%Paper: 
%From: gidi@vesta.TAU.AC.IL (Gideon Lana)
%Date: Fri, 9 Oct 92 11:17:34+020
%Date (revised): Wed, 25 Nov 92 13:43:20+020

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\begin{document}
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\hfill
TAUP 2000-92
\vglue 2.cm
\begin{center} {INSTABILITY OF BUBBLES \\
               \vglue 3pt
               NEAR THE HADRON QUARK-GLUON-PLASMA PHASE TRANSITION}\\
\vglue 5pt
\vglue 1.0cm
{GIDEON LANA }\\
\vglue 0.2cm
{and}\\
\vglue 0.2cm
{BENJAMIN SVETITSKY}\\
\vglue 0.5cm
\baselineskip=14pt
{\it School of Physics and astronomy}\\
\baselineskip=14pt
{\it Raymond and Beverly Sackler Faculty of Exact Sciences}\\
\baselineskip=14pt
{\it Tel Aviv University ,69978 Tel Aviv, Israel}\\

\vglue 0.8cm
{ABSTRACT}
\end{center}
\
\footnotetext[0]{Presented at the Workshop on "QCD Vacuum Structure and its
Applications", June 1-5 1992, Paris, France.}
\vglue 0.3cm
{\rightskip=3pc
 \leftskip=3pc
 \tenrm\baselineskip=12pt
 \noindent
The small surface tension of the interface between hadronic and
quark-gluon-plasma domains, along with a negative curvature tension, implies
that the uniform plasma is unstable against spontaneous formation of hadronic
bubbles. We furthermore show that spherical bubbles are not stable against
small perturbations.}


\vglue 0.8cm
{\bf\noindent 1. Introduction}
\vglue 0.2cm

\normalsize
\baselineskip=14pt

In what follows, we shall adopt the common assumption that the
quark-gluon-plasma phase turns into the hadron phase via a first order
transition. In other words, that at some temperatures the two phases may
coexist.

First order phase transition usually proceeds by nucleation of bubbles.
Consider for example, boiling water at atmospheric pressure near the transition
temperature $T_0=373 K$. The free energy of a bubble of vapor in the water has
a free energy which depends on its radius:

$$
F(R)=\Delta P \, {4\over 3}\pi R^3 + \sigma \, {4\pi R^2}\ +
 \ldots \ . \eqno{(1)}
$$
$\Delta P$ is the difference between the pressures of the two phases (zero at
the transition temperature $T=T_0 \ $, and positive above); $\sigma$ is the
excess of free energy per unit area for a planar interface.

$F(R)$ is shown schematically in figure 1.
For $T<T_0$, both the volume and surface terms act to suppress bubble
formation;
any bubble formed by a fluctuation collapses to $R=0$.
For $T>T_0$, however, the volume term \, $\Delta P \, 4
\pi R^3 / 3 \ $ encourages growth of bubbles
while the surface term \, $\sigma \, {4\pi R^2}\ $ creates a barrier.
The maximum of $F(R)$ occurs at some $R=R_c$.
A bubble formed in the superheated liquid with $R<R_c$ will shrink
away, but a bubble formed with $R>R_c$ will grow until the entire
system is vapor.

\begin{figure}
\vspace{9.0cm}
\caption{ \baselineskip=12pt
{\tenrm Schematic representation of bubble free energy in boiling water,
retaining only volume and area terms, for $T<T_0$ (solid) and $T>T_0$
(dashed).} }
\end{figure}

In Equation 1 we have indicated that there might be more terms beyond
the area term, having in mind an expansion in powers of $1/R$.
In  condensed-matter contexts, a 'curvature' term, $\alpha\cdot 8\pi R$ is
clearly negligible; by dimensional analysis, $\alpha/\sigma \sim \lambda$,
where
$\lambda$
is a microscopic scale, and  $(\alpha R)/(\sigma R^2) \sim
\lambda/R$,
an exceedingly small number since the typical nucleating bubble is
much larger than either the inter-molecular spacing or the
correlation length which are usually of the order of angstroms.
The picture in Figure 1, based on volume and area terms, is then the
whole story.

Note that our picture will be substantially the same for liquid
droplets in the vapor as for vapor bubbles in the liquid, since
$\sigma$ is the same in the two cases.
This is because the surface tension may be measured in planar
interfaces, and it doesn't matter which phase is on which side.



\vglue 0.8cm
{\bf\noindent 2. The QCD Transition}
\vglue 0.2cm


In QCD (at least in the limit of massless quarks) there is only one scale.
Masses, radii, the transition temperature, the latent heat, etc.
are all on the order of 200 MeV or 1 fm.
This is then the only ``microscopic'' size, and 1 fm should also
be the typical size of a bubble, and one cannot ignore additional terms in (1).
Indeed, Monte Carlo measurements $^{2,5,7}$, and MIT bag model
calculations$^{1,3}$, found that for QCD \, $\sigma/T_0^3 \ $ turn out to be a
small number, and one is forced to include additional terms,
$$
F(R)=\Delta P \, {4\over3}\pi R^3 +  \sigma \, {4\pi R^2}\ +
\alpha \, 8\pi R + \ldots\ .
\eqno(2)
$$

 Furthermore, it was found $^3$ that $\alpha$ is sizeable and negative.
Fig. 2. depicts $F(R)$ for a hadron bubble within the quark gluon plasma. It is
qualitatively different than that shown in fig. 1. Indeed, the minimum in the
free energy for $T>T_0$ implies that vacuum bubbles are stable.  The minimum of
$F(R)$ is not a mere artefact of using the expansion (2);  but an exact bag
model calculation $^3$ exhibited a very similar behavior.

 \begin{figure}
\vspace{9.0cm}
\caption{\baselineskip=12pt
{\tenrm
 Free energy of a hadron gas bubble in the plasma for temperatures near  the
transition temperature, fixed at $T_0=150~MeV$. $T=155~MeV$ (solid),
$T=T_0=150~MeV$ (dashed) and $T=147~MeV$ (dotted).  The figures are taken from
a bag model calculation of ref.. 3. } }
\end{figure}

It was further suggested ${3,4}$ that this result might transcend the bag
model:
All that is required is a thin interface, the energy of which is due
primarily to the reaction of the fields on either side.
A lattice calculation of $\alpha$ in the pure glue theory $^5$ apparently
supports this picture.

The fact that one can lower the free energy of the plasma phase by
creating a vacuum bubble seems at first glance to imply that the
equilibrium plasma is a Swiss cheese of bubbles, the density of which
is determined by the (unknown) bubble--bubble interaction.
This would only be true, however, if (1) the bubbles do not affect
each other, and (2) each bubble is stable in itself.
A simple calculation will teach us, however, that
spherical bubbles are unstable against deformation, and that growth
of long, narrow structures is favored.
We do not address the problem of determining the actual equilibrium
configuration of the high-temperature phase, a problem which will
require much more sophisticated analysis.

In order to study the stability of the spherical bubble against small
perturbations, we expand the free energy around the equilibrium spherical
bubble, of radius $R_0$.
Minimizing (2), we find that $R_0$ satisfies
$$
4\pi\Delta P\,R_0^2 + 8\pi\sigma R_0 + 8\pi\alpha=0\ .
\eqno{(3)}
$$
The generalization of (2) to non-spherical shapes is $^6$
$$
F=\Delta P \,V + \sigma\int dS +\alpha\int dS\,\left({1\over
R_1}+{1\over R_2}\right) + \cdots\ ,
\eqno{(4)}
$$
where the integrand in the third term is the local extrinsic curvature,
with $R_1$ and $R_2$ representing the principal radii of curvature.
Let the bubble surface be specified by
$$
R(\theta,\phi)=R_0 + \delR(\theta,\phi) \quad.
\eqno{(5)}
$$

The total $O(\delR)$ contribution to $F$ vanishes identically on the
equilibrium radius $R=R_0$.
The correction to the free energy is then second order and may be written as
$$
\delta F = \int \sin \theta\, d \theta \,d\phi \,\delR\left( A + B L^2 \right)
\delR \ .\eqno{(6)}
$$
$-L^2$ is just the angular part of the Laplacian operator, and
$$ A=\Delta P\, R_0 + \sigma\ ,
\qquad B={\sigma\over 2}+{\alpha\over R_0} \quad. \eqno{(7)}
$$
Recall that we presume $\alpha<0$.
When $R_0$ is determined by (3), it is easy to see that
$A$ is positive and $B$ negative.

Expanding $\delR$ in a multipole expansion,
$$
\delR(\theta,\phi)=\sum_{l,m}c_l^m Y_{lm}(\theta,\phi)\ ,\eqno{(8)}
$$
we have
$$
\delta F = \sum_{l,m}[A+Bl(l+1)]\left|c_l^m\right|^2\ .\eqno{(9)}
$$
Evidently $\delta F>0$ for a monopole perturbation, since (2) has been
minimized;
for a dipole perturbation we obtain $\delta F=0$ , consistent
with the fact that $l=1$ corresponds to a translation of the bubble.
For quadrupole and higher deformations, however, $\delta F$ is always
negative, showing instability against arbitrary changes of shape.


\vglue 0.8cm
{\bf\noindent 3. Discussion}
\vglue 0.4cm

We have here considered only static properties of the quark--hadron interface.
Dynamical processes in the course of the phase transition will
destabilize the interface even more.
For example, the necessity of carrying heat and baryon number away
from a growing vacuum bubble will lead to a negative effective surface
tension and hence to complex phenomena of pattern formation.
On the other hand, interaction of bubbles, which was completely ignored in all
the above treatments, may play a crucial role in stabilizing the bubbles'
surface.

A word of caution: the above analysis relies on a bag model calculation$^3$ .
And although Monte Carlo measurements confirmed the smalleness of $\sigma$ as
well as the sign and magnitude of $\alpha$ of the bag model calculation, it is
still far from actually confirming even qualitatively the appearance of a
minimum in $F(R)$ for a hadron bubble with finite radius surrounded by the
quark gluon plasma, as depicted in fig. 2. As a matter of fact, the lattice
results show that the interface is about 2fm thick, and it is clearly not
negligible compared to the temperature or to $R_0$ , whereas  the stability
analysis is based on a thin wall approximation.



\vskip 24pt
{\bf \noindent 4. Acknowledgements \hfil}
\vglue 0.4cm
This work was supported by a Wolfson Research Award administered by the
Israel Academy of Sciences and Humanities.
\vglue 0.6cm
{\bf\noindent 5. References \hfil}
\vglue 0.4cm

\begin{thebibliography}{9}

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\def\PLB#1{Phys. Lett. B {\bf #1}}
\def\PRC#1{Phys. Rev. C {\bf #1}}
\def\ZPC#1{Z. Phys. C {\bf #1}}
\def\NPA#1{Nucl. Phys. {\bf A#1}}
\def\NPB#1{Nucl. Phys. {\bf B#1}}
\def\AP#1{Ann. Phys. (NY) {\bf #1}}
\def\ibid#1{{\it ibid.\/} {\bf #1}}


\bibitem{FJ}E.~Farhi and R.~L.~Jaffe, \PRD{30} (1984) 2379.
\bibitem{K1}K. Kajantie and L. K\"arkk\"ainen, \PLB{214} (1988) 595;
       K. Kajantie, L. K\"arkk\"ainen, and K.~Rummukainen, \NPB{333}
(1990) 100. \newline
       S.~Huang, J.~Potvin, C.~Rebbi, and S.~Sanielevici, \PRD{42} (1990)
	2864; {\bf 43} (1991) 2056(E).\newline
       R.~Brower, S.~Huang, J.~Potvin, and C.~Rebbi, Boston University
preprint BUHEP-92-3 (unpublished, 1992).
\bibitem{MS}I.~Mardor and B.~Svetitsky, \PRD{44} (1991) 878.
\bibitem{BS}B. Svetitsky,
lecture given at the Workshop on QCD at Finite Temperature and Density,
Brookhaven National Laboratory, August 1991, Tel Aviv preprint TAUP 1909--91
(1991), to appear in proceedings.
\bibitem{K2}K. Kajantie, L. K\"arkk\"ainen, and K.~Rummukainen, Helsinki
	preprint HU-TFT-92-1 (unpublished, 1992).
\bibitem{BB}R.~Balian and C.~Bloch, \AP{60} (1970) 401.
\bibitem{BB} J. Potvin, these Proceedings.

\end{thebibliography}
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1601 68 L
1598 74 L
1598 61 L
1601 68 L
CS [] 0 setdash M
1338 455 M CS [] 0 setdash M
1079 422 M 1146 489 M 1146 422 L
1150 489 M 1150 422 L
1137 489 M 1175 489 L
1185 486 L
1188 483 L
1191 476 L
1191 470 L
1188 463 L
1185 460 L
1175 457 L
1150 457 L
1175 489 M 1182 486 L
1185 483 L
1188 476 L
1188 470 L
1185 463 L
1182 460 L
1175 457 L
1137 422 M 1159 422 L
1166 457 M 1172 454 L
1175 450 L
1185 428 L
1188 425 L
1191 425 L
1195 428 L
1172 454 M 1175 447 L
1182 425 L
1185 422 L
1191 422 L
1195 428 L
1195 431 L
1230 412 M 1228 410 L
1230 408 L
1232 410 L
1232 412 L
1228 416 L
1224 418 L
1218 418 L
1212 416 L
1209 412 L
1207 406 L
1207 402 L
1209 396 L
1212 393 L
1218 391 L
1222 391 L
1228 393 L
1232 396 L
1218 418 M 1214 416 L
1211 412 L
1209 406 L
1209 402 L
CS M
1211 396 L
1214 393 L
1218 391 L
CS [] 0 setdash M
stroke
grestore
showpage
end

%%%inst2.ps
%!PS-Adobe-2.0 EPSF-2.0
%%Creator: SM
%%BoundingBox: 18 144 593 718
%%DocumentFonts: Helvetica
%%EndComments
 20 dict begin
72 300 div dup scale
1 setlinejoin 0 setlinecap
/Helvetica findfont 55 scalefont setfont
/B { stroke newpath } def /F { moveto 0 setlinecap} def
/C { CS M 1 1 3 { pop 3 1 roll 255 div } for SET_COLOUR } def
/CS { currentpoint stroke } def
/CF { currentpoint fill } def
/L { lineto } def /M { moveto } def
/P { moveto 0 1 rlineto stroke } def
/T { 1 setlinecap show } def
errordict /nocurrentpoint { pop 0 0 M currentpoint } put
/SET_COLOUR { pop pop pop } def
 80 600 translate
gsave
CS [] 0 setdash M
CS M 2 setlinewidth
/P { moveto 0 2.05 rlineto stroke } def
 0 0 0 C
CS [] 0 setdash M
CS [] 0 setdash M
255 255 M 2261 255 L
255 255 M 255 324 L
355 255 M 355 289 L
455 255 M 455 289 L
556 255 M 556 289 L
656 255 M 656 289 L
756 255 M 756 324 L
857 255 M 857 289 L
957 255 M 957 289 L
1057 255 M 1057 289 L
1157 255 M 1157 289 L
1258 255 M 1258 324 L
1358 255 M 1358 289 L
1458 255 M 1458 289 L
1559 255 M 1559 289 L
1659 255 M 1659 289 L
1759 255 M 1759 324 L
1859 255 M 1859 289 L
1960 255 M 1960 289 L
2060 255 M 2060 289 L
2160 255 M 2160 289 L
2261 255 M 2261 324 L
223 158 M 251 225 M 242 222 L
235 213 L
232 196 L
232 187 L
235 171 L
242 161 L
251 158 L
258 158 L
268 161 L
274 171 L
277 187 L
277 196 L
274 213 L
268 222 L
258 225 L
251 225 L
245 222 L
242 219 L
239 213 L
235 196 L
235 187 L
239 171 L
242 164 L
245 161 L
251 158 L
258 158 M 264 161 L
268 164 L
271 171 L
274 187 L
274 196 L
271 213 L
268 219 L
264 222 L
258 225 L
724 158 M 743 213 M 750 216 L
759 225 L
759 158 L
756 222 M 756 158 L
743 158 M 772 158 L
1225 158 M 1238 213 M 1242 209 L
1238 206 L
1235 209 L
1235 213 L
1238 219 L
1242 222 L
1251 225 L
1264 225 L
1274 222 L
1277 219 L
1280 213 L
1280 206 L
1277 200 L
1267 193 L
1251 187 L
1245 184 L
1238 177 L
1235 167 L
1235 158 L
1264 225 M 1271 222 L
1274 219 L
1277 213 L
1277 206 L
1274 200 L
1264 193 L
1251 187 L
1235 164 M 1238 167 L
1245 167 L
1261 161 L
1271 161 L
1277 164 L
1280 167 L
1245 167 M 1261 158 L
1274 158 L
1277 161 L
1280 167 L
1280 174 L
1727 158 M 1740 213 M 1743 209 L
1740 206 L
1736 209 L
1736 213 L
1740 219 L
1743 222 L
1753 225 L
1765 225 L
1775 222 L
1778 216 L
1778 206 L
1775 200 L
1765 196 L
1756 196 L
1765 225 M 1772 222 L
1775 216 L
1775 206 L
1772 200 L
1765 196 L
1772 193 L
1778 187 L
1782 180 L
1782 171 L
1778 164 L
1775 161 L
1765 158 L
1753 158 L
1743 161 L
1740 164 L
1736 171 L
1736 174 L
1740 177 L
1743 174 L
1740 171 L
1775 190 M 1778 180 L
1778 171 L
1775 164 L
1772 161 L
1765 158 L
2228 158 M 2267 219 M 2267 158 L
2270 225 M 2270 158 L
2270 225 M 2235 177 L
2286 177 L
2257 158 M 2280 158 L
255 2261 M 2261 2261 L
255 2261 M 255 2192 L
355 2261 M 355 2226 L
455 2261 M 455 2226 L
556 2261 M 556 2226 L
656 2261 M 656 2226 L
756 2261 M 756 2192 L
857 2261 M 857 2226 L
957 2261 M 957 2226 L
1057 2261 M 1057 2226 L
1157 2261 M 1157 2226 L
1258 2261 M 1258 2192 L
1358 2261 M 1358 2226 L
1458 2261 M 1458 2226 L
1559 2261 M 1559 2226 L
1659 2261 M 1659 2226 L
1759 2261 M 1759 2192 L
1859 2261 M 1859 2226 L
1960 2261 M 1960 2226 L
2060 2261 M 2060 2226 L
2160 2261 M 2160 2226 L
2261 2261 M 2261 2192 L
255 255 M 255 2261 L
255 255 M 289 255 L
255 380 M 289 380 L
255 506 M 289 506 L
255 631 M 324 631 L
255 756 M 289 756 L
255 882 M 289 882 L
255 1007 M 289 1007 L
255 1132 M 289 1132 L
255 1258 M 324 1258 L
255 1383 M 289 1383 L
255 1508 M 289 1508 L
255 1634 M 289 1634 L
255 1759 M 289 1759 L
255 1884 M 324 1884 L
255 2010 M 289 2010 L
255 2135 M 289 2135 L
255 2261 M 289 2261 L
77 597 M 90 626 M 148 626 L
177 665 M 171 632 L
177 639 L
187 642 L
196 642 L
206 639 L
212 632 L
216 623 L
216 616 L
212 607 L
206 600 L
196 597 L
187 597 L
177 600 L
174 603 L
171 610 L
171 613 L
174 616 L
177 613 L
174 610 L
196 642 M 203 639 L
209 632 L
212 623 L
212 616 L
209 607 L
203 600 L
196 597 L
177 665 M 209 665 L
177 661 M 193 661 L
209 665 L
161 1224 M 190 1292 M 180 1288 L
174 1279 L
171 1262 L
171 1253 L
174 1237 L
180 1227 L
190 1224 L
196 1224 L
206 1227 L
213 1237 L
216 1253 L
216 1262 L
213 1279 L
206 1288 L
196 1292 L
190 1292 L
184 1288 L
180 1285 L
177 1279 L
174 1262 L
174 1253 L
177 1237 L
180 1230 L
184 1227 L
190 1224 L
196 1224 M 203 1227 L
206 1230 L
209 1237 L
213 1253 L
213 1262 L
209 1279 L
206 1285 L
203 1288 L
196 1292 L
161 1851 M 177 1918 M 171 1886 L
177 1892 L
187 1896 L
196 1896 L
206 1892 L
213 1886 L
216 1876 L
216 1870 L
213 1860 L
206 1854 L
196 1851 L
187 1851 L
177 1854 L
174 1857 L
171 1863 L
171 1867 L
174 1870 L
177 1867 L
174 1863 L
196 1896 M 203 1892 L
209 1886 L
213 1876 L
213 1870 L
209 1860 L
203 1854 L
196 1851 L
177 1918 M 209 1918 L
177 1915 M 193 1915 L
209 1918 L
2261 255 M 2261 2261 L
2261 255 M 2226 255 L
2261 380 M 2226 380 L
2261 506 M 2226 506 L
2261 631 M 2192 631 L
2261 756 M 2226 756 L
2261 882 M 2226 882 L
2261 1007 M 2226 1007 L
2261 1132 M 2226 1132 L
2261 1258 M 2192 1258 L
2261 1383 M 2226 1383 L
2261 1508 M 2226 1508 L
2261 1634 M 2226 1634 L
2261 1759 M 2226 1759 L
2261 1884 M 2192 1884 L
2261 2010 M 2226 2010 L
2261 2135 M 2226 2135 L
2261 2261 M 2226 2261 L
CS [] 0 setdash M
255 1258 M 305 1249 L
355 1240 L
405 1231 L
455 1223 L
505 1215 L
556 1208 L
606 1202 L
CS M
656 1197 L
706 1193 L
756 1190 L
806 1188 L
857 1188 L
907 1190 L
957 1193 L
1007 1198 L
1057 1206 L
1107 1215 L
1157 1227 L
1207 1241 L
1258 1258 L
1308 1277 L
1358 1299 L
1408 1325 L
1458 1353 L
1508 1385 L
1559 1420 L
1609 1458 L
1659 1500 L
1709 1546 L
1759 1596 L
1809 1650 L
1859 1708 L
1910 1771 L
1960 1838 L
2010 1909 L
2060 1986 L
2110 2067 L
2160 2153 L
2210 2244 L
2219 2261 L
255 1258 M 305 1258 L
355 1258 L
405 1258 L
455 1258 L
505 1258 L
556 1258 L
606 1258 L
656 1258 L
706 1258 L
756 1258 L
806 1258 L
857 1258 L
907 1258 L
957 1258 L
1007 1258 L
1057 1258 L
1107 1258 L
1157 1258 L
1207 1258 L
1258 1258 L
1308 1258 L
1358 1258 L
1408 1258 L
1458 1258 L
1508 1258 L
1559 1258 L
1609 1258 L
1659 1258 L
1709 1258 L
1759 1258 L
1809 1258 L
1859 1258 L
1910 1258 L
1960 1258 L
2010 1258 L
2060 1258 L
2110 1258 L
2160 1258 L
2210 1258 L
2261 1258 L
CS [80 24] 0 setdash M
255 1258 M 305 1249 L
355 1240 L
405 1231 L
455 1222 L
505 1213 L
556 1203 L
606 1194 L
656 1185 L
706 1176 L
756 1167 L
806 1158 L
857 1149 L
907 1140 L
957 1131 L
1007 1122 L
1057 1113 L
1107 1104 L
1157 1095 L
1207 1086 L
1258 1077 L
1308 1068 L
1358 1059 L
1408 1050 L
1458 1041 L
1508 1032 L
1559 1023 L
1609 1014 L
1659 1005 L
1709 996 L
1759 987 L
1809 978 L
1859 969 L
1910 960 L
1960 951 L
2010 942 L
2060 933 L
2110 924 L
2160 915 L
2210 906 L
2261 897 L
CS [6 12] 0 setdash M
255 1258 M 305 1249 L
355 1240 L
405 1230 L
455 1221 L
505 1211 L
556 1201 L
606 1190 L
656 1179 L
706 1167 L
756 1155 L
806 1142 L
857 1128 L
907 1113 L
957 1097 L
1007 1080 L
1057 1062 L
1107 1043 L
1157 1022 L
1207 1000 L
1258 977 L
1308 952 L
1358 926 L
1408 898 L
1458 868 L
1508 836 L
1559 803 L
1609 767 L
1659 730 L
1709 690 L
1759 648 L
1809 604 L
1859 558 L
1910 509 L
1960 458 L
2010 404 L
2060 348 L
2110 289 L
2138 255 L
CS [] 0 setdash M
87 975 M 20 991 M 87 991 L
20 995 M 87 995 L
39 1014 M 65 1014 L
20 982 M 20 1033 L
39 1033 L
20 1030 L
52 995 M 52 1014 L
87 982 M 87 1004 L
7 1075 M 13 1068 L
23 1062 L
36 1055 L
52 1052 L
65 1052 L
81 1055 L
93 1062 L
103 1068 L
109 1075 L
13 1068 M 26 1062 L
36 1059 L
52 1055 L
65 1055 L
81 1059 L
90 1062 L
103 1068 L
20 1100 M 87 1100 L
20 1104 M 87 1104 L
20 1091 M 20 1129 L
23 1139 L
26 1142 L
32 1145 L
39 1145 L
45 1142 L
49 1139 L
52 1129 L
52 1104 L
20 1129 M 23 1136 L
26 1139 L
32 1142 L
39 1142 L
45 1139 L
49 1136 L
52 1129 L
87 1091 M 87 1113 L
52 1120 M 55 1126 L
58 1129 L
81 1139 L
84 1142 L
84 1145 L
81 1149 L
55 1126 M 61 1129 L
84 1136 L
87 1139 L
87 1145 L
81 1149 L
77 1149 L
7 1165 M 13 1171 L
23 1177 L
36 1184 L
52 1187 L
65 1187 L
81 1184 L
93 1177 L
103 1171 L
109 1165 L
13 1171 M 26 1177 L
36 1181 L
52 1184 L
65 1184 L
81 1181 L
90 1177 L
103 1171 L
7 1287 M 13 1280 L
23 1274 L
36 1267 L
52 1264 L
65 1264 L
81 1267 L
93 1274 L
103 1280 L
109 1287 L
13 1280 M 26 1274 L
36 1271 L
52 1267 L
65 1267 L
81 1271 L
90 1274 L
103 1280 L
29 1351 M 39 1354 L
20 1354 L
29 1351 L
23 1344 L
20 1335 L
20 1328 L
23 1319 L
29 1312 L
36 1309 L
45 1306 L
61 1306 L
71 1309 L
77 1312 L
84 1319 L
87 1328 L
87 1335 L
84 1344 L
77 1351 L
20 1328 M 23 1322 L
29 1315 L
36 1312 L
45 1309 L
61 1309 L
71 1312 L
77 1315 L
84 1322 L
87 1328 L
61 1351 M 87 1351 L
61 1354 M 87 1354 L
61 1341 M 61 1364 L
61 1383 M 61 1421 L
55 1421 L
49 1418 L
45 1415 L
42 1408 L
42 1399 L
45 1389 L
52 1383 L
61 1380 L
68 1380 L
77 1383 L
84 1389 L
87 1399 L
87 1405 L
84 1415 L
77 1421 L
61 1418 M 52 1418 L
45 1415 L
42 1399 M 45 1392 L
52 1386 L
61 1383 L
68 1383 L
77 1386 L
84 1392 L
87 1399 L
20 1441 M 87 1463 L
20 1444 M 77 1463 L
20 1486 M 87 1463 L
20 1434 M 20 1453 L
20 1473 M 20 1492 L
7 1505 M 13 1511 L
23 1518 L
CS M
36 1524 L
52 1527 L
65 1527 L
81 1524 L
93 1518 L
103 1511 L
109 1505 L
13 1511 M 26 1518 L
36 1521 L
52 1524 L
65 1524 L
81 1521 L
90 1518 L
103 1511 L
CS [6 12] 0 setdash M
CS [] 0 setdash M
845 48 M 913 116 M 913 48 L
916 116 M 916 48 L
903 116 M 942 116 L
951 113 L
954 109 L
958 103 L
958 97 L
954 90 L
951 87 L
942 84 L
916 84 L
942 116 M 948 113 L
951 109 L
954 103 L
954 97 L
951 90 L
948 87 L
942 84 L
903 48 M 926 48 L
932 84 M 938 80 L
942 77 L
951 55 L
954 52 L
958 52 L
961 55 L
938 80 M 942 74 L
948 52 L
951 48 L
958 48 L
961 55 L
961 58 L
983 87 M 983 84 L
980 84 L
980 87 L
983 90 L
990 93 L
1003 93 L
1009 90 L
1012 87 L
1015 80 L
1015 58 L
1019 52 L
1022 48 L
1012 87 M 1012 58 L
1015 52 L
1022 48 L
1025 48 L
1012 80 M 1009 77 L
990 74 L
980 71 L
977 64 L
977 58 L
980 52 L
990 48 L
999 48 L
1006 52 L
1012 58 L
990 74 M 983 71 L
980 64 L
980 58 L
983 52 L
990 48 L
1080 116 M 1080 48 L
1083 116 M 1083 48 L
1080 84 M 1073 90 L
1067 93 L
1060 93 L
1051 90 L
1044 84 L
1041 74 L
1041 68 L
1044 58 L
1051 52 L
1060 48 L
1067 48 L
1073 52 L
1080 58 L
1060 93 M 1054 90 L
1047 84 L
1044 74 L
1044 68 L
1047 58 L
1054 52 L
1060 48 L
1070 116 M 1083 116 L
1080 48 M 1092 48 L
1115 116 M 1112 113 L
1115 109 L
1118 113 L
1115 116 L
1115 93 M 1115 48 L
1118 93 M 1118 48 L
1105 93 M 1118 93 L
1105 48 M 1128 48 L
1150 93 M 1150 58 L
1153 52 L
1163 48 L
1169 48 L
1179 52 L
1185 58 L
1153 93 M 1153 58 L
1157 52 L
1163 48 L
1185 93 M 1185 48 L
1189 93 M 1189 48 L
1141 93 M 1153 93 L
1176 93 M 1189 93 L
1185 48 M 1198 48 L
1246 87 M 1250 93 L
1250 80 L
1246 87 L
1243 90 L
1237 93 L
1224 93 L
1218 90 L
1214 87 L
1214 80 L
1218 77 L
1224 74 L
1240 68 L
1246 64 L
1250 61 L
1214 84 M 1218 80 L
1224 77 L
1240 71 L
1246 68 L
1250 64 L
1250 55 L
1246 52 L
1240 48 L
1227 48 L
1221 52 L
1218 55 L
1214 61 L
1214 48 L
1218 55 L
1327 116 M 1327 48 L
1330 116 M 1330 48 L
1317 116 M 1356 116 L
1365 113 L
1368 109 L
1372 103 L
1372 97 L
1368 90 L
1365 87 L
1356 84 L
1330 84 L
1356 116 M 1362 113 L
1365 109 L
1368 103 L
1368 97 L
1365 90 L
1362 87 L
1356 84 L
1317 48 M 1339 48 L
1346 84 M 1352 80 L
1356 77 L
1365 55 L
1368 52 L
1372 52 L
1375 55 L
1352 80 M 1356 74 L
1362 52 L
1365 48 L
1372 48 L
1375 55 L
1375 58 L
1468 129 M 1461 122 L
1455 113 L
1449 100 L
1445 84 L
1445 71 L
1449 55 L
1455 42 L
1461 32 L
1468 26 L
1461 122 M 1455 109 L
1452 100 L
1449 84 L
1449 71 L
1452 55 L
1455 45 L
1461 32 L
1510 113 M 1506 109 L
1510 106 L
1513 109 L
1513 113 L
1510 116 L
1503 116 L
1497 113 L
1494 106 L
1494 48 L
1503 116 M 1500 113 L
1497 106 L
1497 48 L
1484 93 M 1510 93 L
1484 48 M 1506 48 L
1535 93 M 1535 48 L
1538 93 M 1538 48 L
1538 84 M 1545 90 L
1555 93 L
1561 93 L
1571 90 L
1574 84 L
1574 48 L
1561 93 M 1567 90 L
1571 84 L
1571 48 L
1574 84 M 1580 90 L
1590 93 L
1596 93 L
1606 90 L
1609 84 L
1609 48 L
1596 93 M 1603 90 L
1606 84 L
1606 48 L
1526 93 M 1538 93 L
1526 48 M 1548 48 L
1561 48 M 1583 48 L
1596 48 M 1619 48 L
1635 129 M 1641 122 L
1648 113 L
1654 100 L
1657 84 L
1657 71 L
1654 55 L
1648 42 L
1641 32 L
1635 26 L
1641 122 M 1648 109 L
1651 100 L
1654 84 L
1654 71 L
1651 55 L
1648 45 L
1641 32 L
CS [6 12] 0 setdash M
531 505 M CS [] 0 setdash M
531 477 M 555 533 M 551 529 L
546 522 L
542 513 L
540 502 L
540 493 L
542 482 L
546 473 L
551 466 L
555 462 L
551 529 M 546 520 L
544 513 L
542 502 L
542 493 L
544 482 L
546 475 L
551 466 L
569 509 M 582 477 L
571 509 M 582 482 L
596 509 M 582 477 L
564 509 M 578 509 L
587 509 M 600 509 L
614 504 M 614 502 L
611 502 L
611 504 L
614 507 L
618 509 L
627 509 L
632 507 L
634 504 L
636 500 L
636 484 L
638 480 L
641 477 L
634 504 M 634 484 L
636 480 L
641 477 L
643 477 L
634 500 M 632 498 L
618 495 L
611 493 L
609 489 L
609 484 L
611 480 L
618 477 L
625 477 L
629 480 L
634 484 L
618 495 M 614 493 L
611 489 L
CS M
611 484 L
614 480 L
618 477 L
681 502 M 679 500 L
681 498 L
683 500 L
683 502 L
679 507 L
674 509 L
667 509 L
661 507 L
656 502 L
654 495 L
654 491 L
656 484 L
661 480 L
667 477 L
672 477 L
679 480 L
683 484 L
667 509 M 663 507 L
658 502 L
656 495 L
656 491 L
658 484 L
663 480 L
667 477 L
701 509 M 701 484 L
703 480 L
710 477 L
715 477 L
721 480 L
726 484 L
703 509 M 703 484 L
706 480 L
710 477 L
726 509 M 726 477 L
728 509 M 728 477 L
694 509 M 703 509 L
719 509 M 728 509 L
726 477 M 735 477 L
750 509 M 750 484 L
753 480 L
760 477 L
764 477 L
771 480 L
775 484 L
753 509 M 753 484 L
755 480 L
760 477 L
775 509 M 775 477 L
777 509 M 777 477 L
744 509 M 753 509 L
768 509 M 777 509 L
775 477 M 784 477 L
800 509 M 800 477 L
802 509 M 802 477 L
802 502 M 807 507 L
813 509 L
818 509 L
825 507 L
827 502 L
827 477 L
818 509 M 822 507 L
825 502 L
825 477 L
827 502 M 831 507 L
838 509 L
843 509 L
849 507 L
852 502 L
852 477 L
843 509 M 847 507 L
849 502 L
849 477 L
793 509 M 802 509 L
793 477 M 809 477 L
818 477 M 834 477 L
843 477 M 858 477 L
910 525 M 910 477 L
912 525 M 912 477 L
912 502 M 917 507 L
921 509 L
926 509 L
932 507 L
937 502 L
939 495 L
939 491 L
937 484 L
932 480 L
926 477 L
921 477 L
917 480 L
912 484 L
926 509 M 930 507 L
935 502 L
937 495 L
937 491 L
935 484 L
930 480 L
926 477 L
903 525 M 912 525 L
957 509 M 957 484 L
959 480 L
966 477 L
970 477 L
977 480 L
982 484 L
959 509 M 959 484 L
961 480 L
966 477 L
982 509 M 982 477 L
984 509 M 984 477 L
950 509 M 959 509 L
975 509 M 984 509 L
982 477 M 991 477 L
1006 525 M 1006 477 L
1009 525 M 1009 477 L
1009 502 M 1013 507 L
1018 509 L
1022 509 L
1029 507 L
1033 502 L
1036 495 L
1036 491 L
1033 484 L
1029 480 L
1022 477 L
1018 477 L
1013 480 L
1009 484 L
1022 509 M 1027 507 L
1031 502 L
1033 495 L
1033 491 L
1031 484 L
1027 480 L
1022 477 L
1000 525 M 1009 525 L
1053 525 M 1053 477 L
1056 525 M 1056 477 L
1056 502 M 1060 507 L
1065 509 L
1069 509 L
1076 507 L
1080 502 L
1083 495 L
1083 491 L
1080 484 L
1076 480 L
1069 477 L
1065 477 L
1060 480 L
1056 484 L
1069 509 M 1074 507 L
1078 502 L
1080 495 L
1080 491 L
1078 484 L
1074 480 L
1069 477 L
1047 525 M 1056 525 L
1101 525 M 1101 477 L
1103 525 M 1103 477 L
1094 525 M 1103 525 L
1094 477 M 1110 477 L
1123 495 M 1150 495 L
1150 500 L
1148 504 L
1145 507 L
1141 509 L
1134 509 L
1127 507 L
1123 502 L
1121 495 L
1121 491 L
1123 484 L
1127 480 L
1134 477 L
1139 477 L
1145 480 L
1150 484 L
1148 495 M 1148 502 L
1145 507 L
1134 509 M 1130 507 L
1125 502 L
1123 495 L
1123 491 L
1125 484 L
1130 480 L
1134 477 L
1163 533 M 1168 529 L
1172 522 L
1177 513 L
1179 502 L
1179 493 L
1177 482 L
1172 473 L
1168 466 L
1163 462 L
1168 529 M 1172 520 L
1175 513 L
1177 502 L
1177 493 L
1175 482 L
1172 475 L
1168 466 L
CS [6 12] 0 setdash M
stroke
grestore
showpage
end

