Thermal X-ray radiation (PDF) - SRON

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3.2.7 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 carbon atom described by the ground state configuration. The electron configuration is: 1s 2 2s 2 2p 2 . We have two closed subshells, 1s and 2s, which both give 1 S 0 . 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. The recipe is as follows: m l1 s 1 m l2 s 2 M L M S +1 + 0 + +1 +1 x +1 + −1 + 0 +1 x 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 1. Choose the state with highest M S , 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 M L for this M S . The value will also be the value of L for the term. 3. Cross out one state function for each pair M L , M S which correspond to the possible quantum numbers M L = −L, −L + 1, . . ., 0, . . .,L − 1, L and M S = −S, −S + 1, . . ., 0, . . .,S − 1, S. 4. Repeat the procedure stepwise for remaining functions. Application to the above case: 1. Maximum M S = 1 ⇒ S = 1, thus, this is a triplet. 2. The highest M L for M S = 1 is M L = 1 ⇒ L = 1, thus, it is a P state. The first term is thus 3 P. 3. Cross out a state function for each set of combinations of M L = −1, 0, +1 with M S = −1, 0, +1, for instance the states labeled x in the table above. See table below for what is left over. 13
m l1 s 1 m l2 s 2 M L M S +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 M S = 0. A repetition of the steps 1–3 gives S = 0 and L = 2, therefore we get a 1 D state. Next we cross out a state for each combination of M L = −2, −1, 0, +1, +2 with M S = 0, for instance those indicated with x above. See below what is left: m l1 s 1 m l2 s 2 M L M S −1 + +1 −1 0 0 The remainder is clearly a 1 S state. Thus, we have shown that the allowed terms for the combination 2p 2 are given by 3 P, 1 D and 1 S. exercise 3.1. 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. 3.2.8 Allowed terms The allowed terms for some configurations are shown in Table 3.1 and Table 3.2. Table 3.1: Terms for non-equivalent electrons Configuration s s s p s d p p p d d d Terms 1 S, 3 S 1 P, 3 P 1 D, 3 D 1 S, 1 P, 1 D, 3 S, 3 P, 3 D 1 P, 1 D, 1 F, 3 P, 3 D, 3 F 1 S, 1 P, 1 D, 1 F, 1 G, 3 S, 3 P, 3 D, 3 F, 3 G 14
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: Z El 1s 2s 2p 3s 3p 3d 4s Ground te
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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