Part 3 Higher order multipole moment effects A general theory Some experiments Elaborating our topic hyperfine structure Higher order multipole moment hyperfine effects Ref ID: 810571
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Slide1
Atomic Physics
Hyperfine structure
Part 3
Higher order
multipole
moment effects
A general
theory
Some experiments
Slide2Elaborating our topic – hyperfine structure….
Slide3Higher 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
T
e
(k)
• T
n
(k)
where the T
k
are tensor operators of rank k which operate on the electronic or the nuclear coordinates only.
For k≥1, the Hamiltonian terms are small, and can then yield 1
st
order perturbation energies:
Δ
E(hfs) = <IJF |
H
(hfs) | IJF> =
∑
k
<IJF | T
e
(k)
• T
n
(k)
| IJF>
Slide4The energy is independent of M(J &F) – hence we can use reduced matrix elements Racah – properties of spherical tensor operators – see below
Slide5EvaluationDefinitions:
With the usual definition of K, these become…
M1
Q2
O3
Slide6Evaluation of the matrix elements
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 – 2
5
? (static version not allowed)
And the one after that - 2
6
?
check out
http://physics.unl.edu/~tgay/content/multipoles.html
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:
Slide8Evaluation -3 Since these one-electron matrix elements depend sensitively on the electronic
wavefunctions
near the nucleus, we really need their correct relativistic form
Further 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.html Ref. 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.
Slide9Slide10Doppler 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)
Slide11Beginning the calculation…
Kinetic theory -> atom velocity gives a spectral distribution
Kinetic energy: mv
2
/2 = (3/2)kT ->
Hence
Calculating the detailed profile:
Slide12The Sodium and Rubidium “D-lines”
Which of these structures can be resolved in a room-temperature discharge?
Slide13A “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 90
0
to the two beams
Slide14Example of a doppler-free spectrum for Balmer-
α
in deuterium
Note the “cross-over resonance”:
Challenge 2:
Where does this come from?
Slide15More details of the experimental arrangement:
– 2 probe beams
Slide16More details of the experimental arrangement:
(b) – chopping the probe beam
Slide17Results from a “doppler-free” measurement in rubidium (ND advanced lab)
Doppler broadening
with hfs from probe beam
with chopping frequency
Slide18Slide19Slide20Cs 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
Slide21Results
Slide22See arXiv:0810.5745v2 [
physics.atom-ph
]
5 Dec 2008
Another example – by the same group
Slide23Previous results (1996)
New results (2008)
Slide24The
Octupole
moment is
only just non-zero
!
Slide25Another example – (as simple as possible): Important for (near) level-crossings of states of the SAME parityTheory
: hyperfine interaction is NOT ALWAYS a diagonal
HFS can mix states of the same J, but from different parent levels
N
eutral 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 small
But 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??
Slide26The Hamiltonian for the 1snD states
H
0
defines the energy of the
1
D term,
the other two terms are treated
perturbatively
together
Note
the off-diagonal term between the J=2 states:
(Gets large in JJ coupling…)
Slide27Slide28(Highest & lowest F are diagonal)
Slide29Slide30After matrix diagonalization
Slide31Slide32Beam-foil experiment – result for one transition decay
Slide33Slide34One 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. The
phenomenon was first discussed by
Bowen in 1930
, who
pointed out that the substantial strength that
was observed
in the
6S
2 1
S
0
- 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 6s
2
'So—6s6p
3
Po
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
.
Slide35Slide36Slide37Indelicato
et al – Phys. Rev. A3505 (1989)
Slide38Slide39For more details, see alsoDunford et al – Phys. Rev A 48, 2729 (1993)
Abstract
Slide40A 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 2
3
P0 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
Slide41Note: refs 6 and 7: (see next 2 slides for brief explanation…)
Slide42Ref. 6
Slide43Concerning the possible 1s2 1S0 – 1s2s 1
S
0
2 photon transition
Hyperfine mixing with
3
S
1
state
(see Indelicato et al)Weak interaction mixing with
3P0 state
The weak interaction HamiltonianMixes nearby opposite parity statesPossible 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 10
3
)
Slide44Ground state hyperfine levels
In H-like
Eu
(I=5/2)
Zero-field separation = 1.513 eV
Apply magnetic field and
laser
excite to different Zeeman levels
Ref 7.