Atomic Physics with Stored and Cooled Ions

Size: px

Start display at page:

Download "Atomic Physics with Stored and Cooled Ions"

Cassandra Benson
5 years ago
Views:

1 Lecture #5 Atomic Physics with Stored and Cooled Ions Klaus Blaum Gesellschaft für Schwerionenforschung, GSI, Darmstadt and CERN, Physics Department, Geneva, Switzerland Summer School, Lanzhou, China, 9 17 August Lecture: Test of QED in extreme fields: g-factor of the free and bound electron 1. The g-factor of the leptons 2. The g-factor of the bound electron in highly charged ions 3. New determination of the electron mass

2 History of the magnetic moment of the electron Thompson: discovery of the electron Stern, Gerlach: quantization of magnetic moment Goudsmit, Uhlenbeck: postulate of intrinsic angular momentum (spin) Dirac: g = 2 for the free electron Breit: Dirac equation for the bound electron (relativistic correction) Foley, Kusch (exp), Schwinger (theo): QED correction for the free electron Crane, Rich et al.: g-2 of the free electron in inhomogeneous magnetic fields Grotch, Hegstrom: QED correction for the g-factor of the bound electron Gräff, Klempt, Werth, Dehmelt, Wineland, van Dyck et al.: measurement of g-2 in Penning trap of the free electron Kinoshita: QED corrections up to the order of α 4 for the free electron Persson et al.: QED corrections for the bound electron, higher orders in Zα W. Quint et al.: measurement of the g-factor of the bound electron

Magnetic moment and g-factor of the lepton Dirac: g s = 2; g l = 1 1947: experiment by Foley and Kusch g s 2 30.

3 Magnetic moment and g-factor of the lepton Dirac: g s = 2; g l = : experiment by Foley and Kusch g s December 1947 : relation between magnetic dipole moment and angular momentum Theory by Schwinger g s = 2 (1+ ½ α/π) free lepton: g s = g-factor of the spin 1 st QED correction, order α/π, one internal photon line = emission and absorption of a virtual photon self energy

4 Calculation of the g-factor of the free electron Feynman symbols in- and outgoing free fermions propagating free fermions propagating photons theory vertices

5 QED contributions to the g-factor of the free electron g e = 2 + C 1 α + C 2 α 2 + C 3 α nd order: Schwinger term C 1 α/π = ½ α/π = C 1 = ½ 6 th order C 3 (α/π) 3 C 3 = th order C 2 (α/π) 2 C 2 = th order is calculated

6 plus two spin directions g-2 transition of the free electron

7 Principle of the g-2 determination for the free electron cyclotron frequency Lamor frequency is the revolution frequency of a charged particle in an external magnetic field B: Lorentz force and centrifugal force are equal. ω c = F = m R = ev B = F z ωc 2 L e m B ω c = µ 2 hb B η is the precession frequency of a magnetic moment in an external magnetic field B. It equals the frequency required for flipping the spin. ρ D ρ = µ ω = L S ρ B = g S d µ B hη ρ ds t B Dirac: g s = 2 ω = ω c g s = 2 (1 + a) or a = g s /2 1 L anomaly: a α/(2π) 1/1000

g-factor of the free electron Test of QED: g exp /2 = 1.001 159 652 188 3(42) 4 10-12 g theo /2 = 1.

8 g-factor of the free electron Test of QED: g exp /2 = (42) g theo /2 = (27) CP test : g e + = g e - within exp: theo: van Dyck, Schwinberg, Dehmelt (PRL 59 (1987) 26) Kinoshita

most accurately measured fundamental quantities Most accurately calculated

9 g-2 of the free electron 10-9 measurement of the anomaly (g-2) results into accuracy for the g-factor Best CPT test for leptons One of the most accurately measured fundamental quantities Most accurately calculated fundamental quantity Most accurate determination of the fine structure constant

10 Principle of the g-2 measurement of the muon Prolongation of the lifetime START Ausgangszustand STOP: muon decays Endzustand (Myon zerfällt) B v S ω c = ω L µ + ω c < ω L S µ + v S v µ + Tricks: Prolongation of lifetime Magic momentum Muon polarization Muon detection τ = γ τ(rest) τ(rest) = 2.2 µs γ = 1/(1- β 2 ) 1/2 β = v/c p = GeV/c γ = 29.3 τ = 64 µs

11 Principle of the g-2 measurement of the muon Tricks: Prolongation of lifetime Magic momentum Muon polarization Muon detection µ + π + ν µ Spin 0 µ + e + + νe + ν µ B = 1.4 T δb/b = 10-6 R = 7 m

12 g-2 of the muon at Brookhaven 2002

13 Atomic physics in strong fields H-like Uranium E K = -132 kev <E>= V/cm Z = 92 SELF ENERGY Hydrogen E K = ev <E>= V/cm VACUUM POLARIZATION

14 g-factor of the bound electron g J =? Primary goal: test of QED in strong fields Changes to the fee-electron case: 1. Spin is no longer good quantum number but total angular momentum g J 2. QED has to take into account the effect of binding 1 st order terms 3. Nuclear effects have to be considered

15 From free to bound electrons bound free interaction with the field of the nucleus QED corrections of order (α/π)

16 QED corrections of order (α/π) 2 free: 7 diagramms, calculated in 1957 bound: 50 diagramms, only 10 related Lamb shift diagrams finished in 2001 (V. Yerokhin) Calculation for g J in 12 C 5+ Dirac binding energy finite nuclear size nuclear recoil (1) free QED, order (α/π) bound QED, order (α/π) (9) free QED, orders (α/π) 2 to (α/π) bound QED, order (α/π) 2, (Zα) (4) total theoretical value: (10) (δg/g) theo =

17 QED contributions to the bound electron g bound /g free = 1 - (Zα) 2/3 + α(zα) 2 /4π relativistic motion (Dirac) QED correction (lowest order) Dirac value is exact QED correction can be calculated with high accuracy Nuclear size effect becomes large for uranium, but has only 10-4 uncertainty

18 Principle of the g-factor determination for the bound electron cyclotron frequency Lamor frequency is the revolution frequency of a charged particle in an external magnetic field B: Lorentz force and centrifugal force are equal. is the precession frequency of a magnetic moment in an external magnetic field B. It equals the frequency required for flipping the spin. q ω c = M ion ion B ω L = g S eηh 2m e B ω << ω c L no gain of 3 orders of magnitude as in the case of the free electron! g = ω ω L c q M e m ion e

19 g-factor of the bound electron

Principle of g-factor determination of the bound e - in H-like ions separate functions: homogeneous field for measurement of B and induction of spin flips inhomogeneous field for

20 Principle of g-factor determination of the bound e - in H-like ions separate functions: homogeneous field for measurement of B and induction of spin flips inhomogeneous field for detection of spin direction LHe temperature: single-ion detection small amplitudes extreme ultra-high vaccum long storage time superconducting magnet: high frequencies high stability high accuracy

21 Experimental set-up

Reminder: Resistive cooling of a single 12 C 5+ ion B 7 mm induced image currents: kinetic energy of trapped ions is dissipated in tuned circuit (T = 4 K) End cap

22 Reminder: Resistive cooling of a single 12 C 5+ ion B 7 mm induced image currents: kinetic energy of trapped ions is dissipated in tuned circuit (T = 4 K) End cap Compensation electrode Ring electrode ν z = 350 khz C R L U = I R Resistive Cooling of 12 C 5+ to 4 K Compensation electrode End cap de ion /dt = P cool = -I 2 R R = Qω z L τ cool

23 Resonance detection for 12 C 5+ Cyclotron resonance of a single 12 C 5+ ion

24 Coulomb interactions of 12 C 5+ ions about 100 ions about 20 ions 3 ions 2 ions 1 ion Coulomb interactions and relativistic shifts cause line broadening: With 100 ions present in the trap, the resolving power is only With one ion present in the trap, a resolving power of 10 9 is achieved. Cooling is required for eliminating relativistic mass shifts.

25 g-factor of the bound electron 0.35 g 0 = 2, (1) axial frequency [Hz] spin up spin down spin-flip rate g 0 = time [min] g - 2, [10 9 ] experiment: g( 12 C 5+ ) = (10) (44) theory: g( 12 C 5+ ) = (10) statisitical error: δg/g = total error: δg/g = (limited by the knowledge of m e ) H.Häffner et al., PRL 85 (2000) 5308, updated with new value of fine structure constant

26 Fundamental constants from g-factor experiments: m e theoretical value: (10) experimental value: (10) {44} QED correct m e = (4) u von Dyck et al. (1995) m e = (12) u CODATA (1998) m e = (12) u improvement by a factor of 3

27 Fundamental constants from g-factor experiments: FSK α g Dirac δα α = 2 ( ( Z ) 3 α ( Z 1 α ) 2 δg g 2 { δα/α = C Ca U 91+

28 Fine structure constant versus year

Atomic Physics in Traps

Atomic Physics in Traps QED Fundamental Constants CPT Invariance Wolfgang Quint GSI Darmstadt and Univ. Heidelberg Quantum mechanics, Relativity, and P.A.M. Dirac Quantum mechanics Special Relativity Dirac