Momentum conversion

ERLabPy can calculate momentum coordinates without changing the measured sampling. It can also interpolate the intensity onto a regular momentum grid. Both operations use the same momentum conversion functions. The mapping functions use the complete trigonometric geometry without a small-angle approximation.

Conversion inputs

ARPES data conventions defines the expected names, units, and storage locations.

Input	Role in the mapping functions
configuration	Selects the mapping functions and assigns the analyzer-slit direction to kx or ky
alpha and beta	Supply the measured angular coordinates
xi, chi, and angular offsets	Set the sample orientation and normal emission position; the delta offset rotates the in-plane momentum axes
eV, hv, and sample_workfunction	Determine \(E_k=h\nu-\Phi+E_b\) and the photoelectron momentum magnitude
alpha_scale and beta_scale	Multiply the stored alpha and beta coordinates before evaluation
inner_potential	Enters the free-electron final-state expression for kz in hν-dependent scans; it does not affect fixed-hν in-plane conversion

angle_resolution is not an input to the mapping functions. It is used only to select an automatic interpolation grid, as described below.

Variable experimental configurations

An experimental configuration describes the physical relation between the analyzer slit, deflector, and sample rotation axes during acquisition. It is not a display orientation or a generic correction for loader output.

Most endstations have one fixed configuration, which the loader assigns. Some endstations can rotate the analyzer slit or switch between deflector mapping and a physical sample rotation. One loader can then serve measurements acquired in different configurations.

xarray.DataArray.kspace.as_configuration() performs a semantic translation for these variable-geometry setups.

Changes	Does not change
The configuration attribute	Measured intensity values or sampling
Standard angle-coordinate names, translated by physical role	The physical geometry used during acquisition
A copy of the input object	Arbitrary or endstation-specific names from an incorrect loader

Use Changing the assigned configuration for the concrete ALS BL7 case and the coordinate-name translation.

Normal emission and angular offsets

The normal emission position is the measured angle pair that corresponds to emission along the sample surface normal.

• xarray.DataArray.kspace.set_normal() calculates the angular offsets for the selected configuration from this position.

• xarray.DataArray.kspace.offsets stores the angular offsets used for momentum conversion.

• Momentum conversion does not infer normal emission from the intensity maximum.

• Matrix-element asymmetry can move an intensity maximum away from the correct symmetry position.

The angular offsets are reference angles in the mapping functions. They are relative to the stored angle coordinates, not an absolute position of the sample normal.

• xarray.DataArray.kspace.set_normal() solves the offsets that map a known normal-emission position to zero in-plane momentum.

• Changing an angle coordinate while keeping its offset fixed changes the represented orientation.

• When a sample angle varies with hv, momentum conversion evaluates the varying coordinate together with its fixed reference offset. Replacing the varying coordinate with one constant angle changes the momentum trajectory.

• xarray.DataArray.kspace.offsets permits direct offset assignment when the sign conventions and physical reference angles are already known.

The three panels use the same simulated intensity and the same display limits. The middle panel omits a known \(\xi=3^\circ\) offset. The final panel uses that offset in the mapping functions. The dashed lines mark \(k_x=k_y=0\).

(Source code)

Use Changing the assigned configuration when a variable-geometry measurement uses a different configuration from the loader default. Use Converting to momentum space to set normal emission and perform the conversion.

Coordinates and interpolation

Operation	Sampling and dimensions
convert_coords()	Keeps measured dimensions and intensity values; adds momentum coordinates that can depend on several dimensions
convert()	Creates momentum dimensions and interpolates intensity onto a regular grid

Grid input	Effect
bounds	Sets the momentum limits; omitted limits are calculated from the mapped angular coordinates
resolution	Supplies a target step for each momentum axis; the final step can differ because convert() uses an integer number of intervals between the limits
Explicit kx, ky, or kz arrays	Supplies the exact target coordinates and overrides bounds and resolution for that axis
angle_resolution	Supplies the angular term in the automatic in-plane target-step estimate when no target step or coordinate array is given; the estimate also uses the minimum kinetic energy, the largest absolute scaled angle, and the applicable angle-scale factor

See also Converting measurements to a common momentum grid and Converting coordinates only.

hν–dependent scans

An hν-dependent scan changes the photoelectron kinetic energy. Momentum conversion uses the photon energy to calculate the out-of-plane momentum. For \(h\nu\)–dependent scans:

• eV must already contain binding energy for hν-dependent data.

• inner_potential is a parameter of the free-electron final-state approximation, not an analyzer setting.

• A sample angle that varies with hv produces a curved path through momentum space.

• hv_to_kz() returns calculated coordinates. It does not add measurements at new photon energies.

Use Converting hν–dependent scans for conversion. Use Photon-energy annotations for calculated paths on converted momentum-space intensity.