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Astron. Astrophys. 354, 714-724 (2000)

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3. Density matrix approach

For a description of the Hydrogen atomic system interacting with an external and diffusive radiation field and particles (thermal and non-thermal electrons), according to Berestetskij et al. (1989), Landi Degl'Innocenti (1983), Bommier & Degl'Innocenti (1996), the density matrix [FORMULA] was considered where n is the main quantum number, L is the orbital atomic momentum, J is the total atomic momentum, M - the total angular momentum projected on the magnetic field direction, A stands for `atomic'. The diagonal elements of the density matrix describe the populations of the atomic levels and the non-diagonal elements describe the coherence between two different levels.

As described in Sect. 2.3 a steady state equation for the density matrix with radiation and collisional tensors was solved including all levels of the fine structure above. However, for these levels the radiative transfer equations were not taken into account as their opacity is below unity and the optically thin approach gives a reasonable approximation (Bommier et al. 1991). Instead an averaged diffusive radiation field in the transition [FORMULA] was used from the full non-LTE problem solution for 5 level plus continuum hydrogen atom without fine structure (Zharkova & Kobylinsky 1991). This approach has also been used by other authors (Vogt et al 1997) and gave reliable results.

3.1. Steady state equation

A steady state equation for the density matrix can be written as follows:

[EQUATION]

where [FORMULA] and [FORMULA] are the total probabilities of the transitions [FORMULA] from and to upper level respectively. These, in turn, are determined by the sum of the probabilities of spontaneous [FORMULA], induced radiation [FORMULA], absorption [FORMULA] and collisional transitions [FORMULA], taking into account impacts with both thermal and beam electrons.

The first member of the above equation describes the probability of transitions between sublevels, split in magnetic field, the second member corresponds to the probability of the level excitation and the third - to the probability of its de-excitation.

3.2. Radiative tensor

In order to describe anisotropy of the radiation field, the radiation field tensor for [FORMULA]-line frequencies in a solid angle [FORMULA] in the plane perpendicular to the direction of emitted quanta can be introduced as follows:

[EQUATION]

with

[EQUATION]

where i,j = x, y, z and [FORMULA] is the transition matrix from the electron to emitted photon coordinate systems depending on the azimuthal [FORMULA] and longitudial [FORMULA] angles.

For the calculation in the [FORMULA]-line frequencies a normalised radiative tensor [FORMULA] is used:

[EQUATION]

In [FORMULA] and [FORMULA] lines, owing to their big opacity and assumed detailed balance, the radiative tensor has a form:

[EQUATION]

where [FORMULA] is the Kronecker function.

The local radiation density can be described as:

[EQUATION]

with [FORMULA] being a local intensity in a solid angle [FORMULA] found from the non-LTE radiative transfer equation as described in Sect. 2.3.

3.3. Collisional tensor

The collisional tensor describes the radiation anisotropy caused by collisions with both thermal and beam electrons. Thermal electrons have a Maxwellian distribution in energy and their collisional tensor can be written as follows:

[EQUATION]

whereas for beam electrons the following expression can be used:

[EQUATION]

where [FORMULA] is a normalised dimensionless electron beam distribution function, t - an injection time, [FORMULA] is a depth and [FORMULA] is a pitch-angle. For the calculations a normalised collisional tensor [FORMULA] was used with the normalisation being similar to the radiative tensor above. Also the local collisional density ccan be introduced as follows:

[EQUATION]

where [FORMULA] is a sum of Maxwellian and power law energy distributions for thermal and beam electrons, respectively, that was calculated as described in Sect. 2.2.

3.4. Probabilities of radiative transitions

The probability of spontaneous transition can be described as follows (Bommier 1980):

[EQUATION]

where D is a matrix term presented in the form:

[EQUATION]

with [FORMULA]; [FORMULA].

For induced radiation, the probability will be:

[EQUATION]

and for absorbed radiation, the probability is equal to:

[EQUATION]

with [FORMULA], [FORMULA] and
[FORMULA] being the Einstein coefficients of spontaneous, induced emission and absorption respectively.

3.5. Probabilities of collisional transitions

The probability of the first kind collisional transitions (from lower levels to higher levels) can be written as (Berestetskii et al. 1989):

[EQUATION]

where D is the matrix determined from formula (17).

For the second kind collisional transitions (from higher to lower levels) the probability is:

[EQUATION]

where [FORMULA] is the effective cross-section, which can be determined by the Born formulae. The matrix terms for the transitions from upper to lower level [FORMULA] are described by formula (17) above with m and [FORMULA] having the opposite signs.

3.6. Stokes parameters

The Stokes parameters are linked to the density matrix as follows:

[EQUATION]

Polarisation is considered in the following two directions: the axis [FORMULA] along a magnetic field direction and the axis [FORMULA] perpendicular to it. A viewing angle in this model coincides with a pitch-angle; the flare-to-solar centre direction deviates from a magnetic field direction on this angle. The parameter [FORMULA] describes linear polarisation along these axes [FORMULA] and [FORMULA], the value of [FORMULA] corresponds to full polarisation in the direction parallel to magnetic field lines and [FORMULA] - to full polarisation in the perpendicular direction. The parameter [FORMULA] gives linear polarisation in the direction with the angles of [FORMULA] or [FORMULA] to the observer's line of sight. The parameter [FORMULA] describes circular polarisation with the probabilities [FORMULA] and [FORMULA] of photon having a left-handed or right-handed polarisation. In astrophysical applications these parameters are defined as: [FORMULA], [FORMULA] and [FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: February 9, 2000
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