ANÁLISIS DE LOS SUPUESTOS DE INAPLICACÍON DEL

The line absorption, or bound-bound absorption, takes place when an atomic system makes a transition between two bound states. For a hydrogen-like atom with a transition from the state la> to ib>, the absorption cross-section of the photons with frequency co is formulated as

^ab(®) = fab Fab(®) •

The oscillator strength fab and the line profile function Fab((o) can be calculated according to quantum mechanics.

3.3.3.2 Oscillator strength

The oscillator strength is a correction factor to the classical result. It is determined by the wave functions of the eigenstates la> and lb> involved in the transition and their energy difference cOab=Eb-Ea by the formula

fab = I COab _zla+I 5(la-lb±1 ) frab)^ •

Chapter III The Radiative Opacity

The angular momentum quantum numbers la and ly specify the energy state la> and lb> together with the principal quantum numbers na and ny. The matrix element rab is calculated from the wave functions, i.e., rab=<alrlb> with the radial coordinate r.

The wave functions of a hydrogen-like atom are well known and defined by the nuclear charge and quantum numbers. This is why the hydrogenic approximation is applied to the description of complex atoms. Although the matrix element rab in a Coulomb field can be calculated numerically from the specified wave functions, there are analytical formulae given recently by Carson (1988a).

3.3.3.3 Line Broadening

A number of broadening processes contribute to the determination of the line profile for

bound-bound absorption. They include radiation damping, Doppler effect. Stark effect and

electron impact. Of these processes, the broadening by radiation damping can always be neglected since its effect can never be comparable to the other processes. The Doppler broadening plays an important role in the case of low density and high temperature because its broadening effect only depends on the temperature. It can sometimes be ignored since at the high temperatures and low densities there are very few atomic systems existing. In the conditions of the stellar interior and envelope, the pressure broadening, i.e., from Stark effect and electron impact, are most dominant in the determination of the line profile.

In our opacity calculation we adopt the result by Griem(1959, 1960) for the line broadening by the Stark effect and electron impact. In his study, the broadening by ions is treated by using the quasi-static theory and the impact broadening of overlapping lines is applied to describe the effect of electrons on the line profile. The Holtsmark profile is obtained from the combination of these two kinds of broadening.

3.3.3.4 Line profile function

Chapter HI The Radiative Opacity

The following formulae are obtained from the results of Griem (1960). For hydrogen-like

atoms, the line profile for a transition from the state la> to lb> is defined by

2tcc

T(p,y)

Jab(œ) =_F

qO)^ ^ab

The Holtsmark normal field strength is Fo=2.61n^^^ with the electron number density n^. The coefficient Kab, which relates the wavelength shift to the perturbing field strength, is given by

(nanh)4

Kab = (72)1/3 JtC

with the nuclear charge Z, and principal quantum numbers n& and ny for the states la> and lb>.

The function T(p,y) defines the profile broadened by both ions and electrons. The variable P specifies the Stark effect and is defined by

2tic

Fo Kab

1 _ J_

where cOab is the energy difference between the states ia> and lb>. The variable y is related to the electron impact effect and defined by

2tcc

y = Ao) ---.

The electron damping constant Aco is given by >1)^'^ ne

9

with an integral

Aco = r n^+ng I(Ymm)

Chapter III The Radiative Opacity

I(Ymin) - J,“ Ç d Y_{Ymin ^}

^ -loge ^min " 0.577,

for Ymin « 1, and

Ymin = y 4 ^ ] .

In limit cases, the function T(p,y) can be reduced to simple forms. For large P (P>20), it

reduces to

T(p,Y) = 1.5p-5/2 + | p - 2 ,

which corresponds to the broadening of the Stark effect. The electron impact effect dominates the line broadening for large y (y>10). Thus, a dispersion profile

J(co) ^

Jt (Aco)2+(co-0)ab)^

is used instead of the Holtsmark profile. In the case of y<10 and p<20, the values of T(P,y) are given in the table presented by Griem(1960).

3.3.4 A bsorption by negative hydrogen

Continuous absoiption by negative hydrogen ions (H ) is a major opacity source in a cool stellar atmosphere, especially in a metal-rich gas. The negative hydrogen ions interact with radiation through bound-free transitions and free-free transitions, i.e.,

H" + photon <=> H + e and

H + e + photon <=> H + e.

Chapter III The Radiative Opacity

A detailed study of the free-free absorption and bound-free absorption was given by

Doughty et al. (1966) and Doughty and Fraser (1966). They listed the numerical calculation results in the form of tables. Tsuji (1966) also carried out a study of its absorption coefficients and presented analytical formulae for practical use. A recent study by John (1988) reviews continuous absorption of H' and gives the analytical formulae which permit separate bound-free and free-free absorption coefficients to be generated at a wide range of temperatures and wavelengths. His resulting data are claimed to be accurate to better than 1 % or 2 %.

3.3.5 Free electron scattering

Electrons themselves can interact with photons, which causes a number of scattering processes. Because of the small mass of the electrons, their interaction with photons is much stronger than that of the nuclei. The scattering of photons by the electrons bound in atomic systems is called Rayleigh scattering while that by the free electrons is called Thomson scattering and Compton scattering. In fact, Thomson scattering is the low energy limiting case of Compton scattering.

The cross-section for Compton scattering must be computed quantum mechanically and turns out to be dependent on the frequency of the incoming photons. The expression for the cross-section, known as the Klein-Nishina formula, is

. . 8tc 3 f l + a Ges(m)

l+2a a

+— ln(l+2a)-2a

(l+2a)2j