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Atoms and Nuclei PA 322 Lecture 8 Unit 2: Many-electron atoms (Reminder: http://www.star.le.ac.uk/~nrt3/322) Spectra of many-electron atoms • Topics – L-S (= Russell-Saunders) coupling framework – terms for non-equivalent electrons – terms for equivalent electrons – Hund’s rules for L-S coupling: predicting the ground state configuration – j-j coupling PA322 Lecture 8 2 Many-electron atoms General considerations – typically atoms have more than one active electron – exact solutions to Schrödinger equation formally not possible, but approximate treatments give essentially correct picture – L-S coupling (= Russell-Saunders coupling) provides framework for discussing such atoms in wide variety of circumstances – valid unless individual electron spin-orbit coupling dominates (⇒ j-j coupling) PA322 Lecture 8 3 L-S coupling in many-electron atoms • L-S coupling issues – Effects which produced corrections to energy levels: 1. Spin correlation: electron-electron interaction as for helium atom 2. Orbital angular momentum coupling 3. Spin-orbit coupling – size of effects varies from atom to atom, cannot be trivially predicted – for low Z atoms (and low n): effects order 1 > 2 »3 PA322 Lecture 8 4 L-S coupling framework for many-electron atoms • Spin angular momenta of the electrons couple: – Total spin angular momentum S = Σ si (vector sum) – Related total spin angular momentum quantum number S • Orbital angular momenta of electrons couple: – Total orbital angular momentum L = Σ li (vector sum) – Related total orbital angular momentum quantum number L • Resultant total spin and orbital angular momenta S and L couple to produce total angular momentum vector J = L + S • Valid quantum numbers are now: [s1, s2 ( … sN)], l1, l2 (…lN), L, S, J but not ml1, ms1 … as coupling of l vectors and s vectors destroys their space quantisation but now have analogous new projection quantum number for J called mJ . PA322 Lecture 8 5 Terms for non-equivalent electrons in many-electron atoms • Non-equivalent electrons: those which belong to different (n,l) subshells (e.g. orthohelium 1s2s) • Any (s, l) combination allowed without violating Pauli Exclusion Principle • Terms can be worked out using relationships for L, S and J applying rule for vector addition: – for two active electrons with spin quantum numbers s1 and s2, orbital angular momentum numbers l1 and l2: l2 • S = |s1-s2| … (s1+s2) integer steps • L = |l1-l2| … (l1+l2) integer steps L • J = |L-S| … (L+S) integer steps l1 – equivalent formulae for > 2 electrons (but not considered here) PA322 Lecture 8 6 For le examp • • • Terms for non-equivalent electrons in many-electron atoms S=0 L=1 1P S=0 L=2 1D S=0 L=3 1F S=1 L=1 3P S=1 L=2 3D S=1 L=3 3F Examples for non-equivalent electrons: – npnd, i.e., p & d optically active electrons: – L = |l1-l2| … (l1+l2) but l1=1, l2=2 ⇒ L = 1, 2, 3 1P – S = |s1-s2| … (s1+s2) ⇒ S = 0, 1 (s1=s2=½) S=0 L=1 J=1 1 1 S=0 L=2 J=2 D2 – produces 6 terms: 1P, 1D, 1F, 3P, 3D, 3F 1 S=0 L=3 J=3 F3 3 taking J into account in each case S=1 L=1 J=0,1,2 P0, 3P1, 3P2 3D , 3D , 3D S=1 L=2 J=1,2,3 – J = |L-S| … (L+S) 1 2 3 3F , 3F , 3F S=1 L=3 J=2,3,4 2 3 4 – e.g., for 3F, L=3, S=1 ⇒ J = 2, 3, 4 thus full levels are 3F2, 3F3, 3F4 – in total there are 12 levels for the 6 terms 1P, 1D, 1F, 3P, 3D, 3F for each J there are 2J+1 degenerate mJ values – e.g., npnd configuration: 60 microstates (see next slide) – same number as expected in decoupled representation, i.e., if angular momentum coupling and spin correlation were ignored • e.g., npnd in decoupled representation: (2l1+1) x (2s1+1) x (2l2+1) x (2s2+1) = 3 x 2 x 5 x 2 = 60 ‘microstates’ • same number of microstates as computed from mJ values PA322 Lecture 8 7 For le examp PA322 Σ = 60 Lecture 8 8 Terms for equivalent electrons in many-electron atoms • • • Equivalent electrons: those which belong to same (n,l) sub-shells Allowed combinations must of course be consistent with Pauli Exclusion Principle (all quantum numbers cannot be same) Simple application of L-S coupling rules used for non-equivalent electrons produces some terms that are forbidden – working out valid terms has to be done the hard way • E.g. simple application for 2p2: • l1 = l2 = 1 ⇒ L = 0, 1, 2 & S = 0, 1 ⇒ apparent terms 1S, 3S, 1P, 3P, 1D, 3D • But this would result in too many states! 1S 3S 1P 3P 1D 3D 1 + 3 + 3 + 9 + 5 + 15 = 36 • Total number of apparently allowed microstates (36) ≠ number for decoupled electrons (15) - see next table PA322 Lecture 8 9 Possible quantum numbers for np2 configuration number ml ms ml ms mS mL mJ 1 +1 +½ +1 -½ 0 +2 +2 2 +1 +½ 0 +½ +1 +1 +2 3 +1 +½ 0 -½ 0 +1 +1 4 +1 +½ -1 +½ +1 0 +1 5 +1 +½ -1 -½ 0 0 0 6 +1 -½ 0 -½ -1 +1 0 7 +1 -½ -1 +½ 0 0 0 8 +1 -½ -1 -½ -1 0 -1 9 0 +½ +1 -½ 0 +1 +1 10 0 +½ 0 -½ 0 0 0 11 0 +½ -1 +½ +1 -1 0 12 0 +½ -1 -½ 0 -1 -1 13 -1 +½ 0 -½ 0 -1 -1 14 -1 +½ -1 -½ 0 -2 -2 15 -1 -½ 0 -½ -1 -1 -2 decoupled PA322 coupled Lecture 8 Note, also avoided configurations that are same due to indistinguishability of electrons. must be equivalence between degeneracies (=number of allowed microstates) in the L-S coupled and decoupled representations 10 Terms for equivalent electrons in many-electron atoms • How to decide which terms are allowed and which terms are forbidden? • Staying with application for 2p2: – apparent (i.e. possible w.r.t. s, l values) terms were • 3D, 1D, 3P, 1P, 3S, 1S • in fact only 1S, 1D & 3P are allowed, others are forbidden by application of PEP to decoupled representation • Apply rule: for two equivalent electrons, states with odd values of L+S are not allowed. Easy to see problem with 3D – 2p2 & 3D means • s1= ½, s2= ½, S=1 and thus ms1 = ms2 = ½ (ie spins parallel) • l1=1, l2=1, L = 2 and thus ml1 = ml2 = +1 (to get L = 2 requires l1 and l2 to be parallel) • this is not allowed by PEP in decoupled representation (all qn the same) and is thus not allowed PA322 Lecture 8 11 Terms for equivalent electrons in many-electron atoms: full treatment • Working rule to sort out valid terms: – must be equivalence between degeneracies (=number of allowed microstates) in the L-S coupled and decoupled representations • In practice (see previous table for example) – tabulate all the possible (ml, ms) values for each electron, i.e. for decoupled representation – compute the possible values of mS, mL, mJ – compute apparent terms from s and l values in L-S coupling scheme (i.e., S, L, J values) – check that terms are valid by looking to see if appropriate mS, mL, mJ occur in table • if not, term is not allowed! PA322 Lecture 8 12 Terms for equivalent electrons in many-electron atoms • Further issues for terms of equivalent electrons – closed (full) shells have term 1S since 0 • all electrons must be paired, ie. S = Σ si = 0 • L = Σ li = 0 since orientations of li must be such as to give Σmli = 0 – valid terms for subshells of q electrons are the same as for subshells with N-q electrons where N is the closed (full) subshell complement, • e.g., p2 has same terms as p4 as N=6 for p subshells PA322 Possible terms for equivalent electrons ns0 1S ns1 2S ns2 1S np0 1S np1 np2 2P 1S, 1D np3 np4 2P, 2D 1S, 1D np5 np6 Lecture 8 3P 4S 3P 2P 1S 13 Hund’s rules • Now know which terms are possible, but which correspond to ground configuration? • Hund’s rules give the answer: – rules give lowest energy term and level for many-electron atoms – for L-S coupling – for ground state configuration of atom only PA322 Lecture 8 14 Hund’s rules 1. state with highest spin multiplicity has lowest energy ⇒ Smax 2. if more than 1 term with highest spin multiplicity then term with highest L has lowest energy ⇒ Lmax 3. for terms giving more than 1 level: • lowest J has lowest energy if outermost subshell is less than half full ⇒ Jmin if q < N/2 • highest J has lowest energy if outermost subshell is more than half full ⇒ Jmax if q > N/2 PA322 Lecture 8 15 Examples of application of Hund’s rules For le examp • • PA322 Carbon atom: 1s22s22p2 – terms: 1S 3P0, 1, 2 1D – rule 1 selects 3P? – rule 3 selects 3P0 Oxygen atom: 1s22s22p4 – same terms: 1S 3P0, 1, 2 1D – rule 1 selects 3P? – rule 3 selects 3P2 Lecture 8 16 Examples of application of Hund’s rules For le examp Note, this is excited state. Hund’s rules usually (but not always) work in such cases. PA322 Lecture 8 17 (1983), (1985) Application of Hund’s rules PA322 Atom Terms Configuration H (Z=1) 2S 1s1 2S1/2 He (Z=2) 1S 1s2 1S0 Li (Z=3) 2S 1s2 2s1 2S1/2 Be (Z=4) 1S 1s2 2s2 1S0 B (Z=5) 2P 1s2 2s2 2p1 2P1/2 C (Z=6) 1D, 3P, 1S 1s2 2s2 2p2 3P0 N (Z=7) 4S, 2D, 2P 1s2 2s2 2p3 4S3/2 O (Z=8) 1D, 3P, 1S 1s2 2s2 2p4 3P2 Lecture 8 18 j-j coupling • • • • For high Z (and high n at lower Z) the spin-orbit coupling effects for individual electrons become large L-S coupling no longer valid, instead j-j coupling: – j1 = l1 + s1 for electron 1 – j2 = l2 + s2 for electron 2 …. … and finally J = Σ ji Energy levels different, transition rules different, notation different! j-j coupling example: – 6s6p configuration l1= 0 s1 = ½ ⇒ j1 = ½; l2= 1 s2 = ½ ⇒ j2 = 1/2, 3/2 – notation [j1 j2]J ⇒ [1/2 3/2]2, [1/2 3/2]1, [1/2 1/2]1, [1/2 1/2]0 – if j-j coupling transition 6s6p → 6s2: selection rule ΔJ = 0, ±1 (not 0 → 0) – if L-S coupling: ΔS=0 ΔL=0, ±1 ΔJ=0, ±1 (not 0 → 0) PA322 Lecture 8 19 Reading • Softley Chapter 5, sections 5.1 - 5.3 PA322 Lecture 8 20