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: \[ \sigma = \sqrt{2Dt} \]

    Electron Attachment

    For the detection of the ionization charge in LArTPCs, electron attachment to impurities in LAr (such as water or oxygen) is a source of signal attenuation, given that the drift velocity of ions is about five orders of magnitude lower than that of electrons. The mean lifetime ($\tau_A$) of electron cluster is given by: \[ \tau_A = \frac{1}{k_A n}, \] where $k_A$ is the attachment rate constant and $n$ is the concentration of the impurity.

    The electron attachment rate constant ($k_A$) is an integral over energy of the product of the drifting election energy distribution function and the cross section for the attachment of the electron to an impurity molecule, both of which are functions of the drifting electron energy. Therefore, $k_A$ is a function of the external electric field. For electric fields below ~100 V/cm, the electrons are in thermal equilibrium with the liquid and in this region the attachment rate is independent of the electric field .

    In Ref [2], the electron attachment rate data is fit with a constrained rational polynomial: \[ k_A=10^p \frac{a_1/b_1+a_1E+a_2E^{2}+a_3E^{3}+a_4E^{4}}{1+b_1E+b_2E^{2}+b_3E^{3}+b_4E^{4}}, \] where E is the electric field in units of kV/cm and $k_A$ is the attachment rate constant in units of s$^{-1}$. The best-fit parameters for common impurities can be found in Table 2 of [2].

    Figure taken from [2]. The electron attachment rate constant as a function of the applied electric field for six impurities in LAr at 90K.


    1. Y. Li, et al., "Measurement of Longitudinal Electron Diffusion in Liquid Argon", NIMA 816, 160 (2016). [arXiv]
    2. Y. Li, et al., "Parameterization of Electron Attachment Rate Constants for Common Impurities in LArTPC Detectors", arXiv:2205.06888