\section{Results for the Smooth Graphite Surface}
\label{sec:layr1}

\subsection{Energy Calculations}
In this section we report simulation results for the first layer
obtained by using only the
laterally averaged portion $V_0(z)$ of the helium-graphite potential.
We ignore substrate corrugations.  
This system has recently been studied with the Green's Function Monte Carlo
(GFMC) method,\cite{whitlock98} so we have the opportunity to verify
that the full implementation of our method is in agreement with
the results of these complementary calculations.
The GFMC calculations employ the same helium-graphite potential as we do, 
but use an older form of the helium-helium potential.\cite{aziz79}
The older form was used since the authors wanted to make direct comparisons
with previous work on two-dimensional helium using this
potential.\cite{whitlock88}  The newer potential that 
we use for the helium-helium
interaction is somewhat more attractive (the well depth is about 0.1 K
greater), so we expect the energy per particle to be somewhat lower
near the equilibrium density.  This has
been observed in recent zero-temperature calculations for two dimensional   
liquid helium that employ the newer potential.\cite{giorgini96} 

Guided by previous simulations, we expect the helium film to have
a self-bound liquid phase and to solidify at high densities, before
promotion to the next layer occurs.  Calculations at low and intermediate
densities are performed using a square simulation cell, while solid phase
calculations use a rectangular simulation cell that accommodates
the periodic structure of the triangular solid.  
It is not necessary in PIMC to employ different
forms for the density matrix for the liquid and solid phases.  
Liquid calculations employed 
16 particles, while solid phase calculations used 30 particles.  Both sets of 
calculations were performed at 400 mK.  Calculations at 500 mK are
in agreement with these results, indicating that we have converged to the
zero-temperature limit.  We verified that there were no finite-size
errors in the liquid phase by repeating some of the calculations 
with 32 particles.  The energy values at the two temperatures were
in agreement.  Finite size errors were
found to be negligible for the solid phase also, since the energy value
calculated at 0.0689 atom/$\AA^2$ using the rectangular 
simulation cell (30 particles) agreed with the value 
calculated using the square simulation cell (16 particles).  

These calculations are somewhat different from those we discuss in
the other section of this paper.
The potential between the active helium 
atoms and the underlying substrate
is featureless in the plane of the substrate, so the size of the 
simulation cell may 
be varied continuously.  This allows us to keep the number of particles 
constant for each phase.  In contrast, other calculations we have performed
(including those with substrate corrugations taken into account) 
were with a varying number of particles and a constant simulation cell size.  
The two methods lead to somewhat different forms for the Maxwell 
construction.  In the present case, liquid-solid coexistence will be
characterized by a linear region in the dependence of the energy per
particle on the atomic area (inverse volume).

Figure \ref{fig:engliq} displays our results for the energy of the
first layer liquid.  Also shown are the results obtained from GFMC.
For all points except the near equilibrium, the two calculations 
agree within error bars.  Near the energy minimum, there is some
disagreement, but even here the computed values differ only by about 0.6\%.
This is perhaps attributable to the different helium-helium potentials
used in the two calculations.  Figure \ref{fig:engsolid}
shows our results and the GFMC results for the first layer solid
phase.  Again, there is excellent agreement between the two methods. 

%%%%%%%%%%%%%%%
\begin{figure}[htp]
%\centerline{\psfig{figure=engliq.ps,width=\figwidth}}
\epsfxsize=\figwidth\centerline{\epsffile{engliq.ps}}
\caption{Energy values for the first layer liquid.  The circles are
our results obtained with PIMC.  The squares are 
GFMC results.\cite{whitlock98}  The line is a least squares fit 
of the polynomial, Eq.\ (\ref{eq:polyl}) to our data.
}
\label{fig:engliq}
\end{figure}
%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%
\begin{figure}[htp]
%\centerline{\psfig{figure=engsolid.ps,width=\figwidth}}
\epsfxsize=\figwidth\centerline{\epsffile{engsolid.ps}}
\caption{Energy values for the first layer triangular solid.  The
symbols have the same meaning as in Fig.\ \ref{fig:engsolid}.  The line
is determined from the fit of Eq.\ (\ref{eq:polys}) to our data.
}
\label{fig:engsolid}
\end{figure}
%%%%%%%%%%%%%%%

Following the typical procedure, we have determined the equations of state
for the liquid and solid phases by fitting our energy values to
the polynomials
%%%%%%%%%%%%%%%%
\begin{eqnarray}
&& E/N=e_0 + B(\frac{\rho-\rho_0}{\rho_0})^2 + C(\frac{\rho-\rho_0}{\rho_0})^3,\\
\label{eq:polyl}
&& E/N=\alpha + \beta \rho + \gamma \rho^2 + \delta \rho^3.
\label{eq:polys}
\end{eqnarray}
%%%%%%%%%%%%%%%%
for the liquid and solid phases, respectively.  Values for the 
fitted parameters are given in Table \ref{tab:fit}.  Errors for the liquid
phase are indicated.  The values shown for the solid phase were used in
the plot of Fig.\ \ref{fig:engsolid}.  The values given for
$\beta$,$\gamma$, and $\delta$ are accurate to one significant figure.

For the liquid phase, the 
parameters $e_0$ and $\rho_0$ are the equilibrium energy per particle and the
equilibrium coverage.  The equilibrium density 
we obtain, 0.0450 atom/$\AA^2$, is
in reasonable agreement with the GFMC value of 0.0443.  Below this density,
the system enters gas-liquid coexistence in the thermodynamic limit.
In simulation, phase separation will not occur immediately because of the
finite cost of creating the phase boundary.  The system will instead enter
a ``stretched'', or negative pressure, state.  If one continues to decrease
the coverage, however, the phase separated state will eventually become 
the favored state, and the ``stretched'' state will ``snap'', forming
a droplet plus vacuum.  For a simulation with a constant number of 
particles and variable area, this occurs when the derivative
of the spreading pressure, $P=\rho^2(\partial e(\rho)/\partial \rho)$, is
zero.  At the spinodal point, the sound velocity becomes imaginary and
the compressibility diverges.
From the equation of state, we determine that this occurs 
at 0.034 atom/$\AA^2$.  For comparison, the spinodal density has been
calculated to be between 0.031 and 0.038 atom/$\AA^2$ for two-dimensional
helium.\cite{clements93b,giorgini96,whitlock88}  

%%%%%%%%%%%%%%%%%%
\begin{table}
\begin{center}
\begin{tabular}{|l r | l r|} \hline
Parameter          & Liquid       & Parameter           & Solid  \\ \hline
e$_0$(K)           & -141.689(14) & $\alpha$(K)         & -157.46    \\
$\rho_0$($\AA^{-2}$)  & 0.0450(06)  & $\beta$(K/$\AA^2$)  & -679.83    \\
B(K)               & 2.42(27)     & $\gamma$(K/$\AA^4$) & -10504.77  \\
C(K)		   & 3.29(67)     & $\delta$(K/$\AA^6$)	& 61032.24   \\
$\chi^2/\nu$       & 2.57         & $\chi^2/\nu$        & 1.3   \\ \hline
\end{tabular}
\caption{ Fitted parameters for the liquid and solid equations of state.  The
errors in the last digits of the parameters for the liquid phase are given in 
parenthesis.
}
\label{tab:fit}
\end{center}
\end{table}
%%%%%%%%%%%%%%%%%%

From the polynomial fits for the two layers, we can estimate the regions
of phase coexistence for the two solids by using the Maxwell double
tangent construction.  See Fig.\ \ref{fig:maxwell}.
Since we have a constant number of particles and a variable density, the
Maxwell construction is found from the common tangent of the energy per
particle versus atomic area (inverse coverage).  We determine the
liquid-solid coexistence region to be from 0.0675 to 0.0700 atom/$\AA^2$.
This is comparable to the ranges found for both two-dimensional 
helium\cite{whitlock88,gordillo98} and 
the helium film,\cite{whitlock98} although
the onset of full solidification that we find is at a somewhat lower density.  
The solidification density determined in this manner is subject to 
error, since our fitted parameters are not determined with a high
degree of precision.  

%%%%%%%%%%%%%%%%
\begin{figure}[htp]
%\centerline{\psfig{figure=maxwell.ps,width=\figwidth}}
\epsfxsize=\figwidth\centerline{\epsffile{maxwell.ps}}
\caption{Liquid-solid coexistence regions determined with the 
Maxwell construction.  The dashed-dotted and dashed lines are the 
solid and liquid 
equations of state, respectively.  Circles with error bars are 
calculated energy values for the liquid.  Squares with error bars show
values calculated for the solid phase.
The unbroken line is the coexistence line.
The arrows indicate the beginning and end of the coexistence region. 
}
\label{fig:maxwell}
\end{figure}
%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%
\begin{figure}[htp]
%\centerline{\psfig{figure=rdflayr1.ps,width=\figwidth}}
\epsfxsize=\figwidth\centerline{\epsffile{rdflayr1.ps}}
\caption{Radial distribution function for the liquid phase at
the indicated densities, in atom/$\AA^2$.
}
\label{fig:rdflayr1}
\end{figure}
%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%
\begin{figure}[htp]
%\centerline{\psfig{figure=ssflayr1.ps,width=\figwidth}}
\epsfxsize=\figwidth\centerline{\epsffile{ssflayr1.ps}}
\caption{Static structure function for the liquid phase at
the indicated densities, in atom/$\AA^2$.  The static structure function
in the (01) direction at a typical solid density (top) is also shown.
}
\label{fig:ssflayr1}
\end{figure}
%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%
\begin{figure}[htp]
%\centerline{\psfig{figure=distriblayr1.ps,width=\figwidth}}
\epsfxsize=\figwidth\centerline{\epsffile{distriblayr1.ps}}
\caption{Contour plots of the probability density in the plane of the
substrate.  The densities shown are, top row, left to right, 0.0256
and 0.0421 atom/$\AA^2$; bottom row, left to right,
0.0689, and 0.0918 atom/$\AA^2$.
}
\label{fig:contour1}
\end{figure}
%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%
\begin{figure}[htp]
%\centerline{\psfig{figure=zdistlayr1.ps,width=\figwidth}}
\epsfxsize=\figwidth\centerline{\epsffile{zdistlayr1.ps}}
\caption{Density profiles near second layer promotion.  The coverages, in
atom/$\AA^2$, are indicated.  Normalization is chosen so that integrating
the profiles gives the coverage.
}
\label{fig:zdistlayr1}
\end{figure}
%%%%%%%%%%%%%%%

\subsection{Other properties}
As we have seen, in the absence of corrugations, the first layer
has liquid-gas, self-bound liquid, liquid-solid, 
and solid regions.  The 
gas phase has zero density at zero temperature.  
We expect the liquid-gas phase to have two
regions, one in which phase boundaries between these phases form, and
one in which the liquid phase is artificially stretched.  This is
governed by the energy cost required to create the phase boundary.  The
metastable liquid phase is spatially indistinguishable from the 
equilibrium state, but the formation of droplets can be visualized.
In Fig.\ \ref{fig:rdflayr1} we plot the radial distribution function g(r) 
for the liquid at coverages in the droplet region, near equilibrium, and near
the onset of liquid-solid coexistence.  Here, $r$ is the magnitude of the
projection of the
distance vector between two particles onto the plane of the substrate.
In the droplet region, the
long range tail drops below unity, indicating the system does not uniformly
cover the substrate.  Near equilibrium the first peak has about the same
height as the droplet phase, but the tail goes to unity at long ranges.  The
peaks are located at the same value of $r$, indicating that the average
separation
distance of nearest neighbor particles in the droplet is 
close to the equilibrium value.
The high density liquid has noticeably more correlation.  The peak height is
1.5, compared to 1.2 near equilibrium, and the peak's position has shifted from
4.2 to 3.6 $\AA$, as one would expect from compression caused by increasing
the density.  These values are in agreement with values reported for
the helium liquid in two dimensions,\cite{whitlock88} indicating that the
first layer liquid is very two-dimensional in character.

The results for the static structure factor $S(k)$ for the liquid phase are
shown in Fig.\ \ref{fig:ssflayr1} and can be interpreted similarly.  At the
lowest coverage, $S(k)$ swings upward for low values of $k$ instead of 
going to zero.  This indicates the presence
of droplets and a nonzero compressibility.
For the two uniform liquid coverages, 
$S(k \rightarrow 0) \rightarrow 0$.  The structure function
is more peaked for the high density liquid.  Figure \ref{fig:ssflayr1} also
shows results for a typical solid coverage.  The peak heights for the 
fluid near equilibrium and for the dense fluid are 1.2 and 1.6, respectively.
These values are in reasonable agreement with, but somewhat below, the
two-dimensional values.  


Finally, we show in Fig.\ \ref{fig:contour1} probability contours for the
first layer for the various regions discussed above.  The lowest
coverage shown is below the spinodal point, and as expected 
incompletely covers the substrate.  Near the equilibrium
density, the substrate is uniformly covered.  In the dense liquid, localization
can be observed.  This coverage is in the liquid-solid coexistence region.
Finally the system enters a triangular solid phase at
the highest density.


Promotion to the second layer may be determined by examining the
density profiles for the dense solid.  These are shown in 
Fig.\ \ref{fig:zdistlayr1} at the indicated coverages.  The occupation of
the second layer is clearly visible at the coverages 0.1200 and 0.1270
atom/$\AA^2$, while the coverage 0.1180 shows no evidence of promotion.
Integrating the profiles for the two largest coverages to the minimum
between the peaks (4.5 $\AA$) gives 0.116 and 0.119 atom/$\AA^2$ for
the first layer coverage.  Experimentally, 
estimates\cite{bretz73,bretz78,carneiro81,greywall91} of first layer
completion range from 0.112 to 0.120 atom/$\AA^2$
so our value is in good agreement.  The recent GFMC 
calculations\cite{whitlock98} obtain a value between 0.115 and 0.118 for
the beginning of second layer promotion.











