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
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
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
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
Übergang L-S Kopplung zu J-J Kopplung transition L-S coupling J-J coupling Mittwoch,. Juli
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
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
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
Gesamtdrehimpulse und Termsymbole L S m l m l2 m s m s2 M S M J Term S - + P + - + + + - P + - - - + + +2 + - + + P 2 + - - - - - -2 + + +2 + + 2 - D 2 - - - - -2 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
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
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
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
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
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
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. 2 4 5 6 subshell filling Be B C N O F Ne Mittwoch,. Juli
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). 2 4 5 6 subshell filling Be B C N O F Ne Mittwoch,. Juli
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. 2 4 5 6 subshell filling Be B C N O F Ne Mittwoch,. Juli
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
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
Schalenaufbau shell K L M N O electron # 2 8 8 2 5 subshells (ss) s 2s 2p s p d 4s 4p 4d 4f 5s 5p 5d 5f 5g electrons in filled ss cumulative # of electrons 2 2 6 2 6 2 6 4 2 6 4 8 2 28 6 Mittwoch,. Juli
Mittwoch,. Juli