Lectures 5-6: Magnetic diple mments Sdium D-line dublet Orbital diple mments. Orbital precessin. Grtrian diagram fr dublet states f neutral sdium shwing permitted transitins, including Na D-line transitin at 589 nm. Spin-rbit interactin. Stern-Gerlach experiment. Ttal angular mmentum. Fine structure, hyperfine structure f H and Na. The Lamb shift. D-line is split int a dublet: D 1 = 589.59 nm, D 2 = 588.96 nm. Many lines f alkali atms are dublets. Occur because terms (bar s-term) are split in tw. This fine structure can nly be understd ia magnetic mments f electrn. Na D-line Orbital magnetic diple mments Orbital magnetic diple mments Cnsider electrn ming with elcity () in a circular Bhr rbit f radius r. Prduces a current µ l Magnetic mment can als be written in terms f the Bhr magnetn: where T is the rbital perid f the electrn. Current lp prduces a magnetic field, with a mment L r -e where g l is the rbital g-factr. Gies rati f magnetic mment t angular mmentum (in units f h). In ectr frm, Eqn 2 can be written (1) Specifies strength f magnetic diple. Magnitude f rbital angular mmentum is L = mr = m!r 2. Cmbining with Eqn. 1 => (2) An electrn in the first Bhr rbit with L = h has a magnetic mment defined as = 9.27x10-24 Am 2 Bhr magnetn As The cmpnents f the angular mmentum in the z-directin are L z = m l h where m l = -l, -l+1,, 0,, +l+1, +l. The magnetic mment assciated with the z-cmpnent is crrespndingly
Orbital precessin Electrn spin When magnetic mments is placed in an external magnetic field, it experiences a trque: (3) which tends t align diple with the field. The ptential energy assciated with this frce is Electrn als has an intrinsic angular mmentum, called spin. The spin and its z-cmpnent bey identical relatins t rbital AM: Minimum ptential energy ccurs when µ l!!b. where s = 1/2 is the spin quantum number => If "E = cnst., µ l cannt align with B => µ l precesses abut B. Therefre tw pssible rientatins: (4) => spin magnetic quantum number is ±1/2. But frm Eqn. 3, Fllws that electrn has intrinsic magnetic mments: ˆ S Setting this equal t Eqn. 4 => Called Larmr precessin. Occurs in directin f B. Larmr frequency where g s (=2) is the spin g-factr. ˆ µ s The Stern-Gerlach experiment The Stern-Gerlach experiment This experiment cnfirmed the quantisatin f electrn spin int tw rientatins. Cnclusin f Stern-Gerlach experiment: Ptential energy f electrn spin magnetic mment in magnetic field in z-directin is With field n, classically expect randm distributin at target. In fact find tw bands as beam is split in tw. The resultant frce is There is directinal quantisatin, parallel r antiparallel t B. Atmic magnetic mment has µ z = ±µ B. As g s m s = ±1, The deflectin distance is then, Find same deflectin fr all atms which hae an s electrn in the utermst rbital => all angular mmenta and magnetic mments f all inner electrns cancel. Therefre nly measure prperties f uter s electrn. The s electrn has rbital angular mmentum l = 0 => nly bsere spin.
The Stern-Gerlach experiment Spin-rbit interactin Experiment was cnfirmed using: Fine-structure in atmic spectra cannt be explained by Culmb interactin between nucleus and electrn. Element Electrnic Cnfiguratin H 1s 1 Na {1s 2 2s 2 2p 6 }3s 1 K {1s 2 2s 2 2p 6 3s 2 3p 6 }4s 1 Cu {1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 }4s 1 Ag {1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4p 6 4d 10 }5s 1 Cs {[Ag]5s 2 5p 6 }6s 1 Au {[Cs]5d 10 4f 14 }6s 1 Instead, must cnsider magnetic interactin between rbital magnetic mment and the intrinsic spin magnetic mment. Called spin-rbit interactin. Weak in ne-electrn atms, but strng in multi-electrn atms where ttal rbital magnetic mment is large. In all cases, l = 0 and s = 1/2. Cupling f spin and rbital AM yields a ttal angular mmentum, J ˆ. Nte, shell penetratin is nt shwn abe. Spin-rbit interactin Spin-rbit interactin Cnsider reference frame f electrn: nucleus mes abut electrn. Electrn therefre in current lp which prduces magnetic field. Charged nucleus ming with prduces a current: r +Ze -e We knw that the rientatin ptential energy f magnetic diple mment is but as "E = # ˆ µ s $ B ˆ Accrding t Ampere s Law, this prduces a magnetic field, which at electrn is Transfrming back t reference frame with nucleus, must include the factr f 2 due t Thmas precessin (Appendix O f Eisberg & Resnick): (6) Using Culmb s Law: +Ze r -e This is the spin-rbit interactin energy. => (5) Mre cnenient t express in terms f S and L. As frce n electrn is where c =1/ " 0 µ 0 can write Eqn. 5 as This is the magnetic field experienced by electrn thrugh E exerted n it by nucleus. j B
Spin-rbit interactin Sdium fine structure As Substituting the last expressin fr B int Eqn. 6 gies: Ealuating g s and µ B, we btain: Transitin which gies rise t the Na D-line dublet is 3p"3s. 3p leel is split int states with ttal angular mmentum j=3/2 and j=1/2, where j = l ± s. Fr hydrgenic atms, Substituting int equatin fr "E: Expressin fr spin-rbit interactin in terms f L and S. Nte, " = e 2 /4#$ 0 hc is the fine structure cnstant. (7) General frm Hydrgenic frm In the presence f additinal externally magnetic field, these leels are further split (Zeeman effect). Magnitude f the spin-rbit interactin can be calculated using Eqn. 7. In the case f the Na dublet, difference in energy between the 3p 3/2 and 3p 1/2 subleels is: "E = 0.0021 ev (r 0.597 nm) Hydrgen fine structure Ttal angular mmentum Spectral lines f H fund t be cmpsed f clsely spaced dublets. Splitting is due t interactins between electrn spin S and the rbital angular mmentum L => spin-rbit cupling. H# line is single line accrding t the Bhr r Schrödinger thery. Occurs at 656.47 nm fr H and 656.29 nm fr D (istpe shift, #$~0.2 nm). H# Orbital and spin angular mmenta cuple tgether ia the spinrbit interactin. Internal magnetic field prduces trque which results in precessin f L ˆ and S ˆ abut their sum, the ttal angular mmentum: Called L-S cupling r Russell-Saunders cupling. Maintains fixed magnitude and z-cmpnents, specified by tw quantum numbers j and m j : z ˆ J ˆ L ˆ S Vectr mdel f atm Spin-rbit cupling prduces fine-structure splitting f ~0.016 nm. Crrespnds t an internal magnetic field n the electrn f abut 0.4 Tesla. where m j = -j, -j + 1,, +j - 1, +j. But what are the alues f j? Must use ectr inequality
Ttal angular mmentum Ttal angular mmentum Frm the preius page, we can therefre write Since, s = 1/2, there are generally tw members f series that satisfy this inequality: j = l + 1/2, l - 1/2 Fr l = 0 => j = 1/2 Fr multi-electrn atms where the spin-rbit cupling is weak, it can be presumed that the rbital angular mmenta f the indiidual electrns add t frm a resultant rbital angular mmentum L. Sme examples ectr additin rules This kind f cupling is called L-S cupling r Russell-Saunders cupling. J = L + S, L = 3, S = 1 L + S = 4, L - S = 2, therefre J = 4, 3, 2. L = l 1 + l 2, l 1 = 2, l 2 = 0 l 1 + l 2 = 2, l 1 - l 2 = 2, therefre L = 2 J = j 1 + j 2, j 1 = 5/2, j 2 = 3/2 j 1 + j 2 = 4, j 1 - j 2 = 1, therefre J = 4, 3, 2, 1 Fund t gie gd agreement with bsered spectral details fr many light atms. Fr heaier atms, anther cupling scheme called j-j cupling prides better agreement with experiment. Ttal angular mmentum in a magnetic field Ttal angular mmentum can be isuallised as precessing abut any externally applied magnetic field. Magnetic energy cntributin is prprtinal J z. J z is quantized in alues ne unit apart, s fr the upper leel f the sdium dublet with j=3/2, the ectr mdel gies the splitting in bttm figure. This treatment f the angular mmentum is apprpriate fr weak external magnetic fields where the cupling between the spin and rbital angular mmenta can be presumed t be strnger than the cupling t the external field.