\kappa_i^{m}\omega_{ic(m)}n_i\\
\nonumber
\frac{dn_i}{dt}&=&\sigma
F-\kappa_i\omega_{vi} n_v-(\sum_{m=1}\kappa_i^m\omega_{ic(m)}-\kappa_{is}\omega_{is})n_i,
\end{eqnarray}
where we have neglected vacancy loss at sinks. This is the Regime III
situation discussed earlier. Subtraction of
Eqs.~(\ref{nvtot-scaling-eq}) and elimination of $n_i$, as in
Eq.~(\ref{regIII-eq}), leads to
\begin{equation}
\frac{dn_{vtot}}{dt}\simeq \sigma F -
\frac{\kappa_v}{\kappa_{is}n_s}\frac{dn_{vtot}}{dt}\sum_{m=1}\frac{\kappa_i^m}{\kappa_v}n_c(m).
\end{equation}
At relatively early times, when only small clusters are present,
$\sum_{m=1}\frac{\kappa_m}{\kappa_v}n_c(m)\simeq n_{vtot}$ and
$n_{vtot}$ will initially increase as $n_{vtot}\sim (\kappa_{is}n_s\sigma
F t)^{1/2}$. The subsequent behavior depends on the detailed form of
the capture numbers $\kappa_{is}$.

In the absence of vacancy clustering, Regime III ends when the vacancy
density has become so large that vacancy loss through sinks balances
interstitial loss through sinks. This condition, Eq.~(\ref{regIV-1}),
must hold for a final steady state to appear. In the present simple
model, this condition depends only on the ratio of the diffusivities,
but in a real system, Eq.~(\ref{regIV-1}) also depends on sink
concentrations and capture numbers, which may be different for
interstitial and vacancies. Eventually this condition can also occur
as a result of vacancy clustering, because vacancy clusters act as
additional sinks for the interstitials and remove them symetrically
from the system. At high cluster densities, the low mobile vacancy
concentration can therefore be compensated. In the steady-state regime
IV, we expect the hierarchy
$n_i^{IV}<n_v^{IV}<n_c^{IV}<n_{vtot}^{IV}$. Because of the later entry
into Regime IV, $S=n_{vtot}-n_i$ (the total amount of volume swelling)
will be larger than in the absence of clustering.
\begin{figure}[t]
\includegraphics[width=8cm]{fig2finsm}
\caption{\label{cluster-fig}Interstitial densities $n_i$ (
$\blacksquare$), free vacancy density $n_v$ ($\lozenge$) as well
as the total vacancy density $n_{vtot}$ ($\star$) and vacany cluster
density $n_{c}$ ($\blacktriangle$) as a function of time for (a)
$\Gamma=10^5$, (b) $\Gamma=10^7,$ and (c) $\Gamma=10^{9}$.  The thin
solid lines show the result of direct numerical integration of
Eqs.~(\ref{rate-irr-eq}) with $\kappa^m_i=\kappa_v^m=25$ and a $m_{max}=20$ (see
text).  }
\end{figure}

In Regime IV, we can obtain an approximate scaling relation for the
total cluster density. The density is determined by the competition
between nucleation of clusters out of the vacancy gas and
decomposition of divacancies due to arriving interstitials. Denoting
the fraction of clusters that represent divacancies as $f$, we can
write the rate equation
\begin{equation}
\frac{dnc^{{\text
IV}}}{dt}=\kappa_vn_v^2D_v/a^2-f(t)\kappa_i^2n_c^{\text
IV}n_i\sqrt{\beta}D_i/a^2,
\end{equation}
which implies $n_c^{\text
IV}=\kappa_vn_v^2/f\kappa^2_i\alpha\sqrt{\beta}n_i=\kappa_vn_v/f\kappa_2$,
where we have used Eq.~(\ref{regIV-1}). The cluster density is
therefore of order the free vacancy density with a numerical prefactor
that depends on the mean cluster size.

\subsubsection{Average cluster density}
The effect of irreversible vacancy aggregation on the defect dynamics
as seen through kMC is shown in Fig.~\ref{cluster-fig} for three
values of $\Gamma=10^5$, $\Gamma=10^7$ and $\Gamma=10^{9}$ with all
other parameters as before. In this figure, we also plot the total
cluster density $n_c$ and the total vacancy density $n_{vtot}$. In
Regimes I and II, $n_c$ and $n_i$ are nearly unchanged. The defect
density $n_c$ rises first $\sim t^3$ and then remains constant during
Regime II as discussed above. In Regime III, $n_v$ begins to fall
below the result from Fig.~\ref{time-fig} because of trapping of
vacancies into clusters. At the same time, $n_{vtot}$ rises with an
ideal $t^{1/2}$ law, and Regime III extends to larger times. $n_c$
also rises again due to new cluster nucleation events out of the
vacancy gas. All quantities reach constant values in regime IV. As one
might expect, the total number of vacancies in the system is higher
than in the absence of clustering. However, the general sequence of
regimes remains unchanged.

In order to gain additional insight into the growth dynamics, we also
numerically integrate the full set of rate Eqs.~(\ref{rate-irr-eq})
(solid lines in Fig.~\ref{cluster-fig}). This requires specification
of the capture numbers $\kappa^m$. In the simplest mean field model
for nucleation dynamics, the capture numbers do not depend on cluster
size $m$ and $\kappa^1_{i,v}=\kappa^m_{i,v}=const.$ (i.e., the point cluster
model). Interestingly, we find surprisingly good agreement between the
rate equations (solid lines in Fig.~\ref{cluster-fig}) and the full
kMC simulations when using this simple approximation.  This suggests
that for relatively small $m$, $\kappa^m$ is only very weakly
size-dependent. The rate equations faithfully reproduce the sequence
of regimes, but begin to show deviations at late times when larger
clusters become more prominent and the point cluster assumption is no
longer accurate. Here, $\kappa^m$ begins to show some size
dependence. The success of this comparison shows, however, that a
mapping between kMC and rate equations is possible even with void
nucleation and growth if the $\kappa^m$'s are accurately parametrized.

Note that in the present model, the volume swelling rate $dS/dt$ is
zero in Regime IV. This is possible because we have chosen the same
capture radii $\kappa_s$ for the sinks for both interstitials and
vacancies, i.e., the sinks are symmetric with respect to defect
type. $S\sim t$ would be valid, if, for example, the capture radius
for interstitials is larger than that for vacancies. This situation
can arise with the introduction of dislocation sinks (known
as``dislocation bias'') and has been invoked to explain unusually
large swelling rates (see also Section \ref{conc-sec}) at times later
than considered here \cite{woo96,singh97}.

\subsubsection{Cluster size distribution}
A full description of the evolution of the point defects in the
material includes a characterization of the cluster size
distribution. The growth of these clusters is the result of the
diffusive arrival of interstitial and vacancy defects at already
nucleated clusters. The full dynamics of this process is described by
the last equation in (\ref{rate-irr-eq}). Consider this equation in
the following simplified notation:
\begin{eqnarray}\nonumber
\frac{dn_c(m)}{dt}&=&n_v(\kappa_v^{m-1}n_c(m-1)-\kappa_v^m n_c(m))D_v/a^2\\
&+&n_i(\kappa_i^{m+1} n_c(m+1)-\kappa_i^{m} n_c(m))\sqrt{\beta}D_i/a^2.
\label{clustsize-eq1}
\end{eqnarray}
If $n_i$ and $n_v$ change very slowly in time, the distribution
$n_c(m)$ is given by the steady state solution to
Eq.~(\ref{clustsize-eq1}), i.e., $dn_c(m)/dt=0$. This equation is
supplemented by the detailed balance condition
\begin{equation}
n_v\kappa_v^m n_c(m)D_v/a^2=n_i \kappa_i^{m+1}n_c(m+1)\sqrt{\beta}D_i/a^2,
\end{equation}
which should also hold for $m\ge 2$. From this condition, we can
deduce the distribution $n_c(m)$ by induction. Since $n_v\equiv
n_c(1)$, we have $n_c(2)=n_v^2\kappa_1/(\kappa^2_i\alpha\sqrt{\beta}
n_i)$ and consequently for all $m>1$
\begin{equation}
n_c(m)=\frac{n_v^m}{(\alpha\sqrt{\beta} n_i)^{m-1}}\frac{\prod_{l=1}^{m-1}\kappa_v^l}{\prod_{l=2}^m\kappa_i^l}.
\label{ncdist-eq}
\end{equation}

Equation~(\ref{ncdist-eq}) is tested against the kMC results in
Fig.~\ref{clusterdist-fig}, where we show the cluster size
distributions at four different times. All curves fall along straight
lines in a semilogarithmic plot, and the slope decreases with
increasing mean cluster size. A comparison with Eq.~(\ref{ncdist-eq})
requires, again, knowledge of the capture numbers $\kappa^m$. As in
the previous section, use of the point cluster approximation
$\kappa^l_{i,v}/\kappa^1_{i,v}=1$ yields excellent agreement
between between the kMC data and Eq.~(\ref{ncdist-eq}) upon inserting
the values for $n_i(t)$ and $n_v(t)$ at the appropriate times.

\begin{figure}[t]
\includegraphics[width=8cm]{fig3finsm}
\caption{\label{clusterdist-fig} Plot of the cluster size distribution
$n_{c}(m)$ found from the kMC simulations with $\Gamma=10^5$ of
Fig.~\ref{cluster-fig}(a) at four different times $tD_i/a^2=3\times
10^4$ $(\blacksquare)$, $3\times 10^5$ $(\blacklozenge)$, $3\times
10^6$ $(\star)$, and $3\times 10^7$ $(\blacktriangle)$ in Regimes III
and IV. The straight lines correspond to
$n_v(t)^m/(\alpha\sqrt{\beta}n_i(t))^{m-1}$.}
\end{figure}

Note that the distributions shown in Fig.~\ref{clusterdist-fig} are
not peaked, i.e., the frequency with which different clusters appear
decrease with increasing cluster size and single vacancies occur most
frequently. This situation is in sharp contrast to other cluster
growth situations as, e.g., found in submonolayer island growth during
vapor deposition \cite{amar95}, where the distribution $n_c(m)$ peaks
at the mean cluster size $\langle m \rangle$. In this problem,
however, clusters cannot shrink, since vacancies are absent. One
expects Eq.~(\ref{ncdist-eq}) to hold as long as the assumption of
quasi-stationary values for $n_i$ and $n_v$ is valid.
\begin{figure}[t]
\includegraphics[width=8cm]{fig4afinsm}
\includegraphics[width=8cm]{fig4bfinsm}
\caption{\label{clusterrev-fig}(a) Interstitial density $n_i$
($\blacksquare$), free vacancy density $n_v$ ( $\lozenge$) as
well as the total vacancy density ($(\blacktriangle)$) and vacancy
cluster density $n_{c}$ $(\star)$ as a function of time for
$\Gamma=10^5$, and $10^{7}$ for reversible aggregation, where
$\delta=0.01$. (b) Cluster size distribution $n_{c}(m)$ for the case
of $\Gamma=10^5$ at four different times in Regimes III and IV. The
straight lines correspond to
$n_v(t)^m/(\alpha\sqrt{\beta}n_i(t)+\kappa_d^1/\kappa_i^1\alpha^\prime)^{m-1}$
with $\kappa_d^1/\kappa_i^1=0.25$.}
\end{figure}

\subsection{Reversible attachment}
The above discussion immediately raises the question of whether the
results for $n_c(t)$ and its size distribution Eq.~(\ref{ncdist-eq})
survive in the more realistic situation of reversible cluster growth,
i.e., vacancies have a finite probability to attach to and to leave
the cluster. Let us introduce ``detachment rates'' from a cluster of
size $m$, $\gamma_{det}(m)/a^2$ for all vacancies, independent of
their local environment. Equation~(\ref{clustsize-eq1}) needs to be
generalized by the addition of two terms,
\begin{eqnarray}\nonumber
\frac{dn_c(m)}{dt}&=&n_v(\kappa_v^{m-1}n_c(m-1)-\kappa_v^mn_c(m))D_v/a^2\\
&+&n_i(\kappa_i^{m+1}n_c(m+1)-\kappa_i^mn_c(m))\sqrt{\beta}D_i/a^2,\\
&+&(\kappa^{m+1}_d n_c(m+1)-\kappa^{m}_dn_c(m))\gamma_{det}(m)/a^2,
\label{clustsize-rev-eq}
\end{eqnarray}
where $\kappa^m_d$ represent ``detachment numbers'' in analogy to the
capture numbers $\kappa^m_{i,v}$. Consequently, the condition for detailed
balance now reads
\begin{eqnarray}\nonumber
\kappa_v^{m}n_vn_c(m)D_v/a^2&=&\kappa_i^{m+1}n_in_c(m+1)\sqrt{\beta}D_i/a^2\\
&+&\kappa^{m+1}_dn_c(m+1)\gamma_{det}(m)/a^2
\end{eqnarray}
and Eq.~(\ref{ncdist-eq}) generalizes to
\begin{equation}
n_c(m)=\frac{\prod_{l=1}^{m-1}\kappa_v^l
n_v^m}{\prod_{l=2}^{m}(\kappa_i^l\alpha\sqrt{\beta}
n_i+\kappa_l^d\alpha^\prime(m))},
\label{ncdist-rev-geneq}
\end{equation}
where $\alpha^{\prime}(m)=\gamma_{det}(m)a^2/D_v$. We see that the
general form of the distribution is the same, but the prefactor
changes due to the additional growth and shrinkage probabilities.
From a statistical point of view, interstitial arrival and vacancy
detachment are equivalent. Eq.~(\ref{ncdist-rev-geneq}) can be easily
evaluated for any functional form of the size-depedent capture numbers
and detachment rates.  In the point cluster approximation, where all
$\kappa^m_{i,v,d}/\kappa^1_{i,v,d}=1$ and
$\alpha^\prime=\alpha^{\prime}(m)$ is size-independent, it predicts an
exponential distribution as in the case of irreversible attachment,
\begin{equation}
n_c(m)=\frac{n_v^m}{(\alpha\sqrt{\beta} n_i+\alpha^\prime\kappa^1_d/\kappa_i^1)^{m-1}}.
\label{ncdist-rev-eq}
\end{equation}

Figure~\ref{clusterrev-fig} presents a numerical investigation of
reversible vacancy cluster growth using kMC. Although detachment rates
may be cluster size dependent in general, we only introduce one
size-independent rate $\gamma_{det}/a^2=0.01D_{v}/a^2$ for vacancy
detachment for simplicity. This model would be most relevant for
faceted cluster shapes with one dominant detachment rate from the
faces. The other parameters are that of Fig.~\ref{cluster-fig}. We see
in Fig.~\ref{clusterrev-fig}(a) that the cluster density behaves in a
qualitatively similar manner as in the irreversible case, but $n_c$ is
reduced and the final steady state is reached at earlier times. As in
the irreversible case, the cluster size distribution $n_c(m)$ shown in
Fig.~\ref{clusterrev-fig}(b) is well described by
Eq.~(\ref{ncdist-rev-eq}). The present discussion is of course only
relevant when void coarsening can be neglected.

\begin{figure}[t]
\includegraphics[width=8cm]{fig5finsm}
\caption{\label{intclustimm-fig} Interstitial densities $n_i$ (
$\blacksquare$), free vacancy density $n_v$ ($\lozenge$), total
vacancy density $n_{vtot}$ ($\blacktriangle$), vacany cluster density
$n_{c}$ ($\star$) and density of immobile interstitial cluster
$(\square)$ as a function of time for $\Gamma=10^6$.  }
\end{figure}

\section{Interstitial interactions}
\label{intclust-sec}
One of the fascinating aspects of point defect dynamics in bcc metals
is the high mobility of interstitial clusters \cite{osetsky01}. The
interaction between single interstitial atoms is even stronger than
that between vacancies, and the interstitial clusters, such as
dislocation loops \cite{soneda98}, are stable at all relevant
temperatures.  Unlike the immobile vacancy cluster, however, the
interstitial cluster migrate easily for clusters with up to 50 or 100
interstitial atoms \cite{soneda01,soneda03}. We now modify the
preceding analysis to account for this behavior.

In this analysis, we return to the irreversible aggregation limit,
i.e., interstitials never separate after encounter. This implies a
reduction of the density of mobile interstitials due to nucleation of
interstitial clusters, i.e., interstitials act as sinks for other
interstitials. However, this creates a population of larger clusters
with a larger crosssection and eventually larger capture numbers. The
reaction kinetics will be affected by the competition of these two
effects.

The rate equations (\ref{rate-irr-eq}) can now be expanded to include
terms representing the above processes, but become even more complex.
In our kMC model, we begin by studying the effect of interstitial
cluster formation by considering completely immobile interstitial
clusters in complete analogy to the vacancy cluster. This situation is
rarely realistic, but provides an upper bound on the magnitude of the
effects. Figure~\ref{intclustimm-fig} shows the evolution of the free
interstitial and vacancy densities as well as the interstitial and
vacancy cluster densities for the same parameters as before. As
expected, the immoblization of interstitials increases the total
number of vacancies in the system. In the final steady state,
$n_{vtot}$ is about three times as large as in the situation in
Fig.~\ref{cluster-fig}. The interstitial clusters (open squares)
nucleate earlier than the vacancy cluster, but their density later
drops below that of the vacancy clusters, since the single
interstitial density is much lower than the single vacancy density.

The situation in Fig.~\ref{intclustimm-fig} can be favorably
contrasted with that of completely mobile interstitial clusters, i.e.,
the cluster diffuse with the same rates as the single interstitials
regardless of their size. This case is again not fully realistic,
since interstitial clusters seldom rotate into other $\langle 111
\rangle$ directions once they contain several interstitials (i.e.,
they diffusive one-dimensionally \cite{marian01,soneda01}). For small
clusters, however, the completely mobile interstitial cluster case
represents a good approximation. Figure~\ref{intclustmob-fig} shows
the corresponding results for the various desities introduced
above. As in the case of immobile interstitial clusters, the mobile
interstitial cluster density first rises due to nucleation
events. However, the highly mobile interstitial clusters also collide
with the sinks, and the cluster density decreases rapidly. Since the
free interstitial density also declines, nucleation events become
rare. Once steady state has been achieved, the interstitial density
has become so low that the nucleation of new interstitial clusters due
to diffusion is almost completely absent. Consequently, the vacancy
densities and the total swelling rates are the same as for the case of
non-interacting interstitials.

\begin{figure}[t]
\includegraphics[width=8cm]{fig6finsm}
\caption{\label{intclustmob-fig}Interstitial, vacancy, interstitial
cluster and vacancy cluster densities for $\Gamma=10^5$ and
$\Gamma=10^7$ (symbols as in Fig.~\ref{intclustimm-fig}). Interstitial
clusters $(\square)$ diffuse and rotate with the same rate as single
interstitials. The interstitial cluster density first rises due to
nucleation events, but then rapidly drops as mobile cluster collide
with sinks. }
\end{figure}

\section{Application to vanadium and iron}
\label{real-sec}
In this section, we apply our model to the V and $\alpha-$Fe
cases. Vanadium is particularly interesting, because here the effect
of mixed 1D/3D diffusion is most pronounced. First principles
calculations and classical MD simulations of V have yielded estimates
of $\Delta E_i=0.018$eV\cite{zepeda04}, $\Delta E_r=0.44$eV
\cite{zepeda04} and $\Delta E_v\approx 0.5$eV \cite{han1}.  For Fe,
the different ground state of the self-interstitial ($\langle
110\rangle$ instead of $\langle 111\rangle$) leads to a higher
effective activation barrier for 1D migration, $\Delta E_i$, which
ranges between 0.12 eV \cite{marian01} and 0.17 eV
\cite{soneda01}. The rotation barrier was estimated as $\Delta
E_r=0.16$ eV \cite{soneda01}, and the vacancy migration energy is
assumed to be of the same order \cite{soneda03} as in V (the
prefactors for all processes tend to vary by less than an order of
magnitude). One therefore expects that the self-interstitial
trajectories in $\alpha-$Fe will be much more isotropic than in V.

\begin{table}[b]
\begin{center}
\caption{\label{dimless-table}Dimensionless parameters
$\alpha=D_i/D_v$, $\beta=\gamma_r/D_i$,
$\Gamma=(\sqrt{\beta}D_i+D_v)/\sigma F$, and
$\alpha^\prime=\alpha\sqrt{\beta}$ for V and Fe at 300 K and 600 K.}
\begin{tabular}{|c|c|c|c|c|}
\hline
 & V-300K & V-600K & Fe-300K & Fe-600K\\
\hline
$\alpha$ & $10^{8}$ & $10^4$   & $10^{6}$  & $10^{3}$ \\
$\beta$ & $10^{-8}$ & $10^{-4}$ & $10^{0}$  & $10^{0}$  \\
$\Gamma$ & $10^{11}$ & $10^{13}$ & $10^{11}$  & $10^{13}$  \\
$\alpha^\prime$ & $10^4$ & $10^2$ & $10^{6}$ & $10^{3}$  \\
\hline
\end{tabular}
\end{center}
\end{table}

The two metals are best compared in terms of the relevant
dimensionless parameter $\alpha$, $\beta$ and $\Gamma$.  Table
\ref{dimless-table} summarizes the values for these quantities using
the above energy barriers and the production rate for ion irradiation
$\sigma F=10^{-3}s^{-1}$ for two representative temperatures $T=300$K
and $T=600$K. We first note that $\Gamma$ is typically larger than
$10^{10}$ and would in fact reach $10^{20}$ when typical production
rates for neutron irradiation are used.  A kMC simulation with such
large values of $\Gamma$ would require very long simulation times,
since diffusion and recombination occur much more often than the
introduction of new defects. It also requires large system sizes,
because the defect densities become very small. In addition, the very
small value of $\beta$ implies very long 1D segments of the
interstitial trajectories between rotations. The period of the kMC
simulation box should be several times larger than the typical length
of those segments in order to properly reproduce the continuum theory
values of the encounter rates. If the box period is much smaller than
the 1D segment, the trajectory will wrap around the box many times,
but is not necessary space-filling.

The last of these issues can be addressed by using a result from
Section \ref{simple-sec}, where we showed that the mixed 1D/3D
encounter rates scale like the ideal 3D rates that are reduced by a
factor $\sqrt{\beta}$. We can therefore replace the explicit 1D/3D
trajectories of the interstitials and interstitial cluster with ideal
3D random walks, but reduce the hopping rate by a factor of
$\sqrt{\beta}$. This procedure leaves the reaction kinetics invariant
(up to small corrections from the capture numbers) and implies that
the effective ratio of time scales between interstitial and vacancy
migration is given by
$\alpha^\prime=D_i\sqrt{\beta}/D_v=\sqrt{D_i\gamma_r/a^2}$. Interestingly,
this ratio is very similar for both V and Fe at 300K and 600K (see
Table \ref{dimless-table}), even though the values of $\alpha$ and
$\beta$ are very different. At a given temperature, the
self-interstitial trajectories in $\alpha$-Fe are much more isotropic
than in V, but the encounter rates with vacancies and sinks only
depend on the product of diffusivity $D_i$ and rotation rate
$\gamma_r$.  Within the present model, one therefore expects that the
point defect reaction kinetics in these two metals is very similar.

\begin{figure}[t]
\includegraphics[width=7.7cm]{fig7finsm}
\caption{\label{van-fig}Interstitial, vacancy, interstitial cluster
and vacancy cluster densities (symbols as in
Fig.~\ref{intclustimm-fig}) with parameters $\alpha^\prime=10^2$,
$n_s=10^{-5}$, for (a) $\Gamma=10^9$ (b) $\Gamma=10^{11}$ and (c)
$\Gamma=10^{13}$ representative for V or $\alpha$-Fe at 600K. Thin
solid lines show the result of numerically integrating
Eqs.~(\ref{rate-irr-eq}) again for $\kappa_m=25$ and $m_{max}=20$. The
solid lines have slope 1 and 1/2, respectively.}
\end{figure}

In Fig.~\ref{van-fig}, we use a ratio of time scales representative
for V/$\alpha$-Fe at $T=600$K and show results for $\Gamma=10^9$,
$\Gamma=10^{11}$ and $\Gamma=10^{13}$.  Here, we have multiplied the
time axis with the production rate, which is a more convential
presentation of the data in experimental studies.  Rescaled by the
production rate, all curves initially coincide and start out with
near-constant slopes. For the present parameters, Regime II is
practically absent, and the vacancy density crosses over immediately
from Regime I (where it rises linearly with dose) into Regime III. The
crossover dose is roughly constant and depends on the sink
density. Note that the growth of $n_{vtot}$ in Regime III is well
described by a $(\sigma Ft)^{1/2}$ power law over several decades
before the final steady-state Regime IV is reached. Since the onset of
Regime IV is independent of time (see Section \ref{simple-sec}),
larger values of $\Gamma$ push the beginning of steady state to
smaller doses. Here, steady state is reached at about 1 dpa for
$\Gamma=10^9$ and earlier for $\Gamma=10^{11}$ and
$\Gamma=10^{13}$. There is some initial nucleation of interstitial
clusters at $\Gamma=10^9$, but as discussed before, all of these
rapidly disappear due to fast collisions with sinks. For higher values
of $\Gamma$, the interstitial cluster density becomes negligibly
small.

In the same figure, we also show the predictions of a numerical
integration of the full set of rate equations (\ref{rate-irr-eq})
using the point cluster model (size independent capture numbers). As
in Fig.~\ref{cluster-fig}, the agreement with the ``exact'' kMC
simulation is satisfactory. Since the computational effort for direct
kMC simulations for $\Gamma>10^{10}$ becomes tremendous, the rate
equation approach, when properly parametrized, is clearly preferable.

\label{van-sec}

\section{Conclusions}
\label{conc-sec}
We have studied the early-time evolution of point defect populations
with specific reference to migration mechanisms in bcc metals under
constant irradiation conditions. Only very simplified models that
incorporate the most important processes were employed in order to
identify the rate-limiting events. These models were solved using
scaling arguments, direct numerical integration of kinetic rate
equations and full kMC simulations.

In the simplest case, which only considers homogeneous defect
production, recombination and sink absorption, but no interactions
between defects of the same type, the kMC model and the corresponding
rate equations were shown to be in near perfect agreement. We employed
simple random walk arguments to derive a mixed encounter rate that
describes one-dimensional diffusion with occasional rotations. This
encounter rate scales linearly with target density as the isotropic 3D
encounter rate, but is reduced by the square root of the ratio of
rotation rate and hopping rate. As in ref.~\onlinecite{sizmann68},
four distinct scaling regimes for the point defect density with time
where identified. First, the point defect densities increase linearly
in time due to production (Regime I), then saturate as defect
recombination sets in (Regime II). Sink absorption then begins to
reduce the interstitial density and the vacancy density grows in time
with a characteristic $\sim t^{1/2}$ behavior (regime III). A final
steady state (Regime IV) is reached when all loss processes are taken
into account. The full sequence of Regimes I - IV is most visible when
the production rate is not much smaller than the interstitial hopping
rates. Regimes II and III shrink with decreasing production rate, and
in the limit of very small production rates, Regime I crosses over
directly into Regime IV. These results were based on parameters
typical for bcc metals, but the continuum level reaction kinetics
equally applies to fcc metals or other crystal structures provided
they exhibit similar diffusion mechanisms.

The introduction of vacancy reactions to form immobile vacancy
clusters does not change the general sequence of the scaling regimes,
but has a profound effect on the population dynamics. In steady state,
the density of free vacancies and interstitials is reduced relative to
the non-interacting case, but the total number of vacancies $n_{vtot}$
is enhanced. In Regime III, $n_{vtot}\sim t^{1/2}$, and the steady
state Regime IV is reached at later times. Reasonable agreement
between rate equations and kMC could still be achieved in a point
cluster approximation, where the capture numbers are
size-independent. Of particular interest is the size distribution of
vacancy clusters, which grow and shrink under the diffusive arrival of
interstitials and vacancies. For irreversible vacancy aggregation, we
derived a new expression for the cluster size distribution,
$n_c(m)=\kappa_1 n_v^m/\kappa_m(\alpha\sqrt{\beta} n_i)^{m-1}$.  The
general form of this distribution remains when also allowing vacancy
detachment from the cluster.

Immobile interstitial clusters were shown to further increase the
vacancy population. However, interstitial cluster that are as mobile
as single interstitials decay rapidly in relevant parameter
regimes. We therefore expect that nucleation of interstitial clusters
through diffusion plays a negligible role in the microstructural
damage evolution of pure bcc metals. It can become important, however,
if trapping of self-interstitials near impurities occurs.

Because of their importance in applications and the availablitity of
detailed molecular dynamics studies, we applied the model for
parameters suitable for V and $\alpha$-Fe. Within our model, both
metals exhibit similar defect kinetics and therefore similar swelling
rates at a given temperature. For production rates and diffusivities
in typical experimental situations, our calculations revealed a
sequence of growth regimes for the total vacancy density $n_{vtot}$
$\sim t^1$, $\sim t^{1/2}$ and $\sim t^0$. The volume swelling rate
follows the same scaling sequence in the sub-1 dpa
regime. Interestingly, growth of the vacany cluster density as $\sim
t^{1/2}$ over several decades in time has been observed in the much
more detailed kMC simulation of Ref.~\onlinecite{soneda03}, but the
origin of this growth law had not been previously understood. All our
simulations reach the steady state regime at damage levels of less
than 1 dpa. In the experimentally relevant regime of damage levels
between 1 and 100 dpa, it will therefore be more efficient to use
steady state rate equations rather than explicit kMC to predict the
long time point defect distribution evolution.  Nonetheless, it is
important to understand the onset of damage evolution and the
conditions of applicability of the steady state theory.

The fact that experimental swelling rates of V and $\alpha$-Fe are
very different is of course an indication that our present model does
not yet include all relevant phenomena of radiation damage.  One
obvious refinement of the models developed here would be a more
detailed parametrization of the detachment rates (binding energies) of
interstitial and vacancy cluster. However, the intent of the present
study has been to focus on general trends rather than quantitative
comparisons with experiments, and no new physics is expected to appear
from these additional details. For cascade damage conditions
\cite{bullough75}, however, two other processes not included in this
study are known to have a crucial impact on the defect evolution (in
particular void swelling rates).  Frenkel pair production is only
assumed to be homogeneous for energies right above the displacement
threshold, while higher energies lead to the formation of mobile
interstitial and immobile vacancy clusters during the cascade
phase. Our present model is therefore most relevant to low particle
energies. Inclusion of intracascade clustering processes will change
results qualitatively and quantitatively, but requires reliable
information about the cluster size distribution during cascades from
MD simulations.

In addition to this ``production bias'', ``absorption bias'' also
usually exists in bcc metals.  ``Absorption bias'' leads to a
preferred absorption of self-interstitials at sinks and is known to be
an essential driving force for swelling. The origin of this bias,
which leads to increased capture numbers, are long range elastic
interactions between point defects and sinks such as dislocations
\cite{kamiyama94,bullough70} or grain boundaries \cite{samaras03}.
Extentions of the model to include these effects should provide a
fruitful topic for future work. The combination of kMC and rate theory
can then be used to determine which parameters are most important in
radiation damage, so that atomistic simulation resources can be better
focused.

The authors gratefully acknowledge useful discussions with
L.~Zepeda-Ruiz, B.~Wirth, and N.~Ghoniem as well as the support of the
Office of Fusion Energy Sciences (Grant DE-FG02-01ER54628).

redpajama_set_name: RedPajamaArXiv


We explain in detail the preparation for the trial, its setting in northern Ethiopia, the identification and selection of patients (inclusion and exclusion criterion, identifying and screening of patients at home, enrollment of patients at the health centres and health posts), and randomisation.
We describe the challenges met, together with strategies employed to overcome them.
Examples given in the previous section are contextualised and general principles extracted where possible. We conclude that it is possible to conduct a trial that balances approaches that support internal validity (e.g. careful design of proformas, accurate case identification, control over data quality and high retention rates) with those that favour generalisability (e.g. 'real world' setting and low rates of exclusion). Strategies, such as Rapid Ethical Assessment, that increase researchers' understanding of the study setting and inclusion of hard-to-reach participants are likely to have resource and time implications, but are vital in achieving an appropriate balance.


A coach can be the most influential person in a child's life. In AYSO, Positive Coaching is a fundamental philosophy because of the crucial role that a coach has.
Coaching in AYSO is a chance to spend quality time with your own child but you'll also provide an important role model for all the children on your team. In fact, the joy of coaching is so meaningful, that many AYSO coaches continue coaching long after their own children have moved on from AYSO.
Soccer knowledge is a plus, but AYSO provides training and materials so that even a soccer novice can coach effectively by starting in the younger age divisions. Being a coach involves skills you probably already have in addition to specific soccer knowledge.
How does it work at Carlsbad AYSO?
The City of Carlsbad allocates practice field space most weekdays. We will release the practice field space sign-up sheet in late July or early August.
If you are coaching your child, then you add yourself as a volunteer/coach during the process. If you've already signed up your child, then log in with the same username/password and add yourself as a volunteer/coach.
We will have a coaches meeting in early August where you will receive your equipment.
Go to www.carlsbadayso.org, if you do not have an account then click on the green "Register" button on the top right and follow the process to create an account. If you have a child, then register him/her for the league and you will add yourself as a volunteer coach. If you do not have a child, then just register as a volunteer. If you have an account, then click Login and enter username/password and follow the same instructions.
Log in to AYSOU.ORG using the same username and password you use for this site Complete the Safe Haven and the CDC Concussion Awareness classes, then complete the coaching module for the level you are coaching (i.e. U12).
Some levels require on-field/classroom training. A Local clinic will be conducted in August, we will email you in July when that time and location is determined. If you do not want to wait, then you can register for and attend any other live course in a different region. To do so, go to AYSOU.ORG log in with the same username and password you use for this site. Choose the "Live Courses" menu item, Click the "Sessions" button to the right of the clinic your are looking for, then scroll down through the long list to find the clinic you want to register for. Once you find the clinic you want, click the "Register" button. Although it is rare, if another region charges you for the clinic, get a receipt and we'll reimburse you.


text:
Video: The evolution of Jurgen Klopp's tactics in 1 year at Liverpool FC
Jack Lusby
The Redmen TV's Chris Pajak analysed the tactical changes in Jurgen Klopp's approach over the first year of his time as Liverpool manager.
Klopp's arrival has already seen the Reds hint at future success, both in reaching finals in the Europa League and the League Cup last season, and by sitting in fourth in the Premier League at present.
Liverpool have lost just once in seven league games in 2016/17, winning five times including against Arsenal, Chelsea and Leicester City.
This has marked a major step forward from Klopp's first months on Merseyside, and Pajak's overview of the evolution of his tactics highlights this excellently.
Pajak opens by talking about the improvement in Liverpool's pressing system under Klopp, from their first game against Tottenham Hotspur last October—where there were early signs of this in action, with the Reds swarming opposition within six seconds.
He highlights key areas of the pitch by splitting it first into thirds, and those thirds into fifths—with two wings, two half-spaces and the area in the middle of the park.
With Klopp moving from the 4-2-3-1 to the 4-3-3 this season, this has aided Liverpool against sides employing a low block.
Pajak explains how these sides try to "crush the space in the centre of the park," and where Liverpool previously needed wingers, the attacking fluidity of Klopp's new system is negotiating this.
Projecting the Reds' future, Pajak notes how Klopp began rotating his Borussia Dortmund side when they began enjoying success in the Champions League.
Therefore, if Liverpool are to finish in the top four in the Premier League this season, he believes that for Klopp's tactics to continue to evolve, he will need to sign a genuine backup to both Philippe Coutinho and Sadio Mane.
Tags Jurgen Klopp, The Redmen TV

redpajama_set_name: RedPajamaCommonCrawl


Stuffed Crab is back on the menu!
One off the most awaited times of the year has returned to Santa Cruz!!! This year the Festival started at September 7 and lasts until Octobers 7.
For Only 18,50€ you can have all the Stuffed Crab you can handle! My personal record is 4, not my proudest shape… Anyway come taste this Delight in Our Partners in Santa Cruz!


Did you know that 93% of online experiences now start with a search engine, such as Google? Search has basically replaced the conventional phone book for what people turn to when they need information about local businesses they can rely on for products or services.
Many small businesses struggle to see return on online Search Engine Optimization (also known as SEO). We probably all know at least one business owner who brags about all the Facebook "likes" they've racked up for their page. The reality, though, is that new studies show that Facebook links don't impact Google search rankings at all, according to Search Engine Watch.


Sustainability is about being a positive member of the community and leaving a legacy for our future generations that we can be proud of.
The principle of sustainability guides us in all our activities and practises from project design through to execution and hand-over of projects.
As a member of the 'Green Building Council' our commitment to sustainability is verified.
Health & Safety also fits under this banner with our desire to make sure everyone we engage with in the building process gets home safely every day. Accent hold a current 'Green' Sitewise Prequalification and have a full-time health and safety team dedicated to project and process planning.


When it comes to getting an item like a mattress that we are as well made use of to going to the shop and purchase, we have a tendency to be hesitant often. Below are some of the points that you require to recognize before you buy mattress online.
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Are you looking for mattress sales to give your bed a brand-new life? It is excellent to know that mattress sales are getting extra and also much more regular. Whereas in customer's viewpoint – mattress sales come with unparalleled cost savings and also buyers looking for budget items also, often locate it possible to make a much better and top quality purchase.
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Currently that you know buying a mattress online will certainly not make you regret at the end, let's see the things you can do it make your choice as much better as possible. There are mattresses which are better for back sleepers compared to side sleepers. Know precisely what you are going for, as well as after that begin browsing for the mattress that would certainly match all your requirements.
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text:
Home汪锋组Voice of China Season 4 Episode 6 - Wang Feng's Team, From 12 to 4
Voice of China Season 4 Episode 6 - Wang Feng's Team, From 12 to 4
The cruel PK rounds begin, eliminating eight contestants in one night.
The PK rounds are upon us and in this episode, it is obviously, it is rock-themed (or rather, Wang Feng -themed). Be prepared for quite a lot of heart-thumping, motivational songs. This season truly begins at this point, as some of the contestants appear to have been polished to quite awesome performers.
>>>Spoilers present in this article<<
The Dream Judge assisting Wang Feng is G.E.M Tang, who is only 24 years old this year, but her songs have been crazily popular within the Mandarin Chinese pop industry across all the countries that listen to C-pop... China, Taiwan, Hong Kong, Singapore, Malaysia... who doesn't know her? Her style of judging is pretty much very straightforward, and she doesn't choose contestants to 'console' them when she has to advise Wang Feng on who to choose.
Yang Bao Xin and Zhang Bo Lin's opening performance - Jailhouse Rock didn't manage to highlight both their original standard that they showcased in the blind auditions. There wasn't much chemistry between them but sounded more like they were trying to outdo one another. Picking an English song in which they could hardly articulate the words was a hindrance both to singing and listening.
Zhang Bo Lin wins this round.
Huang Xiao Yun and Wang Fei Xue's very soothing I Believe is the beginning of the upsetting eliminations, because both of them are really good in their own ways. Huang's voice is definitely more contemporary, but Wang's is thicker and more classic. Both voices melded together beautifully, though the choreography of the song is not as interesting as previous seasons.
Huang Xiao Yun wins this round.
The next pair - Xu Zhe and Bie Ri Ke's 曾经的你 reminds me of very good music bar performances, but I really enjoy this pair that really focused more on creating a harmonious performance rather than outdoing one another. First goosebumps of the night!
Xu Zhe wins this round.
Surprisingly, my second favourite pair of the night goes to Huang Yong and Lin Yan, as they sang 從頭再來 which was one of the songs that I would re-play and re-play. Super fun and cutesy song that I totally didn't expect this pair to manage since they aren't thaaat young anymore. But they pulled it off so well. A very difficult decision to make between these two. It's interesting how Wang Feng hardly pair up couples unless absolutely necessary - there aren't exactly the same number of men and women in the team.
Huang Yong wins this round.
The night gets better with 少年錦事 by Zhang Xinxin and Huang Kai, both of whom understand to support each other throughout the song. Zhang Xinxin's voice is significantly better than Huang Kai, which I find that it's a little unfair to pair them up this way, but I guess that's how the competition is supposed to be.
Zhang Xinxin wins this round.
The song that I will keep on replaying till my Youtube videos hang - 我的天空 by Bei Bei and Xiu Er. This is the kind of standard reminisce of the previous seasons. I don't want to say it but this season is slightly lackluster, and finally there's something to show for. Such an unlikely chemistry between the little wild girl Xiu Er and the polished Bei Bei, and I must say while Bei Bei's voice has more potential to go further in the competition, I love Xiu Er's style of performance which is so comfortable to watch. It was very touching to watch the post-performance conversation too.
Bei Bei wins this round.
Loved this post? Read Episode 13 Finale Prelude - best of Season 4, finale song lists, and predictions!
After the Top 6 emerged, they undergo another impromptu pairing as they performed their solo songs. The performances that I really loved included Huang Xiao Yun's 鱼, a song that first appeared on Voice of China by Yao Bei Na. The best of covers happened too - Bei Bei's 存在 is so good, there's really nothing much to fault at all and I was completely in awe! However, when her PK opponent Zhang Xinxin sang 恆星, it is such a crazy feeling - they were both being so good at Wang Feng's songs, they are definitely better than the originals and anyone else who came before them. These three performances are the one to look out for in this episode.
Voice of China 中國好聲音 好声音 汪峰 汪锋组


text:
UK car dealers suffer worst May since 1952 – SMMT
Pramod
FILE PHOTO: Brand new cars are seen on the forecourt of a car dealership in Dartford, east of London. (Photo by DANIEL LEAL-OLIVAS/AFP via Getty Images)
By: PramodKumar
BRITISH new car sales tumbled by an annual 89 per cent in May, only slightly less negative than April's record 97 per cent collapse, as car dealerships remained shuttered by the government's coronavirus lockdown, industry data showed on Thursday (4).
New registrations of 20,247 units represented the weakest May for sales since 1952, the Society of Motor Manufacturers and Traders (SMMT).
Sales were down 51.4 per cent in the first five months of 2020 but the industry is hopeful that a re-opening of dealer showrooms in England this week will help to spur a recovery.
"Early reports suggest there is good business given the circumstances, although it is far too early to tell how demand will pan out over the coming weeks and months," said Mike Hawes, SMMT chief executive.
"Restarting this market is a crucial first step in driving the recovery of Britain's critical car manufacturers and supply chain, and to supporting the wider economy."
The possibility that Britain's transition out of the European Union ends in December with no new trade deal is also likely to weigh on carmakers, some of whom have highly integrated supply chains with the continent.
Nissan's car manufacturing plant in Sunderland, northern England, which employs 7,000 people, is "unsustainable" if Britain leaves the EU without a trade deal, it said on Wednesday (3).
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Sri Lankan President Wickremesinghe calls for 'reconciliation' and 'coexistence' with Tamils & other…
Sri Lankan President Wickremesinghe says debt restructuring talks with India, China 'successful'
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text:
New app: It's what's for dinner
Indy-based startup DinnerCall aims to use technology to bring people together for meals
New app: It's what's for dinner Indy-based startup DinnerCall aims to use technology to bring people together for meals Check out this story on IndyStar.com: http://indy.st/1UPR7m4
Published 6:08 p.m. ET Sept. 13, 2015 | Updated 11:29 a.m. ET Sept. 15, 2015
Gerry Hays is co-founder of DinnerCall.(Photo: Provided by Gerry Hays)
In today's fast-paced, technology-driven world, it's hard to find time to slow down, share a meal with family and friends, and disconnect for an evening.
Now there's an app for that.
DinnerCall, an Indianapolis-based startup that recently launched its app, aims to use technology to help people come together for meals.
"I really like the direction technology can go to help in wellness," said co-founder Gerry Hays. "We rely on the world of health benefits of sitting down and being present with others over a shared meal."
Some of those benefits include lower rates of obesity and depression, reduced divorce rates and better grades in school.
Hays said he got the idea for DinnerCall from his visits with his 97-year-old grandfather, who lives in Chicago.
"I go up there every six to eight weeks, he makes his own dinner, we have a couple glasses of wine and sit at the dinner table for three to four hours," Hays said. "It's a highlight for me, and you just realize everyone should have more of this."
DinnerCall is available on Google Play, Android and Apple.
According to Hays, you can use it to track shared meals with others, track the amount of time you spend and share pictures and recipes with other people using the app.
Hays said the app can help people engage with each other. Under one function, for example, an interesting question will pop up, such as: "If you were going to be alone on a deserted island for a year, who would you take with you?"
"It's a way to try to get everyone in spirit," he said.
Another feature that will come later is a "mobile commerce platform," where families can buy meals through the app and DinnerCall will provide a meal to a family in need via partnerships with local food pantries.
"We want everyone to have a hot, fresh meal for dinner no matter where they are," Hays said.
The goal of the app, Hays said, is to eventually be unnecessary. "It's to help people get back in the mode of just taking a half hour of their day and unplugging with those that we care a lot about. You don't need an app to help you do that."
Follow Star reporter Gabby Ferreira on Twitter: @Its_GabbyF.
Read or Share this story: http://indy.st/1UPR7m4
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Google has made marketers around the world very happy and specifically speaking one in Cape Town, South Africa. They've announced that they will be opening up Data Studios its data analytics and visualisation tool Data Studio to more than 180 countries, with South Africa being one of the lucky recipients.
First introduced as part of its Google Analytics Suite 360 announcement last March, Google Data Studio gives businesses access to a useful data analysis and visualisation tool that integrates with myriad Google products to create interactive reports and dashboards. For example, a user can combine data from Google Sheets, AdWords, Google Analytics, BigQuery, and YouTube, combining a range of raw data to create customised reports. Data Studio was a U.S.-only product until October, when it arrived in 21 new markets including Australia, Brazil, Canada, France, Germany, Japan, Spain, the Republic of Korea, and the U.K. Google hasn't yet revealed the exact 180+ countries it's now available in, but that number pretty much ticks off most of the world's markets.
Today's news comes a little more than a month after Google revealed it was removing the five-report limit from the software's free incarnation — the version aimed at small businesses and individuals. The paid version still offers some additional benefits, including the ability for up to 200 "owners" to edit and manage reports, while it also comes with full customer support. By opening up to more markets, this goes some way toward steering those already within the Google app ecosystem away from competitors such as Microsoft and Power BI, which is already available in well over 100 markets globally.
In addition to the release to over 180 countries, Google has announced a few more features.
Google has removed the five report limit in Data Studio, allowing you to create and share as many reports as needed all for free.
You can now apply Google Analytics segments to your charts. A Google Analytics segment represents a subset of your data, for example, Returning Users. You can now see all your Google Analytics segments in Data Studio and apply them to any chart. And, if you update the definition of your segments in Google Analytics, those changes will apply to the segments in Data Studio.
They've improved several of their most popular data connectors, YouTube, DoubleClick Campaign Manager, and AdWords, by adding many new dimensions and metrics. Some highlights include adding YouTube video title, DoubleClick Campaign Manager revenue and cross-environment conversions broken out by app, AdWords campaign ID, and keyword quality score. For a full list of all the new fields, check out the Data Studio release notes.
They've also announcing tighter integration with the Google Cloud Platform to enable faster data reporting at scale and more powerful functionality.
We know that not all your data resides within SQL or Google databases and Google has made provision for that. Data Studio now has the ability to upload up to 2GB of CSV data for free enabling you to bring in data from anywhere. Have more data? Upload directly into BigQuery using your BigQuery account to take advantage of the scalable processing power of Google's infrastructure.
Not only can you upload data directly into BigQuery, Data Studio now supports Standard SQL in BigQuery for custom queries and partitioned tables. Can you believe All of this packed into a free tool, well, isn't that mighty kind of them?
Reporting not your cup of tea?


Hayley Powell is back in Death of a Bacon Heiress, the seventh installment in the Hayley Powell Food and Cocktails cozy mystery series. This book can be read as a standalone, as the author does a wonderful job of giving backstory so anyone can pick up the book and read it. However, you will definitely want to go and grab the first six in the series! They are equally charming and laugh-out-loud hysterical. I have really enjoyed this series since the very first book, and this book was no different, I really enjoyed this book, The plot itself is well written, well developed and well thought out. The twists in the story kept me guessing and it was a fast-paced read that kept me up late at night reading. I thoroughly enjoyed the new characters introduced and I also enjoyed seeing "new" sides of characters that I have known and loved since the beginning and new character relationship development in process. I can't wait until book eight, Death of a Pumpkin Carver, is released later this year! The combination of magazine columns and recipes add an additional layer of goodness to all of the Hayley Powell books. Overall, Death of a Bacon Heiress was a charming and hysterical read with a fast-paced story line and fun characters, (Hello, Pork Chop anyone?)! I received an ARC of this book, from NetGalley, in exchange for my honest review.
Just too ridiculous for me!
Death of a Bacon Heiress by Lee Hollis is the seventh book in the Hayley Powell Mystery series. Dr. Alvin Foley has been missing for a couple of weeks. It is the talk of the town until Hayley Powell, writer of Island Food and Spirits for The Island Times, is invited to do a cooking demonstration on The Chat where she meets Olivia Redmond. Olivia is the owner of Redmond Meats (a very powerful and rich woman). Olivia asks Hayley to write blog posts for the Redmond Meats website (and Hayley can really use the money). When Hayley stops by Olivia's house, she finds Olivia dead in her yard (with Olivia's pot-bellied pig, Pork Chop nearby). The suspect list is long and includes The Chat hostess, Rhonda Franklin. Hayley will have her work cut out for trying to find the culprit. In the meantime, Hayley has to provide a home for Pork Chop (and Hayley's other animals are not happy) and is dealing with her boyfriend, Dr. Aaron Palmer (local vet). Aaron has been distant lately and is not returning Hayley's phone calls (like he is avoiding her). Is he getting ready to propose? And what is going on with Rhonda Franklin and Hayley's friend, Mona (who has been married for twenty years and has a houseful of unruly children)? They are certainly seeing a lot of each other. Things are lively and about to get worse when the body of Dr. Alvin Foley turns up. I found Death of a Bacon Heiress to be too ridiculous (silly). The mysteries were extremely easy to solve. The writer tries to lead the reader down the wrong path, but the killer is extremely obvious. Death of a Bacon Heiress can easily be read alone (you do not have to read the previous books in the series in order to know what it going on in the novel). There are some bacon recipes included in the back of the book (a bonus). Martha Stewart must be getting a royalty check from this book since she was mentioned four times. I was shocked that the main character thought three hundred thread count sheets were a luxury. I give Death of a Bacon Heiress 3 out of 5 stars. While Death of a Bacon Heiress is not for me, I am sure that many others will enjoy the humor in the novel. I received a complimentary copy of Death of a Bacon Heiress from NetGalley (and Kensington Books) in exchange for an honest evaluation of the novel.


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Posted on : January 1, 2021 By Martha
Ford Motor Co and Mahindra and Mahindra said in separate statements that the two companies have called off their joint venture. As part of the JV, Ford was to cede most of its Indian operations to M&M. The companies attributed the decision to changes in global economic and business conditions caused, in part, by the global Coronavirus (Covid-19) pandemic since the agreement was first announced.
Mahindra and Ford "have mutually, amicably determined they will not complete a previously announced automotive joint venture between their companies," M&M said in a late night notification to the stock exchanges. The action followed passing of the December 31, "longstop", or expiration, date of a definitive agreement the companies had entered into in October 2019, it said.
The latest move will not have any impact on its product plan, Mahindra and Mahindra said, adding it is well positioned in "its core true SUV DNA and product platforms with a strong focus on financial performance." In addition, Mahindra is accelerating its efforts to establish leadership in electric SUVs, the company added.
After two years of intense deliberations, which started in September 2017, M&M and Michigan-based Ford Motor announced a joint venture in which Mahindra controlled the majority 51 per cent stake, whereas Ford was to transfer its entire India business, excluding the engine vertical, to the new entity.
Both the companies expected to achieve cost synergies, pare product development time and cost and leverage each other's distribution reach to boost exports.
The deal was one of the several partnerships formed under Hackett as the Blue Oval sought to reduce costs and increase scale. It comes just months into the leadership of new CEO Jim Farley, who replaced Hackett in October.
US Congress overrides Trump's veto of defence budget, 1st for his admin


A team should work together like clockwork, perfectly coordinated to reach a common goal. But this is not always the case. A variety of personalities, character traits and ambitions can add up to an explosive mixture. How to bring it all together and integrate people around the company's objectives? This is where the RMP-based development programme Management Through Motivation 3.0 can help.
A team should work together like clockwork, perfectly coordinated to reach a common goal. But this is not always the case. A variety of personalities, character traits and ambitions can add up to an explosive mixture. How to bring it all together and integrate people around the company's objectives?
This is where Management Through Motivation 3.0 can help. This motivation approach is based on the assumption that efficient team management is possible only if you first find out the individual needs of each of your employees and determine to what extent they are satisfied at work.
Read the whole article written by Ewa Jochheim for the ManagerPlus website HERE.