2. A SHORT INTRODUCTION TO ATOMIC STRUCTURE

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 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), 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 corresponds to the angular momentum of the electron, and takes discrete values < n. Orbits with = 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 ; 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 - 1/2 (for > 0) and + 1/2. Subshells are subdivided according to their j-value and are designated as n _j. Example: n = 2, = 1, j = 3/2 is designated as 2p_3/2.

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 up to higher values of and from low values of j up to higher values of j:

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 . 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 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. Energy levels of atomic subshells for neutral atoms.

Table 1. Binding energies E, corresponding wavelengths and photoionisation cross sections 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					Z	E
		(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 and photoionisation cross sections at the K-edges of oxygen ions.

Ion	E
	(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	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