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O Orbital Dipole Moments. O Orbital Precession. O Spin-Orbit Interaction

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O Orbital Dipole Moments. O Orbital Precession. O Spin-Orbit Interaction o Orbital dipole moments. o Orbital precession. o Spin-orbit interaction. o Stern-Gerlach experiment. o Total angular momentum. o Fine structure, hyperfine structure of H and Na. o Chapter 8 of Eisberg & Resnick PY3P05 o Grotrian diagram for neutral sodium showing permitted transitions, including Na D-line transition at 589 nm. o D-line is split into a doublet: D1 = 589.59 nm, D2 = 588.96 nm. o Many lines of alkali atoms are doublets. Occur because terms (bar s-term) are split in two. o This fine structure can only be understood via magnetic moments of Na “D-line” electron. PY3P05 o Angular momentum for electron orbiting at distance r and velocity v about nucleus : Lˆ = rˆ " pˆ o Its magnitude is therefore, ! L = mvr o As v = r! => L = mr2 ! o Or, in general L = I ! 2 where I is the moment of inertia I = "miri i ! PY3P05 o Electron moving with velocity (v) in a circular Bohr orbit of radius r produces a current e e# µl i = " = " T 2$ where T is the orbital period of the electron. r e- o Current loop! produces magnetic field, with a moment L v e# 2 1 2 µl = iA = " $r = " e#r (1) 2$ 2 o Specifies strength of magnetic dipole. o Magnitude of orbital angular momentum is L = mvr = m r2. ! ! Combining with Eqn. 1 => e µ = " L l 2m (2) o Electron in the first Bohr orbit with L = ! has magnetic moment e µ B = ! = 9.27x10-24 Am2 Bohr Magneton ! 2m ! PY3P05 ! o Magnetic moment can also be written in terms of the Bohr magneton: glµB µl = L ! where gl is the orbital g-factor. Gives ratio of magnetic moment to angular momentum (in units of ! ). ! glµB ˆ o In vector form, Eqn 2 can be written µˆ l = " L ! ! glµB o As L = l(l +1)! => µl = l(l +1)! = glµB l(l +1) ! ! components of the angular momentum in the z-direction are L = m ! where m = -l, -l +1, …, 0, …, +l - 1, +l. ! z l l o Magnetic moment associated with the z-component is correspondingly g µ g µ ! l B L l B m g m µlz = " z = " l ! = " lµB l ! ! PY3P05 ! o When magnetic moment placed in external magnetic field, it experiences a torque: ˆ "ˆ = µˆ l # B (3) which tends to align dipole with field. Potential energy associated with this force is ˆ "E = #µˆ B $ B o Minimum potential energy occurs when µ !!B. ! l o If "E = const., µl cannot align with! B => µl precesses about B. #L Lsin$#% o Now, " = = = L& sin$ (4) #t #t glµB and from Eqn. 3, " = µl Bsin # = LBsin# (5) ! ! g µ o Equating Eqns. 4 and 5 => l B LBsin" = L# sin" ! ! g µ Larmor => # = l B B ! frequency o Called Larmor precession. Occurs in direction of B. ! PY3P05 o Electron also has an intrinsic angular momentum, called spin. The spin and its z-component obey identical relations to orbital AM: S = s(s +1)! S m z = s! where s = 1/2 is the spin quantum number => S = 1/2(1/2 +1)! = 3 /2! o Therefore two possible! orientations: S 1/2 z = ± ! => spin magnetic quantum number is! ± 1/2. ! o Follows that electron has intrinsic magnetic moments: Sˆ g µ µˆ = " s B Sˆ s ! ! g m µsz = " sµB s µˆ s where gs (=2) is the spin g-factor. ! ! PY3P05 o Confirmed quantisation of electron spin into two orientations. o Potential energy of electron spin magnetic moment in magnetic field in z-direction is E ˆ Bˆ B " = #µ s $ = #µsz = gsµB msB o The resultant force is d(#E) dB ! F = " = "µ g m z z dz B s s dz dB o As g m = ±1, F = ±µ z s s z B dz ! o The deflection distance is then, 2 2 ! 2 F " L% µB L dBz z =1/2at =1/2 = ± m #$ v &' 4KE dz PY3P05 ! o Conclusion of Stern-Gerlach experiment: o With field on, classically expect random distribution at target. In fact find two bands as beam is split in two. o There is directional quantisation, parallel or antiparallel to B. o Atomic magnetic moment has µz = ±µB. o Find same deflection for all atoms which have an s electron in the outermost orbital => all angular momenta and magnetic moments of all inner electrons cancel. Therefore only measure properties of outer s electron. o The s electron has orbital angular momentum l = 0 => only observe spin. PY3P05 o Experiment was confirmed using: Element Electronic Configuration H 1s1 Na {1s22s22p6}3s1 K {1s22s22p63s23p6}4s1 Cu {1s22s22p63s23p63d10}4s1 Ag {1s22s22p63s23p63d104s24p64d10}5s1 Cs {[Ag]5s25p6}6s1 Au {[Cs]5d104f14}6s1 o In all cases, l = 0 and s = 1/2. o Note, shell penetration is not shown above. PY3P05 o Fine-structure in atomic spectra cannot be explained by Coulomb interaction between nucleus and electron. o Instead, must consider magnetic interaction between orbital magnetic moment and the intrinsic spin magnetic moment. o Called spin-orbit interaction. o Weak in one-electron atoms, but strong in multi-electron atoms where total orbital magnetic moment is large. o Coupling of spin and orbital AM yields a total angular momentum, J ˆ. ! PY3P05 v o Consider reference frame of electron: nucleus moves about electron. Electron therefore in current loop which produces +Ze r -e magnetic field. Charged nucleus moving with v produces a current: ˆj = "Zevˆ o According to Ampere’s Law, this produces a magnetic field, which at electron is µ ˆj # rˆ $Zeµ vˆ # rˆ Bˆ = 0 = 0 ! 4" r3 4" r3 -e ˆ Ze rˆ r o Using Coulomb’s Law: E = 3 +Ze 4"#0 r ! 1 => Bˆ = " vˆ # Eˆ (5) c 2 v ! where c =1/ "0µ0 o This! is the magnetic field experienced by electron through E B exerted on it by nucleus. ! j PY3P05 ˆ o We know that the orientation potential energy of magnetic dipole moment is "E = #µˆ s $ B g µ g µ but as µˆ = " s B Sˆ => #E = s B Sˆ $ Bˆ s ! ! ! o Transforming back to reference frame with nucleus, must include the factor of 2 due to Thomas precession (Appendix O of Eisberg & Resnick): ! 1 gsµB ˆ ˆ "E so = S # B (6) 2 ! o This is the spin-orbit interaction energy. o More convenient to express! in terms of S and L. As force on electron is dV(r) rˆ 1 dV(r) rˆ Fˆ = "eEˆ = " => Eˆ = dr r e dr r can write Eqn. 5 as 1 1 dV (r) Bˆ = " vˆ # rˆ ec 2 r dr ! PY3P05 ! 1 1 dV(r) o As Lˆ = rˆ " mvˆ = #mvˆ " rˆ => Bˆ = Lˆ emc 2 r dr gsµB 1 dV (r) ˆ ˆ o Substituting the last expression for B into Eqn. 6 gives: "E so = 2 S # L 2emc ! r dr 1 1 dV (r) !o Evaluating g and µ , we obtain: "E = Sˆ # Lˆ s B so 2m2c 2 r dr o For hydrogenic atoms, Ze2 dV(r!) Ze2 General V(r) = " => = 2 form 4#$0r dr 4#$0r ! 1 1 Ze2 o Substituting into equation for E: ˆ ˆ " "E so = 2 2 2 S % L 2m c r 4#$0r ! e2 Z! Hydrogenic = Sˆ % Lˆ 4 c 2m2cr3 #$0! form Z! ˆ ˆ "E so = # 2 3 S $ L (7) 2m cr ! o Expression for spin-orbit interaction in terms of L and S. Note, " = e2 /4#$ ! c is the fine structure constant. 0 ! PY3P05 ! o Transition which gives rise to the Na D-line doublet is 3p#3s. o 3p level is split into states with total angular momentum j=3/2 and j=1/2, where j = l ± s. o In the presence of additional externally magnetic field, these levels are further split (Zeeman effect). o Magnitude of the spin-orbit interaction can be calculated using Eqn. 7. In the case of the Na doublet, difference in energy between the 3p3/2 and 3p1/2 sublevels is: "E = 0.0021 eV (or 0.597 nm) PY3P05 o Spectral lines of H found to be composed of closely spaced doublets. Splitting is due to interactions between electron spin S and the orbital angular momentum L => spin-orbit coupling. o H# line is single line according to the Bohr or Schrödinger theory. Occurs at 656.47 nm for H and 656.29 nm for D (isotope shift, ~0.2 "$ H# nm). o Spin-orbit coupling produces fine-structure splitting of ~0.016 nm. Corresponds to an internal magnetic field on the electron of about 0.4 Tesla. PY3P05 o Orbital and spin angular momenta couple together via the spin- z orbit interaction. Sˆ Jˆ o Internal magnetic field produces torque which results in Lˆ Sˆ Lˆ precession of and about their sum, the total angular ! momentum: ! Jˆ = Lˆ + Sˆ Vector model of atom o Called! L-S coupling! or Russell-Saunders coupling. Maintains ! fixed magnitude and z-components, specified by two quantum numbers j and mj: ! J = j( j +1)! Jz = m j ! where mj = -j, -j + 1, … , +j - 1, +j. o But what are the! values of j? Must use vector inequality | Lˆ + Sˆ |"|| Lˆ | # | Sˆ || =>| Jˆ |"|| Lˆ | # | Sˆ || PY3P05 ! o From the previous page, we can therefore write j( j l) | l(l l) s(s l) | + ! " + ! # + ! o Since, s = 1/2, there are generally two members of series that satisfy this inequality: j = l + 1/2, l - 1/2 o For l = 0 => j = 1/2! o Some examples vector addition rules o J = L + S, L = 3, S = 1 L + S = 4, |L - S| = 2, therefore J = 4, 3, 2.
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