Lecture 8 Title : Fine Structure : multi-electrn atms Page-0 In this lecture we will cncentrate n the fine structure f the multielectrn atms. As discussed in the previus lecture that the fine structure arises due t spin rbit cupling. Here we will discuss abut the spin rbit cupling in case f mre than ne electrn and their effect n terms transitins.
Page- The Hamiltnian fr the multielectrn atms * H = H + H + HSpin Orbit where H where N N * = i + me i= i= U ( ri ) = + N Ze r ( ) U r N i i= i i< j i e r ij J and H N e = r N i< j ij i< j e r H = A l. s Spin Orbit i i ij Nn-Spherical part nly L The Spin-rbit interactin cuples the ttal spin and ttal rbital angular mmentum (L) t frm the ttal angular mmentum J. S The figure-8. represents the vectr diagram and accrdingly we write l s l J = L + S (fr ne electrn atms). s Because f this cupling, the ne electrn wavefunctins Lm Sm which are the eigenfunctins f l s H CentralField are n mre the eigenfunctins f the ttal Hamiltnian H. + i, j e r ij e H = HCF + + AL. S r i, j l m s m Lm Jm l s L J l m s m Figure 8. l s s ij Sm
Page- The eigenfunctins f the H will be Jm J which are the cupled wavefunctins and can be derived frm the uncupled wavefunctins Lml Sm s, as described in previus lectures. Accrding t the cupling f angular mmentum the value f J = J = L+ S,... L S and the ntatin is S + L where L= 0,,,,...are SPF,,, respectively. S the perturbatin energy E = Jm A LS. Jm J J Nw, J = L + S J = L + S + LS. LS. = J L S J And thus, the energy crrectin due t the spin-rbit interactin is A E = JmJ ALS. JmJ = [ J( J+ ) L( L+ ) S( S+ ) ] (8.)
Page- S + is the knwn as multiplicity. Indicates the degeneracy f the level due t spin. If S = 0 => multiplicity is : singlet term. If S = / => multiplicity is : dublet term. If S = => multiplicity is : triplet term and s n. In the fllwing we will discuss the effect f spin-rbit interactin n terms t terms transitins. Singlet levels S = 0; J = L+ S = L A E = J( J + ) L( L+ ) = 0 P P S S 0 Selectin Rules : l =±, L= 0, ±, S = 0, J = 0, ±
Page- ublet levels Transitin frm t P Fr L=, S =, J = 5, Using equatin-8. ( ) ( ) ( ) A E = + + + = A 5 5 5 ( ) ( ) ( ) A E = + + + = A Fr P L=, S =, J =, ( ) ( ) ( ) A A E P = + + + = ( ) ( ) ( ) A E P = + + + = A S the cnstructin f energy levels 0 A A 5 P A A P P
Page-5 It is imprtant t knw the degeneracy lifted by this spin-rbit interactin. The degeneracy fr level is 0 (L+ )(S+ ) = (. + )(. + ) = 0 When this level splits int 5 and, then the degeneracy is 6 because J = 5, m 5,,,,, 5 J =. All m J levels will 5 be degenerate. The m J degeneracy will nt be lifted until the external magnetic field is applied. Similarly, the degeneracy is because J =, m,,, J =. The degeneracy fr P level is 6 (L+ )(S + ) = (. + )(. + ) = 6 When this level splits int P and P P the degeneracy is because J =, m,,, J =. All m J levels will be degenerate. The m J degeneracy will nt be lifted until the external magnetic field is applied. Similarly, P the degeneracy is because J =, m, J =. We will discuss the later the effect f magnetic field n these levels. Transitins : as shwn in the adjacent figure-8. A Fr transitin = 0 + A Fr transitin = 0 A + A A Fr transitin = 0 A 0 Figure 8. In general A > A s transitin pattern will be > >.
Page-6 P transitin Fr term S = L = J =,, S,,, Fr P term S = L = J =,, 0 p, p, p0 Nw, let us calculate the perturbatin energies fr these levels using equatin-8.. We take the spin-rbit cnstants A and A fr and P respectively. A E ( ) = [ 6 ] = A, E ( ) ( ) A E = 6 = A A E ( P) = [ 6 ] = A, E ( P) A ( ) [ ] E P = 0 = A A = 6 6 = A A = = A Transitin energies are = + A A 0 = A A 0 = A + A 0 = A A 0 = A + A 5 0 P 0 5 6 P P = A + A 6 0 P 0
Page- The Observed transitin f Calcium is given in this figure belw. 5 6 Figure-8. Observed transitins Observed transitins (nm) Observed transitins (cm - ) Calculated transitins (cm - ) = + A A 5. 8.0.6 0 = A A 5.5.6.6 0 = A + A.50 5. 5. 0 = A A 5.66 8.6 8.6 0 = A + A.5 5.6 5.8 5 0 = A + A.5 56.68 5.6 6 0
Page-8 Frm transitins and E0 A A =.6 E0 A + A = 5. A = 5..6 = 05. A = 5.85 cm Frm transitins and E0 A A = 8.6 E0 A A =.6 A =.6 8.6.5 A = =.65 cm And 0 = 6.8 cm The ther transitins calculated frm these values are given in the table.
Page- Let us take anther example, the transitins f Cpper [d sp] F [d s5s] Fr term S = L = S, J =, 5,,, 5,, Fr F term S = L = J =,, 5, F, F, F5, F Fr F A 5 EJ ( = ) =... A = A 6 6 EJ= = = E ( ) [ ] 0 5 A 5 6 A 8 ( J = ) = [ ] =. = A A 5 6 A 8 E( J = ) = [ ] =. = 6A Fr A 6 A E ( J = ) = [ ] =. = A 5 E J = ) = A 5 [ ] = A. = A ( A 5 A E( J = ) = [ ] =. = A E A A 6 ( J = ) = [ ] =. = A
Page-0 The cnstructed energy level diagram is shwn in this figure belw. F F F F 5 F 0 5 Selectin rules: derived frm Wigner-Eckart therem discussed earlier l =±, S = 0, J = 0, ±. The J = 0 J = 0is frbidden. The transitin energies are () v(f ) = A + A () v(f ) = A A () v(f 5 ) = + () v(f5 ) = A A
Page- A (5) v(f5 5 ) = A+ (6) v(f5 ) = A+ A A () v(f 5 ) = 6A+ (8) v(f ) = 6A+ A () v(f ) = 6A+ A The experimental set up fr recrding the cpper F spectrum. Cu electrdes Cu arc Fiber ptic cupled spectrmeter High vltage pwer supply The cpper F spectrum. Intensity (arb. units) 80000 60000 0000 0000 65. 5.8 60.86 58. 0.5 5.0 Cpper (F -> ).0 0 50 60 0 80 0 500 Wavelength (nm)
Page- Assigning the bserved three transitins as fllws. Transitins Observed wavelength (nm) Observed transitin energy (cm - ) F 58.0. 5 F 65.. F 0.5 50.8 S we have three equatins belw t slve fr btaining three parameters. A + A =. A = 50.8 A + =. Frm these abve equatins, we get A + =. A = 50.8 A = 50. 50.X 5. A = = cm = 50.8 + *5. = A =. 50.8 =..* A 5. = = cm
Page- Transitins Transitin energy Calculated transitin energy (cm - ) Calculated transitins (nm) bserved transitins (nm) F A + A. 65. 65. F A 50.8 0.5 0.5 A F 5 +. 58.0 58.0 F 5 A A 06..8.0 A F5 5 A + 60.5 6. 60.86 F 5 A + A 8.5 5.80 5.8 A F 5 6A + 6.8 65.88 ----- F 6A + A 850.8 5.65 ------ F 6A + A 08. 5.8 5.0
Page- Summary f Atmic Energy levels Grss structure f the atmic energy levels: It cvers largest interactins within the atm: (a) Kinetic energy f electrns in their rbits. (b) Attractive electrstatic ptential between psitive nucleus and negative electrns (c) Repulsive electrstatic interactin between electrns in a multi-electrn atm. These interactins give energies in the -0 ev range and upwards. It determine the spectrum range whether a phtn is IR, visible, UV r X-ray. Fine structure: Spectral lines ften cme as multiplets. E.g., Hα line. This smaller interactins within atm, called spin-rbit interactin. The rigin f this interactin is Electrns in rbit abut nucleus give rise t magnetic mment which interacts with spin f the electrns. This intrduces splitting f the energy levels and prduces small shift in energy. Hund s rule : () Of the terms given by equivalent electrns, thse with greatest multiplicity lie deepest, and f these the lwest is that with the greatest L () Multiplets frmed equivalent electrns are regular when less than half the shell is ccupied, but inverted when mre than half shell is ccupied.
Page-5 Recap In this lecture we have seen the effect f spin rbit cupling n the fine structure f the multielectrn atms. We have discussed the transitins between the varius spin multiplicity energy levels, smetime called as multiplets. We have als gne thrugh the experiments in the cpper transitins and als learnt hw t assign the transitins. We als learnt hw t evaluate the spin-rbit cnstant term fr a level frm the bserved transitins.