LS Kopplung. = a ij l i l j. W li l j. = b ij s i s j. = c ii l i s i. W li s j J = L + S. L = l i L = L(L + 1) J = J(J + 1) S = s i S = S(S + 1)

Transkript

1 LS Kopplung in many electron systems there are many li and si the coupling to for total momentum J depends on the energetic ordering of the interactions orbital momenta interaction W li l j = a ij l i l j spin spin interaction W li l j = b ij s i s j µ li = µ B l i larger than µ si = g s µ B s i spin orbit interaction W li s j = c ii l i s i L-S coupling L = l i L = L(L + ) S = s i S = S(S + ) J = L + S J = J(J + ) observed for small Z light atoms Mittwoch,. Juli

2 L-S Kopplung energy levels in L-S coupling E J = E(n, L, S) + CL S J 2 = (L + S) 2 = L 2 + S 2 + L S state labeling n 2S+ L J CL S = 2 C[J(J + ) L(L + ) S(S + )] 2 Mittwoch,. Juli

3 fine structure constant C is largest for small n with L! and S! magnitude of multiplet splitting decreases with increasing n L-S coupling is valid mainly for light atoms with small Z QM calculations show that C is propotional to C Z 4 n but energy separation between levels with different L only increases with Z 2 n for large Z, fine structure constant C is comparable with the separation of levels with different L, validity of L-S coupling breaks down Mittwoch,. Juli

4 J-J Kopplung spin orbit interaction W li s j = c ii l i s i large compared to orbital momenta interaction W li l j = a ij l i l j spin spin interaction W li l j = b ij s i s j observed for large Z heavy atoms j i = l i + s i J = j i there are no longer S,P,D... levels! only good quantum number is J Mittwoch,. Juli

5 Übergang L-S Kopplung zu J-J Kopplung transition L-S coupling J-J coupling Mittwoch,. Juli

6 Vektormodelle zur Drehimpulskopplung (a) l l 2 L + s S (b) l + j l 2 j 2 s 2 s s 2 L = J j = J S j 2 Abbildung 7.5: Vektormodell der L-S-Kopplung (a) und der j-j-kopplung (b). Mittwoch,. Juli

7 Kopplungsmöglichkeiten zweier Bahndrehimpulse (a) l 2 = L = l = L = 2 l = l2 = L = l = l 2 = (b) s = ½ s 2 = ½ S = s = ½ S = s2 = ½ Abbildung 7.6: (a) Kopplungsmöglichkeiten (schematisch) zweier Bahndrehimpulse l = l 2 = (p 2 - Konfiguration) und (b) zweier Elektronenspins s = s 2 = /2. Mittwoch,. Juli

8 Gesamtdrehimpulse und Termsymbole Elektronenkonfiguration s Drehimpulsquantenzahlen L S J ½ ½ Spektroskopische Symbole ²S /2 s² S S sp,,2 P P P P 2 p² 2 2,,2 2,2, S P P P 2 D 2 S P D,2, Tabelle 7.: Mögliche Gesamtdrehimpulse und spektroskopische Symbole für verschiedene Elektronenkonfigurationen. Die rot markierten Terme sind aufgrund des Pauli-Prinzips für n = n 2 verboten. Mittwoch,. Juli

9 Gesamtdrehimpulse und Termsymbole L S m l m l2 m s m s2 M S M J Term S - + P P P D Tabelle 7.2: Mögliche Zustände der Konfiguration np 2 mit Quantenzahlen L,S,m l,m l2,m s,m s2,m S = m s + m s2 und M J = m l + m l2 + m s + m s2 für gleiche Hauptquantenzahlen n = n 2 der beiden p-elektronen. Mittwoch,. Juli

10 S S = P D 2 p² S P S P 2 P S = P D D D 2 D Abbildung 7.7: Mögliche Atomterme der p 2 -Konfiguration. Die rot markierten Terme (gestrichelte Linien) sind aufgrund des Pauli-Prinzips verboten. Mittwoch,. Juli

11 Termschema des He Atoms total wavefunction is product of spatial and spin part total = ( r, r 2 ) (S, M S ) state symbol n S+ L J L = l + l 2 J = L + S Mittwoch,. Juli

12 Energieniveaus Helium energy levels of the helium atom first excited state E = E s + E 2s + J ± K Coulomb integral exchange integral spin orbit splitting of the triplet Mittwoch,. Juli

13 Übergänge Helium optical transitions and selection rules l = ± m l =, ± j =, ± j i = j k = s = only one electron is influenced total angular momenta L = ± M L =, ± S = Mittwoch,. Juli

14 Hundsche Regeln filling the shells S the p-shells are first filled with parallel spins first (spatial wave function is asymmetric, with electron farther apart) Mittwoch,. Juli

15 Hundsche Regeln rule # Terms with larger total spin are energetically favoured. Electrons with parallel spin tend to avoid each other..5. for parallel spins, the spatial wavefunction is asymmetric S.5 larger separation of electrons subshell filling Be B C N O F Ne Mittwoch,. Juli

16 Hundsche Regeln rule #2 For a given multiplicity, the term with the largest value of L lies lowest in energy..5 L..5 the repulsion of electrons is less if they orbit in the same direction (they meet less often) subshell filling Be B C N O F Ne Mittwoch,. Juli

17 Hundsche Regeln rule # For atoms with less than half filled shells, the level with the lowest value of J lies lowest in energy. 2. J = L S J = L + S.5 J..5 for subshells less than half filled we have J= L-S, for subshells more than half filled we have J= L+S for the ground state subshell filling Be B C N O F Ne Mittwoch,. Juli

18 Schalenaufbau radial distribution P (r) = r 2 R n,l (r) 2 n l= (2l + ) = n 2 time averaged charge distribution e n 2 = e = Ce l m l n,l,m l 2 l R n,l (r) 2 is always spherically symmetric these spherically symmetric charge distributions are called electron shells Mittwoch,. Juli

19 subshells shell names: n=: K-shell n=4: N-shell n=2: L-shell n=5: O-shell n=: M-shell n=6: P-shell each of these shells has including the spin 2n 2 states (n, l, m l, m s ) According to the Pauli principle each electron shell can be occupied by at most 2n 2 electrons. For each value of n there are n different values of l and therefore n subshells. Mittwoch,. Juli

20 Schalenaufbau shell K L M N O electron # subshells (ss) s 2s 2p s p d 4s 4p 4d 4f 5s 5p 5d 5f 5g electrons in filled ss cumulative # of electrons Mittwoch,. Juli

21 Mittwoch,. Juli