Three-body recombination into deep bound states\\ in a Bose gas with large scattering length
Eric Braaten and H.-W. Hammer
An effective field theory for the three-body system with large scattering length $a$ is applied to three-body recombination into deep bound states in a Bose gas. The recombination constant $\alpha$ is calculated to first order in the short-distance interactions that allow the recombination. For $a<0$, the dimensionless combination $m\alpha/(\hbar a^4)$ is a periodic function of $\ln |a|$ that exhibits resonances at values of $a$ that differ by multiplicative factors of 22.7. This dramatic behavior should be observable near a Feshbach resonance when $a$ becomes large and negative.
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\begin{document}
\title
{Three-body recombination into deep bound states\\
in a Bose gas with large scattering length}

\author{Eric Braaten and H.-W. Hammer}

\address{Department of Physics,
 The Ohio State University, Columbus, OH 43210, USA}
\date{June 4, 2001}

\maketitle

\begin{abstract}
An effective field theory for the 
three-body system with large scattering length $a$
is applied to three-body recombination
into deep bound states in a Bose gas.
The recombination constant $\alpha$ is calculated to first
order in the short-distance interactions that allow the recombination. 
For $a<0$, the dimensionless combination
$m\alpha/(\hbar a^4)$ is a periodic function of $\ln |a|$ that
exhibits resonances 
at values of $a$ that differ by multiplicative factors of 22.7.
This dramatic behavior should be observable near a Feshbach
resonance when $a$ becomes large and negative.
\end{abstract}

\setcounter{page}{1}
\vskip 0.5cm
%
One important factor limiting the achievable density in Bose-Einstein 
condensates (BEC's) of trapped atoms is the loss of atoms through 3-body 
recombination \cite{SGD98,RCC00}. 
Such losses occur when three atoms scatter to form a molecular bound 
state and a third atom.
The kinetic energy of the final-state particles allows them  
to escape from the trapping potential. This
process provides a unique window on 3-body dynamics of cold atoms.
In a BEC, 3-body loss rates could also reveal
new collective mechanisms involving a molecular condensate \cite{molcond}.

The recombination event rate
can be parametrized as $\nu_{rec}=\alpha n^3$, where $\alpha$
is the recombination constant and $n$ the density of the gas.
The calculation of $\alpha$ in general is a complicated problem,
because it is sensitive to the detailed behavior of the interaction
potential. If the $s$-wave
scattering length $a$ is large compared to the size of the atoms,
however, the problem simplifies.
Assuming that $a$ is the only important length scale,
dimensional analysis implies $\alpha={\cal C} \hbar a^4/m$, 
where $m$ is the mass of the atoms and ${\cal C}$ is dimensionless.
For $a>0$, there can be 3-body recombination into a shallow $s$-wave 
state with binding energy $\hbar^2 /(ma^2)$.
For either sign of $a$, there can also be recombination into
deep molecular bound states. The coefficient $\alpha$
can then be split into two contributions $\alpha_S$ and $\alpha_D$
coming from recombination into the shallow and deep bound states,
respectively.

The 3-body recombination into the shallow bound state has been 
studied in Refs. \cite{Fed96,NiM99,EGB99,BBH00}. For the $s$-wave 
contribution, ${\cal C}$ is an oscillatory function of $\ln a$: 
\begin{equation}
\label{resalphaS}
\alpha_S \approx 67.9 \cos^2 [s_0 \ln(a\Lambda_*)+ 1.74]\,
 \frac{\hbar a^4}{m}\qquad (a>0)\,,
\end{equation}
where $s_0\approx 1.0064$ and $\Lambda_*$ is the 3-body
parameter introduced in Ref.~\cite{BBH00}.
In the adiabatic hyperspherical approximation, the oscillatory
behavior can be explained by interference between paths
connecting two hyperspherical potentials \cite{NiM99,EGB99}.
Within the framework of effective field theory, this unique behavior
is due to scaling violations created by a 3-body interaction 
required for consistent renormalization \cite{BBH00}.
For $a<0$, there is no shallow bound state (there is a low-energy virtual
state instead) and $\alpha_S \equiv 0$.

Effective field theory (EFT) is a powerful method for describing
systems composed of particles with wave number $k$ much smaller
than the inverse of the characteristic range $R$ of their interaction.
EFT focuses on the universal long-distance aspects of the problem,
by modelling the interactions as pointlike \cite{gospel}.
For wave numbers $k\ll 1/R$, one can expand
in powers of the small variable $k R$.
Generically, the scattering length $a$
is comparable to $R$, and the expansion is effectively in powers of $ka$.
In the case of large scattering length $a \gg R$,
the dependence on $ka$ must be treated nonperturbatively.
In the EFT, a single 3-body parameter is necessary and sufficient 
to renormalize the  ultraviolet divergences
at leading order in $kR$ \cite{BHK99}. The scattering length and this 3-body
parameter describe all low-energy 3-body observables
up to errors of order $R/a$.

In this Letter, we use EFT methods to study
the recombination constant $\alpha_D$ for deep bound states.
The deep bound states are outside the range of the effective theory 
and cannot be treated explicitly. Nevertheless, their contribution
to 3-body recombination can be calculated within
the EFT by using the optical theorem (cf.~Fig.~\ref{fig1}(a)).
%
\begin{figure}[htb]
%\epsfxsize=8.cm
\epsfxsize=10.cm
\centerline{\epsffile{deepfig1.ps}}
\vspace*{0.3cm}
\caption{(a) Illustration of the optical theorem.
Two-body bound states are indicated by the shaded double line.
(b) Diagrams for the contribution of the deep bound states.
The single (double) lines represent the exact propagators for
the field $\psi$ ($d$) and the triangle denotes an insertion of $ih'$.}
\label{fig1}
\end{figure}
%
The 3-body recombination channels generate
imaginary parts in the $3\to 3$ scattering amplitude. 
The imaginary part from recombination into the shallow bound state is
dynamically generated in the EFT. 
The imaginary part from 3-body recombination into deep bound states
can be taken into account by adding a local interaction term 
to the EFT. 

A similar approach has been used successfully to describe
the effects of the annihilation decays of positronium in QED and heavy
quarkonium in QCD within the framework of nonrelativistic EFT's
\cite{BBL95}. The annihilation of positronium into
photons involves intermediate states with relativistic electrons 
and positrons that are outside the range of validity of the
EFT. However, the effects of the annihilation process on 
$e^+ e^-$-scattering at low energies can be taken into 
account systematically by adding local 4-fermion operators with 
imaginary coefficients to the effective Lagrangian.
In the case of recombination into a molecule
with binding energy $B \gg \hbar^2/(m a^2)$, 
the molecule and the recoiling atom
will emerge with large momenta of order $\sqrt{mB}$
which is outside the domain of the EFT.
Up to corrections of order $p^2/(mB)$, where $p$ is the scale of 
the momentum of the three atoms, the effect of the recombination process
on the $3\to 3$ scattering amplitude can be taken into
account with a local 3-body interaction term whose coefficient
has an imaginary part. The interaction is local, 
because the three atoms have to approach to within
a distance of order $\hbar/\sqrt{mB}\ll a$ to recombine.
The imaginary 3-body coefficient
accounts collectively for recombination into all deep bound states.
It could in principle be calculated if the 2- and 3-body potentials
describing the interactions between atoms were known with sufficient
accuracy. In practice, it has to be determined from experimental data. 
Once this parameter is fixed, the dependence
of $\alpha_D$ on $a$ and $\Lambda_*$ can be predicted.

In Ref.~\cite{BBH00}, we have calculated $\alpha_S$ for a large, 
positive scattering length $a$.
In the following, we extend this work to calculate $\alpha_D$ for
large $a$ of either sign. For simplicity, we now set $\hbar=1$. 
We start with a local Lagrangian for a nonrelativistic 
boson field $\psi$ and an auxiliary field $d$ with the quantum numbers 
of two bosons \cite{BBH00,BHK99}:
\begin{eqnarray}
\label{lagt}
{\cal L}  &=&  \psi^\dagger
\bigg(i \frac{\partial}{\partial t} +\frac{\vec{\nabla}^{2}}{2m}\bigg)\psi
  + d^\dagger d \\
  &-&\frac{g}{\sqrt{2}} (d^\dagger \psi\psi +\psi^\dagger \psi^\dagger d)
   +(h+ih')\, d^\dagger d \psi^\dagger\psi +\ldots\,,\nonumber
\end{eqnarray}
\noindent
where $h'$ is the free parameter accounting for recombination
into the deep states.
The dots denote terms with more derivatives and/or fields;
those with more fields do not contribute to the 3-body amplitude,
while those with more derivatives are suppressed at low energy.
The couplings $g$ and $h$ in Eq. (\ref{lagt}) 
can be eliminated in favor of $a$ and $\Lambda_*$, respectively.
The atom propagator has the usual nonrelativistic
form $i/(\omega-p^2/2m)$. 
The 2-atom scattering amplitude for incoming atoms with 
momenta $\pm \vec{k}$ has the form $(-1/a-ik)^{-1}$. 
The exact propagator for $d$ has a pole at 
$\omega=-1/(ma^2)+\vec{p}^{\,2}/(4m)$ corresponding to the 
shallow bound state \cite{BBH00,BHK99}.

Before describing the calculation of $\alpha_D$, we
summarize the calculation of $\alpha_S$ in Ref.~\cite{BBH00}.
The coefficient for recombination into the shallow bound state can 
be expressed as
$\alpha_S= 512 \pi^2 \left|t(p_f)\right|^2 /(\sqrt{3}m)$,
where $t(p)$ is the amplitude for the transition between three atoms at rest
and a final state consisting of an atom and a shallow $s$-wave state
with momentum $p$ in their center-of-momentum frame.
This amplitude is evaluated on shell at the 
value $p_f=2/(\sqrt{3}a)$ prescribed by energy
conservation. However, $t(p)$ is also defined at off-shell values of $p$
as the solution of an integral equation that contains
the 2- and 3-body interaction terms \cite{BBH00}. 
Note that $t$ is related to the amplitude $T$ of Ref.~\cite{BBH00} via 
$t(p)=mT(p)/(48 \pi^{3/2} \sqrt{a})$.
The integral equation is regularized by a momentum cutoff $\Lambda$.
All observables can be made independent of the cutoff by adjusting the
3-body coefficient $h(\Lambda)$ as a function of $\Lambda$.
The evolution of $h$ with $\Lambda$  approaches a limit cycle:
$\Lambda^2 h$ varies periodically between $+\infty$ and $-\infty$ 
as $\Lambda$ increases by multiplicative factors of $\exp(\pi/s_0)\approx
22.7$ \cite{BHK99}. Since all observables are cutoff independent,
we can choose a $\Lambda$ for which $h(\Lambda)=0$.
The 3-body parameter $\Lambda_*$ then appears in the upper limit of the 
integral and we obtain a renormalized equation \cite{HaM00}, 
\begin{eqnarray}
\label{inteq}
& &t(p)=\frac{2}{p^2} + 
       \frac{2}{\pi}\int_0^{\Lambda_n}
      \frac{dq\,q^2\, t(q)}{-1/a+\sqrt{3}q/2-i\epsilon}\\
& &\hspace*{3cm}\times
    \,\frac{1}{pq}\ln\left|\frac{q^2+p q+p^2}{q^2-q p+p^2}\right|
    \,,\nonumber
\end{eqnarray}
with $\Lambda_n=\Lambda_* \exp[(n\pi+\arctan(1/s_0))/s_0 ]$ and
$s_0\approx 1.0064$. The integer 
$n$ should be large enough that errors of ${\cal O}(1/
(\Lambda_n a))$ can be neglected. In practice, one can often choose $n=1$.
Solving the integral equation for $t(p)$ and evaluating the solution
as described above, we obtain the result in Eq.~(\ref{resalphaS}).
The integral equation (\ref{inteq}) also has a solution for $a<0$,
but there is no shallow bound state in this case and $\alpha_S =0$.

In order to describe the recombination into
the deep bound states, we use the optical theorem as discussed
above and include an imaginary part $h'$ in the coefficient of the 
3-body contact interaction. We assume that the $h'$ term 
can be treated as a perturbation
at first order. However, the external legs have to be dressed with 
initial and final state interactions to all orders.
We have to calculate the diagrams shown in Fig.~\ref{fig1}(b),
where the blob represents the solution of Eq.~(\ref{inteq})
and the triangle represents an insertion of $ih'$. This leads to
\begin{eqnarray}
\alpha_D=\frac{288 \pi^2 a^2}{m}\frac{h_1}{\Lambda_n^2}\left|1+\frac{2}{\pi}
\int_0^{\Lambda_n}\!\!\frac{dq\,q^2\, t(q)}{-1/a+\sqrt{3}q/2-i\epsilon}
\right|^2,
\label{loop}
\end{eqnarray}
where we have defined $h'/(mg^2)=h_1/\Lambda_n^2$.
Eq.~(\ref{loop}) gives a renormalized expression for $\alpha_D$ up to 
corrections of ${\cal O}(1/(\Lambda_n a))$.
Since $t(q)$ is known numerically from solving Eq.~(\ref{inteq}),
the calculation of $\alpha_D$ entails the numerical
evaluation of the integral in Eq.~(\ref{loop}).

%
\begin{figure}[ht]
%\epsfxsize=8.0cm
\epsfxsize=8.0cm
\centerline{\epsffile{deepfig2.ps}}
\vspace*{0.3cm}
\caption{The recombination coefficients
$\alpha_D$ (solid line) and  $\alpha_S$ (dashed line)
as functions of $|a|\Lambda_*$ for $a>0$ and $a<0$. The free
parameter $h_1$ in $\alpha_D$ was chosen such that
$\alpha_D=10$ for $a>0$.}
\label{fig2}
\end{figure}
%
There are now two cases to consider: $a>0$ and $a<0$.
We first consider the case $a>0$.
Evaluating Eq.~(\ref{loop}), we find that the amplitude is dominated by
a linear divergence in the integral and is proportional to $\Lambda_n a$.
The factors of $\Lambda_n$ cancel and the result is proportional to $a^4$
with a coefficient that, within the numerical accuracy,
is independent of $\Lambda_* a$:
\begin{equation}
\alpha_D=0.041 \,h_1\,\frac{\hbar a^4}{m}\qquad (a>0)\,.
\end{equation}
In Fig.~\ref{fig2}(a), we show both $\alpha_S$ and $\alpha_D$ for
$a > 0$, with $h_1$ adjusted such that $\alpha_D=10$.
The recombination into the deep bound states fills in the
zeroes of $\alpha_S$, so that they become at best local minima of
the recombination rate.

We now turn to the more interesting case $a<0$. Evaluating Eq.~(\ref{loop}), 
we find that the amplitude is again dominated by a linear
divergence in the integral. However, the coefficient of
$a^4/m$ is not a pure number but exhibits scaling violations.
$\alpha_D$ has the remarkable dependence
on $\Lambda_* |a|$ shown in Fig.~\ref{fig2}(b). The coefficient
diverges at a sequence of values of $\Lambda_* |a|$ that are
equally spaced on a logarithmic scale.
To a high numerical accuracy, the results can be described by the formula
\begin{equation}
\alpha_D=
\frac{5.61\,h_1}{\sin^2[s_0 \ln(\Lambda_* |a|)+1.77]}\,\frac{\hbar a^4}{m}
\qquad (a<0)\,.
\end{equation}
The oscillatory dependence on $\ln|a|$ is in accord with a general
scaling law by Efimov \cite{Efi71}.
There are divergences whenever $s_0 \ln(\Lambda_* |a|)+1.77$ is an integer
multiple of $\pi$. They occur because a 3-body Efimov state \cite{Efi71}
has been tuned to threshold by the variation of $a$.
The divergences are artifacts of the first order perturbation
theory in $h_1$, which treats the Efimov states below the 3-body
threshold as having sharp energies. However, these Efimov states 
aquire a width that depends on $h_1$ from recombination into the deep
states \cite{Estates}. This will change the divergences into resonances 
at which the recombination rate is enhanced by a factor $(\hbar^2/\Gamma 
m a^2)^2$, where $\Gamma$ is the width of an Efimov state at the
3-body threshold. First-order
perturbation theory in $h_1$ is justified if $\Gamma \ll \hbar^2/(m
a^2)$, except within $\Gamma$ of a resonance.
The determination of the height of the resonance peaks
requires a calculation that is nonperturbative in $h_1$.
Note that the elastic 3-body scattering amplitude will also
exhibit these resonances if $a<0$.
For $a>0$, an Efimov state at the 3-body threshold has a width
of order $\hbar^2/(ma^2)$ from recombination 
into the shallow 2-body bound state.
Consequently, there should be no resonances in the recombination rate.
%or the elastic 3-body scattering amplitude.


The EFT results for $\alpha$ depend on two parameters $\Lambda_*$
and $h_1$ that are determined by interatomic interactions at distances
much smaller than $|a|$. They have predictive power only if $a$ can be
varied as a function of some external parameter and if the dependence
of $\Lambda_*$ and $h_1$ on that parameter is known. One case in which
there is predictive power is near a Feshbach resonance, where
$a$ is varied by tuning a molecular bound state close
to the threshold of two atoms by using an external magnetic field $B$.
Short-distance parameters must be smooth functions of $B$. Over the
narrow interval of a Feshbach resonance, they can be approximated
by a linear or even constant function of $B$. Linear dependence
of the short-distance parameter $g$ in (\ref{lagt}) corresponds to
$a(B)=a_0+\Delta/(B-B_0)$, where $a_0$ is the off-resonant scattering length
and $\Delta$ is the width of the Feshbach resonance.
The short distance parameters $\Lambda_*$
and $h_1$ can be approximated by constants near the resonance at $B_0$.

%
\begin{figure}[htb]
%\epsfxsize=9.cm
\epsfxsize=10.0cm
\centerline{\epsffile{deepfig3.ps}}
\vspace*{0.3cm}
\caption{The atom loss rate $K_3=3 \alpha$ for a gas of ultracold 
$^{85}$Rb-atoms. The dots (triangles) represent data (upper bounds)
for $K_3$ from Ref.~\protect\cite{RCC00}.
The EFT predictions for $K_{3S}$ (solid lines) and $K_{3D}$
(dashed lines) are shown for $\Lambda_*=5$/a.u. (black curves) and 
$\Lambda_*=20$/a.u. (grey curves). 
The arbitrary normalization factor in $K_{3D}$ is set to $h_1=24.7$.} 
\label{fig3}
\end{figure}
%
In Fig.~\ref{fig3} we compare our theory with measurements
of the atom loss rate coefficient $K_3=3\alpha$ in an ultracold gas of 
$^{85}$Rb-atoms by Roberts {\it et al.} \cite{RCC00}.
In $^{85}$Rb, there is a Feshbach resonance with
width $\Delta=11.5$ G at a magnetic field $B_0 =155.2$ G \cite{RCC00}.
In Fig.~\ref{fig3}, the scattering length is negative
below the Feshbach resonance at $B_0=155.2$ G and positive above.
Consequently, the shallow bound state contributes to $K_3$
only above the resonance. The free parameter $h_1=24.7$ has been fixed 
to reproduce the overall magnitude of $K_3$ below the resonance.
For $a>0$, this value of $h_1$ leads to a contribution of the deep bound 
states to the loss rate ($K_{3D}$) that is much smaller than the 
contribution of the shallow bound state ($K_{3S}$) except 
near the minima of $K_{3S}$. One might be able to
determine $h_1$ experimentally from the depth of these minima.
In Fig.~\ref{fig3}, we have plotted $K_{3D}$ and $K_{3S}$ for
two values of the 3-body parameter: 
$\Lambda_*=5/\mbox{a.u.}$ and $\Lambda_*=20/\mbox{a.u.}$.
The parameter $\Lambda_*$ would be determined very accurately if
a local minimum or maximum in the loss rate were observed.

Divergences in the 3-body recombination rate for $a<0$ have been
observed previously by Esry {\it et al.}~\cite{EGB99}. They
calculated $\alpha$ numerically for various 2-body potentials,
most of them with a single bound state (either shallow or deep). 
They varied $a$ from large negative values to large positive
values by tuning the depth of the potential. For $a>0$,
$\alpha/a^4$ increases to a maximum at some value $a_{\rm max}$
and then decreases to an interference minimum as predicted by
Eq.~(\ref{resalphaS}). For $a<0$, $\alpha/a^4$ diverges at some value
$a_{\rm div}$. The ratio $|a_{\rm div}|/a_{\rm max}$ ranged from
1.6 to 2 for the potentials studied. 
Since the maxima of $\alpha_S/a^4$ and the divergences of $\alpha_D/a^4$ 
in Fig.~\ref{fig2} appear at the same values of $|a|$, the EFT predicts
$|a_{\rm div}|/a_{\rm max}=1$. The discrepancy between the results of 
Ref.~\cite{EGB99} and the EFT prediction can be explained by a variation 
in the effective short-distance parameters $\Lambda_*$ and $h_1$
as the depth of the potential is tuned to vary $a$. Near a Feshbach
resonance, the short distance behavior of the potential is essentially 
unchanged as $B$ is varied. Thus $\Lambda_*$ and $h_1$ should be 
nearly constant and the EFT prediction should hold as long as $|a|$ is 
much larger than its off-resonance value.

We have shown that the EFT approach to the 3-body system with
large scattering length can be applied to recombination into deep
bound states. For $a>0$, this contribution fills in the interference
minima of the recombination rate into shallow bound states, transforming
them into local minima of the total recombination rate. For $a<0$,
the EFT approach makes the dramatic prediction that the recombination
rate for zero-momentum atoms should exhibit resonances due to
Efimov states at a sequence of values 
for $a$ that differ by multiplicative factors of $\exp(\pi/s_0)
\approx 22.7$.
Provided the widths of these Efimov states are numerically small
compared to $\hbar^2/(ma^2)$, the resonances should be
observable as sharp peaks in the recombination rate near a 
Feshbach resonance.

We thank C.H. Greene and J.P. Burke for useful discussions
and for providing us with numerical results of their calculations. 
This research was supported by NSF grant PHY-9800964 
and DOE grant DE-FG02-91-ER4069.

%\vspace*{-0.2cm}

\begin{thebibliography}{50}
 
%\vspace{-1.cm}

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 257 (1999);
 J.~Stenger {\it et al.}, Phys.~Rev.~Lett. {\bf 82}, 2422 (1999).

\bibitem{RCC00}J.L.~Roberts, N.R.~Claussen, S.L.~Cornish, and
 C.E.~Wieman, Phys.~Rev.~Lett. {\bf 85}, 728 (2000).

\bibitem{molcond}
F.A.~van Abeelen and B.J.~Verhaar, Phys.~Rev.~Lett.~{\bf 83},
1550 (1999); V.~Yurowsky~{\it et al.}, Phys.~Rev.~A~{\bf 60},
R765 (1999); E.~Timmermans~{\it et al.}, {\tt [cond-mat/9805323]}.

\bibitem{Fed96}P.O. Fedichev, M.W. Reynolds, and G.V. Shlyapnikov,
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\bibitem{NiM99}E.~Nielsen and J.H.~Macek, Phys.~Rev.~Lett. {\bf 83},
1566 (1999).

\bibitem{EGB99}B.D.~Esry, C.H.~Greene, and J.P.~Burke, 
Phys.~Rev.~Lett. {\bf 83}, 1751 (1999).

\bibitem{BBH00}P.F.~Bedaque, E.~Braaten, and H.-W.~Hammer,
Phys. Rev. Lett. {\bf 85}, 908 (2000).

\bibitem{gospel}
 G.P.~Lepage, in {\it From Actions to Answers, TASI'89},
                 ed. T.~DeGrand and D.~Toussaint,
                 (World Scientific, Singapore, 1990);
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\bibitem{BHK99}P.F.~Bedaque, H.-W.~Hammer, and U.~van Kolck, Nucl. Phys. A
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\bibitem{BBL95}G.T.~Bodwin, E.~Braaten, and G.P.~Lepage, Phys.~Rev.~D 
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\bibitem{HaM00}H.-W.~Hammer and T.~Mehen, Nucl.~Phys.~A (in print) 
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\bibitem{Efi71}V. Efimov, Sov. J. Nucl. Phys. {\bf 12}, 589 (1971);
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\bibitem{Estates}E.~Nielsen and B.D.~Esry, Bull.~Am.~Phys.~Soc {\bf 45},
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(eds.), North Holland, Amsterdam, 1976.

\end{thebibliography}

\end{document}





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%%%                        be \epsfxsize and its height will be scaled
%%%                        proportionately (or the height will be
%%%                        \epsfysize and its width will be scaled
%%%                        proportionately).
%%%
%%%                        The width (and height) is restored to zero
%%%                        after each use, so \epsfxsize or \epsfysize
%%%                        must be specified before EACH use of
%%%                        \epsfbox.
%%%
%%%                        A more general facility for sizing is
%%%                        available by defining the \epsfsize macro.
%%%                        Normally you can redefine this macro to do
%%%                        almost anything.  The first parameter is
%%%                        the natural x size of the PostScript
%%%                        graphic, the second parameter is the
%%%                        natural y size of the PostScript graphic.
%%%                        It must return the xsize to use, or 0 if
%%%                        natural scaling is to be used.  Common uses
%%%                        include:
%%%
%%%                           \epsfxsize  % just leave the old value alone
%%%                           0pt         % use the natural sizes
%%%                           #1          % use the natural sizes
%%%                           \hsize      % scale to full width
%%%                           0.5#1       % scale to 50% of natural size
%%%                           \ifnum #1>\hsize\hsize\else#1\fi
%%%                                       % smaller of natural, hsize
%%%
%%%                        If you want TeX to report the size of the
%%%                        figure (as a message on your terminal when
%%%                        it processes each figure), say
%%%                        `\epsfverbosetrue'.
%%%
%%%                        If you only want to get the bounding box
%%%                        extents, without producing any output boxes
%%%                        or \special{}, then say
%%%                        \epsfgetbb{filename}.  The extents will be
%%%                        saved in the macros \epsfllx \epsflly
%%%                        \epsfurx \epsfury in PostScript units of
%%%                        big points.
%%%
%%%                        Revision history:
%%%
%%%                        ---------------------------------------------
%%%                        epsf.tex macro file:
%%%                        Originally written by Tomas Rokicki of
%%%                        Radical Eye Software, 29 Mar 1989.
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Don Knuth, 3 Jan 1990.
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Tomas Rokicki, 18 Jul 1990.
%%%                        Accept bounding boxes with no space after
%%%                        the colon.
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Nelson H. F. Beebe
%%%                        <beebe@math.utah.edu>, 03 Dec 1991 [2.0].
%%%                        Add version number and date typeout.
%%%
%%%                        Use \immediate\write16 instead of \message
%%%                        to ensure output on new line.
%%%
%%%                        Handle nested EPS files.
%%%
%%%                        Handle %%BoundingBox: (atend) lines.
%%%
%%%                        Do not quit when blank lines are found.
%%%
%%%                        Add a few percents to remove generation of
%%%                        spurious blank space.
%%%
%%%                        Move \special output to
%%%                        \epsfspecial{filename} so that other macro
%%%                        packages can input this one, then change
%%%                        the definition of \epsfspecial to match
%%%                        another DVI driver.
%%%
%%%                        Move size computation to \epsfsetsize which
%%%                        can be called by the user; the verbose
%%%                        output of the bounding box and scaled width
%%%                        and height happens here.
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Nelson H. F. Beebe
%%%                        <beebe@math.utah.edu>, 05 May 1992 [2.1].
%%%                        Wrap \leavevmode\hbox{} around \vbox{} with
%%%                        the \special so that \epsffile{} can be
%%%                        used inside \begin{center}...\end{center}
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Nelson H. F. Beebe
%%%                        <beebe@math.utah.edu>, 09 Dec 1992 [2.2].
%%%                        Introduce \epsfshow{true,false} and
%%%                        \epsfframe{true,false} macros; the latter
%%%                        suppresses the insertion of the PostScript,
%%%                        and instead just creates an empty box,
%%%                        which may be handy for rapid prototyping.
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Nelson H. F. Beebe
%%%                        <beebe@math.utah.edu>, 14 Dec 1992 [2.3].
%%%                        Add \epsfshowfilename{true,false}.  When
%%%                        true, and \epsfshowfalse is specified, the
%%%                        PostScript file name will be displayed
%%%                        centered in the figure box.
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Nelson H. F. Beebe
%%%                        <beebe@math.utah.edu>, 20 June 1993 [2.4].
%%%                        Remove non-zero debug setting of \epsfframemargin,
%%%                        and change margin handling to preserve EPS image
%%%                        size and aspect ratio, so that the actual
%%%                        box is \epsfxsize+\epsfframemargin wide by
%%%                        \epsfysize+\epsfframemargin high.
%%%                        Reduce output of \epsfshowfilenametrue to
%%%                        just the bare file name.
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Nelson H. F. Beebe
%%%                        <beebe@math.utah.edu>, 13 July 1993 [2.5].
%%%                        Add \epsfframethickness for control of
%%%                        \epsfframe frame lines.
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Nelson H. F. Beebe
%%%                        <beebe@math.utah.edu>, 02 July 1996 [2.6]
%%%                        Add missing initialization \epsfatendfalse;
%%%                        the lack of this resulted in the wrong
%%%                        BoundingBox being picked up, mea culpa, sigh...
%%%                        ---------------------------------------------
%%%
%%%                        ---------------------------------------------
%%%                        Revised by Nelson H. F. Beebe
%%%                        <beebe@math.utah.edu>, 25 October 1996 [2.7]
%%%                        Update to match changes in from dvips 5-600
%%%                        distribution: new user-accessible macros:
%%%                        \epsfclipon, \epsfclipoff, \epsfdrafton,
%%%                        \epsfdraftoff, change \empty to \epsfempty.
%%%                        ---------------------------------------------
%%%                        
%%%                        Modified to avoid verbosity, give help.
%%%                        --kb@cs.umb.edu, for Texinfo.
%%%  }
%%% ====================================================================
%
\ifx\epsfannounce\undefined \def\epsfannounce{\immediate\write16}\fi
 \epsfannounce{This is `epsf.tex' v2.7k <10 July 1997>}%
%
\newread\epsffilein    % file to \read
\newif\ifepsfatend     % need to scan to LAST %%BoundingBox comment?
\newif\ifepsfbbfound   % success?
\newif\ifepsfdraft     % use draft mode?
\newif\ifepsffileok    % continue looking for the bounding box?
\newif\ifepsfframe     % frame the bounding box?
\newif\ifepsfshow      % show PostScript file, or just bounding box?
\epsfshowtrue          % default is to display PostScript file
\newif\ifepsfshowfilename % show the file name if \epsfshowfalse specified?
\newif\ifepsfverbose   % report what you're making?
\newdimen\epsfframemargin % margin between box and frame
\newdimen\epsfframethickness % thickness of frame rules
\newdimen\epsfrsize    % vertical size before scaling
\newdimen\epsftmp      % register for arithmetic manipulation
\newdimen\epsftsize    % horizontal size before scaling
\newdimen\epsfxsize    % horizontal size after scaling
\newdimen\epsfysize    % vertical size after scaling
\newdimen\pspoints     % conversion factor
%
\pspoints = 1bp        % Adobe points are `big'
\epsfxsize = 0pt       % default value, means `use natural size'
\epsfysize = 0pt       % ditto
\epsfframemargin = 0pt % default value: frame box flush around picture
\epsfframethickness = 0.4pt % TeX's default rule thickness
%
\def\epsfbox#1{\global\def\epsfllx{72}\global\def\epsflly{72}%
   \global\def\epsfurx{540}\global\def\epsfury{720}%
   \def\lbracket{[}\def\testit{#1}\ifx\testit\lbracket
   \let\next=\epsfgetlitbb\else\let\next=\epsfnormal\fi\next{#1}}%
%
% We use \epsfgetlitbb if the user specified an explicit bounding box,
% and \epsfnormal otherwise.  Because \epsfgetbb can be called
% separately to retrieve the bounding box, we move the verbose
% printing the bounding box extents and size on the terminal to
% \epsfstatus.  Therefore, when the user provided the bounding box,
% \epsfgetbb will not be called, so we must call \epsfsetsize and
% \epsfstatus ourselves.
%
\def\epsfgetlitbb#1#2 #3 #4 #5]#6{%
   \epsfgrab #2 #3 #4 #5 .\\%
   \epsfsetsize
   \epsfstatus{#6}%
   \epsfsetgraph{#6}%
}%
%
\def\epsfnormal#1{%
    \epsfgetbb{#1}%
    \epsfsetgraph{#1}%
}%
%
\newhelp\epsfnoopenhelp{The PostScript image file must be findable by
TeX, i.e., somewhere in the TEXINPUTS (or equivalent) path.}%
%
\def\epsfgetbb#1{%
%
%   The first thing we need to do is to open the
%   PostScript file, if possible.
%
    \openin\epsffilein=#1
    \ifeof\epsffilein
        \errhelp = \epsfnoopenhelp
        \errmessage{Could not open file #1, ignoring it}%
    \else                       %process the file
        {%                      %start a group to contain catcode changes
            % Make all special characters, except space, to be of type
            % `other' so we process the file in almost verbatim mode
            % (TeXbook, p. 344).
            \chardef\other=12
            \def\do##1{\catcode`##1=\other}%
            \dospecials
            \catcode`\ =10
            \epsffileoktrue         %true while we are looping
            \epsfatendfalse     %[02-Jul-1996]: add forgotten initialization
            \loop               %reading lines from the EPS file
                \read\epsffilein to \epsffileline
                \ifeof\epsffilein %then no more input
                \epsffileokfalse %so set completion flag
            \else                %otherwise process one line
                \expandafter\epsfaux\epsffileline:. \\%
            \fi
            \ifepsffileok
            \repeat
            \ifepsfbbfound
            \else
                \ifepsfverbose
                    \immediate\write16{No BoundingBox comment found in %
                                    file #1; using defaults}%
                \fi
            \fi
        }%                      %end catcode changes
        \closein\epsffilein
    \fi                         %end of file processing
    \epsfsetsize                %compute size parameters
    \epsfstatus{#1}%
}%
%
% Clipping control:
\def\epsfclipon{\def\epsfclipstring{ clip}}%
\def\epsfclipoff{\def\epsfclipstring{\ifepsfdraft\space clip\fi}}%
\epsfclipoff % default for dvips is OFF
%
% The special that is emitted by \epsfsetgraph comes from this macro.
% It is defined separately to allow easy customization by other
% packages that first \input epsf.tex, then redefine \epsfspecial.
% This macro is invoked in the lower-left corner of a box of the
% width and height determined from the arguments to \epsffile, or
% from the %%BoundingBox in the EPS file itself.
%
% This version is for dvips:
\def\epsfspecial#1{%
     \epsftmp=10\epsfxsize
     \divide\epsftmp\pspoints
     \ifnum\epsfrsize=0\relax
       \special{PSfile=\ifepsfdraft psdraft.ps\else#1\fi\space
                llx=\epsfllx\space
                lly=\epsflly\space
                urx=\epsfurx\space
                ury=\epsfury\space
                rwi=\number\epsftmp
                \epsfclipstring
               }%
     \else
       \epsfrsize=10\epsfysize
       \divide\epsfrsize\pspoints
       \special{PSfile=\ifepsfdraft psdraft.ps\else#1\fi\space
                llx=\epsfllx\space
                lly=\epsflly\space
                urx=\epsfurx\space
                ury=\epsfury\space
                rwi=\number\epsftmp
                rhi=\number\epsfrsize
                \epsfclipstring
               }%
     \fi
}%
%
% \epsfframe macro adapted from the TeXbook, exercise 21.3, p. 223, 331.
% but modified to set the box width to the natural width, rather
% than the line width, and to include space for margins and rules
\def\epsfframe#1%
{%
  \leavevmode                   % so we can put this inside
                                % a centered environment
  \setbox0 = \hbox{#1}%
  \dimen0 = \wd0                                % natural width of argument
  \advance \dimen0 by 2\epsfframemargin         % plus width of 2 margins
  \advance \dimen0 by 2\epsfframethickness      % plus width of 2 rule lines
  \vbox
  {%
    \hrule height \epsfframethickness depth 0pt
    \hbox to \dimen0
    {%
      \hss
      \vrule width \epsfframethickness
      \kern \epsfframemargin
      \vbox {\kern \epsfframemargin \box0 \kern \epsfframemargin }%
      \kern \epsfframemargin
      \vrule width \epsfframethickness
      \hss
    }% end hbox
    \hrule height 0pt depth \epsfframethickness
  }% end vbox
}%
%
\def\epsfsetgraph#1%
{%
   %
   % Make the vbox and stick in a \special that the DVI driver can
   % parse.  \vfil and \hfil are used to place the \special origin at
   % the lower-left corner of the vbox.  \epsfspecial can be redefined
   % to produce alternate \special syntaxes.
   %
   \leavevmode
   \hbox{% so we can put this in \begin{center}...\end{center}
     \ifepsfframe\expandafter\epsfframe\fi
     {\vbox to\epsfysize
     {%
        \ifepsfshow
            % output \special{} at lower-left corner of figure box
            \vfil
            \hbox to \epsfxsize{\epsfspecial{#1}\hfil}%
        \else
            \vfil
            \hbox to\epsfxsize{%
               \hss
               \ifepsfshowfilename
               {%
                  \epsfframemargin=3pt % local change of margin
                  \epsfframe{{\tt #1}}%
               }%
               \fi
               \hss
            }%
            \vfil
        \fi
     }%
   }}%
   %
   % Reset \epsfxsize and \epsfysize, as documented above.
   %
   \global\epsfxsize=0pt
   \global\epsfysize=0pt
}%
%
%   Now we have to calculate the scale and offset values to use.
%   First we compute the natural sizes.
%
\def\epsfsetsize
{%
   \epsfrsize=\epsfury\pspoints
   \advance\epsfrsize by-\epsflly\pspoints
   \epsftsize=\epsfurx\pspoints
   \advance\epsftsize by-\epsfllx\pspoints
%
%   If `epsfxsize' is 0, we default to the natural size of the picture.
%   Otherwise we scale the graph to be \epsfxsize wide.
%
   \epsfxsize=\epsfsize{\epsftsize}{\epsfrsize}%
   \ifnum \epsfxsize=0
      \ifnum \epsfysize=0
        \epsfxsize=\epsftsize
        \epsfysize=\epsfrsize
        \epsfrsize=0pt
%
%   We have a sticky problem here:  TeX doesn't do floating point arithmetic!
%   Our goal is to compute y = rx/t. The following loop does this reasonably
%   fast, with an error of at most about 16 sp (about 1/4000 pt).
%
      \else
        \epsftmp=\epsftsize \divide\epsftmp\epsfrsize
        \epsfxsize=\epsfysize \multiply\epsfxsize\epsftmp
        \multiply\epsftmp\epsfrsize \advance\epsftsize-\epsftmp
        \epsftmp=\epsfysize
        \loop \advance\epsftsize\epsftsize \divide\epsftmp 2
        \ifnum \epsftmp>0
           \ifnum \epsftsize<\epsfrsize
           \else
              \advance\epsftsize-\epsfrsize \advance\epsfxsize\epsftmp
           \fi
        \repeat
        \epsfrsize=0pt
      \fi
   \else
     \ifnum \epsfysize=0
       \epsftmp=\epsfrsize \divide\epsftmp\epsftsize
       \epsfysize=\epsfxsize \multiply\epsfysize\epsftmp
       \multiply\epsftmp\epsftsize \advance\epsfrsize-\epsftmp
       \epsftmp=\epsfxsize
       \loop \advance\epsfrsize\epsfrsize \divide\epsftmp 2
       \ifnum \epsftmp>0
          \ifnum \epsfrsize<\epsftsize
          \else
             \advance\epsfrsize-\epsftsize \advance\epsfysize\epsftmp
          \fi
       \repeat
       \epsfrsize=0pt
     \else
       \epsfrsize=\epsfysize
     \fi
   \fi
}%
%
% Issue some status messages if the user requested them
%
\def\epsfstatus#1{% arg = filename
   \ifepsfverbose
     \immediate\write16{#1: BoundingBox:
                  llx = \epsfllx\space lly = \epsflly\space
                  urx = \epsfurx\space ury = \epsfury\space}%
     \immediate\write16{#1: scaled width = \the\epsfxsize\space
                  scaled height = \the\epsfysize}%
   \fi
}%
%
%   We still need to define the tricky \epsfaux macro. This requires
%   a couple of magic constants for comparison purposes.
%
{\catcode`\%=12 \global\let\epsfpercent=%\global\def\epsfbblit{%BoundingBox}}%
\global\def\epsfatend{(atend)}%
%
%   So we're ready to check for `%BoundingBox:' and to grab the
%   values if they are found.
%
%   If we find a line
%
%   %%BoundingBox: (atend)
%
%   then we ignore it, but set a flag to force parsing all of the
%   file, so the last %%BoundingBox parsed will be the one used.  This
%   is necessary, because EPS files can themselves contain other EPS
%   files with their own %%BoundingBox comments.
%
%   If we find a line
%
%   %%BoundingBox: llx lly urx ury
%
%   then we save the 4 values in \epsfllx, \epsflly, \epsfurx, \epsfury.
%   Then, if we have not previously parsed an (atend), we flag completion
%   and can stop reading the file.  Otherwise, we must keep on reading
%   to end of file so that we find the values on the LAST %%BoundingBox.
\long\def\epsfaux#1#2:#3\\%
{%
   \def\testit{#2}%             % save second character up to just before colon
   \ifx#1\epsfpercent           % then first char is percent (quick test)
       \ifx\testit\epsfbblit    % then (slow test) we have %%BoundingBox
            \epsfgrab #3 . . . \\%
            \ifx\epsfllx\epsfatend % then ignore %%BoundingBox: (atend)
                \global\epsfatendtrue
            \else               % else found %%BoundingBox: llx lly urx ury
                \ifepsfatend    % then keep parsing ALL %%BoundingBox lines
                \else           % else stop after first one parsed
                    \epsffileokfalse
                \fi
                \global\epsfbbfoundtrue
            \fi
       \fi
   \fi
}%
%
%   Here we grab the values and stuff them in the appropriate definitions.
%
\def\epsfempty{}%
\def\epsfgrab #1 #2 #3 #4 #5\\{%
   \global\def\epsfllx{#1}\ifx\epsfllx\epsfempty
      \epsfgrab #2 #3 #4 #5 .\\\else
   \global\def\epsflly{#2}%
   \global\def\epsfurx{#3}\global\def\epsfury{#4}\fi
}%
%
%   We default the epsfsize macro.
%
\def\epsfsize#1#2{\epsfxsize}%
%
%   Finally, another definition for compatibility with older macros.
%
\let\epsffile=\epsfbox
\endinput

cond-mat0103331.extracted_bib
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%\vspace{-1.cm}

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