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Lectures 7-8: Fine and hyperfine structure of hydrogen

Lectures 7-8: Fine and hyperfine structure of hydrogen. Fine structure Spin-orbit interaction. Relativistic kinetic energy correction Hyperfine structure The Lamb shift. Nuclear moments. Spin-orbit coupling in H-atom. Fine structure of H-atom is due to spin-orbit interaction:

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Lectures 7-8: Fine and hyperfine structure of hydrogen

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  1. Lectures 7-8: Fine and hyperfine structure of hydrogen • Fine structure • Spin-orbit interaction. • Relativistic kinetic energy correction • Hyperfine structure • The Lamb shift. • Nuclear moments. PY3P05

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

  3. Spin-orbit effects in H fine structure • For practical purposes, convenient to express spin-orbit coupling as where is the spin-orbit coupling constant. Therefore, for the 2p electron: j = 3/2 +1/2a Angular momenta aligned E 2p1 j = 1/2 -a Angular momenta opposite PY3P05

  4. Spin-orbit coupling in H-atom • The spin-orbit coupling constant is directly measurable from the doublet structure of spectra. • If we use the radius rnof the nthBohr radius as a rough approximation for r (from Lectures 1-2): • 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). • Evaluating the quantum mechanical form, • Therefore, using this and s = 1/2: PY3P05

  5. Term Symbols • Convenient to introduce shorthand notation to label energy levels that occurs in the LS coupling regime. • Each level is labeled by L, S and J: 2S+1LJ • L = 0 => S • L = 1 => P • L = 2 =>D • L = 3 =>F • If S = 1/2, L =1 => J = 3/2 or 1/2. This gives rise to two energy levels or terms, 2P3/2 and 2P1/2 • 2S + 1 is the multiplicity. Indicates the degeneracy of the level due to spin. • If S = 0 => multiplicity is 1: singlet term. • If S = 1/2 => multiplicity is 2: doublet term. • If S = 1 => multiplicity is 3: triplet term. • Most useful when dealing with multi-electron atoms. PY3P05

  6. Term diagram for H fine structure • The energy level diagram can also be drawn as a term diagram. • Each term is evaluated using: 2S+1LJ • For H, the levels of the 2P term arising from spin-orbit coupling are given below: +1/2a Angular momenta aligned 2P3/2 E 2p1 (2P) -a Angular momenta opposite 2P1/2 PY3P05

  7. Hydrogen fine structure • 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. • 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 nm). • 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. H PY3P05

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

  9. Total fine structure correction • For H-atom, the spin-orbit and relativistic corrections are comparable in magnitude, but much smaller than the gross structure. Enlj = En + EFS • Gross structure determined by Enfrom Schrödinger equation. The fine structure is determined by • As En = -Z2E0/n2, where E0 = 1/22mc2, we can write • Gives the energy of the gross and fine structure of the hydrogen atom. PY3P05

  10. Fine structure of hydrogen • Energy correction only depends on j, which is of the order of 2 ~ 10-4 times smaller that the principle energy splitting. • All levels are shifted down from the Bohr energies. • For every n>1and l, there are two states corresponding to j = l ± 1/2. • 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. • Using incorrect assumptions, this fine structure was derived by Sommerfeld by modifying Bohr theory => right results, but wrong physics! PY3P05

  11. Hyperfine structure: Lamb shift • Spectral lines give information on nucleus. Main effects are isotope shift and hyperfine structure. • According to Schrödinger and Dirac theory, states with same n and j but different l are degenerate. However, Lamb and Retherford showed in 1947 that 22S1/2 (n = 2, l = 0, j = 1/2) and 22P1/2 (n = 2, l = 1, j = 1/2) of H-atom are not degenerate. • Experiment proved that even states with the same total angular momentum J are energetically different. PY3P05

  12. Hyperfine structure: Lamb shift • Excite H-atoms to 22S1/2 metastable state by e- bombardment. Forbidden to spontaneuosly decay to 12S1/2 optically. • Cause transitions to 22P1/2state using tunable microwaves. Transitions only occur when microwaves tuned to transition frequency. These atoms then decay emitting Ly line. • Measure number of atoms in 22S1/2 state from H-atom collisions with tungsten (W) target. When excitation to 22P1/2, current drops. • Excited H atoms (22S1/2 metastable state) cause secondary electron emission and current from the target. Dexcited H atoms (12S1/2 ground state) do not. PY3P05

  13. Hyperfine structure: Lamb shift • 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. • Effect is explained by the theory of quantum electrodynamics, in which the electromagnetic interaction itself is quantized. • For further info, see http://www.pha.jhu.edu/~rt19/hydro/node8.html PY3P05

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

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

  16. Hyperfine structure: Nuclear moments • The energy splitting of the hyperfine interaction is given by where a is the hyperfine structure constant. • E.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 2S1/2 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). • For ground state of the hydrogen atom (n=1), the energy separation between the states of F = 1 and F = 0 is 5.9 x 10-6 eV. F = 1 F = 0 21 cm radio map of the Milky Way PY3P05

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

  18. Summary of Atomic Energy Scales • Gross structure: • Covers largest interactions within the atom: • Kinetic energy of electrons in their orbits. • Attractive electrostatic potential between positive nucleus and negative electrons • Repulsive electrostatic interaction between electrons in a multi-electron atom. • Size of these interactions gives energies in the 1-10 eV range and upwards. • Determine whether a photon is IR, visible, UV or X-ray. • Fine structure: • Spectral lines often come as multiplets. E.g., H line. => smaller interactions within atom, called spin-orbit interaction. • Electrons in orbit about nucleus give rise to magnetic moment of magnitude B, which electron spin interacts with. Produces small shift in energy. • Hyperfine structure: • Fine-structure lines are split into more multiplets. • Caused by interactions between electron spin and nucleus spin. • Nucleus produces a magnetic moment of magnitude ~B/2000 due to nuclear spin. • E.g., 21-cm line in radio astronomy. PY3P05

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