Atomic Physics

Hyperfine structurePart 3

Higher order multipole moment effectsA general theory

Some experiments

Elaborating our topic – hyperfine structure….

The National Science Museum, Delhi

Higher order multipole moment hyperfine effects

Ref: Theory of hyperfine structure: Schwartz, Phys. Rev. 97, 380 (1955)

The previous work on the E0, M1 and E2 interactions between the nucleus and the electron(s) suggests that the general nuclear interaction can be written in the form:

H(hfs) = ∑k Te(k) • Tn

(k)

where the Tk are spherical tensor operators of rank k which operate each on the electronic or the nuclear coordinates only.

For k≥1, the Hamiltonian terms are small, and can then yield 1st order perturbation energies:

ΔE(hfs) = <IJF | H(hfs) | IJF> = ∑k <IJF | Te(k) • Tn

(k) | IJF>

The energy is independent of M(J &F) – hence we can use reduced matrix elements Racah – properties of spherical tensor operators – see below

EvaluationDefinitions:

With the usual definition of K, these become…

M1

Q2

O3

Evaluation of the matrix elements1. Expand the electrostatic interaction due to charge distributions:

next terms in spherical harmonic expansion used for the Q2 interaction – next term is the hexadecapole term E16 (never observed in hfs, but observed in nuclear charge distributions)

Question:What is the name of the next multipole – 25 ? (static version not allowed)And the one after that - 26?

check out http://physics.unl.edu/~tgay/content/multipoles.html

2. Expand the nuclear current for the magnetic interactions: requires expansion in vector spherical harmonics – see Schwartz paper, plus an E&M text such as Jackson….Yielding working definitions for the magnetic and electric nuclear multipole

moments

Evaluation - 2

The multipole expansions of the potentials are then:

Evaluation -3Since these one-electron matrix elements depend sensitively on the electronic

wavefunctions near the nucleus, we really need their correct relativistic formFurther topics– We need to learn how to solve the Dirac equation for the relativistic hydrogen atom. - Second part of the Schwartz paper evaluates these equations, and also generalizes the hyperfine interactions to 2-electron systems, off-diagonal interactions (needed for some second order interaction cases).- For more information on irreducible tensor operators, see:-Ref. 1 – see Fischer et al, appendix (word version on Berry website)- http://atoms.vuse.vanderbilt.edu/Elements/CompMeth/HF/node24.htmlRef. 2 – see Breinig - Rotations, spherical tensor operators and the Wigner-Eckart Theorem (word version on Berry website) at

http://electron6.phys.utk.edu/qm2/modules/m4/wigner.htm

Now we turn to the only measurement of the hyperfine structure due to an octupole moment.

Doppler broadening

Challenge #1: You have a discharge containing rubidium:Calculate the width of the spectral line of the resonance transition in rubidium at room temperature.

Procedure:

1 - Work in pairs

2 – (a) understand the problem (b) develop a set of steps to reach the solution(c) estimate the answer

3 - What parameter values do you need to get an answer?(Use symbols for these parameters until you get to the end of the problem)

Beginning the calculation…

Kinetic theory -> atom velocity gives a spectral distribution

Kinetic energy: mv2/2 = (3/2)kT ->

Hence

Calculating the detailed profile:

The Sodium and Rubidium “D-lines”

Which of these structures can be resolved in a room-temperature discharge?

A “Doppler-free” technique

In case (b) the second beam sees fewer ground state atoms ready to excite:

Consider the resonant frequencies for these atoms: only those atoms whose velocities match in both cases will be affected – hence OPPOSING PUMP and PROBE laser beams will only affect atoms with velocities at 900 to the two beams

Example of a doppler-free spectrum for Balmer-α in deuterium

Note the “cross-over resonance”:

Challenge 2:Where does this come from?

More details of the experimental arrangement:

(a) – 2 probe beams

More details of the experimental arrangement:

(b) – chopping the probe beam

Results from a “doppler-free” measurement in rubidium (ND advanced lab)

(a) Doppler broadening(b) with hfs from probe beam(c) with chopping frequency

Cs vapor is heated in an oven to 170 C. The atoms effuse through a nozzle constructed from an array of stainless steel tubes to produce a dense atomic beam, collimated with a stack of microscope cover slips. The resulting atomic beam has an angular divergence less than 13.6 mrad in the horizontal plane confirmed by the experimentally determined 2.3(1) MHz residual Doppler width of the spectral lines. The Cs atoms are excited with a probe laser perpendicular to the atomic beam. The fluorescence intensity is detected with a large area photodiode placed below the laser-atom interaction region. A second detector monitors the transmission of the probe beam.

The crossed-beam experiment

Results

See arXiv:0810.5745v2 [physics.atom-ph] 5 Dec 2008

Another example – by the same group

Previous results (1996)

New results (2008)

The Octupole moment is only just non-zero!

A beam-foil example – (as simple as possible): Important for (near) level-crossings of states of the SAME parity

Theory: hyperfine interaction is NOT ALWAYS a diagonalHFS can mix states of the same J, but from different parent levels

Neutral helium, isotope-3 nuclear spin ½ (i.e. only magnetic dipole)Almost completely LS-structure, with small JJ mixing (all states)Expect almost no hfs in singlet states!!!

1snd Triplet D and singlet D states: fine structure is smallBut 1s hyperfine structure is large!Hence, off-diagonal matrix elements…

See Brooks et al, Nucl. Instr. Meths. 202, 113 (1982)[Expt and theory]Q? Are there any “heavy ion” examples –

eg in close-lying ground states?? Or excited states??

The Hamiltonian for the 1snD states

H0 defines the energy of the 1D term,the other two terms are treated perturbatively together

Note the off-diagonal term between the J=2 states:(Gets large in JJ coupling…)

(Highest & lowest F are diagonal)

After matrix diagonalization

Beam-foil experiment – result for one transition decay

One of the most interesting phenomena in the theory of highly forbidden transitions is the effect of hyperfine quenching, whereby mixing by the hyperfine interaction can significantly alter the lifetimes of the levels. Thephenomenon was first discussed by Bowen in 1930, who pointed out that the substantial strength that was observed in the 6s2 1S0 – 6s6p 3P2 line at 2270 A in the spectrum of Hg I was primarily due to E1 radiation caused by coupling with the nuclear spin and not to possible higher-order multipole radiation as had been suggested.

Hyperfine quenching in atomic spectra ref: Dunford et al Phys. Rev. A44, 764 (1991)

Bowen's conclusion was confirmed in 1937 by Mrozowski [3], who experimentally observed the 6s2 'So—6s6p 3Po line at 2656 A in Hg I. This transition would be rigorously forbidden to all multipole orders of single-photon decay in an atom with a spinless nucleus by the J=0~J=0 selection rule of angular-momentum conservation.

Indelicato et al – Phys. Rev. A3505 (1989)

For more details, see alsoDunford et al – Phys. Rev A 48, 2729 (1993)

Abstract

A Bondarevskaya et al

Abstract. The hyperfine quenching (HFQ) mechanism of metastable states in polarized He-like heavy ions is considered. The lifetime dependence of these states on the ion polarization in an external magnetic field is established. This dependence is presented for the 23P0 state of the europium (Z = 63) ion and is proposed as a method for the measurement of the ion polarization in the experiments for the search of parity violating effects.

The hyperfine quenching of polarized two-electron ions in an external magnetic field

Note: refs 6 and 7: (see next 2 slides for brief explanation…)

Ref. 6

Concerning the possible 1s2 1S0 – 1s2s 1S02 photon transition

1. Hyperfine mixing with 3S1 state (see Indelicato et al)

1. Weak interaction mixing with 3P0 state

The weak interaction HamiltonianMixes nearby opposite parity states

Possible experiment requires an electron polarized beam:One possibility is a tilted thin foil target -Then, observations should see a small 180 degree anisotropy(note: anisotropy is very small – less than 1 in 103)

Ground state hyperfine levelsIn H-like Eu (I=5/2)

Zero-field separation = 1.513 eV

Apply magnetic field and laser excite to different Zeeman levels

Ref 7.

Note 1 – chair seat springsNote 2 – as the radii go to zero, the magnetic and electrostatic multipole fields go to zeroNote 3 – the toroidal multipole fileds do NOT go to zero

the t-dipole moment is called the anapole moment

See

Complete electromagnetic multipole expansion including toroidal moments

A. Gongora T. And E. Ley-Koo

REVISTA MEXICANA DE FISICA E 52 (2) 188–197 2006

The electromagnetic multipole expansion presented in this paper is complete on two accounts: i) It is valid for all points in space, and ii) it recognizes the existence of toroidal moments. The electromagnetic field due to alternating poloidal currents in a toroidal solenoid is evaluated exactly via the solution of the inhomogeneous vector Helmholtz equations, using the outgoing wave Green function technique andthe Debye potentials for sources and fields. The physical meaning of the toroidalmoments can be appreciated when they are compared with the familiar electric and magnetic moments; the analysis of the long-wavelength limit of the exact results also explains the previous neglect of the toroidal moments. The magnetostatic limit and the point source limit are also physically and didactically interesting

Next week:

1 – Trapping

2 – Incomplete stories about nothing and infinity