Lecture #7: Many e Atoms. energly levels eigenstates transition intensities

Transcription

1 5.76 Lecture #7 //94 Page Outline of approach: Configuration which L S J states? H eff [ζ, F k, G k ] alternative Limiting Case Coupling Schemes L S J vs. j j J off-diagonal ζ vs. F Lecture #7: Many e Atoms energly levels eigenstates transition intensities. list of orbital occupancies e.g. s s p. Which L S terms result? * list all spin-orbitals p: 0 0 (overbar means m S = / or β ) * list all Slater determinants (anti-symmetrized microstates) consistent with Exclusion Principle e.g. M L = M S = 0 standard order of spin-orbitals * Classify Slaters according to M L and M S * Method of crossing out of microstates: M L, M S ranges for each L S term. Find the linear combinations of Slaters that are eigenstates in either representation: LM L SM S OR LSJMJ J These cause H eff to be block diagonalized. 4. Compute matrix elements of H in selected basis set. SO * H = ξ( r i ) i s i ME of e operator z nl ζ(nl ) * = e r ME of e operator f k F k (nl, n l ) + g k G k (nl, n l ) H ee i> j ij nl k,nl, n l H eff expressed in terms of fit parameters: spin-orbit ζ direct exchange F k G k ζ, F k, G k orbital radial integrals fit parameters z n, f k, g k are exactly calculable ANGULAR INTEGRALS Would take or lectures to explain how to compute: * e and e operator matrix elements of Slater determinants * how to work out z n, f k, g k factors My goal here is to expose you to the atomic H eff models.

2 5.76 Lecture #7 //94 Page Often the relevant /r ij and spin-orbit matrices can be found in a book like Condon and Shortley. Read Tinkham pages for methods of evaluating these matrix elements yourselves. Examine the n = and n = levels of H atom explicitly and compare against the observed spectrum. n = can have = 0 and n = has l = 0,, j = / j = / and / n j s E /hc H SO /hc j = 5/, / 0 / / / / / / / / / / / / / / / / See Figures Energy Level Diagram Spectrum (Theoretical from C & S) Compare P / P / energy level splitting for hydrogenic systems vs. Z H I 0.66 cm Li III 0 cm Na XI 5400 cm

3 5.76 Lecture #7 //94 Page Energy level Diagram for H I n= and n = n = / s p 5/ / / / d f g d e a b c n = / / / none Transition Label CALC OBS energy order a = = b = = c = = highest d = = " e = = f = = g = = lowest parity selection rule = ± vector coupling = j favored (propensity rule)

4 5.76 Lecture #7 //94 Page 4 Image of The Theory of Atomic Spectra by E.U. Condon and G.H. Shortley removed due to copyright restrictions. Figure removed due to copyright restrictions. Please see: Condon, E. E. and Shortley, G. H. The Theory of Atomic Spectra. Cambridge, UK: Cambridge University Press, 95. This illustrates the spin-orbit fine structure of the H atom n = n = spectrum. Transitions are labeled (a) (g) following the table on the previous page. The lengths of the lines represent the calculated relative intensities (assuming equal populations in all m j components of the n = levels). Orbital-Based Periodic Correlations Lowest L S J state? Center of gravity of entire configuration? Degeneracy d 0 D d 9 (L + )(S + ) = 0 d 0 9 = 45 G, F, D, P, S d 8 d = 0 H, G, 4 F, F, D, D, 4 P, P d 7 d = 0 incredible d6 d = 4 unbelievable d5 Yet all is given by ζ(nd), F 0, F, F 4 (no G k s for p or any set of identical n orbitals) Massively complicated spectra for d series. No corresponding states for d N d N + Magic decoder is ζ, F 0, F, F 4 in effective Hamiltonian. We know how each of these parameters should scale vs. Z for isoelectronic series or across row as Z Z +, or as N N + vs. n for Rydberg series Example ζ(d) for d N 4s of Sc Cu Goes as Z 4 eff (imperfect shielding of one d by others) as Z increases. The plot of ζ(ls) is for the lowest L S term (MAX S, MAX L)

5 5.76 Lecture #7 //94 Page 5 Figure removed due to copyright restrictions. Please see: Figure 6. in Moore, C. E. Atomic Energy Levels. Natl. Bur. Standards, Circ Vols. I and II, 949 and 95. property of d orbital eff ( Z d ) 4 eff scaling. Z increases in steps of 0.8. property of lowest L S term of (d) (4s) configuration. No scaling. Hund s rd Rule. p Example LS jj Coupling p configuration D, P,,0, S 0 See Condon and Shortley, pages 98, 68, 74-5, 94

6 5.76 Lecture #7 //94 Page 6 The Russell-Saunders Case: Energy Levels removed due to copyright restrictions.

7 5.76 Lecture #7 //94 Page 7 H ee ( p )= H SO ( p )= S 0 D P P P 0 S 0 P 0 P D P F 0 +0F F 0 + F F 0 5F F 0 5F F 0 5F J = 0 J = J = 0 / / / / / / ζ( np) 5 5 is fully diagonal 5 5, in d ifferent order than above H ee, factors into,, and Add H ee + H SO to get secular equations for J =,, 0. These matrices were evaluated in L S J basis set. Could have used LM L SM S. More work, same results. To get secular equations into most convenient form * subtract out center of gravity (C of G) * put into form ² E V = 0 ² V E Eigenvalues E ± = ±[ /4 + V ] / J = 0 5 S 0 F o +0F / ζ P 0 / ζ F o 5F ζ = F + 5 o F ζ + F + ζ / ζ 5 sym F ζ So E ± (J = 0) = F F 5 ζ± 4 F + 5 / F ζ+ 4 ζ + ζ J = E(J = ) = F 0 5F - ζ/ J = F D 0 + F / ζ P / ζ F 0 5F + ζ = F 0 F + F 4 ζ+ 4 ζ / ζ / ζ F + 4 ζ E ± (J = ) = F 0 F + 4 ζ± 9F F ζ+ 6 ζ + / ζ

8 5.76 Lecture #7 //94 Page 8 Note that these matrices have ζ off-diagonal and E differences dominated by F. There are two convenient limits for intraconfigurational energy level patterns. For p : ζ = 0 L S coupling J = L + r S F = 0 j j coupling J = j + j 0F S ζ (/, /) 0 (/, /) fine 9F /ζ F D coarse structure /ζ (/, /) (/, /) fine 6F /ζ 5F P 0,, fine structure ζ (/, /) 0 possible values for j and j for p =, s = / are / and / total possible values of J degeneracy 4 (/, /) 0 4 (/, /) (/, /) 0 exclusion principle exclusion NOT OBVIOUS exclusion

9 5.76 Lecture #7 //94 Page 9 LS (j, j ) J Coupling Patterns LS Ge(4p) Pb(6p) S 0 S 0 S 0 ( P 0 ) jj S 0 ~ P 0 D ~ P electrostatic spin-orbit F = 07 cm ζ = 880 cm F = 9 cm ζ = 794 cm (/, /) 0 and (/, /) D ( P ) D P ~ D P D (/, /) and (/, /) P ( D ) P P P P 0,, P 0 ( S 0 ) P 0 ~ S 0 P 0 See Condon and Shortley, pages 74-5, plotted on scale to keep E MAX E MIN = constant. (/, /) 0 S 0 P 0 P D P F 0 +0F / ζ Matrix in L S J BASIS SET / ζ F 0 5F ζ F 0 5F ζ/ F 0 + F / ζ / ζ F 0 5F +ζ/

10 5.76 Lecture #7 //94 Page 0 Perturbation Theory in ζ/f limit o S 0 = E S o P 0 = E P o P = E P o P = E P ζ + 5F +ζ ζ ζ/ +ζ/+ ζ 5F +ζ Landé Interval Rule ζ 6F ζ negligible Perturbation Theory works when perturbation theory if / ζ 5F +ζ H ij E i o E j o and. Thus, our L S J basis set matrix for H is suitable for / ζ 6F ζ are J = 0 J = (i.e. ζ F ) Alternatively, we can transform to the jj basis set using the transformation given on page 94 of Condon and Shortley. ( /,/ ) = / P + / ( /,/ ) = / P ( /,/) = P ( /,/ ) = 0 / S 0 / ( /,/ ) = / 0 S 0 + D / D P 0 / P 0

11 5.76 Lecture #7 //94 Page And the matrices are given by H ( /,/),( /,/) = H + + / P P H D D H P D H ( /,/),/,/ ( ) = H + / P P H D D H P D H ( /,/),( /,/) = H P P H D D H P D MATRICES TRANSFORMED TO j j J BASIS SET F +ζ / F (/, /) H (J = ) = / F F ζ (/, /) H (J = ) = 5F ζ/ (/, /) 5F +ζ 5 F (/, /) 0 H (J = 0) = 5 F ζ (/, /) 0 Goes to limiting j j pattern when F 0. Note: ζ is off-diagonal in L S J basis F is off-diagonal in j j J basis

12 5.76 Lecture #7 //94 Page Transitions p sp (j, j ) J S 0 (/, /) 0 (/, /) ² = ± ²J = ²L strong ²S = 0 ²j = ² J or ²j = ² J D P ²J = 0, ± but J = 0 (/, /) (/, /) P P 0 (/, /) 0 (/, /) P (/, /) P P P 0 (/, /) (/, /) 0 p p s active e = ± j = 0 J = L strong j = 0, ± S = 0 J = 0, ± J = 0 / J = 0 J = j strong

13 5.76 Lecture #7 //94 Page Intermediate coupling text removed due to copyright restrictions.

14 5.76 Lecture #7 //94 Page 4 Table of matrices of spin-orbit interaction removed due to copyright restrictions.

15 5.76 Lecture #7 //94 Page 5 Transformations in the theory of complex spectra text removed due to copyright restrictions.