4. Line Emission

When an ion is excited to a state involving in inner-shell vacancy it may decay either by autoionization or by discrete radiative transitions. The first process is called the Auger effect. An electron from an excited ion becomes unbound leaving an ion in the next higher ionization stage. The second process is fluorescence. The probability that an excited ion will decay by fluorescence is the fluorescence yield $\omega$.

The Doppler width of a spectral line of frequency $\nu_{0}^{}$ is given by

$$\Delta \nu {\nu_0\over c} \sqrt{2kT\over m_i}$$ (29)

where T is the ion temperature and m_i is the ion mass. For the 6.4 keV iron $\rm K_\alpha$
line this means that

$$\Delta \over$$ (30)

where T₈ is the ion temperature in units of 10⁸ K. There is also collisional or pressure broadening which results in a Lorentz profile with width $\Gamma$ = 2$\nu_{\rm col}^{}$ where $\nu_{\rm col}^{}$ is the collision frequency. If the line comes from the inner accretion disk or magnetosphere it will be broadened by orbital motion. For the purposes of spectral fitting in this work line profiles are assumed to be Gaussian. This is appropriate if thermal Doppler broadening dominates. However at high densities collisional broadening becomes important and the resulting line profile is characterized by the Voigt function

$$\equiv {a\over\pi} \int_{-\infty}^{\infty} {e^{-y^2}\,dy\over a^2 + (u-y)^2}$$ (31)

where

$$\equiv {\Gamma\over4\pi\Delta\nu_D}$$ (32)

and

$$\equiv {\nu - \nu_0\over\Delta\nu_D}$$ (33)

The parameter a is a measure of the relative importance of collisional and thermal broadening. For a $\ll$ 1, thermal broadening dominates and the line profile is Gaussian. For a $\gg$ 1, collisional broadening dominates and the line profile is Lorentzian.

The highest resolution spectrometer used in this work is the ASCA SIS (see section 3.2) which has a spectral resolution of about 130 eV FWHM at the Fe K$_\alpha$
line energy. Thus it is safe to fit emission lines with Gaussians.