atoMEC: an open-source average-atom Python code

Micro-summary

Average-atom models are a crucial tool in modelling the warm dense matter regime
Average-atom models are computationally efficient and (relatively) simple to develop
We present atoMEC: an open-source average-atom Python code for Matter under Extreme Conditions
The aim of atoMEC is to make average-atom models more accessible
We discuss some background, the code structure, and some example capabilities

Poster hint: change zoom to optimize layout

What is warm dense matter?

Figure depicting the temperatures and densities spanned by the warm dense matter regime, and some examples of wardm dense matter such as white dwarfs

Warm dense matter (WDM) lies somewhere between condensed matter physics (CMP) and plasma physics
It is characterized by moderate to high temperatures and densities ranging from dense gases to highly compressed solids
There are many astrophysical and technological (e.g. inertial confinement fusion (ICF)) examples of warm dense matter

What are average-atom models?

Figure visualizing the average-atom concept

Figure depecting a many-body system of interacting electrons and nuclei

Average-atom models map the many-body system of electrons and nuclei to a single atom immersed in a plasma
This massively reduces the computational complexity: typically, from hundreds of core hours to less than one
Drag the slider to visualize this mapping

Average-atom calculation: theory

The main step in an average-atom calculation is the minimization of the grand free energy
This minimization follows an iterative process known as a self-consistent field procedure
This gives access to the electronic wave-functions and associated quantities such as their energies and the total electron density
These can be used either directly or with some post-processing to calculate properties of interest such as the pressure or the ionization degree

Average-atom calculation: considerations

The minimization of the grand free energy requires the following considerations:

Physical conditions

What is the temperature and the density of my material?

Model and approximations

For example: What boundary conditions should I use? How should I treat delocalized (unbound) electrons?

Algorithms and numerics

For example: How large should my grid be? How should I solve the differential equations?

Physical conditions: `atoMEC.Atom`

In atoMEC, the choice of density and temperature is handled by the atoMEC.Atom class.

Consider Aluminium at its normal metallic density (2.7 g cm\(^{-3}\)) and room temperature (300 K):

  
Al_atom = atoMEC.Atom(
    "Al",
    300,
    density=2.7,
    units_temp="K",
)

We use the mendeleev library to get the atomic properties.

Model and approximations: `atoMEC.models`

An "ion-sphere" model is invoked with the atoMEC.models.ISModel class.

Suppose we choose the "neumann" boundary condition and "quantum" treatment for delocalized (unbound) electrons:


model = atoMEC.models.ISModel(
    Al_atom,
    bc="neumann",
    unbound="quantum",
)

The choice of exchange-correlation functionals (see corresponding paper) is handled by the LIBXC library.

Algorithms and numerics: `ISModel.CalcEnergy`

The minimization of the grand free energy is handled by the ISModel.CalcEnergy function.

Suppose we want the 3 lowest-energy orbitals in the "n" and "l" directions (these denote quantum numbers), with a grid size of 1500 points:

from atoMEC import config
config.numcores = -1  # parallelize over all cores
SCF_out = model.CalcEnergy(
    3,
    3,
    grid_params={"ngrid": 1500},
)

We use SciPy (scipy.sparse) to find the eigenvalues, joblib for parallelization, and SciPy and NumPy for various other tasks (root finding, integration, ...).

Case-study: metallization of helium

In the outer layers of white dwarfs, Helium is expected to transition from insulating to metallic behaviour
The conditions for this transition (\(\sim 10-100\)kK, \(\sim 1-30\) g cm\(^{-3}\)) are a typical example of warm dense matter
We can model this transition with an average-atom model
The key quantity is the so-called electronic band-gap: a gap in the permitted energy range for the electrons

Helium metallization: density-of-states

Density: g cm\(^{-3}\)

The density-of-states shows the permitted electron energy ranges
Drag the slider to see the band-gap narrow as density increases (at fixed temperature 50 kK)
Insulator-to-metallic transition happens when band-gap vanishes

Helium metallization: band-gap and ionization

At 50 kK, the band-gap decreases linearly with density and closes at 5.5 g cm\(^{-3}\)
The ionization fraction - the fraction of delocalized (free) electrons - increases with density
This also correlates with higher conductivity

Useful links

atoMEC source code and documentation: contributions welcome!
Papers:
1. Corresponding paper for this poster: arXiv:2206.01074
2. First-principles derivation of an average-atom model: 10.1103/physrevresearch.4.023055
3. Mean ionization state comparisons: arXiv:2203.05863
Introductory notebook for atoMEC
Repository for the helium band-gap calculations
More information about warm dense matter and our research at CASUS