### 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?

• 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?  • 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

• 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