Lecture 7: Fine and hyperfine structure of one-electron Relativistic kinetic energy correction atoms o It was shown in Lecture 6 that fine-structure due to spin-orbit effects is: o Fine structure (1)
o Relativistic kinetic energy correction o But, from Lectures 1-2, the velocity of the electron is (2)
o The Lamb shift 4 4 o Therefore as the spin-orbit correction (!Eso) is of the order of " => !Eso ~ v .
o Hyperfine structure o The KE used when solving the Schrodinger equation had the form
o Nuclear moments which, using Eqn 2 = > KE ~ "2.
o If KE is expressed in its relativistic form:
o Second term is of order p4 => Relativistic KE will contribute of order "4
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Relativistic kinetic energy correction Relativistic kinetic energy correction o In non-relativistic Schrodinger theory, we wrote o The total energy correction correction for spin-orbit and relativistic effects can be obtained by combining Eqns. 1 and 3 and using the fact that j = l -1/2, l + 1/2: where o Including relativistic effects, H is replaced with
o Energy correction depends on n and j. where Hrel includes relativistic effects: o Using o Is of the order of 10-4 times smaller than principle energy splitting . o Using perturbation theory, it can be shown that o All levels are shifted down from Bohr energies. (3)
PY3004 PY3004 Gross and fine structure of hydrogen atom Fine structure of hydrogen o For H-atom, the spin-orbit and relativistic corrections are comparable in magnitude, o Energy correction only depends on j, which is but much smaller than the gross structure. of the order of 10-4 times smaller that the principle energy splitting.
o All levels are shifted down from the Bohr energies. 2 2 o Gross structure determined by En from Schrödinger equation. As En = -Z E0/n , 2 2 where E0 = 1/2" mc , we can write 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 Gives the energy of the gross and fine structure of the hydrogen atom. o Using incorrect assumptions, this fine structure was derived by Sommerfeld by modifying Bohr theory => right results, but wrong physics!
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Fine structure: Lamb shift Fine structure: Lamb shift
2 - 1. Excite H-atoms to 2 S1/2 metastable state by e bombardment. Forbidden to spontaneuosly o Spectral lines can give info on nucleus via isotope shift and hyperfine structure. 2 decay to 1 S1/2 optically. o According to Schrödinger and Dirac theory, states with same n and j but different l 2 2. Cause transitions to 2 P1/2 state using tunable microwaves. Transitions only occur when 2 are degenerate. However, Lamb and Retherford showed in 1947 that 2 S1/2 (n = 2, l microwaves tuned to transition frequency. These atoms then decay emitting H" line. 2 = 0, j = 1/2) and 2 P1/2 (n = 2, l = 1, j = 1/2) of H-atom are not degenerate. 2 3. Measure number of atoms in 2 S1/2 state from H-atom collisions with tungsten (W) target. 2 o Experiment proved that even states with the same total angular momentum J are When excitation to 2 P1/2, current drops. energetically different. 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.
PY3004 PY3004 Fine structure: Lamb shift Hyperfine structure: Nuclear moments
o Hyperfine structure can also result from magnetic interaction between the electron’s 2 o Experiment showed that the P1/2 state is slightly total angular momentum (J) and the nuclear spin (I). 2 lower than the S1/2 state. o Angular momentum of electron creates a magnetic field at the nucleus which is o The Lamb shift is due to the quantised EM field - a proportional to J. quantum electrodynamic effect. o Interaction energy is therefore 2 o Using " E"t # mc "t $ h the range over which force operates is: 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 o The proton-electron force is slightly weakened fine structure. when the electron is very close to nucleus => causes 2 2 the S1/2 level to be higher in energy than P1/2 level.
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Hyperfine structure: Nuclear moments Hyperfine structure: Nuclear moments o Like electron, the proton has a spin angular momentum and an associated intrinsic o The energy splitting of the hyperfine interaction dipole moment is given by F = 1 o The proton dipole moment is weaker than the electron dipole moment by M/m ~ F = 0 2000 and hence the effect is small. where a is the hyperfine structure constant. o Total angular momentum including nuclear spin, orbital angular momentum and o E.g., consider the ground state of H-atom. electron spin is Nucleus consists of a single proton, so I = 1/2. 2 The hydrogen ground state is the 1s S1/2 term, where 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 o The quantum number f has possible values f = j + 1/2, j - 1/2 since the proton has (1420 MHz). spin 1/2. 21 cm radio map of the Milky Way o For ground state of the hydrogen atom (n=1), o Hence every energy level associated with a particular set of quantum numbers n, l, the energy separation between the states of F = and j will be split into two levels of slightly different energy, depending on the 1 and F = 0 is 5.9 x 10-6 eV. relative orientation of the proton magnetic dipole with the electron state.
PY3004 PY3004 Gross, fine and hyperfine structure of hydrogen Selection rules for transitions
o Selection rules determine the allowed transitions between terms.
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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 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, called spin-orbit interaction. o Relativistic effects have similar magnitude. o Hyperfine structure: o Fine-structure lines are split into more multiplets. o Caused by interaction between electron spin and nucleus spin. o E.g., 21-cm line in radio astronomy.
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