1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<!--Converted with LaTeX2HTML 2019.2 (Released June 5, 2019) -->
<HTML lang="EN">
<HEAD>
<TITLE>A..3 Scalar-relativistic case</TITLE>
<META NAME="description" CONTENT="A..3 Scalar-relativistic case">
<META NAME="keywords" CONTENT="pseudo-gen">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=utf-8">
<META NAME="viewport" CONTENT="width=device-width, initial-scale=1.0">
<META NAME="Generator" CONTENT="LaTeX2HTML v2019.2">
<LINK REL="STYLESHEET" HREF="pseudo-gen.css">
<LINK REL="next" HREF="node19.html">
<LINK REL="previous" HREF="node17.html">
<LINK REL="next" HREF="node19.html">
</HEAD>
<BODY >
<!--Navigation Panel-->
<A
HREF="node19.html">
<IMG WIDTH="37" HEIGHT="24" ALT="next" SRC="next.png"></A>
<A
HREF="node15.html">
<IMG WIDTH="26" HEIGHT="24" ALT="up" SRC="up.png"></A>
<A
HREF="node17.html">
<IMG WIDTH="63" HEIGHT="24" ALT="previous" SRC="prev.png"></A>
<A ID="tex2html134"
HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALT="contents" SRC="contents.png"></A>
<BR>
<B> Next:</B> <A
HREF="node19.html">A..4 Numerical solution</A>
<B> Up:</B> <A
HREF="node15.html">A. Atomic Calculations</A>
<B> Previous:</B> <A
HREF="node17.html">A..2 Fully relativistic case</A>
<B> <A ID="tex2html135"
HREF="node1.html">Contents</A></B>
<BR>
<BR>
<!--End of Navigation Panel-->
<H2><A ID="SECTION00053000000000000000">
A..3 Scalar-relativistic case</A>
</H2>
<P>
<EM>The full relativistic KS equations
is be transformed into an equation for the large component only
and averaged over spin-orbit components. In atomic units
(Rydberg: <!-- MATH
$\hbar=1, m=1/2, e^2=2$
-->
<IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="img62.png"
ALT="$\hbar$"> = 1, <I>m</I> = 1/2, <I>e</I><SUP>2</SUP> = 2):
</EM>
<BR>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{eqnarray}
-{d^2 R_{nl}(r)\over dr^2}
+\left( {l(l+1)\over r^2} + M(r)\left(V(r)-\epsilon\right)
\right) R_{nl}(r) \qquad \nonumber \\\mbox{} -
{\alpha^2\over 4M(r)} {dV(r)\over dr}
\left({dR_{nl}(r)\over dr} +
\langle\kappa\rangle {R_{nl}(r)\over r}\right)= 0,
\end{eqnarray}
-->
<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">- <IMG STYLE="height: 5.07ex; vertical-align: -1.69ex; " SRC="img16.png"
ALT="$\displaystyle {d^2 R_{nl}(r)\over dr^2}$"> + <IMG STYLE="height: 5.60ex; vertical-align: -2.30ex; " SRC="img63.png"
ALT="$\displaystyle \left(\vphantom{ {l(l+1)\over r^2} + M(r)\left(V(r)-\epsilon\right)
}\right.$"><IMG STYLE="height: 4.95ex; vertical-align: -1.69ex; " SRC="img30.png"
ALT="$\displaystyle {l(l+1)\over r^2}$"> + <I>M</I>(<I>r</I>)<IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="img64.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="img65.png"
ALT="$\displaystyle \left.\vphantom{V(r)-\epsilon}\right)$"><IMG STYLE="height: 5.60ex; vertical-align: -2.30ex; " SRC="img66.png"
ALT="$\displaystyle \left.\vphantom{ {l(l+1)\over r^2} + M(r)\left(V(r)-\epsilon\right)
}\right)$"><I>R</I><SUB>nl</SUB>(<I>r</I>) </TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"> - <IMG STYLE="height: 5.60ex; vertical-align: -2.27ex; " SRC="img67.png"
ALT="$\displaystyle {\alpha^2\over 4M(r)}$"><IMG STYLE="height: 4.95ex; vertical-align: -1.69ex; " SRC="img68.png"
ALT="$\displaystyle {dV(r)\over dr}$"><IMG STYLE="height: 5.60ex; vertical-align: -2.30ex; " SRC="img69.png"
ALT="$\displaystyle \left(\vphantom{{dR_{nl}(r)\over dr} +
\langle\kappa\rangle {R_{nl}(r)\over r}}\right.$"><IMG STYLE="height: 4.95ex; vertical-align: -1.69ex; " SRC="img70.png"
ALT="$\displaystyle {dR_{nl}(r)\over dr}$"> + 〈<I>κ</I>〉<IMG STYLE="height: 4.95ex; vertical-align: -1.69ex; " SRC="img11.png"
ALT="$\displaystyle {R_{nl}(r)\over r}$"><IMG STYLE="height: 5.60ex; vertical-align: -2.30ex; " SRC="img71.png"
ALT="$\displaystyle \left.\vphantom{{dR_{nl}(r)\over dr} +
\langle\kappa\rangle {R_{nl}(r)\over r}}\right)$"> = 0,</TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(12)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL">
<EM>
where <!-- MATH
$\alpha=1/137.036$
-->
<I>α</I> = 1/137.036 is the fine-structure constant,
<!-- MATH
$\langle\kappa\rangle=-1$
-->
〈<I>κ</I>〉 = - 1 is the degeneracy-weighted average value
of the Dirac's <I>κ</I> for the two spin-orbit-split levels, <I>M</I>(<I>r</I>) is
defined as
</EM>
<P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{equation}
M(r)= 1 - {\alpha^2\over 4} \left(V(r)-\epsilon\right).
\end{equation}
-->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP>
<I>M</I>(<I>r</I>) = 1 - <IMG STYLE="height: 5.01ex; vertical-align: -1.69ex; " SRC="img72.png"
ALT="$\displaystyle {\alpha^2\over 4}$"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="img64.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="img65.png"
ALT="$\displaystyle \left.\vphantom{V(r)-\epsilon}\right)$">.
</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(13)</TD></TR>
</TABLE>
</DIV><EM>
The charge density is defined as in the nonrelativistic case:
</EM>
<P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{equation}
n(r) = \sum_{nl} \Theta_{nl}{R^2_{nl}(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: 5.01ex; vertical-align: -3.07ex; " SRC="img36.png"
ALT="$\displaystyle \sum_{{nl}}^{}$"><I>Θ</I><SUB>nl</SUB><IMG STYLE="height: 5.07ex; vertical-align: -1.69ex; " SRC="img37.png"
ALT="$\displaystyle {R^2_{nl}(r)\over 4\pi r^2}$">.
</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14)</TD></TR>
</TABLE>
</DIV>
<P>
<BR><HR>
</BODY>
</HTML>
|
