Prof. Dr. I. Nasser atomic and molecular physics -551 (T-112) February 20, 2012 Spin_orbit.doc. The Fine Structure of the Hydrogen Atom

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1 Pof. D. I. Nasse atomic ad molecula physics -55 (T-) Febuay 0, 0 Spi_obit.doc The Fie Stuctue of the Hydoge Atom Whilst the pedictios of the quatum model of hydoge ae a vey good appoximatio to eality, it tus out that i high esolutio specta of hydoge that the pedicted lies ae i fact split ito sets of lies. This is the so called fie stuctue of hydoge ad meas that we must have missed out somethig fom the model we have witte dow. Whe we wote dow the quatum model of the hydoge atom we used the Schödige equatio. The Schödige equatio is the quatum equivalet to Newto s equatio of motio i as much as it is oelativistic. Just as with Newto s equatios o-elativistic quatum mechaics is a good appoximatio ude may cicumstaces. Howeve, it is kow to fail ude othe cicumstaces. The extesio of quatum mechaics to make it elativistic was made by Diac who eplaced the Schödige equatio with the Diac equatio. We will ot ty to solve the Diac equatio. Istead, as i the small velocity limit the Diac equatio teds towads the Schödige equatio, we will teat the diffeece betwee the Diac ad Schödige equatios as a seies of petubatios to the Schödige equatio. Fo histoic easos the diffeet petubatios have bee amed. Spi-Obit Couplig A chage movig ceates a cuet. A cuet ceates a magetic field. Fom the electo s poit of efeece, the ucleus appeas to be movig; theefoe, the ucleus ceates a magetic field. Recall the electo with its spi has a magetic momet. - i.e., it behaves like a ba maget. The magetic momet of the electo ties to alig itself with the magetic field that comes via the obitig of the ucleus. This effect is the oigi of spi-obit couplig. Ze µ 0 Eso s l ξ ( ) s l 8πme whee s is the spi agula mometum ad l is the obital agula mometum. µ 0 - pemittivity of fee space. The spi-couplig tem affects the eegy levels of the electos i a atom. Sometimes the eegy levels that wee thought to be degeeate ae split ito sepaate eegy levels because of spi-obit couplig, e.g., p z α ad p z β ae ot degeeate whe spi-obit couplig is take ito accout. Commet: We have to teat the tem E so as petubatio o the hydoge atom states. The fist thig to ote is that the petubatio beaks two fudametal symmeties of the Hamiltoia. These ae the symmeties associated with chagig the sig of the agula obital mometum ad the sig of the spi agula mometum. I the ew model, chagig the sig of oe o the othe o thei ow chages the sig of the petubatio, ad it is oly if both ae chaged simultaeously that the eegy is uchaged. This less fomal agumet is bo out by the fact that the petubatio o loge commutes with the opeatos associated with the z compoets of the obital o spi agula mometa. This symmety beakig meas we eed to seek the coect combiatio of the degeeate states give by the oigial simple calculatio to use i the petubatio calculatio. It tus out that we have to ivet a ew type of agula mometum called the total agula mometum which is give by J L S J + L+ S L + S + L S ( ) Coside the Hamiltoia Z Z e H H + H + ζ( ) L S, ζ( ) m m c H.W. Examie the followig elatios: 0 SO

2 Pof. D. I. Nasse atomic ad molecula physics -55 (T-) Febuay 0, 0 Spi_obit.doc ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ z z z z z z S, L L, L L, S L, S S, L S, S 0 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ H,,,,, ˆ, ˆ o L Ho S Ho L z Ho S z Ho J Ho J z 0 L S, L L S, S L S, J L S, J z 0 But L S, L z 0, L S, S z 0 H.W. discuss the oigi of the tem ξ ( ). Coclusios: - Fo oe o -electos system, both ucouple epesetatio, m, s, m, m s, m ad the coupled epesetatio,,, i i i si i i i si s m eigefuctios of H o ad it is immateial which epesetatio is used. -, m, s, ms will ot ecessaily be a eigefuctios of LS. - L, S, J, mj is a simultaeous eigefuctios of ˆ J, ˆL, ˆ S ad LS. ae Tem Symbols Defiitios Total Spi Agula Mometum S the sum of spi agula mometa fo all of the electos i the atom. S s + s + s + S s - spi agula mometa added as vectos. i i Multiplicity the umbe of possible spi states fo a atom. Total spi of siglet 0 multiplicity Total spi of doublet / multiplicity Total spi of tiplet multiplicity I geeal, oe ca calculate the multiplicity if oe kow the total spi ad vice vesa multiplicity S + Total Obital Agula Mometum L the sum of obital agula mometa fo all of the electos i the atom. L l+ l + l + L i - obital agula mometa added as vectos. i Fo two electos, L l+ l, l+ l, l+ l,, l l +, l l Total agula mometum J the sum of the idividual total agula mometa (Spi ad obital) fo all of the electos i the atom The total agula mometum, J, comes fom the vecto additio of L ad S. J L + S Thus J L+ S,L+ S,L+ S,, L S The magitudes ae give by: L L(L + ), S S(S + ), J J(J + ) Ad thei z-compoet ae quatized accodig to:

3 Pof. D. I. Nasse atomic ad molecula physics -55 (T-) Febuay 0, 0 Spi_obit.doc Lz M L, Sz M S, Jz MJ The z-compoet quatum umbes ae elated to those of the idividual electos by the followig additio ules: M (m ), M (m ), M M + M KowigM, L M S, ad L i S s i J L S i i M J, oe ca ife L, S, ad J fom the coditios: ML L,L, 0,, ( L ), L MS S,S, 0,, ( S ), S M J,J, 0,, J, J J ( ) Tem symbols ae abbeviated way of descibig a electoic state. Detemiig the electoic state is impotat because diffeet states will have diffeet eegies. The diffeeces i the eegies come pimaily fom couplig of agula mometa such as spi-obit couplig. The tem symbol has the followig ifomatio..) multiplicity.) total agula mometum, J.) total obital agula mometum, L Multiplicity S+ L J Total obital agula mometum Total agula mometum The obital agula mometum is give as lette symbol. L etc symbol S P D F G H etc Selectio ules fo the allowed tasitios betwee eegy levels Chage of the picipal quatum umbe: 0, ±, ±, ±, Chage of the azimuthal (obital) quatum umbe: L 0, ± Chage of the Spi quatum umbe: S 0 Chage of the total agula quatum umbe: J 0, ± (The tasitio J 0 J 0 is ot allowed) Chage of the magetic quatum umbe MJ 0, ± (But if J 0, the the tasitio MJ 0 MJ 0 is ot allowed) These ae electic dipole tasitios; othe tasitios (quadupole, octupole,..) occu but they ae foud to be much weake. If the electic dipole tasitio ivolves ust oe electo, L 0. The tasitios with 0 ae vey weak because they equie a evesal of the diectio of the spi elative to the obital agula mometum of the electo. But the foce that would poduce such spi evesal is the spi-obit iteactio, which is a elatively weak foce.

4 Pof. D. I. Nasse atomic ad molecula physics -55 (T-) Febuay 0, 0 Spi_obit.doc Example: Calculate the allowed values of fo p-electo. Aswe: Fo p-electo we have, the the allowed values of fo a p-electo ae: l + s + l s Eegy levels with Eegy levels with tem s l tem eglected s l icluded Example: Calculate the possible values of LS fo a p-electo. Aswe: Use the elatio: J L+ S J ( L+ S) L + S + L S L S J L S [ J(J + ) L(L + ) S(S + ) ] Fo J L S ( ) ( ) ( ) Fo J L S ( ) ( ) ( ) H.W. Calculate the value of ξ ( ) i the equatio: Es ξ L S ( ) The followig figue illustates the elative oietatio of the thee vectos. fo the two sodium D lies ( A o, A o ). Remembe that the uits of is ev. s 7 [As: ξ ( ).8 0 ] ev. s 4

5 Pof. D. I. Nasse atomic ad molecula physics -55 (T-) Febuay 0, 0 Spi_obit.doc Note: The tasitio p s will be allowed but with low itesity compaed with the tasitio p s ; Itesity (s p) Itesity (s p) 0.66 Itesity (s p) / Itesity (s p) /

6 Pof. D. I. Nasse atomic ad molecula physics -55 (T-) Febuay 0, 0 Spi_obit.doc H.W. Fo a electo i P-state, calculate the matix elemets of the opeato: H () a L ˆ. S ˆ, a costat i- i the couplig epesetatio. Fo couplig states, m, use Lˆ. Sˆ { Jˆ Lˆ Sˆ } ( LS ˆ. ˆ ),,,,,,,,,,,, b- I the ucouplig epesetatio, l m ms, use Lˆ. Sˆ Lˆ ˆ ( ˆ ˆ ˆ ˆ zsz + L+ S + L S+ ) 6

7 Pof. D. I. Nasse atomic ad molecula physics -55 (T-) Febuay 0, 0 Spi_obit.doc ( LS ˆ. ˆ ),,,, 0, 0,,, 0, 0, ,,

8 Pof. D. I. Nasse atomic ad molecula physics -55 (T-) Febuay 0, 0 Spi_obit.doc Oe-electo coectios of eegy The Hamiltoia p Z H o m has the ucoupled wave fuctio, m, s, ms, m s, ms which idetify the agula ad spi pats of the wave fuctio. m is the poectio quatum umbe associated with ad m s is the poectio quatum umbe associated with s satisfies the elatios: ˆ, m, s, m L, m, s, m ( + ) δ δ δ δ s s ss mm mm s s m s m Lˆ m s m m δ δ δ,,,,,, s z s ss mm mm s s m s m Sˆ m s m s s +,,,,,, ( ) s s ss mm mm s s m s m Sˆ m s m m δ δ δ,,,,,, s z s s ss mm mm s s δ δ δ δ Aslo, the wave fuctio, s,, m i LS-couplig has simila elatios: ˆ, s,, m L, s,, m ( + ) δ δ δ δ ss m m ˆ + δ δss δ δ m m ˆ + δ δss δ δ mm ˆ z δ δss δ δ m m, s,, m S, s,, m s( s ), s,, m J, s,, m ( ), s,, m J, s,, m m I which J L + S, ad ˆ ˆ ˆ Jˆ Jˆ ˆ ˆ ˆ ˆ. ˆ ˆ ˆ ˆ ˆ ˆ ˆ x + J y + Jz L + S + L S L + S + LzSz + L+ S + L S+, Note that, s,, m ae ot eigefuctios of L ˆz o S ˆz., s,, m coupled epesetatio. δ δ ae said to be i the Example: Fo oe-electo atoms wok out the eegy coectios due to the followig tems: elativistic ( H ˆ ), spi obit ( H ˆ LS ), Dawi ( H ˆ D ). Give a example of the cotibutios of each tem to the splittig of the level of hydoge. Hˆ Hˆ ˆ ˆ ˆ o + H + H LS + H D; ˆ 4 ˆ P H, 8mc ˆ ˆ ˆ Z e H ( ). ˆ. ˆ LS ξ LS LS, mc ˆ Ze Ze H D ( ) [ 4 πδ ( )] mc mc 8

9 Pof. D. I. Nasse atomic ad molecula physics -55 (T-) Febuay 0, 0 Spi_obit.doc Spi-Obit Couplig ad Fie stuctue This is explai the easo that the eegy level with > 0, split ito two compoets, which could ot be explaied by the Schödige theoy. Sice this splittig is vey small ad ca be oly esolved with high esolutio spectogaphs, whee the hydoge lies appea as a fie substuctue, it was amed fie stuctue. ˆ Ze ˆ Z α E H sˆ., { } LS LS ) mc a o ( + ( + ) 4 Z e ( + ) ( + ) s( s + ) { } 4 amc o ( + ) ( + ) 4 α Z ( + ) ( + ) s( s + ) { } Ry ( + ( + ) 4 α Z + E LS { } ( + ( + ) ( + ) spi dow whee α is the fie stuctue costat ad is give by: e 5 α α 5. 0, c 7 Ad ) spi up ) e - o a, Ry mc.605 ev 09,77. cm me c 4 4 α Z 5 Z ELS ELS ( + ) E ( ) 5. 0 Ry LS ( + ) ( + ) Example: Fo the p levels of the H atom, we have Z, ad. Fom the above expessio we theefoe obtai fo the fie stuctue splittig as: E(Ry) E Ry(cm ) (Ry) / / -0. / / / / LS.4E-6 (0.) -.9E-6 (-0.4) E LS E LS.E E-7 0. (cm ) 4.6E

10 Pof. D. I. Nasse atomic ad molecula physics -55 (T-) Febuay 0, 0 Spi_obit.doc This shows that: - Fie stuctue splittig ae obseved oly fo levels with > 0, i.e. fo p, d, f, levels, ot fo s-levels with 0. - The splittig is vey small compaed to the eegy of the levels ad ustifies the ame fie stuctue. - The fie stuctue splittig deceases with iceasig quatum umbes ad, but it iceases popotioally to Z, o the poduct Z E. Sice the eegies E of the levels with picipal quatum umbe follow the elatio E Z /. 4- The eegy levels split, depedig o the oietatio of the spi, ito the two compoets with + ad. 5- The fie stuctue may be egaded as Zeema splittig due to the iteactio of the magetic spi momet with the iteal magetic field geeated by the obital motio of the electo. Relativistic Mass Coectio The Relativistic Coectio Movig fom the o-elativistic fomula fo the eegy of a electo to the elativistic fomula we make the chage Taylo expadig the squae oot aoud, we fid 0

11 Pof. D. I. Nasse atomic ad molecula physics -55 (T-) Febuay 0, 0 Spi_obit.doc So we have ou ext ode coectio tem. Notice that was ust the lowest ode coectio to. What about the ``educed mass poblem? The poto is vey o-elativistic so oly the electo tem is impotat ad the educed mass is vey close to the electo mass. We ca theefoe eglect the small coectio to the small coectio ad use Accodig to special elativity, the kietic eegy (i.e., the diffeece betwee the total eegy ad the est mass eegy) of a paticle of est mass m ad mometum p is 4 4 p p T p c + m c mc m 8m c + I the o-elativistic limit p<< mc, we ca expad the squae-oot i the above expessio to give p p T + m 4 mc The secod tem i the expasio of the kietic eegy is called the elativistic mass coectio. Notes: Neglectig the uclea motio, the Hamiltoia with elativistic coectio of the Hydoge atom is give by: ˆ ˆ ˆ ˆ ˆ P Ze P H Ho + H me mec me With the otatio: ˆ ˆ P Ze (0) H o E me

12 Pof. D. I. Nasse atomic ad molecula physics -55 (T-) Febuay 0, 0 Spi_obit.doc Put Usig The Pˆ me (0) Ze E +, whee E (0) Z Ry, the the fist ode coectio will be: ˆ ˆ P E lm H lm lm lm mc e m e (0) Ze (0) Ze + + mc e lm E E lm ; (0) Ze (0) Ze (0) (0) Ze Ze (0) Ze E + E + E + E + E + E [ lm E lm + ( Ze lm E lm (0) (0) mc e (0) + + ( ) lm E lm ) Ze lm lm ] Fo the tems; the fist oe will be: (0) (0) lm E ( ) lm E δm m The secod ad thid tems will be: (0) (0) (0) lm E lm lm E lm E l l δ The fouth tem will be: lm lm l l δ mm The fial coectio is: (0) ( E ) Ze (0) ( Ze ) E E l l l l mc e mc e mc e But we kow: Z Z l l Ry, l l Ry a o ao ( l + ) Theefoe: 4 Z α E [ ] Ry 4 ( + ) m m Commets: - The elativistic eegy shift is maximum fo the goud state of atoms ( ad 0 ). - The coectio depeds o both quatum umbes ad. The ( ) -fold degeeacy of states(,), deduced fom the Schödige theoy is lifted by the elativistic coectio.

13 Pof. D. I. Nasse atomic ad molecula physics -55 (T-) Febuay 0, 0 Spi_obit.doc - At a give value of, the electo comes closest the ucleus (ad theefoe acquies the lagest velocity fo small values of (the Sommefeld obits ae the ellipses with lage ecceticity). The icease i elativistic mass is the becomes maximum, which deceases the eegy tem value. Fo the maximum allowed ( ) the obit is cicula ad the velocity of the electo has a costat medium value. the elativistic mass coectio is the miimum. Example: Fo H atom with, 0 the magitude of the elativistic coectio is: E E α ev 7. cm 4 Fo, 0 we obtai: E Eα.5 0 ev.0 cm 6, we obtai: 7 E Eα.6 0 ev 0. cm 48 Fo As the umeical examples show, the elativistic mass coectio oly amouts to less tha 0-4 of the Coulomb eegy. l E(cm ) 0 / -/4 0 / -5/64 /,/ -5/64, -/64 Sum of Relativistic Fie Stuctue Coectios Take togethe the thee elativistic coectios discussed above give a oveall fist ode eegy shift of

14 Pof. D. I. Nasse atomic ad molecula physics -55 (T-) Febuay 0, 0 Spi_obit.doc Dawi Coectio E E 4 α Z + ELS { }Ry, + 4 The thid coectio has o classical aalogue is due to the highly o-classical dyamics of electos ea the ucleus. It ca be show that to fist ode it gives a coectio to the eegy of l 0. ˆ Ze Ze H D ( ) [ 4 ( )] πδ mc mc Z δ( ) 00 δ( ) 00 ψ(0) π ao 4 ˆ α Z ED H D Ry 4

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