---
author:
- 'M. Derouich'
date: 'Received 01 June 2012 / Accepted 23 July 2012'
title: 'Tensorial depolarization of alkali atoms by isotropic collisions with neutral hydrogen. ' 
---

\

Context, notations and numerical results
========================================

Context
-------

Atomic polarization consists in an unbalance of the populations of the Zeeman sublevels pertaining to a given atomic level and in the presence of coherences between the sublevels themselves.  The scattering polarization is the observational signature of the atomic polarization (e.g. Landi Degl’Innocenti & Landolfi 2004; Trujillo Bueno 2001).  In real plasmas like the solar atmosphere, emitting atoms suffer the effects of wide variety of collisions with nearby abundant particles.  Sometimes the information encoded in accurate observations would be inaccessible if the effect of the collisions is misunderstood or ignored.  Scattering polarization is sensitive to radiation, to magnetic and electric fields, to collisions, etc.  As the scattering polarization is usually rather small in the problems of practical interest (e.g. scattering polarization of the order of 1% in Fraunhofer lines in the photosphere and chromosphere of the Sun), careful and precise modelling of that polarization is of fundamental importance for learning especially about a weak and even unresolved magnetic field by its Hanle effect.  For this purpose, it is of interest to obtain detailed statistical equilibrium equations (SEE) including all the tensorial components of the collisional rates. 

Notations
---------

Let us consider the atomic states ($\alpha$ $j$) of alkali atoms where $j$ is the total angular momentum and $\alpha$ represents the other quantum numbers necessary to define the electronic level.  The atomic states ($\alpha$ $j$) are quantified by the spherical tensor components $\rho_{q}^{k} (\alpha j)$ of the density matrix, $k$ is the tensorial order and $q$ represents the coherences between the levels (e.g. Fano 1963, Omont 1977, Sahal-Bréchot 1977; Landi Degl’Innocenti & Landolfi 2004).  It is useful to note that for isotropic processes the coupling terms implying $k \ne k'$ and transfer of coherence $q$ to $q'$ are zero, and that the collisional depolarization and polarization transfer rates are $q$-independent.  At the solar photosphere where the second solar spectrum is formed and at the low chromosphere, the dominant collisions with neutral hydrogen are isotropic, so that the non-diagonal components of the collisional depolarization matrix are zero. 

In the case of collisions of neutral atoms with neutral hydrogen, the inelastic and super-elastic excitation between two different electronic levels are negligible.  Therefore, the indice $\alpha$ is omitted from now on for the sake of brevity.  We apply the general theory developed by Derouich et al. ( 2003, 2005) and the associated numerical code to provide, the all non zero tensorial collisional rates of any $s$ and $p$ level.  We denote by $C^{k}_E$ the collisional rates due to elastic collisions and by $D^{k}(j)=C^{0}_E(j)  - C^{k}_E(j)$ the usual depolarizing rate; the expression of $D^{k}$ is given for example by equations (7) and (9) of Derouich et al. ( 2003).  $C^k_I(j,j_l)$ and $C^k_S(j,j_u)$ are the collisional transfer rates due to inelastic and super-elastic collisions respectively (see Derouich et al.  2007; Landi Degl’Innocenti & Landolfi 2004).  The indices $l$ and $u$ denote any level of energy respectively lower or higher than the energy of the level ($j$).  The contribution of isotropic collisions in the SEE is given by Equation (5) of Derouich et al. ( 2007) and Equation (7.101) of Landi Degl’Innocenti & Landolfi (2004). 

Numerical results
-----------------

The present work provides new complementary numerical results indispensable for rigorous analysis of the polarization profiles.  Derouich et al. ( 2003) determined solely the depolarization rates $D^{k=2}$ of P-states.  In fact, Table 4 of Derouich et al. ( 2003) gives the alignment ($k=2$) depolarization rates as a function of $n^*$ for the temperatures of 5000 K and 6000 K. The collisional rates $C^{k}_E$, $C^{k}_I$ and $C^{k}_S$ were not given.  This paper attempts to fill this lacuna by providing $C^{k}_E$, $C^{k}_I$ and $C^{k}_S$ as a power function of $n^*$ and $T$. In addition, compact analytical power laws of the $C^{k}_E$ rates associated to S-states are derived from the collisional method presented in Derouich et al. ( 2005). 

Grids of collisional rates are computed for the effective principal quantum number $n^*$ of the s and p-states in the interval \[1.5, 3\].  We adopt a step size of 0.1 and we obtain laws behaviours with $n^*$ which are, with correlation coefficients $R >0.9$, fit by:

—

:   **Spherical tensor components of the level $3s$ $^2S_{j=\frac{1}{2}}$** $$\begin{aligned}
      \label{eq_11}
    C^{0}_E(\frac{1}{2})  &=&  1.838 \times 10^{-9} \;  n_{\textrm {\scriptsize H}} \times   n^{*^{3.675}} \left(\frac{T}{5000}\right)^{0.416} \nonumber \\
    C^{1}_E(\frac{1}{2})  &=&   0.958 \times 10^{-9} \;  n_{\textrm {\scriptsize H}} \times   n^{*^{4.011}}   \left(\frac{T}{5000}\right)^{0.416}   
    $$

—

:   **Spherical tensor components of the level $3p$ $^2P_{j=\frac{1}{2}}$** $$\begin{aligned}
      \label{eq_11}
    C^{0}_E(\frac{1}{2})  &=&  1.301 \times 10^{-9} \;  n_{\textrm {\scriptsize H}} \times   n^{*^{2.293}} \left(\frac{T}{5000}\right)^{0.406} \nonumber \\
    C^{1}_E(\frac{1}{2})  &=&   0.753 \times 10^{-9} \;  n_{\textrm {\scriptsize H}} \times   n^{*^{2.279}}   \left(\frac{T}{5000}\right)^{0.420}  
    $$

—

:   **Spherical tensor components of the level $3p$ $^2P_{j=\frac{3}{2}}$** $$\begin{aligned}
      \label{eq_11}
    C^{0}_E(\frac{3}{2})  &=&  1.685 \times 10^{-9} \;  n_{\textrm {\scriptsize H}} \times   n^{*^{2.671}} \left(\frac{T}{5000}\right)^{0.420} \nonumber \\
    C^{1}_E(\frac{3}{2})  &=&  0.953 \times 10^{-9} \;  n_{\textrm {\scriptsize H}} \times   n^{*^{2.773}} \left(\frac{T}{5000}\right)^{0.390} \nonumber \\
    C^{2}_E(\frac{3}{2})  &=&  0.472 \times 10^{-9} \;  n_{\textrm {\scriptsize H}} \times   n^{*^{2.400}} \left(\frac{T}{5000}\right)^{0.380} \nonumber \\
    C^{3}_E(\frac{3}{2})  &=&  0.482 \times 10^{-9} \;  n_{\textrm {\scriptsize H}} \times   n^{*^{2.670}} \left(\frac{T}{5000}\right)^{0.400}   \\
    C^{0}_I(\frac{3}{2}, \frac{1}{2})  &=&  0.975 \times 10^{-9} \;  n_{\textrm {\scriptsize H}} \times   n^{*^{2.705}} \left(\frac{T}{5000}\right)^{0.400} \nonumber \\
    C^{1}_I(\frac{3}{2}, \frac{1}{2})  &=& - 0.095 \times 10^{-9} \;  n_{\textrm {\scriptsize H}} \times   n^{*^{2.687}} \left(\frac{T}{5000}\right)^{0.410} \nonumber\end{aligned}$$ Only the excitation collisional transfer rates $C^{0}_I(\frac{3}{2}, \frac{1}{2})$ and $C^{1}_I(\frac{3}{2}, \frac{1}{2}) $ are given.  However, it is straightforward to retrieve the values of the deexcitation collisional rates $C^{0}_S(\frac{1}{2}, \frac{3}{2}) $ and $C^{1}_S(\frac{1}{2}, \frac{3}{2}) $ by applying the detailed balance relation: $$\begin{aligned}
      \label{eq_11}
    C^k_S(j,j_u) &=&     \frac{2j+1}{2j_u+1}  \exp \left(\frac{E_{j_u}-E_{j}}{k_BT}\right) \; \; C^k_I(j_u,j)\end{aligned}$$ with $E_{j}$ being the energy of the level ($j $) and $k_B$ the Boltzmann constant. 

  --------------- ------- ------- ------- ------- ------- -- -- -- -- -- -- --
                   Li I    Na I     K I    Rb I    Cs I                     
  n$^*$ ($^2$S)    1.588   1.627   1.770   1.804   1.869                    
  n$^*$ ($^2$P)    1.960   2.117   2.234   2.285   2.329                    
  --------------- ------- ------- ------- ------- ------- -- -- -- -- -- -- --

  : Effective quantum number n$^*$ of the alkali atoms.[]{data-label="V-nondiagonal"}

We notice that all the collisional rates are given in s$^{-1}$, $n_{\textrm {\scriptsize H}}$ is the neutral hydrogen density in cm$^{-3}$ and the temperature $T$ is in Kelvins.  Table 1 gives the effective principal quantum number $n^*$ of the s and p-states of the alkali atoms Li I, Na I, K I, Rb I and Cs I. In particular, since $n^*$=1.627 for ground state of Na I, the destruction of orientation rate ($k=1$) is $$\begin{aligned}
  \label{eq_11}
D^1(\frac{1}{2}) &=&  C^{0}_E(\frac{1}{2}) -C^{1}_E(\frac{1}{2})  \nonumber \\
&=& \left[1.838 \times   n^{*^{3.675}} - 0.958 \times   n^{*^{4.011}}\right] \nonumber \\ && \hspace{1cm} \times 10^{-9} \;  n_{\textrm {\scriptsize H}}   \left(\frac{T}{5000}\right)^{0.416}   \\
  &=&   4.246  \times 10^{-9} \;  n_{\textrm {\scriptsize H}} \times    \left(\frac{T}{5000}\right)^{0.416}    \nonumber \end{aligned}$$ which is very close to the fully quantum rate given in the erratum of Kerkeni et al. ( 2000) (At $T$=5000K, they found a rate of 4.32 $\times$ 10$^{-9}$ $n_{\textrm {\scriptsize H}}$ s$^{-1}$). 

Hyperfine structure
===================

In typical solar conditions, the hyperfine splitting is usually much lower than the inverse of the typical time duration of a collision and therefore one can assume that the nuclear spin is conserved during the collision[^1].  This is the frozen nuclear spin approximation implying that the hyperfine collisional rates are given as a linear combination of the fine rates $C^{k}_E$, $C^{k}_I$ and $C^{k}_S$ given in this paper (e.g. Nienhuis 1976 and Omont 1977). 

Conclusion
==========

Tensorial colisional components are given in this work and could be implemented in the numerical simulations of the scattering polarization in a way similar to radiative rates.  Hyperfine and fine collisional rates may be derived from simple power laws provided in this work. 

Our results are valid for collisions without external magnetic field or for a sufÞciently weak field.  In fact, under solar conditions, to the best of the author’s knowledge, there have been no study which includes magnetic fields explicitly in the calculations of collisional rates.  Depolarizing collisional rates commonly encountred in the literature should be used in the Hanle effect regime but are not valid in the Paschen-Back effect regime.  It remains a challenge to take into account the effect of the magnetic fields in the calculation of the collisional cofficients . 

Derouich M., Sahal-Bréchot S., Barklem P.S., & O’Mara, B. J. 2003, A&A, 404, 763

Derouich M., Barklem P.S., & Sahal-Bréchot S., 2005, A&A, 441, 395

Derouich, M., Trujillo Bueno, J., Manso Sainz, R., 2007, A&A, 472, 269

Fano,U. 1963, Phys.  Rev.,131, 259

Kerkeni B., Spielfiedel A., Feautrier N., 2000, A&A, 364, 937

Landi Degl’Innocenti E., & Landolfi, M. 2004, Polarization in Spectral Lines (Dordrecht: Kluwer)

Nienhuis G., 1976, J. Phys.  B: Atom.  Molec.  Phys.  9, 167

Omont A., 1977, Prog.  Quantum Electronics, 5, 69

Sahal-Bréchot S., 1977, ApJ., 213, 887

Trujillo Bueno, J. 2001, in Advanced Solar Polarimetry, ed.  M. Sigwarth, ASP Conf.  Series Vol.  236, 161

[^1]: It is important, however, not to confuse this condition with the fact that the SEE must be solved for the hyperfine levels when the inverse of the lifetime of the level is lower than the hyperfine splitting, i.e. the hyperfine levels are separated.