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2.1. The Bohr atom

The electrons in an atom have orbits with discrete energy levels and quantum numbers. The principal quantum number n corresponds to the energy In of the orbit (in the classical Bohr model for the hydrogen atom the energy In = EH n-2 with EH = 13.6 eV the Rydberg energy), and it takes discrete values n = 1, 2, 3, …. An atomic shell consists of all electrons with the same value of n.

The second quantum number ell corresponds to the angular momentum of the electron, and takes discrete values ell < n. Orbits with ell = 0, 1, 2, 3 are designated as s, p, d, and f orbits. A subshell consists of all electrons with the same value of n and ell; they are usually designated as 1s, 2s, 2p, etc.

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 ell - 1/2 (for ell > 0) and ell + 1/2. Subshells are subdivided according to their j-value and are designated as n ellj. Example: n = 2, ell = 1, j = 3/2 is designated as 2p3/2.

There is also another notation that is commonly used in X-ray spectroscopy. Shells with 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 ell up to higher values of ell and from low values of j up to higher values of j:

1s 2s 2p1/2 2p3/2 3s 3p1/2 3p3/2 3d3/2 3d5/2 4s etc.
2 2 2 4 2 2 4 4 6 2  

The third row in this table indicates the maximum number of electrons that can be contained in each subshell. This so-called statistical weight is simply 2j + 1.

Atoms or ions with multiple electrons in most cases have their shells filled starting from low to high n and ell. For example, neutral oxygen has 8 electrons, and the shells are filled like 1s22s22p4, 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 O2+ for doubly ionised oxygen. In astronomical spectroscopic practice, more often one starts to count with the neutral atom, such that O2+ is designated as O III. 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), e.g. C V, N VI and O VII.

2.2. Level notation in multi-electron systems

For ions or atoms with more than one electron, the atomic structure is determined by the combined quantum numbers of the electrons. These quantum numbers have to be added according to the rules of quantum mechanics. We will not go into detail here, but refer to textbooks such as Herzberg (1944). The determination of the allowed quantum states and the transitions between these states can be rather complicated. Important here is to know that for each electron configuration (for example 1s 2p) there is a number of allowed terms or multiplets, designated as 2S+1L, with S the combined electron spin (derived from the individual s values of the electrons), and L represents the combined angular momentum (from the individual ell values). For L one usually substitutes the alphabetic designations similar to those for single electrons, namely S for L = 0, P for L = 1, etc. The quantity 2S + 1 represents the multiplicity of the term, and gives the number of distinct energy levels of the term. The energy levels of each term can be distinguished by J, the combined total angular momentum quantum number j of the electrons. Terms with 2S + 1 equals 1, 2 or 3 are designated as singlets, doublets and triplets, etc.

For example, a configuration with two equivalent p electrons (e.g., 3p2), has three allowed terms, namely 1S, 1D and 3P. The triplet term 3P has 2S + 1 = 3 hence S = 1 and L = 1 (corresponding to P), and the energy levels of this triplet are designated as 3P0, 3P1 and 3P2 according to their J values 0, 1 and 2.

2.3. Binding energies

The binding energy I of K-shell electrons in neutral atoms increases approximately as I ~ Z2 with Z being the nuclear charge (see Table 1 and Fig. 1). Also for other shells the energy increases strongly with increasing nuclear charge Z.

Figure 1

Figure 1. Energy levels of atomic subshells for neutral atoms.

Table 1. Binding energies E, corresponding wavelengths lambda and photoionisation cross sections sigma at the edges of neutral atoms (nuclear charge Z) for abundant elements. Only edges in the X-ray band are given. Note that 1 Mbarn = 10-22 m2.

Z E lambda sigma          Z E lambda sigma
    (eV) (Å) (Mbarn)            (eV) (Å) (Mbarn)

1s shell: 2s shell:
H 1 13.6 911.8 6.29        O 8 16.6 747.3 1.37
He 2 24.6 504.3 7.58        Si 14 154 80.51 0.48
C 6 288 43.05 0.96        S 16 232 53.44 0.37
N 7 403 30.77 0.67        Ar 18 327 37.97 0.29
O 8 538 23.05 0.50        Ca 20 441 28.11 0.25
Ne 10 870 14.25 0.29        Fe 26 851 14.57 0.15
Mg 12 1308 9.48 0.20        Ni 28 1015 12.22 0.13
Si 14 1844 6.72 0.14        2p1/2 shell:
S 16 2476 5.01 0.096        S 16 169 73.26 1.61
Ar 18 3206 3.87 0.070        Ar 18 251 49.46 1.45
Ca 20 4041 3.07 0.060        Ca 20 353 35.16 0.86
Fe 26 7117 1.74 0.034        Fe 26 726 17.08 0.42
Ni 28 8338 1.49 0.029        Ni 28 877 14.13 0.35
                 2p3/2 shell:
                 S 16 168 73.80 3.24
                 Ar 18 249 49.87 2.97
                 Ca 20 349 35.53 1.74
                 Fe 26 713 17.39 0.86
                 Ni 28 860 14.42 0.73

For ions of a given element the ionisation energies decrease with decreasing ionisation stage: for lowly ionised ions, a part of the Coulomb force exerted on an electron by the positively charged nucleus is compensated by the other electrons in the ion, thereby allowing a wider orbit with less energy. An example is given in Table 2.

Table 2. Binding energies E, corresponding wavelengths lambda and photoionisation cross sections sigma at the K-edges of oxygen ions.

Ion E lambda sigma
  (eV) (Å) (Mbarn)
      (10-22 m2)

O I 544 22.77 0.50
O II 565 21.94 0.45
O III 592 20.94 0.41
O IV 618 20.06 0.38
O V 645 19.22 0.35
O VI 671 18.48 0.32
O VII 739 16.77 0.24
O VIII 871 14.23 0.10

2.4. Abundances

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 for a few of the key elements.

Table 3. 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 ident 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 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.

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