arpes.analysis.derivative.curvature

arpes.analysis.derivative.curvature(arr, dims=('phi', 'eV'), alpha=1)

Provides “curvature” analysis for band locations.

Keep it for just compatilitiby

Defined via

\[C(x,y) = \frac{([C_0 + (df/dx)^2]\frac{d^2f}{dy^2} - 2 \frac{df}{dx}\frac{df}{dy} \frac{d^2f}{dxdy} + [C_0 + (\frac{df}{dy})^2]\frac{d^2f}{dx^2})}{ (C_0 (\frac{df}{dx})^2 + (\frac{df}{dy})^2)^{3/2}}\]

of in the case of inequivalent dimensions \(x\) and \(y\)

\[C(x,y) = \frac{[1 + C_x(\frac{df}{dx})^2]C_y \frac{d^2f}{dy^2} - 2 C_x C_y \frac{df}{dx}\frac{df}{dy}\frac{d^2f}{dxdy} + [1 + C_y (\frac{df}{dy})^2] C_x \frac{d^2f}{dx^2}}{ (1 + C_x (\frac{df}{dx})^2 + C_y (\frac{df}{dy})^2)^{3/2}}\]

(Eq. (14) in Rev. Sci. Instrum. 82, 043712 (2011).)

where

\[C_x = C_y (\frac{dx}{dy})^2\]

The value of \(C_y`\) can reasonably be taken to have the value

\[(\frac{df}{dx})_\text{max}^2 + \left|\frac{df}{dy}\right|_\text{max}^2 C_y = (\frac{dy}{dx}) (\left|\frac{df}{dx}\right|_\text{max}^2 + \left|\frac{df}{dy}\right|_\text{max}^2) \alpha\]

for some dimensionless parameter \(\alpha\).

Parameters:

• arr (xr.DataArray) – ARPES data

• dims (tuple[str, str]) – Dimension for apply the maximum curvature

• alpha (float) – regulation parameter, chosen semi-universally, but with no particular justification

Returns:

The curvature of the intensity of the original data.