Section 2 Atomic Spectra (Lectures 2-3 ish)

Transcription

1 Section Atomic Spectra (Lectures -3 ish) Previously: Atomic Structure: IC yr Quantum theory of atoms / molecules H-Atom: Wavefunctions, quantum numbers and energy levels Spectral transitions: Rydberg series and selection rules Spin and spin-orbit coupling Alkali Metal Atoms (e.g., Na): Penetration and shielding, quantum defects Larger spin-orbit coupling He Atom Singlets and triplets Electron correlation General Russell-Saunders and j-j coupling Atoms in external fields

2 Lecture : Atomic Spectroscopy. Revision: The H-atom (& other e - atoms) sin r r r sin r sin Solutions: n,l,m (r, q, j) = R n,l (r)y l,m (, ) Laguerre functions Quantum numbers: Principal quantum number, n Orbital angular momentum quantum number, l Orbital angular momentum projection quantum number, m l (or magnetic ) e.g., n = 3: l = : m l = l = : m l =, ± l = : m l =, ±, ± Spherical harmonics (complex) Permitted values: n =,, 3,... l =,,, 3, 4,... n- m l = -l, -l +,... +l one of 3s orbital three of 3p orbitals five of 3d orbitals

3 .. Atomic Quantum Numbers The principal quantum number, n: Ionization potential determines the energies of orbitals: n.b., E ns = E np = E nd... y y 4 3 and the mean radius 4 Leads to radial wavefunctions / distribution functions: y y P(r) s s p e s s p y(r) has n - l - nodes

4 .. Atomic Quantum Numbers Orbital angular momentum quantum number, l & orb. ang. mom. projection (or magnetic) quantum number, m l The magnitude of the angular momentum come from l: l = ħ {l(l+)} / and the projection of l on the z-axis is provided by m l : l z = m l ħ l z l Lead to the angular wavefunctions and shapes of orbitals (e - wavefunctions): s nd s np (n.b. Real linear combinations plotted)

5 . H-Atom Spectrum: Grotrian Diagrams Selection Rules: Dn unrestricted Dl = ± (Laporte) Dml =, ± absorption Transitions observed at wavenumbers: y Multiple series of lines (different n) Graph of n vs /n is straight with intercept = ionization energy IR UV emission

6 . 3 Origin of selection rules I Selection Rules: Dn unrestricted Dl = ± Dml =, ± A physical picture: Conservation of momentum The photon has intrinsic angular momentum of l ph Total angular momentum must be conserved in the absorption/ emission process vectorially: l l F I But lf is also quantised, l F lf lf I F only specific relative orientations are permitted It is easily shown that the maximum and minimum lf are given by lf max li i.e. l lf min li i.e. l Which is not quite our selection rule... I F l I F l

7 .3 Origin of selection rules II Selection Rules: All dipole allowed selection rules ultimately originate from the need for a non-zero transition dipole moment: F I F I i i Dn unrestricted Dl = ± Dml =, ± i To be non-zero the integrand must be totally symmetric under the symmetry operations of the group (the full rotation group, R3 for atoms). Consider the operation i: An atomic orbital with quantum number l has parity (-)l Since i changes the parity, yf and yi must have opposite parity Hence Dl for reasons of symmetry Dml =, ± arises due to the helicity (s = ±) of the photon

8 .5 More than one electron For any atom other than the H-atom, the Schrödinger equation becomes: i i e i Electron K.E in ij i j Nuclear attractions Electron repulsions The total energy, E, is the energy of all electrons and all interactions in i ij i j The electron repulsion term, Vij, renders the Schrödinger equation insoluble. The wavefunction, y: For one electron: y =y(x, y, z)yspin = y(x, y, z, ms) For two electrons: y =y(x, y, z, (ms), x, y, z, (ms)) In general we need: 3N spatial and N spin coordinates

9 .6 The orbital approximation In order to proceed, assume Yspace is the product of n, one-electron wavefunctions (or orbitals) Yspace= ja(r)jb(r)jc(r3)jd(r4)... jz(rn) For each orbital i e in i j ij i i e- e- repulsion averaged over positions of all other electrons i.e.., electron i experiences the mean field of all other electrons Solve using numerically by e.g., Self Consistent Field approach (Hartree) to yield orbital energies. Including exchange yields Hartree-Fock orbitals. This method is flawed because in order to satisfy the Pauli principle we must really write Yspace as linear combinations of orbitals with defined symmetry with respect to electron permutation. Oh, and it also neglects spin and correlation (see.8,.9). But apart from that...

10 .7 The spectra of alkali metal atoms Ground state electron configurations [RG] ns i.e., closed shell cores We are only interested here in valence excitations (core excitations at higher energies). Within the orbital approximation only one electron can change orbital per photon. Energy Levels: The effects of penetration and shielding Consider the radial distribution functions of 3s, 3p 3p A 3s electron spends more of its time penetrating the core region and experiencing the full nuclear attraction (or equivalently, 3p is more shielded) Na atom 3p 3s 3d 3s H atom n=3 The Energy of a level now depends on n and l quantum numbers with Ens< Enp< End

11 .8 The spectrum of the Na atom S F D P We account for penetration, shielding with a quantum defect, dnl for each level: nl fundamental series sharp series y nl y nl eff y For a given l, dnl is only weakly dependent on n Selection Rules: diffuse series H-Atom energy levels principal series Dn unrestricted Dl = ± Dml =, ± Dj =, ± Absorption: 3s np Emission: ns np np ns nd np nf nd sharp principal diffuse fundamental

12 .9 Spin-Orbit Coupling I A minor effect for the H-atom SOC becomes increasingly important with atomic number The electron has spin quantum number and spin projection quantum number s=½ ms = ±½ Associated with any angular momentum of a charged particle is a magnetic moment via which orbital and spin angular momenta interact leading to spectral fine structure. The orbital angular momentum l has (if l > ) an associated magnetic moment to which the spin angular momentum s can couple, yielding a total angular momentum, j. Vectorially: j The permitted range of quantum numbers, j, are given by a Clebsch-Gordon series: j = l+s, l+s-, l+s-,... l s Hence, for a single electron, j = l ± ½:

13 .9. Spin-Orbit Coupling II: Quantitative The spin-orbit Hamiltonian, j l But Hence: l SO j l l l l Substituting for eigenvalues: l The energy of a given l, s, j> level is l,s, j l l Where A is the spin-orbit coupling constant which, for one electron atoms is given by 3 4 n.b. spin-orbit coupling increases rapidly with atomic number l l Spin-Orbit Coupling constants: H(p) =.43 cm- H(3p) =.7 cm- Li(p) =.3 cm- Na(3p) = 7. cm- K(4p) = 57.7 cm- Rb(5p) = 37.6 cm- Cs(6p) = 554.cm-

14 . Spectral Fine Structure Spin-orbit coupling gives rise to splittings in spectra called fine structure. j = 3/ mj = ± /, ± 3/ j = / mj = ± / l =, s = ½ Selection Rule Dj =, ± (but not ) arises from conservation of angular momentum. D e.g., e.g., 5/ P 4d 3/ 3p D P 3/ / Na S 3s / Na D-lines 7 cm- Na 3p P 3/ P /

15 . Atomic Term Symbols Just a succint notation for the angular momentum coupling in an atom Term symbols contain 3 pieces of information: Spin multiplicity = S+ (Where S is the total spin quantum number for the atom) Gives L, the total orbital angular momentum quantum number for the atom: S: P: D: F: L= L= L= L = 3, etc. 3/ J, the total angular momentum quantum number for the atom. i.e., how L and S are coupled

16 . Atomic Term Symbols: one unpaired electron For atoms with a single unpaired electron: A. s = ½ and therefore S = ½ S+ = and all terms are doublets B. l ( and therefore L) is dependent on the orbital the electron is in: l = (s), (p), (d), give rise to S, P and D terms, respectively C. As we ve seen, J depends on how L and S are coupled but the possible quantum numbers are given by the Clebsch-Gordan series: J = L+S, L+S-, L+S-,... L-S e.g., So, the [Ne]3s ground state configuration of Na is S/ The excited [Ne]3p confguration yields P 3/ and P/ terms. n.b., In the absence of external fields, each J level has (J+) degeneracy arising from mj states: mj 3/ / -/ -3/

17 .3 Atomic Term Symbols: General Russell-Saunders Coupling Term symbols really become useful when dealing with many-electron atoms. In Russell-Saunders Coupling: A. The spins of all electrons, si couple to yield a total spin angular momentum S. The possible values of the corresponding S quantum number arise from a Clebsch-Gordan Series. S = s + s, s + s -,... s s for two electrons B. The orbital angular momenta of all electrons, li couple similarly to yield a total orbital angular momentum L. The possible values of the corresponding L quantum number arise from a Clebsch-Gordan Series. L = l + l, l + l -,... l l for two electrons e.g., coupling of two electron spins to give S=, e.g., coupling of orbital angular momenta l =,:

18 .3 Atomic Term Symbols: General Russell-Saunders Coupling C. L and S can then couple to yield the total angular momentum, J The possible values of the corresponding J quantum number arise from a Clebsch-Gordan Series. J = L+S, L+S-,... L-S e.g., 3D terms S L =, S = S J L 3D 3 L S J 3D L J 3D n.b., the above coupling scheme is appropriate in the limit of weak spin-orbit coupling (small Z) and is known as Russell-Saunders Coupling

19 .4 The spectrum of the He atom The simplest example of a valence electron atom: The ground state is simple: configuration s, electrons spin-paired, hence a S term. This satisfies the Pauli Exclusion Principle as formulated: No two electrons may have identical quantum numbers - Since n, l, ml and s are the same for both electrons they must have different ms. This lies at the heart of the aufbau principle of electronic configurations. Singly excited configurations, snl (n ) are more interesting: Now the electrons may be spin-paired (antiparallel) or spins-aligned (parallel) These differ in their overall spin angular momentum (S = and S =, respectively) denoted singlet and triplet states. Why singlet and triplet? We need to understand the Spin wavefunctions and to do that we must consider the more general form of the Pauli Principle:

20 .5 The Pauli Principle Any acceptable wavefunction must be anti-symmetric with respect to the exchange of two identical fermions and totally symmetric with respect to the exchange of identical bosons Applies to the total wavefunction. We will see several manifestations of this throughout this course but for the exchange of two electrons (Fermions, s = ½): Ytot = YspaceYspin. Consider Yspin for He: Each electron can have either ms = ½ (spin up, or a state) or ms = -½ (spin-down, b) Four combinations can arise: a()a(), b()b(), a()b(), a()b() The first two are clearly symmetric with respect to exchange but the latter two are neither symmetric, a()b() a()b(), nor anti-symmetric, a()b() a()b().

21 .6 The Pauli Principle The two electrons are, however, indistinguishable and we can take linear combinations of these to form eigenfunctions of the S and Sz operators:, and, s+(,) is clearly symmetric with respect to exchange, s-(,) anti-symmetric So our four spin wavefunctions are:,, Of which the st three are symmetric wrt exchange and the final one anti-symmetric. These need pairing with Yspace to give anti-symmetric total wavefunctions Ytot.

22 .7 Picturing spin states SINGLET (spin paired): Complete cancellation of s,s to give S =, S =, Ms = a single arrangement TRIPLET: three arrangements with S = [ S = ()/ħ] S S S Ms= Ms=+ n.b., ms, ms are no longer well-defined Ms=-

23 .8 The Energy of Singlet and Triplet states: Electron Correlation For states arising from the same configuration, the triplet state lies lower in energy than the singlet. To understand why, consider the space wavefunctions: Consider the ss configuration: Yspace= js() js() or js() js() Which, again are neither symmetric nor anti-symmetric wrt exchange. Form linear combinations of these: space s s s s Of which Y+ is symmetric under exchange and Y- anti-symmetric. As the spin wavefunctions for the triplet states are always symmetric wrt exchange these must be paired with Y-. Consider exchange for Y-: space When r = r space s s space s s space which can only be true if space

24 .9 The Energy of Singlet and Triplets: Fermi Heaps & Fermi Holes There is zero probability of finding two electrons in the same region of space if they are described by Y- (and triplet states are). Conversely there is a slight maximum in the probability of finding two electrons described by Y+ in exactly the same place. The electrons are correlated the location of each depends intimately on the other. Fermi Hole Y- Y+ Triplet r = r is maximum electron repulsion (i.e., high energy) Lower energy r-r Fermi Heap Singlet r-r Higher Energy The degeneracy of singlet and triplet is lifted by the electron repulsion. A more detailed treatment (e.g., MQM) shows the energy difference to be DE = K where K is the exchange integral: s s s s

25 . The Grotrian diagram of the He atom (finally) Singlet S s3s P s3p Triplet D s3d 3S s3s sp 3P s3p 3D s3d sp ss ss Absorption: s (S) snp (P) s For each configuration, every triplet state is lower in energy than the corresponding singlet Selection Rules: Single e changes Dn unrestricted Dl = ± (= DL) DS= NEW! Singlet Triplet forbidden Emission: sns (S) sn p (P) sns (3S) sn p (3P) snp (P) sn s (S) snp (3P) sn s (3S) snp (P) sn d (D) snp (3P) sn d (3D) etc.,

26 . Configurations, Terms and Levels A given electron configuration may give rise, within Russell-Saunders coupling, to several Terms (different L, S combinations). These in turn are split by spin-orbit coupling into Levels (unique J) which comprise J+ degenerate quantum states (mj). For example: J+= 3 P P Jz = mjħ mj = J ns np 3P 3P configuration Interactions: 3P 3P terms Electrostatic / Spin correlation levels magnetic states ( in total) - J+ components (degenerate in zero field)

27 . But beware the Pauli principle... Consider the C atom ground state, ssp and which terms arise. Naively: s = ½, s = ½ and thus S =, (i.e., singlet and triplet terms) l =, l = and thus L =,, (i.e., S, P and D terms) i.e., we expect But think carefully about the 3D term: L = hence it must have ML = component which must mean ml = and ml = S = hence it must have MS = component which must mean ms = ½ and ms = ½ But l = l = and n = n = These two electrons would have to have identical quantum numbers which would violate the Pauli Exclusion Principle. It is possible to invoke group theory to tell if a given term is allowed (MQM p38) or we can consider all possible combinations of ML and MS

28 .3 Microstate tables P: l=l= ml = -,, s = s = ½ ms= -½, ½ Draw up a table of every possible combination of ms and ml permitted by Pauli ml - Sml Sms The largest value of Sml = which must correspond to a D term. This only occurs with Sms =. Hence a D term with all it s 5 ML states (ML = ±, ±, ) must exist. Strike 5 such states from the table: ml - Sml Sms - - -

29 .3 Microstate tables ml - Sml Sms The largest remaining Sml = which must correspond to a P term. This occurs with Sms =, ±, i.e., triplet terms. Hence a 3P term with all it s 9 states must exist. ml - Sml Sms Leaving only a S term remaining. Hence a p configuration gives rise to D, 3P and S. Interestingly, a p4 configuration gives the same terms.

30 .4 Energetic ordering: Hund s Rules Friedrich Hund Within Russell-Saunders (LS) Coupling, for a given configuration:.. 3. The term with largest S is lowest in Energy For a given S the term with largest L is lowest in Energy For a term with several levels: if the sub-shell is less than half full the lowest J level is lowest in Energy if the sub-shell is more than half full the highest J level is lowest in Energy Assumes spin-correlation >> orbital angular momm coupling >> spin-orbit coupling Strictly applies only to finding ground states of atoms. As we ve seen, the C atom ground state p configuration yields D, 3P, S terms. Lowest energy term (Rules, ) Lowest energy level (Rule 3) 3P,3P, 3P

31 .5 LS (Russell-Saunders) versus j-j coupling Russell-Saunders (LS) Coupling: l s + + l s + + l3... L s S J Good quantum numbers: L, S, J Selection Rules: Dn unrestricted Dl = ± DL =, ± DJ =, ± (not ) DS=

32 .5 LS (Russell-Saunders) versus j-j coupling But if spin-orbit coupling is strong (large Z) then the spin, s, of an electron prefers to couple to its own orbital angular momentum, l, to give j: magnetic j-j Coupling: Good quantum numbers: l l l ji = li ± ½ J = j + j +..., j + j , etc.. s s s3 n.b. L, S undefined j3... J Selection Rules: Dn unrestricted Dl = ±, Dj =, ± for one electron Dl =, Dj = for other electrons DJ =, ± (not ) Si Ge Sn intermediate Pb j-j j + j + electrostatic e.g., down Group IV: C Coupling: LS

33 .6 Atoms in External Fields: lifting degeneracy As we ve seen S and L give rise to magnetic moments. These can interact with an externally applied B field:.6. The Normal Zeeman Effect (when S = ) In the absence of external fields the 3 ML components of an P term are degenerate. But the magnetic moment associated with the orbital angular momentum can interact with any external B field to lift this degeneracy. Classically, the interaction energy: E = m.b = gelzb [ ge is the gyromagnetic (or magnetogyric) ratio] B Lz ML = L Hence, quantum mechanically: E = geml ħb = mbmlb e e B e = the Bohr magneton lifts the field-free degeneracy resulting in splittings in the spectrum -

34 .6. The Normal Zeeman Effect (when S = ) cont. B Selection Rule: DML =, ± L E = mbmlb B> B= ML = ML - - D P DML = - B Lz ML = L -

35 .6. The Anomalous Zeeman Effect (when S ) Of course the spin angular momentum also gives rise to a magnetic moment and when both L and S are non-zero the splittings are more complex. B Now, E = ge(l + S).B J But we need consider only projections of L and S on B. e.g., J J J e J L S e Where the Landé g-factor, J J g J L,S J J J J S S L L e.g., gj =.5 for a 3P level =. for 3S level (normal spin g-factor) n.b., when S =, J = L and gj = Normal Zeeman Effect

36 .6. The Anomalous Zeeman Effect cont. So the levels are split into J + components but the splitting itself is dependent on J, L, and S. Selection Rule: DMJ =, ± B> B= D MJ 3/ / -/ -3/ 3/ / P / -/ DMJ = - E = -gegjħmjb i.e., splitting proportional to mj, MJ & B