Experiments Lamb shift, hyperfine structure, quasimolecules and MO spectra

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1 Modern Atomic Physics: Experiment and Theory Preliminary plan of the lectures Experiments Lamb shift, hyperfine structure, quasimolecules and MO spectra Lecture 3 April 9 th, Preliminary Discussion / Introduction Experiments (discovery of the positron, formation of antihydrogen,...) Experiments (Lamb shift, hyperfine structure, quasimolecules and MO spectra) Theory (from Schrödinger to Dirac equation, solutions with negative energy) Theory (photon, quantum field theory (just few words), Feynman diagrams, QED corrections) Theory (matrix elements and their evaluation, radiative decay and absorption) Experiment (photoionization, radiative recombination, ATI, HHG...) Theory (single and multiple scattering, energy loss mechanisms, channeling regime) Experiment (Kamiokande, cancer therapy,.) Experiment (Auger decay, dielectronic recombination, double ionization) Theory (interelectronic interactions, extension of Dirac (and Schrödinger) theory for the description of many-electron systems, approximate methods) Theory (atomic-physics tests of the Standard Model, search for a new physics) Experiment (Atomic physics PNC experiments (Cs,...), heavy ion PV research) 1

2 One needs three quantum numbers to define the state of a hydrogen (hydrogen-like) atom: n = 1,, 3 (principal) l = 0, n-1 (orbital) m = -l,. +l (magnetic) The energy depends only on the principal quantum number: E n Z n i.e. in nonrelativistic theory the states are degenerate (l, m)! Hydrogenic spectrum 0 ( r) ( r,, ) R ( r) Y (, ) Energy (au) -1/18-1/8-1/ 3s (n=3, l=0) s (n=, l=0) 1s (n=1, l=0) nl p (n=, l=1) lm Electron is free Electron is bound to ion.... 3p (n=3, l=1) 3d (n=3, l=) Why set (n,l,m) is good for the description of bound hydrogenic states? Let us re-consider the set of (nonrelativistic) quantum numbers one more time! Prediction of anti-matter E +m 0 c 0 -m 0 c positive continuum bound states negative continuum Dirac, Anderson, the Positron and the anti-matter. In his famous equation Paul Dirac combined (199) the fundamental equation of quantum mechanics, the Schrödinger-equation with the theory of special relativity. He did not discard the negative energy solutions of his equation as unphysical but interpreted them as states of the antiparticle of the electron (positron, having the same mass but opposite charge). In 193 Carl Anderson discovered the positron the first time in the cosmic radiation. This was the proof of the existence of 'anti-matter', with incalculable consequences for the future of physics.

3 Electron-positron pair production The first confirmation of a positron In order to produce electron-positron pairs we would need: at least two times the rest mass of the elecreon!!! E +m 0 c electron energy of about mc 1 MeV The first 'fingerprint' of anti-matter. Anderson discovers the trace of a positron in his cloud chamber (in the middle one can see a lead-plate of 6mm thickness). 0 -quantum with energy ћ to induce pair production 1. The upper part of the bending gives information about the incoming-direction.. The lower part gives the positive charge of the particle by its bending-direction. 3. By analyzing the radius of curvature before and after the transition the momentum can be estimated -m 0 c hole (positron) In case a whole in the negative continuum excits, an electron will immediately fill the vacancy and two 511 kev quanta are emitted. 3

4 Experiments Lamb shift in one and two-electron ions, hyperfine structure Content QED: Radiative correction viewed by an experimentalist Experiments on Hydrogen Strong fields: High-Z ions versus exotic atomic systems The lamb shift in uranium The hyperfine structure at high-z 4

5 + mc 0 - mc E nj mc Positive Continuum bound states Negative Energy Continuum DIRAC Theory 1 n j 1/ e - e+ Z ( j 1/ ) ( Z) n: principle quantum number j: total angular momentum α: fine structure constant- α = 1/ ) mc : electron rest mass (511 kev) Dirac-Theorie: (complete relativistic treatment of single electron systems) All levels with the same n and j are degenerated. Critical electromagnetic fields Let us come back to Dirac energy of a single hydrogen-like ion: What happens if we increase the nuclear charge Z? If nuclear charge of the ion is greater than Z crit the ionic levels can dive into Dirac s negative continuum. Physical vacuum becomes unstable: creation of pairs may take place! Z E1 s mc 1 5

6 Relativistic quantum numbers and spectroscopic notations n j parity spectroscopic notation 1 1/ + 1s 1/ Supercritical fields: Formation of Quasi-Molecules Merged Beams U 91+ 1/ + s 1/ 1/ - p 1/ 3/ - p 3/ U 9+ Finally, we know how to characterize bound states of (relativistic) hydrogen. What are the energies of these states? A. Artemyev, et al. J. Phys. B 43 (010)

7 The Lamb shift Experiment: Lamb-Retherford 1947; Theoretical Explanation: Bethe 1947 This experiment proves that for hydrogen the s1/ and the p1/ are not degenerated Lamb shift: idea of radiation corrections From the end of 1930 th : interaction of electron with radiation field. But which field? s 1/ => In contradiction to Dirac-Theory p 3/ Ly 1 (E1) p 1/ Lamb-Shift deviations from the Dirac theory for a point like nucleus Ideas: electron may interact with its own field. The Coulomb potential is therefore perturbed by a small amount and the degeneracy of the two energy levels is removed. But: problems with divergence of results! E1 1s 1/ Ly (E1) Level sheme for hydrogen The Lamb shift (LS) is essentially a consequence of the interaction of the bound electron with its own radiation field In 1947 Hans Bethe has shown how to identify the divergent terms and to substract them from the theoretical expression. Development of Quantum Electrodynamics (QED) and Quantum Field Theory (QFT) Hans Bethe 7

8 Feynman diagrams The effects of QED increase with the intensity of the electrical field, the electrons are exposed to. That means the s electrons are most affected. Feynman diagrams are a nice tool to represent exchange forces and, hence, to perform calculations of scattering processes. Richard Feynman Lamb-Shift (three effects) QED self-energy Self Energy Electron-electron repulsion real electrons virtual photon (carrier of interaction) Vacuum-polarization Vacuum Polarization time Z size of the nucleus From: 8

9 Lamb Shift: An effect of strong fields QED corrections E Z 4 /n 3 Z: nuclear charge n: principal quantum number r (r) nucleus 1s s r = Z/n in atomic units. With the normalization condition 3 follows: Appendix: probability density of the electrons at the nucleus (non-relativistic) (r) d r 1 Z ( r) n 3/ What is the electron density at the nucleus (r=0)? p 1/ 10-4 p 3/ for l = 0 Z ( r 0) n , radius [fm] 9

10 Self energy Self-energy: Describes the emission and reabsorption of a virtual photon (Bethe 1947) The Self-energy decreases the binding energy. virtual photon Vacuum Polarization (VP) electron in the outer field (z.b. nucleus-coulombfield) Massrenormalization m me m m e m: measurable electron mass. It already includes the radiation correction. The self-energy is the most important radiation correction for light systems and decreases the binding energy. Here the electron emits and reabsorbs one and the same virtual photon. In every classical theory the consequence would be an infinitely high mass corresponding to an infinitely high self energy. The first calculations on the self-energy which explains the lamb shift of Hydrogen were made by Bethe. His success is based on an idea of Kramer, called mass-renormalization. The interaction with the virtual photon leads to a modification of the electron mass. This effect is always present and the measured electron mass m already includes the radiation correction. Therefore the self-energy correction for a bound state can be regarded as the mass difference between the bound electron and a free electron of the mass m e that is not liable to any interaction with the electromagnetic field at the time t m electron mass of the free electron without interaction with a field. Virtual creation and annihilation of an e + e - -pair Charge renormalization VP acts attractive 10

11 10-1 Summary of the Lamb Shift Effects (Z) 4 F(Z) m e c Self Energy Vacuum Polarization Bound-State QED: 1s Lamb Shift at High-Z Sum of all corrections, leading to deviations from the Dirac theory for a point like nucleus Self energy Vacuum polarization 1s Lamb Shift, E / Z 4 [mev] self energy Lamb- Shift (-) vacuum polarization nuclear size nuclear charge number, Z The size of the individual corrections in detail p 1/ 1s 1/ Finestructure sls 1sLS Hydrogen, H H-like uranium, U ev 45.4 μev 4.4 μev 35.6 μev 97.6 kev 4.56 kev 75.3 ev ev Goal: U 9+ SE VP NS ev ev ev (Z) 4 F(Z) m e c 1 ev Low Z-Regime: Z << 1 F(Z): series expansion in Z High Z-Regime: Z 1 F(Z): series expansion in Z not appropriate 11

12 QED correction in second order SE SE The Lamb-Shift The effects of QED increase with the intensity of the electrical field, that the electrons are exposed to. That means the s electrons are most affected. QED-corrections E Z 4 /n 3 Z: nuclear charge n: principal quantum number VP VP s 1s SE VP <E> [V/cm] field strength: 1s electron in ground state s p 1/ S(VP)E Nuclear Charge, Z p 3/ 1

13 Self Energy Test of Bound-State QED at high-z Experiments 1s Lamb Shift eqed for He-like ions s-p transitions in Li-like heavy ions hyperfine-structure Vacuum Polarization Content QED: Radiative correction viewed by an experimentalist Experiments on Hydrogen Strong fields: High-Z ions versus exotic atomic systems The lamb shift in uranium The hyperfine structure at high-z g-factor of bound electrons 13

14 Hydrogen Atom ( f(1s-s) = (34) Hz Laser Spectroscopy of Hydrogen Rydberg Constant R = (84) m -1 Lamb Shift L1S = () MHz relative accuracy laser spectroscopy frequency measurements year 14

15 Laser spectroscopy s f0 Laser frequency: f excitation frequency: f 0 s f f0 1s 1s v << c f f 0 v f f0(1 ) c v f f0(1 ) c Large Doppler broadening by thermal distribution 15

16 Two-photon laser spectroscopy Real precision is achieved by relative measurements!!! f 0 = x f Laser f s Laser f Assumption: The corrections (relativistic, QED etc.) are much better known for high n (much smaller!!!) than for n=1 and n= Detection of fluorescence-radiation as a function of the laser frequency 1s f (1 f (1 0 f1 0 f f f 1 f v ) c v ) c Doppler free-spectroscopy Condition for Precision-spectroscopy Idea: Compare Balmer-β radiation with Lyman-α Balmer-β: Lyman-: E E Effectively it is: 1 Ry 1 4 Ba 1 Ry 1 Ly 4 EBa 1ELy C 4E 1 Ba E Ly classically correct!!! E Ba and E Ly are measurement categories in the experiment C is the parameter for the ground state, one has to determine for checking the theories 16

17 Applications: Bose-Einstein of atoms

18 Content QED: Radiative correction viewed by an experimentalist Experiments on Hydrogen Strong fields: High-Z ions versus exotic atomic systems The Lamb shift of uranium (Z=9) The hyperfine structure at high-z Extremly Strong Coulomb Fields <E> [V/cm] s Z = 9 Intense Laser Hydrogen E K = ev Z = 1 <E>= V/cm Nuclear Charge, Z H-like Uranium E K = ev <E>= V/cm 18

19 Higly charged ions p 3/ s 1/ p1/ 1s 1/ H-like ions Muonic atoms vs. HCI E n r 0 Z m n 1 mz Muonic atoms s 1/ 1s 1/ muonic hydrogen p 3/ p 1/ 1s Lamb shift in muonic atoms and HCI Muonic hydrogen (-.5 kev) Lamb shift: -1.9 ev Self energy 0.4% Vacuum Polarization 97.9% Nuclear size 1.6% Muonic nitrogen (-136 kev) Z H-like silicon (-.6 kev) Lamb shift: 0.48 ev Self energy 93.6% Vacuum Polarization 5.9% Nuclear size 0.5% H-like uranium (-13 kev) Z >> 1 Relativistic corrections QED effects (self-energy) Electron-electron interaction High quantity production (ECRIS, EBIT, storage rings, ) m >>m electron QED effects (vacuum polarization) Nuclear structure probe Diffcult to produce Obtained using intense pion beam (TRIUMF, PSI, ) Lamb shift: 199 ev Self energy 0.7% Vacuum Polarization 4.8% Nuclear size 56.5% Lamb shift: 46 ev Self energy 54.3% Vacuum Polarization 13.8% Nuclear size 30.4% Complementary systems!! 19

20 More exotic bound systems... Pionic and kaonic atoms H-like systems EM and strong interaction Nucleon-meson interaction (Z=1,,..) Nuclear probe (high Z)...and more! Antiprotonic atoms EM and strong interaction Nucleon-antiproton interaction (Z=1,,..) -> nucleon-nucleon interaction Nuclear probe (high Z) Recoil and QED corrections Soon available in high quantities at FAIR Obtained at LEAR (Gotta 1999) strong interaction electromagnetic interaction u d Pion (-) or Kaon (-) u s 1S QED KeV ph(3p-1s) 1. bar T=30K H (10 bar) 1S d u u strong interaction d u u electromagnetic interaction 0

21 Content QED: Radiative correction viewed by an experimentalist Experiments on Hydrogen The Structure of One-Electron Systems s 1/ M1 p 3/ Ly 1 (E1) Ly (E1) p 1/ QED corrections E Z 4 /n 3 Z: nuclear charge number n: prinzipal quantum number Strong fields: High-Z ions versus exotic atomic systems The Lamb shift of uranium (Z=9) The hyperfine structure at high-z 1s 1/ Atomic systems at high-z Large relativistic effects on energy levels and transition rates (e.g. shell and subshell splitting) Large QED corrections Transition energies close to 100 kev 1

22 GSI-Accelerator Facility X-Ray Spectroscopy at the ESR Storage Ring Injection Energy MeV/U UNILAC ESR 11.4 MeV/u U 73+ M1 Ly 1 Ly MeV/u U 9+ up to 1000 MeV/u U 9+ SIS Circumference: 108 m Number of Ions: 10 8 Frequency: /s 1s 1/ At the ESR, production of characteristic x-rays by electron capture into the bare ions (electron cooler or jet-target)

23 ESR Storage Ring Storage Rings: Cooled Ion Beams electron collector electron collector electron gun high voltage platform electron gun magnetic field electron beam high voltage platform ion beam magnetic field electron beam ion beam 3

24 The Electron Cooler The Effect of Cooling electron collector high voltage platform electron gun electron collector electron gun before cooling after cooling magnetic field electron beam ion beam high voltage platform magnetic field electron beam ion beam ion intensity Electron Cooler.5 m interaction zone Voltage: 5 to 00 kv Current: 10 to 1000 ma rel. ion velocity v/v 0 Ions interact /s with a collinear beam of cold electrons Properties of the cold ions Momentum spread p/p : Diameter mm 4

25 The Experimental Challenge X-Ray Spectroscopy at the ESR Storage Ring Relativistic Doppler-Transformation E lab Eproj γ (1βcosθ lab E lab : Photon energy in the laboratory system E proj : Photon energy in the emitter system ) E lab /E proj MeV/u ( =0.69) 0 MeV/u ( =0.59) 68 MeV/u ( =0.36) 49 MeV/u ( =0.31) Injection Energy 400 MeV/u deceleration Doppler-Correction: Strong dependence on velocity and the observation angle LAB observation anglel, lab [deg] 10 MeV/u γ 1 ; β 1β v c Excited states are produced by electron capture (jet target) / recombination (electron target) 5

26 Principle of LS Experiments Experiments at the Jet-Target 1. Production of bare ions. Cooling Coincidence Particle Counter (MWPC) 3. Electron capture in excited states (jet-target or electron cooler) 4. Detection of x-rays 5. Doppler correction 6. Comparison with Dirac theory 1s Lamb Shift Dipole Magnet Gas Jet Ge Detector Stored Ions (Z=Q) Ereignisse K-REC Ly Ly ω K E KIN L Ly 1 Z Electron transfer from the target atom into the HCI -REC M-REC L-REC 8 Q Z e (Q1) K-REC Photonenenergie (kev) ω... 6

27 U 9+ Lamb Shift-Experiment at the Jet-Target U mm 5 mm 48R 48 O segmented Ge(i) detector (x,y) Gasjet 48 O 48L(7) 90 O 90(7) U O Counts Ly 1 (p 3/ 1s ½ ) ev Ereignisse Ly Result of the jet-target experiment using decelerated uranium ions Geometry Fit Ly 1 (p 3/ 1s ½ ) 8.5 ev.6 ev 9.7 ev ev Ly 1 The 1s-Lamb Shift Experiment: 468 ev 13 ev Theory: 466 ev* *Theory: T. Beier et al., 1998 Simultaneous observation at various angles Forward/Backward symmetry Left/Right symmetry Energy (kev) 1s-Lamb Shift 3.0 deg Experiment: 468 ev 13 ev Theory: 466 ev* Energie [kev] 3% Sensitivity to Lamb Shift (466 ev) 4% Sensitivity to Self Energy (355 ev) 15% Sensitivity to Vacuum Polarization (-88.6 ev) 7

28 0 o Spectroscopy at the Electron Cooler The Ground State Lamb Shift in H-like Uranium 00 Ly counts H-like Uranium Balmer L-RR j=1/ j=3/ Ly Ly K-RR Energy [kev] Dipole Magnet Blue shift has its maximum 0.9 E lab 1.43 E proj LAB not critical, almost no Doppler width Uncertainty caused by has its maximum counts s Lamb shift in U 91+ statistical Ly Ly photon energy [kev] 460.±.3±3.5 ev 4.6 ev K-RR uncertainty in the 8

29 Test of Quantum Electrodynamics (1s-LS) counts Ly 1 Ly The 1s-LS in H-like Uranium 1s-Lamb Shift Experiment: ev 4.6 ev K-RR Theory: ev s 1/ M1 p 3/ p 1/ photon energy [kev] 1s Lamb shift 1s 1/ Ly 1 (E1) Ly (E1) H. Beyer, et al Scanning technique versus Bragg-Laue micro-strip Relation detectors λ (pm) z λ d sinθ d R Scanning FOCAL Spectrometer: ε Events per Hour Micro-Strip Germanium Detector Timing, Energy and Position Resolution Research Highlights Nature 435, (16 June 005) Lamb Shift [ev] U 91+ Gasjet Cooler Decelerated Ions: Jet Decelerated Ions: Cooler (our exp.) Year Theory A. Gumberidze PhD thesis 003, PRL 94, 3001 (005) 00 Strips x 00 μm E 1.6 kev τ 50 ns z (mm) microstrip Alexandre Gumberidze Wavelength standards in the x-ray region, Helmholtz-Institut Jena,

30 Prototype D STRIP X-Ray Detector There is a need for D detectors! D STRIP planar detector systems for precision x-ray spectroscopy experiments (FOCAL) energy resolution timing - D position sensitiviy p + -contact (18 strips) fast moving source +0.6 x -0.6 [mm] z-direction [mm] stationary source Alexandre Gumberidze Wavelength standards in the x-ray region, Helmholtz-Institut Jena, a-ge-contact (48 strips) 11 mm front: 18 strips pitch ~50µm back: 48 strips pitch ~1167µm equivalent to 6144 pixel STRIP detector developed by Alexandre Gumberidze Wavelength standards in the x-ray region, Helmholtz-Institut Jena,

31 The FOCAL setup April 01 Preliminary results Au 79+ Au 78+ gasjet target supersonic Argon-jet density: ~10 1 atoms/cm width: ~5 mm ion-beam Au 79+ ion beam energy:15 MeV/u Ar Raw D spectrum Au 78+ Au 79+ Au 79+ dipole magnet particledetector Au 78+ Target EC p 3/ Ly 1 D spectrum with energy and time condition 3 5 counts/h coincidence between particle- and x-ray detector K-REC 1s Alexandre Gumberidze Wavelength standards in the x-ray region, Helmholtz-Institut Jena, 1/ Alexandre Gumberidze Wavelength standards in the x-ray region, Helmholtz-Institut Jena,

32 Content Energy levels of hydrogen atom: From Schrödinger to Dirac to QED QED: Radiative correction viewed by an experimentalist Experiments on Hydrogen Strong fields: High-Z ions versus exotic atomic systems Do we expect some more effects which can influence the level structure? The Lamb shift of uranium (Z=9) The hyperfine structure at high-z Actually yes! 3

33 Hyperfine interaction Hyperfine interaction Hyperfine structure is a perturbation in the energy levels of atoms/ions due to the magnetic dipoledipole interaction, arising from the interaction of the nuclear magnetic moment with the magnetic field of the electron. I J For the case of ground 1s 1/ state (j=1/) of hydrogen, HF interaction results in splitting of energy level into two levels. F = I +1/ F = I -1/ Coupling of electron and nuclear momenta give rise to the total angular momentum of the system: The HF splitting is: F J I From: One can observe transition between two HF levels: famous 1 cm line in astrophysics! From: E HF a F( F 1) I( I 1) J ( J 1) Levels are spitted one more time! 1 cm radiation is used, for example, to measure radial velocities of the spiral arms of the Milky Way. From 33

34 Hyperfine Structure at High-Z Hydrogen: () cm 10 7 years Hydrogen Atom: Groundstate E HFS 4 1 A E HFS A for j=1/, s p =1/, F=1 for j=1/, s p =1/, F=0 Eta Carinae j=1/ F=1 (F+1)=3 ΔE HFS ev (F+1)=1 F=0 34

35 Hyperfine Structure at High-Z HFS: Test of QED in strong magnetic fields (high-z) 1 8 hc I Z 3 3 n F 1 1 ( j, Z) QED??? Fermiformula for the HFS 3 Ry g relativistic correction I M m Nuclear-Charge distribution (Breit-Rosental-Corwford-Effect) current distribution (Bohr-Weisskopf-Effect) strong magnetic fields 10 1 Gauss e 35

36 Hyperfine Structure at high-z The Candidates 1 8 hc I Z 3 3 n F 1 1 ( j, Z) QED??? Fermi Formula for HFS 3 Ry g Relativistic Correction I M m Nuclear Charge Distribution (Breit-Rosental-Corwford-Effekt) Magnetisation Distribution (Bohr-Weisskopf-Effekt) e E 5 ev 36

37 Laser Experiments at the ESR Bi 8+ Bi 8+ Klaft et al.,

38 We have a problem! 09 Bi Pb 81+ In-ring spectroscopy of Li-like Bi RMS radius fm fm Magn.Mom.(corr.) n.m n.m. Point nucl.(dirac) 1.30 nm nm Breit-Schawlow (50) nm (1) nm Ext. Nucleus (50) nm (1) nm Bohr-Weisskopf (330) nm + 9.5(0) nm Theory, no QED 43.91(38) nm (1) nm Vac. polarisation nm nm Self energy +.86 nm nm Total QED + 1. nm nm Theory incl.qed 45.13(58) nm 104.(5)nm Experimental 43.87() nm ()? laser rot ( 10 m) t 0 Reference Ref.-Bunch t 1/ rot Signal MHz Signal-Bunch First observation of Hyperfine-Splitting in a Li-like heavy ion accumulated counts result of 14 accumulated scans ~ 0. nm Zoran Anđelković wavelength [a.u.] Zoran Andelkovic, Christopher Geppert, Matthias Lochmann, Wilfried Nörtershäuser, et. al. to be published AP 38

39 DIRAC Theory (relativistic formulation of quantum mechanics) E E nj mc 1 n j 1/ Z ( j 1/ ) ( Z) +m 0 c bound states positive continuum n: principal quantum number j: total angular momentum α: Finestructure- or Sommerfeldconstant α = 1/ ) mc : electron mass at rest (511 kev) 0 -m 0 c negative continuum Dirac-Theory: Relativistic complete description of quantum mechanics All states with same n and j are degenerated. 39