When the total energy of an ionizing particle, $E_0$, is absorbed in the detector medium, the number of the ion pairs $N_i$ is given as $E_0/W$, where W is an average energy required to form an ion pair (called $W$-value). The r.m.s. fluctuation of the number of ion pairs can be calculated to be $\sqrt{FN_i}$, where $F$ is the so-called "Fano-Factor" [2].

Considering the excitation (requiring $W_e$) and ionization (requiring $W_i$) in the atom, we have \[ E_0 = N_i \cdot W = N_i \cdot W_i + N_e \cdot W_e. \] Due to the actual much smaller $W_e$ than $W_i$, every "collision" will most likely cause excitation while the ionization must be combined with the excitation to dissipate precisely the total energy. Define $\sigma_x = \sigma(N_x)$ and $\sigma_e = \sqrt{N_e}$ ($N_e$ can be well approximated by a Poisson distribution), it can be derived that \[ \sigma_i \cdot W_i = \sigma_e \cdot W_e = \sqrt{N_e} \cdot W_e \] The fano factor then can be expressed as the last term in the following formula, \[ \sigma_i = \sqrt{N_i} \cdot \sqrt{\frac{W_e}{W_i}(\frac{W}{W_i}-1)}. \] In reality, three impacts are involved in the estimation of fano factor, which are ionization, excitation, and nuclear scattering, respectively. The contribution of nuclear scattering can be neglected.

The ultimate number of ionization electrons has to consider the recombination (discussed below) to the ions which serves as the survival probabilty of the ionization electrons liberated from the atoms. The excitation and recombination will contribute to the scintillation photons of liquid argon.

The recombination factor $R_c$ describes the efficiency in converting $dE/dx$ into $dQ/dx$, the visible energy available to be collected by the sense wires before attenuation. The ionized electron has about 5 eV energy with a distance about 0.5 nm from the ion. In about 2 ns, the ionized electron would go through 10,000 collisions before reaching thermal energy (~0.01 eV). At this point, the average separation distance is about 2.5 $\mu$m [3,4].

There are two common methods to model the recombination effect.

Birks Model: \[ R_c = \frac{dQ/dx}{dE/dx} = \frac{A_{3t}}{1+k_{3t}/\varepsilon \times dE/dx} \] , where $A_{3t} = 0.8$, $k_{3t} = 0.0486 (g/MeV cm^2)(kV/cm)$ and $\varepsilon = E \rho$ is the product of electric field and density.

Modified Box model: \[ R_c = \frac{dQ/dx}{dE/dx} = \frac{\ln(A+B/\varepsilon \times dE/dx)}{B/\varepsilon \times dE/dx} \] , where $A = 0.930$, $B = 0.212 (g/MeV cm^2)(kV/cm)$ and $\varepsilon = E \rho$ is the product of electric field and density.

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There are three microscopic theories describing the recombination: Germinate, Bulk, and Columnar. In particular, the prediction from Columnar theory suggests that the recombination has a dependence on the track angle with respect to the drift direction. Such an effect has been studied in ArgoNeuT [3]. While there are some hints regarding the angular dependence, the effect seems to be much smaller than that predicted by Columnar theory (see Fig. 9 from Ref [5]).