Content-Type: multipart/mixed; boundary="-------------0404262127805" This is a multi-part message in MIME format. ---------------0404262127805 Content-Type: text/plain; name="04-130.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-130.comments" For more information, you may send me an e-mail: yjkim@feynman.uprm.edu. ---------------0404262127805 Content-Type: text/plain; name="04-130.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-130.keywords" Superconductvity, Josephson effect ---------------0404262127805 Content-Type: application/x-tex; name="jos.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="jos.tex" %%%%%%%%%%%%%%%%%%%%%% MANUSCRIPT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentstyle[aps,preprint,prb]{revtex} %\documentstyle[aps,preprint,prb] \begin{document} \tightenlines %\pagestyle{empty} \def\vk{\vec k} \def\br{{\bf r}} \title{\bf A New Approach to the Josephson Effect } \author{Yong-Jihn Kim} \address{Department of Physics, University of Puerto Rico$^{\dagger}$ \\ Mayaguez, PR 00681\\ Department of Physics, Bilkent University\\ 06533 Bilkent, Ankara, Turkey} \maketitle \begin{abstract} We introduce a new approach to the Josephson effect in SIS tunnel junctions. The Josephson coupling energy is calculated from the overlap of real space Cooper pair wavefunctions in two superconductors through an insulating barrier. It is shown that the Josephson tunneling is limited by the size of the Cooper pair and its shrinking during the tunneling. Therefore, the Josephson coupling energy and the critical current become extremely small in high $T_{c}$ superconductors, including $MgB_{2}$. This shrinking also causes the observed DC supercurrent in low $T_{c}$ superconductors, such as Pb and Sn, to fall off much faster than $1/R_{n}$ for tunneling resistance $R_{n}$ above several ohms. Consequently there is a material-dependent threshold resistance, above which the supercurrent decreases much faster with increasing resistance. The impurity-induced shrinking is also shown to limit the critical current. Furthermore, the (weak) temperature dependence of the Cooper pair size is found to contribute to the temperature dependence of the DC supercurrent. \end{abstract} \vskip 3pc $\dagger$ Present Address PACS numbers: 74.50.+r, 74.20Fg, 74.25.Sv \newpage \section{\bf Introduction} The Josephson effect may be the most fascinating and intriguing property of superconductors.$^{1,2}$ In 1962 Josephson predicted that supercurrents can flow through the insulating barrier between two superconductors due to the Cooper pair tunneling.$^{1}$ The supercurrent depends on the relative phase of two superconductors, where the phase is associated with the order parameter of the superconducting condensed state. The prediction was confirmed in experiments quickly.$^{3,4}$ It is interesting that the experiment by Nicol et al.$^{5}$ showed already the possible DC supercurrent, although its reality was not noticed. This discovery led to the fabrication of the Josephson junction devices and the high precision measurement of fundamental constants.$^{6}$ Now it seems that our understanding of the Josephson effect is on firm ground.$^{6,7,8}$ Josephson$^{1,2}$ used the tunneling Hamiltonian$^{9}$ to calculate tunneling currents between two superconductors and noticed that even at zero applied voltages, a DC supercurrent can occur. At finite voltages V, he found an AC supercurrent of the same amplitude as that of the DC supercurrent with frequency 2eV/h. The amplitude of the supercurrent was calculated by Anderson$^{7}$ and Ambegaokar and Baratoff.$^{10}$ At $T=0K$ their result of the DC supercurrent, j, is \begin{equation} j=j_{1}sin\phi, \end{equation} with \begin{equation} j_{1}={\pi\over 2e}{\Delta\over R_{n}}. \end{equation} Here $\phi$ is the phase difference of the two superconductors with the same energy gap $\Delta$. Note that the amplitude of the supercurrent is proportional to the energy gap and inversely proportional to the tunneling resistance. However, there are still some fundamental experiments which remain puzzling.$^{11,12,13,14,15}$ For instance, the pair-quasi particle interference term was shown to be negative in experiments, whereas the theory predicts positive sign.$^{11}$ For AC Josephson effect, Likharev even claimed that {\sl there are more questions than answers}.$^{12}$ Furthermore, high $T_{c}$ cuprate Josephson junctions are mainly SNS type junctions, since SIS junctions do not show the Josephson effect.$^{13}$ This result is not consistent with the above theory, which predicts bigger Josephson current for higher $T_{c}$.$^{1,2,7,8,10}$ Recently, this behavior has also been found in $MgB_{2}$ SIS junctions.$^{14,15}$ In particular, the Josephson supercurrent was not observed for the big gap ($\sim 7meV$) of $MgB_{2}$. We stress that this behavior is closely related to the anomalous dependence of the maximum DC Josephson current on the tunneling resistance $R_{n}$, found in low $T_{c}$ superconductors, such as Pb and Sn:$^{16,17,18,19}$ the Josephson current decreases much faster than $1/R_{n}$ above several $\Omega$. In this rapid communication we present Cooper pair wavefunction approach to the Josephson effect. It is shown that the Josephson coupling energy is determined by the overlap of the Cooper pair wavefunctions of two superconductors divided by a thin insulating layer. (In fact, this idea was suggested by Josephson in his original paper.$^{1,10}$ However, it was not pursued thoroughly.) The critical current is, then, calculated from the coupling energy $E_{J}$ in the usual way.$^{2}$ Since the Cooper pair tunneling can be understood more easily from the Cooper pair wavefunction, this method is advantageous to deal with the Josephson effect in high $T_{c}$ superconductors including Mg$B_{2}$ and for the insulating barriers with high tunneling resistance $R_{n}$. We have found that the Cooper pair tunneling is strongly limited by the size of the Cooper pair and its reduction during the tunneling. Since the Cooper pair size is also reduced by the impurity potential scattering, ordinary impurity also limits the supercurrent. As a result, the Josephson current is much smaller than expected in higher $T_{c}$ SIS junctions and falls off much faster than $1/R_{n}$ for thicker and/or higher insulating barriers, in good agreement with the experimental findings. The preliminary result was reported before.$^{20}$ \section{\bf Cooper pair wavefunction approach to Josephson Tunneling : T=0K} \subsection{\bf Josephson Coupling Energy} The approximate expression of the supercurrent suggested by Josephson$^{1}$ is \begin{equation} j\cong{1\over 2}j_{1}\psi^{*}_{l}\psi_{r}+{1\over 2}j_{1}\psi_{r}^{*}\psi_{l}, \end{equation} where $\psi_{l}$ and $\psi_{r}$ are the effective superconducting wavefunctions on the left and right sides, respectively.$^{21,22}$ Actually, it can be exact and it is better to write first the Josephson coupling energy $E_{J}$ in terms of the Cooper pair wavefunctions and then calculate the supercurrent j from $E_{J}$:$^{2}$ \begin{equation} j={2e\over \hbar}{\partial E_{J}\over \partial \phi}. \end{equation} Figure 1 shows the real space Cooper pair wavefunctions of two superconductors with the same energy gap near the insulating barrier. The solid lines denote the Cooper pair wavefunctions in the absence of the barrier, whereas the thick lines denote the change of the Cooper pair wavefunctions due to the insulating barrier. The Josephson coupling energy is, therefore, determined by the overlap of the Cooper pair wavefunctions, connected by the phonon Green's function.$^{23,24}$ Previously, this idea was employed in some cases.$^{25,26}$ For instance, for a pure superconductor without a barrier with the Einstein phonons, the coupling energy or the pairing energy is given by \begin{eqnarray} E_{int}&=&V\int\int F^{*}(x,y)F(x,y)\delta(x-y)dxdy\nonumber\\ &=& V\sum_{k}\sum_{k'}u_{k}v_{k}u_{k'}v_{k'}, \end{eqnarray} where the (effective) Cooper pair wavefunctions $F(x,y)(=\sum_{k}u_{k}v_{k}e^{i{\bf k}\cdot ({\bf r}-{\bf y})}$) and $F^{*}(x,y)$ are coupled by the Einstein phonon Green's function, i.e., the Dirac delta function, $\delta(x-y)$. Here V is the phonon-mediated matrix element. In the same way, the Josephson coupling energy $E_{J}$ is given by \begin{eqnarray} E_{J}&=&V \int\int F_{r}^{*}(x,y)F_{l}(x,y)\delta(x-y)dxdy +V \int\int F_{l}^{*}(x,y)F_{r}(x,y)\delta(x-y)dxdy\nonumber\\ &=& V\int F_{r}^{*}(x)F_{l}(x)dx + V\int F_{l}^{*}(x)F_{r}(x)dx, \end{eqnarray} with $F_{l}$ and $F_{r}$ being the effective Cooper pair wavefunctions in left and right sides. Observe that this expression has the same form as the approximate one, Eq. (3) except the integration. Figure 1 shows that it is essential to calculate the tail of the Cooper pair wavefunctions to determine the Josephson coupling energy. It is necessary to emphasize that Eq. (6) leads to the familiar result in k-space. From the tunneling Hamiltonian$^{9}$ \begin{equation} H_{T}=\sum_{kk's}(T_{kk'}c_{k's}^{r+}c_{ks}^{l}+H.C.) \end{equation} one finds the Josephson pair tunneling Hamiltonian, $H_{J}$,$^{27,28,29}$ \begin{equation} H_{J}=-\sum_{kq}(J_{kq}c^{r+}_{k\uparrow}c^{r+}_{-k\downarrow}c^{l}_{-q\downarrow}c^{l}_{q\uparrow}+H.C.). \end{equation} Then, the coupling energy is \begin{eqnarray} E_{J}&=&-J(\sum_{k}u^{r}_{k}v_{k}^{r*}\sum_{q}u_{q}^{l}v_{q}^{l}+ \sum_{k}u^{r}_{k}v_{k}^{r}\sum_{q}u_{q}^{l}v_{q}^{l*})\nonumber\\ &=&-J(\sum_{kq}F_{k}^{r*}F_{q}^{l}+\sum_{kq}F_{k}^{r}F_{q}^{l*}) \end{eqnarray} with $J_{kq}\equiv J$. Comparing Eqs. (6) and (9), it is evident that J is proportional to the phonon-mediated matrix element V, i.e., \begin{equation} J\propto V. \end{equation} Accordingly, it is crucial to note that $J_{kq}$ is the pair scattering matrix element across the barrier due to both the tunneling and the electron-phonon interaction. If we define \begin{eqnarray} \Psi_{r}&\equiv &\sum_{k}F_{k}^{r}=|\Psi_{r}|e^{i\phi_{r}}\nonumber\\ \Psi_{l}&\equiv &\sum_{q}F_{q}^{l}=|\Psi_{l}|e^{i\phi_{l}} \end{eqnarray} Eq. (9) is rewritten \begin{equation} E_{J}=-J(\Psi_{l}^{*}\Psi_{r}+\Psi_{l}\Psi_{r}^{*})=-2J|\Psi_{l}||\Psi_{r}|cos(\phi_{l}-\phi_{r}). \end{equation} It should be noticed that the result is formally the same as the internal Josephson coupling energy in two-band superconductors,$^{27,28}$ and believed to reproduce the Josephson coupling energy $E_{J}$ in conventional SIS junctions.$^{27,28,29}$ However, the exact expression of $J_{kq}\equiv J$ is missing, although the (2nd order) tunneling-induced pair transfer term ($J^{T}_{kq}$), \begin{equation} J^{T}_{kq}={|T_{kq}|^{2}\over E_{k}^{l}+E_{q}^{r}}, \end{equation} was confused with J.$^{27,29}$ Here $E_{k}^{l}$ and $E_{q}^{r}$ are quasi-particle energies. It would be dangerous indeed to suppose that $J_{kq}\equiv J$ (phonon-mediated pair-scattering matrix element across the barrier) and $J^{T}_{kq}$ (tunneling-induced only pair-scattering matrix element) are the same when they are not. Note that $J_{kq}^{T}$ is not proportional to V, unlike J, so Eq. (9) or Eq. (12) with $J^{T}_{kq}$ does not reproduce the previous result, Eq. (2), which remains puzzling.$^{29}$ \subsection{\bf k-space Approach } Now we need to compute Eq. (6) or Eq. (9) to find the Josephson coupling energy $E_{J}$ and the Josephson current j. It is clear that both equations should lead to the same result. First, we consider Eq. (9), which requires the calculation of the pair-scattering matrix element $J_{kq}$. For simplicity, we assume two superconductors with the same gap, $\Delta$, which will be generalized later. From Eq. (8) we obtain \begin{equation} J_{kq}=^{r}^{l} \end{equation} where $V_{e-ph}$ is the phonon-mediated electron-electron interaction. Tunneling effect is included in the states of the left and right sides. It is noteworthy that this situation is much the same as the calculation of the matrix element between scattered-state pairs in Anderson's theory of dirty superconductors,$^{30,31}$ where impurity effect is included in the scattered states and the electron-phonon interaction couples the states. Using the Tunneling Hamiltonian, Eq. (7), the one-particle wavefunctions are given by \begin{eqnarray} \psi^{l}_{q}(r)&=& N_{q}[\phi_{q}^{l}(r)+\sum_{k}{T_{kq}\over \epsilon_{q}-\epsilon_{k}}\phi_{k}^{r}(r)]\nonumber\\ \psi^{r}_{k}(r)&=& N_{k}[\phi_{k}^{r}(r)+\sum_{q}{T_{qk}\over \epsilon_{k}-\epsilon_{q}}\phi_{q}^{l}(r)] \end{eqnarray} where $\phi_{q}^{l}$ and $\phi_{k}^{r}$ are the left- and right-hand states$^{32}$ with $\epsilon_{q}$ and $\epsilon_{k}$ being the electron energies. $N_{k}$ and $N_{q}$ are the normalization constants. Prange$^{32}$ proposed the {\sl nonorthogonal almost confined states} for the left- and right-hand states, whereas Bardeen$^{33}$ introduced the WKB wavefunction, \begin{eqnarray} \phi_{m}^{l}&=& Cp_{x}^{-1/2}e^{i(p_{y}y+p_{z}z)} sin(p_{x}x+\gamma), \quad \ \ x{N_{F}\pi^{2}\over 2\hbar v_{F}}\xi_{0} = <|T_{kq}|^{2}>{N_{F}\pi\over 2\Delta}\nonumber\\ {1\over \Omega_{l}}\sum_{q}{|T_{qk}|^{2}\over (\epsilon_{k}-\epsilon_{q})^{2}}&\approx & <|T_{qk}|^{2}>{N_{F}\pi^{2}\over 2\hbar v_{F}}\xi_{0} = <|T_{qk}|^{2}>{N_{F}\pi\over 2\Delta} \end{eqnarray} and \begin{equation} \equiv J\approx V<|T|^{2}>{N_{F}\pi\over \Delta}. \end{equation} Here $N_{F}$ is the density of states at the Fermi level and the angular brackets indicate the average value over the states. Consequently, substituting Eq. (21) to Eq. (12) the Josephson coupling energy is written \begin{equation} E_{J}\approx - {2\pi\over \lambda} <|T|^{2}>N_{F}^{2}\Delta\ cos(\phi_{l}-\phi_{r}), \end{equation} where $\lambda=N_{F}V$. Accordingly, the Josephson supercurrent is \begin{equation} j={1\over \lambda}{\Delta\over eR_{n}}\ sin(\phi_{l}-\phi_{r}), \end{equation} with \begin{equation} j_{1}={1\over \lambda}{\Delta\over eR_{n}}. \end{equation} Although this equation for the maximum DC supercurrent is similar to the previous one, Eq. (2), physically it is significantly different. The factor $1/\lambda$ shows that the Josephson coupling energy and the supercurrent depend on the superconductor. (Notice that this factor also appears in the initial $T_{c}$ decreases due to magnetic impurities$^{34}$ and weak localization.$^{31}$) As a result, if we include the reduction of the Cooper pair size during the tunneling, which is important especially for superconductors with high $T_{c}$, the resulting supercurrent is not similar at all. When the superconductors have different energy gaps, $\Delta_{1}$ and $\Delta_{2}$, the pair-scattering matrix elements are \begin{equation} \equiv J\approx V_{1}<|T|^{2}>{N_{F,1}\pi\over 2\Delta_{2}} + V_{2}<|T|^{2}>{N_{F,2}\pi\over 2\Delta_{1}} \end{equation} where $N_{F,1}$ and $N_{F,2}$ are the density of states of left and right sides, respectively. Subsequently, one finds \begin{equation} E_{J}= - \pi <|T|^{2}>N_{1,F}N_{F,2}\Delta_{1}\Delta_{2}({1\over \lambda_{1}\Delta_{1}}+{1\over \lambda_{2}\Delta_{2}}) cos(\phi_{l}-\phi_{r}), \end{equation} and \begin{equation} j={2\pi\over \hbar}e<|T|^{2}>N_{F,1}N_{F,2}\Delta_{1}\Delta_{2} ({1\over \lambda_{1}\Delta_{1}}+{1\over \lambda_{2}\Delta_{2}}) sin(\phi_{l}-\phi_{r}), \end{equation} with \begin{equation} j_{1}={1\over 2e}{1\over R_{n}}\Delta_{1}\Delta_{2} ({1\over \lambda_{1}\Delta_{1}}+{1\over \lambda_{2}\Delta_{2}}) . \end{equation} Here $\lambda_{1}$ and $\lambda_{2}$ are the BCS coupling constants for two superconductors. \subsection{\bf Cooper Pair Wavefunction Approach} We consider now Eq. (6), the Josephson coupling energy in terms of the Cooper pair wavefunctions. This method is more appropriate for high $T_{c}$ superconductors, including $MgB_{2}$, which have tightly-bound Cooper pairs with small size. It is also desirable for the insulating barrier with high tunneling resistance $R_{n}$, since the Cooper pair size shrinks significantly during the tunneling in that case. A similar reduction of the Cooper pair size caused by the ordinary impurity scattering requires this approach, too. The Cooper pair wavefunctions are expressed in terms of the one-particle wavefunctions, Eq. (15): \begin{eqnarray} F_{l}(x)&=& \sum_{q}u_{q}v_{q}\psi_{q}^{l}(x)\psi_{-q}^{l}(x)\\ F_{r}(x)&=& \sum_{k}u_{k}v_{k}\psi_{k}^{r}(x)\psi_{-k}^{r}(x). \end{eqnarray} Substituting Eqs. (29) and (30) into Eq. (6), it is straightforward to show that \begin{equation} E_{J}=V( {1\over \Omega_{r}}\sum_{k}{|T_{kq}|^{2}\over (\epsilon_{q}-\epsilon_{k})^{2}}+ {1\over \Omega_{l}}\sum_{q}{|T_{qk}|^{2}\over (\epsilon_{k}-\epsilon_{q})^{2}})(\sum_{kq}u_{k}^{r}v_{k}^{r*}u_{q}^{l}v_{q}^{l}+\sum_{kq}u_{q}^{l}v_{q}^{l*}u_{k}^{r}v_{k}^{r}) \end{equation} which is indeed the same as that obtained by the k-space approach, i.e., Eq. (9) with Eq. (19). Whereas this real space approach allows us to take into account the change of the Cooper pair size during the tunneling through the Cooper pair wavefunctions. If we insert the Bardeen's WKB wavefunction into Eq. (29), the left-side Cooper pair wavefunction shows the exponential decay in the barrier region: \begin{equation} F_{l}(x)={C^{2}\over 4}(\sum_{p}u_{p}v_{p})|p^{0}_{x}|^{-1}e^{-2\kappa(x-x_{a})} \quad x_{a}x_{b}. \end{equation} We have used the fact that only ($\phi_{k}^{r}$, $\phi_{-k}^{r}$) pair contributes to the coupling energy, Eq. (31). (It is obvious that $|T_{kq}|^{2}\propto e^{-2\kappa d}$, with the barrier thickness $d=x_{b}-x_{a}$.$^{35,36}$) However, since $\phi_{k}^{r}$ and $\phi_{-k}^{r}$ are basically plane-wave-like, Eq. (33) shows not the presumed exponential tail but a constant amplitude. Of course, the divergence in the sum implies this behavior needs to be corrected according to the bounded Cooper pair wavefunction. Therefore, we assume \begin{equation} F_{l}(x)\approx <|T_{kq}|^{2}>\sum_{q}u_{q}v_{q}{N_{F}\pi^{2}\over 2\hbar v_{F}}e^{-(x-x_{b})/\xi_{0}} \quad x>x_{b}. \end{equation} (In fact, $x$ denotes the center of mass coordinate of the Cooper pair rather than the relative coordinate. Since the phonon-mediated attraction between the electrons in the left-hand side can propagate only up to the range of the Cooper pair size $\xi_{0}$ in the right-hand side, this assumption seems to be reasonable.) Inserting Eq. (34) into the first integral in Eq. (6) we obtain (in the right side) \begin{eqnarray} V\int^{r} F_{r}^{*}(x)F_{l}(x)dx &=& V<|T|^{2}>\sum_{k}u_{k}^{r}v_{k}^{r*}\sum_{q}u^{l}_{q}v^{l}_{q}{N_{F}\pi^{2}\over 2\hbar v_{F}}\int_{x_{b}}^{\infty}e^{-(x-x_{b})/\xi_{0}}dx\nonumber\\ &=& V<|T|^{2}>\sum_{k}u_{k}^{r}v_{k}^{r*}\sum_{q}u^{l}_{q}v^{l}_{q}{N_{F}\pi\over 2\Delta} \end{eqnarray} where $\int^{r}$ denotes the integration over the right side and $x_{a}=0$. $F_{r}(x)$ will have the same exponential tail in the left side. As expected, the Cooper pair wavefunction method leads to the same result for the Josephson coupling energy and the supercurrent: \begin{eqnarray} E_{J}&=& -{2\pi\over \lambda} <|T|^{2}>N_{F}^{2}\Delta \ cos(\phi_{l}-\phi_{r})\\ j &=&{1\over \lambda}{\Delta\over eR_{n}}\ sin(\phi_{l}-\phi_{r}), \end{eqnarray} where \begin{equation} j_{1}={1\over \lambda}{\Delta\over eR_{n}}. \end{equation} \subsection{\bf Shrinking of the Cooper Pair Size Due to the Insulating Barrier} In the previous section we assumed that the tail of $F_{l}(x)$ in the right side, Eq. (34), is controlled by the same Cooper pair size, $\xi_{0}$, of the left side. In other words, we disregarded the effect of the insulating barrier potential on the Cooper pair size. However, it is well-known that the Cooper pair size is reduced due to the impurity potentials.$^{37,38}$ Therefore, it is obvious that this assumption is not appropriate especially for high $T_{c}$ superconductors, including $MgB_{2}$, and for the insulating barriers with high tunneling resistance, $R_{n}$, since the Cooper pair will have so much difficulty in tunneling in these cases. We estimate the reduction of the Cooper pair size due to the barrier potential. Equation (32) implies that the decay length of the Cooper pair wavefunction in the insulating barrier region is $\sim 1/2\kappa$. Whereas, after tunneling the Cooper pair (of size $\xi_{0}$) will have the memory of the phonon-mediated attraction up to $\xi_{0}-d$. So we suppose that the Cooper pair have size $1/2\kappa$ with the weight $d/\xi_{0}$ and $\xi_{0}$ with the weight $(\xi_{0}-d)/\xi_{0}$. Accordingly, the effective Cooper pair size $\xi_{eff}$ may be written as \begin{equation} {1\over \xi_{eff}}= {1\over \xi_{0}}{\xi_{0}-d\over \xi_{0}}+2\kappa {d\over\xi_{0}}. \end{equation} Since $d<<\xi_{0}$ in most cases, we find the effective Cooper pair size in the right side, \begin{equation} \xi_{eff} \cong {\xi_{0}\over 1+2\kappa d}. \end{equation} Then, the effective Cooper pair wavefunction of $F_{\ell}(x)$ in the right side is \begin{equation} F_{l}(x)\propto e^{-{x\over \xi_{eff}}}=e^{-{1+2\kappa d\over \xi_{0}}x} \quad\quad x\geq x_{a}=0. \end{equation} Now, it is important to notice that this reduction of the Cooper pair size is fully meaningful only when the exponential factor in Eq. (41) is $\sim e^{-1}$ just after tunneling, i.e., \begin{equation} F_{l}(d)\propto e^{-{d\over \xi_{eff}}}=e^{-{1+2\kappa d\over \xi_{0}}d}\sim e^{-1}. \end{equation} Thus, the Cooper pair size shrinks significantly when $d\sim \sqrt{\xi_{0}/2\kappa}$, i.e., for high $T_{c}$ superconductors, such as $MgB_{2}$ and for the insulating barriers with high $R_{n}$. To put it another way, there is a (sample-dependent) threshold tunneling resistance, $R_{th}$, above which the supercurrent decreases much faster, although it is hard to determine the exact value of $R_{th}$. The overlap of the Cooper pair wavefunctions is given by \begin{eqnarray} V\int^{r} F_{r}^{*}(x)F_{l}(x)dx &\cong & V<|T|^{2}>\sum_{k}u_{k}^{r}v_{k}^{r*}\sum_{q}u^{l}_{q}v^{l}_{q}{N_{F}\pi^{2}\over 2\hbar v_{F}}\int_{x_{b}}^{\infty} e^{-{1+2\kappa d\over \xi_{0}}x} dx\nonumber\\ &\cong & V<|T|^{2}>\sum_{k}u_{k}^{r}v_{k}^{r*}\sum_{q}u^{l}_{q}v^{l}_{q}{N_{F}\pi\over 2\Delta}{1\over 1+2\kappa d}e^{-{1+2\kappa d\over \xi_{0}}d}. \end{eqnarray} Consequently, for $d\sim \sqrt{\xi_{0}/2\kappa}$ the Josephson coupling energy and the supercurrent are written as \begin{equation} E_{J}= - {2\pi\over \lambda} <|T|^{2}>N_{F}^{2}\Delta {1\over 1+2\kappa d} e^{-{1+2\kappa d\over \xi_{0}}d}\ cos(\phi_{l}-\phi_{r}). \end{equation} and \begin{equation} j= {1\over \lambda}{\Delta\over eR_{n}}{1\over 1+2\kappa d} e^{-{1+2\kappa d\over \xi_{0}}d}\ sin(\phi_{l}-\phi_{r}), \end{equation} with \begin{equation} j_{1}= {1\over \lambda}{\Delta\over eR_{n}}{1\over 1+2\kappa d} e^{-{1+2\kappa d\over \xi_{0}}d}. \end{equation} Since the DC Josephson current is very small in this case, fabrication of $MgB_{2}$ SIS junctions requires very accurate nanometer scale manipulation of the insulating barriers. For instance, if $\xi_{0}\sim 100\AA$, we need an insulating barrier of thickness d smaller than $\sim 10\AA$ (i.e., $R_{th}\sim 0.01-0.1\Omega)$ to see the Josephson supercurrent, which explains why the previous experiments couldn't see the DC supercurrent corresponding to the big gap of $MgB_{2}$.$^{14,15}$ \subsection{\bf $R_{n}$ Dependence of the DC Supercurrent } We have found the maximum DC supercurrent: \begin{eqnarray} j_{1}&=& {1\over \lambda}{\Delta\over eR_{n}}\hspace{1in} \quad\quad\ \ \rm{for}\ \rm{ low}\ R_{n}\\ j_{1}&=&{1\over \lambda}{\Delta\over eR_{n}}{1\over 1+2\kappa d} e^{-{1+2\kappa d\over \xi_{0}}d} \quad\ \ \rm{for}\ \rm{ high}\ R_{n}\ (d \sim \sqrt{\xi_{0}/2\kappa}) \end{eqnarray} where \begin{equation} R_{n}^{-1}= {4\pi e^{2}\over \hbar}N_{F}^{2}<|T|^{2}>. \end{equation} For low $R_{n}$ the DC supercurrent is inversely proportional to $R_{n}$, which may be called {\sl linear region}, whereas for high $R_{n}$, corresponding to $d \sim \sqrt{\xi_{0}/2\kappa}$, the supercurrent decreases more quickly with increasing $R_{n}$, which may be called {\sl steep region}, although the boundary is not that sharp. This behavior agrees with the experiments.$^{15-19}$ (Presumably in the {\sl steep region} the supercurrent may drop to zero exponentially. This is the reason why we kept the exponential factor in the previous section.) One needs an interpolation formula that smoothly connects both limits, for low $R_{n}$ and for high $R_{n}$. (Nevertheless, high $R_{n}$ formula Eq. (48) is valid in a rather narrow range where $d\sim \sqrt{\xi_{0}/2\kappa}$ and the exponential factor $\sim e^{-1}$.) For that purpose, we may multiply $2\kappa d$ in Eq. (48) by the approximate function for the Heaviside unit step function,$^{39}$ i.e., \begin{equation} S_{n}(\kappa d)={1\over 2}[1+tanh\ n(2\kappa d-2\kappa d_{c})] \end{equation} where $2\kappa d_{c}$ may be chosen to give the exponential factor equal to $e^{-1}$, while n is selected to have the resistance spread of factor of 10, according to the experimental data. (In this case $2\kappa d_{c}$ gives much higher resistance than the threshold resistance $R_{th}$.) In the presence of ordinary impurities, the Cooper pair size also decreases from $\xi_{0}$ to $\tilde{\xi_{0}}$, defined by$^{21,31,38}$ \begin{equation} {1\over \tilde{\xi_{0}}}= {1\over \xi_{0}}+{1\over \ell}, \end{equation} where $\ell$ is the mean free path. (It seems that Eq. (51) works better than the other expression, i.e., $\sqrt{\ell\xi_{0}}$.$^{37}$) The effective Cooper pair size is then replaced by \begin{equation} \xi_{eff} \cong {\ell\xi_{0}\over (1+2\kappa d)(\ell+\xi_{0})}. \end{equation} It is remarkable that ordinary impurities also decrease the supercurrent by reducing the Cooper pair size. Figure 2 shows the comparison of our theoretical calculations with the experimental data for Pb-Pb$\rm{O_{x}}$-Pb (PPP) at 4.2K by Schwidtal and Finnegan,$^{16}$ and Sn-SnO-Pb (SSP) at 1.4K by Tinkham's group (Danchi et. al.).$^{19}$ The coherence lengths $\xi_{0}$ for Pb and Sn are $\sim 820\AA$ and $\sim 1800\AA$, respectively.$^{40}$ Note that above $R_{n}\sim 40\Omega$ the supercurrent of Sn-SnO-Pb junction becomes larger than that of Pb-Pb$\rm{O_{x}}$-Pb junction, due to the bigger Cooper pair size of Sn. For {\sl linear region} $j_{1}=1.5/R_{n}(mA)$ for PPP and $j_{1}=1.2/R_{n}(mA)$ for SSP. McMillan and Rowell$^{41}$ observed that for a typical junction with a resistance of 30$\Omega$, the exponential factor is roughly $exp(-20)$. So the tunneling resistance is assumed to be $R_{n}=30\times exp(2\kappa d-20)(\Omega)$. The threshold resistance for PPP is $R_{th} \sim 30\Omega$, (i.e., $d_{th}\sim 20\AA$), whereas for SSP, there are two threshold resistances for Sn and PB, leading to (average) value, $R_{th} \sim 50\Omega$. For PPP junction we used $Sn(\kappa d)={1/2}[1+tanh\ 1.9(\kappa d-21.25)]$, whereas for SSP junction we used $Sn(\kappa d)={1/2}[1+tanh\ 1.0(\kappa d-21.25)]$ for Pb and $Sn(\kappa d)={1/2}[1+tanh\ 1.0(\kappa d-22.75)]$ for Sn, respectively. Unfortunately, the mean free path $\ell$ is not available for those experimental data. However, since the thickness of the film is about $2,000\AA$,$^{16}$ we suppose $\ell\sim 1000\AA$. If we assume $\ell=1000\AA$ for Pb-Pb$\rm{O_{x}}$-Pb junction, we obtain the impurity-limited coherence length $\tilde \xi_{0}=\ell\xi_{0}/(\ell+\xi_{0})=450.6\AA$, leading to $d_{c}\sim 21.23\AA$, in agreement with the above $R_{n}(\kappa d)$. For SSP junction, the coherence length of Sn, $\sim 1800\AA$ requires $\ell\sim 800\AA$ to get $d_{c}\cong 23.5\AA$, which is also consistent with the $R_{n}$ used. Considering the uncertainty in $\ell$, the perfect fit with the experimental data is rather fortuitous. Nevertheless, our approach explains clearly why the Cooper pair of Sn can tunnel through the barrier more easily than that of Pb. \section{\bf Josephson Effect at Finite Temperature} Now we consider the Josephson coupling energy and the supercurrent at finite temperatures for low tunneling resistance $R_{n}$. The Bogoliubov-Valatin transformations are defined by$^{42}$ \begin{eqnarray} c_{k\uparrow}^{r}&=&u_{k}^{r}e_{k\uparrow}+v_{k}^{r}e^{+}_{-k\downarrow}\nonumber\\ c_{k\downarrow}^{r}&=&u_{k}^{r}e_{k\downarrow}-v_{k}^{r}e^{+}_{k\uparrow}\\ c_{q\uparrow}^{l}&=&u_{q}^{l}f_{q\uparrow}+v_{q}^{l}f^{+}_{-q\downarrow}\nonumber\\ c_{q\downarrow}^{l}&=&u_{q}^{l}f_{q\downarrow}-v_{q}^{l}f^{+}_{q\uparrow}. \end{eqnarray} Consequently, the coupling energy is written \begin{eqnarray} E_{J}&=&-J\sum_{kq}[u^{r}_{k}v_{k}^{r*}(1-2f_{k})u_{q}^{l}v_{q}^{l}(1-2f_{q})+ u^{r}_{k}v_{k}^{r}(1-2f_{k})u_{q}^{l}v_{q*}^{l}(1-2f_{q})]\nonumber\\ &=&-2J\sum_{kq}{\Delta_{1}(T)\Delta_{2}(T)cos(\phi_{l}-\phi_{r})\over 2E_{k}^{r}\times 2E_{q}^{l}}(1-2f_{k})(1-2f_{q}). \end{eqnarray} At $T=0K$ this expression leads to Eq. (12). After summations over k and q, we obtain \begin{equation} E_{J}=-2J{\Delta_{1}(T)\Delta_{2}(T)\over V_{1}V_{2}}cos(\phi_{l}-\phi_{r}). \end{equation} Notice that the pair-scattering matrix element, J, has rather weak temperature dependence through the effective Cooper pair sizes $\xi_{eff,1}(T)$ and $\xi_{eff,2}(T)$: \begin{equation} J= V_{1}<|T|^{2}>{N_{F,1}\pi^{2}\over 2\hbar v_{F}}\xi_{eff, 2}(T) + V_{2}<|T|^{2}>{N_{F,2}\pi^{2}\over 2\hbar v_{F}}\xi_{eff, 1}(T). \end{equation} It is well-known that the pair-correlation amplitude, or the Cooper pair wavefunction falls exponentially:$^{21,26}$ \begin{eqnarray} F(r)&\propto& exp(-{r\over \pi\xi_{0}})\quad\quad \quad T=0K\\ F(r)&\propto& exp(-{2r\over 3.5\xi_{0}})\quad \quad\ T\rightarrow T_{c} \end{eqnarray} leading to the decrease of the effective Cooper pair size $\xi_{eff}(T)$ from $\pi\xi_{0}/2$ at $T=0K$ to $3.5\xi_{0}/4$ near $T_{c}$. (In the previous sections we used $\xi_{0}$ instead of $\pi\xi_{0}/2$ at $T=0K$.) The limiting behavior of the effective Cooper pair size can be shown to be$^{43}$ \begin{eqnarray} \xi_{eff}(T)&\cong& {\pi\xi_{0}\over 2}(1-{\pi^{2}T^{2}\over 2\Delta^{2}}) \quad \quad\quad\quad\ \rm{near }\ \ T=0K\\ \xi_{eff}(T) &\cong&{1\over 2} {\hbar v_{F}\over \pi T }(1-{\Delta^{2}\over 2\pi^{2}T^{2}})\quad\quad\quad \rm{near}\ \ T_{c}, \end{eqnarray} where $\xi_{eff}(T)\cong {\pi\Delta_{0}\xi_{0}\over 2\sqrt{\pi^{2}T^{2}+\Delta^{2}}}$ at any temperature. Unlike $\Delta(T)$, near $T=0K$, $\xi_{eff}(T)$ decreases with increasing temperature in a parabolic manner. Accordingly, for a symmetric junction the Josephson coupling energy and the supercurrent are written as \begin{eqnarray} E_{J}&=&-2|T|^{2}{\pi^{2}\over \hbar v_{F}}{\xi_{eff}(T)\over \lambda}N_{F}^{2}\Delta^{2}(T)cos\phi\\ j&=&{1\over \lambda}{\pi\xi_{eff}(T)\over \hbar v_{F}}{\Delta^{2}(T)\over eR_{n}}sin\phi, \end{eqnarray} with \begin{equation} j_{1}={1\over \lambda}{\pi\xi_{eff}(T)\over \hbar v_{F}}{\Delta^{2}(T)\over eR_{n}}. \end{equation} The reduced dc Josephson current is, then, given by \begin{eqnarray} {j_{1}(T)\over j_{1}(0K)}&=&{\xi_{eff}(T)\over {{\pi\over 2}\xi_{0}}}{\Delta^{2}(T)\over \Delta_{0}^{2}}\\ &=& {\Delta^{2}(T)\over \Delta_{0}\sqrt{\pi^{2}T^{2}+\Delta^{2}(T)}}. \end{eqnarray} For an asymmetric junction, one finds the coupling energy and the supercurrent: \begin{equation} E_{J}=- \pi <|T|^{2}>N_{F,1}N_{F,2}\Delta_{1}(T)\Delta_{2}(T) ({\pi\xi_{eff,1}(T)\over \lambda_{1}\hbar v_{F,1}}+{\pi\xi_{eff,2}(T)\over \lambda_{2}\hbar v_{F,2}}) cos(\phi_{l}-\phi_{r}), \end{equation} and \begin{equation} j={2\pi\over \hbar}e<|T|^{2}>N_{F,1}N_{F,2}\Delta_{1}(T)\Delta_{2}(T)({\pi\xi_{eff,1}(T)\over \lambda_{1}\hbar v_{F,1}}+{\pi\xi_{eff,2}(T)\over \lambda_{2}\hbar v_{F,2}}) sin(\phi_{l}-\phi_{r}), \end{equation} with \begin{equation} j_{1}={1\over 2e}{1\over R_{n}}\Delta_{1}(T)\Delta_{2}(T) ({\pi\xi_{eff,1}(T)\over \lambda_{1}\hbar v_{F,1}}+{\pi\xi_{eff,2}(T)\over \lambda_{2}\hbar v_{F,2}}). \end{equation} Therefore, the reduced DC Josephson current for an asymmetrical junction is \begin{equation} {j_{1}(T)\over j_{1}(0K)}={\Delta_{1}(T)\Delta_{2}(T)\over \Delta_{1}(0)\Delta_{2}(0)} {[{\xi_{eff,1}(T)\over \lambda_{1}}+{\xi_{eff,2}(T)\over \lambda_{2}}]\over {\pi\over 2}[{\xi_{0,1}\over \lambda_{1}}+{\xi_{0,2}\over \lambda_{2}}]}. \end{equation} Figure 3 shows the temperature dependence of the maximum DC supercurrent $j_{1}(T)$ for a Sn/Sn and a Pb/Sn junction compared with our theoretical calculation from Eqs. (64) and (68). Data are from Fiske.$^{44}$ As in Figure 2, we used $\ell=800\AA$ and $\xi_{0}=820\AA$ for Pb and $\xi_{0}=1800\AA$ for Sn, respectively. The approximate temperature dependence of $\xi_{eff}(T)$ is given by $\xi_{eff}(T)={\pi\Delta_{0}\xi_{0}\over 2\sqrt{\pi^{2}T^{2}+\Delta^{2}}}$, which is almost identical to the accurate numerical calculation of $\xi_{eff}(T)$.$^{43}$ As can be seen, the agreement between theory and experiment is fairly good. \section{\bf Discussion } It is clear that more study is needed to understand the intriguing properties of the Josephson effect. In particular, the sign of pair-quasi particle interference term, generalization of this approach to weak links and SNS junctions, and explanation of many unsolved problems in AC Josephson effect may be interesting problems. For AC Josephson effect an investigation based on this approach will be published separately.$^{45}$ The Josephson effect in $MgB_{2}$ requires more careful study. It is highly desirable to fabricate clear-cut SIS junction with very low tunneling resistance ($R_{n}\sim 0.01-0.1\Omega$) for the detection of the Josephson current for the big gap. The insulating layer should have thickness not larger than $10\AA$ and a small band gap. Very recently, a similar problem has been found in the flux quantization of superconducting cylinders.$^{46}$ Whereas most previous approaches focused on the phase of the effective Cooper pair wavefunction in the presence of the magnetic flux, flux quantization turned out to be due to the flux dependence of the pairing energy.$^{46}$ \section{\bf Conclusion } We have introduced Cooper pair wavefunction approach to the Josephson effect in SIS tunnel junctions. We have found that the Josephson tunneling depends on the size of the Cooper pair and its shrinking during the tunneling. Accordingly there is a material-dependent threshold of tunneling resistance above which the DC Josephson current decreases much faster with increasing the tunneling resistance. High $T_{c}$ superconductors, including $MgB_{2}$, have tightly-bound Cooper pairs with small size, which can not tunnel through the insulating barrier easily, leading to extremely small critical current, in agreement with experimental findings. This understanding also explains why the observed DC supercurrent of low $T_{c}$ superconductors, such as Sn and Pb, decreases much faster than $1/R_{n}$ above the tunneling resistances $R_{n}$ in excess of several ohms. It is of interest that ordinary impurities limit the supercurrent by reducing the Cooper pair size, too. We have also shown that the (weak) temperature dependence of the Cooper pair size contribute to the temperature dependence of DC supercurrent. \vspace{2pc} \centerline{\bf ACKNOWLEDGMENTS} Special thanks are due to Faculty of Arts and Sciences at UPR-Mayaguez for the release time. I am grateful to Emin Yeltepe, Miguel A. Morales, Prof. Ju H. Kim and Prof. C. Bulutay for extensive discussions. I also thank Attila Altay and Profs. B. Tanatar, C. Pabon, P. Rapp, and R. Ramos for discussions. I take this opportunity to thank the members of Physics Department of Bilkent University for the hospitality extended to me during my stay. \begin{references} \bibitem{1} B. D. Josephson, Phys. Lett. {\bf 1}, 251 (1962). \bibitem{2} B. D. Josephson, Rev. Mod. Phys. {\bf 36}, 216 (1964). \bibitem{3} P. W. Anderson and J. M. Rowell, Phys. Rev. Lett. {\bf 10}, 230 (1963). \bibitem{4} S. Shapiro, Phys. Rev. Lett. {\bf 11}, 80 (1963). \bibitem{5} J. Nicol, S. Shapiro, and P. H. Smith, Phys. Rev. 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Walecka, {\sl Quantum Theory of Many-Particle Systems} (McGraw-Hill, New York, 1971), p.401. \bibitem{25} A. J. Leggett, Rev. Mod. Phys. {\bf 47}, 331 (1975). \bibitem{26} P. W. Anderson and W. F. Brinkman, in {\sl The Helium Liquids}, ed. J. G. M. Armitage and I. E. Farquhar, (Academic Press, New York, 1975), p. 315. \bibitem{27} P. R. Wallace and M. J. Stavn, Can. Jour. Phys. {\bf 43}, 411 (1965). \bibitem{28} A. J. Leggett, Prog. Theo. Phys. {\bf 36}, 901 (1966). \bibitem{29} P. W. Anderson, {\sl The Theory of Superconductivity in the High-Tc Cuprates} (Princeton University Press, Princeton, 1997), p. 309. \bibitem{30} P. W. Anderson, J. Phys. Chem. Solids {\bf 11}, 26 (1959). \bibitem{31} Mi-Ae Park and Yong-Jihn Kim, Phys. Rev. B {\bf 61}, 14733 (2000). \bibitem{32} R. E. Prange, Phys. Rev. {\bf 131}, 1083 (1994). \bibitem{33} J. Bardeen, Phys. Rev. Lett. {\bf 6}, 57 (1961). \bibitem{34} Yong-Jihn Kim and A. W. Overhauser, Phys. Rev. B {\bf 49}, 15799 (1994). \bibitem{35} W. A. Harrison, Phys. Rev. {\bf 123}, 85 (1961). \bibitem{36} R. E. Prange, {\sl Lectures on the Many-body Problem}, Vol.2, Ed. E. R. Caianiello (Academic Press, 1964), p. 137. \bibitem{37} P. G. de Gennes, {\sl Superconductivity of Metals and Alloys} (Benjamin, New York, 1966) PP.220 and 225. \bibitem{38} Ref. 21, p. 337. \bibitem{39} G. Arfken, {\sl Mathematical Methods for Physicists}, (Academic Press, Orlando, 1985), p. 490. \bibitem{40} C. P. Poole, H. A. Farach, and R. J. Creswick, {\sl Superconductivity}, (Academic Press, San Diego, 1995). \bibitem{41} W. L. McMillan and J. M. Rowell, in {\sl Superconductivity}, ed. Parks, (Marcel Dekker, New York, 1969), Vol. 2, p. 574. \bibitem{42} H. Suhl, B. T. Matthias, and L. R. Walker, Phys. Rev. Lett. {\bf 3}, 552 (1959). \bibitem{43} Miguel A. Morales and Yong-Jihn Kim, unpublished. \bibitem{44} M. D. Fiske, Rev. Mod. Phys. {\bf 36}, 221 (1964). \bibitem{45} Miguel A. Morales and Yong-Jihn Kim, unpublished. \bibitem{46} Mi-Ae Park and Yong-Jihn Kim, Supercond. Sci. Technol. {\bf 17}, L10 (2004). \end{references} \vfill\eject \begin{figure} \caption{ Cooper pair wavefunctions near the insulating barrier. The solid lines denote the Cooper pair wavefunctions without the barrier, while the thick lines show the change of the Cooper pair wavefunctions due to the barrier.} \end{figure} \begin{figure} \caption{ The maximum DC supercurrent vs the tunneling resistance $R_{n}$ for Pb-$\rm{PbO_{x}}$-Pb junction (at 4.2K) and for Sn-SnO-Pb junction (at 1.4K). Data are from Schwidtal and Finnegan, Ref. 16 and Danchi et. al., Ref. 19. Notice the crossing of the supercurrents near $R_{n}\sim 40\Omega$. The solid lines are our theoretical calculation from Eqs. (47) and (48).} \end{figure} \begin{figure} \caption{ Temperature dependence of the DC supercurrent of a Sn/Sn and a Pb/Sn junction in comparison with our theoretical calculation. 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884.83 1339.33 920.5 1280 c 956.17 1220.67 974 1157.67 974 1091 c 974 1027.67 957 970 923 918 c 889 866 838.67 824.67 772 794 c 858.67 774 926 732.5 974 669.5 c 1022 606.5 1046 527.67 1046 433 c 1046 305 999.33 196.5 906 107.5 c 812.67 18.5 694.67 -26 552 -26 c 423.33 -26 316.5 12.33 231.5 89 c 146.5 165.67 98 265 86 387 c closepath } bind def /s52 { 662 0 m 662 351 662 0 662 351 c 26 351 662 351 26 351 c 26 516 26 351 26 516 c 695 1466 26 516 695 1466 c 842 1466 695 1466 842 1466 c 842 516 842 1466 842 516 c 1040 516 842 516 1040 516 c 1040 351 1040 516 1040 351 c 842 351 1040 351 842 351 c 842 0 842 351 842 0 c 662 0 842 0 662 0 c closepath 662 516 m 662 1177 662 516 662 1177 c 203 516 662 1177 203 516 c 662 516 203 516 662 516 c closepath } bind def /s69 { 162 0 m 162 1466 162 0 162 1466 c 1222 1466 162 1466 1222 1466 c 1222 1293 1222 1466 1222 1293 c 356 1293 1222 1293 356 1293 c 356 844 356 1293 356 844 c 1167 844 356 844 1167 844 c 1167 672 1167 844 1167 672 c 356 672 1167 672 356 672 c 356 173 356 672 356 173 c 1256 173 356 173 1256 173 c 1256 0 1256 173 1256 0 c 162 0 1256 0 162 0 c closepath } bind def end /BuildGlyph { exch dup /Metrics get 2 index get 0 setcharwidth /CharProcs get exch 2 copy known not{pop/.notdef}if get exec fill}bind def /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec } bind def currentdict end /MC783_Arial exch definefont pop %%EndFont %%BeginFont: /MC784_Arial 12 dict begin /FontInfo 8 dict dup begin /ItalicAngle 0 def /UnderlinePosition -217.00 def /UnderlineThickness 150.00 def /StrikeoutPosition 530.00 def /StrikeoutThickness 102.00 def /isFixedPitch false def end readonly def /FontName /MC784_Arial def /FontType 3 def /FontMatrix [1 2048 div 0 0 1 2048 div 0 0] def /FontBBox [-1361 -665 4096 2060] def /Encoding 256 array def 0 1 255{Encoding exch/.notdef put}for Encoding 32/s32 put Encoding 40/s40 put Encoding 41/s41 put Encoding 45/s45 put Encoding 79/s79 put Encoding 80/s80 put Encoding 82/s82 put Encoding 83/s83 put Encoding 98/s98 put Encoding 110/s110 put Encoding 120/s120 put /BoundingBoxes 12 dict def BoundingBoxes begin /s32 [0 -434 569 1854] def /s40 [0 -434 682 1854] def /s41 [0 -434 682 1854] def /s45 [0 -434 682 1854] def /s79 [0 -434 1593 1854] def /s80 [0 -434 1366 1854] def /s82 [0 -434 1479 1854] def /s83 [0 -434 1366 1854] def /s98 [0 -434 1139 1854] def /s110 [0 -434 1139 1854] def /s120 [0 -434 1024 1854] def end /Metrics 12 dict def Metrics begin /s32 569 def /s40 682 def /s41 682 def /s45 682 def /s79 1593 def /s80 1366 def /s82 1479 def /s83 1366 def /s98 1139 def /s110 1139 def /s120 1024 def end /CharProcs 12 dict def CharProcs begin /.notdef {} def /s40 { 479 -431 m 379.67 -305.67 295.67 -159 227 9 c 158.33 177 124 351 124 531 c 124 689.67 149.67 841.67 201 987 c 261 1155.67 353.67 1323.67 479 1491 c 608 1491 479 1491 608 1491 c 527.33 1352.33 474 1253.33 448 1194 c 407.33 1102 375.33 1006 352 906 c 323.33 781.33 309 656 309 530 c 309 209.33 408.67 -111 608 -431 c 479 -431 608 -431 479 -431 c closepath } bind def /s41 { 253 -431 m 124 -431 253 -431 124 -431 c 323.33 -111 423 209.33 423 530 c 423 655.33 408.67 779.67 380 903 c 357.33 1003 325.67 1099 285 1191 c 259 1251 205.33 1351 124 1491 c 253 1491 124 1491 253 1491 c 378.33 1323.67 471 1155.67 531 987 c 582.33 841.67 608 689.67 608 531 c 608 351 573.5 177 504.5 9 c 435.5 -159 351.67 -305.67 253 -431 c closepath } bind def /s45 { 65 440 m 65 621 65 440 65 621 c 618 621 65 621 618 621 c 618 440 618 621 618 440 c 65 440 618 440 65 440 c closepath } bind def /s79 { 99 714 m 99 957.33 164.33 1147.83 295 1285.5 c 425.67 1423.17 594.33 1492 801 1492 c 936.33 1492 1058.33 1459.67 1167 1395 c 1275.67 1330.33 1358.5 1240.17 1415.5 1124.5 c 1472.5 1008.83 1501 877.67 1501 731 c 1501 582.33 1471 449.33 1411 332 c 1351 214.67 1266 125.83 1156 65.5 c 1046 5.17 927.33 -25 800 -25 c 662 -25 538.67 8.33 430 75 c 321.33 141.67 239 232.67 183 348 c 127 463.33 99 585.33 99 714 c closepath 299 711 m 299 534.33 346.5 395.17 441.5 293.5 c 536.5 191.83 655.67 141 799 141 c 945 141 1065.17 192.33 1159.5 295 c 1253.83 397.67 1301 543.33 1301 732 c 1301 851.33 1280.83 955.5 1240.5 1044.5 c 1200.17 1133.5 1141.17 1202.5 1063.5 1251.5 c 985.83 1300.5 898.67 1325 802 1325 c 664.67 1325 546.5 1277.83 447.5 1183.5 c 348.5 1089.17 299 931.67 299 711 c closepath } bind def /s80 { 158 0 m 158 1466 158 0 158 1466 c 711 1466 158 1466 711 1466 c 808.33 1466 882.67 1461.33 934 1452 c 1006 1440 1066.33 1417.17 1115 1383.5 c 1163.67 1349.83 1202.83 1302.67 1232.5 1242 c 1262.17 1181.33 1277 1114.67 1277 1042 c 1277 917.33 1237.33 811.83 1158 725.5 c 1078.67 639.17 935.33 596 728 596 c 352 596 728 596 352 596 c 352 0 352 596 352 0 c 158 0 352 0 158 0 c closepath 352 769 m 731 769 352 769 731 769 c 856.33 769 945.33 792.33 998 839 c 1050.67 885.67 1077 951.33 1077 1036 c 1077 1097.33 1061.5 1149.83 1030.5 1193.5 c 999.5 1237.17 958.67 1266 908 1280 c 875.33 1288.67 815 1293 727 1293 c 352 1293 727 1293 352 1293 c 352 769 352 1293 352 769 c closepath } bind def /s82 { 161 0 m 161 1466 161 0 161 1466 c 811 1466 161 1466 811 1466 c 941.67 1466 1041 1452.83 1109 1426.5 c 1177 1400.17 1231.33 1353.67 1272 1287 c 1312.67 1220.33 1333 1146.67 1333 1066 c 1333 962 1299.33 874.33 1232 803 c 1164.67 731.67 1060.67 686.33 920 667 c 971.33 642.33 1010.33 618 1037 594 c 1093.67 542 1147.33 477 1198 399 c 1453 0 1198 399 1453 0 c 1209 0 1453 0 1209 0 c 1015 305 1209 0 1015 305 c 958.33 393 911.67 460.33 875 507 c 838.33 553.67 805.5 586.33 776.5 605 c 747.5 623.67 718 636.67 688 644 c 666 648.67 630 651 580 651 c 355 651 580 651 355 651 c 355 0 355 651 355 0 c 161 0 355 0 161 0 c closepath 355 819 m 772 819 355 819 772 819 c 860.67 819 930 828.17 980 846.5 c 1030 864.83 1068 894.17 1094 934.5 c 1120 974.83 1133 1018.67 1133 1066 c 1133 1135.33 1107.83 1192.33 1057.5 1237 c 1007.17 1281.67 927.67 1304 819 1304 c 355 1304 819 1304 355 1304 c 355 819 355 1304 355 819 c closepath } bind def /s83 { 92 471 m 275 487 92 471 275 487 c 283.67 413.67 303.83 353.5 335.5 306.5 c 367.17 259.5 416.33 221.5 483 192.5 c 549.67 163.5 624.67 149 708 149 c 782 149 847.33 160 904 182 c 960.67 204 1002.83 234.17 1030.5 272.5 c 1058.17 310.83 1072 352.67 1072 398 c 1072 444 1058.67 484.17 1032 518.5 c 1005.33 552.83 961.33 581.67 900 605 c 860.67 620.33 773.67 644.17 639 676.5 c 504.33 708.83 410 739.33 356 768 c 286 804.67 233.83 850.17 199.5 904.5 c 165.17 958.83 148 1019.67 148 1087 c 148 1161 169 1230.17 211 1294.5 c 253 1358.83 314.33 1407.67 395 1441 c 475.67 1474.33 565.33 1491 664 1491 c 772.67 1491 868.5 1473.5 951.5 1438.5 c 1034.5 1403.5 1098.33 1352 1143 1284 c 1187.67 1216 1211.67 1139 1215 1053 c 1029 1039 1215 1053 1029 1039 c 1019 1131.67 985.17 1201.67 927.5 1249 c 869.83 1296.33 784.67 1320 672 1320 c 554.67 1320 469.17 1298.5 415.5 1255.5 c 361.83 1212.5 335 1160.67 335 1100 c 335 1047.33 354 1004 392 970 c 429.33 936 526.83 901.17 684.5 865.5 c 842.17 829.83 950.33 798.67 1009 772 c 1094.33 732.67 1157.33 682.83 1198 622.5 c 1238.67 562.17 1259 492.67 1259 414 c 1259 336 1236.67 262.5 1192 193.5 c 1147.33 124.5 1083.17 70.83 999.5 32.5 c 915.83 -5.83 821.67 -25 717 -25 c 584.33 -25 473.17 -5.67 383.5 33 c 293.83 71.67 223.5 129.83 172.5 207.5 c 121.5 285.17 94.67 373 92 471 c closepath } bind def /s98 { 301 0 m 134 0 301 0 134 0 c 134 1466 134 0 134 1466 c 314 1466 134 1466 314 1466 c 314 943 314 1466 314 943 c 390 1038.33 487 1086 605 1086 c 670.33 1086 732.17 1072.83 790.5 1046.5 c 848.83 1020.17 896.83 983.17 934.5 935.5 c 972.17 887.83 1001.67 830.33 1023 763 c 1044.33 695.67 1055 623.67 1055 547 c 1055 365 1010 224.33 920 125 c 830 25.67 722 -24 596 -24 c 470.67 -24 372.33 28.33 301 133 c 301 0 301 133 301 0 c closepath 299 539 m 299 411.67 316.33 319.67 351 263 c 407.67 170.33 484.33 124 581 124 c 659.67 124 727.67 158.17 785 226.5 c 842.33 294.83 871 396.67 871 532 c 871 670.67 843.5 773 788.5 839 c 733.5 905 667 938 589 938 c 510.33 938 442.33 903.83 385 835.5 c 327.67 767.17 299 668.33 299 539 c closepath } bind def /s110 { 135 0 m 135 1062 135 0 135 1062 c 297 1062 135 1062 297 1062 c 297 911 297 1062 297 911 c 375 1027.67 487.67 1086 635 1086 c 699 1086 757.83 1074.5 811.5 1051.5 c 865.17 1028.5 905.33 998.33 932 961 c 958.67 923.67 977.33 879.33 988 828 c 994.67 794.67 998 736.33 998 653 c 998 0 998 653 998 0 c 818 0 998 0 818 0 c 818 646 818 0 818 646 c 818 719.33 811 774.17 797 810.5 c 783 846.83 758.17 875.83 722.5 897.5 c 686.83 919.17 645 930 597 930 c 520.33 930 454.17 905.67 398.5 857 c 342.83 808.33 315 716 315 580 c 315 0 315 580 315 0 c 135 0 315 0 135 0 c closepath } bind def /s120 { 15 0 m 403 552 15 0 403 552 c 44 1062 403 552 44 1062 c 269 1062 44 1062 269 1062 c 432 813 269 1062 432 813 c 462.67 765.67 487.33 726 506 694 c 535.33 738 562.33 777 587 811 c 766 1062 587 811 766 1062 c 981 1062 766 1062 981 1062 c 614 562 981 1062 614 562 c 1009 0 614 562 1009 0 c 788 0 1009 0 788 0 c 570 330 788 0 570 330 c 512 419 570 330 512 419 c 233 0 512 419 233 0 c 15 0 233 0 15 0 c closepath } bind def end /BuildGlyph { exch dup /Metrics get 2 index get 0 setcharwidth /CharProcs get exch 2 copy known not{pop/.notdef}if get exec fill}bind def /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec } bind def currentdict end /MC784_Arial exch definefont pop %%EndFont %%BeginFont: /MC785_Arial 12 dict begin /FontInfo 8 dict dup begin /ItalicAngle 0 def /UnderlinePosition -217.00 def /UnderlineThickness 150.00 def /StrikeoutPosition 530.00 def /StrikeoutThickness 102.00 def /isFixedPitch false def end readonly def /FontName /MC785_Arial def /FontType 3 def /FontMatrix [1 2048 div 0 0 1 2048 div 0 0] def /FontBBox [-1361 -665 4096 2060] def /Encoding 256 array def 0 1 255{Encoding exch/.notdef put}for Encoding 32/s32 put Encoding 40/s40 put Encoding 41/s41 put Encoding 44/s44 put Encoding 49/s49 put Encoding 65/s65 put Encoding 67/s67 put Encoding 68/s68 put Encoding 83/s83 put Encoding 99/s99 put Encoding 101/s101 put Encoding 106/s106 put Encoding 109/s109 put Encoding 110/s110 put Encoding 112/s112 put Encoding 114/s114 put Encoding 116/s116 put Encoding 117/s117 put /BoundingBoxes 19 dict def BoundingBoxes begin /s32 [0 -434 569 1854] def /s40 [0 -434 682 1854] def /s41 [0 -434 682 1854] def /s44 [0 -434 569 1854] def /s49 [0 -434 1139 1854] def /s65 [-3 -434 1366 1854] def /s67 [0 -434 1479 1854] def /s68 [0 -434 1479 1854] def /s83 [0 -434 1366 1854] def /s99 [0 -434 1024 1854] def /s101 [0 -434 1139 1854] def /s106 [-94 -434 455 1854] def /s109 [0 -434 1706 1854] def /s110 [0 -434 1139 1854] def /s112 [0 -434 1139 1854] def /s114 [0 -434 682 1854] def /s116 [0 -434 569 1854] def /s117 [0 -434 1139 1854] def end /Metrics 19 dict def Metrics begin /s32 569 def /s40 682 def /s41 682 def /s44 569 def /s49 1139 def /s65 1366 def /s67 1479 def /s68 1479 def /s83 1366 def /s99 1024 def /s101 1139 def /s106 455 def /s109 1706 def /s110 1139 def /s112 1139 def /s114 682 def /s116 569 def /s117 1139 def end /CharProcs 19 dict def CharProcs begin /.notdef {} def /s40 { 479 -431 m 379.67 -305.67 295.67 -159 227 9 c 158.33 177 124 351 124 531 c 124 689.67 149.67 841.67 201 987 c 261 1155.67 353.67 1323.67 479 1491 c 608 1491 479 1491 608 1491 c 527.33 1352.33 474 1253.33 448 1194 c 407.33 1102 375.33 1006 352 906 c 323.33 781.33 309 656 309 530 c 309 209.33 408.67 -111 608 -431 c 479 -431 608 -431 479 -431 c closepath } bind def /s41 { 253 -431 m 124 -431 253 -431 124 -431 c 323.33 -111 423 209.33 423 530 c 423 655.33 408.67 779.67 380 903 c 357.33 1003 325.67 1099 285 1191 c 259 1251 205.33 1351 124 1491 c 253 1491 124 1491 253 1491 c 378.33 1323.67 471 1155.67 531 987 c 582.33 841.67 608 689.67 608 531 c 608 351 573.5 177 504.5 9 c 435.5 -159 351.67 -305.67 253 -431 c closepath } bind def /s44 { 182 0 m 182 205 182 0 182 205 c 387 205 182 205 387 205 c 387 0 387 205 387 0 c 387 -75.33 373.67 -136.17 347 -182.5 c 320.33 -228.83 278 -264.67 220 -290 c 170 -213 220 -290 170 -213 c 208 -196.33 236 -171.83 254 -139.5 c 272 -107.17 282 -60.67 284 0 c 182 0 284 0 182 0 c closepath } bind def /s49 { 763 0 m 583 0 763 0 583 0 c 583 1147 583 0 583 1147 c 539.67 1105.67 482.83 1064.33 412.5 1023 c 342.17 981.67 279 950.67 223 930 c 223 1104 223 930 223 1104 c 323.67 1151.33 411.67 1208.67 487 1276 c 562.33 1343.33 615.67 1408.67 647 1472 c 763 1472 647 1472 763 1472 c 763 0 763 1472 763 0 c closepath } bind def /s65 { -3 0 m 560 1466 -3 0 560 1466 c 769 1466 560 1466 769 1466 c 1369 0 769 1466 1369 0 c 1148 0 1369 0 1148 0 c 977 444 1148 0 977 444 c 364 444 977 444 364 444 c 203 0 364 444 203 0 c -3 0 203 0 -3 0 c closepath 420 602 m 917 602 420 602 917 602 c 764 1008 917 602 764 1008 c 717.33 1131.33 682.67 1232.67 660 1312 c 641.33 1218 615 1124.67 581 1032 c 420 602 581 1032 420 602 c closepath } bind def /s67 { 1204 514 m 1398 465 1204 514 1398 465 c 1357.33 305.67 1284.17 184.17 1178.5 100.5 c 1072.83 16.83 943.67 -25 791 -25 c 633 -25 504.5 7.17 405.5 71.5 c 306.5 135.83 231.17 229 179.5 351 c 127.83 473 102 604 102 744 c 102 896.67 131.17 1029.83 189.5 1143.5 c 247.83 1257.17 330.83 1343.5 438.5 1402.5 c 546.17 1461.5 664.67 1491 794 1491 c 940.67 1491 1064 1453.67 1164 1379 c 1264 1304.33 1333.67 1199.33 1373 1064 c 1182 1019 1373 1064 1182 1019 c 1148 1125.67 1098.67 1203.33 1034 1252 c 969.33 1300.67 888 1325 790 1325 c 677.33 1325 583.17 1298 507.5 1244 c 431.83 1190 378.67 1117.5 348 1026.5 c 317.33 935.5 302 841.67 302 745 c 302 620.33 320.17 511.5 356.5 418.5 c 392.83 325.5 449.33 256 526 210 c 602.67 164 685.67 141 775 141 c 883.67 141 975.67 172.33 1051 235 c 1126.33 297.67 1177.33 390.67 1204 514 c closepath } bind def /s68 { 158 0 m 158 1466 158 0 158 1466 c 663 1466 158 1466 663 1466 c 777 1466 864 1459 924 1445 c 1008 1425.67 1079.67 1390.67 1139 1340 c 1216.33 1274.67 1274.17 1191.17 1312.5 1089.5 c 1350.83 987.83 1370 871.67 1370 741 c 1370 629.67 1357 531 1331 445 c 1305 359 1271.67 287.83 1231 231.5 c 1190.33 175.17 1145.83 130.83 1097.5 98.5 c 1049.17 66.17 990.83 41.67 922.5 25 c 854.17 8.33 775.67 0 687 0 c 158 0 687 0 158 0 c closepath 352 173 m 665 173 352 173 665 173 c 761.67 173 837.5 182 892.5 200 c 947.5 218 991.33 243.33 1024 276 c 1070 322 1105.83 383.83 1131.5 461.5 c 1157.17 539.17 1170 633.33 1170 744 c 1170 897.33 1144.83 1015.17 1094.5 1097.5 c 1044.17 1179.83 983 1235 911 1263 c 859 1283 775.33 1293 660 1293 c 352 1293 660 1293 352 1293 c 352 173 352 1293 352 173 c closepath } bind def /s83 { 92 471 m 275 487 92 471 275 487 c 283.67 413.67 303.83 353.5 335.5 306.5 c 367.17 259.5 416.33 221.5 483 192.5 c 549.67 163.5 624.67 149 708 149 c 782 149 847.33 160 904 182 c 960.67 204 1002.83 234.17 1030.5 272.5 c 1058.17 310.83 1072 352.67 1072 398 c 1072 444 1058.67 484.17 1032 518.5 c 1005.33 552.83 961.33 581.67 900 605 c 860.67 620.33 773.67 644.17 639 676.5 c 504.33 708.83 410 739.33 356 768 c 286 804.67 233.83 850.17 199.5 904.5 c 165.17 958.83 148 1019.67 148 1087 c 148 1161 169 1230.17 211 1294.5 c 253 1358.83 314.33 1407.67 395 1441 c 475.67 1474.33 565.33 1491 664 1491 c 772.67 1491 868.5 1473.5 951.5 1438.5 c 1034.5 1403.5 1098.33 1352 1143 1284 c 1187.67 1216 1211.67 1139 1215 1053 c 1029 1039 1215 1053 1029 1039 c 1019 1131.67 985.17 1201.67 927.5 1249 c 869.83 1296.33 784.67 1320 672 1320 c 554.67 1320 469.17 1298.5 415.5 1255.5 c 361.83 1212.5 335 1160.67 335 1100 c 335 1047.33 354 1004 392 970 c 429.33 936 526.83 901.17 684.5 865.5 c 842.17 829.83 950.33 798.67 1009 772 c 1094.33 732.67 1157.33 682.83 1198 622.5 c 1238.67 562.17 1259 492.67 1259 414 c 1259 336 1236.67 262.5 1192 193.5 c 1147.33 124.5 1083.17 70.83 999.5 32.5 c 915.83 -5.83 821.67 -25 717 -25 c 584.33 -25 473.17 -5.67 383.5 33 c 293.83 71.67 223.5 129.83 172.5 207.5 c 121.5 285.17 94.67 373 92 471 c closepath } bind def /s99 { 828 389 m 1005 366 828 389 1005 366 c 985.67 244 936.17 148.5 856.5 79.5 c 776.83 10.5 679 -24 563 -24 c 417.67 -24 300.83 23.5 212.5 118.5 c 124.17 213.5 80 349.67 80 527 c 80 641.67 99 742 137 828 c 175 914 232.83 978.5 310.5 1021.5 c 388.17 1064.5 472.67 1086 564 1086 c 679.33 1086 773.67 1056.83 847 998.5 c 920.33 940.17 967.33 857.33 988 750 c 813 723 988 750 813 723 c 796.33 794.33 766.83 848 724.5 884 c 682.17 920 631 938 571 938 c 480.33 938 406.67 905.5 350 840.5 c 293.33 775.5 265 672.67 265 532 c 265 389.33 292.33 285.67 347 221 c 401.67 156.33 473 124 561 124 c 631.67 124 690.67 145.67 738 189 c 785.33 232.33 815.33 299 828 389 c closepath } bind def /s101 { 862 342 m 1048 319 862 342 1048 319 c 1018.67 210.33 964.33 126 885 66 c 805.67 6 704.33 -24 581 -24 c 425.67 -24 302.5 23.83 211.5 119.5 c 120.5 215.17 75 349.33 75 522 c 75 700.67 121 839.33 213 938 c 305 1036.67 424.33 1086 571 1086 c 713 1086 829 1037.67 919 941 c 1009 844.33 1054 708.33 1054 533 c 1054 522.33 1053.67 506.33 1053 485 c 261 485 1053 485 261 485 c 267.67 368.33 300.67 279 360 217 c 419.33 155 493.33 124 582 124 c 648 124 704.33 141.33 751 176 c 797.67 210.67 834.67 266 862 342 c closepath 271 633 m 864 633 271 633 864 633 c 856 722.33 833.33 789.33 796 834 c 738.67 903.33 664.33 938 573 938 c 490.33 938 420.83 910.33 364.5 855 c 308.17 799.67 277 725.67 271 633 c closepath } bind def /s106 { 134 1257 m 134 1466 134 1257 134 1466 c 314 1466 134 1466 314 1466 c 314 1257 314 1466 314 1257 c 134 1257 314 1257 134 1257 c closepath -94 -412 m -60 -259 -94 -412 -60 -259 c -24 -268.33 4.33 -273 25 -273 c 61.67 -273 89 -260.83 107 -236.5 c 125 -212.17 134 -151.33 134 -54 c 134 1062 134 -54 134 1062 c 314 1062 134 1062 314 1062 c 314 -58 314 1062 314 -58 c 314 -188.67 297 -279.67 263 -331 c 219.67 -397.67 147.67 -431 47 -431 c -1.67 -431 -48.67 -424.67 -94 -412 c closepath } bind def /s109 { 135 0 m 135 1062 135 0 135 1062 c 296 1062 135 1062 296 1062 c 296 913 296 1062 296 913 c 329.33 965 373.67 1006.83 429 1038.5 c 484.33 1070.17 547.33 1086 618 1086 c 696.67 1086 761.17 1069.67 811.5 1037 c 861.83 1004.33 897.33 958.67 918 900 c 1002 1024 1111.33 1086 1246 1086 c 1351.33 1086 1432.33 1056.83 1489 998.5 c 1545.67 940.17 1574 850.33 1574 729 c 1574 0 1574 729 1574 0 c 1395 0 1574 0 1395 0 c 1395 669 1395 0 1395 669 c 1395 741 1389.17 792.83 1377.5 824.5 c 1365.83 856.17 1344.67 881.67 1314 901 c 1283.33 920.33 1247.33 930 1206 930 c 1131.33 930 1069.33 905.17 1020 855.5 c 970.67 805.83 946 726.33 946 617 c 946 0 946 617 946 0 c 766 0 946 0 766 0 c 766 690 766 0 766 690 c 766 770 751.33 830 722 870 c 692.67 910 644.67 930 578 930 c 527.33 930 480.5 916.67 437.5 890 c 394.5 863.33 363.33 824.33 344 773 c 324.67 721.67 315 647.67 315 551 c 315 0 315 551 315 0 c 135 0 315 0 135 0 c closepath } bind def /s110 { 135 0 m 135 1062 135 0 135 1062 c 297 1062 135 1062 297 1062 c 297 911 297 1062 297 911 c 375 1027.67 487.67 1086 635 1086 c 699 1086 757.83 1074.5 811.5 1051.5 c 865.17 1028.5 905.33 998.33 932 961 c 958.67 923.67 977.33 879.33 988 828 c 994.67 794.67 998 736.33 998 653 c 998 0 998 653 998 0 c 818 0 998 0 818 0 c 818 646 818 0 818 646 c 818 719.33 811 774.17 797 810.5 c 783 846.83 758.17 875.83 722.5 897.5 c 686.83 919.17 645 930 597 930 c 520.33 930 454.17 905.67 398.5 857 c 342.83 808.33 315 716 315 580 c 315 0 315 580 315 0 c 135 0 315 0 135 0 c closepath } bind def /s112 { 135 -407 m 135 1062 135 -407 135 1062 c 299 1062 135 1062 299 1062 c 299 924 299 1062 299 924 c 337.67 978 381.33 1018.5 430 1045.5 c 478.67 1072.5 537.67 1086 607 1086 c 697.67 1086 777.67 1062.67 847 1016 c 916.33 969.33 968.67 903.5 1004 818.5 c 1039.33 733.5 1057 640.33 1057 539 c 1057 430.33 1037.5 332.5 998.5 245.5 c 959.5 158.5 902.83 91.83 828.5 45.5 c 754.17 -0.83 676 -24 594 -24 c 534 -24 480.17 -11.33 432.5 14 c 384.83 39.33 345.67 71.33 315 110 c 315 -407 315 110 315 -407 c 135 -407 315 -407 135 -407 c closepath 298 525 m 298 388.33 325.67 287.33 381 222 c 436.33 156.67 503.33 124 582 124 c 662 124 730.5 157.83 787.5 225.5 c 844.5 293.17 873 398 873 540 c 873 675.33 845.17 776.67 789.5 844 c 733.83 911.33 667.33 945 590 945 c 513.33 945 445.5 909.17 386.5 837.5 c 327.5 765.83 298 661.67 298 525 c closepath } bind def /s114 { 133 0 m 133 1062 133 0 133 1062 c 295 1062 133 1062 295 1062 c 295 901 295 1062 295 901 c 336.33 976.33 374.5 1026 409.5 1050 c 444.5 1074 483 1086 525 1086 c 585.67 1086 647.33 1066.67 710 1028 c 648 861 710 1028 648 861 c 604 887 560 900 516 900 c 476.67 900 441.33 888.17 410 864.5 c 378.67 840.83 356.33 808 343 766 c 323 702 313 632 313 556 c 313 0 313 556 313 0 c 133 0 313 0 133 0 c closepath } bind def /s116 { 528 161 m 554 2 528 161 554 2 c 503.33 -8.67 458 -14 418 -14 c 352.67 -14 302 -3.67 266 17 c 230 37.67 204.67 64.83 190 98.5 c 175.33 132.17 168 203 168 311 c 168 922 168 311 168 922 c 36 922 168 922 36 922 c 36 1062 36 922 36 1062 c 168 1062 36 1062 168 1062 c 168 1325 168 1062 168 1325 c 347 1433 168 1325 347 1433 c 347 1062 347 1433 347 1062 c 528 1062 347 1062 528 1062 c 528 922 528 1062 528 922 c 347 922 528 922 347 922 c 347 301 347 922 347 301 c 347 249.67 350.17 216.67 356.5 202 c 362.83 187.33 373.17 175.67 387.5 167 c 401.83 158.33 422.33 154 449 154 c 469 154 495.33 156.33 528 161 c closepath } bind def /s117 { 831 0 m 831 156 831 0 831 156 c 748.33 36 636 -24 494 -24 c 431.33 -24 372.83 -12 318.5 12 c 264.17 36 223.83 66.17 197.5 102.5 c 171.17 138.83 152.67 183.33 142 236 c 134.67 271.33 131 327.33 131 404 c 131 1062 131 404 131 1062 c 311 1062 131 1062 311 1062 c 311 473 311 1062 311 473 c 311 379 314.67 315.67 322 283 c 333.33 235.67 357.33 198.5 394 171.5 c 430.67 144.5 476 131 530 131 c 584 131 634.67 144.83 682 172.5 c 729.33 200.17 762.83 237.83 782.5 285.5 c 802.17 333.17 812 402.33 812 493 c 812 1062 812 493 812 1062 c 992 1062 812 1062 992 1062 c 992 0 992 1062 992 0 c 831 0 992 0 831 0 c closepath } bind def end /BuildGlyph { exch dup /Metrics get 2 index get 0 setcharwidth /CharProcs get exch 2 copy known not{pop/.notdef}if get exec fill}bind def /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec } bind def currentdict end /MC785_Arial exch definefont pop %%EndFont %%BeginFont: /MC786_Symbol 12 dict begin /FontInfo 8 dict dup begin /ItalicAngle 0 def /UnderlinePosition -223.00 def /UnderlineThickness 100.00 def /StrikeoutPosition 530.00 def /StrikeoutThickness 102.00 def /isFixedPitch false def end readonly def /FontName /MC786_Symbol def /FontType 3 def /FontMatrix [1 2048 div 0 0 1 2048 div 0 0] def /FontBBox [-1 -450 2279 2059] def /Encoding 256 array def 0 1 255{Encoding exch/.notdef put}for Encoding 32/s32 put /BoundingBoxes 3 dict def BoundingBoxes begin /s32 [0 -450 512 2059] def /s61527 [0 -450 0 2059] def end /Metrics 3 dict def Metrics begin /s32 512 def /s61527 0 def end /CharProcs 3 dict def CharProcs begin /.notdef {} def /s61527 { 1517 0 m 950 0 1517 0 950 0 c 974 279 950 0 974 279 c 1068 318.33 1137.17 374.33 1181.5 447 c 1225.83 519.67 1248 621.67 1248 753 c 1248 934.33 1203.83 1073.5 1115.5 1170.5 c 1027.17 1267.5 916 1316 782 1316 c 652.67 1316 543.33 1266.83 454 1168.5 c 364.67 1070.17 320 936.33 320 767 c 320 633.67 342.83 527.33 388.5 448 c 434.17 368.67 503 312.33 595 279 c 620 0 595 279 620 0 c 56 0 620 0 56 0 c 56 279 56 0 56 279 c 91 279 56 279 91 279 c 93 238.33 103.67 210.67 123 196 c 153 172.67 198.67 161 260 161 c 530 161 260 161 530 161 c 530 231 530 161 530 231 c 390 271.67 281.83 341.33 205.5 440 c 129.17 538.67 91 655 91 789 c 91 957 153.17 1098.67 277.5 1214 c 401.83 1329.33 570 1387 782 1387 c 912 1387 1031.5 1362.5 1140.5 1313.5 c 1249.5 1264.5 1333.5 1193 1392.5 1099 c 1451.5 1005 1481 901.67 1481 789 c 1481 651 1440.5 530.83 1359.5 428.5 c 1278.5 326.17 1172.67 260.33 1042 231 c 1034 161 1042 231 1034 161 c 1309 161 1034 161 1309 161 c 1370.33 161 1413 170.33 1437 189 c 1461 207.67 1475 237.67 1479 279 c 1517 279 1479 279 1517 279 c 1517 0 1517 279 1517 0 c closepath } bind def end /BuildGlyph { exch dup /Metrics get 2 index get 0 setcharwidth /CharProcs get exch 2 copy known not{pop/.notdef}if get exec fill}bind def /BuildChar { 1 index /Encoding get exch get 1 index /BuildGlyph get exec } bind def currentdict end /MC786_Symbol exch definefont pop %%EndFont 0 0 576 751.18 rectpath cl np /clipproc{clippath{[{/m cvx}{/l cvx}{/c cvx}{/cp cvx}pathforall}stopped {cleartomark[pathbbox exch 3 i sub exch 2 i sub/rectpath cvx}if]np}bd /iclmtx mtx cmtx def/iclpath clipproc xd/clinit{pathproc initclip np iclpath cl np exec}bd /clmove{pathproc 3 1 roll clippath pathproc 2 i neg 2 i neg tr initclip np exec cl tr np exec}bd /clrect[gs np iclpath{pathbbox}stopped{0 0 576 751.18 }if gr]def /cloper{mtx cmtx iclmtx smtx exch exec smtx}bd /cland{{exch cvx exec}cloper}bd /clcopy{{exch clinit cvx exec}cloper}bd /cldiff{{exch pathbbox 4 -1 roll clrect 0 get 2 copy gt{exch}if pop 4 1 roll 3 -1 roll clrect 1 get 2 copy gt{exch}if pop 3 1 roll 2 -1 roll clrect 2 get 2 copy lt{exch}if pop 2 1 roll 1 -1 roll clrect 3 get 2 copy lt{exch}if pop 1 1 roll exch 3 i sub exch 2 i sub rectpath cvx exec}cloper}bd /icl{{clinit}cloper}bd 0.270933 0.270928 sc /initmtx mtx cmtx def gs gs [0.885827 0 0 -0.885827 0 2772.62] concat 1 -1 sc /MC783_Arial 69 self 665 -2722 m (1)[38 ] xs 1 -1 sc gr gs [0.885827 0 0 -0.885827 0 2772.62] concat 1 -1 sc /MC783_Arial 69 self 1132 -2722 m (10)[38 38 ] xs 1 -1 sc gr gs [0.885827 0 0 -0.885827 0 2772.62] concat 1 -1 sc /MC783_Arial 69 self 1598 -2722 m (100)[38 38 38 ] xs 1 -1 sc gr gs [0.885827 0 0 -0.885827 0 2772.62] concat 1 -1 sc /MC783_Arial 69 self 228 -2630 m (1E-4)[38 46 23 38 ] xs 1 -1 sc gr gs [0.885827 0 0 -0.885827 0 2772.62] concat 1 -1 sc /MC783_Arial 69 self 228 -2160 m (1E-3)[38 46 23 38 ] xs 1 -1 sc gr gs [0.885827 0 0 -0.885827 0 2772.62] concat 1 -1 sc /MC783_Arial 69 self 240 -1689 m (0.01)[38 19 38 38 ] xs 1 -1 sc gr gs [0.885827 0 0 -0.885827 0 2772.62] concat 1 -1 sc /MC783_Arial 69 self 278 -1219 m (0.1)[38 19 38 ] xs 1 -1 sc gr gs [0.885827 0 0 -0.885827 0 2772.62] concat 1 -1 sc /MC783_Arial 69 self 335 -748 m (1)[38 ] xs 1 -1 sc gr [0.885827 0 0 -0.885827 0 2772.62] concat 430 2628 m 430 2611 l 7 sw 1 sj 1 scap st 491 2628 m 491 2611 l st 538 2628 m 538 2611 l st 576 2628 m 576 2611 l st 609 2628 m 609 2611 l st 637 2628 m 637 2611 l st 662 2628 m 662 2611 l st 684 2645 m 684 2611 l st 830 2628 m 830 2611 l st 916 2628 m 916 2611 l st 976 2628 m 976 2611 l st 1024 2628 m 1024 2611 l st 1062 2628 m 1062 2611 l st 1094 2628 m 1094 2611 l st 1123 2628 m 1123 2611 l st 1148 2628 m 1148 2611 l st 1170 2645 m 1170 2611 l st 1316 2628 m 1316 2611 l st 1402 2628 m 1402 2611 l st 1462 2628 m 1462 2611 l st 1509 2628 m 1509 2611 l st 1548 2628 m 1548 2611 l st 1580 2628 m 1580 2611 l st 1608 2628 m 1608 2611 l st 1633 2628 m 1633 2611 l st 1655 2645 m 1655 2611 l st 1802 2628 m 1802 2611 l st 1887 2628 m 1887 2611 l st 1948 2628 m 1948 2611 l st 1995 2628 m 1995 2611 l st 2033 2628 m 2033 2611 l st 2066 2628 m 2066 2611 l st 430 2611 m 2066 2611 l st 396 2611 m 430 2611 l st 413 2469 m 430 2469 l st 413 2387 m 430 2387 l st 413 2328 m 430 2328 l st 413 2282 m 430 2282 l st 413 2245 m 430 2245 l st 413 2213 m 430 2213 l st 413 2186 m 430 2186 l st 413 2162 m 430 2162 l st 396 2141 m 430 2141 l st 413 1999 m 430 1999 l st 413 1916 m 430 1916 l st 413 1857 m 430 1857 l st 413 1812 m 430 1812 l st 413 1774 m 430 1774 l st 413 1743 m 430 1743 l st 413 1716 m 430 1716 l st 413 1692 m 430 1692 l st 396 1670 m 430 1670 l st 413 1528 m 430 1528 l st 413 1446 m 430 1446 l st 413 1387 m 430 1387 l st 413 1341 m 430 1341 l st 413 1304 m 430 1304 l st 413 1272 m 430 1272 l st 413 1245 m 430 1245 l st 413 1221 m 430 1221 l st 396 1200 m 430 1200 l st 413 1058 m 430 1058 l st 413 975 m 430 975 l st 413 916 m 430 916 l st 413 871 m 430 871 l st 413 833 m 430 833 l st 413 802 m 430 802 l st 413 775 m 430 775 l st 413 751 m 430 751 l st 396 729 m 430 729 l st 413 587 m 430 587 l st 413 505 m 430 505 l st 413 446 m 430 446 l st 413 400 m 430 400 l st 413 363 m 430 363 l st 430 2611 m 430 363 l st 430 363 m 430 380 l st 491 363 m 491 380 l st 538 363 m 538 380 l st 576 363 m 576 380 l st 609 363 m 609 380 l st 637 363 m 637 380 l st 662 363 m 662 380 l st 684 363 m 684 397 l st 830 363 m 830 380 l st 916 363 m 916 380 l st 976 363 m 976 380 l st 1024 363 m 1024 380 l st 1062 363 m 1062 380 l st 1094 363 m 1094 380 l st 1123 363 m 1123 380 l st 1148 363 m 1148 380 l st 1170 363 m 1170 397 l st 1316 363 m 1316 380 l st 1402 363 m 1402 380 l st 1462 363 m 1462 380 l st 1509 363 m 1509 380 l st 1548 363 m 1548 380 l st 1580 363 m 1580 380 l st 1608 363 m 1608 380 l st 1633 363 m 1633 380 l st 1655 363 m 1655 397 l st 1802 363 m 1802 380 l st 1887 363 m 1887 380 l st 1948 363 m 1948 380 l st 1995 363 m 1995 380 l st 2033 363 m 2033 380 l st 2066 363 m 2066 380 l st 430 363 m 2066 363 l st 2066 2611 m 2032 2611 l st 2066 2469 m 2049 2469 l st 2066 2387 m 2049 2387 l st 2066 2328 m 2049 2328 l st 2066 2282 m 2049 2282 l st 2066 2245 m 2049 2245 l st 2066 2213 m 2049 2213 l st 2066 2186 m 2049 2186 l st 2066 2162 m 2049 2162 l st 2066 2141 m 2032 2141 l st 2066 1999 m 2049 1999 l st 2066 1916 m 2049 1916 l st 2066 1857 m 2049 1857 l st 2066 1812 m 2049 1812 l st 2066 1774 m 2049 1774 l st 2066 1743 m 2049 1743 l st 2066 1716 m 2049 1716 l st 2066 1692 m 2049 1692 l st 2066 1670 m 2032 1670 l st 2066 1528 m 2049 1528 l st 2066 1446 m 2049 1446 l st 2066 1387 m 2049 1387 l st 2066 1341 m 2049 1341 l st 2066 1304 m 2049 1304 l st 2066 1272 m 2049 1272 l st 2066 1245 m 2049 1245 l st 2066 1221 m 2049 1221 l st 2066 1200 m 2032 1200 l st 2066 1058 m 2049 1058 l st 2066 975 m 2049 975 l st 2066 916 m 2049 916 l st 2066 871 m 2049 871 l st 2066 833 m 2049 833 l st 2066 802 m 2049 802 l st 2066 775 m 2049 775 l st 2066 751 m 2049 751 l st 2066 729 m 2032 729 l st 2066 587 m 2049 587 l st 2066 505 m 2049 505 l st 2066 446 m 2049 446 l st 2066 400 m 2049 400 l st 2066 363 m 2049 363 l st 2066 2611 m 2066 363 l st gs pathproc 430 363 1637 2249 np rectpath /eocl cland np exec np 558 530 18 0 360 a cp gs 1 1 1 rgb eofi gr 3 sw st np 569 539 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 640 608 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 690 650 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 728 687 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 784 742 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 816 772 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 898 855 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 987 950 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 1006 962 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 1018 982 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 1088 1038 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 1170 1131 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 1202 1162 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 1304 1261 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 1387 1367 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 1496 1550 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 1572 1789 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 1585 1812 18 0 360 a cp gs 1 1 1 rgb eofi gr st np 1598 1909 18 0 360 a cp gs 1 1 1 rgb eofi gr st gr gs pathproc 430 363 1637 2249 np rectpath /eocl cland np exec 537 508 m 537 508 m 558 528 l 579 548 l 600 569 l 621 589 l 642 610 l 663 630 l 684 651 l 705 671 l 726 692 l 748 712 l 769 733 l 790 753 l 811 773 l 832 794 l 853 814 l 874 835 l 895 855 l 916 876 l 937 896 l 958 917 l 980 937 l 1001 958 l 1022 978 l 1043 999 l 1064 1019 l 1085 1040 l 1106 1060 l 1127 1081 l 1148 1101 l 1169 1122 l 1191 1142 l 1212 1163 l 1233 1184 l 1254 1206 l 1275 1227 l 1296 1249 l 1317 1272 l 1338 1296 l 1359 1321 l 1380 1349 l 1402 1380 l 1423 1414 l 1444 1453 l 1465 1498 l 1486 1550 l 1507 1608 l 1528 1673 l 1549 1744 l 1570 1819 l 1591 1897 l 1612 1976 l 1634 2053 l 3 sw st gr gs pathproc 430 363 1637 2249 np rectpath /eocl cland np exec np 768 796 18 0 360 a cp gs eofi gr 3 sw st np 873 858 18 0 360 a cp gs eofi gr st np 873 894 18 0 360 a cp gs eofi gr st np 936 925 18 0 360 a cp gs eofi gr st np 954 947 18 0 360 a cp gs eofi gr st np 980 1028 18 0 360 a cp gs eofi gr st np 1048 1063 18 0 360 a cp gs eofi gr st np 1284 1322 18 0 360 a cp gs eofi gr st np 1218 1429 18 0 360 a cp gs eofi gr st np 1351 1465 18 0 360 a cp gs eofi gr st np 1740 2054 18 0 360 a cp gs eofi gr st np 1775 2044 18 0 360 a cp gs eofi gr st np 1937 2277 18 0 360 a cp gs eofi gr st np 1943 2406 18 0 360 a cp gs eofi gr st gr gs pathproc 430 363 1637 2249 np rectpath /eocl cland np exec 558 570 m 558 570 m 568 580 l 579 590 l 589 600 l 600 611 l 610 621 l 621 631 l 632 641 l 642 651 l 653 662 l 663 672 l 674 682 l 684 692 l 695 703 l 705 713 l 716 723 l 726 733 l 737 744 l 748 754 l 758 764 l 769 774 l 779 785 l 790 795 l 800 805 l 811 815 l 821 826 l 832 836 l 842 846 l 853 856 l 864 867 l 874 877 l 885 887 l 895 897 l 906 908 l 916 918 l 927 928 l 937 939 l 948 949 l 958 959 l 969 970 l 980 980 l 990 990 l 1001 1001 l 1011 1011 l 1022 1022 l 1032 1032 l 1043 1042 l 1053 1053 l 1064 1063 l 1075 1074 l 1085 1085 l 1096 1095 l 1106 1106 l 1117 1117 l 1127 1127 l 1138 1138 l 1148 1149 l 1159 1160 l 1169 1171 l 1180 1182 l 1191 1193 l 1201 1204 l 1212 1215 l 1222 1226 l 1233 1238 l 1243 1249 l 1254 1261 l 1264 1272 l 1275 1284 l 1285 1296 l 1296 1307 l 1307 1319 l 1317 1331 l 1328 1344 l 1338 1356 l 1349 1368 l 1359 1381 l 1370 1393 l 1380 1406 l 1391 1418 l 1402 1431 l 1412 1444 l 1423 1457 l 1433 1471 l 1444 1484 l 1454 1498 l 1465 1511 l 1475 1525 l 1486 1540 l 1496 1554 l 1507 1569 l 1518 1583 l 1528 1599 l 1539 1614 l 1549 1630 l 1560 1646 l 1570 1663 l 1581 1680 l 1591 1697 l 1602 1715 l 1612 1733 l 1623 1751 l 1634 1771 l 1644 1790 l 1655 1810 l 1665 1830 l 1676 1851 l 1686 1872 l 1697 1894 l 1707 1916 l 1718 1938 l 1728 1961 l 1739 1984 l 1750 2007 l 1760 2031 l 1771 2055 l 1781 2079 l 1792 2103 l 1802 2127 l 1813 2152 l 1823 2176 l 1834 2201 l 1845 2225 l 1855 2249 l 1866 2274 l 1876 2298 l 1887 2322 l 1897 2346 l 1908 2369 l 1918 2392 l 1929 2415 l 1939 2438 l 3 sw st gr 1628 2325 m 1793 2249 l 5 sw st 1793 2249 m 1768 2269 l 1765 2262 l 1761 2255 l cp gs eofi gr st gs 1 -1 sc /MC784_Arial 84 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