o Spin-orbit interaction.
o Relativistic kinetic energy correction
o The Lamb shift.
o Nuclear moments.
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o Fine structure of H-atom is due to spin-orbit interaction: Jˆ is a max Z! "E = # Sˆ $ Lˆ so 2m2cr3 Lˆ Sˆ o If L is parallel to S => J is a maximum => high energy ! configuration. +Ze ! ! -e ! o Angular momenta are described in terms of quantum numbers, s, l and j: Jˆ is a min Jˆ = Lˆ + Sˆ Lˆ Jˆ 2 = (Lˆ + Sˆ )(Lˆ + Sˆ ) = Lˆ Lˆ + Sˆ Sˆ + 2Sˆ " Lˆ 1 ! Sˆ " Lˆ = (Jˆ " Jˆ # Lˆ " Lˆ # Sˆ " Sˆ ) +Ze 2 ! -e !2 => Sˆ " Lˆ = [ j( j +1) # l(l +1) # s(s +1)] ˆ 2 S Z!3 1 "#E so = $ 2 3 [ j( j +1) % l(l +1) % s(s +1)] 4m c r ! ! PY3P05
! o For practical purposes, convenient to express spin-orbit coupling as a "E so = j( j +1) # l(l +1) # s(s +1) 2 [ ] 2 2 2 3 where a = Ze µ 0 ! /8"m r is the spin-orbit coupling constant. Therefore, for the 2p electron: ! a* 3# 3 & 1# 1 &- 1 "E so = , % +1( )1(1+1) ) % +1(/ = + a ! 2 + 2$ 2 ' 2$ 2 '. 2 a* 1# 1 & 1# 1 &- "E so = , % +1( )1(1+1) ) % +1(/ = )a 2 + 2$ 2 ' 2$ 2 '.
j = 3/2 +1/2a !E 2p1 Angular momenta aligned
j = 1/2 -a Angular momenta opposite
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o The spin-orbit coupling constant is directly measurable from the doublet structure of spectra.
th o If we use the radius rn of the n Bohr radius as a rough approximation for r (from Lectures 1-2): 2 2 n ! r = 4"# 0 mZe2 Z 4 => a ~ 6 n o Spin-orbit coupling increases sharply with Z. Difficult for observed for H-atom, as
Z = 1: 0.14 Å (H!), 0.08 Å (H"), 0.07 Å (H#). ! Z 4 o Evaluating the quantum mechanical form, a ~ n 3[l(l +1)(2l +1)] o Therefore, using this and s = 1/2: 2 4 Z # 2 [ j( j +1) $ l(l +1) $ 3/4] "E so = 3 mc ! 2n l(l +1)(2l +1)
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! o Convenient to introduce shorthand notation to label energy levels that occurs in the LS coupling regime.
2S+1 o Each level is labeled by L, S and J: LJ
o L = 0 => S o L = 1 => P o L = 2 =>D o L = 3 =>F
2 2 o If S = 1/2, L =1 => J = 3/2 or 1/2. This gives rise to two energy levels or terms, P3/2 and P1/2 o 2S + 1 is the multiplicity. Indicates the degeneracy of the level due to spin. o If S = 0 => multiplicity is 1: singlet term. o If S = 1/2 => multiplicity is 2: doublet term. o If S = 1 => multiplicity is 3: triplet term. o Most useful when dealing with multi-electron atoms.
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o The energy level diagram can also be drawn as a term diagram.
2S+1 o Each term is evaluated using: LJ o For H, the levels of the 2P term arising from spin-orbit coupling are given below:
+1/2a 2P E 3/2 Angular momenta aligned 2p1 (2P)
-a 2 P1/2 Angular momenta opposite
PY3P05 o Spectral lines of H 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.
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o According to special relativity, the kinetic energy of an electron of mass m and velocity v is: p2 p4 T = " + ... 2m 8m3c 2 o The first term is the standard non-relativistic expression for kinetic energy. The second term is the lowest-order! relativistic correction to this energy. o Therefore, the correction to the Hamiltonian is
1 "H = # p4 rel 8m3c 2 p2 o Using the fact that = E " V we can write 2m ! 1 "H = # E 2 # 2EV +V 2 rel 2mc 2 ( ) !
PY3P05 ! o With V = -Z2e / r , applying first-order perturbation theory to previous Hamiltonian reduces the problem of finding the expectation values of r -1 and r -2.
Z 2$ 4 % 1 3 ( "E = # mc 2' # * rel n 3 & 2l +1 8n) o Produces an energy shift comparable to spin-orbit effect.
! 4 4 -8 o Note that !Erel ~ " => (1/137) ~ 10 o A complete relativistic treatment of the electron involves the solving the Dirac equation.
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o For H-atom, the spin-orbit and relativistic corrections are comparable in magnitude, but much smaller than the gross structure.
Enlj = En + $EFS
o Gross structure determined by En from Schrödinger equation. The fine structure is determined by Z 4$ 4 % 1 3 ( "E = "E + "E = # mc 2' # * FS so rel 2n 3 & 2l +1 8n)
2 2 2 2 o As En = -Z E0/n , where E0 = 1/2! mc , we can write ! 2 $ 2 2 $ '' Z E 0 Z # 1 3 E H "atom = " 2 &1 + & " )) n % n % j +1/2 4n((
o Gives the energy! of the gross and fine structure of the hydrogen atom.
PY3P05 o Energy correction only depends on j, which is of the order of !2 ~ 10-4 times smaller that the principle energy splitting. o All levels are shifted down from the Bohr energies. o For every n>1 and l, there are two states corresponding to j = l ± 1/2. o States with same n and j but different l, have the same energies (does not hold when Lamb shift is included). i.e., are degenerate. o Using incorrect assumptions, this fine structure was derived by Sommerfeld by modifying Bohr theory => right results, but wrong physics!
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o Spectral lines give information on nucleus. Main effects are isotope shift and hyperfine structure. o According to Schrödinger and Dirac theory, states with same n and j but different l are 2 degenerate. However, Lamb and Retherford showed in 1947 that 2 S1/2 (n = 2, l = 0, j = 1/2) 2 -6 and 2 P1/2 (n = 2, l = 1, j = 1/2) of are not degenerate. Energy difference is ~4.4 x 10 eV. o Experiment proved that even states with the same total angular momentum J are energetically different.
PY3P05 2 - 1. Excite H-atoms to 2 S1/2 metastable state by e bombardment. Forbidden to spontaneuosly 2 decay to 1 S1/2 optically.
2 2. Cause transitions to 2 P1/2 state using tunable microwaves. Transitions only occur when microwaves tuned to transition frequency. These atoms then decay emitting Ly! line.
2 3. Measure number of atoms in 2 S1/2 state from H-atom collisions with tungsten (W) target. 2 When excitation to 2 P1/2, current drops.
2 4. Excited H atoms (2 S1/2 metastable state) cause secondary electron emission and current from 2 the target. Dexcited H atoms (1 S1/2 ground state) do not.
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o According to Dirac and Schrödinger theory, states with the same n and j quantum numbers but different l quantum numbers ought to be degenerate. Lamb and Retherford showed that 2 S1/2 (n=2, l=0, j=1/2) and 2P1/2 (n=2, l=1, j=1/2) states of hydrogen atom were not degenerate, but that the S state had slightly higher energy by E/h = 1057.864 MHz. o Effect is explained by the theory of quantum electrodynamics, in which the electromagnetic interaction itself is quantized.
o For further info, see http://www.pha.jhu.edu/~rt19/hydro/node8.html
PY3P05 Energy scale Energy (eV) Effects Gross structure 1-10 electron-nuclear attraction Electron kinetic energy Electron-electron repulsion Fine structure 0.001 - 0.01 Spin-orbit interaction Relativistic corrections Hyperfine structure 10-6 - 10-5 Nuclear interactions
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o Hyperfine structure results from magnetic interaction between the electron’s total angular momentum (J) and the nuclear spin (I). o Angular momentum of electron creates a magnetic field at the nucleus which is proportional to J.
ˆ ˆ ˆ o Interaction energy is therefore "E hyperfine = #µˆ nucleus $ Belectron % I $ J o Magnitude is very small as nuclear dipole is ~2000 smaller than electron (µ~1/m). ! o Hyperfine splitting is about three orders of magnitude smaller than splitting due to fine structure.
PY3P05 o Like electron, the proton has a spin angular momentum and an associated intrinsic dipole moment e µˆ = g Iˆ p p M o The proton dipole moment is weaker than the electron dipole moment by M/m ~ 2000 and hence the effect is small. g e2 o Resulting energy correction can! be shown to be: "E = p Iˆ # Jˆ p mMc 2r3 o Total angular momentum including nuclear spin, orbital angular momentum and electron spin is ˆ ˆ ˆ F = I + J ! where F = f ( f +1)! F = m ! z f o ! The quantum number f has possible values f = j + 1/2, j - 1/2 since the proton has spin 1/2,. o !Hence every energy level associated with a particular set of quantum numbers n, l, and j will be split into two levels of slightly different energy, depending on the relative orientation of the proton magnetic dipole with the electron state.
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o The energy splitting of the hyperfine interaction is given by F = 1 a "E HFS = f ( f +1) # i(i +1) # j( j +1) 2 [ ] F = 0 where a is the hyperfine structure constant.
! o E.g., consider the ground state of H-atom. Nucleus consists of a single proton, so I = 1/2. The hydrogen 2 ground state is the 1s S1/2 term, which has J = 1/2. Spin of the electron can be parallel (F = 1) or antiparallel (F = 0). Transitions between! these levels occur at 21 cm (1420 MHz). o For ground state of the hydrogen atom (n=1), the 21 cm radio map of the Milky Way energy separation between the states of F = 1 and F = 0 is 5.9 x 10-6 eV. Prove this!
PY3P05 o Selection rules determine the allowed transitions between terms.
$n = any integer $l = ±1 $j = 0, ±1 $f = 0, ±1
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o Gross structure: o Covers largest interactions within the atom: o Kinetic energy of electrons in their orbits. o Attractive electrostatic potential between positive nucleus and negative electrons o Repulsive electrostatic interaction between electrons in a multi-electron atom. o Size of these interactions gives energies in the 1-10 eV range and upwards. o Determine whether a photon is IR, visible, UV or X-ray. o Fine structure: o Spectral lines often come as multiplets. E.g., H! line. => smaller interactions within atom, called spin-orbit interaction. o Electrons in orbit about nucleus give rise to magnetic moment
of magnitude µB, which electron spin interacts with. Produces small shift in energy. o Hyperfine structure: o Fine-structure lines are split into more multiplets. o Caused by interactions between electron spin and nucleus spin. o Nucleus produces a magnetic moment of magnitude
~µB/2000 due to nuclear spin. o E.g., 21-cm line in radio astronomy.
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