When a charged particle goes through liquid argon (LAr), it takes about 23.6 eV energy to ionize an electron in average. For minimal ionizing particle (heavy ionizing particle), the mean energy loss is about 2.1 (25) MeV/cm leading to about 100 (10) nm average separation between two adjacent ions.

The **mean rate** of energy loss by moderately relativistic charged heavy particles is well-described by the Bethe formula. Considering the actual energy transfers might be restricted to $T \le T_{cut} \le T_{max}$, this formula can be expressed in a unified way,
\[
-< \frac{dE}{dx} > = Kz^2 \frac{Z}{A}\frac{1}{\beta^2}[\frac{1}{2}ln\frac{2m_ec^2\beta^2\gamma^2T_{cut}}{I^2}-\frac{\beta^2}{2}(1+\frac{T_{cut}}{T_{max}})-\frac{\delta(\beta\gamma)}{2}].
\]
Details of the parameters can be seen in PDG.
Note that the mean $dE/dx$ has no dependency on the thickness of the absorbers. Calculations regarding the LAr are shown in the central plot on the right side (from Milind Diwan).

The flucatuations in enegy loss can be adequately described by the highly-skewed Landau or Landau-Vavilov distribution (see the upper pannel of the right plot), though the Landau distribution fails to describe energy loss in thin absorbers.
The **most probable energy loss** can be calculated with the equation,
\[
\Delta_p = {\xi} [ln\frac{2mc^2\beta^2\gamma^2}{I} + ln\frac{{\xi}}{I} +j -\beta^2 - \delta(\beta\gamma)],
\]
where $\xi = (K/2)< Z/A >(x/\beta^2)$ MeV for a detector with a thickness $x$ in $g~cm^{-2}$. Details can be seen in Ref. [1], especailly the density effect $\delta$ which was not included in Landau's or Vavilov's work. Note that the most probable energy loss is generally smaller than the mean value and do have dependency on the thickness of the absorbers as indicated by $\xi$. MPV vs. muon momentum (or thickness) are shown in the central (or lower) plot on the right side.

**Useful numbers for liquid argon**: LAr density 1.38/1.40 g/cm$^3$ at 89/87 K. The minimum ionizing muon (about 260 MeV kinetic energy) has the mean energy loss of 2.1 MeV/cm and 5 GeV $\mu$ in 1 (0.03) cm LAr corresponds to a most probable (MPV) energy loss of 1.8 (1.5) MeV/cm.

Click the following links to retrieve the relevent plots:

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.

Click the following links to retrieve the relevent plots:

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]).

- H. Bichsel, "The Density Effect for the Ionization Loss in Various Materials", Rev. Mod. Phys. 60, 663 (1988)
- T. Doke, A. Hitachi, et al., "Estimation of fano factors in liquid argon, krypton, xenon and xenon-doped liquid argon", Nuclear Instruments and Methods, Volume 134, Issue 2, 15 April 1976
- Michal Jaskolski and Mariusz Wojcik, "Electron Recombination in Ionized Liquid Argon: A Computational Approach Based on Realistic Models of Electron Transport and Reactions", J. Phys. Chem. A, 2011, 115 (17), pp 4317–4325.
- Ulrich Sowada, John M. Warman, and Matthijs P. de Haas, "Hot-electron thermalization in solid and liquid argon, krypton, and xenon", Phys. Rev. B 25, 3434(R)
- R. Acciarri et al, "A study of electron recombination using highly ionizing particles in the ArgoNeuT Liquid Argon TPC", Journal of Instrumentation, Volume 8, August 2013