Understanding ARPES: Momentum Conversion

The geometry we discussed in the introduction gives a complete picture of the experiment, but is not practical. For an experimenter, the situation is more complicated: while for a single Energy Distribution Curve (EDC) these equations can be directly solved, calculating the emission angle is not always straightforward because of the number of degrees of freedom on the six-axis sample goniometers which have become the standard for photoemission. Essentially all photoemission experiments now record very high dimensional datasets either due to capable analyzers, or by scanning goniometer angles. As a result, to transform evenly-spaced volumetric data in angle to evenly-spaced data in momentum, we actually require the inverse transform from momentum-space to angle-space, at which point we can interpolate the data based on the local structure in the recorded angle-resolved photocurrent.

In the most general case we record the electron velocity with three angles (you can read more about how PyARPES sets angle conventions and about its data format here): \(\phi\), the angle along the analyzer slit in the case of hemispheres, \(\psi\) the angle perpendicular to a hemispherical analyzer’s slit, and \(\alpha\) a rotation angle about the spectrometer axis. In our convention, horizontal slit hemispherical analyzers measure always with \(\alpha = 0\), while vertical slit analyzers measure with \(\alpha = \pi/2\). The photoelectron velocity that the analyzer records, \(\textbf{v}_\text{a}\) can be written as

\[\begin{split}\textbf{v}_\text{a} = \left[\begin{matrix} \cos\alpha\cos\psi\sin\phi - \sin\alpha\sin\psi \\ \sin\alpha\cos\psi\sin\phi + \cos\alpha\sin\psi \\ \cos\phi\cos\psi \end{matrix}\right]\end{split}\]

We need the velocity in the sample coordinate system, as these are the ones that can be related to the crystal momentum. A six-axis ARPES goniometer implements three rotations about the sample normal (\(\chi\)), about an axis perpendicular to the cryostat (\(\beta\)), and finally one around the cryostat axis (\(\theta\)). Depending on the design of the manipulator, the order of the \(\beta\) and \(\theta\) rotations may be reversed. Finally, rotations from the cryostat to sample normal coordinates \((\chi', \beta',\theta')\) must be performed if they arise due to unintended or intentional offsets of the crystal and cryostat normal vectors. Because these are often small we will absorb them into \((\chi, \beta,\theta)\) by the small angle approximation.

\[\textbf{v}_\text{s} = \text{R}(\chi,\hat{\textbf{z}})\text{R}(\beta,\hat{\textbf{x}})\text{R} (\theta,\hat{\textbf{y}})\textbf{v}_\text{a}\]

Finally, depending on the coordinates underlying the recorded ARPES data, these equations must be inverted from the appropriate velocities to the scanned analyzer. As an example, this can be done directly after small angle approximation in the case of a hemispherical analyzer which has perpendicular electron deflectors with \(\alpha = 0\)

\[\begin{split}\begin{aligned} \left(\phi - \theta\right) &\approx \arcsin\left(\frac{ \left(\text{R}\left(\chi,\hat{\textbf{z}}\right)\hat{\textbf{k}}\right)\cdot\hat{\textbf{x}}} {\sqrt{1 - \left(\left(\text{R}\left(\chi, \hat{\textbf{z}}\right)\hat{\textbf{k}}\right)\cdot\hat{\textbf{y}}\right)^2}}\right) \\ \left(\psi - \beta\right) &\approx \arcsin\left(\left(\text{R}\left(\chi, \hat{\textbf{z}}\right)\hat{\textbf{k}}\right) \cdot\hat{\textbf{y}}\right) \\ \end{aligned}\end{split}\]

The inverse transforms can be calculated for the remainder of the geometries, using one or both in-plane equations for fixed photon energy and using the out-of-plane expression when the incident photon energy is varied. So long as the hardware for a particular experiment does not change dramatically, these equations once calculated can be applied directly to data.

You can find a relatively complete implementation of these transforms in the small angle approximation picture in this Mathematica notebook.