Lectures 5-6: Magnetic dipole moments Sodium D-line doublet
o Grotrian diagram for doublet states of o Orbital dipole moments. neutral sodium showing permitted transitions, including Na D-line transition at 589 nm. o Orbital precession.
o Spin-orbit interaction. o D-line is split into a doublet: D1 = 589.59 nm, D2 = 588.96 nm. o Stern-Gerlach experiment. o Many lines of alkali atoms are doublets. o Total angular momentum. Occur because terms (bar s-term) are split in two. o Fine structure, hyperfine structure of H and Na. o This fine structure can only be Na “D-line” o The Lamb shift. understood via magnetic moments of electron.
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Orbital magnetic dipole moments Orbital magnetic dipole moments o Consider electron moving with velocity (v) in a circular Bohr orbit o Magnetic moment can also be written in terms of the Bohr magneton: of radius r. Produces a current µl
where gl is the orbital g-factor. Gives ratio of magnetic moment to angular momentum (in where T is the orbital period of the electron. r units of h ). -e o Current loop produces a magnetic field, with a moment v o In vector form, Eqn 2 can be written L ! (1) o As o Specifies strength of magnetic dipole. o The components of the angular momentum in the z-direction are 2 o Magnitude of orbital angular momentum is L = mvr = m!r . Lz = m l h where ml = -l, -l+1, …, 0, …, +l+1, +l. Combining with Eqn. 1 => (2) o The magnetic moment associated with the z-component is correspondingly o An electron in the first Bohr orbit with L = h has a magnetic moment defined as ! = 9.27x10-24 Am2 Bohr magneton ! PY3004 PY3004 Orbital precession Electron spin o When magnetic moments is placed in an external magnetic field, it experiences a torque: o Electron also has an intrinsic angular momentum, called spin. The (3) spin and its z-component obey identical relations to orbital AM: which tends to align dipole with the field. The potential energy associated with this force is
where s = 1/2 is the spin quantum number => o Minimum potential energy occurs when µl !!B.
o Therefore two possible orientations: o If "E = const., µl cannot align with B => µl precesses about B.
(4) => spin magnetic quantum number is ±1/2. o But from Eqn. 3, o Follows that electron has intrinsic magnetic moments: Sˆ o Setting this equal to Eqn. 4 =>
! Larmor ˆ frequency µs where gs (=2) is the spin g-factor. o Called Larmor precession. Occurs in direction of B.
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The Stern-Gerlach experiment The Stern-Gerlach experiment o This experiment confirmed the quantisation of electron spin into two orientations. o Conclusion of Stern-Gerlach experiment: o Potential energy of electron spin magnetic moment in magnetic field in z-direction is 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 The resultant force is
o Atomic magnetic moment has µz = ±µB.
o Find same deflection for all atoms which have an s electron in the outermost o As gsms = ±1, orbital => all angular momenta and magnetic moments of all inner electrons cancel. Therefore only measure properties of outer s electron. o The deflection distance is then,
o The s electron has orbital angular momentum l = 0 => only observe spin.
PY3004 PY3004 The Stern-Gerlach experiment Spin-orbit interaction o Experiment was confirmed using: o Fine-structure in atomic spectra cannot be explained by Coulomb interaction between nucleus and electron. Element Electronic Configuration H 1s1 o Instead, must consider magnetic interaction between orbital magnetic moment and Na {1s22s22p6}3s1 the intrinsic spin magnetic moment. K {1s22s22p63s23p6}4s1 Cu {1s22s22p63s23p63d10}4s1 o Called spin-orbit interaction. Ag {1s22s22p63s23p63d104s24p64d10}5s1 Cs {[Ag]5s25p6}6s1 o Weak in one-electron atoms, but strong in multi-electron atoms where total orbital magnetic moment is large. Au {[Cs]5d104f14}6s1 o In all cases, l = 0 and s = 1/2. o Coupling of spin and orbital AM yields a total angular momentum, Jˆ . o Note, shell penetration is not shown above. !
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Spin-orbit interaction Spin-orbit interaction
v ˆ o Consider reference frame of electron: nucleus moves about o We know that the orientation potential energy of magnetic dipole moment is "E = #µˆ s $ B electron. Electron therefore in current loop which produces +Ze r -e magnetic field. Charged nucleus moving with v produces a but as current: ! o Transforming back to reference frame with nucleus, must include the factor of 2 due to o According to Ampere’s Law, this produces a magnetic field, Thomas precession (Appendix O of Eisberg & Resnick): which at electron is (6) -e o Using Coulomb’s Law: +Ze r o This is the spin-orbit interaction energy.
o More convenient to express in terms of S and L. As force on electron is => (5) v
where c =1/ "0µ0 can write Eqn. 5 as o This is the magnetic field experienced by electron through E B ! exerted on it by nucleus. j
PY3004 PY3004 Spin-orbit interaction Sodium fine structure o As o Transition which gives rise to the Na D-line doublet is 3p"3s. o Substituting the last expression for B into Eqn. 6 gives: o 3p level is split into states with total angular o Evaluating gs and µB, we obtain: momentum j=3/2 and j=1/2, where j = l ± s.
General o For hydrogenic atoms, o In the presence of additional externally magnetic form field, these levels are further split (Zeeman effect). o Substituting into equation for "E: o Magnitude of the spin-orbit interaction can be Hydrogenic calculated using Eqn. 7. In the case of the Na
form doublet, difference in energy between the 3p3/2 and 3p1/2 sublevels is: (7) "E = 0.0021 eV (or 0.597 nm) o Expression for spin-orbit interaction in terms of L and S. Note, 2 is the fine " = e /4#$0hc structure constant.
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Hydrogen fine structure Total angular momentum o Spectral lines of H found to be composed of o Orbital and spin angular momenta couple together via the spin- z closely spaced doublets. Splitting is due to orbit interaction. Sˆ interactions between electron spin S and the Jˆ orbital angular momentum L => spin-orbit o Internal magnetic field produces torque which results in coupling. Lˆ Sˆ Lˆ precession of and about their sum, the total angular ! momentum: ! o H# line is single line according to the Bohr or Vector model of atom Schrödinger theory. Occurs at 656.47 nm for o Called! L-S coupling! or Russell-Saunders coupling. Maintains ! H and 656.29 nm for D (isotope shift, ~0.2 #$ H# fixed magnitude and z-components, specified by two quantum nm). numbers j and mj: o Spin-orbit coupling produces fine-structure splitting of ~0.016 nm. Corresponds to an internal magnetic field on the electron of where mj = -j, -j + 1, … , +j - 1, +j. about 0.4 Tesla. o But what are the values of j? Must use vector inequality
PY3004 PY3004 Total angular momentum Total angular momentum o From the previous page, we can therefore write o For multi-electron atoms where the spin-orbit coupling is weak, it can be presumed that the orbital angular momenta of the individual o Since, s = 1/2, there are generally two members of series that satisfy this inequality: electrons add to form a resultant orbital angular j = l + 1/2, l - 1/2 momentum L. o For l = 0 => j = 1/2 o Some examples vector addition rules o This kind of coupling is called L-S coupling or Russell-Saunders coupling. o J = L + S, L = 3, S = 1 L + S = 4, |L - S| = 2, therefore J = 4, 3, 2. o Found to give good agreement with observed spectral details for many light atoms. o L = l1 + l2, l1 = 2, l2 = 0 l + l = 2, | l - l | = 2, therefore L = 2 1 2 1 2 o For heavier atoms, another coupling scheme o J = j + j , j = 5/2, j = 3/2 called j-j coupling provides better agreement 1 2 1 2 with experiment. j1 + j2 = 4, | j1 - j2 | = 1, therefore J = 4, 3, 2, 1
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Total angular momentum in a magnetic field o Total angular momentum can be visuallised as precessing about any externally applied magnetic field.
o Magnetic energy contribution is proportional Jz.
o Jz is quantized in values one unit apart, so for the upper level of the sodium doublet with j=3/2, the vector model gives the splitting in bottom figure. o This treatment of the angular momentum is appropriate for weak external magnetic fields where the coupling between the spin and orbital angular momenta can be presumed to be stronger than the coupling to the external field.
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