## Electron Transportation

### Electron mobility in LAr

The electron mobility data set used in our global data fit are from Ref. [29, 45–58] in Ref. [1]. LAr temperature used in this global fit is 89 K. The data reported in the references are all scaled to this temperature with the temperature dependence of $T^{−3/2}$ (Ref. [59] in Ref. [1]). The fitting function is a rational polynomial expressed as: $\mu=\frac{a_0+a_1E+a_2E^{3/2}+a_3E^{5/2}}{1+(a_1/a_0)E+a_4E^2+a_5E^3}\left(\frac{T}{T_0}\right)^{-3/2},$ where $E$ is the electric field divided by 1 kV/cm, $T$ is the LAr temperature, and $a_0 = 551.6$ cm$^2$/V/s is the electron mobility at zero field with temperature of $T_0$ = 89 K. The fitting parameters are given bys $a_0 = 551.6, \quad a_1 = 7158.3, \quad a_2 = 4440.43, \quad a_3 = 4.29, \quad a_4 = 43.63, \quad a_5 = 0.2053$ Note that $a_1$ in the above is adjusted down by a factor of 0.9 from Ref. [1] to match the two recent precise measurements from MicroBooNE (1.101 mm/$\mu$s at 273 V/cm and 89 K [2]) and ProtoDUNE-SP (1.560 mm/$\mu$s at 486.7 V/cm and 87.7 K [3]).

### Electron drift velocity in LAr

Electron drift velocity is a product of electron mobility and the electric field. $v = \mu E$

### Effective longitudinal electron energy

We introduce a parameterization of the effective electron energy for the convenience of application. Both the data in Ref. [1] and ICARUS's data at low field are included. The parameterization is also in a form of rational polynomial $\epsilon_L=\frac{b_0+b_1E+b_2E^2}{1+(b_1/b_0)E+b_3E^2}\left(\frac{T}{T_1}\right)$ where $E$ is the the electric field divided by 1 kV/cm, $T$ is the LAr temperature, and $b_0=0.0075$ eV is the electron energy at $T_1$ = 87 K under zero field. The parameterization can be applied to other temperatures with a linear temperature dependence of $T$. The fitting parameters are given bys $b_0 = 0.0075, \quad b_1 = 742.9, \quad b_2 = 3269.6, \quad b_3 = 31678.2, \quad$

### Longitudinal and transverse diffusion coefficients

The longitudinal diffusion coefficients $D_L$ in the range of 0.1 to 1.5 kV/cm can be expressed as defined by the Einstein relation $D_L=\frac{\mu\epsilon_L}{e}=\left(\frac{a_0+a_1E+a_2E^{3/2}+a_3E^{5/2}}{1+(a_1/a_0)E+a_4E^2+a_5E^3}\right)\left(\frac{b_0+b_1E+b_2E^2}{1+(b_1/b_0)E+b_3E^2}\right)\left(\frac{T}{T_0}\right)^{-3/2}\left(\frac{T}{T_1}\right),$ with the parameters given previously.
The transverse diffusion coefficients $D_T$ is related to $D_L$ by: $\frac{D_L}{D_T} = 1+ \frac{E}{\mu}\frac{\partial \mu}{\partial E}$ The corresponding diffusion length can be calculated from the drift time $t$ by: $\sigma = \sqrt{2Dt}$

## References

1. Y. Li, et al., "Measurement of Longitudinal Electron Diffusion in Liquid Argon", NIMA 816, 160 (2016). [arXiv]