epsproc.MFPAD module

ePSproc MFPAD functions

29/10/20 v2 Updated with Wigner delay routine.: Tidied up To Do list items. Tidied up docstring.
05/08/19 v1 Initial python version.: Based on original Matlab code ePS_MFPAD.m

Structure

Calculate MFPAD on a grid from input ePS matrix elements. Use fast functions, pre-calculate if possible. Data in Xarray, use selection functions and multiplications based on relevant quantum numbers, other axes summed over.

Choices for functions…

Moble’s spherical functions (quaternion based)

Provides fast wignerD, 3j and Ylm functions, uses Numba.

Install with:
```
>>> conda install -c conda-forge spherical_functions
```
Scipy special.sph_harm

To Do

More efficient computation? Use Xarray group by?

Formalism

MF-PADs numerical expansion

This is implemented numerically in epsproc.MFPAD.mfpad(). For direct computation of $beta_{LM}$ parameters (rather than full observable expansion) see epsproc.MFBLM() (original looped version) or epsproc.geomFunc.mfblmGeom() (geometric tensor version).

From ePSproc: Post-processing suite for ePolyScat electron-molecule scattering calculations.

\[ \begin{align}\begin{aligned}I_{\mu_{0}}(\theta_{\hat{k}},\phi_{\hat{k}},\theta_{\hat{n}},\phi_{\hat{n}})=\frac{4\pi^{2}E}{cg_{p_{i}}}\sum_{\mu_{i},\mu_{f}}|T_{\mu_{0}}^{p_{i}\mu_{i},p_{f}\mu_{f}}(\theta_{\hat{k}},\phi_{\hat{k}},\theta_{\hat{n}},\phi_{\hat{n}})|^{2}\label{eq:MFPAD}\\T_{\mu_{0}}^{p_{i}\mu_{i},p_{f}\mu_{f}}(\theta_{\hat{k}},\phi_{\hat{k}},\theta_{\hat{n}},\phi_{\hat{n}})=\sum_{l,m,\mu}I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)Y_{lm}^{*}(\theta_{\hat{k}},\phi_{\hat{k}})D_{\mu,\mu_{0}}^{1}(R_{\hat{n}})\label{eq:TMF}\\I_{l,m,\mu}^{p_{i}\mu_{i},p_{f}\mu_{f}}(E)=\langle\Psi_{i}^{p_{i},\mu_{i}}|\hat{d_{\mu}}|\Psi_{f}^{p_{f},\mu_{f}}\varphi_{klm}^{(-)}\rangle\label{eq:I}\end{aligned}\end{align} \]

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 $p_{i}$.

For more details 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

(MF) Wigner Delays

This is implemented numerically in epsproc.MFPAD.mfWignerDelay().

The partial wave, or channel resolved, Wigner delay can be given as the energy-derivative of the continuum wavefunction phase, $arg(psi_{lm}^{*})$

\[\begin{equation} \tau_{w}(k,\theta,\phi)=\hbar\frac{d\arg(\psi_{lm}^{*})}{d\epsilon} \end{equation}\]

And the total, or group, delay from the (coherent) sum over these channels.

\[\begin{equation} \tau_{w}^{g}(k,\theta,\phi)=\hbar\frac{d\arg(\sum_{l,m}\psi_{lm}^{*})}{d\epsilon}\label{eq:tauW_mol_sum} \end{equation}\]

Here the full wavefunction $sum_{l,m}psi_{lm}^{*}$ is essentially equivalent to the phase of the (conjugate) $T_{mu_{0}}^{p_{i}mu_{i},p_{f}mu_{f}*}(theta_{hat{k}},phi_{hat{k}},theta_{hat{n}},phi_{hat{n}})$ functions given above (and by epsproc.MFPAD.mfpad()).

For more details see, for example:

Hockett, Paul, Eugene Frumker, David M Villeneuve, and Paul B Corkum. “Time Delay in Molecular Photoionization.” Journal of Physics B: Atomic, Molecular and Optical Physics 49, no. 9 (May 2016): 095602. https://doi.org/10.1088/0953-4075/49/9/095602.
Carvalho, C.A.A. de, and H.M. Nussenzveig. “Time Delay.” Physics Reports 364, no. 2 (June 2002): 83–174. https://doi.org/10.1016/S0370-1573(01)00092-8.
Wigner, Eugene. “Lower Limit for the Energy Derivative of the Scattering Phase Shift.” Physical Review 98, no. 1 (April 1955): 145–47. https://doi.org/10.1103/PhysRev.98.145.

epsproc.MFPAD.mfWignerDelay(dataIn, pType='phaseUW')[source]

Compute MF Wigner Delay as phase derivative of continuum wavefunction.

Parameters

dataIn (Xarray) – Contains set(s) of wavefunctions to use, as output by epsproc.MFPAD.mfpad().
pType (str, optional, default = 'phaseUW') – Used to set data conversion options, as implemented in epsproc.plotTypeSelector() - ‘phase’ use np.angle() - ‘phaseUW’ unwapped with np.unwrap(np.angle())

epsproc.MFPAD.mfpad(dataIn, thres=0.01, inds={'Type': 'L', 'it': 1}, res=50, R=None, p=0)[source]

Parameters

dataIn (Xarray) – Contains set(s) of matrix elements to use, as output by epsproc.readMatEle().
thres (float, optional, default 1e-2) – Threshold value for matrix elements to use in calculation.
ind (dictionary, optional.) – Used for sub-selection of matrix elements from Xarrays. Default set for length gauage, single it component only, inds = {‘Type’:’L’,’it’:’1’}.
res (int, optional, default 50) – Resolution for output (theta,phi) grids.
R (list or Xarray of Euler angles or quaternions, optional.) – Define LF > MF polarization geometry/rotations. For default case (R = None), calls epsproc.setPolGeoms(), where 3 geometries are calculated, corresponding to z-pol, x-pol and y-pol cases. Defined by Euler angles (p,t,c) = [0 0 0] for z-pol, [0 pi/2 0] for x-pol, [pi/2 pi/2 0] for y-pol.
p (int, optional.) – Defines LF polarization state, p = -1…1, default p = 0 (linearly pol light along z-axis). TODO: add summation over p for multiple pol states in LF.

Returns

Ta – Xarray (theta, phi, E, Sym) of MFPADs, summed over (l,m)
Tlm – Xarray (theta, phi, E, Sym, lm) of MFPAD components, expanded over all (l,m)