Included contents of ebfield.tex from MFG.

Author: fnevgeny

doc/physics.tex

@@ -682,7 +682,7 @@ \subsection{Factorized Collision Strength}
 points. The calculation of $<\psi_0||Z^k||\psi_1>$ is the same as that
 involved in the radiative transition rates, see Sec.~\ref{sec:structure}.
 
-\subsection{Solution of the Dirac Equation for The Continuum}
+\subsection{Solution of the Dirac Equation for the Continuum}
 In FAC, the continuum orbitals are obtained by solving the Dirac equations with
 the same central potential as that for bound orbitals. However, in obtaining the
 potential, one may optionally add a high lying subshell to the mean
@@ -1721,3 +1721,155 @@ \subsection{Introduction}
 energies should be used with caution for resonances below a few tens of
 eV. Sometimes, shifting the resonance energies for an entire series according
 to the experimental transition energies of the core prove to be practical.
+
+\section{External Electric and Magnetic Fields}
+The electric and magnetic fields are considered to be in the weak-field regime,
+in the sense that the  Zeeman and Stark energy splitting of atomic ions is much
+smaller than the Coulomb interaction between the electrons and the nucleus.
+However, the magnitude of electron-electron correlation energies may be
+comparable or even smaller than the interaction with the external fields.
+Therefore, the two effects must be taken into account in the theoretical
+treatment with the same level of detail. The natural method of solving this
+problem is the configuration interaction approximation, where the effects of
+both electron correlation and interaction with the external fields are included
+in all order within a limited configuration space.
+
+For atomic ions in a field-free environment, the total angular  momentum, $J$,
+and parity, $\pi$, are good quantum numbers. Therefore, the total  Hamiltonian
+is block diagonal when basis states of definite $J\pi$ symmetries are used. The
+magnetic field breaks the spherical symmetry, although $\pi$ and the angular
+momentum projection on the field direction are still conserved. The electric
+field in a direction different from that of the magnetic field further destroys
+the cylindrical and mirror symmetries, and no quantum number other than the
+energy can be considered conserved. The Hamiltonian matrix in the external
+fields is therefore typically much larger than in the zero-field case.
+
+Relativistic effects can be important for highly charged ions, and we will start
+from the Dirac Hamiltonian to solve the atomic structure
+\begin{equation}
+H = H_0 + H_B^{(1)} + H_B^{(2)} + H_E,
+\end{equation}
+where
+\begin{eqnarray}
+H_B^{(1)} &=& \sum_i \mu_B\left(2\vec{S}_i + \vec{L}_i\right)\cdot\vec{B}
+\nonumber\\
+H_B^{(2)} &=& \sum_i \frac{1}{2}\mu_B^2\left|\vec{B}\times\vec{r}_i\right|^2
+\nonumber\\
+H_E &=& \sum_i \vec{E}\cdot\vec{r}_i.
+\end{eqnarray}
+The summation in the above equations is over all electrons, $H_0$ is the
+field-free Hamiltonian, $H_B^{(1)}$ is the linear Zeeman term, $H_B^{(2)}$ is
+the diamagnetic Zeeman term, $H_E$ is the interaction with the electric field,
+$\vec{S}_i$, $\vec{L}_i$, $\vec{r}_i$, are the spin angular momentum, orbital
+angular momentum, and position operators of the $i$-th electron, $\mu_B =
+5.788\times 10^{-5}$~eV/T is the Bohr magneton, and $\vec{E}$ and $\vec{B}$ are
+the electric and magnetic field vectors.
+
+In the configuration interaction approximation, the wavefunction of the system
+is assumed to be $\psi = \sum_i b_i \phi_i$, where $\phi_i$ is the
+antisymmetrized product of single-electron Dirac wavefunctions for any given
+electronic configuration. These single-electron wavefunctions are normally
+derived from self-consistent Dirac-Fock calculations without consideration of
+external fields. The mixing coefficients, $b_i$, and the energy value
+associated with the total wavefunction $\psi$ are the eigenvalue solution of
+the Hamiltonian matrix in the representation of $\phi_i$ basis. The matrix
+elements of the Hamiltonian are given by $H_{ij} = <\phi_i|H|\phi_j>$.
+
+The first step in solving the atomic structure is therefore to construct the
+Hamiltonian matrix with a suitable set of basis wavefunctions. Because we are
+only concerned with the weak-field regime in this project, the basis
+wavefunctions are derived using the same method used as in the zero-field
+approximation. The calculation of Hamiltonian matrix elements of $H_0$ is also
+the same as in the zero-field case. Therefore, we will only need to develop new
+codes to obtain matrix elements of $H_B^{(1)}$, $H_B^{(2)}$, and $H_E$. These
+matrix elements are more easily calculated by converting the vector products
+into spherical tensor form, and separating the radial and angular integrations
+using the angular momentum theory.
+
+Using the spherical tensor components of a vector
+\begin{eqnarray}
+T_1 &=& -\frac{1}{\sqrt{2}}\left(V_x+iV_y\right) \nonumber\\
+T_0 &=& V_z \nonumber\\
+T_{-1} &=& \frac{1}{\sqrt{2}}\left(V_x-iV_y\right).
+\end{eqnarray}
+The $H_E$ term is rewritten as
+\begin{eqnarray}
+\label{eq:he}
+H_E &=& \sum_q (-1)^q E_q r_iC^1_{-q}(i) \nonumber\\
+     &=& \sum_q (-1)^q E_q \sum_{\alpha\beta}
+     Z^1_{-q}(\alpha\beta)<\alpha||C^1||\beta>r,
+\end{eqnarray}
+where $q=-1$,0, or 1, $C^k_q$ is the normalized spherical harmonics defined as
+\begin{equation}
+C^k_q = \left(\frac{4\pi}{2k+1}\right)^{1/2}Y_{kq}(\theta,\phi),
+\end{equation}
+and $Z^k_q(\alpha\beta)$ is the second quantized form of the angular
+operator, and $\alpha$ and $\beta$ are the single-electron Dirac
+orbitals present
+in the basis states. The reduced matrix elements of $C^k$ are given by
+\begin{equation}
+<\alpha||C^k||\beta> =
+(-1)^{j_\alpha+1/2}\left[(2j_\alpha+1)(2j_\beta+1)\right]^{1/2}
+\threej{j_\alpha}{k}{j_\beta}{\frac{1}{2}}{0}{-\frac{1}{2}},
+\end{equation}
+where \threej{a}{b}{c}{d}{e}{f} represents the Wigner $3j$ symbol.
+This standard way of separating radial and angular
+integration is used throughout FAC to compute matrix elements of
+various operators, including, but not limited to, the Hamiltonian.
+
+After some algebraic manipulation, the angular reduction of the $H_B^{(1)}$
+term results in
+\begin{eqnarray}
+\label{eq:hb1}
+H_B^{(1)} &=& \mu_B\sum_q (-1)^q B_q\sum_{\alpha\beta}
+Z^1_{-q}(\alpha\beta)<\alpha||J^1 + S^1||\beta> \nonumber\\
+&=& \mu_B\sum_q (-1)^q B_q\sum_{\alpha\beta}Z^1_{-q}(\alpha\beta)
+\Bigg\{\delta_{j_\alpha
+   j_\beta}\left[j_\alpha(j_\alpha+1)(2j_\alpha+1)\right]^{1/2} \nonumber\\
+&+&
+(-1)^{j_\alpha+l_\alpha-1/2}\left(\frac{3}{2}\right)^{1/2}
+\left[(2j_\alpha+1)(2j_\beta+1)\right]^{1/2}
+\sixj{l_\alpha}{\frac{1}{2}}{j_\alpha}{1}{j_\beta}{\frac{1}{2}}\Bigg\},
+\end{eqnarray}
+where $l_\alpha$, $j_\alpha$ are the orbital and total angular momentum of
+the Dirac orbital $\alpha$, and \sixj{a}{b}{c}{d}{e}{f} represents the Wigner
+$6j$ symbol, which comes as part of the reduced matrix elements of the spin
+operator.
+
+The reduction of the $H_{B}^{(2)}$ term is more complicated, as it involves
+the cross product, and the quadratic term causes a rank-2 tensor to
+appear in the expression. It can be shown that
+\begin{eqnarray}
+\label{eq:hb2}
+H_B^{(2)} &=& \sqrt{3}\mu_B^2r^2\sum_{q_1p_1}
+\threej{1}{1}{0}{q_1}{p_1}{0}B_{q_1}B_{p_1}\sixj{1}{1}{0}{1}{1}{1}
+\sum_{\alpha\beta}Z^0_0(\alpha\beta)<\alpha||C^0||\beta>
+\nonumber \\
+&-&\sqrt{30}\mu_B^2r^2\sum_{qq_1p1}\threej{1}{1}{2}{q_1}{p_1}{q}B_{q_1}B_{p_1}
+\sixj{1}{1}{2}{1}{1}{1}
+  \sum_{\alpha\beta}Z^2_q(\alpha\beta)<\alpha||C^2||\beta>.
+\end{eqnarray}
+
+Using Equations~\ref{eq:he}, \ref{eq:hb1} and \ref{eq:hb2}, the construction
+of the Hamiltonian matrix becomes a simple matter of evaluating the angular
+coefficients $<\phi_i|Z^k(\alpha\beta)|\phi_j>$ and the average values of the
+radial operators $r$ and $r^2$. These quantities are already calculated in FAC
+in the zero-field atomic structure theory, and existing code can be used.
+
+After the Hamiltonian matrix is constructed, a standard linear algebra
+algorithm is used to solve the eigenvalue problem to obtain the field-modified
+energy levels and mixing coefficients, $b_i$, of the wavefunctions.
+
+The calculation of radiative transition rates proceeds as in the zero-field
+theory, except that the
+wavefunctions now do not have a definite total angular momentum and
+parity. For example, the line strength of E1 transitions can be calculated as
+\begin{equation}
+S_{fi} = \sum_M\left|\sum_{\mu\nu}b_{f\mu}b_{i\nu}\sum_{\alpha\beta}
+<\phi_\mu||Z^1_M(\alpha,\beta)||\phi_\nu><\alpha||C^1||\beta>
+M^1_{\alpha\beta}\right|^2 ,
+\end{equation}
+where $M^1_{\alpha\beta}$ is the radial part of the relativistic
+single-electron E1 transition operator, as defined by \citet{grant74}. The
+oscillator strength and transition rates are proportional to the line
+strengths.