Thermal X-ray radiation (PDF) - SRON

with
τ(λ) = τ 0 ϕ(λ) (3.83)
where ϕ(λ) is the line profile and τ 0 is the opacity at the line centre λ 0 , given by:
τ 0 = αhλfN i
2 √ 2πm e σ v
. (3.84)
Apart from the fine structure constant α and Planck’s constant h, the optical
depth also depends on the properties of the absorber, namely the ionic column
density N i and the velocity dispersion σ v . Furthermore, it depends on the oscillator
strength f which is a dimensionless quantity that is different for each transition and
is of order unity for the strongest transitions.
In the simplest approximation, a Gaussian profile ϕ(λ) = exp −(λ−λ 0 ) 2 /b 2 ) can
be adopted, corresponding to pure Doppler broadening for a thermal plasma. Here
b = √ 2σ with σ the normal Gaussian root-mean-square width. The full width at
half maximum of this profile is given by √ ln 256σ or approximately 2.35σ. One may
even include a turbulent velocity σ t into the velocity width σ v , such that
σ 2 v = σ 2 t + kT/m i (3.85)
with m i the mass of the ion (we have tacitly changed here from wavelength to
velocity units through the scaling ∆λ/λ 0 = ∆v/c).
The equivalent width W of the line is calculated from
W = λσ c
∫ ∞
[1 − exp (−τ 0 e −y2 /2 )]dy. (3.86)
−∞
For the simple case that τ 0 ≪ 1, the integral can be evaluated as
τ 0 ≪ 1 :
W = αhλ2 fN i
2m e c
= 1 2 α hν
m e c 2 fλ2 N i . (3.87)
A Gaussian line profile is only a good approximation when the Doppler width
of the line is larger than the natural width of the line. The natural line profile for
an absorption line is a Lorentz profile ϕ(λ) = 1/(1 + x 2 ) with x = 4π∆ν/A. Here
∆ν is the frequency difference ν − ν 0 and A is the total transition probability from
the upper level downwards to any level, including all radiative and Auger decay
channels.
Convolving the intrinsic Lorentz profile with the Gaussian Doppler profile due
to the thermal or turbulent motion of the plasma, gives the well-known Voigt line
profile
ϕ = H(a, y) (3.88)
where
a = Aλ/4πb (3.89)
and y = c∆λ/bλ. The dimensionless parameter a (not to be confused with the a in
Eqn. 3.27) represents the relative importance of the Lorentzian term (∼ A) to the
Gaussian term (∼ b). It should be noted here that formally for a > 0 the parameter
τ 0 defined by (3.84) is not exactly the optical depth at line centre, but as long as
a ≪ 1 it is a fair approximation. For a ≫ 1 it can be shown that H(a, 0) → 1/a √ π.
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