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<H2><A ID="SECTION00052000000000000000">
A..2 Fully relativistic case</A>
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<P>
<EM>The relativistic KS equations are
Dirac-like equations for a spinor with a ``large'' <!-- MATH
$R_{nlj}(r)$
-->
<I>R</I><SUB>nlj</SUB>(<I>r</I>) and
a ``small'' <!-- MATH
$S_{nlj}(r)$
-->
<I>S</I><SUB>nlj</SUB>(<I>r</I>) component:
</EM>
<BR>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{eqnarray}
c\left({d \over dr} + {\kappa\over r}\right)R_{nlj}(r) & = &
\left(2mc^2 - V(r) + \epsilon \right)S_{nlj}(r)\\
c\left({d \over dr} - {\kappa\over r}\right)S_{nlj}(r) & = &
\left( V(r) + \epsilon \right) R_{nlj}(r)
\end{eqnarray}
-->
<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><I>c</I><IMG STYLE="height: 5.60ex; vertical-align: -2.30ex; " SRC="img50.png"
ALT="$\displaystyle \left(\vphantom{{d \over dr} + {\kappa\over r}}\right.$"><IMG STYLE="height: 4.84ex; vertical-align: -1.69ex; " SRC="img51.png"
ALT="$\displaystyle {d \over dr}$"> + <IMG STYLE="height: 4.25ex; vertical-align: -1.69ex; " SRC="img52.png"
ALT="$\displaystyle {\kappa\over r}$"><IMG STYLE="height: 5.60ex; vertical-align: -2.30ex; " SRC="img53.png"
ALT="$\displaystyle \left.\vphantom{{d \over dr} + {\kappa\over r}}\right)$"><I>R</I><SUB>nlj</SUB>(<I>r</I>)</TD>
<TD WIDTH="10" ALIGN="CENTER" NOWRAP>=</TD>
<TD ALIGN="LEFT" NOWRAP><IMG STYLE="height: 2.80ex; vertical-align: -0.91ex; " SRC="img54.png"
ALT="$\displaystyle \left(\vphantom{2mc^2 - V(r) + \epsilon }\right.$">2<I>mc</I><SUP>2</SUP> - <I>V</I>(<I>r</I>) + <I>ε</I><IMG STYLE="height: 2.80ex; vertical-align: -0.91ex; " SRC="img55.png"
ALT="$\displaystyle \left.\vphantom{2mc^2 - V(r) + \epsilon }\right)$"><I>S</I><SUB>nlj</SUB>(<I>r</I>)</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(9)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><I>c</I><IMG STYLE="height: 5.60ex; vertical-align: -2.30ex; " SRC="img56.png"
ALT="$\displaystyle \left(\vphantom{{d \over dr} - {\kappa\over r}}\right.$"><IMG STYLE="height: 4.84ex; vertical-align: -1.69ex; " SRC="img51.png"
ALT="$\displaystyle {d \over dr}$"> - <IMG STYLE="height: 4.25ex; vertical-align: -1.69ex; " SRC="img52.png"
ALT="$\displaystyle {\kappa\over r}$"><IMG STYLE="height: 5.60ex; vertical-align: -2.30ex; " SRC="img57.png"
ALT="$\displaystyle \left.\vphantom{{d \over dr} - {\kappa\over r}}\right)$"><I>S</I><SUB>nlj</SUB>(<I>r</I>)</TD>
<TD WIDTH="10" ALIGN="CENTER" NOWRAP>=</TD>
<TD ALIGN="LEFT" NOWRAP><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="img58.png"
ALT="$\displaystyle \left(\vphantom{ V(r) + \epsilon }\right.$"><I>V</I>(<I>r</I>) + <I>ε</I><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="img59.png"
ALT="$\displaystyle \left.\vphantom{ V(r) + \epsilon }\right)$"><I>R</I><SUB>nlj</SUB>(<I>r</I>)</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(10)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL">
<EM>
where <I>j</I> is the total angular momentum (<I>j</I> = 1/2 if <I>l</I> = 0,
<!-- MATH
$j=l+1/2,l-1/2$
-->
<I>j</I> = <I>l</I> + 1/2, <I>l</I> - 1/2 otherwise); <!-- MATH
$\kappa=-2(j-l)(j+1/2)$
-->
<I>κ</I> = - 2(<I>j</I> - <I>l</I> )(<I>j</I> + 1/2) is the Dirac
quantum number (<I>κ</I> = - 1 is <I>l</I> = 0, <!-- MATH
$\kappa=-l-1,l$
-->
<I>κ</I> = - <I>l</I> - 1, <I>l</I> otherwise);
and the charge density is given by
</EM>
<P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{equation}
n(r) = \sum_{nlj} \Theta_{nlj}{R^2_{nlj}(r)+S^2_{nlj}(r)\over 4\pi r^2}.
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP>
<I>n</I>(<I>r</I>) = <IMG STYLE="height: 5.30ex; vertical-align: -3.37ex; " SRC="img60.png"
ALT="$\displaystyle \sum_{{nlj}}^{}$"><I>Θ</I><SUB>nlj</SUB><IMG STYLE="height: 5.36ex; vertical-align: -1.69ex; " SRC="img61.png"
ALT="$\displaystyle {R^2_{nlj}(r)+S^2_{nlj}(r)\over 4\pi r^2}$">.
</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(11)</TD></TR>
</TABLE>
</DIV>
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