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Methods

07/02/20

Each notebook on ePSdata shows various aspects of the results from an ePolyScat computation. The main sections of the notebooks are self-describing. Some additional details on the methods used are given here.

Formalism

ePolyScat matrix elements:

\begin{equation} I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)=\langle\Psi_{i}^{p_{i},\mu_{i}}|\hat{d_{\mu}}|\Psi_{f}^{p_{f},\mu_{f}}\varphi_{klm}^{(-)}\rangle\label{eq:I} \end{equation}

MFPADs are given by:

\begin{equation} T_{\mu_{0}}^{p_{i}\mu_{i},p_{f}\mu_{f}}(\theta_{\hat{k}},\phi_{\hat{k}},\theta_{\hat{n}},\phi_{\hat{n}})=\sum_{l,m,\mu}I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)Y_{lm}^{*}(\theta_{\hat{k}},\phi_{\hat{k}})D_{\mu,\mu_{0}}^{1}(R_{\hat{n}})\label{eq:TMF} \end{equation}

\begin{equation} I_{\mu_{0}}(\theta_{\hat{k}},\phi_{\hat{k}},\theta_{\hat{n}},\phi_{\hat{n}})=\frac{4\pi^{2}E}{cg_{p_{i}}}\sum_{\mu_{i},\mu_{f}}|T_{\mu_{0}}^{p_{i}\mu_{i},p_{f}\mu_{f}}(\theta_{\hat{k}},\phi_{\hat{k}},\theta_{\hat{n}},\phi_{\hat{n}})|^{2}\label{eq:MFPAD} \end{equation}

In this formalism:

  • \(I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)\) is the radial part of the dipole matrix element, determined from the initial and final state electronic wavefunctions \(\Psi_{i}^{p_{i},\mu_{i}}\)and \(\Psi_{f}^{p_{f},\mu_{f}}\), photoelectron wavefunction \(\varphi_{klm}^{(-)}\) and dipole operator \(\hat{d_{\mu}}\). Here the wavefunctions are indexed by irreducible representation (i.e. symmetry) by the labels \(p_{i}\) and \(p_{f}\), with components \(\mu_{i}\) and \(\mu_{f}\) respectively; \(l,m\) are angular momentum components, \(\mu\) is the projection of the polarization into the MF (from a value \(\mu_{0}\) in the LF). Each energy and irreducible representation corresponds to a calculation in ePolyScat.

  • \(T_{\mu_{0}}^{p_{i}\mu_{i},p_{f}\mu_{f}}(\theta_{\hat{k}},\phi_{\hat{k}},\theta_{\hat{n}},\phi_{\hat{n}})\) is the full matrix element (expanded in polar coordinates) in the MF, where \(\hat{k}\) denotes the direction of the photoelectron \(\mathbf{k}\)-vector, and \(\hat{n}\) the direction of the polarization vector \(\mathbf{n}\) of the ionizing light. Note that the summation over components \(\{l,m,\mu\}\) is coherent, and hence phase sensitive.

  • \(Y_{lm}^{*}(\theta_{\hat{k}},\phi_{\hat{k}})\) is a spherical harmonic.

  • \(D_{\mu,\mu_{0}}^{1}(R_{\hat{n}})\) is a Wigner rotation matrix element, with a set of Euler angles \(R_{\hat{n}}=(\phi_{\hat{n}},\theta_{\hat{n}},\chi_{\hat{n}})\), which rotates/projects the polarization into the MF .

  • \(I_{\mu_{0}}(\theta_{\hat{k}},\phi_{\hat{k}},\theta_{\hat{n}},\phi_{\hat{n}})\) is the final (observable) MFPAD, for a polarization \(\mu_{0}\) and summed over all symmetry components of the initial and final states, \(\mu_{i}\) and \(\mu_{f}\). Note that this sum can be expressed as an incoherent summation, since these components are (by definition) orthogonal.

  • \(g_{p_{i}}\) is the degeneracy of the state \(p_{i}\).

See: Toffoli, D., Lucchese, R. R., Lebech, M., Houver, J. C., & Dowek, D. (2007). Molecular frame and recoil frame photoelectron angular distributions from dissociative photoionization of NO2. The Journal of Chemical Physics, 126(5), 054307. https://doi.org/10.1063/1.2432124

From the matrix elements other quantities can be computed. The \(\beta_{L,M}\) parameters (MF) are defined as:

:nbsphinx-math:`begin{eqnarray} beta_{L,-M}^{mu_{i},mu_{f}} & = & sum_{l,m,mu}sum_{l’,m’,mu’}(-1)^{M}(-1)^{m}(-1)^{(mu’-mu_{0})}left(frac{(2l+1)(2l’+1)(2L+1)}{4pi}right)^{1/2}left(begin{array}{ccc} l & l’ & L\ 0 & 0 & 0 end{array}right)left(begin{array}{ccc} l & l’ & L\ -m & m’ & -M end{array}right)nonumber \

& times & sum_{P,R’,R}(2P+1)(-1)^{(R’-R)}left(begin{array}{ccc}

1 & 1 & P\ mu & -mu’ & R’ end{array}right)left(begin{array}{ccc} 1 & 1 & P\ mu_{0} & -mu_{0} & R end{array}right)D_{-R’,-R}^{P}(R_{hat{n}})I_{l,m,mu}^{p_{i}mu_{i},p_{f}mu_{f}}(E)I_{l’,m’,mu’}^{p_{i}mu_{i},p_{f}mu_{f}*}(E) end{eqnarray}`

See the ePSproc docs on ``ep.mfblm()` <https://epsproc.readthedocs.io/en/latest/modules/epsproc.MFBLM.html>`__ for further details, and this demo notebook.