Thermal X-ray radiation (PDF) - SRON

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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 <strong>radiation</strong> 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
Page 1 and 2: High Energy Astrophysics Jelle Kaas
Page 3 and 4: 3.1 Introduction Thermal X-ray radi
Page 5 and 6: Figure 3.2: The observed X-ray spec
Page 7 and 8: Figure 3.6: X-ray spectrum of the S
Page 9 and 10: series of alkaline spectra). The no
Page 11 and 12: • The first atomic shell (n = 1)
Page 13 and 14: Z El 1s 2s 2p 3s 3p 3d 4s Ground te
Page 15 and 16: m l1 s 1 m l2 s 2 M L M S +1 + +1
Page 17 and 18: 3.3 Energy levels We have seen the
Page 19 and 20: 3.4 Transitions between different l
Page 21 and 22: Figure 3.10: Transitions between a
Page 23 and 24: 3.5 Abundances of the elements With
Page 25 and 26: 3.6 Basic processes In order to be
Page 27 and 28: leading to Here the constant S 0 is
Page 29: 3.6.3 Radiative transition probabil
Page 32 and 33: Figure 3.12: Fluorescence yield as
Page 34 and 35: The formula of Younger While the Lo
Page 36 and 37: 3.6.6 Excitation-Autoionisation In
Page 38 and 39: 3.6.7 Photoionisation Figure 3.20:
Page 42 and 43: 3.6.10 Dielectronic recombination T
Page 44 and 45: 3.7 The ionisation balance In order
Page 46 and 47: of computing time. Therefore one us
Page 48 and 49: 3.8 Continuum emission processes Af
Page 50 and 51: decays back to the 1s orbit by a ra
Page 52 and 53: Inner-shell ionisation The third li
Page 54 and 55: 3.9.3 Line width For most X-ray spe
Page 56 and 57: Table 3.6: The strongest emission l
Page 58 and 59: Table 3.8: Some important iron line
Page 60 and 61: with τ(λ) = τ 0 ϕ(λ) (3.83) wh
Page 62 and 63: Table 3.9: The most important X-ray
Page 64 and 65: where the hydrogen column density N
Page 66 and 67: source itself. Fortunately, with hi
Page 68 and 69: Figure 3.35: Stability for the ioni
Page 70: Table 3.10: Wavelengths in Å of th

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