Variation of Fundamental Constants from Big Bang to Atomic Clocks

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Variation ofFundamental Constants from Big Bang
to Atomic Clocks

• Theory and observations

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• V.V. Flambaum
• School of Physics, UNSW, Sydney, Australia and
  Argonne National Laboratory, USA
• Co-authors
• Atomic and molecular calculations
  V.Dzuba,M.Kozlov,E.Angstmann,J.Berengut,M.Marchenk
  o,Cheng Chin,S.Karshenboim,A.Nevsky
• Nuclear and QCD calculations
• E.Shuryak,V.Dmitriev,D.Leinweber,A.Thomas,R.Young
  ,A.Hoell,
• P.Jaikumar,C.Roberts,S.Wright,A.Tedesco,W.Wiringa
• Cosmology
• J.Barrow
• Quasar data analysis
• J.Webb,M.Murphy,M.Drinkwater,W.Walsh,P.Tsanavaris,
  S.Curran
• Quasar observations
• C.Churchill,J.Prochazka,A.Wolfe, Wiklind, Comb,
  thanks to W.Sargent,R.Simcoe

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Motivation

• Extra space dimensions (Kaluza-Klein, Superstring
  and M-theories). Extra space dimensions is a
  common feature of theories unifying gravity with
  other interactions. Any change in size of these
  dimensions would manifest itself in the 3D world
  as variation of fundamental constants.
• Scalar fields . Fundamental constants depend on
  scalar fields which vary in space and time
  (variable vacuum dielectric constant e0 ). May
  be related to dark energy and accelerated
  expansion of the Universe.

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Motivation

• Fine tuning of fundamental constants is needed
  for humans to exist. Example low-energy
  resonance in production of carbon from helium in
  stars (HeHeHeC). Slightly different coupling
  constants no resonance - no life.
• Variation of coupling constants in
  space provide natural explanation of the fine
  tuning we appeared in area of the Universe
  where values of fundamental constants are
  suitable for our existence.

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Search for variation of fundamental constants

• Big Bang Nucleosynthesis
• Quasar Absorption Spectra 1
• Oklo natural nuclear reactor
• Atomic clocks 1

Dcgt0?
Dcgt0?
1 Based on analysis of atomic spectra
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Which Constants?

• Since variation of dimensional constants
  cannot be distinguished from variation of units,
  it only makes sense to consider variation of
  dimensionless constants.
• Fine structure constant ae2/hc1/137.036
• Electron or quark mass/QCD strong interaction
  scale, me,q/LQCD
• a strong (r)const/ln(r LQCD /ch)
• me,q are proportional to Higgs vacuum (weak
  scale)

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Relation between variations of different coupling
constants

• Grand unification models (Calmet,Fritzch
  Langecker, Segre, Strasser)

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• a 3 -1(m)a strong -1 (m)b3ln(m /LQCD )
• a -1(m)5/3 a 1 -1(m) a 2 -1(m)

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Dependence on quark mass

• Dimensionless parameter is mq/LQCD . It is
  convenient to assume LQCD const, i.e. measure mq
  in units of LQCD
• mp is proportional to (mqLQCD)1/2
  Dmp/mp0.5Dmq/mq
• Other meson and nucleon masses remains finite for
  mq0. Dm/mK Dmq/mq
• Argonne K are calculated for p,n,r,w,s.

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Nuclear magnetic moments depends on p-meson mass
mp
Nucleon magnetic moment
p
n
p
p
Spin-spin interaction between valence and core
nucleons
p
n
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• Nucleon magnetic moment

Nucleon and meson masses
QCD calculations lattice, chiral perturbation
theory,cloudy bag model, Dyson-Schwinger and
Faddeev equations, semiempirical. Nuclear
calculations meson exchange theory of strong
interaction. Nucleon mass in kinetic energy p2/2M
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Deuterium bottleneck

• At temeperature Tlt0.3 Mev all abundances follow
  deuteron abundance
• (no other nuclei produced if there are no
  deuterons)
• Reaction g d n p , exponentially small number
  of energetic photons, e-( Ed/T)
• Exponetilal sensitivity to deuteron binding
  energy Ed , Ed2 Mev ,
• Freezeout temeperure Tf 30 KeV

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New BBN result

• Dent, Stern, Wetterich dependence of BBN on
  energies of 2,3H,3,4He,6,7Li ,7Be
• Flambaum,Wiringa dependence of binding energies
  of 2,3H,3,4He,6,7Li, 7,8Be on nucleon and meson
  masses,
• Flambaum,Holl,Jaikumar,Roberts,Write,
• Maris dependence of nucleon and meson masses on
  light quark mass mq.

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Big Bang Nucleosynthesis Dependence on mq/ LQCD

• 2H 17.7x1.07(15) x0.009(19)
• 4He 1-0.95x1.005(36) x-0.005(38)
• 7Li 1-50x0.33(11) x0.013(02)
• Final result (Flambaum,Wiringa 2007)
• xDXq/Xq 0.013 (02), Xqmq/ LQCD
• Dominated by 7Li abundance (3 times
  difference), consistent with 2H,4He

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Alkali Doublet Method(Bahcall,SargentVarshalovic
h, Potekhin, Ivanchik, et al)

• Fine structure interval
• DFS E(p3/2) - E(p1/2) A(Za)2
• If Dz is observed at red shift z and D0 is FS
  measured on Earth then

Ivanchik et al, 1999 Da/a -3.3(6.5)(8) x
10-5. Murphy et al, 2001 Da/a -0.5(1.3) x
10-5.
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Variation of fine structure constant a Dzuba,
Flambaum,Webb
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Many Multiplet Method(Dzuba,Flambaum, Webb)
p3/2
p3/2
p1/2
p1/2
dw gtgt dDFS !
w
w
s1/2

s1/2
a1
a2

• Advantages
• Order of magnitude gain in sensitivity
• Statistical all lines are suitable for analysis
• Observe all unverse (up to z4.2)
• Many opportunities to study systematic errors

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Quasar absorption spectra
Gas cloud
Quasar
Earth
Light
a
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Quasar absorption spectra
Gas cloud
Quasar
Earth
Light
One needs to know E(a2) for each line to do the
fitting
a
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• Use atomic calculations to find w(a).
• For a close to a0 w w0 q(a2/a02-1)
• q is found by varying a in computer codes
• q dw/dx w(0.1)-w(-0.1)/0.2, xa2/a02-1

a e2/hc0 corresponds to non-relativistic limit
(infinite c).
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Correlation potential method
Dzuba,Flambaum,Sushkov (1989)

• Zeroth-order relativistic Hartree-Fock.
  Perturbation theory in difference between exact
  and Hartree-Fock Hamiltonians.
• Correlation corrections accounted for by
  inclusion of a correlation potential ?

In the lowest order ? is given by

• External fields included using Time-Dependent
  Hartree-Fock (RPAE core polarization)correlation
  s

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The correlation potential
Use the Feynman diagram technique to include
three classes of diagrams to all orders
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The correlation potential
Use the Feynman diagram technique to include
three classes of diagrams to all orders
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Atoms of interest
1Nve number of valence electrons
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Methods of Atomic Calculations
These methods cover all periodic system of
elements

• They were used for many important problems
• Test of Standard Model using Parity Violation in
  Cs, Tl
• Predicting spectrum of Fr (accuracy 0.1), etc.

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Relativistic shifts-doublets
Energies of normal fine structure doublets as
functions of a2
DEA(Za)2
0 (a/a0)2
1
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Relativistic shifts-triplets
Energies of normal fine structure triplets as
functions of a2
DEA(Za)2
0 (a/a0)2
1
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Fine structure anomalies and level crossing
Energies of strongly interacting states as
functions of a2
DEA(Za)2
1D2
3P0,1,2
0 (a/a0)2
1
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Implications to study of a variation

• Not every energy interval behaves like
  DEAB(Za)2 .
• Strong enhancement is possible (good!).
• Level crossing may lead to instability of
  calculations (bad!).

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Problem level pseudo crossing
Energy levels of Ni II as functions of a2
Values of qdE/da2 are sensitive to the
position of level crossing
0 (a/a0)2
1
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Problem level pseudo crossing
Energy levels of Ni II as functions of a2

• Values of qdE/da2 are sensitive to the
  position of level crossing

Solution matching experimental g-factors
0 (a/a0)2
1
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Results of calculations (in cm-1)
Negative shifters
Anchor lines
Positive shifters
Also, many transitions in Mn II, Ti II, Si IV, C
II, C IV, N V, O I, Ca I, Ca II, Ge II, O II, Pb
II
Different signs and magnitudes of q provides
opportunity to study systematic errors!
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• Murphy et al, 2003 Keck telescope, 143 systems,
  23 lines, 0.2ltzlt4.2
• Da/a-0.54(0.12) x 10-5

• Quast et al, 2004 VL telescope, 1 system, Fe II,
  6 lines, 5 positive q-s, one negative q, z1.15
• Da/a-0.4(1.9)(2.7) x 10-6

• Srianand et al, 2004 VL telescope, 23 systems,
  12 lines, Fe II, Mg I, Si II, Al II, 0.4ltzlt2.3
• Da/a-0.06(0.06) x 10-5
• Murphy et al 2007 Da/a-0.64(0.36) x 10-5
• Further revision may be necessary.

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Request for laboratory measurements shopping
list physics/0408017

• More accurate measurements of UV transition
  frequencies
• Measurements of isotopic shifts
• Cosmological evolution of isotope abundances in
  the Universe
• a). Systematics for the variation of a
• b). Test of theories of nuclear reactions in
  stars and supernovae

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Two sets of line pairs

• 1.dalt0 imitated by compression of the spectrum
• 2. dalt0 imitated by expansion of the spectrum
• Both sets give dalt0 !

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Spatial variation (Steinhardt list update)

• 10
  5 Da/a
• Murphy et al
• North hemisphere -0.66(12)
• South (close to North) -0.36(19)
• Strianand et al (South) -0.06(06)??
• Murphy et al (South) -0.64(36)

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hyperfinea2 gp me / Mp atomic units
Rotationme/Mp atomic units
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Measurements me / Mp or me / LQCD

• Tsanavaris,Webb,Murphy,Flambaum,
• Curran PRL 2005
• Hyperfine H/optical , 9 quasar absorption systems
  with Mg,Ca,Mn,C,Si,Zn,Cr,Fe,Ni
• Measured Xa2 gp me / Mp
• DX/X0.6(1.0)10-5 No variation

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Best limit from ammonia NH3Flambaum, Kozlov
PRL2007

• Inversion spectrum exponentially smallquantum
  tunneling frequency winvW exp(-S)
• S(me / Mp )-0.5 f(Evibration/Eatomic) ,
  Evibration/Eatomic const (me / Mp )-0.5
• winv is exponentially sensitive to me / Mp
• First enhanced effect in quasar spectra, 5 times
• D(me / Mp )/ (me / Mp)-0.6(1.9)10-6 No
  variation
• z0.68, 6.5 billion years ago, -1(3)10-16 /year
• Combined with Hg(opt)/Cs clocks Fortier et al
  2007-
• best limit on variation of a -0.8(0.8)10-16
  /year

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Measurements me / Mp or me / LQCD

• Reinhold,Buning,Hollenstein,Ivanchik,
• Petitjean,Ubachs PRL 2006 , H2 molecule, 2
  systems
• D(me / Mp )/ (me / Mp)-2.4(0.6)10-5 Variation
  4 s !
• Space-time variation? Grand Unification model?

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Oklo natural nuclear reactor

• n149Sm capture cross section is dominated by
• Er 0.1 eV resonance
• ShlyakhterDamour,DysonFujii et al
• Limits on variation of alpha
• Flambaum,Shuryak 2002,2003 Dmitriev,Flambaum 2003
• DEr 100 MeV DXs/Xs- 10 MevDXq/Xq 1 MeV Da/a
• Xsms/ LQCD , enhancement 100 MeV/0.1 eV109
• 2006 Gould et al, Petrov et al DEr lt0.1eV ,
• DX/X lt10-9 two billion years ago, 10-18
  /year

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Oklo natural nuclear reactor

• 1.8 billion years ago
• n149Sm capture cross section is dominated by
  Er 0.1 eV resonance
• ShlyakhterDamour,DysonFujii et al
• DEr 1 MeV Da/a
• Limits on variation of alpha

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Oklo limits on Xqmq/ LQCD

• Flambaum,Shuryak 2002,2003 Dmitriev,Flambaum 2003
• Flambaum,Wiringa 2007
• 150Sm DEr 10 MeV DXq/Xq - 1 MeV Da/a
• Limits on xDXq/Xq - 0.1 Da/a from
• Fujii et al DErlt0.02 eV xlt2.10-9
• Petrov et al DErlt0.07 eV xlt8. 10-9
• Gould et al DErlt0.026 eV xlt3. 10-9
  , lt1.6 10-18 y-1
• There is second, non-zero solution x1.0(1)
  10-8

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Atomic clocks

• Cesium primary frequency standard

F4 F3
HFS of 6s
n 9 192 631 770 Hz
Also Rb, Cd, Ba, Yb, Hg, etc.
E.g. n(Hg) 40 507 347 996.841 59(14)(41) Hz
(D. J. Berkeland et al, 1998).
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Optical frequency standards
Also H, Al, Sr, Ba, Yb, Hg, Hg, Tl, Ra, etc.
Accuracy about 10-15 can be further improved to
10-18!
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Atomic clocks

• Comparing rates of different clocks over long
  period of time can be used to study time
  variation of fundamental constants!

Optical transitions a Microwave
transitions a, (me, mq )/LQCD
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Advantages

• Very narrow lines, high accuracy of measurements.
• Flexibility to choose lines with larger
  sensitivity to variation of fundamental
  constants.
• Simple interpretation (local time variation).

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Calculations to link change of frequency to
change of fundamental constants

• Optical transitions atomic calculations (as for
  quasar absorption spectra) for many narrow lines
  in Al II, Ca I, Sr I, Sr II, In II, Ba II, Dy I,
  Yb I, Yb II, Yb III, Hg I, Hg II, Tl II, Ra II
• w w0 q(a2/a02-1)

• Microwave transitions hyperfine frequency is
  sensitive to a and to nuclear magnetic moments

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We performed atomic, nuclear and QCD calculations

• of powers k ,b for H,D,He,Rb,Cd,Cs,Yb,Hg
• VC(Ry)(me/Mp)a2k (mq/LQCD)b , Dw/wDV/V
• 133Cs k 0.83, b0.009
• Cs standard is insensitive to variation of
  nuclear magnetic
• g-factor!
• 87Rb k 0.34, b-0.016
• 171Yb k 1.5, b-0.085
• 199Hg k 2.28, b-0.088
• 1H k 0, b-0.100
• Complete Table in Flambaum, Tedesco PRC 2006

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Results for variation of fundamental constants
aassuming mq/LQCD Const
Combined results d/dt lna -0.3(0.3) x 10-15
yr-1 d/dt
ln(mq/LQCD) 8(22) x10-15 yr-1
me /Mp or me/LQCD
1.5(1.7)x10-15 yr -1
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Dysprosium miracle

• Dy 4f105d6s E19797.96 cm-1 , q 6000
  cm-1
• 4f95d26s E19797.96 cm-1 , q -23000
  cm-1
• Interval Dw 10-4 cm-1
• Dzuba, Flambaum Enhancement factor K 108
  (!), i.e. Dw/w0 108 Da/a

Measurements (Berkeley,Los Alamos) dlna/dt
-2.7(2.6)x 10-15 yr-1
Problem states are not narrow!
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More suggestions
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Enhancement in molecular clocks

• DeMille 2004 enhancement in Cs2 cancellation
  between electron excitation and vibration
  energies
• Flambaum 2006 Cancellations between rotational
  and hyperfine intervals in very narrow microwave
  transitions in LaS, LaO, LuS,LuO, YbF, etc.
• w0 Erotational -E hyperfine E hyperfine
  /100-1000
• Dw/w0 K Da/a Enhancement K 102 -103

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Cancellation between fine structure and vibrations

• Flambaum, Kozlov PRL2007 K 104 -105,
• SiBr, Cl2 microwave transitions between
  narrow excited states, sensitive to a and
  mme/Mp
• w0 E fine - Evibrational E fine /K
• Dw/w0 K (Da/a -1/4 Dm/m)
• Enhancement K 104 -105
• E fine is proportional to Z2a2
• Evibrational nw is proportional to nm0.5 ,
  n1,2,
• Enhancement for all molecules along the lines
  Z(m,n)
• Shift 0.03 Hz for Da/a10-15 width 0.01 Hz
  
• Compare with Cs/Rb hyperfine shift 10-5 Hz
• HfF K 103 shift 1 Hz

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Cancellation between fine structure and rotation
in light molecules

• Bethlem,Bunning,Meijer,Ubach 2007
• OH,OD,CN,CO,CH,LiH,
• E fine is proportional to Z2a2
• Erotational is proportional to Lm , L0,1,2,
• mme/Mp
• Enhancement for all molecules along the lines
  Z(m,L)

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Nuclear clocks(suggested by Peik,Tamm 2003)

• Very narrow UV transition between first excited
  and ground state in 229 Th nucleus
• Energy 7.6(5) eV, width 10-4 Hz
• Flambaum PRL2006
• Nuclear/QCD estimate Enhancement 105 ,
• Dw/w0 105 ( 0.1Da/a DXq/Xq)
• Xqmq/ LQCD ,
• Shift 105 Hz for Da/a10-15
• Compare with atomic clock shift 1 Hz
• 235 U energy 76 eV, width 6 10-4 Hz

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Nuclear clocks(suggested by Peik,Tamm 2003)

• Very narrow UV transition between first excited
  and ground state in 229 Th nucleus
• Energy 7.6(5) eV, width 10-4 Hz
• Flambaum PRL2006
• Nuclear/QCD estimate Enhancement 105 ,
• Dw/w0 105 ( 0.1Da/a 0.5DXq/Xq-5DXs/Xs )
• Xqmq/ LQCD , Xsms/ LQCD
• Shift 105 Hz for Da/a10-15
• Compare with atomic clock shift 1 Hz
• 235 U energy 76 eV, width 6 10-4 Hz

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Why enhancement is so large?

• Total Coulomb energy 103 MeV in 229 Th
• Difference of moments of inertia between ground
  and excited states 4 (Feldmaier)
• If this is due to the difference in deformation,
  the Coulomb energy would change by Q26 MeV
• Neutron removal Q1.3 Mev
• Upper estimate for the enhancement
• Q/w0 lt 1.3 x106 eV / 7 eV 2x105

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Enhancement factors in 229Th

• a Xqmq/ LQCD
• Flambaum 2006 105 0.5 105
  estimate
• Hayes,Frier 2007 0 impossible arguments
• He,Ren 2007 0.04 105 0.8 105
  rel.mean field
• Main effect (dependence of deformation on a)
  missed, change of mean-field potential only
• Dobaczewski
• et al 2007 0.15 105
  Hartree-Fock
• 
  preliminary

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229Th Flambaum,Wiringa 2007

• wEpkEso 7.6 eV huge cancellations!
• Eso ltVs L Sgtspin-orbit-1.04 MeV
• Epk potentialkinetic1 MeV
• Extrapolation from light nuclei
• DEpk/Epk-1.4 Dmq/mq
• DEso/Eso-0.24 Dmq/mq
• Dw/w0 1.6 105 DXq/Xq

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229Th Flambaum,Wiringa 2007

• wEpkQEso 7.6 eV huge cancellations!
• QCoulomb105 KeV, Dobaczewski et al
• Eso ltVs L Sgtspin-orbit-1.04 MeV
• Epk potentialkinetic1 MeV
• Extrapolation from light nuclei
• DEpk/Epk-1.4 Dmq/mq
• DEso/Eso-0.24 Dmq/mq
• Dw/w0 105 ( 0.15 Da/a 1.6 DXq/Xq )

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Experimental progress in 229Th

• Transition energy measured in Livermore
• 7.6 (5) eV instead of 3.5(1.0) eV
• Intensive search for direct radiation
• Argonne
• Peik et al,
• Habs et al,

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Ultracold atomic and molecular collisions (in
Bose condensate). Cheng Chin, Flambaum PRL2006

• Enhancement near Feshbach resonance.
• Variation of scattering length
• a/aK Dm/m , K102 1012
• mme/Mp
• Hart,Xu,Legere,Gibble Nature 2007
• Accuracy in scattering length 10-6

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Evolution fundamental constants and their
dependence on gravitational potential

• Fundamental constants depend on scalar field f -
  dark energy, Higgs, dilaton, distance between
  branes, size of extra dimensions.
• Cosmological evolution of f in space and time is
  linked to evolution of matter.
• Changes of Universe equation of state
• Radiation domination, cold matter domination,
  dark energy domination-
• Change of f - change of a(f)

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Scalar charge-source of f

• Massive bodies have scalar charge S proportional
  to the number of particles
• Scalar field fS/r , proportional to
  gravitational potential GM/r -
• Variation of a proportional to gravitational
  potential
• da/aKa d(GM/rc2)
• Neutron star, white/brown dwarfs, galaxy, Earth,
  Sun compare spectra, w(a)

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Dependence of fundamental constants on
gravitational potential

• Projects atomic clocks at satellites in space or
  close to Sun
• Earth orbit is elliptic,3 change in distance to
  Sun
• Fortier et al Hg(opt)/Cs , Ashby et al -H/Cs
• Flambaum,Shuryak limits on dependence of a, me/
  LQCD and mq/ LQCD on gravity
• Ka 0.17Ke-3.5(6.0) 10-7
• Ka 0.13 Kq2(17) 10-7

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Dysprosium da/aKa d(GM/rc2)

• Dy 4f105d6s E19797.96 cm-1 , q 6000
  cm-1
• 4f95d26s E19797.96 cm-1 , q -23000
  cm-1
• Interval Dw 10-4 cm-1
• Enhancement factor K 108 , i.e. Dw/w0 108
  Da/a

Measurements (Berkeley-Los Alamos-Yale-Sydney) Ka
-8.7(6.6) 10-6 Ke4.9(3.9) 10-6 Kq6.6(5.2)
10-6
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Sr/Cs comparison Boulder-Paris-Tokyo-Sydney

• New best limits

Ka-2.3(3.1) 10-6 Ke1.1(1.7) 10-6
Kq1.7(2.7) 10-6
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Conclusions

• Quasar data MM method provided sensitivity
  increase 100 times. Anchors, positive and
  negative shifters-control of systematics. Keck-
  variation of a, VLT-?. Systematics or spatial
  variation.
• me /Mp hyperfineH/optical, NH3 no variation,
  H2 - variation 4 s . Space-time variation?
  Grand Unification model?
• Big Bang Nucleosynthesis may be interpreted as a
  variation of
• mq/ LQCD ?
• Oklo sensitive to mq/ LQCD ,, effect lt3 10-9
• Atomic clocks present time variation of a , m/
  LQCD
• Transitions between narrow close levels in atoms,
  molecules and nuclei huge enhancement!
• Dependence of fundamental constants on
  gravitational potential
• No variation for small redshift, hints for
  variation at high red shift

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More suggestions
E. J. Angstmann et al, submitted to J. Phys. B
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Publications

• V. A. Dzuba, V. V. Flambaum, J, K. Webb, PRL 82,
  888 (1999).
• V. A. Dzuba, V. V. Flambaum, J, K. Webb, PRA 59,
  230 (1999).
• V. A. Dzuba, V. V. Flambaum, PRA 61, 034502
  (2000).
• V. A. Dzuba, V. V. Flambaum, M. T. Murphy, J, K.
  Webb, LNP 570, 564 (2001).
• J. K. Webb et al , PRL 87, 091301 (2001).
• V. A. Dzuba, V. V. Flambaum, M. T. Murphy, J, K.
  Webb, PRA 63, 042509 (2001).
• M. M. Murphy et al, MNRAS, 327, 1208 (2001).
• V. A. Dzuba et al, PRA, 66, 022501 (2002).
• V. A. Dzuba, V. V. Flambaum, M. V. Marchenko, PRA
  68, 022506 (2003).
• E. J. Angstmann, V. A. Dzuba, V. V. Flambaum, PRA
  70, 014102 (2004).
• J. C. Berengat et al, PRA 70, 064101 (2004).
• M. M. Murphy et al, LNP, 648, 131 (2004).
• V. A. Dzuba, PRA, 71, 032512 (2005).
• V. A. Dzuba, V. V. Flambaum, PRA, 71, 052509
  (2005).
• V. A. Dzuba, V. V. Flambaum, PRA, 72, 052514
  (2005).
• V. A. Dzuba, PRA, 71, 062501 (2005).
• S. G. Karshenboim et al, physics/0511180.

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Alkali Doublet Method(Bahcall,Sargent,Varshalovic
h, Potekhin, Ivanchik, et al)

• Fine structure interval
• DFS E(p3/2) - E(p1/2) A(Za)2
• If DZ is observed at red shift Z and D0 is FS
  measured on Earth then

Ivanchik et al, 1999 Da/a -3.3(6.5)(8) x
10-5. Murphy et al, 2001 Da/a -0.5(1.3) x
10-5.
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Many Multiplet Method(Flambaum, Webb, Murphy, et
al)
p3/2
p3/2
p1/2
p1/2
dw gtgt dDFS !
w
w
s1/2

s1/2
a1
a2

• Advantages
• Order of magnitude gain in sensitivity
• Statistical all lines are suitable for analysis
• Many opportunities to study systematic errors

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Atoms of interest
1Nve number of valence electrons
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Fine structure unomalies and level crossing
Energies of normal fine structure doublets as
functions of a2
DEA(Za)2
0 (a/a0)2
1
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Fine structure unomalies and level crossing
Energies of strongly interacting states as
functions of a2
DEA(Za)2
1D2
3P0,1,2
0 (a/a0)2
1
141
Implications to study of a variation

• Not every fine structure interval can be used in
  the analysis based on formula DEA(Za)2 (not
  good!).
• Strong enhancement is possible (good, but for
  atomic clocks only).
• Level crossing may lead to instability of
  calculations (bad!).

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Problem level pseudo crossing
Energy levels of Ni II as functions of a2
Values of qdE/da2 are sensitive to the
position of level crossing
0 (a/a0)2
1
143
Pb II g-factors dont help
Energy levels of Pb II as functions of a2

• Two 3D3/2 states are strongly mixed, but
  g-factors do not depend on mixing.

2D3/2
2D5/2
2D3/2
2D5/2
2S1/2
Solution perform calculations with extremely
high accuracy.
4P1/2
4P5/2
4P3/2
0 (a/a0)2
1