Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.astro.spbu.ru/staff/afk/AtDatCentre/AtDatCat/Chapter1.pdf Дата изменения: Fri Nov 19 16:06:58 2010 Дата индексирования: Tue Oct 2 04:12:40 2012 Кодировка: Поисковые слова: m 103

Chapter 1 Main knowledge ab out atomic sp ectroscopy

1.1 The classification of levels Standard scheme of classification of atomic state levels is the single configuration approximation (SCA) and the LS - coupling of the angular momenta. In the single configuration approximation one assumes that every electron is moving in the effective central-symmetric field generated by the nucleus and by all other electrons. To all electrons one can ascrib e definite quantum numb ers, starting with the principle quantum numb er (n = 1, 2, 3, ...) and the orbital quantum numb er (l = 0, 1, ..., n - 1). The orbital angular momentum of an electron nl equals to lЇ , where Ї = h/2 and h is the Planck constant. The h h electrons with the same n and l values differ by the values of pro jections of orbital angular momentum ml and of the electron spin momentum ms . The feasible values for ml and ms are ml = -l, -l +1, ..., l - 1,l, and ms = - 1/2, +1/2. Electrons which have the orbital quantum numb er values l = 0, 1, 2, 3... are denoted in the atomic sp ectroscopy resp ectively by latin minuscules s, p, d, f and further in the alphab etic order. The atomic electrons with equal values of n and l are named to b e equivalent ones. A set q of equivalent electrons builds up an electron shell (nl)q in which the maximal numb er of electrons is 2(2l +1). If q = 2(2l + 1) then such a shell is called to b e filled. Distribution of electrons in electron shells is termed as electron configuration. The energetically lowest state in which all electrons of the atom or of the ion have minimal p ossible values of n and l is called the ground configuration, all other configurations are called the excited ones. The p opulation of electron shells undergoes definite rules. First, the electron shell n = 1 will b e p opulated, thereafter electron shells with n = 2, etc. At a given value of n first will b e filled

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the shells with l = 0 (s - shell), thereafter l = 1 (p - shell), l = 2 (d - shell) etc. The order of p opulation of electron shells with n 3 is the following: 1s2 2s2 2p6 3s2 3p6 3d10 ... . For instance the ground configuration of carb on atom is 1s2 2s2 2p2 . Here 1s2 and 2s2 are the filled shells, but the shell 2p2 is unfilled. The regularity in shell filling order is violated for d - shells and f shells. So, for K and Ca atoms first will b e filled the 4s - shell and thereafter the 3d - shell. The electron configurations of atomic and ion ground states are given, for example, in monographs by Moore (1949), Sob elman (1977), Allen (1973) and in a manual by Radtsig & Smirnov (1986). For classification of atomic states of the same configuration several approximate methods, so-called coupling schemes of momenta, have b een used, among which the widest use has the LS - coupling. According to this scheme the atomic states have different values of L (the total atomic orbital momentum) and S (the total atomic spin). The rule of momentum addition we shall illustrate by the example of two electrons. The values of total momentum L cover range from | l1 - l2 | to | l1 + l2 |. Similar rule holds for total spin : if s1 = s2 =1/2 then the total value of spin can b e 0 or 1. The sum J = L + S (called the total atomic momentum) according to general rules of addition of momenta can take values |L - S | J L + S . A definite value of J b elongs to a definite energy level LS J . The statistical weight corresp onding to J is given by g(J ) = 2J + 1 and it is the numb er of atomic states with the same energy, but different values of pro jection MJ . In the case of 3 and more electrons the addition of momenta must b e carried out step by step ­ first for two electrons, thereafter the third electron will b e added etc. The set of levels b elonging to one configuration with given values of L and S forms a sp ectral term (notation
2S +1

LJ , where 2S + 1 sp ecifies the term multiplicity, J describ es its fine structure

and LS J ­ the energy level. If S L then the numb er of term energy levels equals to its multiplicity, but if S > L then the term has 2L + 1 levels. Each level consists of 2J +1 states. The values of L are sp ecified by latin ma juscules: L = 0 (S - term), 1 (P - term), 2 (D - term), 8


3 (F - term) etc. The terms which have 2S +1 = 1, 2, 3, 4, 5 ... are named resp ectively singlets, doublets, triplets, quartets, quintets etc. In addition to values of LS the terms differ by configuration parity = (-1) is the algebraic sum of all electron orbital momenta, i.e.
li

, where

li

li = l1 + l2 + ... + lN . If = -1

(odd terms) then sup erscript " o " is added to the term notation on its right hand side. For three and more electrons with different values of nl in order to give unambiguous description of energy levels the additional quantum numb ers are needed. Usually the genealogy of term, i.e. the intermediary values of L and S , are given. For instance, the excited configuration 1s2 2s2pnl of CI I can have two groups of terms: 2s2p(1P 0 )nlLS and 2s2p(3P 0 )nlLS . The last group of terms has energy values ab out 6 eV lower than the first group. A sp ecial case are shells with equivalent electrons (nl)q . The numb er of terms of the (nl)q shell is limited by the Pauli principle. Thus, for configuration 2p2 the p ossible three terms are
3P 0 , 1S 3D

and 1D, while ignoring the Pauli principle for this configuration also terms 1P 0 , 3S and

were the p ossible ones. The terms of shells (nl)q are given, for example, in monograph by

Sob elman (1977). In the case of shells dq and f q there are various terms with equal L and S values. To discriminate them from each other the seniority quantum numb er v = 1, 2, 3 ... (left underscript to the term notation) is added. The real atomic states LS J due to approximate nature of single configuration and LS - coupling concepts are in fact some mixtures of the pure states with equal and J values, which b elong to different configurations and terms. In many cases instead of LS - coupling the other typ es of coupling, namely jj , LS0 and LK (see the monograph by Nikitin & Rudzikas (1983), Rudzikas, Nikitin & Kholtygin (1990)) are used.

1.2 The radiative transitions. Line strengths and oscillator strengths. Transition probabilities

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The most imp ortant characteristic of radiative transition in atoms is its probability Ali defined in such a manner, that nl Ali is the numb er of transitions in sp ectral line l - i p er unit volume and p er unit time. The value of Ali is strongly dep endent on the transition typ e. If in transition l - i there would b e emitted a photon having momentum k (in h units) relative to Ї the atom, where k = 1, 2, ... and with parity = (-1)k , then we term it as the electric (E ) radiative transition of multip olity k (E k - radiation), but in the case of parity = (-1)k+1 we term the transition as the magnetic (M ) radiative transition of multip olity k (M k - radiation). The probabilities of M k and E k + 1 transitions are ab out 10-4 ­ 10-6 times less than these of corresp onding E k transitions. Thus, the largest transition probabilities have electric dip ole (E 1) transitions for which Ali 108 s
-1

which are followed by electric quadruple (E 2) transitions 102 ­ 104 s
-1

and magnetic dip ole transitions for b oth of which Ali

. The transitions of higher

multip olity in the sp ectra of astrophysical ob jects have not b een observed. The transition probabilities can b e expressed by the use of line strengths Sli . Namely holds (see Levinson & Nikitin (1962), Sob elman (1977)) gl AE li
1,M 1

= 2.67 · 109 e3 Sli ,

gl AE 2 = 1.78 · 103 e5 Sli , li

(1.1)

where the ratio e = Eli /Ry is the transition energy expressed in Rydb ergs (Ry = 13.606 eV). The line strengths are connected with dimensionless quantities -- the oscillator strengths fil by gi f
E 1,M 1 il

=

1 eSli , 3

gi f

E2 il

= 2.22 · 10-7 e3 Sli ,

(1.2)

Thus, the transition probabilities can b e expressed by oscillator strengths in the form gl AE li
1,M 1

= 8.01 · 109 e2 gi fil ,

gl AE 2 = 8.02 · 109 e2 gi fil , li

(1.3)

Taking into account the conservation laws of momentum, angular momentum and parity, it follows that each process of typ e E 1,E 2 and M 1 can take place only if there hold definite 10


selection rules sp ecifying the p ossible differences of quantum numb ers in the initial and final states of the transition. Let us consider now the selection rules for E k and M 1 transitions in the LS - coupling and single configuration approximation. For E k- transition nlLS J n l L S J the selection rules are J = 0, ±1, ..., ±k, J + J k; L = 0, ±1, ..., ±k, L + L k; S = 0,

(1.4)

where only the quantum numb ers of one definite electron undergo changes, and are the additional quantum numb ers needed to describ e the levels J and J for which holds the selection rule = . Further, for E 1 -transitions holds l = ±1 and for E 2 -transitions l = 0, ±2. For M 1 - transition LS J L S J the selection rules are J = 0, ±1; L = 0, S = 0, l = 0. (1.5)

There has b een taken into account that the magnetic dip ole transitions can take place only b etween the levels of the same term (l = 0). Due to approximate nature of LS - coupling and single configuration assumptions the selection rules given ab ove are not the exact ones, i.e. these selection rules can b e violated in transitions which have essentially smaller probabilities. In the atomic sp ectroscopy the transitions in which the selection rules hold, are termed the allowed transitions, but in the opp osite case they are termed the forbidden transitions. In the table added here we give the classification of radiative transitions used in astrophysics in the case of selection rules violation. The classification differs somewhat from the one used in atomic sp ectroscopy. For instance, all E 2 transitions in astrophysics are treated as the forbidden ones indep endent of whether the selection rules hold or not. The typical values of the transition probabilities for the transitions under consideration are given in column 3 of the table. In the first column there are presented also the transition typ e notations (p - p ermitted, f - forbidden,

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i - intercombination, 2e - dielectronic). The given typical values of Ali corresp ond to transitions in the visible and near infrared regions of sp ectra of light elements and low-charge ions.

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Classification of the transition typ es Transition typ e E 1, p E 1, i E 1, 2e E 2, E 2, M 1, M 1, f f f f Selection rule violation no S = 0 quantum numb er change of two(three) electrons no S = 0 no transition b etween levels of different terms Aki 107 102 106 (s-1 ) - 109 - 104 - 108 102 -1 10 -1

1- 10-4 1- 10-4

For illustration of several typ es of electron transitions we give the scheme of lower levels of OI I I . In Fig. 1 the wavelengths of most imp ortant observed sp ectral lines together with the transition typ e are given. Mark denotes that the transition is intercombinational. For atoms with Z 50 and for multiple ions the selection rules do not hold exactly due to relativistic effects. For such atoms and ions the difference b etween the allowed and forbidden transitions weakens and can even b e vanishing.

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