Thermal X-ray radiation (PDF) - SRON

High Energy Astrophysics
Jelle Kaastra & Frank Verbunt
January 12, 2010
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Chapter 3
Thermal X-ray radiation
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3.1 Introduction
Thermal X-ray radiation is an important diagnostic tool for studying cosmic sources
where high-energy processes are important. One can think of the hot corona of the
Sun and of stars, solar and stellar flares, supernova remnants, cataclysmic variables,
accretion disks in binary stars and around black holes (galactic and extragalactic),
the diffuse interstellar medium of our Galaxy or external galaxies, the outer parts of
active galactic nuclei (AGN), the hot intracluster medium, the diffuse intercluster
medium. In all these cases there is thermal X-ray emission or absorption.
We will see in this chapter that it is possible to derive many different physical
parameters from an X-ray spectrum: temperature, density, chemical abundances,
plasma age, degree of ionisation, irradiating continuum, geometry etc. In this chapter
we focus on X-ray emission and absorption in optically thin plasma’s. For optically
thick plasmas one needs to take account of the full radiation transport in
order to understand these plasmas. In that case stellar atmosphere models become
relevant. However, we will not treat those cases here and restrict our discussion to
plasmas with τ 1.
The power of high-resolution X-ray spectroscopy is shown in Fig. 3.1. This shows
a simulated spectrum of Capella based on old low-resolution observations with the
EXOSAT satellite. That the predictions work is shown in Fig. 3.2. This shows the
success of the plasma emission codes developed at SRON by Rolf Mewe (1935–2004)
and his colleagues.
The strength of spectral analysis lies often in the details. While there are many
spectral lines, some lines contain more information than others. Fig. 3.3 shows an
example of this. It is a part of the spectrum of a flare star. The three lines shown –
the famous Ovii triplet that we will encounter more often during this course – have
an enormous diagnostic power.
This triplet occurs in many other sources, with often totally different intensity
ratio. See for example Fig. 3.4 and compare the intensities of the triplet with those
in Fig. 3.3. While EQ Peg is in collisional ionisation equilibrium, NGC 1068 is in
photoionisation equilibrium. These terms will be explained later.
Up to now you only saw examples of emission spectra. A nice example of an
absorption spectrum is shown in Fig. 3.5.
In order to see thermal imprints on a spectrum sometimes requires the skills
of a spectroscopist. See for example Fig. 3.6. There has been a fierce debate in
the literature whether the sharp change in the spectrum near 18 Å is due to an
absorption edge (lower flux to the left) or a broad emission line (higher flux to the
right). Here we only note that this course is meant to train you as a spectroscopist
so that you can judge such issues yourself.
For the proper calculation of an X-ray spectrum, one should consider three different
steps:
1. the determination of the ionisation balance
2. the determination of the emitted spectrum
3. possible absorption of the photons on their way towards Earth
However, before discussing these processes we will first give a short summary of
atomic structure and radiative transitions, as these are needed to understand the
basic processes.
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Figure 3.1: The predicted X-ray spectrum of Capella. The picture illustrates the
remarkable increase in sensitivity and resolution afforded by the Reflection Grating
Spectrometer (RGS) of XMM-Newton by comparing the predicted RGS spectrum
of the star Capella with that obtained with the EXOSAT gratings (inset). The
throughput is three orders of magnitude, and the spectral resolution more than
a factor of 20, better than achieved with EXOSAT. ( c○ G. Branduardi-Raymont,
www.mssl.ucl.ac.uk/www astro/rgs/rgs impact.html)
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Figure 3.2: The observed X-ray spectrum of Capella. The spectrum is taken with
the Reflection Grating Spectrometer (RGS) of XMM-Newton. The plot shows the
wealth of emission lines in this source. From Audard et al., A&A, 365, L329 (2001).
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Figure 3.3: The observed X-ray spectrum
with the RGS of XMM-Newton
of the flare star EQ Pegasi. The plot
shows the so-called Ovii triplet, consisting
of the resonance, intercombination
and forbidden line. The relative
line ratio’s of this triplet contain a
wealth of physical information. Image
courtesy of J. Schmitt, Hamburger
Sternwarte, Germany and ESA.
Figure 3.4: X-ray spectrum of the
Seyfert 2 galaxy NGC 1068. From
Kinkhabwala et al., ApJ, 575, 732
(2002).
Figure 3.5: X-ray spectrum of the Seyfert 1 galaxy IRAS 13349+2438, obtained
with the RGS of XMM-Newton. From: Sako et al., A&A 365, L168.
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Figure 3.6: X-ray spectrum of the Seyfert 1 galaxy MCG 6-30-15, obtained with the
RGS of XMM-Newton.
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3.2 Summary of spectroscopic notations
3.2.1 Atoms or ions with one valence electron
The electrons in an atom have orbits with discrete energy levels and quantum numbers.
The valence electron or light-electron in an atom or charged ion is the electron
in the outermost atomic shell which is responsible for the emission of light. The
energy configuration can be described by four quantum numbers:
n - principal quantum number, characteristic for the relevant shell. It takes discrete
values n =1, 2, 3, . . .. An atomic shell consists of all electrons with the same
value of n.
l - quantum number of angular momentum of orbital motion of the electron: l =
0, 1, 2, . . . n − 1.
s - quantum number of spin of electron: s = ±1/2 (for l > 0) or s = 1/2 (for
l = 0).
j - quantum number of total angular momentum of electron: j = l±1/2 (for l > 0)
or j = 1/2 (for l = 0).
All angular momenta are measured in units of = h/2π (h is Planck’s constant).
The magnitude of the angular momentum vectors is given by
| ⃗ l| = √ l(l + 1) ≈ l (l ≫ 1).
Note that ⃗j = ⃗ l + ⃗s.
For hydrogenic orbits (Coulomb field) in the theory of Bohr, the semi-major
axis r of an elliptical orbit in shell n is given by r = n 2 a 0 /Z, where Z is the
nuclear charge and the Bohr radius a 0 equals 5.291772108 × 10 −11 m (∼0.529 Å).
Note that a 0 = α/4πR ∞ with α the fine structure constant and R ∞ the Rydberg
constant R ∞ = m e cα 2 /2h; R ∞ hc = 1 2 α2 m e c 2 is the Rydberg energy (13.6056923 eV,
2.17987209 × 10 −18 J). Thus, heavy nuclei are more compact, and also the orbits
with small n are more compact than orbits with large n.
The principal quantum number n corresponds to the energy I n of the orbit.
In the classical Bohr model for the hydrogen atom the energy I n = E H n −2 with
E H = 13.6 eV the Rydberg energy. Thus, electrons with large value of n have small
energies (and therefore are most easily lost from an atom by collisons).
Further, the characteristic velocity of the electron in the Bohr model is given
by v = αZc/n. Electrons withsmall n or in atoms with large Z have the highest
velocities. These velocities are sometimes useful to see if an interaction is important
or not. For instance, if a free electron or proton collides with an atom, it will have the
longest and most intense interaction with those electrons that have similar orbital
velocity as the velocity of the invader.
The following notation for electron orbits is frequently used:
l = 0 1 2 3 4 5 6 7 8 9
s p d f g h i k l m electrons
and further alphabetic (with no duplication of s and p, or course!). The characters
s, p, d and f originate from sharp, principal, diffuse and fundamental series (the line
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series of alkaline spectra). The notation is complicated, but could have been worse
if for example the Chinese alphabet would have been used.
Example 3.1. Take the hydrogen atom. An electron in the ground state has n = 1
and l = 0; this is called a 1s orbit. For n = 2, l = 0 we have a 2s orbit; for n = 2,
l = 1 we have a 2p orbit. Which are the orbits and quantum numbers for n = 3?
The spin quantum number s of an electron can take values s = ±1/2, and the
combined total angular momentum j has a quantum number with values between
l − 1/2 (for l > 0) and l + 1/2. Subshells are subdivided according to their j-value
and are designated as nl j . Example: n = 2, l = 1, j = 3/2 is designated as 2p 3/2 .
3.2.2 Atoms or ions with more than one electron
Above we summarised the situation where only one electron plays a role in the
formation of spectra. Now we consider a multi-electron system. For each of the
electrons one may, in analogy with the single electron configuration, give the orbital
and spin angular momentum of the electrons, like ⃗ l 1 , ⃗ l 2 , ⃗ l 3 , . . . and ⃗s 1 , ⃗s 2 , ⃗s 3 , . . .. Similar
to the single electron system, we need to compose the total angular momentum ⃗ J
from individual angular momenta. Now one needs to consider the coupling between
the electrons. In general, two different types of coupling are used, the so-called jj
coupling and the Russell-Saunders or LS coupling:
1. jj coupling:
⃗j i = l ⃗ i + ⃗s i
∑
⃗j i = J ⃗
This coupling may occur in complicated energy configurations.
i
2. LS coupling: relevant for simple configurations, for example the light elements,
alkaline and earth-alkaline elements (groups I and II of the periodic system).
As the LS coupling is applicable in many situations, we will give a more detailed
treatment here. The coupling is determined by first combining the individual orbital
angular momenta and spins, and then adding the total angular momentum to the
total spin:
∑
i
⃗ li
∑
⃗s i
i
⃗L + ⃗ S =
= ⃗ L
= ⃗ S
⃗ J
In these equations, ⃗ L is the total orbital angular momentum, ⃗ S the total spin, and
⃗J the total angular momentum.
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The corresponding quantum number L is an integer: L = 0, 1, 2, . . .. Further,
S is integer (for an even number of electrons), or broken (for an odd number of
electrons), because the individual ⃗s i must be either parallel or anti-parallel.
Example 3.2. For helium, with two electrons, we have S = 0 or S = 1.
Finally, J is also integer or broken, for an even or odd number of electrons,
respectively. The allowed values for J are:
J = |L − S|, |L − S| + 1, . . ., |L + S|
Example 3.3. Below are a few illustrative cases:
L S allowed values for J
2 1 1,2,3
3 1/2 5/2, 7/2
1 0 1
Similar to the single electron notation, there is a standard notation for the orbital
angular momentum of the multi-electron configurations:
L = 0 1 2 3 4 5 6 7 8 9
S P D F G H I K L M levels
Note the capitals in the notation!
For L ≥ S there are r = 2S + 1 levels for a given value of L. These levels are
only distinguished by a different magnetic interaction of ⃗ L and ⃗ S, and sometimes
they have the same energy (so-called degeneration). Such a group with r levels are
called a term with multiplicity r.
Although for L < S the multiplicity is given by 2L + 1 ≠ r, even in those cases
we call r the multiplicity.
Example 3.4. In Helium or any other two electron system, for the situation with
one 1s electron and another electron in the 2s orbit, the S = 1 state has L = 0
(because both electrons have l = 0). Therefore, only J = 1 is allowed here, and the
multiplicity of 1 is less than 2S + 1.
The following notation for multiplicity is used:
S = 0 1/2 1 3/2 2 5/2 3 7/2
r = 1 2 3 4 5 6 7 8
singlet doublet triplet quartet quintet sextet septet octet
3.2.3 Building electron configurations
A multi-electron atom or ion can be build by starting from a bare nucleus and adding
subsequently electrons one by one. In doing this, we should take account of the Pauli
principle, which states that no two identical electron states can occur. Therefore in a
single atom there cannot exist two electrons with the same combination of quantum
numbers, for example n, l, j and m j (for m j see §3.2.5). Electrons with the same
value of n and l but with different value of j are called equivalent electrons.
How is then such a complex ion build?
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• The first atomic shell (n = 1) has l = 0, and there are two possible orientations
of the spin ⃗s, hence there are two 1s orbits. If there is one electron in this
shell, the designation is 1s. If the shell is full, with two electrons, we write 1s 2
for a closed shell.
• The second shell (n = 2) has as one particular possibility l = 0, s = ±1/2,
and these orbits are called 2s orbits. But however for n = 2 we also can have
l = 1.
• For l = 1, the orbital angular momentum can have three orientations, given
by m l = −1, 0, 1, and again the electron spins can have two different values,
s = ±1/2. This then leads to 6 different p-orbits, and since n = 2, these are
2p orbits. The full n = 2 shell is now designated as 1s 2 2s 2 2p 6 . Frequently one
omits the filled shells, so that for example 1s 2 2s 2 2p 5 is abbreviated as simply
2p 5 .
• The third closed shell is 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 , etc. There are 10 d-orbits
because there we have l = 2 and hence m l = −2, −1, 0, 1, 2 which can be
combined with two different electron spin values.
• The sequence of filling subshells obeys the following rules: first fill orbits in
order of increasing n + l, and where n + l has the same value, fill the orbits
in order of increasing n. Therefore, the order of filling is 1s, 2s, 2p, 3s, 3p, 4s,
3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, . . .. The 4f orbit is an inner orbit, that is
filled only at relatively high nuclear charge (the rare earth elements).
A closed shell has no net contribution to the angular momentum:
∑
⃗l i = 0
i
∑
⃗s i = 0
i
∑
⃗j i = 0
i
∑
l i = even
i
∑
s i = integer.
Each given electron configuration corresponds to a certain group of terms which
all have approximately similar energies (see also §3.4.3).
3.2.4 Term notation
In general, a term with quantum numbers LSJ is designated as r (L-symbol) J . We
can illustrate this with an example.
Example 3.5. Take the electron configuration 1s 2 2s 2 2p 6 3s 2 3p 2 , with L = 1,
S = 1, J = 2. This is designated as 1s 2 2s 2 2p 6 3s 2 3p 2 3 P 2 , or shortly with omission
of the closed shells, as 3p 2 3 P 2 (pronounce as ”three p two triplet p two”). This is
a triplet term.
In general we omit all shells from the designation that are spectroscopically
irrelevant. Whenever the algebraic sum ∑ l i = even or odd, the term is even or
i
odd. For example, the doublet term 2 D is even, while the triplet term 3 P o is odd.
Therefore sometimes the index o from ”odd” is added just like in the triplet P term
above. In the table below we give as examples the ground state configuration of a
few atoms.
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Z Element configuration term
1 H 1s 2 S 1/2
2 He 1s 2 1 S 0
3 Li 2s 2 S 1/2
4 Be 2s 2 1 S 0
5 B 2p 2 P 1/2
6 C 2p 2 3 P 0
All noble gases have a closed shell with L = 0, S = 0 and as a consequence their
ground state is always 1 S 0 . The ground term of a shell that is filled only half is
always an S term, for example for nitrogen 2p 3 4 S 3/2 .
3.2.5 Statistical weight
The statistical weight of an energy level with quantum number J is the number
of directions that the vector ⃗ J can take with respect to some preferred direction,
for example the magnetic field. This is also the allowed number of sublevels. The
projection of ⃗ J on the preferred direction is called MJ (or m j for a single electron
with angular momentum ⃗j). In the case of a magnetic field M J is the socalled
magnetic quantum number (Zeeman effect). M J can take any value between
−J, −J + 1,...,0,...,J − 1, J. This gives 2J + 1 possible orientations of ⃗ J with
respect to the preferred direction. Therefore the statistical weight g J = 2J + 1.
Example 3.6. The 3 D 2 level has g 2 = 5. Note that for 3 D we have L = 2,
S = 1. The total statistical weight of the 3 D term (levels with J = 3, 2, and 1) is
g = 7 + 5 + 3 = 15.
In a strong magnetic field the vectors ⃗ L and ⃗ S are decoupled ( ⃗ J does not exist
any more) and they are arranged independently with respect to ⃗ B. Instead of the
4 quantum numbers n, L, J and M J (Zeeman effect), one then takes n, L, M L and
M S , with M L and M S the projections of ⃗ L and ⃗ S on ⃗ B (Paschen Back effect). In that
case we find for the 3 D term M L = −2, −1, 0, 1, 2, M S = −1, 0, 1 → g = 5 × 3 = 15.
3.2.6 The periodic system
With the knowledge obtained before we now consider the periodic system. The
table shows how the electronic subshells are being filled. Note also that there are
sometimes small irregularities, such as for Mn.
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Z El 1s 2s 2p 3s 3p 3d 4s Ground term
1 H 1 2 S 1/2
2 He 2 1 S 0
3 Li 2 1 2 S 1/2
4 Be 2 2 1 S 0
5 B 2 2 1 2 P 1/2
6 C 2 2 2 3 P 0
7 N 2 2 3 4 S 3/2
8 O 2 2 4 3 P 2
9 F 2 2 5 2 P 3/2
10 Ne 2 2 6 1 S 0
11 Na 2 2 6 1 2 S 1/2
12 Mg 2 2 6 2 1 S 0
13 Al 2 2 6 2 1 2 P 1/2
14 Si 2 2 6 2 2 3 P 0
15 P 2 2 6 2 3 4 S 3/2
16 S 2 2 6 2 4 3 P 2
17 Cl 2 2 6 2 5 2 P 3/2
18 Ar 2 2 6 2 6 1 S 0
19 K 2 2 6 2 6 1 2 S 1/2
20 Ca 2 2 6 2 6 2 1 S 0
21 Sc 2 2 6 2 6 1 2 2 D 3/2
22 Ti 2 2 6 2 6 2 2 3 F 2
23 V 2 2 6 2 6 3 2 4 F 3/2
24 Cr 2 2 6 2 6 5 1 7 S 3
25 Mn 2 2 6 2 6 5 2 6 S 5/2
26 Fe 2 2 6 2 6 6 2 5 D 4
27 Co 2 2 6 2 6 7 2 4 F 9/2
28 Ni 2 2 6 2 6 8 2 3 F 4
29 Cu 2 2 6 2 6 10 1 2 S 1/2
30 Zn 2 2 6 2 6 10 2 1 S 0
Atoms or ions with multiple electrons in most cases have their shells filled starting
from low to high n and l. For example, neutral oxygen has 8 electrons, and the shells
are filled like 1s 2 2s 2 2p 4 , where the superscripts denote the number of electrons in
each shell. Ions or atoms with completely filled subshells (combining all allowed
j-values) are particularly stable. Examples are the noble gases neutral helium, neon
and argon, and more general all ions with 2, 10 or 18 electrons. In chemistry, it is
common practice to designate ions by the number of electrons that they have lost,
like O 2+ for doubly ionised oxygen. In astronomical spectroscopic practice, more
often one starts to count with the neutral atom, such that O 2+ is designated as Oiii.
As the atomic structure and the possible transitions of an ion depend primarily on
the number of electrons it has, there are all kinds of scaling relationships along
so-called iso-electronic sequences. All ions on an iso-electronic sequence have the
same number of electrons; they differ only by the nuclear charge Z. Along such
a sequence, energies, transitions or rates are often continuous functions of Z. A
well known example is the famous 1s−2p triplet of lines in the helium iso-electronic
sequence (2-electron systems).
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3.2.7 How to determine allowed terms
In the example below we illustrate how it is possible to determine the allowed terms
for a given electron configuration. We will determine the terms that may arise for a
carbon atom described by the ground state configuration.
The electron configuration is: 1s 2 2s 2 2p 2 . We have two closed subshells, 1s and
2s, which both give 1 S 0 . We may therefore concentrate on the p-shell with two
equivalent p-electrons. In this we include only states that are allowed with respect
to the Pauli principle. Of course, we omit also states that are symmetric, i.e. a
situation with electron 1 in state a and electron 2 in state b is the same as electron
1 in state b and electron 2 in state a.
The recipe is as follows:
m l1 s 1 m l2 s 2 M L M S
+1 + 0 + +1 +1 x
+1 + −1 + 0 +1 x
0 + −1 + −1 +1 x
+1 + +1 − +2 0
+1 + 0 − +1 0 x
+1 − 0 + +1 0
+1 + −1 − 0 0 x
0 + 0 − 0 0
−1 + +1 − 0 0
−1 + 0 − −1 0 x
−1 − 0 + −1 0
−1 + −1 − −2 0
+1 − 0 − +1 −1 x
+1 − −1 − 0 −1 x
−1 − 0 − −1 −1 x
1. Choose the state with highest M S , in this case 1. This will also be the value
of S for the term. Hence, the multiplicity defined by r = 2S + 1 = 3.
2. Determine the highest M L for this M S . The value will also be the value of L
for the term.
3. Cross out one state function for each pair M L , M S which correspond to the
possible quantum numbers M L = −L, −L + 1, . . ., 0, . . .,L − 1, L and M S =
−S, −S + 1, . . ., 0, . . .,S − 1, S.
4. Repeat the procedure stepwise for remaining functions.
Application to the above case:
1. Maximum M S = 1 ⇒ S = 1, thus, this is a triplet.
2. The highest M L for M S = 1 is M L = 1 ⇒ L = 1, thus, it is a P state. The
first term is thus 3 P.
3. Cross out a state function for each set of combinations of M L = −1, 0, +1 with
M S = −1, 0, +1, for instance the states labeled x in the table above. See table
below for what is left over.
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m l1 s 1 m l2 s 2 M L M S
+1 + +1 − +2 0 x
+1 − 0 + +1 0 x
0 + 0 − 0 0 x
−1 + +1 − 0 0
−1 − 0 + −1 0 x
−1 + −1 − −2 0 x
We see that there are only states left with M S = 0. A repetition of the steps
1–3 gives S = 0 and L = 2, therefore we get a 1 D state. Next we cross out a state
for each combination of M L = −2, −1, 0, +1, +2 with M S = 0, for instance those
indicated with x above. See below what is left:
m l1 s 1 m l2 s 2 M L M S
−1 + +1 −1 0 0
The remainder is clearly a 1 S state. Thus, we have shown that the allowed terms
for the combination 2p 2 are given by 3 P, 1 D and 1 S.
exercise 3.1. Show that the total statistical weight for these three configurations
is 15. It is equal to the number of combinations that we had in our original table,
as it should be.
3.2.8 Allowed terms
The allowed terms for some configurations are shown in Table 3.1 and Table 3.2.
Table 3.1: Terms for non-equivalent electrons
Configuration
s s
s p
s d
p p
p d
d d
Terms
1 S, 3 S
1 P, 3 P
1 D, 3 D
1 S, 1 P, 1 D, 3 S, 3 P, 3 D
1 P, 1 D, 1 F, 3 P, 3 D, 3 F
1 S, 1 P, 1 D, 1 F, 1 G, 3 S, 3 P, 3 D, 3 F, 3 G
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Table 3.2: Terms for equivalent electrons
Configuration
s 2
p 2
p 3
p 4
p 5
p 6
d 2
d 3
d 4
d 5
Terms
1 S
1 S, 1 D, 3 P
2 P, 2 D, 4 S
1 S, 1 D, 3 P
2 P
1 S
1 S, 1 D, 1 G, 3 P, 3 F
2 P, 2 D(2), 2 F, 2 G, 2 H, 4 P, 4 F
1S(2), 1 D(2), 1 F, 1 G(2), 1 I, 3 P(2), 3 D, 3 F(2), 3 G, 3 H, 5 D
2 S, 2 P, 2 D(3), 2 F(2), 2 G(2), 2 H, 2 I, 4 P, 4 D, 4 F, 4 G, 6 S
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3.3 Energy levels
We have seen the notation for the various electronic shells in an ion being designated
as 1s, 2p etc. Except for this notation, there is yet another notation that is commonly
being used in X-ray spectroscopy. Shells with principal quantum number n =1, 2,
3, 4, 5, 6 and 7 are indicated with K, L, M, N, O, P, Q. A further subdivision is
made starting from low values of l up to higher values of l and from low values of
J up to higher values of J:
1s 2s 2p 1/2 2p 3/2 3s 3p 1/2 3p 3/2 3d 3/2 3d 5/2 4s etc.
K L I L II L III M I M II M III M IV M V N I
The binding energy I of K-shell electrons in neutral atoms increases approximately
as I ∼ Z 2 (see Table 3.3 and Figs. 3.7–3.8). Also for other shells the energy increases
strongly with increasing nuclear charge Z.
Figure 3.7: Energy levels of atomic
subshells for neutral atoms.
Figure 3.8: Energy levels of atomic
subshells for neutral atoms.
The binding energy of more highly ionised ions is in general larger than for neutral
atoms (see Table 3.3). For example, neutral iron has a K-shell energy of 7.1 keV,
Fexxvi has I = 9.3 keV. In general, if there is an ionised layer of gas between us
and any X-ray source, the position of absorption edges tells you immediately which
ions are present.
It is common practice in X-ray spectroscopy to designate ions also according to
the number of electrons, thus Fexxvi is hydrogen-like iron, Fexxv is helium-like,
etc. The atomic physics for ions with the same number of electrons is very similar,
and often the relevant parameters are smooth functions of Z, allowing interpolation
along the so-called iso-electronic sequences.
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Table 3.3: Binding energies (in eV) of 1s electrons in the K-shell of free atoms and
ions in the ground state
Z el I II III IV V VI VII VIII IX X XI XII XIII
1 H 13.6
2 He 24.59 54.4
3 Li 58 76 122
4 Be 115 126 154 218
5 B 192 206 221 259 340
6 C 288 306 325 344 392 490
7 N 403 426 448 471 494 552 667
8 O 538 565 592 618 645 671 739 871
9 F 694 724 754 785 815 846 876 954 1103
10 Ne 870 903 937 971 1005 1039 1074 1108 1196 1362
11 Na 1075 1101 1139 1177 1215 1253 1291 1329 1367 1465 1649
12 Mg 1308 1333 1360 1402 1444 1486 1528 1570 1612 1654 1762 1963
13 Al 1564 1590 1618 1646 1692 1738 1784 1830 1876 1922 1968 2086 2304
14 Si 1844 1872 1901 1930 1959 2009 2059 2109 2160 2210 2260 2310 2438
15 P 2148 2178 2208 2238 2269 2300 2354 2408 2462 2517 2571 2625 2679
16 S 2476 2508 2540 2571 2603 2636 2668 2726 2785 2843 2902 2960 3018
17 Cl 2829 2862 2896 2929 2962 2996 3030 3064 3126 3189 3252 3315 3377
18 Ar 3206 3241 3276 3311 3346 3381 3417 3452 3488 3554 3621 3688 3755
19 K 3610 3644 3681 3718 3754 3791 3828 3866 3902 3940 4010 4081 4152
20 Ca 4041 4075 4110 4149 4187 4225 4264 4303 4342 4380 4420 4494 4569
21 Sc 4494 4530 4567 4604 4644 4684 4724 4764 4805 4846 4886 4928 5006
22 Ti 4970 5008 5047 5086 5125 5167 5209 5251 5292 5335 5378 5420 5464
23 V 5470 5511 5551 5592 5633 5674 5717 5761 5805 5848 5893 5938 5982
24 Cr 5995 6038 6080 6122 6165 6208 6250 6295 6341 6387 6432 6479 6526
25 Mn 6544 6589 6633 6677 6721 6766 6810 6854 6901 6949 6997 7045 7094
26 Fe 7117 7164 7210 7256 7301 7348 7394 7440 7486 7535 7585 7636 7686
27 Co 7715 7763 7811 7859 7906 7954 8002 8050 8098 8146 8198 8250 8303
28 Ni 8338 8387 8436 8486 8535 8585 8635 8685 8735 8785 8835 8889 8943
29 Cu 8986 9035 9086 9137 9188 9240 9294 9344 9396 9448 9500 9552 9609
30 Zn 9663 9708 9760 9813 9866 9919 9973 10028 10082 10136 10190 10244 10298
Z el XIV XV XVI XVII XVIII XIX XX XXI XXII XXIII XXIV XXV XXVI
14 Si 2673
15 P 2817 3070
16 S 3076 3224 3494
17 Cl 3439 3501 3658 3946
18 Ar 3821 3887 3953 4121 4426
19 K 4223 4293 4363 4433 4611 4934
20 Ca 4644 4719 4793 4867 4941 5129 5470
21 Sc 5085 5164 5243 5321 5399 5477 5675 6034
22 Ti 5546 5629 5712 5795 5877 5959 6041 6249 6626
23 V 6028 6114 6201 6288 6375 6461 6547 6633 6851 7246
24 Cr 6572 6621 6711 6802 6893 6984 7074 7164 7254 7482 7895
25 Mn 7143 7191 7242 7336 7431 7526 7621 7715 7809 7903 8141 8572
26 Fe 7737 7788 7838 7891 7989 8088 8187 8286 8384 8482 8580 8828 9278
27 Co 8355 8408 8461 8513 8569 8671 8774 8877 8980 9082 9184 9286 9544
28 Ni 8998 9053 9108 9163 9217 9275 9381 9488 9595 9702 9808 9914 10020
29 Cu 9665 9722 9779 9836 9893 9949 10009 10119 10230 10341 10452 10562 10672
30 Zn 10357 10415 10474 10533 10592 10651 10709 10772 10886 11001 11116 11231 11345
17

3.4 Transitions between different levels
3.4.1 Relative ordering of terms with the same configuration
First we discuss the rules of Hund 1 for terms that correspond with one electron
configuration in LS coupling:
1. Of all terms with the same value of L the term with the highest multiplicity
(i.e., maximum S, spin vectors ↑↑) has the lowest energy. For example triplet
terms have a lower energy than the corresponding singlet terms.
2. Of all terms with the same multiplicity the one with the highest L value (i.e.,
orbital angular momenta ↑↑) has the lowest energy.
3. Of all levels of one term the lowest energy is due to the state with the minimum
value of J if the corresponding outer electron shell is filled with less than half of
the maximum number of electrons, and due to the maximum value of J if the
outer shell is filled with more than half of the maximum number of electrons.
Example 3.7. Calcium in a 3dnp state has terms 3 F, 3 D, 3 P, 1 F, 1 D and 1 P; of
all these terms, the 3 F term has the lowest energy.
Example 3.8. See Fig. 3.9.
Figure 3.9: Illustration of Hund’s rules and Landé’s interval rule.
Furthermore, there is the interval rule of Landé 2 , which is strictly speaking only
valid in LS coupling: the energy interval between two subsequent levels (J and J+1)
in a multiplet is proportional to the highest quantum level (J + 1).
Example 3.9. In a 4 D term (J = 1, 3
2
3:5:7.
, 5
and 7
2 2 2
) the separations have the ratio
1 Named after Friedrich Hund (1896-1997)
2 named after Alfred Landé (1888-1976)
18

3.4.2 Selection rules
The transition or combination between two terms corresponds to a spectral line.
However, not all transitions between levels are possible. We will consider here only
electric dipole radiation, which is the most important process. Magnetic dipole
radiation is 5 orders of magnitude weaker and electric quadrapole radiation is 8
orders of magnitude weaker.
We start first with some general rules that are valid for both LS and JJ coupling.
1. There is no restriction on n, ∆n = integer and can be ∆n = 0, 1, 2, 3, . . ..
2. Rule of Laporte 3 : only transitions between even and odd terms and vice-versa
are allowed: ∆( ∑ l i ) = ±1, ±3, . . .. Thus in principle there can be transitions
with more than one electron involved, although these will be weaker. For
example for the two-electron transition: ∆l 1 = ±1, ∆l 2 = δ (and reversed),
with δ even. In most cases δ = −2, 0, 2 (1-electron jump: ∆l = ±1).
3. ∆J = 0, ±1 and J = 0 cannot be combined with J = 0.
4. ∆M J = 0, ±1 and M J = 0 cannot combine with M J = 0, in case ∆J = 0.
For LS coupling only the additional rules apply:
1. ∆L = 0, ±1
2. ∆S = 0 (intercombinations are forbidden). This is for example strictly valid
for helium (the singlet system does not combine with the triplet system), but
not for Hg (deviation from LS coupling), and also not for higher ions.
Finally for JJ coupling only the following condition applies:
1. for 2 electron jumps: ∆j 1 = 0, ∆j 2 = 0, ±1 (and of course the other way
around).
3.4.3 Examples of combinations between terms.
Because a single term can have more than one level, in general we talk about multiplet
terms or simply multiplets. The combination between two multiplets gives rise
to composite spectral lines; those composites are usually also called multiplets. The
few rules that are applicable to combinations of lines have already been discussed
above.
It is in general also possible to make statements about the relative intensity of
all lines belonging to a multiplet. The famous sum rule of Ornstein, Burger and
Dorgelo 4 states that the sum of the intensities of all lines in a multiplet that belong
to the same lower or upper level is proportional to the statistical weight (g J = 2J+1)
of that level.
When we combine this with the selection rules it follows that
1. The strongest transitions in a multiplet are those for which J and L change
in the same way.
2. Transitions corresponding to large values of J are stronger than those with
smaller values of J.
19

Figure 3.10: Transitions between a 2 D and 2 P term.
Example 3.10. We combine a doublet D term with a doublet P term. In the
spectrum, this gives a triplet (see Fig. 3.10):
a: 2 P 3/2 – 2 D 3/2
b: 2 P 3/2 – 2 D 5/2
c: 2 P 1/2 – 2 D 3/2
Note that the 2 P 1/2 – 2 D 5/2 transition is not allowed (why?). Applying the sum rule
we find:
I b : (I a + I c ) = 6:4
(I a + I b ) : I c = 4:2
and this implies
I a : I b : I c = 1 : 9 : 5 (3.1)
Figure 3.11: Illustration of the relative line intensities for the 2p 2 P − 3d 2 D triplet
of Fexxiv.
Example 3.11. We illustrate this transition with a practical example. Consider
Fexxiv. This is Fe 23+ ; because an iron nucleus has Z = 26 protons, this ion has
26−23 = 3 electrons. Hence, it has a similar electronic structure as lithium (Z = 3).
For most transitions, the core of two 1s electrons can be ignored, and then only the
outer valence electron is relevant. The ground state of this ion is a 2 S 1/2 state (why?),
3 Named after Otto Laporte (1902-1971); paper in JOSA 11, 459 (1925)
4 L.S. Ornstein & H.C. Burger, Z.Phys. 24, 41 (1924)
20

ut of course the ion can also be in a higher state, for instance the valence electron
can be in a 2p orbit or a 3d orbit. Suppose now that it is excited to a 3d orbit.
This orbit has n = 3 and L = 2. The single electron produces S = 1/2. Therefore,
the multiplicity r = 2S + 1 = 2, hence we have a 2 D term. Allowed values of J
are 3/2 and 5/2. If the electron falls back to one of the n = 2 states, than it can
be seen that only a transition to L = 1 is allowed (why?). It is easy to show that
this is the 2p 2 P 1/2 , 2 P 3/2 doublet. Therefore, we will get the radiative transition
triplet 2 P− 2 D as treated above. The three lines can be found easily in a simulated
spectrum (Fig. 3.11). The lines are found at 11.029, 11.171 and 11.188 Å. The
spectral region shows several other lines from other (mainly iron) ions, so knowing
the predicted line ratios is helpfull in identifying these triplets. Verify the relative
intensity of these lines.
A more complex example is given by the following.
Example 3.12. Transition from 3 D to 3 P. In the spectrum, this gives a sextet:
a: 3 D 1 – 3 P 2
b: 3 D 2 – 3 P 2
c: 3 D 3 – 3 P 2
d: 3 D 1 – 3 P 1
e: 3 D 2 – 3 P 1
f: 3 D 1 – 3 P 1
Applying the sum rule we find:
I c : (I b + I e ) : (I a + I d + I f ) = 7:5:3
(I a + I b + I c ) : (I d + I e ) : I f = 5:3:1
and this implies
I a : I b : I c : I d : I e : I f = 1.2 : 17.9 : 100 : 17.9 : 53.6 : 23.8 (3.2)
exercise 3.2. For the above example, make a plot similar to Fig. 3.10. Use Hund’s
law to plot a realistic relative distance between the terms of the upper and lower
multiplet.
The spectroscopic notation is based on the LS-coupling, because that applies
in the most simple cases. LS-coupling dominates whenever the distance between
two multiplets (determined by the electrostatic interaction of the electrons) is much
larger than the multiplet-splitting (= largest distance within one multiplet, caused
by the spin-orbit interaction), which is the case for a low nuclear charge Z (the
light elements). Because the spin-orbit interaction increases faster with Z than
the electrostatic interaction, for the heavier elements in the end JJ-coupling will
dominate, in particular for complicated term configurations, where there are many
terms close together. In an excited state (with large principle quantum number
n), where the electrostatic interaction decreases rapidly, already for lower values of
Z JJ-coupling can occur. For example, for Pb, Sn and Ge for the ground term
LS-coupling applies, while the higher excited states have JJ-coupling.
21

3.5 Abundances of the elements
With high spectral resolution and sensitivity, optical spectra of stars sometimes show
spectral features from almost all elements of the Periodic Table, but in practice only
a few of the most abundant elements show up in X-ray spectra of cosmic plasmas.
In several situations the abundances of the chemical elements in an X-ray source are
similar to (but not necessarily equal to) the abundances for the Sun or a constant
fraction of that. There have been many significant changes over the last several
years in the adopted set of standard cosmic abundances. A still often used set of
abundances is that of Anders & Grevesse (1989), but a more recent one is the set of
proto-Solar abundances of Lodders (2003), that we list in Table 3.4 for a few of the
key elements.
Table 3.4: Proto-Solar abundances for the 15 most common chemical elements.
Abundances A are given with respect to hydrogen. Data from Lodders (2003).
Element abundance Element abundance Element abundance
H ≡ 1 Ne 89.1 × 10 −6 S 18.2 × 10 −6
He 0.0954 Na 2.34 × 10 −6 Ar 4.17 × 10 −6
C 288 × 10 −6 Mg 41.7 × 10 −6 Ca 2.57 × 10 −6
N 79.4 × 10 −6 Al 3.47 × 10 −6 Fe 34.7 × 10 −6
O 575 × 10 −6 Si 40.7 × 10 −6 Ni 1.95 × 10 −6
In general, for absorption studies the strength of the lines is mainly determined
by atomic parameters that do not vary much along an iso-electronic sequence, and
the abundance of the element. Therefore, in the X-ray band the oxygen lines are the
strongest absorption lines, as oxygen is the most abundant metal. The emissivity
of ions in the X-ray band often increases with a strong power of the nuclear charge.
For that reason, in many X-ray plasmas the strongest iron emission lines are often
of similar strength to the strongest oxygen lines, despite the fact that the cosmic
abundance of iron is only 6 % of the cosmic oxygen abundance.
Without being complete or giving too many details, we give here a short list of
situations where the abundances differ from proto-Solar, and where X-ray spectroscopic
observations are key to determine these abundances:
• Stellar coronae: here due to the (inverse) FIP effect (FIP = First Ionisation
potential), abundances can be different from Proto-Solar. Elements with high
FIP, for instance noble gases line Ne, remain longer neutral in the transition
zones between photosphere and corona. Neutral particles are not tied to the
magnetic field in stellar flares, hence segregration between neutral and ionised
species is possible, leading to deviant abundances.
• Some hot stars show strong effects of the CNO cycle, leading e.g. to enhnced
nitrogen abundances in some O-stars like ζ Pup.
• In several young supernova remnants (e.g. Cas A) one sees directly the freshly
produced chemical elements lighten-up in X-rays
22

• In older supernova remnants, emission is dominated by swept-up Interstellar
matter. This offers a unique opportunity to sample the ISM of for instance
metal-poor galaxies like the LMC and SMC, by observing their old supernova
remnants.
• In clusters of galaxies, the hot gas is a fair sample of the chemical composition
of the Universe as a whole, which differs from the Solar composition.
• One of the poorest known environments is the cosmic web between clusters
of galaxies. depending upon the galacitic evolution scenario of the galaxies
in that web, one predicts models with low metallicity (say 0.1 time Solar), or
models with strongly varying abundances, ranging from 0.001 to 1 times Solar.
• Active Galactic Nuclei sample the composition of galactic cores, which due
to higher stellar birth rates in those environments is higher than in the Solar
neighbourhood.
23

3.6 Basic processes
In order to be able to calculate X-ray spectra, it is necessary to determine first the
concentrations of the various ions. There are several ionisation and recombination
processes that are important for determining these concentrations. Some of these
processes also play directly a role in spectral line formation processes. We will
first introduce the basic processes in some detail, then we will discuss how the ion
concentrations can be determined, and then we return to the continuum and line
formation processes. The relevant basic processes that we discuss here are:
1. excitation processes:
(a) collisional excitation
(b) collisional de-excitation
(c) radiative excitation
2. Radiative transitions
3. Auger processes and fluorescence
4. ionisation processes:
(a) collisional (direct) ionisation
(b) Excitation-Autoionisation
(c) Photoionisation
(d) Compton ionisation
5. recombination processes:
(a) Radiative recombination
(b) Dielectronic recombination
6. charge transfer processes:
(a) Charge Transfer ionisation
(b) Charge Transfer recombination
All these processes will be treated below.
24

3.6.1 Collisional excitation
A bound electron in an ion can be brought into a higher, excited energy level through
a collision with a free electron or by absorption of a photon. The latter will be
discussed in more detail in Sect. 3.10. Here we focus upon excitation by electrons.
The cross section Q ij for excitation from level i to level j for this process can be
conveniently parametrised by
Q ij (U) = πa2 0
w i
E H Ω(U)
E ij U , (3.3)
where U = E/E ij with E ij the excitation energy from level i to j, E the energy of
the exciting electron, E H the Rydberg energy (13.6 eV), a 0 the Bohr radius and w i
the statistical weight of the lower level i. The dimensionless quantity Ω(U) is the socalled
collision strength. For a given transition on an iso-electronic sequence, Ω(U)
is not a strong function of the atomic number Z, or may be even almost independent
of Z.
Mewe (1972) introduced a convenient formula that can be used to describe most
collision strengths, written here as follows:
Ω(U) = A + B U + C U + 2D + F lnU, (3.4)
2 U3 where A, B, C, D and F are parameters that differ for each transition. The expression
can be integrated analytically over a Maxwellian electron distribution, and the
result can be expressed in terms of exponential integrals.
The Maxwell velocity distribution is given by
f(v) = ( 2 v 2
π )1/2 e −v2 /2ve, 2 (3.5)
where v 2 e ≡ kT/m e and m e is the electron mass. Transforming this to energy units
using E = 1 2 m ev 2 gives
f(E) =
v 3 e
2E 0.5
√ π(kT)
3/2 e−E/kT , (3.6)
exercise 3.3. Verify that
∞∫
0
f(v)dv = 1. Also verify that
∞∫
0
f(E)dE = 1.
After averaging over a Maxwellian electron distribution the number of excitations
can be determined as:
∞∫
√
8πkT a 2 0
S ij = Q ij (E)f(E)v(E)dE =
E ∫∞
H
y 2 Ω(U)e −Uy dU, (3.7)
m e w i E ij
E 1
ij
where y ≡ E ij /kT.
We now define the average collision strength ¯Ω(y) as:
∫∞
¯Ω(y) ≡ ye y
1
25
Ω(U)e −Uy dU (3.8)

leading to
Here the constant S 0 is given by
S ij = S 0 ¯Ω(y)T −1/2 e −y , (3.9)
S 0 ≡ √ 8π/m e k a 2 0 E H = h 2 (2πm e ) −3/2 k −1/2 ≃ 8.629 × 10 −12 K 0.5 m 3 s −1 (3.10)
(using E H = 1 2 α2 m e c 2 and a 0 = h/2παm e c).
For the special choice (3.4) it follows that
¯Ω(y) = A + (By − Cy 2 + Dy 3 + F)e y E 1 (y) + (C + D)y − Dy 2 . (3.11)
In the above, E 1 (x) is the exponential integral, defined by (see for example
Abramowitz & Stegun, chapter 5):
E 1 (x) ≡
∫ ∞
e −t
x
t
dt. (3.12)
It is possible to show that the asymptotic behaviour of E 1 (x) is given by:
x ≪ 1 :
x ≫ 1 :
E 1 (x) = −γ − ln x −
E 1 (x) = e−x
x
where γ = 0.5772156649 is the constant of Euler.
With (3.13) and (3.14) we find (check!):
∞∑
n=0
∞∑
n=1
(−x) n
nn!
(3.13)
n!
(−x) n (3.14)
kT ≪ E ij , y ≫ 1 : S ij ≃ S 0 (A + B + C + 2D)T −1 2 e
−E ij /kT
(3.15)
kT ≫ E ij , y ≪ 1 : S ij ≃ S 0 FT −1 2 ln(kT/Eij ) (3.16)
For low T there are exponentially few excitations (only the tail of the Maxwellian has
sufficient energy for excitation); for high T the number of excitations also approaches
zero, because of the small collision cross section.
exercise 3.4. Derive (3.15) and (3.16).
Not all transitions have this asymptotic behaviour, however. For instance, socalled
forbidden transitions have F = 0 and hence have much lower excitation rates
at high energy. So-called spin-forbidden transitions even have A = B = F = 0.
exercise 3.5. Derive the high-temperature limit equivalent of (3.16) for forbidden
and spin-forbidden transitions.
In most cases, the excited state is stable and the ion will decay back to the
ground level by means of a radiative transition, either directly or through one or
more steps via intermediate energy levels. Only in cases of high density or high
radiation fields, collisional excitation or further radiative excitation to a higher level
may become important, but for typical cluster and ISM conditions these processes
are not important in most cases. Therefore, the excitation rate immediately gives
the total emission line power.
26

3.6.2 Collisional de-excitation
A collision between an ion and a free electron can either bring an electron into a
higher orbit or into a lower orbit. When a bound electron ends up in a lower energy
orbit, the process is called collisional de-excitation. Of course, de-excitation can
only occur when the ion is already in an excited state. Therefore, de-excitation is
in practice important only for high density plasmas, where due to collisions higher
levels are populated, or for so-called metastable levels (levels that have only a small
probability to decay radiatively to a lower level).
The rate coefficient S ′ ji of de-excitation j → i is related to S ij, the rate coefficient
of the inverse excitation process i → j. The relation can be derived from the
principles of detailed balance, and is given by:
S ′ ji = w i
w j
e E ij/kT Sij . (3.17)
27

3.6.3 Radiative transition probabilities
It is well known from quantum mechanics that the occurence of radiative transitions
from one upper level j to a lower level i cannot be predicted in advance. However,
the transition probability can be calculated, and it is usualy denoted by A ji . It has
units of s −1 . The transition probability is related to the dimensionless absorption
oscillator strength f ij that we encounter later (see Sect. 3.10) as
A ji = 4παw if ij E 2 ij
w j m e c 2 h = 4.33 × 1013 s −1 w i
w j
f ij E 2 ij,keV, (3.18)
where E ij is the energy difference between level i and j (the energy of the emitted
line).
28

• Auger channels
[1s]µ K−MM(p)
−→
K−MM(s)
−→
K−LM(p)
−→
K−LM(s)
−→
K−LL
−→
⎧
⎨
⎩
⎧
⎨
µ −2 + e −
[3p]µ −1 + e −
[3s] µ −1 + e −
⎫
⎬
⎭
[3p] 2 µ + e −
[3s] [3p]µ + e −
⎩
[3s] 2 µ + e −
{ }
[2s] µ −1 + e −
[2p] µ −1 + e −
⎧
⎪⎨
⎪⎩
⎧
⎨
⎩
⎫
⎬
⎭
(3.22)
(3.23)
(3.24)
[2s] [3p]µ + e
⎫⎪ −
⎬
[2p][3p] µ + e −
[2s] [3s] µ + e −
(3.25)
⎪ ⎭
[2p][3s]µ + e −
[2s] 2 µ + e −
[2s] [2p]µ + e −
[2p] 2 µ + e −
⎫
⎬
⎭
(3.26)
where the negative exponent of µ stands for the number of electrons that have
been extracted from the outer subshells. In the radiative channels, forbidden and
two-electron transitions have been excluded as it has been confirmed by calculation
that they display very small transition probabilities (log A r < 11). Therefore, the
two main photo-decay pathways are the result of the 2p → 1s and 3p → 1s single
electron jumps that give rise respectively to the Kα (∼ λ1.94) and Kβ (∼ λ1.75)
arrays.
Let us focus now upon a useful example. If for example neutral Fe (Fei) looses a
K-shell electron by photoionisation, there is a hole in the K-shell. This state should
formally be considered of course as a (core-excited) state of Feii, since now there are
not 26 but 25 electrons left. This hole will be filled. For Fe, in approximately 30 %
of all cases the hole is filled by a radiative transition from the L 2 or L 3 shell (2p),
and in the other 70 % by an Auger transition, usually a K–L 2,3 L 2,3 transition (L 2,3
means here either L 2 or L 3 , i.e. 2p). Therefore, in 30 % of all cases a single vacancy
in the L 2,3 shells is produced, and in 70 % of all cases there are two vacancies in
the L 2,3 shell. These L 2,3 vacancies are in turn filled in most cases by a L 2,3 –MM
transition, etc. In this way one single photoionisation can release several electrons.
For neutral iron one finds that a single K-shell ionisation leads to on average 5.7
free electrons (including the first photo-electron). One L 1 -shell ionisation even leads
to 5.8 electrons, an L 2,3 -shell ionisation to 3.3 electrons, a M 1 -shell ionisation to 2.9
electrons and a M 2,3 -shell ionisation to 2 electrons.
The fluorescence yield ω is defined as the probability of a radiative transition
(fluorescence) to fill a vacancy in an inner shell of an atom or ion. It depends upon
the nuclear charge (see Fig. 3.12).
In general the fluorescence yield is increasing with nuclear charge, and largest for
the innermost shells. For the most abundant elements ω L is small (for Fe < 0.5 %).
For the labelling of fluorescent lines there is a distinct naming convention, invented
by Manne Siegbahn 5 in the 1920’s. See Table 3.5. The notation is based
5 Nobel Prize in Physics in 1924 for his discoveries and research in the field of X-ray spectroscopy
30

Figure 3.12: Fluorescence yield as a
function of atomic number Z.
Figure 3.13: Probability of fluorescence,
Coster-Kronig and Auger transitions
for a vacancy in the L 1 subshell.
Table 3.5: Siegbahn notation for inner shell transitions. The table gives the recommended
notation by the IUPAC.
Siegbahn IUPAC Siegbahn IUPAC Siegbahn IUPAC Siegbahn IUPAC
Kα 1 K–L 3 Lα 1 L 3 –M 5 Lγ 1 L 2 –N 4 Mα 1 M 5 –N 7
Kα 2 K–L 2 Lα 2 L 3 –M 4 Lγ 2 L 1 –N 2 Mα 2 M 5 –N 6
Kβ 1 K–M 3 Lβ 1 L 2 –M 4 Lγ 3 L 1 –N 3 Mβ M 4 –N 6
K I β 2 K–N 3 Lβ 2 L 3 –N 5 Lγ 4 L 1 –O 3 Mγ M 3 –N 5
K II β 2 K–N 2 Lβ 3 L 1 –M 3 Lγ 4 ′ L 1 –O 2 Mζ
Kβ 3 K–M 2 Lβ 4 L 1 –M 2 Lγ 5 L 2 –N 1
M 4,5 –N 2,3
K I β 4 K–N 5 Lβ 5 L 3 –O 4,5 Lγ 6 L 2 –O 4
K II β 4 K–N 4 Lβ 6 L 3 –N 1 Lγ 8 L 2 –O 1
Kβ 4x K–N 4 Lβ 7 L 3 –O 1 Lγ 8 ′ L 2 –N 6(7)
K I β 5 K–M 5 Lβ 7 ′ L 3 –N 6,7 Lη L 2 –M 1
K II β 5 K–M 4 Lβ 9 L 1 –M 5 Ll L 3 –M 1
Lβ 10 L 1 –M 4 Ls L 3 –M 3
Lβ 15 L 3 –N 4 Lt L 3 –M 2
Lβ 17 L 2 –M 3 Lu L 3 –N 6,7
Lv L 2 –N 6(7)
upon the relative intensity of lines from different series. The system is still widely
used but incomplete in particular for the M and N series. For this reason the IUPAC
(International Union of Pure and Applied Chemistry) recommends another, more
logical notation (see Table 3.5).
The intensity ratio between Kα 1 and Kα 2 is 2:1. exercise 3.6. Argue why the
above ratio is 2:1. In general Kβ is weaker than Kα. For neutral iron for example
Kβ has about 12 % of the intensity of Kα.
31

3.6.5 Collisional ionisation
Collisional ionisation occurs when during the interaction of a free electron with an
atom or ion the free electron transfers a part of its energy to one of the bound
electrons, which is then able to escape from the ion. A necessary condition is that
the kinetic energy E of the free electron must be larger than the binding energy I
of the atomic shell from which the bound electron escapes.
Q (m 2 )
0 2×10 −26 4×10 −26 6×10 −26
Fe I
Fe XXV
Fe XXVI
10 100 1000
Energy (keV)
Figure 3.14: Illustration of the Lotz formula for three iron ions. Shown is the K-shell
ionisation cross section. Note that Fei and Fexxv have two electrons in this shell,
while Fexxvi only has one (therefore its cross section is half that of Fexxv).
The Lotz formula
There are several formulae that describe the effective cross section Q as a function
of E. A formula that gives a correct order of magnitude estimate of the cross section
σ of this process and that has the proper asymptotic behaviour (first calculated by
Bethe and Born) is the formula of Lotz (1968):
where
and further
uI 2 Q = C ln u, (3.27)
u ≡ E/I (3.28)
C = an s , (3.29)
where n s is the number of electrons in the shell and the normalisation a = 4.5 ×
10 −24 m 2 keV 2 . See Fig. 3.14 for an illustration of the shape and scaling of this
formula.
exercise 3.7. Show that for E → I, Q → 0 and in particular Q ∼ E−I. Determine
the constant of proportionality.
It is further evident that for E → ∞ the cross section Q ∼ (ln E)/E → 0. This
shows that energetic electrons are not very efficient in ionising an atom or ion. They
pass too fast.
32

The formula of Younger
While the Lotz formula in general gives the right order of magnitude for the cross
section, it is not the most accurate and flexible formula for representation of the
direct ionisation cross section Q. A better formula was obtained by Younger (1981):
uI 2 Q = A(1 − 1 u ) + B(1 − 1 u )2 + C ln u + D ln u
u . (3.30)
The constants A, B, C and D differ per ion and shell. Note that the Lotz formula
is a special case of this equation.
Figure 3.15: Example of a fit to direct ionisation cross sections. From Arnaud &
Rothenflug (1985).
In general it is not always simple to calculate Q theoretically, and sometimes
discrepancies between theory and observations remain. However, experimental data
are also not always available, or sometimes the experiments suffer from ”contamination”,
for instance because of contamination from so-called metastable levels. In
general an uncertainty of 5 −10 % is not uncommon in the cross sections. The same
holds by the way for many of the other processes that we describe elsewhere in this
chapter. See Fig. 3.15 for an example of a fit to direct ionisation cross sections.
Integration over an electron distribution
You may have wondered why we have expressed the left hand side of eqns. (3.27)
and (3.30) in the strange form uI 2 Q. The reason is that we need to average the
effective cross section over a Maxwellian electron distribution:
f(v) = n e ( 2 v 2
π )1/2 e −v2 /2ve, 2 (3.31)
where v 2 e ≡ kT/m e and m e is the electron mass. The total number of direct ionisations
per unit volume per unit time is then given by:
∫∞
C DI = n i
0
v 3 e
vf(v)Q(v)dv. (3.32)
33

In this equation n i is the ion density of the ion that is being considered. It is easy
to show (see exercise below) that this implies
C DI =
2√ ∫∞
2n e n i
√ (uI 2 Q)e −uy du, (3.33)
πme (kT) 3
1
where
y ≡ I/kT. (3.34)
exercise 3.8. Show that by substituting E = 1 2 m ev 2 and u = E/I you arrive at
eqn. (3.33).
We see that the direct ionisation rate follows immediately from the Laplace
transform of uI 2 Q.
It can be shown that by substitution of the Lotz formula (3.27) into (3.33) the
integral can be expressed as:
C DI = 2√ 2n e n i C E
√ 1 (y)
, (3.35)
πme (kT) 3 y
where E 1 (x) is the exponential integral (see 3.12).
We apply the asymptotic approximations (3.13) and (3.14) now to eqn. (3.35)
and we see that
kT ≪ I :
C DI ≃ ( 2√ 2C
√ πme
) n en i
√
kTe
−I/kT
I 2 (3.36)
and
kT ≫ I :
C DI ≃ ( 2√ 2C
√ πme
) n en i ln(kT/I)
I √ kT
(3.37)
For low temperatures, the ionisation rate therefore goes exponentially to zero. This
can be understood simply, because for low temperatures only the electrons from the
exponential tail of the Maxwell distribution have sufficient energy to ionise the atom
or ion. For higher temperatures the ionisation rate also approaches zero, but this
time because the cross section at high energies is small.
For each ion the direct ionisation rate per atomic shell can now be determined.
The total rate follows immediately by adding the contributions from the different
shells. Usually only the outermost two or three shells are important. That is because
of the scaling with I −2 and I −1 in (3.36) and (3.37), respectively.
34

3.6.6 Excitation-Autoionisation
In the previous section we showed how a collision of a free electron with an ion can
lead to ionisation. In principle, if the free electron has insufficient energy (E < I),
there will be no ionisation. However, even in that case it is sometimes possible to
ionise, but in a more complicated way. The process is as follows. The collision
can bring one of the electrons in the outermost shells in a higher quantum level
(excitation). In most cases, the ion will return to its ground level by a radiative
transition. But in some cases the excited state is unstable, and a radiationless Auger
transition can occur. Later on we will treat the Auger process in more detail, but
shortly speaking the vacancy that is left behind by the excited electron is being filled
by another electron from one of the outermost shells, while the excited electron is
able to leave the ion (or a similar, process with the role of both electrons reversed).
Because of energy conservation, this process only occurs if the excited electron
comes from one of the inner shells (the ionisation potential barrier must be taken
anyhow). The process is in particular important for Li-like and Na-like ions, and
for several other individual atoms and ions.
Figure 3.16: The Excitation-Autoionisation process for a Li-like ion.
As an example we treat here Li-like ions (see Fig. 3.16). In that case the most
important contribution comes from a 1s–2p excitation. The effective cross section
Q for excitation from 1s to 2p followed by auto-ionisation can be approximated by
(E H = 13.6 eV):
Q = πa 2 0 2B
Z2
Z 2 eff (1s) E H
E Ω H (3.38)
in which a 0 is the Bohr radius, and Z eff = Z − 0.43 is the effective charge that
a 1s electron experiences because of the presence of the other 1s electron and the
2s electron in the Li-like ion. Just as before, E is the energy of the free electron
that causes the excitation. Further, B is the so-called branching ratio. It gives
the probability that after the excitation there will be an auto-ionisation (the other
possibility was a radiative transition). The following approximation for B can be
used:
1
B ≃
(3.39)
1 + (Z/17) 3
Finally, the scaled hydrogenic collision strength is given by:
Z 2 Ω H ≃ 2.22 lnǫ + 0.67 − 0.18
ǫ
35
+ 1.20
ǫ 2 (3.40)

where ǫ ≡ E/E ex and the excitation energy E ex ≡ I 1s − I 2p . Similar to the case of
direct ionisation, it is straightforward to average the excitation-autoionisation cross
section over a Maxwellian electron distribution.
Figure 3.17: Collisional ionisation
cross section for Ovi.
Figure 3.18: Collisional ionisation
cross section for Fexvi.
In Figs. 3.17 and 3.18 we show the total collisional ionisation cross section (direct
+ excitation-autoionisation) for an ion in the Li electronic sequence and the Na
isoelectronic sequence. Clearly, for Fexvi the excitation autoionisation process is
very important. The increasing importance of excitation autoionisation relative to
direct ionisation is also illustrated in Fig. 3.19.
Figure 3.19: Ratio between the Excitation-Autoionisation cross section to the Direct
Ionisation cross section for Na-like ions. From Arnaud & Rothenflug (1985).
36

3.6.7 Photoionisation
Figure 3.20: Photoionisation cross section in barn (10 −28 m −2 ) for Fei (left) and
Fexvi (right). The p and d states (dashed and dotted) have two lines each because
of splitting into two sublevels with different j.
This process is very similar to direct ionisation. The difference is that in the case
of photoionisation a photon instead of an electron is causing the ionisation. Further,
the effective cross section is in general different from that for direct ionisation. As
an example Fig. 3.20 shows the cross section for neutral iron and Na-like iron. It
is evident that for the more highly ionised iron the so-called ”edges” (ionisation
potentials) are found at higher energies than for neutral iron. Contrary to the case
of direct ionisation, the cross section at threshold for photoionisation is not zero. The
effective cross section just above the edges sometimes changes rapidly (the so-called
Cooper minima and maxima).
Contrary to the direct ionisation, all the inner shells have now the largest cross
section. For the K-shell one can approximate for E > I
σ(E) ≃ σ 0 (I/E) 3 (3.41)
where as before I is the edge energy. For a given ionising spectrum F(E) (photons
per unit volume per unit energy) the total number of photoionsations follows as
C PI = c
For hydrogenlike ions one can write:
∫ ∞
0
n i σ(E)F(E)dE. (3.42)
σ PI = 64πng(E, n)αa2 0
3 √ 3Z 2 ( I E )3 , (3.43)
where n is the principal quantum number, α the fine structure constant and A 0 the
Bohr radius. The gaunt factor g(E, n) is of order unity and varies only slowly. It
has been calculated and tabulated by Karzas & Latter (1961). The above equation
is also applicable to excited states of the atom, and is a good approximation for all
excited atoms or ions where n is larger than the corresponding value for the valence
electron.
37

3.6.8 Compton ionisation
Scattering of a photon on an electron generally leads to energy transfer from one
of the particles to the other. In most cases only scattering on free electrons is
considered. But Compton scattering also can occur on bound electrons. If the energy
transfer from the photon to the electron is large enough, the ionisation potential can
be overcome leading to ionisation. This is the Compton ionisation process.
Figure 3.21: Compton ionisation cross section for neutral atoms of H, He, C, N, O,
Ne and Fe.
In the Thomson limit the differential cross section for Compton scattering is
given by
dσ
dΩ = 3σ T
16π (1 + cos2 θ), (3.44)
with θ the scattering angle and σ T the Thomson cross section (6.65 × 10 −29 m −2 ).
The energy transfer ∆E during the scattering is given by (E is the photon energy):
∆E =
E 2 (1 − cosθ)
m e c 2 + E(1 − cosθ) . (3.45)
Only those scatterings where ∆E > I contribute to the ionisation. This defines a
critical angle θ c , given by:
cosθ c = 1 −
m ec 2 I
E 2 − IE . (3.46)
For E ≫ I we have σ(E) → σ T (all scatterings lead in that case to ionisation) and
further for θ c → π we have σ(E) → 0. Because for most ions I ≪ m e c 2 , this last
condition occurs for E ≃ √ Im e c 2 /2 ≫ I. See Fig. 3.21 for an example of some
cross sections. In general, Compton ionisation is important if the ionising spectrum
contains a significant hard X-ray contribution, for which the Compton cross section
is larger than the steeply falling photoionisation cross section.
One sees immediately that for E ≫ I we have σ(E) → σ T (all scatterings lead
in that case to ionisation) and further that for θ c → π we have σ(E) → 0. Because
for most ions I ≪ m e c 2 , this last condition occurs for E ≃ √ Im e c 2 /2 ≫ I. See
Fig. 3.21 for an example of some cross sections.
exercise 3.9. Estimate for which energy Compton ionisation becomes more important
than photoionisation for neutral iron.
38

3.6.9 Radiative recombination
Radiative recombination is the reverse process of photoionisation. A free electron is
captured by an ion while emitting a photon. The released radiation is the so-called
free-bound continuum emission. It is relatively easy to show that there is a simple
relation between the photoionisation cross section σ bf (E) and the recombination
cross section σ fb , namely the Milne-relation:
σ fb (v) = E2 g n σ bf (E)
m e c 2 m e v 2 (3.47)
where g n is the statistical weight of the quantum level into which the electron is captured
(for an empty shell this is g n = 2n 2 ). By averaging over a Maxwell distribution
one gets the recombination-coefficient to level n:
∫∞
R n = n i n e
Of course there is energy conservation, so E = 1 2 m ev 2 + I.
0
vf(v)σ fb (v)dv. (3.48)
exercise 3.10. Demonstrate that for the photoionisation cross section (3.43) and
for g = 1, constant and g n = 2n 2 :
R n = 128√ 2πn 3 αa 2 0I 3 e I/kT n i n e
3 √ 3m e kTZ 2 kTm e c 3 E 1 (I/kT) (3.49)
With the asymptotic relations (3.13) and (3.14) it is seen that
kT ≪ I : R n ∼ T −1 2 (3.50)
kT ≫ I : R n ∼ ln(I/kT)T −3/2 (3.51)
Therefore for T → 0 the recombination coefficient approaches infinity: a cool plasma
is hard to ionise. For T → ∞ the recombination coefficient goes to zero, because of
the Milne relation (v → ∞) and because of the sharp decrease of the photoionisation
cross section for high energies.
As a rough approximation we can use further that I ∼ (Z/n) 2 . Substituting
this we find that for kT ≪ I (recombining plasma’s) R n ∼ n −1 , while for kT ≫ I
(ionising plasma’s) R n ∼ n −3 . In recombining plasma’s therefore in particular many
higher excited levels will be populated by the recombination, leading to significantly
stronger line emission (see Sect. 3.9). On the other hand, in ionising plasmas (such
as supernova remnants) recombination mainly occurs to the lowest levels. Note
that for recombination to the ground level the approximation (3.49) cannot be used
(the hydrogen limit), but instead one should use the exact photoionisation cross
section of the valence electron. By adding over all values of n and applying an
approximation Seaton (1959) found for the total radiative recombination rate α RR
(in units of m −3 s −1 ):
α RR ≡ ∑ n
R n = 5.197 × 10 −20 Zλ 1/2 {0.4288 + 0.5 lnλ + 0.469λ −1/3 } (3.52)
39

with λ ≡ E H Z 2 /kT and E H the Rydberg energy (13.6 eV). Note that this equation
only holds for hydrogen-like ions. For other ions usually an analytical fit to numerical
calculations is used:
α RR ∼ T −η (3.53)
where the approximation is strictly speaking only valid for T near the equilibrium
concentration. The approximations (3.50) and (3.51) raise suspicion that for T → 0
or T → ∞ (3.53) could be a poor choice.
40

3.6.10 Dielectronic recombination
This process is more or less the inverse of excitation-autoionisation. Now a free
electron interacts with an ion, by which it is caught (quantum level n ′′ l ′′ ) but at the
same time it excites an electron from (nl) → (n ′ l ′ ). The doubly excited state is in
general not stable, and the ion will return to its original state by auto-ionisation.
However there is also a possibility that one of the excited electrons (usually the
electron that originally belonged to the ion) falls back by a radiative transition to
the ground level, creating therefore a stable, albeit excited state (n ′′ l ′′ ) of the ion.
In particular excitations with l ′ = l+1 contribute much to this process. In order to
calculate this process, one should take account of many combinations (n ′ l ′ )(n ′′ l ′′ ).
The final transition probability is often approximated by
α DR = A
T 3/2e−T 0/T (1 + Be −T 1/T ) (3.54)
where A, B, T 0 and T 1 are adjustable parameters. Note that for T → ∞ the
asymptotic behaviour is identical to the case of radiative recombination. For T → 0
however, dielectronic recombination can be neglected; this is because the free electron
has insufficient energy to excite a bound electron. Dielectronic recombination
is a dominant process in the Solar corona, and also in other situations it is often
very important.
41

3.6.11 Charge transfer ionisation and recombination
In most cases ionisation or recombination in collisionally ionised plasmas is caused
by interactions of an ion with a free electron. At low temperatures (typically below
10 5 K) also charge transfer reactions become important. During the interaction
of two ions, an electron may be transferred from one ion to the other; it is usually
captured in an excited state, and the excited ion is stabilised by one or more radiative
transitions. As hydrogen and helium outnumber by at least a factor of 100 any other
element, in practice only interactions between those elements and heavier nuclei are
important. Reactions with Hi and Hei lead to recombination of the heavier ion,
and reactions with Hii and Heii to ionisation.
Figure 3.22: The effects of charge transfer reactions on the ionisation balance.
We show the role of charge transfer processes on the ionisation balance in Fig. 3.22.
The electron captured during charge transfer recombination of an oxygen ion
(for instance Ovii, Oviii) is usually captured in an intermediate quantum state
(principal quantum number n = 4 − 6). This leads to enhanced line emission from
that level as compared to the emission from other principal levels, and this signature
can be used to show the presence of charge transfer reactions. Another signature –
actually a signature for all recombining plasmas – is of course the enhancement of
the forbidden line relative to the resonance line in the Ovii triplet (Sect. 3.9).
An important example is the charge transfer of highly charged ions from the
Solar wind with the neutral or weakly ionised Geocorona. Whenever the Sun is
more active, this process may produce enhanced thermal soft X-ray emission in
addition to the steady foreground emission from our own Galaxy. See Bhardwaj et
al. (2006) for a review of X-rays from the Solar System. Care should be taken not
to confuse this temporary enhanced Geocoronal emission with soft excess emission
in a background astrophysical source.
42

3.7 The ionisation balance
In order to calculate the X-ray emission or absorption from a plasma, apart from the
physical conditions also the ion concentrations must be known. These ion concentrations
can be determined by solving the equations for ionisation balance (or in more
complicated cases by solving the time-dependent equations). A basic ingredient in
these equations are the ionisation and recombination rates, that we discussed in the
previous section. Here we consider three of the most important cases: collisional
ionisation equilibrium, non-equilibrium ionisation and photoionisation equilibrium.
3.7.1 Collisional Ionisation Equilibrium (CIE)
The simplest case is a plasma in collisional ionisation equilibrium (CIE). In this case
one assumes that the plasma is optically thin for its own radiation, and that there
is no external radiation field that affects the ionisation balance.
Photo-ionisation and Compton ionisation therefore can be neglected in the case
of CIE. This means that in general each ionisation leads to one additional free electron,
because the direct ionisation and excitation-autoionisation processes are most
efficient for the outermost atomic shells. The relevant ionisation processes are collisional
ionisation and excitation-autoionisation, and the recombination processes
are radiative recombination and dielectronic recombination. Apart from these processes,
at low temperatures also charge transfer ionisation and recombination are
important.
We define R z as the total recombination rate of an ion with charge z to charge
z −1, and I z as the total ionisation rate for charge z to z +1. Ionisation equilibrium
then implies that the net change of ion concentrations n z should be zero:
z > 0 : n z+1 R z+1 − n z R z + n z−1 I z−1 − n z I z = 0 (3.55)
and in particular for z = 0 one has
n 1 R 1 = n 0 I 0 (3.56)
(a neutral atom cannot recombine further and it cannot be created by ionisation).
Next an arbitrary value for n 0 is chosen, and (3.56) is solved:
n 1 = n 0 (I 0 /R 1 ). (3.57)
This is substituted into (3.55) which now can be solved. Using induction, it follows
that
n z+1 = n z (I z /R z+1 ). (3.58)
Finally everything is normalised by demanding that
Z∑
n z = n element (3.59)
z=0
where n element is the total density of the element, determined by the total plasma
density and the chemical abundances.
Examples of plasmas in CIE are the Solar corona, coronae of stars, the hot
intracluster medium, the diffuse Galactic ridge component. Fig. 3.23 shows the ion
fractions as a function of temperature for two important elements.
43

Figure 3.23: Ion concentration of oxygen ions (left panel) and iron ions (right panel)
as a function of temperature in a plasma in Collisional Ionisation Equilibrium (CIE).
Ions with completely filled shells are indicated with thick lines: the He-like ions Ovii
and Fexxv, the Ne-like Fexvii and the Ar-like Feix; note that these ions are more
prominent than their neighbours.
3.7.2 Non-Equilibrium Ionisation (NEI)
The second case that we discuss is non-equilibrium ionisation (NEI). This situation
occurs when the physical conditions of the source, like the temperature, suddenly
change. A shock, for example, can lead to an almost instantaneous rise in temperature.
However, it takes a finite time for the plasma to respond to the temperature
change, as ionisation balance must be recovered by collisions. Similar to the CIE
case we assume that photoionisation can be neglected. For each element with nuclear
charge Z we write:
1
n e (t)
d
⃗n(Z, t) = A(Z, T(t))⃗n(Z, t) (3.60)
dt
where ⃗n is a vector of length Z + 1 that contains the ion concentrations, and which
is normalised according to Eqn. 3.59. The transition matrix A is a (Z +1) ×(Z +1)
matrix given by
⎛
A =
⎜
⎝
⎞
−I 0 R 1 0 0 . . .
I 0 −(I 1 + R 1 ) R 2 0
0 I 1 . . . . . .
.
. .. . . . .
. . . R Z−1 0
⎟
. . . 0 I Z−2 −(I Z−1 + R Z−1 ) R Z
⎠
. . . 0 I Z−1 −R Z
We can write the equation in this form because both ionisations and recombinations
are caused by collisions of electrons with ions. Therefore we have the uniform scaling
with n e .
In general, the set of equations (3.60) must be solved numerically. Because the
characteristic time constants of the ions can have quite different values, solution
through standard methods (for example Runge-Kutta integration) will take a lot
44
.

of computing time. Therefore one uses a trick. First, the eigenvalues (λ) of A
are determined, and also the corresponding eigenvectors ⃗ V . These are put into a
matrix V. We now transform the concentrations ⃗n to the eigenvector basis; the
concentrations on the eigenvector basis will be denoted by ⃗m. By definition, we
then have
⃗n = V⃗m. (3.61)
We now define a diagonal matrix Λ which has the eigenvalues on its diagonal. We
then have
AV = ΛV. (3.62)
Using this we rewrite (3.60) as
1 d
V⃗m = ΛV⃗m. (3.63)
n e dt
For small steps in T the transition matrix A does not vary much. Therefore, for
any time interval where T changes by only a small amount we can assume A and
thereby Λ and V to be constant. In that case, one can interchange the order of
matrix multiplication and differentiation in the left hand side of (3.63). It is then
simple to solve this equation. One finds as a solution
⃗m(t) = V −1 PV⃗m(t 1 ) (3.64)
where the solution is valid for t ≥ t 1 and as long as T(t) − T(t 1 ) is small. Here P is
a diagonal matrix with elements e λ∆u where
∆u ≡
∫ t
t 1
n e (t)dt. (3.65)
For each interval in T one can now determine the eigenvectors and eigenvalues
once and for all and store these for example in the computer memory. By systematically
subdividing the range of T(t) in small intervals ∆T, one finds the final solution
simply by applying repeatedly simple exponentiation and matrix multiplication.
The time evolution of the plasma can be described in general well by the parameter
∫
U = n e dt. (3.66)
The integral should be done over a co-moving mass element. Typically, for most
ions equilibrium is reached for U ∼ 10 18 m −3 s. We should mention here, however,
that the final outcome also depends on the temperature history T(t) of the mass
element, but in most cases the situation is simplified to T(t) = constant.
3.7.3 Photoionisation Equilibrium (PIE)
The third case that we treat are the photoionised plasmas. Usually one assumes
equilibrium (PIE), but there are of course also extensions to non-equilibrium photoionised
situations. Apart from the same ionisation and recombination processes that
play a role for plasmas in NEI and CIE, also photoionisation and Compton ionisation
are relevant. Because of the multiple ionisations caused by Auger processes, the
45

equation for the ionisation balance is not as simple as (3.55), because now one needs
to couple more ions. Moreover, not all rates scale with the product of electron
and ion density, but the balance equations also contain terms proportional to the
product of ion density times photon density. In addition, apart from the equation
for the ionisation balance, one needs to solve simultaneously an energy balance
equation for the electrons. In this energy equation not only terms corresponding
to ionisation and recombination processes play a role, but also several radiation
processes (Bremsstrahlung, line radiation) or Compton scattering. The equilibrium
temperature must be determined in an iterative way. A classical paper describing
such photoionised plasmas is Kallman & McGray (1982).
46

3.8 Continuum emission processes
After we have determined the ionisation balance, and solved for the ion concentrations,
it is possible to calculate the X-ray spectrum of a plasma. We distinguish
continuum radiation and line radiation. Here we discuss briefly the continuum emission
processes: Bremsstrahlung, free-bound emission and two-photon emission.
3.8.1 Bremsstrahlung
Bremsstrahlung is caused by a collision between a free electron and an ion. The
emissivity ǫ ff (photonsm −3 s −1 J −1 ) can be written as:
ǫ ff = 2√ 2ασ T cn e n i Zeff
2 ( me c 2 ) 1
2
√ g ff e −E/kT , (3.67)
3πE kT
where α is the fine structure constant, σ T the Thomson cross section, n e and n i the
electron and ion density, and E the energy of the emitted photon. The factor g ff is
the so-called Gaunt factor and is a dimensionless quantity of order unity. Further,
Z eff is the effective charge of the ion, defined as
Z eff =
( n
2 )1
r I r 2
E H
(3.68)
where E H is the ionisation energy of hydrogen (13.6 eV), I r the ionisation potential of
the ion after a recombination, and n r the corresponding principal quantum number.
Example 3.14. For the Bremsstrahlung of Feii (Fe + ) one should take the ionisation
potential of neutral iron (7.87 eV); the outermost electron for neutral iron has a 4s
orbit, implying n r = 4 and hence for Feii we find Z eff = 3.04.
Figure 3.24: The Gaunt factor for Bremsstrahlung. Note that ζ is Euler’s constant
(1.781 . . .).
It is also possible to write (3.67) as ǫ ff = P ff n e n i with
P ff = 3.031 × 10−21 Z 2 eff g effe −E/kT
E keV T 1/2
keV
47
, (3.69)

where in this case P ff is in photons ×m 3 s −1 keV −1 and E keV is the energy in keV.
The total amount of radiation produced by this process is given by
W tot = 4.856 × 10−37 Wm 3
√
TkeV
∫∞
0
Z 2 eff g ffe −E/kT dE keV . (3.70)
From (3.67) we see immediately that the Bremsstrahlung spectrum (expressed in
Wm −3 keV −1 ) is flat for E ≪ kT, and for E > kT it drops exponentially. In order
to measure the temperature of a hot plasma, one needs to measure near E ≃ kT. The
Gaunt factor g ff can be calculated analytically; there are both tables and asymptotic
approximations available. In general, g ff depends on both E/kT and kT/Z eff . See
Fig. 3.24.
For a plasma (3.67) needs to be summed over all ions that are present in order
to calculate the total amount of Bremsstrahlung radiation. For cosmic abundances,
hydrogen and helium usually yield the largest contribution. Frequently, one defines
an average Gaunt factor G ff by
G ff = ∑ ( ni
)
Zeff,i 2 g eff,i . (3.71)
n
i e
3.8.2 Free-bound emission
Free-bound emission occurs during radiative recombination (Sect. 3.6.9).
exercise 3.11. Why is there no free-bound emission for dielectronic recombination?
The energy of the emitted photon is at least the ionisation energy of the recombined
ion (for recombination to the ground level) or the ionisation energy that
corresponds to the excited state (for recombination to higher levels). From the
recombination rate (see Sect. 3.6.9) the free-bound emission is determined immediately:
ǫ fb = ∑ i
n e n i R r . (3.72)
Also here it is possible to define an effective Gaunt factor G fb . Free-bound emission
is in practice often very important. For example in CIE for kT = 0.1 keV, free-bound
emission is the dominant continuum mechanism for E > 0.1 keV; for kT = 1 keV
it dominates above 3 keV. For kT ≫ 1 keV Bremsstrahlung is always the most
important mechanism, and for kT ≪ 0.1 keV free-bound emission dominates. See
also Fig. 3.25.
Of course, under conditions of photoionisation equilibrium free-bound emission
is even more important, because there are more recombinations than in the CIE
case (because T is lower, at comparable ionisation).
3.8.3 Two photon emission
This process is in particular important for hydrogen-like or helium-like ions. After
a collision with a free electron, an electron from a bound 1s shell is excited to the
2s shell. The quantum-mechanical selection rules do not allow that the 2s electron
48

decays back to the 1s orbit by a radiative transition. Usually the ion will then be
excited to a higher level by another collision, for example from 2s to 2p, and then it
can decay radiatively back to the ground state (1s). However, if the density is very
low (n e ≪ n e,crit , Eqn. 3.74–3.75), the probability for a second collision is very small
and in that case two-photon emission can occur: the electron decays from the 2s
orbit to the 1s orbit while emitting two photons. Energy conservation implies that
the total energy of both photons should equal the energy difference between the 2s
and 1s level (E 2phot = E 1s −E 2s ). From symmetry considerations it is clear that the
spectrum must be symmetrical around E = 0.5E 2phot , and further that it should be
zero for E = 0 and E = E 2phot . An empirical approximation for the shape of the
spectrum is given by:
√
F(E) ∼ sin(πE/E 2phot ). (3.73)
An approximation for the critical density below which two photon emission is important
can be obtained from a comparison of the radiative and collisional rates from
the upper (2s) level, and is given by Mewe, Gronenschild & van den Oord (1986):
H − like : n e,crit = 7 × 10 9 m −3 Z 9.5 (3.74)
He − like : n e,crit = 2 × 10 11 m −3 (Z − 1) 9.5 . (3.75)
For example for carbon two photon emission is important for densities below 10 17 m −3 ,
which is the case for many astrophysical applications. Also in this case one can determine
an average Gaunt factor G 2phot by averaging over all ions. Two photon
emission is important in practice for 0.5 kT 5 keV, and then in particular for
the contributions of C, N and O between 0.2 and 0.6 keV. See also Fig. 3.25.
3.9 Line radiation
Apart from continuum radiation, line radiation plays an important role for thermal
plasmas. In some cases the flux, integrated over a broad energy band, can be
completely dominated by line radiation (see Fig. 3.25). The production process of
line radiation can be broken down in two basic steps: the excitation and the emission
process.
3.9.1 Line radiation: excitation
Collisional excitation
A bound electron in an ion can be brought into a higher, excited energy level through
a collision with a free electron. Before we already discussed the case where the
excited state is unstable, which sometimes can lead to auto-ionisation. In most
cases, however, the excited state is stable and the ion will decay back to the ground
level by means of a radiative transition, either directly or through one or more steps
through intermediate energy levels.
We have seen before that the excitation rate from level i to level j can be written
as (Eq. 3.9):
S ij = S 0 ¯Ω(Eij /kT)T −1/2 e −E ij/kT , (3.76)
For sufficient low density every excitation is followed immediately by a radiative
de-excitation to the ground level (or an intermediate level). Apart of a small
49

Figure 3.25: Emission spectra of plasmas with solar abundances. The histogram
indicates the total spectrum, including line radiation. The spectrum has been binned
in order to show better the relative importance of line radiation. The thick solid
line is the total continuum emission, the thin solid line the contribution due to
Bremsstrahlung, the dashed line free-bound emission and the dotted line two-photon
emission. Note the scaling with Ee E/kT along the y-axis.
correction factor C 1 for cascades from higher excited levels to level k, and possibly
a correction factor B for decay to intermediate levels or for autoionisation, (3.9)
gives the line power from level k to the ground level. The factor B is the so-called
branching ratio.
Radiative recombination
When a free electron is captured by an ion, continuum radiation is produced (freebound
emission). The captured electron not always reaches the ground level immediately.
We have seen before that in particular for cool plasma’s (kT ≪ I) the
higher excited levels are frequently populated. In order to get to the ground state,
one or more radiative transitions are required. Apart from cascade corrections from
and to higher levels the recombination line radiation is essentially given by (3.49).
A comparison of recombination with excitation tells that in particular for low temperatures
(compared to the line energy) recombination radiation dominates, and for
high temperatures excitation radiation.
50

Inner-shell ionisation
The third line radiation process is inner–shell ionisation. An electron from one of
the inner shells can be removed from an ion not only by photoionisation, but also
through a collision with electrons. This last process is in particular important under
NEI conditions, for example in supernova remnants. There the plasma temperature
can be much higher than the typical ionisation energy of the ions that are present.
By means of fluorescence (discussed before) the hole in the shell can be filled again.
Obviously, the line power is proportional to the ionisation cross section times the
fluorescence yield.
Dielectronic recombination
Dielectronic recombination produces more than one line photon. Consider for example
the dielectronic recombination of a He-like ion into a Li-like ion:
e + 1s 2 → 1s2p3s → 1s 2 3s + hν 1 → 1s 2 2p + hν 2 → 1s 2 2s + hν 3 (3.77)
The first arrow corresponds to the electron capture, the second arrow to the
stabilising radiative transition 2p→1s and the third arrow to the radiative transition
3s→2p of the captured electron. This last transition would have also occurred if the
free electron was caught directly into the 3s shell by normal radiative recombination.
Finally, the electron has to decay further to the ground level and this can go through
the normal transitions in a Li-like ion (fourth arrow). This single recombination thus
produces three line photons.
Because of the presence of the extra electron in the higher orbit, the energy hν 1
of the 2p→1s transition is slightly different from the energy in a normal He-like ion.
The stabilising transition is therefore also called a satellite line. Because there are
many different possibilities for the orbit of the captured electron, one usually finds
a forest of such satellite lines surrounding the normal 2p→1s excitation line in the
He-like ion (or analogously for other iso-electronic sequences). Fig. 3.26 gives an
example of these satellite lines.
3.9.2 Line radiation: emission
It does not matter by whatever process the ion is brought into an excited state j,
whenever it is in such a state it may decay back to the ground state or any other
lower energy level i by emitting a photon. The probability per unit time that this
occurs is given by the spontaneous transition probability A ij (units: s −1 ) which is a
number that is different for each transition. The total line power P ij (photons per
unit time and volume) is then given by
P ij = A ij n j (3.78)
where n j is the number density of ions in the excited state j. For the most simple
case of excitation from the ground state g (rate S gj ) followed by spontaneous
emission, one can simply approximate n g n e S gj = n j A gj . From this equation, the
relative population n j /n g ≪ 1 is determined, and then using (3.78) the line flux
is determined. In realistic situations, however, things are more complicated. First,
51

Figure 3.26: Spectrum of a plasma in collisional ionisation equilibrium with kT =
2 keV, near the Fe-K complex. Lines are labelled using the most common designations
in this field. The Fexxv ”triplet” consists of the resonance line (w),
intercombination line (actually split into x and y) and the forbidden line (z). All
other lines are satellite lines. The labelled satellites are lines from Fexxiv, most of
the lines with energy below the forbidden (z) line are from Fexxiii. The relative
intensity of these satellites is a strong indicator for the physical conditions in the
source.
the excited state may also decay to other intermediate states if present, and also
excitations or cascades from other levels may play a role. Furthermore, for high densities
also collisional excitation or de-excitation to and from other levels becomes
important. In general, one has to solve a set of equations for all energy levels of
an ion where all relevant population and depopulation processes for that level are
taken into account. For the resulting solution vector n j , the emitted line power is
then simply given by Eqn. (3.78).
Note that not all possible transitions between the different energy levels are
allowed. There are strict quantum mechanical selection rules that govern which
lines are allowed; see for instance Herzberg (1944) or Mee (1999). Sometimes there
are higher order processes that still allow a forbidden transition to occur, albeit with
much smaller transition probabilities A ij . But if the excited state j has no other
(fast enough) way to decay, these forbidden lines occur and the lines can be quite
strong, as their line power is essentially governed by the rate at which the ion is
brought into its excited state j.
One of the most well known groups of lines is the He-like 1s–2p triplet. Usually
the strongest line is the resonance line, an allowed transition. The forbidden line
has a similar strength as the resonance line, for the low density conditions in the
ISM and intracluster medium, but it can be relatively enhanced in recombining
plasmas, or relatively reduced in high density plasmas like stellar coronal loops. In
between both lines is the intercombination line. In fact, this intercombination line
is a doublet but for the lighter elements both components cannot be resolved. But
see Fig. 3.26 for the case of iron.
52

3.9.3 Line width
For most X-ray spectral lines, the line profile of a line with energy E can be approximated
with a Gaussian exp (−∆E 2 /2σ 2 ) with σ given by σ/E = σ v /c where the
velocity dispersion is
σ 2 v = σ2 t + kT i/m i . (3.79)
Here T i is the ion temperature (not necessarily the same as the electron temperature),
and σ t is the root mean squared turbulent velocity of the emitting medium. For large
ion temperature, turbulent velocity or high spectral resolution this line width can
be measured, but in most cases the lines are not resolved for CCD type spectra.
3.9.4 Resonance scattering
Resonance scattering is a process where a photon is absorbed by an atom and then
re-emitted as a line photon of the same energy into a different direction. As for
strong resonance lines (allowed transitions) the transition probabilities A ij are large,
the time interval between absorption and emission is extremely short, and that is
the reason why the process effectively can be regarded as a scattering process. We
discuss the absorption properties in Sect. 3.10.3, and have already discussed the
spontaneous emission in Sect. 3.9.2.
Resonance scattering of X-ray photons is potentially important in the dense cores
of some clusters of galaxies for a few transitions of the most abundant elements, as
first shown by Gilfanov et al. (1987). The optical depth for scattering can be written
conveniently as (cf. also Sect. 3.10.3):
τ =
4240 fN 24
(
ni
n Z
) (
nZ
n H
) ( M
T keV
) 1/2
E keV
{
1 + 0.0522Mv2 100
T keV
} 1/2
, (3.80)
where f is the absorption oscillator strength of the line (of order unity for the
strongest spectral lines), E keV the energy in keV, N 24 the hydrogen column density
in units of 10 24 m −2 , n i the number density of the ion, n Z the number density of the
element, M the atomic weight of the ion, T keV the ion temperature in keV (assumed
to be equal to the electron temperature) and v 100 the micro-turbulence velocity in
units of 100 km/s. Resonance scattering in clusters causes the radial intensity profile
on the sky of an emission line to become weaker in the cluster core and stronger in
the outskirts, without destroying photons. By comparing the radial line profiles of
lines with different optical depth, for instance the 1s−2p and 1s−3p lines of Ovii
or Fexxv, one can determine the optical depth and hence constrain the amount of
turbulence in the cluster.
Another important application was proposed by Churazov et al. (2001). They
show that for WHIM filaments the resonance line of the Ovii triplet can be enhanced
significantly compared to the thermal emission of the filament due to resonant scattering
of X-ray background photons on the filament. The ratio of this resonant line
to the other lines of the triplet therefore can be used to estimate the column density
of a filament.
53

3.9.5 Some important line transitions
In Tables 3.6–3.7 we list the 100 strongest emission lines under CIE conditions.
Note that each line has its peak emissivity at a different temperature. In particular
some of the H-like and He-like transitions are strong, and further the so-called Fe-L
complex (lines from n = 2 in Li-like to Ne-like iron ions) is prominent. At longer
wavelengths, the L-complex of Ne Mg, Si and S gives strong soft X-ray lines. At
very short wavelengths, there are not many strong emission lines: between 6–7 keV,
the Fe-K emission lines are the strongest spectral features.
We conclude here with mentioning some important iron lines.
First, in hydrogen-like ions the Lyman series (np→1s) and the Balmer series
(n → 2) are important. For He-like ions in particular the 2 → 1 transitions are
very prominent, and thee lines from the He-like ions are often stronger than for
hydrogen-like ions. Also thee Ne-like ions often have a very pronounced spectrum.
Some important lines from iron are mentioned in Table 3.8.
54

Table 3.6: The strongest emission lines for a plasma with proto-solar abundances (Lodders 2003) in the X-ray
band 43 Å < λ < 100 Å. At longer wavelengths sometimes a few lines from the same multiplet have been added.
All lines include unresolved dielectronic satellites. T max (K) is the temperature at which the emissivity peaks,
Q max = P/(n en H ), with P the power per unit volume at T max, and Q max is in units of 10 −36 Wm 3 .
E λ − log log ion iso-el. lower upper
(eV) (Å) Q max T max seq. level level
126.18 98.260 1.35 5.82 Neviii Li 2p 2 P 3/2 3d 2 D 5/2
126.37 98.115 1.65 5.82 Neviii Li 2p 2 P 1/2 3d 2 D 3/2
127.16 97.502 1.29 5.75 Nevii Be 2s 1 S 0 3p 1 P 1
128.57 96.437 1.61 5.47 Siv Ne 2p 1 S 0 3d 1 P 1
129.85 95.483 1.14 5.73 Mgvi N 2p 4 S 3/2 3d 4 P 5/2,3/2,1/2
132.01 93.923 1.46 6.80 Fexviii F 2s 2 P 3/2 2p 2 S 1/2
140.68 88.130 1.21 5.75 Nevii Be 2p 3 P 1 4d 3 D 2,3
140.74 88.092 1.40 5.82 Neviii Li 2s 2 S 1/2 3p 2 P 3/2
147.67 83.959 0.98 5.86 Mgvii C 2p 3 P 3d 3 D, 1 D, 3 F
148.01 83.766 1.77 5.86 Mgvii C 2p 3 P 2 3d 3 P 2
148.56 83.457 1.41 5.90 Feix Ar 3p 1 S 0 4d 3 P 1
149.15 83.128 1.69 5.69 Sivi F 2p 2 P 3/2 ( 3 P)3d 2 D 5/2
149.38 83.000 1.48 5.74 Mgvi N 2p 4 S 3/2 4d 4 P 5/2,3/2,1/2
150.41 82.430 1.49 5.91 Feix Ar 3p 1 S 0 4d 1 P 1
154.02 80.501 1.75 5.70 Sivi F 2p 2 P 3/2 ( 1 D) 3d 2 D 5/2
159.23 77.865 1.71 6.02 Fex Cl 3p 2 P 1/2 4d 2 D 5/2
165.24 75.034 1.29 5.94 Mgviii B 2p 2 P 3/2 3d 2 D 5/2
165.63 74.854 1.29 5.94 Mgviii B 2p 2 P 1/2 3d 2 D 3/2
170.63 72.663 1.07 5.76 Svii Ne 2p 1 S 0 3s 3 P 1,2
170.69 72.635 1.61 6.09 Fexi S 3p 3 P 2 4d 3 D 3
171.46 72.311 1.56 6.00 Mgix Be 2p 1 P 1 3d 1 D 2
171.80 72.166 1.68 6.08 Fexi S 3p 1 D 2 4d 1 F 3
172.13 72.030 1.44 6.00 Mgix Be 2p 3 P 2,1 3s 3 S 1
172.14 72.027 1.40 5.76 Svii Ne 2p 1 S 0 3s 1 P 1
177.07 70.020 1.18 5.84 Sivii O 2p 3 P 3d 3 D, 3 P
177.98 69.660 1.36 6.34 Fexv Mg 3p 1 P 1 4s 1 S 0
177.99 69.658 1.57 5.96 Siviii N 2p 4 S 3/2 3s 4 P 5/2,3/2,1/2
179.17 69.200 1.61 5.71 Sivi F 2p 2 P 4d 2 P, 2 D
186.93 66.326 1.72 6.46 Fexvi Na 3d 2 D 4f 2 F
194.58 63.719 1.70 6.45 Fexvi Na 3p 2 P 3/2 4s 2 S 1/2
195.89 63.294 1.64 6.08 Mgx Li 2p 2 P 3/2 3d 2 D 5/2
197.57 62.755 1.46 6.00 Mgix Be 2s 1 S 0 3p 1 P 1
197.75 62.699 1.21 6.22 Fexiii Si 3p 3 P 1 4d 3 D 2
198.84 62.354 1.17 6.22 Fexiii Si 3p 3 P 0 4d 3 D 1
199.65 62.100 1.46 6.22 Fexiii Si 3p 3 P 1 4d 3 P 0
200.49 61.841 1.29 6.07 Siix C 2p 3 P 2 3s 3 P 1
203.09 61.050 1.06 5.96 Siviii N 2p 4 S 3/2 3d 4 P 5/2,3/2,1/2
203.90 60.807 1.69 5.79 Svii Ne 2p 1 S 0 3d 3 D 1
204.56 60.610 1.30 5.79 Svii Ne 2p 1 S 0 3d 1 P 1
223.98 55.356 1.00 6.08 Siix C 2p 3 P 3d 3 D, 1 D, 3 F
234.33 52.911 1.34 6.34 Fexv Mg 3s 1 S 0 4p 1 P 1
237.06 52.300 1.61 6.22 Sixi Be 2p 1 P 1 3s 1 S 0
238.43 52.000 1.44 5.97 Siviii N 2p 4 S 3/2 4d 4 P 5/2,3/2,1/2
244.59 50.690 1.30 6.16 Six B 2p 2 P 3/2 3d 2 D 5/2
245.37 50.530 1.30 6.16 Six B 2p 2 P 1/2 3d 2 D 3/2
251.90 49.220 1.45 6.22 Sixi Be 2p 1 P 1 3d 1 D 2
252.10 49.180 1.64 5.97 Arix Ne 2p 1 S 0 3s 3 P 1,2
261.02 47.500 1.47 6.06 Six O 2p 3 P 3d 3 D, 3 P
280.73 44.165 1.60 6.30 Sixii Li 2p 2 P 3/2 3d 2 D 5/2
283.46 43.740 1.46 6.22 Sixi Be 2s 1 S 0 3p 1 P 1
55

Table 3.7: As Table 3.6, but for λ < 43 Å.
E λ − log log ion iso-el. lower upper
(eV) (Å) Q max T max seq. level level
291.52 42.530 1.32 6.18 Sx N 2p 4 S 3/2 3d 4 P 5/2,3/2,1/2
298.97 41.470 1.31 5.97 Cv He 1s 1 S 0 2s 3 S 1 (f)
303.07 40.910 1.75 6.29 Sixii Li 2s 2 S 1/2 3p 2 P 3/2
307.88 40.270 1.27 5.98 Cv He 1s 1 S 0 2p 1 P 1 (r)
315.48 39.300 1.37 6.28 Sxi C 2p 3 P 3d 3 D, 1 D, 3 F
336.00 36.900 1.58 6.19 Sx N 2p 4 S 3/2 4d 4 P 5/2,3/2,1/2
339.10 36.563 1.56 6.34 Sxii B 2p 2 P 3/2 3d 2 D 5/2
340.63 36.398 1.56 6.34 Sxii B 2p 2 P 1/2 3d 2 D 3/2
367.47 33.740 1.47 6.13 Cvi H 1s 2 S 1/2 2p 2 P 1/2 (Lyα)
367.53 33.734 1.18 6.13 Cvi H 1s 2 S 1/2 2p 2 P 3/2 (Lyα)
430.65 28.790 1.69 6.17 Nvi He 1s 1 S 0 2p 1 P 1 (r)
500.36 24.779 1.68 6.32 Nvii H 1s 2 S 1/2 2p 2 P 3/2 (Lyα)
560.98 22.101 0.86 6.32 Ovii He 1s 1 S 0 2s 3 S 1 (f)
568.55 21.807 1.45 6.32 Ovii He 1s 1 S 0 2p 3 P 2,1 (i)
573.95 21.602 0.71 6.33 Ovii He 1s 1 S 0 2p 1 P 1 (r)
653.49 18.973 1.05 6.49 Oviii H 1s 2 S 1/2 2p 2 P 1/2 (Lyα)
653.68 18.967 0.77 6.48 Oviii H 1s 2 S 1/2 2p 2 P 3/2 (Lyα)
665.62 18.627 1.58 6.34 Ovii He 1s 1 S 0 3p 1 P 1
725.05 17.100 0.87 6.73 Fexvii Ne 2p 1 S 0 3s 3 P 2
726.97 17.055 0.79 6.73 Fexvii Ne 2p 1 S 0 3s 3 P 1
738.88 16.780 0.87 6.73 Fexvii Ne 2p 1 S 0 3s 1 P 1
771.14 16.078 1.37 6.84 Fexviii F 2p 2 P 3/2 3s 4 P 5/2
774.61 16.006 1.55 6.50 Oviii H 1s 2 S 1/2 3p 2 P 1/2,3/2 (Lyβ)
812.21 15.265 1.12 6.74 Fexvii Ne 2p 1 S 0 3d 3 D 1
825.79 15.014 0.58 6.74 Fexvii Ne 2p 1 S 0 3d 1 P 1
862.32 14.378 1.69 6.84 Fexviii F 2p 2 P 3/2 3d 2 D 5/2
872.39 14.212 1.54 6.84 Fexviii F 2p 2 P 3/2 3d 2 S 1/2
872.88 14.204 1.26 6.84 Fexviii F 2p 2 P 3/2 3d 2 D 5/2
896.75 13.826 1.66 6.76 Fexvii Ne 2s 1 S 0 3p 1 P 1
904.99 13.700 1.61 6.59 Neix He 1s 1 S 0 2s 3 S 1 (f)
905.08 13.699 1.61 6.59 Neix He 1s 1 S 0 2s 3 S 1 (f)
916.98 13.521 1.35 6.91 Fexix O 2p 3 P 2 3d 3 D 3
917.93 13.507 1.68 6.91 Fexix O 2p 3 P 2 3d 3 P 2
922.02 13.447 1.44 6.59 Neix He 1s 1 S 0 2p 1 P 1 (r)
965.08 12.847 1.51 6.98 Fexx N 2p 4 S 3/2 3d 4 P 5/2
966.59 12.827 1.44 6.98 Fexx N 2p 4 S 3/2 3d 4 P 3/2
1009.2 12.286 1.12 7.04 Fexxi C 2p 3 P 0 3d 3 D 1
1011.0 12.264 1.46 6.73 Fexvii Ne 2p 1 S 0 4d 3 D 1
1021.5 12.137 1.77 6.77 Nex H 1s 2 S 1/2 2p 2 P 1/2 (Lyα)
1022.0 12.132 1.49 6.76 Nex H 1s 2 S 1/2 2p 2 P 3/2 (Lyα)
1022.6 12.124 1.39 6.73 Fexvii Ne 2p 1 S 0 4d 1 P 1
1053.4 11.770 1.38 7.10 Fexxii B 2p 2 P 1/2 3d 2 D 3/2
1056.0 11.741 1.51 7.18 Fexxiii Be 2p 1 P 1 3d 1 D 2
1102.0 11.251 1.73 6.74 Fexvii Ne 2p 1 S 0 5d 3 D 1
1352.1 9.170 1.66 6.81 Mgxi He 1s 1 S 0 2p 1 P 1 (r)
1472.7 8.419 1.76 7.00 Mgxii H 1s 2 S 1/2 2p 2 P 3/2 (Lyα)
1864.9 6.648 1.59 7.01 Sixiii He 1s 1 S 0 2p 1 P 1 (r)
2005.9 6.181 1.72 7.21 Sixiv H 1s 2 S 1/2 2p 2 P 3/2 (Lyα)
6698.6 1.851 1.43 7.84 Fexxv He 1s 1 S 0 2p 1 P 1 (r)
6973.1 1.778 1.66 8.17 Fexxvi H 1s 2 S 1/2 2p 2 P 3/2 (Lyα)
56

Table 3.8: Some important iron lines
Ion E (keV) transition remark
Fexxvi 6.97 2p–1s Lyα
Fexxvi 8.21 3p–1s Lyβ
Fexxvi 1.29 3p–2s Hα
Fexxv 6.70 2p–1s 1 P 1 , resonance line
Fexxv 6.67 2p–1s 3 P 2,1 , intercombination line
Fexxv 6.63 2s–1s 3 S 1 , forbidden line
Fexxv 7.90 3p–1s 1 P 1
Fexxv 1.22 3p–2s 1 P 1
Feii 6.40 2p–1s Kα (from ionisation of Fei)
Feii 7.06 3p–1s ”Kβ” (from ionisation of Fei)
Feii 0.79 3p–2s ”Lβ” (from ionisation of Fei)
57

3.10 Absorption of X-ray radiation
3.10.1 Continuum versus line absorption
X-rays emitted by cosmic sources do not travel unattenuated to a distant observer.
This is because intervening matter in the line of sight absorbs a part of the X-
rays. With low-resolution instruments, absorption can be studied only through
the measurement of broad-band flux depressions caused by continuum absorption.
However, at high spectral resolution also absorption lines can be studied, and in
fact absorption lines offer more sensitive tools to detect weak intervening absorption
systems. We illustrate this in Fig. 3.27. At an Oviii column density of 10 21 m −2 , the
absorption edge has an optical depth of 1 %; for the same column density, the core
of the Lyα line is already saturated and even for 100 times lower column density,
the line core still has an optical depth of 5 %.
Figure 3.27: Continuum (left) and Lyα (right) absorption spectrum for a layer
consisting of Oviii ions with column densities as indicated.
3.10.2 Continuum absorption
Continuum absorption can be calculated simply from the photoionisation cross sections,
that we discussed already in Sect. 3.6.7. The total continuum opacity τ cont
can be written as
τ cont (E) ≡ N H σ cont (E) = ∑ N i σ i (E) (3.81)
i
i.e, by averaging over the various ions i with column density N i . Accordingly,
the continuum transmission T(E) of such a clump of matter can be written as
T(E) = exp (−τ cont (E)). For a worked out example see also Sect. 3.10.6.
3.10.3 Line absorption
When light from a background source shines through a clump of matter, a part of
the radiation can be absorbed. We discussed already the continuum absorption.
The transmission in a spectral line at wavelength λ is given by
T(λ) = e −τ(λ) (3.82)
58

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.
59

3.10.4 Some important X-ray absorption lines
There are several types of absorption lines. The most well-known are the normal
strong resonance lines, which involve electrons from the outermost atomic shells.
Examples are the 1s–2p line of Ovii at 21.60 Å, the Oviii Lyα doublet at 18.97 Å
and the well known 2s–2p doublet of Ovi at 1032 and 1038 Å. See Richter et al.
(2008) for an extensive discussion on these absorption lines in the WHIM.
The other class are the inner-shell absorption lines. In this case the excited
atom is often in an unstable state, with a large probability for the Auger process
(Sect. 3.6.4). As a result, the parameter a entering the Voigt profile (Eqn. 3.89) is
large and therefore the lines can have strong damping wings.
In some cases the lines are isolated and well resolved, like the Ovii and Oviii
1s–2p lines mentioned above. However, in other cases the lines may be strongly
blended, like for the higher principal quantum number transitions of any ion in case
of high column densities. Another important group of strongly blended lines are the
inner-shell transitions of heavier elements like iron. They form so-called unresolved
transition arrays (UTAs); the individual lines are no longer recognisable. The first
detection of these UTAs in astrophysical sources was in a quasar (Sako et al. 2001).
In Table 3.9 we list the 70 most important X-ray absorption lines for λ < 100 Å.
The importance was determined by calculating the maximum ratio W/λ that these
lines reach for any given temperature.
Note the dominance of lines from all ionisation stages of oxygen in the 17–24 Å
band. Furthermore, the Feix and Fexvii lines are the strongest iron features. These
lines are weaker than the oxygen lines because of the lower abundance of iron; they
are stronger than their neighbouring iron ions because of somewhat higher oscillator
strengths and a relatively higher ion fraction (cf. Fig. 3.23).
Note that the strength of these lines can depend on several other parameters,
like turbulent broadening, deviations from CIE, saturation for higher column densities,
etc., so some care should be taken when the strongest line for other physical
conditions are sought.
3.10.5 Curve of growth
In most practical situations, X-ray absorption lines are unresolved or poorly resolved.
As a consequence, the physical information about the source can only be retrieved
from basic parameters such as the line centroid (for Doppler shifts) and equivalent
width W (as a measure of the line strength).
As outlined in Sect. 3.10.3, the equivalent width W of a spectral line from an
ion i depends, apart from atomic parameters, only on the column density N i and
the velocity broadening σ v . A useful diagram is the so-called curve of growth. Here
we present it in the form of a curve giving W versus N i for a fixed value of σ v .
The curve of growth has three regimes. For low column densities, the optical
depth at line centre τ 0 is small, and W increases linearly with N i , independent of σ v
(Eqn. 3.87). For increasing column density, τ 0 starts to exceed unity and then W
increases only logarithmically with N i . Where this happens exactly, depends on σ v .
In this intermediate logarithmic branch W depends both on σ v and N i , and hence by
comparing the measured W with other lines from the same ion, both parameters can
be estimated. At even higher column densities, the effective opacity increases faster
(proportional to √ N i ), because of the Lorentzian line wings. Where this happens
60

Table 3.9: The most important X-ray absorption lines with λ < 100 Å. Calculations
are done for a plasma in CIE with proto-solar abundances (Lodders 2003) and only
thermal broadening. The calculations are done for a hydrogen column density of
10 24 m −2 , and we list equivalent widths for the temperature T max (in K) where it
reaches a maximum. Equivalent widths are calculated for the listed line only, not
taking into account blending by other lines. For saturated, overlapping lines, like
the Oviii Lyα doublet near 18.97 Å, the combined equivalent width can be smaller
than the sum of the individual equivalent widths.
λ ion W max log λ ion W max log
(Å) (mÅ) T max (Å) (mÅ) T max
9.169 Mgxi 1.7 6.52 31.287 N i 5.5 4.04
13.447 Neix 4.6 6.38 33.426 C v 7.0 5.76
13.814 Nevii 2.2 5.72 33.734 C vi 12.9 6.05
15.014 Fexvii 5.4 6.67 33.740 C vi 10.3 6.03
15.265 Fexvii 2.2 6.63 34.973 C v 10.3 5.82
15.316 Fexv 2.3 6.32 39.240 S xi 6.5 6.25
16.006 O viii 2.6 6.37 40.268 C v 17.0 5.89
16.510 Feix 5.6 5.81 40.940 C iv 7.2 5.02
16.773 Feix 3.0 5.81 41.420 C iv 14.9 5.03
17.396 O vii 2.8 6.06 42.543 S x 6.3 6.14
17.768 O vii 4.4 6.11 50.524 Six 11.0 6.14
18.629 O vii 6.9 6.19 51.807 S vii 7.6 5.68
18.967 O viii 8.7 6.41 55.305 Siix 14.0 6.06
18.973 O viii 6.5 6.39 60.161 S vii 12.2 5.71
19.924 O v 3.2 5.41 61.019 Siviii 11.9 5.92
21.602 O vii 12.1 6.27 61.070 Siviii 13.2 5.93
22.006 O vi 5.1 5.48 62.751 Mgix 11.1 5.99
22.008 O vi 4.1 5.48 68.148 Sivii 10.1 5.78
22.370 O v 5.4 5.41 69.664 Sivii 10.9 5.78
22.571 O iv 3.6 5.22 70.027 Sivii 11.3 5.78
22.739 O iv 5.1 5.24 73.123 Sivii 10.7 5.78
22.741 O iv 7.2 5.23 74.858 Mgviii 17.1 5.93
22.777 O iv 13.8 5.22 80.449 Sivi 13.0 5.62
22.978 O iii 9.4 4.95 80.577 Sivi 12.7 5.61
23.049 O iii 7.1 4.96 82.430 Feix 14.0 5.91
23.109 O iii 15.1 4.95 83.128 Sivi 14.2 5.63
23.350 O ii 7.5 4.48 83.511 Mgvii 14.2 5.82
23.351 O ii 12.6 4.48 83.910 Mgvii 19.8 5.85
23.352 O ii 16.5 4.48 88.079 Neviii 15.2 5.80
23.510 O i 7.2 4.00 95.385 Mgvi 15.5 5.67
23.511 O i 15.0 4.00 95.421 Mgvi 18.7 5.69
24.779 N vii 4.6 6.20 95.483 Mgvi 20.5 5.70
24.900 N vi 4.0 5.90 96.440 Siv 16.8 5.33
28.465 C vi 4.6 6.01 97.495 Nevii 22.8 5.74
28.787 N vi 9.8 6.01 98.131 Nevi 16.7 5.65
depends on the Voigt parameter a (Eqn. 3.89). In this range of densities, W does
not depend any more on σ v .
In Fig. 3.28 we show a few characteristic examples. The examples of Oi and
Ovi illustrate the higher W for UV lines as compared to X-ray lines (cf. Eqn. 3.87)
as well as the fact that inner shell transitions (in this case the X-ray lines) have
61

Figure 3.28: Equivalent width versus column density for a few selected oxygen
absorption lines. The curves have been calculated for Gaussian velocity dispersions
σ = b/ √ 2 of 10, 50 and 250 kms −1 , from bottom to top for each line. Different
spectral lines are indicated with different line styles.
larger a-values and hence reach sooner the square root branch. The example of
Ovii shows how different lines from the same ion have different equivalent width
ratios depending on σ v , hence offer an opportunity to determine that parameter.
The last frame shows the (non)similarties for similar transitions in different ions.
The innershell nature of the transition in Ovi yields a higher a value and hence an
earlier onset of the square root branch.
3.10.6 Galactic foreground absorption
All radiation from X-ray sources external to our own Galaxy has to pass through
the interstellar medium of our Galaxy, and the intensity is reduced by a factor of
e −τ(E) with the optical depth τ(E) = ∑ i σ i(E) ∫ n i (l)dl, with the summation over
all relevant ions i and the integration over the line of sight dl. The absorption
cross section σ i (E) is often taken to be simply the (continuum) photoionisation
cross section, but for high-resolution spectra it is necessary to include also the line
opacity, and for very large column densities also other processes such as Compton
ionisation or Thomson scattering.
For a cool, neutral plasma with cosmic abundances one often writes
τ = σ eff (E)N H (3.90)
62

where the hydrogen column density N H ≡ ∫ n H dx. In σ eff (E) all contributions to
the absorption from all elements as well as their abundances are taken into account.
Figure 3.29: Left panel: Neutral interstellar absorption cross section per hydrogen
atom, scaled with E 3 . The most important edges with associated absorption lines
are indicated, as well as the onset of Thompson scattering around 30 keV. Right
panel: Contribution of the various elements to the total absorption cross section as
a function of energy, for solar abundances.
The relative contribution of the elements is also made clear in Fig. 3.29. Below
0.28 keV (the carbon edge) hydrogen and helium dominate, while above 0.5 keV in
particular oxygen is important. At the highest energies, above 7.1 keV, iron is the
main opacity source.
Figure 3.30: Left panel: Column density for which the optical depth becomes unity.
Right panel: Distance for which τ = 1 for three characteristic densities.
Yet another way to represent these data is to plot the column density or distance
for which the optical depth becomes unity (Fig. 3.30). This figure shows that in
particular at the lowest energies X-rays are most strongly absorbed. The visibility
in that region is thus very limited. Below 0.2 keV it is extremely hard to look outside
our Milky Way.
63

exercise 3.12. The hydrogen column density to the Galactic pole is about 10 24 m −2 .
How large is the transmission for photons of 0.1, 0.2, 1 and 10 keV?
Complexity of the ISM
Figure 3.31: Left panel: Simulated 100 ks absorption spectrum as observed with the
Explorer of Diffuse Emission and Gamma-ray Burst Explosions (EDGE), a mission
proposed for ESA’s Cosmic Vision program. The parameters of the simulated source
are similar to those of 4U 1820−303 (Yao et al. 2006). The plot shows the residuals
of the simulated spectrum if the absorption lines in the model are ignored. Several
characteristic absorption features of both neutral and ionised gas are indicated.
Right panel: Simulated spectrum for the X-ray binary 4U 1820−303 for 100 ks with
the WFS instrument of EDGE. The simulation was done for all absorbing oxygen in
its pure atomic state, the models plotted with different line styles show cases where
half of the oxygen is bound in CO, water ice or olivine. Note the effective shift of
the absorption edge and the different fine structure near the edge. All of this is well
resolved by EDGE, allowing a determination of the molecular composition of dust
in the line of sight towards this source. The absorption line at 0.574 is due to highly
ionised Ovii.
The interstellar medium is by no means a homogeneous, neutral, atomic gas. In
fact, it is a collection of regions all with different physical state and composition.
This affects its X-ray opacity. We briefly discuss here some of the most important
features.
The ISM contains cold gas (< 50 K), warm, neutral or lowly ionised gas (6000
− 10000 K) as well as hotter gas (a few million K). We have seen before that for
ions the absorption edges shift to higher energies for higher ionisation. Thus, with
instruments of sufficient spectral resolution, the degree of ionisation of the ISM can
be deduced from the relative intensities of the absorption edges. Cases with such
high column densities are rare, however (and only occur in some AGN outflows), and
for the bulk of the ISM the column density of the ionised ISM is low enough that only
the narrow absorption lines are visible (see Fig. 3.27). These lines are only visible
when high spectral resolution is used (see Fig. 3.31). It is important to recognise
these lines, as they should not be confused with absorption line from within the
64

source itself. Fortunately, with high spectral resolution the cosmologically redshifted
absorption lines from the X-ray source are separated well from the foreground hot
ISM absorption lines. Only for lines from the Local Group this is not possible, and
the situation is here complicated as the expected temperature range for the diffuse
gas within the Local Group is similar to the temperature of the hot ISM.
Another ISM component is dust. A significant fraction of some atoms can be
bound in dust grains with varying sizes, as shown below (from Wilms et al. 2000):
H He C N O Ne Mg Si S Ar Ca Fe Ni
0 0 0.5 0 0.4 0 0.8 0.9 0.4 0 0.997 0.7 0.96
The numbers represent the fraction of the atoms that are bound in dust grains.
Noble gases like Ne and Ar are chemically inert hence are generally not bound in
dust grains, but other elements like Ca exist predominantly in dust grains. Dust has
significantly different X-ray properties compared to gas or hot plasma. First, due to
the chemical binding, energy levels are broadened significantly or even absent. For
example, for oxygen in most bound forms (like H 2 O) the remaining two vacancies in
the 2p shell are effectively filled by the two electrons from the other bound atom(s)
in the molecule. Therefore, the strong 1s–2p absorption line at 23.51 Å (527 eV)
is not allowed when the oxygen is bound in dust or molecules, because there is
no vacancy in the 2p shell. Transitions to higher shells such as the 3p shell are
possible, however, but these are blurred significantly and often shifted due to the
interactions in the molecule. Each constituent has its own fine structure near the
K-edge (Fig. 3.31b). This fine structure offers therefore the opportunity to study
the (true) chemical composition of the dust, but it should be said that the details of
the edges in different important compounds are not always (accurately) known, and
can differ depending on the state: for example water, crystalline and amorphous ice
all have different characteristics. On the other hand, the large scale edge structure,
in particular when observed at low spectral resolution, is not so much affected. For
sufficiently high column densities of dust, self-shielding within the grains should be
taken into account, and this reduces the average opacity per atom.
Finally, we mention here that dust also causes scattering of X-rays. This is in
particular important for higher column densities. For example, for the Crab nebula
(N H = 3.2×10 25 m −2 ), at an energy of 1 keV about 10 % of all photons are scattered
in a halo, of which the widest tails have been observed out to a radius of at least
half a degree; the scattered fraction increases with increasing wavelength.
3.10.7 Absorption from ionised gas
The absorption properties become different when the plasma is ionised. Fig. 3.32
gives the effective cross section for plasmas in CIE.
For 10 4 K, σ is essentially at the neutral limit. For 2 × 10 4 K hydrogen is fully
ionised, and therefore σ for energies of a few hundred eV is decreasing; at 10 5 K also
He is ionised and σ is in particular small below the C-edge. Around the O-edge (the
region between 0.5–1 keV) we see that by increasing the ionisation of oxygen the
edge shifts from 533 eV (Oi) to 870 eV (Oviii). Below the O-edge σ becomes very
small for increasing temperatures, because the most important absorbers (H, He)
are fully ionised. But unless oxygen is fully ionised (3 × 10 6 K) σ above the oxygen
K-edge is decreasing only slowly. Because of this the contrast around the K-edge of
65

Figure 3.32: The optical depth per hydrogen atom times E 3 for a plasma in CIE.
Figure 3.33: The optical depth per hydrogen atom times E 3 for a plasma in PIE.
oxygen increases for increasing T. Similar features are seen around the K-edge of
Si (∼ 2 keV), S (∼ 2.5 keV) and in particular Fe (7–8.5 keV) between 10 5 K and
3 × 10 8 K.
Therefore we see that for T > 10 7 K the plasma is almost transparent below
2 keV. between 10 7 and 10 8 K, σ between 2-6 keV is an order of magnitude smaller
than for neutral plasma’s, while above the K-edge of iron the effective cross section
hardly changes, unless T > 10 8 K.
For plasmas in photoionisation one finds qualitatively a similar behaviour (Fig. 3.33).
The parameter Ξ is the so-called ionisation parameter:
Ξ ≡ L/4πcr 2 n H kT. (3.91)
Essentially, it represents the ratio between radiation pressure and gas pressure. In
general, Ξ depends on the precise form of the incoming X-ray spectrum. For a
power law spectrum with energy index 1, or photon index 2 (dF/dE ∼ E −1 ) the
equilibrium solution for the temperature as a function of Ξ is shown in Fig. 3.34.
66

Figure 3.35: Stability for the ionisation
equilibrium.
Figure 3.34: Equilibrium temperature
versus ionisation parameter.
Arrows indicate the points
that are shown in Fig. 3.33. From
Krolik & Kallman 1984.
Note that in general for a single value of Ξ more temperature solutions are possible
(in particular for Ξ close to 10). Only those solutions where dT/dΞ > 0 are stable:
in that case a small increase in temperature brings the material to a region where
cooling is stronger than heating and hence the temperature will decrease again (see
Fig. 3.35). In the opposite case temperature variations will become stronger, hence
instability.
We see that for Ξ < 0.03 the absorption properties are almost identical to those
for neutral matter. Further, there are significant differences with the CIE case: in
photoionisation equilibrium a temperature of 10 5 K is already sufficient to reduce
σ eff to the same level as for CIE at 10 6 K. Plasmas in photoionisation equilibrium
therefore are much more ”transparent” than CIE plasmas.
67

3.11 The He-like triplets
Now you have seen most of the important radiation mechanisms, it is time to have
a closer look to the He-like triplets. This triplet has very important applications,
and therefore is important to understand.
First, we speak about a triplet, but in fact there are four spectral lines involved.
Two of these lines often have very similar wavelengths, hence are sometimes considered
to be one spectral line.
Figure 3.36: Simplified level scheme for helium-like ions. w (or r), x; y (or i), and
z (or f): resonance, intercombination, and forbidden lines, respectively. Full upward
arrows: collisional excitation transitions, broken arrows: radiative transitions
(including photo-excitation from n = 2 3 S 1 to n = 2 3 P 0,1,2 levels, and 2-photon continuum
from n = 2 1 S 0 to the ground level), and thick skew arrows: recombination
(radiative and dielectronic) plus cascade processes. From Porquet et al. 2001.
These lines correspond to transitions between the n = 2 shell and the n = 1
ground-state shell (see Fig. 3.36). Gabriel & Jordan (1969) were the first to show
that the line ratios can be used to estimate temperature and electron density:
R(n e ) =
z
x + y
(also designated as f/i), (3.92)
G(T e ) = z + x + y (also designated as (f + i)/r). (3.93)
w
In the low-density limit, all n = 2 states are populated directly, or via upperlevel
radiative cascades by electron impact from the He-like ground state and/or
by (radiative and dielectronic) recombination of H-like ions. These states decay
radiatively directly or by cascades to the ground level. The relative intensities of
the three intense lines are then independent of density. As n e increases from the
low-density limit, some of these states (1s2s 3 S 1 and 1 S 0 ) are depleted by collisions
to the nearby states where n crit C ∼ A, with C being the collisional coefficient rate,
68

Table 3.10: Wavelengths in Å of the triplet lines for some elements
Ion resonance intercombination intercombination forbidden
w x y z
Cv 40.268 40.728 40.731 41.472
Nvi 28.787 29.082 29.084 29.534
Ovii 21.602 21.804 21.807 22.101
Neix 13.447 13.550 13.553 13.699
Mgxi 9.1685 9.2279 9.2310 9.3141
Sixiii 6.6477 6.6848 6.6879 6.7400
Fexxv 1.8505 1.8554 1.8595 1.8682
A being the radiative transition probability from n = 2 to n = 1 (ground state),
and n crit being the critical density. Collisional excitation depopulates first the 1s2s
3 S 1 level (upper level of the forbidden line) to the 1s2p 3 P 0,1,2 levels (upper levels
of the intercombination lines). The intensity of the forbidden line decreases while
those of the intercombination lines increase, hence implying a reduction of the ratio
R (according to Eq. 3.92), over approximately two or three decades of density. For
much higher densities, 1s2s 1 S 0 is also depopulated to 1s2p 1 P 1 , and the resonance
line becomes sensitive to the density (this has been nicely illustrated by Gabriel &
Jordan (1972) in their Fig. 4.6.9).
As the critical density for each ion is different, using more than one triplet allows
one to cover a broad range of densities.
However, there are other ways to modify the triplet ratio. If the source has a
strong UV radiation field, the upper level of the forbidden line can be photo-excited
to the upper level of the intercombination lines, and therefore a strong UV field will
also decrease the R ratio.
Further, we note that the triplet is also an excellent indicator of the ionisation
balance. For collisionally ionised plasmas, the resonance line has a similar intensity
as the sum of the forbidden and intercombination lines. However, for a photoionised
plasma, in general these last two lines contain most power. This is because
recombination is proportional to the statistical weight of the level towards which
the recombination occurs. While the upper level of the resonance line (a singlet)
has w = 3, while that of the intercombination line has w = 9 and the forbidden
line w = 3. Hence, as a rough approximation G = (9 + 3)/3 = 4 for a photoionised
plasma.
Finally, because the oscillator strength for the intercombination and forbidden
line are much smaller than for the resonance line, if one observes the triplet in absorption,
one usually only sees the resonance line. Only for high nuclear charge (like
iron, Z = 26) the intercombination line also starts contributing to the absorption.
69