Lectures 7 8 Fine and hyperfine structure of

Lectures 7 -8: Fine and hyperfine structure of hydrogen o Fine structure o Spin-orbit interaction. o Relativistic kinetic energy correction o Hyperfine structure o The Lamb shift. o Nuclear moments. PY 3 P 05

Spin-orbit coupling in H-atom o Fine structure of H-atom is due to spin-orbit interaction: o If L is parallel to S => J is a maximum => high energy configuration. is a max +Ze -e o Angular momenta are described in terms of quantum numbers, s, l and j: is a min +Ze -e PY 3 P 05

Spin-orbit effects in H fine structure o For practical purposes, convenient to express spin-orbit coupling as where electron: is the spin-orbit coupling constant. Therefore, for the 2 p E j = 3/2 +1/2 a Angular momenta aligned j = 1/2 -a Angular momenta opposite 2 p 1 PY 3 P 05

Spin-orbit coupling in H-atom o The spin-orbit coupling constant is directly measurable from the doublet structure of spectra. o If we use the radius rn of the nth Bohr radius as a rough approximation for r (from Lectures 1 -2): 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 ). o Evaluating the quantum mechanical form, o Therefore, using this and s = 1/2: PY 3 P 05

Term Symbols o Convenient to introduce shorthand notation to label energy levels that occurs in the LS coupling regime. o Each level is labeled by L, S and J: o o 2 S+1 L J L = 0 => S L = 1 => P L = 2 =>D L = 3 =>F o If 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 o 2 S + 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. PY 3 P 05

Term diagram for H fine structure o The energy level diagram can also be drawn as a term diagram. o Each term is evaluated using: 2 S+1 LJ o For H, the levels of the 2 P term arising from spin-orbit coupling are given below: 2 P E 3/2 +1/2 a Angular momenta aligned 2 p 1 (2 P) 2 P 1/2 -a Angular momenta opposite PY 3 P 05

Hydrogen fine structure 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 nm). o H 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. PY 3 P 05

Relativistic kinetic energy correction o According to special relativity, the kinetic energy of an electron of mass m and velocity v is: 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 Using perturbation theory, it can be show that o Produces an energy shift comparable to spin-orbit effect. o A complete relativistic treatment of the electron involves the solving the Dirac equation. PY 3 P 05

Total fine structure correction 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 o As En = -Z 2 E 0/n 2, where E 0 = 1/2 2 mc 2, we can write o Gives the energy of the gross and fine structure of the hydrogen atom. PY 3 P 05

Fine structure of hydrogen 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! PY 3 P 05

Hyperfine structure: Lamb shift 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 degenerate. However, Lamb and Retherford showed in 1947 that 22 S 1/2 (n = 2, l = 0, j = 1/2) and 22 P 1/2 (n = 2, l = 1, j = 1/2) of H-atom are not degenerate. o Experiment proved that even states with the same total angular momentum J are energetically different. PY 3 P 05

Hyperfine structure: Lamb shift 1. Excite H-atoms to 22 S 1/2 metastable state by e- bombardment. Forbidden to spontaneuosly decay to 12 S 1/2 optically. 2. Cause transitions to 22 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 22 S 1/2 state from H-atom collisions with tungsten (W) target. When excitation to 22 P 1/2, current drops. 4. Excited H atoms (22 S 1/2 metastable state) cause secondary electron emission and current from the target. Dexcited H atoms (12 S 1/2 ground state) do not. PY 3 P 05

Hyperfine structure: Lamb shift 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 S 1/2 (n=2, l=0, j=1/2) and 2 P 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 = 1057. 864 MHz. o Effect is explained by theory of quantum electrodynamics, in which the electromagnetic interaction itself is quantized. o For further info, see http: //www. pha. jhu. edu/~rt 19/hydro/node 8. html PY 3 P 05

Hyperfine structure: Nuclear moments 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. o Interaction energy is therefore o Magnitude is very small as nuclear dipole is ~2000 smaller than electron ( ~1/m). o Hyperfine splitting is about three orders of magnitude smaller than splitting due to fine structure. PY 3 P 05

Hyperfine structure: Nuclear moments o Like electron, the proton has a spin angular momentum and an associated intrinsic dipole moment o The proton dipole moment is weaker than the electron dipole moment by M/m ~ 2000 and hence the effect is small. o Resulting energy correction can be shown to be: o Total angular momentum including nuclear spin, orbital angular momentum and electron spin is where 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. PY 3 P 05

Hyperfine structure: Nuclear moments o The energy splitting of the hyperfine interaction is given by 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 ground state is the 1 s 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). o 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 e. V. F=1 F=0 21 cm radio map of the Milky Way PY 3 P 05

Selection rules o Selection rules determine the allowed transitions between terms. n = any integer l = ± 1 j = 0, ± 1 f = 0, ± 1 PY 3 P 05

Summary of Atomic Energy Scales 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 e. V 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 B, 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 ~ B/2000 due to nuclear spin. o E. g. , 21 -cm line in radio astronomy. PY 3 P 05