PY3P05 Lectures 7-8: Fine and hyperfine structure of hydrogen oFine structure oSpin-orbit interaction. oRelativistic kinetic energy correction oHyperfine structure oThe Lamb shift. oNuclear moments.

PY3P05 Spin-orbit coupling in H-atom oFine structure of H-atom is due to spin-orbit interaction: oIf L is parallel to S => J is a maximum => high energy configuration. oAngular momenta are described in terms of quantum numbers, s, l and j: +Ze -e is a max +Ze -e is a min

PY3P05 Spin-orbit effects in H fine structure oFor practical purposes, convenient to express spin-orbit coupling as where is the spin-orbit coupling constant. Therefore, for the 2p electron: E 2p 1 j = 3/2 j = 1/2 +1/2a Angular momenta aligned -a Angular momenta opposite

PY3P05 Spin-orbit coupling in H-atom oThe spin-orbit coupling constant is directly measurable from the doublet structure of spectra. oIf we use the radius r n of the n th Bohr radius as a rough approximation for r (from Lectures 1-2): oSpin-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  ). oEvaluating the quantum mechanical form, oTherefore, using this and s = 1/2:

PY3P05 Term Symbols oConvenient to introduce shorthand notation to label energy levels that occurs in the LS coupling regime. oEach level is labeled by L, S and J: 2S+1 L J oL = 0 => S oL = 1 => P oL = 2 =>D oL = 3 =>F oIf S = 1/2, L =1 => J = 3/2 or 1/2. This gives rise to two energy levels or terms, 2 P 3/2 and 2 P 1/2 o2S + 1 is the multiplicity. Indicates the degeneracy of the level due to spin. oIf S = 0 => multiplicity is 1: singlet term. oIf S = 1/2 => multiplicity is 2: doublet term. oIf S = 1 => multiplicity is 3: triplet term. oMost useful when dealing with multi-electron atoms.

PY3P05 Term diagram for H fine structure oThe energy level diagram can also be drawn as a term diagram. oEach term is evaluated using: 2S+1 L J oFor H, the levels of the 2 P term arising from spin-orbit coupling are given below: E 2p 1 ( 2 P) 2 P 3/2 2 P 1/2 +1/2a Angular momenta aligned -a Angular momenta opposite

PY3P05 Hydrogen fine structure oSpectral 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. oH  line is single line according to the Bohr or Schrödinger theory. Occurs at nm for H and nm for D (isotope shift,  ~0.2 nm). oSpin-orbit coupling produces fine-structure splitting of ~0.016 nm. Corresponds to an internal magnetic field on the electron of about 0.4 Tesla. HH

PY3P05 Relativistic kinetic energy correction oAccording to special relativity, the kinetic energy of an electron of mass m and velocity v is: oThe first term is the standard non-relativistic expression for kinetic energy. The second term is the lowest-order relativistic correction to this energy. oUsing perturbation theory, it can be show that oProduces an energy shift comparable to spin-orbit effect. oA complete relativistic treatment of the electron involves the solving the Dirac equation.

PY3P05 Total fine structure correction oFor H-atom, the spin-orbit and relativistic corrections are comparable in magnitude, but much smaller than the gross structure. E nlj = E n +  E FS oGross structure determined by E n from Schrödinger equation. The fine structure is determined by oAs E n = -Z 2 E 0 /n 2, where E 0 = 1/2  2 mc 2, we can write oGives the energy of the gross and fine structure of the hydrogen atom.

PY3P05 Fine structure of hydrogen oEnergy correction only depends on j, which is of the order of  2 ~ times smaller that the principle energy splitting. oAll levels are shifted down from the Bohr energies. oFor every n>1 and l, there are two states corresponding to j = l ± 1/2. oStates with same n and j but different l, have the same energies (does not hold when Lamb shift is included). i.e., are degenerate. oUsing incorrect assumptions, this fine structure was derived by Sommerfeld by modifying Bohr theory => right results, but wrong physics!

PY3P05 Hyperfine structure: Lamb shift oSpectral lines give information on nucleus. Main effects are isotope shift and hyperfine structure. oAccording to Schrödinger and Dirac theory, states with same n and j but different l are degenerate. However, Lamb and Retherford showed in 1947 that 2 2 S 1/2 (n = 2, l = 0, j = 1/2) and 2 2 P 1/2 (n = 2, l = 1, j = 1/2) of H-atom are not degenerate. oExperiment proved that even states with the same total angular momentum J are energetically different.

PY3P05 Hyperfine structure: Lamb shift 1.Excite H-atoms to 2 2 S 1/2 metastable state by e - bombardment. Forbidden to spontaneuosly decay to 1 2 S 1/2 optically. 2.Cause transitions to 2 2 P 1/2 state using tunable microwaves. Transitions only occur when microwaves tuned to transition frequency. These atoms then decay emitting Ly  line. 3.Measure number of atoms in 2 2 S 1/2 state from H-atom collisions with tungsten (W) target. When excitation to 2 2 P 1/2, current drops. 4.Excited H atoms (2 2 S 1/2 metastable state) cause secondary electron emission and current from the target. Dexcited H atoms (1 2 S 1/2 ground state) do not.

PY3P05 Hyperfine structure: Lamb shift oAccording 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 S 1/2 (n=2, l=0, j=1/2) and 2P 1/2 (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 = MHz. oEffect is explained by the theory of quantum electrodynamics, in which the electromagnetic interaction itself is quantized. oFor further info, see

PY3P05 Hyperfine structure: Nuclear moments oHyperfine structure results from magnetic interaction between the electron’s total angular momentum (J) and the nuclear spin (I). oAngular momentum of electron creates a magnetic field at the nucleus which is proportional to J. oInteraction energy is therefore oMagnitude is very small as nuclear dipole is ~2000 smaller than electron (  ~1/m). oHyperfine splitting is about three orders of magnitude smaller than splitting due to fine structure.

PY3P05 Hyperfine structure: Nuclear moments oLike electron, the proton has a spin angular momentum and an associated intrinsic dipole moment oThe proton dipole moment is weaker than the electron dipole moment by M/m ~ 2000 and hence the effect is small. oResulting energy correction can be shown to be: oTotal angular momentum including nuclear spin, orbital angular momentum and electron spin is where oThe quantum number f has possible values f = j + 1/2, j - 1/2 since the proton has spin 1/2,. oHence 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.

PY3P05 Hyperfine structure: Nuclear moments oThe energy splitting of the hyperfine interaction is given by where a is the hyperfine structure constant. oE.g., consider the ground state of H-atom. Nucleus consists of a single proton, so I = 1/2. The hydrogen ground state is the 1s 2 S 1/2 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). oFor ground state of the hydrogen atom (n=1), the energy separation between the states of F = 1 and F = 0 is 5.9 x eV. F = 1 F = 0 21 cm radio map of the Milky Way

PY3P05 Selection rules oSelection rules determine the allowed transitions between terms.  n = any integer  l = ±1  j = 0, ±1  f = 0, ±1

PY3P05 Summary of Atomic Energy Scales oGross structure: oCovers largest interactions within the atom: oKinetic energy of electrons in their orbits. oAttractive electrostatic potential between positive nucleus and negative electrons oRepulsive electrostatic interaction between electrons in a multi-electron atom. oSize of these interactions gives energies in the 1-10 eV range and upwards. oDetermine whether a photon is IR, visible, UV or X-ray. oFine structure: oSpectral lines often come as multiplets. E.g., H  line. => smaller interactions within atom, called spin-orbit interaction. oElectrons in orbit about nucleus give rise to magnetic moment of magnitude  B, which electron spin interacts with. Produces small shift in energy. oHyperfine structure: oFine-structure lines are split into more multiplets. oCaused by interactions between electron spin and nucleus spin. oNucleus produces a magnetic moment of magnitude ~  B /2000 due to nuclear spin. oE.g., 21-cm line in radio astronomy.