Thermal X-ray radiation (PDF) - SRON

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3.2 Summary of spectroscopic notations 3.2.1 Atoms or ions with one valence electron The electrons in an atom have orbits with discrete energy levels and quantum numbers. The valence electron 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 - principal quantum number, 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 angular momentum of orbital motion of the electron: l = 0, 1, 2, . . . n − 1. s - quantum number of spin 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| = √ l(l + 1) ≈ l (l ≫ 1). Note that ⃗j = ⃗ l + ⃗s. For hydrogenic orbits (Coulomb field) in the theory of Bohr, the semi-major axis r of an elliptical orbit in shell n is given by r = n 2 a 0 /Z, where Z is the nuclear charge and the Bohr radius a 0 equals 5.291772108 × 10 −11 m (∼0.529 Å). Note that a 0 = α/4πR ∞ with α the fine structure constant and R ∞ the Rydberg constant R ∞ = m e cα 2 /2h; R ∞ hc = 1 2 α2 m e c 2 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 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. 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 withsmall 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 proton 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 = 0 1 2 3 4 5 6 7 8 9 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 7
series of alkaline spectra). The notation is complicated, but could have been worse if for example the Chinese alphabet would have been used. Example 3.1. Take the hydrogen atom. An electron in the ground state 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 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 l − 1/2 (for l > 0) and l + 1/2. Subshells are subdivided according to their j-value and are designated as nl j . Example: n = 2, l = 1, j = 3/2 is designated as 2p 3/2 . 3.2.2 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 ⃗s 1 , ⃗s 2 , ⃗s 3 , . . .. Similar 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 + ⃗s i ∑ ⃗j i = J ⃗ This coupling may occur in complicated energy configurations. i 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: ∑ i ⃗ li ∑ ⃗s i i ⃗L + ⃗ S = = ⃗ L = ⃗ S ⃗ J In these equations, ⃗ L is the total orbital angular momentum, ⃗ S the total spin, and ⃗J the total angular momentum. 8
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: Figure 3.6: X-ray spectrum of the S
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 40 and 41: 3.6.9 Radiative recombination Radia
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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