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documentation:standard_operators:spin_orbit_coupling [2016/10/06 08:23] – created Maurits W. Haverkortdocumentation:standard_operators:spin_orbit_coupling [2017/02/23 17:27] (current) – Maurits W. Haverkort
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 +{{indexmenu_n>4}}
 +====== Spin orbit coupling operator (l.s) ======
  
 +###
 +The spin-orbit interaction is defined as:
 +\begin{equation}
 +\xi \sum_i l_i \cdot s_i,
 +\end{equation}
 +with $l_i$ and $s_i$ the one electron orbital and spin operators respectively and the sum over $i$ summing over all electrons. The prefactor $\xi$ is an atom dependent constant, which is to a good approximation material independent and given as:
 +\begin{equation}
 +\xi = \left\langle R(r) \left| \frac{1}{2m^2c^2}\frac{1}{r}\frac{\mathrm{d}V(r)}{\mathrm{d} r} \right| R(r) \right\rangle.
 +\end{equation}
 +The derivative of the potential multiplied by $1/r$ is only contributing close to the nucleus where electrons have relativistic speeds. We therefore can make the approximation that the potential has a spherical form and one can separate the radial and angular parts of the wave-function. Using these approximations one can derive the equation above starting from the Dirac equation and using perturbation theory.
 +###
 +
 +###
 +In second quantization the spin-orbit operator becomes:
 +\begin{eqnarray}
 +\sum_i l_i \cdot s_i &=& \sum_i l_z^i s_z^i + \frac{1}{2} (l_i^+ s_i^- + l_i^-s_i^+)\\
 +\nonumber &=& \sum_{m=-l}^{m=l} \sum_{\sigma} m \sigma a^{\dagger}_{m\sigma}a^{\phantom{\dagger}}_{m\sigma} \\
 +\nonumber &+&  \sum_{m=-l}^{m=l-1}\frac{1}{2} \sqrt{l+m+1}\sqrt{l-m}(a^{\dagger}_{m+1,\downarrow}a^{\phantom{\dagger}}_{m,\uparrow}+a^{\dagger}_{m,\uparrow}a^{\phantom{\dagger}}_{m+1,\downarrow}).
 +\end{eqnarray}
 +The equivalent operator in Quanty is created by:
 +<code Quanty Example.Quanty>
 +Oppldots = NewOperator("ldots", NF, IndexUp, IndexDn)
 +</code>
 +###
 +
 +
 +===== Table of contents =====
 +{{indexmenu>.#1|msort nsort}}

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