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Variation ofFundamental Constants from Big Bang

to Atomic Clocks

- Theory and observations

- 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

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.

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

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)

Relation between variations of different coupling

constants

- Grand unification models (Calmet,Fritzch

Langecker, Segre, Strasser)

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

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.

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

- 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.

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

Quasar absorption spectra

Gas cloud

Quasar

Earth

Light

a

Quasar absorption spectra

Gas cloud

Quasar

Earth

Light

One needs to know E(a2) for each line to do the

fitting

a

- 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

The correlation potential

Use the Feynman diagram technique to include

three classes of diagrams to all orders

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

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.

Relativistic shifts-doublets

Energies of normal fine structure doublets as

functions of a2

DEA(Za)2

0 (a/a0)2

1

Relativistic shifts-triplets

Energies of normal fine structure triplets as

functions of a2

DEA(Za)2

0 (a/a0)2

1

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

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!).

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

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

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 !

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)

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

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

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

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

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

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).

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!

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

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).

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

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

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

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!

More suggestions

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

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

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)

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

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

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

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

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

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 )

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,

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

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)

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

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

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

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

More suggestions

E. J. Angstmann et al, submitted to J. Phys. B

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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Text

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

Atoms of interest

1Nve number of valence electrons

Fine structure unomalies and level crossing

Energies of normal fine structure doublets as

functions of a2

DEA(Za)2

0 (a/a0)2

1

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

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!).

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

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