atomic number vs ion density

[ddudt]

Presently, we do not allow the user to specify the ion density profile directly. Instead, we input the atomic_number $Z_{eff}$ and then compute the ion density as:

$n_i = \frac{n_e}{Z_{eff}}$

However, the effective atomic number is commonly computed as:

$Z_{eff} = \frac{\Sigma n_i Z_i^2}{n_e}$

This will result in an incorrect ion density profile. What we really mean is the "average" atomic number defined as:

$Z_{avg} = \frac{\Sigma n_i Z_i}{n_e}$

I see two solutions to resolve this issue:

1. Update the documentation for the existing atomic_number profile and "Zeff" compute quantity to clarify that it is really the "average" atomic number.
2. Add a new equilibrium profile for ion_density. This would be an optional profile that can be used instead of atomic_number, similar to how we handle the option for either a current or iota profile.

I suggest that we implement both solutions.

[ddudt]

We require the atomic_number profile as an input to enforce that the plasma is quasi-neutral. So we do not want to add an ion_density profile input option.

[ddudt]

References indicating that our definition of $Z_{eff}$ is incorrect:

• Fusion Wiki
• TRANSP docs
• Wesson's Tokamaks textbook, equation 2.16.6

[ddudt]

The only place in the existing code where $Z_{eff}$ is used is in the bootstrap current computations. SIMSOPT also assumes that $n_i = n_e / Z_{eff}$ and notes that $Z_{eff}$ is "the average impurity charge", consistent with our usage. But Redl et al. does not define $Z_{eff}$ and Landreman et al. assumes $Z_{eff}=1$ in their results.

If the definition of $Z_{eff}$ in the bootstrap current equations is the "average" atomic number, then this issue will be resolved by clarifying our documentation. But if the Redl formula uses the definition I cited above, then we might need to input the atomic number and ion density profiles independently.

@landreman can you comment on this? Or @f0uriest can you discuss it with him?

@dpanici can you look into how V3FIT uses their profile inputs?