Anisotropy Parameter

For linearly polarized light the angular distribution of photodetached electrons from negative-ions is given by:

\[I(\epsilon, \theta) = \frac{\sigma_{\rm total}(\epsilon)}{4\pi} [ 1 + \beta(\epsilon) P_2(\cos\theta)]\]

where \(\beta(\epsilon)\) is the electron kinetic energy (\(\epsilon\)) dependent anisotropy parameter, that varies between -1 and +2, and \(P_2(\cos\theta)\) is the 2nd order Legendre polynomial in \(\cos\theta\). \(\sigma_{\rm total}\) is the total photodetachment cross section. The anisotropy parameter provides phase information about the dynamics of the photon process [1].

Methods

PyAbel provides two methods to determine the anisotropy parameter \(\beta\):

Method 1: linbasex evaluates \(\beta\) directly, available as the class attribute Beta[1]

This method fits spherical harmonic functions to the velocity-image to directly determine the anisotropy parameter as a function of the radial coordinate. This parameter has greater uncertainty in radial regions of low intensity, and so it is commonly plotted as the product \(I \times \beta\). See examples/example_linbasex.py
example_linbasex output image

Method 2: using abel.tools.vmi.radial_integration()

This method determines the anisotropy parameter from the inverse Abel transformed image, by extracting intensity vs angle for each specified radial range (tuples) and then fitting the intensity formula given above. This method is best applied to the radial ranges the correspond to strong spectral (intensity) in the image. It has the advantage of providing the least-squares fit error estimate for the parameter(s).

Example of both methods

See examples/example_anisotropy_parameter.py. In this case the anisotropy parameter is determined using each method. Note:

  1. Method 1 the filter parameter threshold=0.2 is set to a larger value so as to exclude evaluation in regions of weak intensity.
  2. Method 2 evaluates the anisotropy parameter for particular radial regions, of strong intensity.

(Source code, png, hires.png, pdf)

_images/example_anisotropy_parameter.png