A&A 414, 373-376 (2004)
M. Derouich 1 - S. Sahal-Bréchot 1 - P. S. Barklem 2
1 - Observatoire de Paris-Meudon, LERMA UMR CNRS 8112, 5 place Jules Janssen, 92195 Meudon Cedex, France
2 - Department of Astronomy and Space Physics, Uppsala University, Box 515, S 751 20 Uppsala, Sweden
Received 1 August 2003 / Accepted 30 September 2003
The theory of collisional depolarization of spectral lines by atomic hydrogen (Derouich et al. 2003a; Derouich et al. 2003b) is extended to f-atomic levels (l=3). Depolarization rates, polarization and population transfer rates are calculated and results are given. Each cross section as a function of the effective quantum number for a relative velocity of 10 km s-1 is given together with an exponent , if it exists, with the assumption that the cross section varies with velocity as . The general trends of depolarization rates, polarization transfer rates and population transfer rates are discussed.
Key words: Sun: atmosphere - atomic processes - line: formation - polarization
The observation of the so-called "second solar spectrum'' (a term first suggested by V. V. Ivanov; see Stenflo & Keller 1997; Stenflo et al. 2000; Stenflo 2001; Gandorfer 2000; Gandorfer 2002), which is the spectrum of the linear polarization observed near the limb, is due to the scattering of the underlying anisotropic radiation. The atomic polarization may be modified by several factors, in particular the magnetic field (Hanle effect), and also the isotropic collisions with the neighboring particles of hydrogen. Therefore the depolarization rates, polarization and population transfer rates by collisions with hydrogen are needed in order to quantitatively interpret the observed polarization in terms of magnetic fields in solar quiet regions.
In Derouich et al. (2003a) and Derouich et al. (2003b) (hereafter Papers I and II, respectively), a semi-classical theory for the calculation of depolarization rates, polarization and population transfer rates has been developed and applied to p (l=1) and d (l=2) atomic states. In the present paper we extend this theory to f-atomic levels (l=3). This paper presents the first calculations of the depolarization and the collisional transfer rates for f-atomic states.
Our semi-classical theory is not specific for a given atom and its application is possible even to heavy atoms (Ti, Fe, ...), for which there are no data available for depolarization rates, transfer of polarization and population rates. The extension of this method permits calculation of depolarization and collisional transfer rates of p (l=1), d (l=2) and f (l=3) atomic levels. It should now be possible to rapidly obtain the large amount of data needed for the interpretation of the second solar spectrum. Using our method, the general trends of all rates for p (l=1), d (l=2) and f (l=3) atomic levels with orbital angular momentum quantum number l can be discussed for the first time.
As for the p and d atomic state calculations, in most cases the behaviour of the cross sections with the relative velocity v obeys a power law of the form:
Figure 1 shows the alignment depolarization rates (k=2) as a function of the local temperature T and n* for l=3. The population transfer rates (k=0) and the linear polarization transfer rates (k=2) as a function of T and n* are displayed in Figs. 2 and 3. All these rates increase with temperature. For a temperature K, the destruction rate of alignment , and . The population transfer rate and the linear polarization transfer rate . These numerical values are given for which includes most of the lines of interest.
Table 1: Variation of the cross sections, for the relative velocity of , with the effective principal number. Cross sections are in atomic units.
Table 2: Velocity exponents corresponding to the cross sections of Table 1.
|Figure 1: Depolarization rates per unit H-atom density as a function of temperature T and n*. For l=3, each figure: and ; S=0 and J=3; and . Depolarization rates are given in 10-14 .|
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|Figure 2: Population transfer rate per unit H-atom density (k=0) as a function of temperature T and n*. l=3, , and . Population transfer rate is given in 10-14 .|
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|Figure 3: Linear polarization transfer rate per unit H-atom density (k=2) as a function of temperature T and n*. l=3, , and . Linear polarization transfer rate is given in 10-15 .|
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For a given effective quantum number n*, and for the cases l=1, l=2, and l=3, destruction rates of alignment are such that < < . A similar result has been obtained for the broadening of spectral lines. In fact, Barklem et al. (1998) have previously shown that, also for a given n*, lines with upper p-states (l=1) are more broadened than lines with upper d-states (l=2), and similarly lines with upper d-states are more broadened than lines with upper f-states (l=3). This effect is similar to that first seen observationally in the solar spectrum by Carter (1949). In general, when the orbital angular momentum quantum number l increases, the depolarization rates and transfer of polarization and population rates decrease for a given value of the energy of the state of the valence electron .
For f-states, when J=7/2 and J'=5/2 we have: > > . We recall that is a linear combination of (Eq. (3) in Paper II). The population transfer rate is the greater transfer rate because for k = 0 the coefficients of this linear combination are positive. These coefficients are constant and equal to which leads to a which is proportional to (Eq. (5) in Paper II). However, the sign of the coefficients of the linear combination for transfer rates of rank is sometimes positive and sometimes negative. For example, these coefficients have the sign of for orientation transfer rates (k=1) and the sign of for alignment transfer rates. The other coefficients of the linear combination for k > 2 may be obtained on request from the authors. We conclude that, for , the collisional transfer rates may be positive or negative as a function of transition probabilities between the Zeeman sublevels which depend on n*. The depolarization rates are usually positive.
All rates were found to increase with temperature T. The functional form may be accurately fitted to the depolarization rates and the population transfer rates (Paper I, Paper II). However, sometimes the collisional transfer rates with , cannot be fitted by the power-law and so is not reported (Table 2 in the present paper and Table 2 in Paper II). This is due to the fact that these collisional transfer rates are the sum of incoherent contributions from the states and . We notice that the above remarks are valid also for p and d-atomic states.
Unfortunately, there are neither experimental nor quantum chemistry depolarization and collisional transfer rates for f-states for comparison. We expect that the main differences between the RSU potentials and those from quantum chemistry, which are considered as more realistic, occur for the short-range interactions. We have verified that these close collisions do not influence the computed depolarization and collisional transfer rates for f-states. The decisive contribution to the depolarization and collisional transfer rates occurs at intermediate-range interactions. In Paper I, which is concerned with p-states, comparison with quantum chemistry results in Kerkeni (2002) gives depolarization rates in agreement to better than 20%. Extrapolating our results obtained for p and d states (Paper I, Paper II), we expect a rather good agreement (relative difference less than 20% at solar temperatures) between our rates obtained for f-states and a full quantum mechanical treatment.