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<TITLE>A. Appendix: Electron-phonon coefficients</TITLE>
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<H1><A ID="SECTION00080000000000000000">
A. Appendix: Electron-phonon coefficients</A>
</H1>
<P>
The electron-phonon coefficients <I>g</I>
are defined as
<P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{equation}
g_{{\bf q}\nu}({\bf k},i,j) =\left({\hbar\over 2M\omega_{{\bf q}\nu}}\right)^{1/2}
\langle\psi_{i,{\bf k}}| {dV_{SCF}\over d {\hat u}_{{\bf q}\nu} }\cdot
\hat \epsilon_{{\bf q}\nu}|\psi_{j,{\bf k}+{\bf q}}\rangle.
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP>
<I>g</I><SUB><IMG STYLE="height: 1.05ex; vertical-align: -0.40ex; " SRC="img6.png"
ALT="$\scriptstyle \bf q$"><I>ν</I></SUB>(<IMG STYLE="height: 1.63ex; vertical-align: -0.10ex; " SRC="img7.png"
ALT="$\displaystyle \bf k$">, <I>i</I>, <I>j</I>) = <IMG STYLE="height: 5.83ex; vertical-align: -2.42ex; " SRC="img8.png"
ALT="$\displaystyle \left(\vphantom{{\hbar\over 2M\omega_{{\bf q}\nu}}}\right.$"><IMG STYLE="height: 5.48ex; vertical-align: -2.34ex; " SRC="img9.png"
ALT="$\displaystyle {\hbar\over 2M\omega_{{\bf q}\nu}}$"><IMG STYLE="height: 6.29ex; vertical-align: -2.52ex; " SRC="img10.png"
ALT="$\displaystyle \left.\vphantom{{\hbar\over 2M\omega_{{\bf q}\nu}}}\right)^{{1/2}}_{}$">〈<I>ψ</I><SUB>i,<IMG STYLE="height: 1.11ex; vertical-align: -0.10ex; " SRC="img11.png"
ALT="$\scriptstyle \bf k$"></SUB>|<IMG STYLE="height: 5.48ex; vertical-align: -2.34ex; " SRC="img12.png"
ALT="$\displaystyle {dV_{SCF}\over d {\hat u}_{{\bf q}\nu}}$">⋅<IMG STYLE="height: 2.56ex; vertical-align: -0.97ex; " SRC="img13.png"
ALT="$\displaystyle \hat{\epsilon}_{{{\bf q}\nu}}^{}$">| <I>ψ</I><SUB>j,<IMG STYLE="height: 1.11ex; vertical-align: -0.10ex; " SRC="img11.png"
ALT="$\scriptstyle \bf k$">+<IMG STYLE="height: 1.05ex; vertical-align: -0.40ex; " SRC="img6.png"
ALT="$\scriptstyle \bf q$"></SUB>〉.
</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(1)</TD></TR>
</TABLE>
</DIV>
The phonon linewidth <!-- MATH
$\gamma_{{\bf q}\nu}$
-->
<I>γ</I><SUB><IMG STYLE="height: 1.05ex; vertical-align: -0.40ex; " SRC="img6.png"
ALT="$\scriptstyle \bf q$"><I>ν</I></SUB> is defined by
<P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{equation}
\gamma_{{\bf q}\nu} = 2\pi\omega_{{\bf q}\nu} \sum_{ij}
\int {d^3k\over \Omega_{BZ}} |g_{{\bf q}\nu}({\bf k},i,j)|^2
\delta(e_{{\bf q},i} - e_F) \delta(e_{{\bf k}+{\bf q},j} - e_F),
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP>
<I>γ</I><SUB><IMG STYLE="height: 1.05ex; vertical-align: -0.40ex; " SRC="img6.png"
ALT="$\scriptstyle \bf q$"><I>ν</I></SUB> = 2<I>πω</I><SUB><IMG STYLE="height: 1.05ex; vertical-align: -0.40ex; " SRC="img6.png"
ALT="$\scriptstyle \bf q$"><I>ν</I></SUB><IMG STYLE="height: 4.60ex; vertical-align: -2.91ex; " SRC="img14.png"
ALT="$\displaystyle \sum_{{ij}}^{}$"><IMG STYLE="height: 4.31ex; vertical-align: -1.67ex; " SRC="img15.png"
ALT="$\displaystyle \int$"><IMG STYLE="height: 5.36ex; vertical-align: -2.04ex; " SRC="img16.png"
ALT="$\displaystyle {d^3k\over \Omega_{BZ}}$">| <I>g</I><SUB><IMG STYLE="height: 1.05ex; vertical-align: -0.40ex; " SRC="img6.png"
ALT="$\scriptstyle \bf q$"><I>ν</I></SUB>(<IMG STYLE="height: 1.63ex; vertical-align: -0.10ex; " SRC="img7.png"
ALT="$\displaystyle \bf k$">, <I>i</I>, <I>j</I>)|<SUP>2</SUP><I>δ</I>(<I>e</I><SUB><IMG STYLE="height: 1.05ex; vertical-align: -0.40ex; " SRC="img6.png"
ALT="$\scriptstyle \bf q$">, i</SUB> - <I>e</I><SUB>F</SUB>)<I>δ</I>(<I>e</I><SUB><IMG STYLE="height: 1.11ex; vertical-align: -0.10ex; " SRC="img11.png"
ALT="$\scriptstyle \bf k$">+<IMG STYLE="height: 1.05ex; vertical-align: -0.40ex; " SRC="img6.png"
ALT="$\scriptstyle \bf q$">, j</SUB> - <I>e</I><SUB>F</SUB>),
</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2)</TD></TR>
</TABLE>
</DIV>
while the electron-phonon coupling constant <!-- MATH
$\lambda_{{\bf q}\nu}$
-->
<I>λ</I><SUB><IMG STYLE="height: 1.05ex; vertical-align: -0.40ex; " SRC="img6.png"
ALT="$\scriptstyle \bf q$"><I>ν</I></SUB> for
mode <I>ν</I> at wavevector <IMG STYLE="height: 1.52ex; vertical-align: -0.55ex; " SRC="img1.png"
ALT="$\bf q$"> is defined as
<P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{equation}
\lambda_{{\bf q}\nu} ={\gamma_{{\bf q}\nu} \over \pi\hbar N(e_F)\omega^2_{{\bf q}\nu}}
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP>
<I>λ</I><SUB><IMG STYLE="height: 1.05ex; vertical-align: -0.40ex; " SRC="img6.png"
ALT="$\scriptstyle \bf q$"><I>ν</I></SUB> = <IMG STYLE="height: 5.13ex; vertical-align: -2.57ex; " SRC="img17.png"
ALT="$\displaystyle {\gamma_{{\bf q}\nu} \over \pi\hbar N(e_F)\omega^2_{{\bf q}\nu}}$">
</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(3)</TD></TR>
</TABLE>
</DIV>
where <I>N</I>(<I>e</I><SUB>F</SUB>) is the DOS at the Fermi level.
The spectral function is defined as
<P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{equation}
\alpha^2F(\omega) = {1\over 2\pi N(e_F)}\sum_{{\bf q}\nu}
\delta(\omega-\omega_{{\bf q}\nu})
{\gamma_{{\bf q}\nu}\over\hbar\omega_{{\bf q}\nu}}.
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP>
<I>α</I><SUP>2</SUP><I>F</I>(<I>ω</I>) = <IMG STYLE="height: 5.30ex; vertical-align: -2.27ex; " SRC="img18.png"
ALT="$\displaystyle {1\over 2\pi N(e_F)}$"><IMG STYLE="height: 4.43ex; vertical-align: -2.72ex; " SRC="img19.png"
ALT="$\displaystyle \sum_{{{\bf q}\nu}}^{}$"><I>δ</I>(<I>ω</I> - <I>ω</I><SUB><IMG STYLE="height: 1.05ex; vertical-align: -0.40ex; " SRC="img6.png"
ALT="$\scriptstyle \bf q$"><I>ν</I></SUB>)<IMG STYLE="height: 4.90ex; vertical-align: -2.34ex; " SRC="img20.png"
ALT="$\displaystyle {\gamma_{{\bf q}\nu}\over\hbar\omega_{{\bf q}\nu}}$">.
</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(4)</TD></TR>
</TABLE>
</DIV>
The electron-phonon mass enhancement parameter <I>λ</I>
can also be defined as the first reciprocal momentum of
the spectral function:
<P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{equation}
\lambda = \sum_{{\bf q}\nu} \lambda_{{\bf q}\nu} =
2 \int {\alpha^2F(\omega) \over \omega} d\omega.
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP>
<I>λ</I> = <IMG STYLE="height: 4.43ex; vertical-align: -2.72ex; " SRC="img19.png"
ALT="$\displaystyle \sum_{{{\bf q}\nu}}^{}$"><I>λ</I><SUB><IMG STYLE="height: 1.05ex; vertical-align: -0.40ex; " SRC="img6.png"
ALT="$\scriptstyle \bf q$"><I>ν</I></SUB> = 2<IMG STYLE="height: 4.31ex; vertical-align: -1.67ex; " SRC="img15.png"
ALT="$\displaystyle \int$"><IMG STYLE="height: 5.07ex; vertical-align: -1.69ex; " SRC="img21.png"
ALT="$\displaystyle {\alpha^2F(\omega) \over \omega}$"><I>dω</I>.
</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(5)</TD></TR>
</TABLE>
</DIV>
<P>
Note that a factor <I>M</I><SUP>-1/2</SUP> is hidden in the definition of
normal modes as used in the code.
<P>
McMillan:
<P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{equation}
T_c = {\Theta_D \over 1.45} \mbox{exp} \left [
{-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}\right ]
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP>
<I>T</I><SUB>c</SUB> = <IMG STYLE="height: 4.84ex; vertical-align: -1.69ex; " SRC="img22.png"
ALT="$\displaystyle {\Theta_D \over 1.45}$">exp<IMG STYLE="height: 5.83ex; vertical-align: -2.42ex; " SRC="img23.png"
ALT="$\displaystyle \left[\vphantom{
{-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}}\right.$"><IMG STYLE="height: 5.48ex; vertical-align: -2.27ex; " SRC="img24.png"
ALT="$\displaystyle {-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}$"><IMG STYLE="height: 5.83ex; vertical-align: -2.42ex; " SRC="img25.png"
ALT="$\displaystyle \left.\vphantom{
{-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}}\right]$">
</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(6)</TD></TR>
</TABLE>
</DIV>
or (better?)
<P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{equation}
T_c = {\omega_{log}\over 1.2} \mbox{exp} \left [
{-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}\right ]
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP>
<I>T</I><SUB>c</SUB> = <IMG STYLE="height: 4.20ex; vertical-align: -1.69ex; " SRC="img26.png"
ALT="$\displaystyle {\omega_{log}\over 1.2}$">exp<IMG STYLE="height: 5.83ex; vertical-align: -2.42ex; " SRC="img23.png"
ALT="$\displaystyle \left[\vphantom{
{-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}}\right.$"><IMG STYLE="height: 5.48ex; vertical-align: -2.27ex; " SRC="img24.png"
ALT="$\displaystyle {-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}$"><IMG STYLE="height: 5.83ex; vertical-align: -2.42ex; " SRC="img25.png"
ALT="$\displaystyle \left.\vphantom{
{-1.04(1+\lambda)\over \lambda(1-0.62\mu^*)-\mu^*}}\right]$">
</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(7)</TD></TR>
</TABLE>
</DIV>
where
<P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{equation}
\omega_{log} = \mbox{exp} \left [ {2\over\lambda} \int {d\omega\over\omega}
\alpha^2F(\omega) \mbox{log}\omega \right ]
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP>
<I>ω</I><SUB>log</SUB> = exp<IMG STYLE="height: 5.83ex; vertical-align: -2.42ex; " SRC="img27.png"
ALT="$\displaystyle \left[\vphantom{ {2\over\lambda} \int {d\omega\over\omega}
\alpha^2F(\omega) \mbox{log}\omega }\right.$"><IMG STYLE="height: 4.78ex; vertical-align: -1.69ex; " SRC="img28.png"
ALT="$\displaystyle {2\over\lambda}$"><IMG STYLE="height: 4.31ex; vertical-align: -1.67ex; " SRC="img15.png"
ALT="$\displaystyle \int$"><IMG STYLE="height: 4.84ex; vertical-align: -1.69ex; " SRC="img29.png"
ALT="$\displaystyle {d\omega\over\omega}$"><I>α</I><SUP>2</SUP><I>F</I>(<I>ω</I>)log<I>ω</I><IMG STYLE="height: 5.83ex; vertical-align: -2.42ex; " SRC="img30.png"
ALT="$\displaystyle \left.\vphantom{ {2\over\lambda} \int {d\omega\over\omega}
\alpha^2F(\omega) \mbox{log}\omega }\right]$">
</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(8)</TD></TR>
</TABLE>
</DIV>
<P>
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