Momentum conversion¶

Momentum conversion in ERLabPy is exact with no small angle approximation, but is also very fast, thanks to the numba-accelerated trilinear interpolation in erlab.analysis.interpolate.

Nomenclature¶

Momentum conversion in ERLabPy follows the nomenclature from Ishida and Shin [3].

All experimental geometry can be classified into two configurations, Type 1 and Type 2, based on the relative position of the rotation axis and the analyzer slit. These can be further divided into 4 configurations depending on the use of photoelectron deflectors (DA).

Definition of angles differ for each geometry, but in all cases, \(\delta\) is the azimuthal angle that indicates in-plane rotation, \(\alpha\) is the angle detected by the analyzer, and \(\beta\) is the angle along which mapping is performed.

For instance, imagine a typical Type 1 setup with a vertical slit that acquires maps by rotating about the z axis in the lab frame. In this case, the polar angle (rotation about z) is \(\beta\), and the tilt angle becomes \(\xi\).

The following table summarizes angle conventions for commonly encountered configurations.

Analyzer slit orientation	Mapping angle	Configuration	Polar	Tilt	Deflector	Azimuth	Analyzer
Vertical	Polar	1 (Type 1)	`beta`	`xi`		`delta`	`alpha`
Horizontal	Tilt	2 (Type 2)	`xi`	`beta`
Vertical	Deflector	3 (Type 1 + DA)	`chi`	`xi`	`beta`
Horizontal	Deflector	4 (Type 2 + DA)	`chi`	`xi`	`beta`

Note

Analyzers that can measure two-dimensional angular information simultaneously (e.g. time-of-flight analyzers) can be treated like hemispherical analyzers equipped with a deflector.

import matplotlib.pyplot as plt
import numpy as np

import erlab.plotting as eplt

Momentum conversion¶

Note

For momentum conversion to work properly, the data must follow the conventions listed here.

Setting parameters¶

Parameters that are needed for momentum conversion are:

The information about the experimental configuration
Work function of the system
The inner potential \(V_0\) (for photon energy dependent data)
Angle offsets

These parameters are all stored as data attributes. The kspace accessor provides various ways to access and modify these parameters.

Experimental configuration¶

The first step is to set the experimental configuration. Most of the time, this information is already recorded in the data file by the data loader plugin. If not, it can be set manually using xarray.DataArray.kspace.configuration:

data.kspace.configuration = erlab.constants.AxesConfiguration.Type1DA

This will set the configuration to Type 1 with a deflector. The configuration can also be set using numbers:

data.kspace.configuration = 3

Sometimes, the automatically determined configuration may be incorrect. For example, plugins for setups equipped with an electrostatic deflector will assign configuration 3 or 4 to the data, but the data may have been acquired without the deflector, in which case configuration 1 or 2 should be used. Also, some setups (e.g. ALS BL7) have variable slit orientation by allowing the analyzer to be rotated about the lens axis. As such, it is not always possible to determine the configuration from the data alone. In these cases, the configuration can be converted with xarray.DataArray.kspace.as_configuration() which takes a configuration number or an enum as an argument. For example, consider a data taken at ALS BL7 with configuration 2 (horizontal slit, tilt map). By default, the loader will assign configuration 3 (vertical slit, DA map) to the data, which is incorrect. To convert the data to configuration 2, you can do:

data = data.kspace.as_configuration(2)

which returns a copy of the data with the configuration set to 2, and the coordinates renamed accordingly.

Note

The method assumes a typical ARPES setup with a vertical cryostat. For complex setups, the user should manually set the configuration attribute and rename the coordinates.

Work function of the system¶

The work function of the system can be set using xarray.DataArray.kspace.work_function:

data.kspace.work_function = 4.5

Inner potential \(V_0\)¶

The inner potential (for photon energy dependent data) can be set using xarray.DataArray.kspace.inner_potential:

data.kspace.inner_potential = 10.0

Angle offsets¶

Angle offsets can be set using xarray.DataArray.kspace.offsets. For demonstration, let’s generate some example data, this time in angle coordinates.

To view the currently set angle offsets:

dat.kspace.offsets

delta	0.0
xi	0.0
beta	0.0

Since we haven’t set any offsets, they are all zero. Now, if we want to set the azimuthal angle to 60 degrees and the polar offset to 30 degrees, we can update the offsets as follows:

dat.kspace.offsets.update(delta=60.0, beta=30.0)

delta	60.0
xi	0.0
beta	30.0

The offsets behave like a dictionary, and you can also reset or override all offsets at once using a dictionary:

dat.kspace.offsets = dict(delta=30.0)

dat.kspace.offsets

delta	30.0
xi	0.0
beta	0.0

See xarray.DataArray.kspace.offsets for more ways to access and modify offsets.

Note

Offsets can be easily determined with the interactive momentum conversion tool.

Interactive conversion¶

For three dimensional momentum conversion like maps or photon energy scans, an interactive window can be opened where you can adjust the parameters and see the effect right away.

There are three ways to invoke the GUI. The first one is to call xarray.DataArray.kspace.interactive():

data.kspace.interactive()

The second option is to invoke the GUI with erlab.interactive.ktool(). This option is recommended because the name of the input data will be automatically detected and applied to the generated code that is copied to the clipboard.

import erlab.interactive as eri

eri.ktool(data)

The final option is to trigger the GUI from the ImageTool with the “Open in ktool” menu in the View menu. The button will be disabled if the data is not compatible with erlab.interactive.ktool().

The GUI is divided into two tabs.

The first tab is for setting momentum conversion parameters. The image is updated in real time as you change the parameters. Clicking the “Copy code” button will copy the code for conversion to the clipboard. The “Open in ImageTool” button performs a full three-dimensional conversion and opens the result in the ImageTool.

The second tab provides visualization options. You can overlay Brillouin zones and high symmetry points on the result, adjust colors, apply binning, and more. The “Add Circle ROI” button allows you to add a circular region of interest to the image, which can be edited by dragging or right-clicking on it.

You can pass some parameters to customize the GUI. For example, you can set the Brillouin zone size/orientation and the colormap like this:

data.kspace.interactive(
    avec=np.array([[-3.485, 6.03], [6.97, 0.0]]), rotate_bz=30.0, cmap="viridis"
)

See the documentation of erlab.interactive.ktool() for more information.

Momentum conversion¶

Nomenclature¶

Momentum conversion¶

Setting parameters¶

Experimental configuration¶

Work function of the system¶

Inner potential \(V_0\)¶

Angle offsets¶

Converting to momentum space¶

Converting coordinates only¶

\(k_z\)-dependent data¶

Annotating the photon energy¶

Interactive conversion¶