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2.7.6 Excitation-Autoionisation in the Previous Section We Showed How a Collision of a Free Electron with an Ion Can Lead to Ionisation

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2.7.6 Excitation-Autoionisation in the Previous Section We Showed How a Collision of a Free Electron with an Ion Can Lead to Ionisation 2.7.6 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 2.16: The Excitation-Autoionisation process for a Li-like ion. As an example we treat here Li-like ions (see Fig. 2.16). 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 (EH = 13.6 eV): 2 2 Z EH Q = πa0 2B 2 ΩH (2.38) Zeff (1s) E in which a0 is the Bohr radius, and Zeff = 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 (2.39) ≃ 1+(Z/17)3 Finally, the scaled hydrogenic collision strength is given by: 0.18 1.20 Z2Ω 2.22 ln ǫ +0.67 + (2.40) H ≃ − ǫ ǫ2 49 where ǫ E/E and the excitation energy E I I . Similar to the case of ≡ ex ex ≡ 1s − 2p direct ionisation, it is straightforward to average the excitation-autoionisation cross section over a Maxwellian electron distribution. Figure 2.17: Collisional ionisation Figure 2.18: Collisional ionisation cross section for O vi. cross section for Fe xvi. In Figs. 2.17 and 2.18 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 Fe xvi the excitation autoionisation process is very important. The increasing importance of excitation autoionisation relative to direct ionisation is also illustrated in Fig. 2.19. Figure 2.19: Ratio between the Excitation-Autoionisation cross section to the Direct Ionisation cross section for Na-like ions. From Arnaud & Rothenflug (1985). 50 2.7.7 Photoionisation Figure 2.20: Photoionisation cross section in barn (10−28 m−2) for Fe i (left) and Fe xvi (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. 2.20 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 (I/E)3 (2.41) ≃ 0 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 ∞ CPI = c niσ(E)F (E)dE. (2.42) Z 0 For hydrogenlike ions one can write: 2 64πng(E, n)αa0 I 3 σPI = ( ) , (2.43) 3√3Z2 E where n is the principal quantum number, α the fine structure constant and A0 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. 51 2.7.8 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 2.21: 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 3σ = T (1 + cos2 θ), (2.44) dΩ 16π with θ the scattering angle and σ the Thomson cross section (6.65 10−29 m−2). T × The energy transfer ∆E during the scattering is given by (E is the photon energy): E2(1 cos θ) ∆E = − . (2.45) m c2 + E(1 cos θ) e − Only those scatterings where ∆E>I contribute to the ionisation. This defines a critical angle θc, given by: m c2I cos θ =1 e . (2.46) c − E2 IE − For E I we have σ(E) σT (all scatterings lead in that case to ionisation) and ≫ → 2 further for θc π we have σ(E) 0. Because for most ions I mec , this last condition occurs→ for E Im c2→/2 I. See Fig. 2.21 for an≪ example of some ≃ e ≫ cross sections. In general, Comptonp 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 (all scatterings lead ≫ → T in that case to ionisation) and further that for θc π we have σ(E) 0. Because for most ions I m c2, this last condition occurs→ for E Im c2→/2 I. See ≪ e ≃ e ≫ Fig. 2.21 for an example of some cross sections. p exercise 2.9. Estimate for which energy Compton ionisation becomes more im- portant than photoionisation for neutral iron. 52 2.7.9 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 σbf (E) and the recombination cross section σfb, namely the Milne-relation: 2 E gnσbf (E) σfb(v)= 2 2 (2.47) mec mev where gn is the statistical weight of the quantum level into which the electron is cap- 2 tured (for an empty shell this is gn =2n ). By averaging over a Maxwell distribution one gets the recombination-coefficient to level n: ∞ Rn = nine vf(v)σfb(v)dv. (2.48) Z 0 1 2 Of course there is energy conservation, so E = 2 mev + I. exercise 2.10. Demonstrate that for the photoionisation cross section (2.43) and 2 for g = 1, constant and gn =2n : 128√2πn3αa2I3eI/kT n n R = 0 i e E (I/kT ) (2.49) n 2 3 1 3√3mekTZ kT mec With the asymptotic relations (2.13) and (2.14) it is seen that − 1 kT I : R T 2 (2.50) ≪ n ∼ kT I : R ln(I/kT )T −3/2 (2.51) ≫ n ∼ 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 ∼ −1 this we find that for kT I (recombining plasma’s) Rn n , while for kT I ≪−3 ∼ ≫ (ionising plasma’s) Rn n . 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. 2.10). 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 (2.49) cannot be used (the hydrogen limit), but instead one should use the exact photoionisation cross section of the valence electron.
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