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Chapter 2

Thermal -ray radiation

14 2.1 Introduction

Figure 2.1: 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)

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 ex- tragalactic), 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 chap- ter we focus on X-ray emission and absorption in optically thin plasma’s. For op- tically 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

15 Figure 2.2: 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).

16 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 is shown in Fig. 2.1. 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. 2.2. This shows the success of the plasma emission codes developed at SRON by Rolf Mewe (1935–2004) and his colleagues.

Figure 2.3: The observed X-ray spec- Figure 2.4: X-ray spectrum of the trum with the RGS of XMM-Newton Seyfert 2 galaxy NGC 1068. From of the flare star EQ Pegasi. The plot Kinkhabwala et al., ApJ, 575, 732 shows the so-called O vii triplet, con- (2002). sisting of the resonance, intercombi- nation and forbidden line. The rela- tive line ratio’s of this triplet contain a wealth of physical information. Image courtesy of J. Schmitt, Hamburger Sternwarte, Germany and ESA.

The strength of spectral analysis lies often in the details. While there are many spectral lines, some lines contain more information than others. Fig. 2.3 shows an example of this. It is a part of the spectrum of a flare star. The three lines shown – the famous O vii 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. 2.4 and compare the intensities of the triplet with those in Fig. 2.3. 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. 2.5. In order to see thermal imprints on a spectrum sometimes requires the skills of a spectroscopist. See for example Fig. 2.6. There has been a fierce debate in the literature whether the sharp change in the spectrum near 18 A˚ 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.

17 Figure 2.5: 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.

Figure 2.6: X-ray spectrum of the Seyfert 1 galaxy MCG 6-30-15, obtained with the RGS of XMM-Newton.

For the proper calculation of an X-ray spectrum, one should consider three dif- ferent steps:

1. the determination of the ionisation balance

2. the determination of the emitted spectrum

3. possible absorption of the photons on their path 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.

18 2.2 Summary of spectroscopic notations

2.2.1 or ions with one The in an have orbits with discrete energy levels and num- bers. The 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 - , 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 of orbital of the electron: ℓ = 0, 1, 2, ... n − 1. s - quantum number of 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| = pℓ(ℓ + 1)~ ≈ ℓ~ (ℓ ≫ 1).

Note that ~j = ~ℓ + ~s. For hydrogenic orbits (Coulomb field) in the theory of Bohr, the semi-major 2 axis r of an elliptical orbit in shell n is given by r = n a0/Z, where Z is the −11 nuclear and the Bohr radius a0 equals 5.291772108 × 10 m (∼0.529 A).˚ Note that a0 = α/4πR∞ with α the fine structure constant and R∞ the Rydberg 2 1 2 2 constant R∞ = mecα /2h; R∞hc = 2 α mec 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 In of the orbit. −2 In the classical for the atom the energy In = EHn with EH = 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 with small 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 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 =0123456789 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

19 series of alkaline spectra). The notation is complicated, but could have been worse if for example the Chinese alphabet would have been used.

Example 2.1. Take the . An electron in the 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 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 nℓj. Example: n = 2, ℓ = 1, j =3/2 is designated as 2p3/2.

2.2.2 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 ~s1, ~s2, ~s3,.... Sim- ilar 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 + ~si ~ ~ X ji = J i

This coupling may occur in complicated energy configurations.

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:

~ ~ X li = L i ~ X ~si = S i L~ + S~ = J~

In these equations, L~ is the total orbital angular momentum, S~ the total spin, and J~ the total angular momentum.

20 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 ~si must be either parallel or anti-parallel.

Example 2.2. 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 2.3. 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 =0123456789 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 is called a term with r. Although for L

Example 2.4. 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 S = 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 5678 singlet doublet triplet quartet quintet sextet septet octet

2.2.3 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 mj (for mj see §2.2.5). 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?

21 • 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 1s2 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 mℓ = −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 1s2 2s2 2p6. Frequently one omits the filled shells, so that for example 1s2 2s2 2p5 is abbreviated as simply 2p5.

• The third closed shell is 1s2 2s2 2p6 3s2 3p6 3d10, etc. There are 10 d-orbits because there we have ℓ = 2 and hence mℓ = −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:

~ ~ X ℓi =0 X ~si =0 X ji =0 X ℓi = even X si = integer. i i i i i Each given electron configuration corresponds to a certain group of terms which all have approximately similar energies (see also §2.5.3).

2.2.4 Term notation

r In general, a term with quantum numbers LSJ is designated as (L-symbol)J . We can illustrate this with an example.

Example 2.5. Take the electron configuration 1s2 2s2 2p6 3s2 3p2, with L = 1, 2 2 6 2 2 3 S = 1, J = 2. This is designated as 1s 2s 2p 3s 3p P2, or shortly with omission 2 3 of the closed shells, as 3p P2 (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 P ℓi = even or odd, the term is even or i odd. For example, the doublet term 2D is even, while the triplet term 3Po 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.

22 Z Element configuration term 2 1 H 1s S1/2 2 1 2 He 1s S0 2 3 Li 2s S1/2 2 1 4 Be 2s S0 2 5 B 2p P1/2 2 3 6 C 2p P0

All noble gases have a closed shell with L = 0, S = 0 and as a consequence their 1 ground state is always S0. The ground term of a shell that is filled only half is 3 4 always an S term, for example for nitrogen 2p S3/2.

2.2.5 Statistical weight The statistical weight of an 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 mj for a single elec- tron with angular momentum ~j). In the case of a magnetic field MJ is the so- called (Zeeman effect). MJ 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 gJ =2J + 1.

3 3 Example 2.6. The D2 level has g2 = 5. Note that for D we have L = 2, S = 1. The total statistical weight of the 3D 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 MJ (Zeeman effect), one then takes n, L, ML and MS, with ML and MS the projections of L~ and S~ on B~ (Paschen Back effect). In that 3 case we find for the D term ML = −2, −1, 0, 1, 2, MS = −1, 0, 1 → g =5 × 3 = 15.

2.2.6 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.

23 Z El 1s 2s 2p 3s 3p 3d 4s Groundterm 2 1H 1 S1/2 1 2 He 2 S0 2 3Li 2 1 S1/2 1 4Be 2 2 S0 2 5B 22 1 P1/2 3 6C 22 2 P0 4 7N 22 3 S3/2 3 8O 22 4 P2 2 9F 225 P3/2 1 10 Ne 2 2 6 S0 2 11Na 2 2 6 1 S1/2 1 12Mg 2 2 6 2 S0 2 13Al 2 2 6 2 1 P1/2 3 14Si 2 2 6 2 2 P0 4 15P 22 6 2 3 S3/2 3 16S 22 6 2 4 P2 2 17Cl 2 2 6 2 5 P3/2 1 18Ar 2 2 6 2 6 S0 2 19K 22626 1 S1/2 1 20Ca22626 2 S0 2 21Sc 2 2 6 2 6 1 2 D3/2 3 22Ti 2 2 6 2 6 2 2 F2 4 23V 22626 32 F3/2 7 24Cr 2 2 6 2 6 5 1 S3 6 25Mn2 2 6 2 6 5 2 S5/2 5 26Fe 2 2 6 2 6 6 2 D4 4 27Co 2 2 6 2 6 7 2 F9/2 3 28Ni 2 2 6 2 6 8 2 F4 2 29Cu 2 2 6 2 6 10 1 S1/2 1 30Zn 2 2 6 2 6 10 2 S0

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 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 , 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).

24 2.2.7 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 atom described by the ground state configuration. The electron configuration is: 1s2 2s2 2p2. We have two closed subshells, 1s and 1 2s, which both give S0. 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.

mℓ1 s1 mℓ2 s2 ML MS +1 + 0 + +1 +1 x +1 + −1 + 0 +1x 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

The recipe is as follows:

1. Choose the state with highest MS, 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 ML for this MS. The value will also be the value of L for the term. In this example it is 1.

3. Cross out one state function for each pair ML, MS which correspond to the possible quantum numbers ML = −L, −L +1,..., 0,...,L − 1, L and MS = −S, −S +1,..., 0,...,S − 1,S. 4. Repeat the procedure stepwise for remaining functions. Application to the above case:

1. Maximum MS =1 ⇒ S = 1, thus, this is a triplet.

2. The highest ML for MS = 1 is ML = 1 ⇒ L = 1, thus, it is a P state. The first term is thus 3P .

3. Cross out a state function for each set of combinations of ML = −1, 0, +1 with MS = −1, 0, +1, for instance the states labeled x in the table above. See table below for what is left over.

25 mℓ1 s1 mℓ2 s2 ML MS +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 MS = 0. A repetition of the steps 1–3 gives S = 0 and L = 2, therefore we get a 1D state. Next we cross out a state for each combination of ML = −2, −1, 0, +1, +2 with MS = 0, for instance those indicated with x above. See below what is left:

mℓ1 s1 mℓ2 s2 ML MS −1 + +1 −1 0 0

The remainder is clearly a 1S state. Thus, we have shown that the allowed terms for the combination 2p2 are given by 3P , 1D and 1S. exercise 2.1. 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.

2.2.8 Allowed terms The allowed terms for some configurations are shown in Table 2.1 and Table 2.2.

Table 2.1: Terms for non-equivalent electrons

Configuration Terms s s 1S, 3S s p 1P, 3P s d 1D, 3D p p 1S, 1P, 1D, 3S, 3P, 3D p d 1P, 1D, 1F, 3P, 3D, 3F d d 1S, 1P, 1D, 1F, 1G, 3S, 3P, 3D, 3F, 3G

26 Table 2.2: Terms for equivalent electrons

Configuration Terms s2 1S p2 1S, 1D, 3P p3 2P, 2D, 4S p4 1S, 1D, 3P p5 2P p6 1S d2 1S, 1D, 1G,3P, 3F d3 2P, 2D(2), 2F,2G, 2H, 4P, 4F d4 1S(2), 1D(2), 1F, 1G(2), 1I, 3P(2), 3D, 3F(2), 3G, 3H, 5D d5 2S, 2P, 2D(3), 2F(2), 2G(2), 2H, 2I, 4P, 4D, 4F, 4G, 6S

27 2.3 Energy levels

It is common practice in X-ray spectroscopy to designate ions also according to the number of electrons, thus Fe xxvi is hydrogen-like iron, Fe xxv 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. We have seen the notation for the various electronic shells in an ion being des- ignated 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 subdi- vision 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 2p1/2 2p3/2 3s 3p1/2 3p3/2 3d3/2 3d5/2 4s etc. K LI LII LIII MI MII MIII MIV MV NI The binding energy I of K-shell electrons in neutral atoms increases approximately as I ∼ Z2 (see Table 2.3 and Figs. 2.7–2.8). Also for other shells the energy increases strongly with increasing nuclear charge Z.

Figure 2.7: Energy levels of atomic Figure 2.8: Energy levels of atomic subshells for neutral atoms. subshells for neutral atoms.

The binding energy of more highly ionised ions is in general larger than for neutral atoms (see Table 2.3). For example, neutral iron has a K-shell energy of 7.1 keV, Fe xxvi 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.

28 Table 2.3: 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

29 2.4 How to calculate energy levels: Slater’s rules

Slater (1930) found an elegant way to calculate energy levels and from that spectral line energies in any given electron configuration. Essentially, it is a recipe to calculate how much a given electron is shielded by the other electrons from the attractive force from the nucleus. Slater provides a way to calculate the effective charge Zeff = Z − s where Z is the true nuclear charge of the nucleus, and s is the screening constant. The electron + + then experiences not the Coulomb force Ze but Zeff e . First the electrons are arranged into a sequence of groups in order of increasing n, and for equal n in order of increasing l, except that s and p orbitals are kept together: [1s] [2s,2p] [3s,3p] [3d] [4s,4p] [4d] [4f] [5s, 5p] [5d] etc. Each group gets a different shielding constant depending upon the number and types of electrons in the preceding groups. The shielding constant for each group is calculated as the sum of the following contributions:

1. An amount of 0.35 from each other electron within the same group except for the [1s] group, where the other electron contributes only 0.30.

2. If the group is of the [ns,np] type, an amount of 0.85 from each electron with principal quantum number n−1, and an amount of 1.00 for each electron with principal quantum number n − 2 or less.

3. If the group is of the [d] or [f] type, an amount of 1.00 for each electron ”closer” to the atom than the group. This includes both i) electrons with a smaller principal quantum number than n and ii) electrons with principal quantum number n and a smaller l.

group other electrons other electrons (′) other electron (′) other electron (′) in same group n′ = n, but ℓ′ < ℓ n′ = n − 1 n′ ≤ n − 2 1s 0.30 – – – ns,np 0.35 – 0.85 1.00 nd or nf 0.35 1.00 1.00 1.00

Example 2.7. How to calculate the effective nuclear charge in nitrogen for the 2p electron. The electron configuration is 1s22s22p3. We group this according to the recipe into (1s2), (2s2,2p3). The 2p electron has 4 other electrons in the same group, and 2 electrons in the group with n = 1. Thus, the screening constant s =4 × 0.35+2 × 0.85=3.10. The effective nuclear charge (nitrogen has 7 ) is therefore Zeff = Z − s = 7 − 3.10 = 3.90. The corresponding energy is dus 2 2 Zeff /2 × 13.6 eV or 52 eV.

If a line energy is needed for an emission or absorption line between an upper level and a lower level, one thus determines the energy of the upper level by adding all energies of all electrons in the upper level (apply Slater’s rule for each electron), and the same for the lower level. The difference in the summed energies for upper and lower level is then the energy of the line.

30 Example 2.8. Consider the transition in Li-like Fe: 1s23p → 1s22s. One first calcu- lates the energy of the upper state 1s23p using Slater’s rule for both 1s electrons and the 3p electron (screening constants of 0.30, 0.30 and 2.00, adding the corresponding energies gives (in Rydberg units of 13.6 eV) (26 − 0.3)2/12 + (26 − 0.3)2/12 + (26 − 2.0)2/32 = 1384.98. For the lower state one gets screening constants of 0.30, 0.30 and 1.90, hence an energy of (26−0.3)2/12+(26−0.3)2/12+(26−1.9)2/22 = 1466.18. The energy difference (in eV) is thus 1104 eV. An exact calculation (not using Slater’s rules) gives about 1165 eV.

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