Positron theoretical prediction

Transcription

1 Positron theoretical prediction Schrödinger equation: ˆ 2 p x, t Vx, t x, t i 22 m tt non-relativistic equation of motion for electron Erwin Schrödinger 1933 Nobel prize

2 Positron theoretical prediction 2 x, t ˆ i tt Dirac equation: αpc mc x, t relativistic equation of motion for electron solutions with positive energy: normal electrons Paul Adrien Maurice Dirac 1933 Nobel prize P.A.M. Dirac, Proc. R. Soc. Lond. A 117, (1928) P.A.M. Dirac, Proc. R. Soc. Lond. A 133, (1931)

3 Positron theoretical prediction 2 x, t ˆ i tt Dirac equation: αpc mc x, t relativistic equation of motion for electron solutions with positive energy: normal electrons solutions with negative energy energy of a free particle E 1 mv p 2m (classical) Paul Adrien Maurice Dirac 1933 Nobel prize P.A.M. Dirac, Proc. R. Soc. Lond. A 117, (1928) P.A.M. Dirac, Proc. R. Soc. Lond. A 133, (1931)

4 Positron theoretical prediction 2 x, t ˆ i tt Dirac equation: αpc mc x, t relativistic equation of motion for electron solutions with positive energy: normal electrons solutions with negative energy relativistic energy E 2 m 2 c 4 p 2 c 2 E m 2 c 4 p 2 c 2 Paul Adrien Maurice Dirac 1933 Nobel prize P.A.M. Dirac, Proc. R. Soc. Lond. A 117, (1928) P.A.M. Dirac, Proc. R. Soc. Lond. A 133, (1931)

5 Positron theoretical prediction 2 x, t ˆ i tt Dirac equation: αpc mc x, t relativistic equation of motion for electron solutions with positive energy: normal electrons vacuum is a see of electrons with negative energy E > 0 mc 2 E = 0 -mc 2 E < 0 ray Paul Adrien Maurice Dirac 1933 Nobel prize P.A.M. Dirac, Proc. R. Soc. Lond. A 117, (1928) P.A.M. Dirac, Proc. R. Soc. Lond. A 133, (1931)

6 Positron theoretical prediction 2 x, t ˆ i tt Dirac equation: αpc mc x, t relativistic equation of motion for electron solutions with positive energy: normal electrons vacuum is a see of electrons with negative energy positron is a hole in vacuum E > 0 mc 2 E = 0 -mc 2 E < 0 + electron Paul Adrien Maurice Dirac 1933 Nobel prize pair production positron P.A.M. Dirac, Proc. R. Soc. Lond. A 117, (1928) P.A.M. Dirac, Proc. R. Soc. Lond. A 133, (1931)

7 Positron theoretical prediction 2 x, t ˆ i tt Dirac equation: αpc mc x, t relativistic equation of motion for electron solutions with positive energy: normal electrons vacuum is a see of electrons with negative energy positron is a hole in vacuum E > 0 mc 2 E = 0 -mc 2 + E < 0 ray ray anihilation Paul Adrien Maurice Dirac 1933 Nobel prize P.A.M. Dirac, Proc. R. Soc. Lond. A 117, (1928) P.A.M. Dirac, Proc. R. Soc. Lond. A 133, (1931)

8 Discovery of positron discovery of positron 1932 Lorentz force F ev B B e + 23 MeV 6 mm Pb foil e + 63 MeV Carl David Anderson 1936 Nobel prize

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10 Discovery of positron discovery of positron 1932 B = 1.7 T P = 425 kw m >3 t

11 Positron positron positron = antiparticle of electron rest mass: m e charge: +e spin: 1/2

12 Positron sources Cosmic rays 90 % protons 9 % -particles 1 % heavier nuclei & other particles (e -, e +, p - ) interaction with atmosphere p + NO N,

13 Positron sources Cosmic rays 90 % protons 9 % -particles 1 % heavier nuclei & other particles (e -, e +, p - ) interaction with atmosphere p + NO N,

14 Positron sources Cosmic rays 90 % protons 9 % -particles 1 % heavier nuclei & other particles (e -, e +, p - ) interaction with atmosphere p + NO N, e e e e

15 Positron sources Cosmic rays 90 % protons 9 % -particles 1 % heavier nuclei & other particles (e -, e +, p - ) interaction with atmosphere p + NO N, e e e e

16 Positron sources Cosmic rays 90 % protons 9 % -particles 1 % heavier nuclei & other particles (e -, e +, p - ) interaction with atmosphere p + NO N, e e e e

17 Positron sources Cosmic rays 90 % protons 9 % -particles 1 % heavier nuclei & other particles (e -, e +, p - ) interaction with atmosphere p + NO N, e e e e e e

18 Positron sources - decay - decay: e A Z A Z e X X ' 1 e e p n e u d e e Ba Cs decay: e A Z A Z e X X ' 1 e e u d e e n p N 22 N 22 e e Ne Na

19 Positron sources - decay - decay: A Z X n Cs p A Z X' e e e Bae e e + decay: A Z X p A Z 1 ne X' e e e Na 22 Nee e

20 Interaction of e + with solid Positron implantation profile Mg Cu E ~ 250 kev p kev depth (m) + mean Free positron lifetimes penetration depth Mg Cu m ps Al Nb m ps Cu Al m ps Nb Mg m ps Thermalization t < 1110 Na ps 10Ne e l ~ 50 m E kt ( ev) 1/2 = 3.7 ps Ne Difusion t Na ~ 100 ps l ~ 100 nm T 1/2 = 2.6 year 90.4 %, EC 9.5 % 1274 kev 0.06 % kev 511 kev Anihilation start signal

21 Positron lifetime N(t) - probability that e + is alive at time t : dn( t) N ( t) N ( 0) 1 dt positron annihilation rate: r 0 c n ( r) n ( r) dr N ( t) e t positron lifetime spectrum: - d N ( t ) dt e t mean time of positron life: dn( t) dt t d t 0 free positron lifetime: t t t 1 1 t te dt te 0 e dt 0 0 e 0 t 1

22 Positron lifetime spectrum positron lifetime spectrum: S id - dn ( t ) t e t Cu 114 ps Mg 233 ps intensity time (ps)

23 Positron lifetime spectrum positron lifetime spectrum: S id - dn ( t) t e t 1e-2 Cu 114 ps Mg 233 ps 1e-3 slope: 1 inten nsity 1e-4 1e-5 1e background time (ps) real positron lifetime spectrum: S real ( t) S id ( t ) R( )d B resolution function of spectrometer

24 Positron trapping Cu: fcc lifetime B = 114 ps 2.0 (001) plane 10] [ e-5 2e-5 3e-5 4e-5 5e-5 6e-5 7e-5 8e-5 (001) plane y (a) 1.0 1e-4 8e n + (a.u.) 6e-5 4e x (a) [100] 2e [010] 1.0 y (a) x (a) [100]

25 Positron trapping Cu: fcc vacancy in [1/2,1/2,0] position lifetime 1v = 180 ps 2.0 (001) plane [010 0] 1.5 2e-3 4e-3 6e-3 8e-3 1e-2 (001) plane y (a) n + (a.u.) x (a) [100] [010] 1.0 y (a) x (a) [100]

26 Positron lifetime spectrum 1e+6 Intensity 1e+5 1e+4 Cu: c 1v = at% 1 = 84 ps I 1 = 48% 2 = 180 ps I 2 = 62% 1e+3 1e t (ps) decomposition of PL spectrum: lifetimes i type of defects present intensities I i defect densities

27 Simple positron trapping model: Cu with vacancies E free positron N f ( t ) f thermalization annihilation rate B = 1/ B binding energy ~ 1 ev trapping rate K 1V K 1v 1v c trapped positron N 1v ( t) 1v annihilation rate 1v = 1/ 1v Cu vacancy: 1v annihilation B = 114 ps s -1 at. 1v = 180 ps two-component spectrum: 1 K 1 I 1 1 I 2 B 1v 2 S 1 K1v I2 K 1v B 1v id 1 I 1 1 e t 1 1 free positrons 1v 2 I 2 e t 2 positrons trapped at vacancies

28 Positronium hydrogen-like bound state of positron and electron parapositronium p-ps 1 S 0, singlet state, antiparallel spins, lifetime p-ps = 125 ps, 2- orthopositronium o-ps 3 S 1, triplet state, parallel spins, lifetime o-ps = 142 ns, 3- formed in large open volumes: e.g. in polymers in solids pick-off annihilation of o-ps o-ps reduced to several ns polycarbonate iron

29 Positron lifetime spectrometer 511 kev STOP detector START detector 511 kev 1274 kev Detector photomultiplier p BaF 2 scintillator

30 Positron lifetime spectrometer 511 kev STOP detector START detector 511 kev 1274 kev Detector BaF 2 scintillator Fast component: 1 = 220 nm, = 0.6 ns Slow component: 2 = 310 nm, = 630 ns

31 Positron lifetime spectrometer 511 kev STOP detector START detector 511 kev 1274 kev Detector Linearly focused PMT

32 Positron lifetime spectrometer 511 kev STOP detector START detector 511 kev 1274 kev CFD constant-fraction discrimination inversion+attenuation crossover CFD delay sum amplitude selection lower level upper level

33 Positron lifetime spectrometer STOP detector 511 kev START detector 511 kev 1274 kev CFD CFD START STOP delay STOP TAC START ADC PC

34 Positron lifetime spectrometer detectors t fast-fast PL spectrometer timing resolution 160 ps (FWHM 22 Na) coincidence count rate 100 s counts in spectrum F. Bečvář et al., Nucl. Instr. Meth. A 443, 557 (2000) source-sample sandwich

35 Hydrogen in Niobium Nb: bcc structure t a = (1) 3033(1) Å [PDF-2] H in Nb interstitial solid solution size (r Nb ): N i /cell 6 12 N i /M 3 6 H x H = N H /M number of hydrogen atoms per metal atoms

36 Hydrogen in Niobium Equilibrium phase diagram of Nb-H system two-phase field ( + ) x H = (atom ratio H/Nb): single phase solid solution (-phase), bcc H fills tetrahedral interstitial positions

37 Hydrogen in Niobium Equilibrium phase diagram of Nb-H system two-phase field ( + ) -phase (NbH) orthorhombic a = Å b = Å c = Å c z y x x H = (atom ratio H/Nb): single phase solid solution (-phase), bcc H fills tetrahedral interstitial positions b a

38 Samples defect-free Nb - bulk Nb (99.9%) - annealing 1000 o C / 1h to anneal out all existing defects electron irradiated Nb - bulk Nb (99.9%) 9%) - annealing 1000 o C / 1h to anneal out all existing defects - 10 MeV electron irradiation, F = m -2, T irr 100 o C all samples: Pd cap (thickness 30 nm) prevent oxidation facilitate H absorption R. Kircheim et al., Acta Metall. 30, 1059 (1982)

39 Hydrogen loading Electrochemical H loading R. Kircheim, Prog. Mat. Sci. 32 (1988), p. 261 hydrogen concentration ti Faraday s law i = 0.3 ma/cm 2 V impedance changer timer + current source + Ag/AgCl electrode - Evolution of defects was studied as a function of gradually increased hydrogen concentration in sample. Pt electrolyte: sample H 3 PO 4 (85%) + glycerin (85%) 1 : 2

40 H-induced lattice expansion: X-ray diffraction relative lattice expansion: a a 0 x a 0 H a 0 a lattice constant for virgin sample lattice constant t for hydrogen-loaded d d sample for Nb : H. Peisl, in:. Hydrogen in Metals I, Springer-Verlag, g, Berlin (1978), p a / a = 0.058(1) x H

41 Hydrogen-induced vacancies well annealed Nb (1000 o C / 1h): single component PL spectrum 1 = ( ) ps calculated l dbulk lknblif lifetime (ATSUP): B = (126 1) ps defect-free material TEM no dislocations observed, grain size > 10 m

42 Hydrogen-induced vacancies well annealed Nb (1000 o C / 1h) hydrogen loading hydrogen induced defects 2 = ( ) ps hydrogen-induced volume expansion elastic process dislocations calculated lifetime for Nb vacancy (ATSUP): V = (222 1) ps vacancies surrounded by hydrogen shortening of positron lifetime lifetime (ps s) I 2 (%) x H x H

43 1e+0 e+71e+ 1e+0 Hydrogen-induced vacancies Effective medium theory J. Nørskov, Phys. Rev. B 26, 2875 (1982) E hom (r) in (001) plane vacancy in 1,1,0 1e+6 1e+7 1e+8 1e+5 1e+5 1e+4 1e+4 1e+3 1e+6 1e+7 1e+8 1e+3 1e+4 1e+5 1e+6 1e+3 1e+2 1e+2 1e+1 1e+2 1e e+1 1 1e+0 1e+0 n e-1 1e e+1e+6+8 1e +4 1e+1 1e e+51e1e [010] 1e +2 1e e 1e-1 1e 1e 1e+3 1e+4 1e+ e+5 1e+ e+6 1e-1 1e [010] 1e+0 1e e+2 1e+1 1e+2 1e+0 1e+5 1e+4 1e+3 1e+3 1e+4 1e+1 1e+5 1e+6 1e+2 1e+6 1e [100] 1e+4 [100] H positions: [0.64,1,0],, [1,0.64,0],, [1,1,0.36], [1.36,1,0], [1,1.36,0], [1,1,-0.36]

44 1e+0 Hydrogen-induced vacancies Effective medium theory Stott, Zaremba, Nørskov, Lang e+6 1e+7 1e e+5 1e+5 1e+4 1e+4 1e+3 1e+6 1e+7 1e+8 1e+3 1e+4 1e+6 1e+5 1e+3 1e+2 1e+2 1e e+2 1e+1 Nb 1e+1 1e+0 1e e-1 1e e e+3 1e 1e+1 1e e+51e1e+6 octahedral position [0,1,0.5] H e+71e+ [010] 1e +2 1e e 1e-1 1e 1e 1e+3 1e+4 1e+ e+5 1e+ e+6 1e-1 1e [0.64,1,0], 1e+0 1e+0 1e+1 1e+2 1e+1 1e+2 1e+0 1e+5 1e+6 1e+4 1e+3 1e+3 1e+4 1e+1 1e+5 1e+2 1e+6 1e+3 1e+4 [100] H positions: [0.64,1,0],, [1,0.64,0],, [1,1,0.36], [1.36,1,0], [1,1.36,0], [1,1,-0.36] displacement = 0.46(7) Å

45 ATSUP calculations Nb vacancy in [1,1,0] and H in [0.64,1,0] electron density in (001) plane positron density in (001) plane n [010] [100] [010] [100] [010] [100] positron lifetime v-h = 204(1) ps

46 ATSUP calculations complexes (vacancy H) vacancy + 2H vacancy + 3H [010] [010] = 182(1) ps [100] 0.25 = 169(1) ps [100]

47 ATSUP calculations complexes (vacancy H) vacancy + 4H vacancy + 5H [010] [010] = 151(1) ps [100] 0.25 = 142(1) ps [100]

48 ATSUP calculations complexes (vacancy H) 2.00 vacancy + 6H [010] n [100] [010] [100] = 127(1) ps

49 Hydrogen-induced vacancies calculated positron lifetime for vacancy is surrounded by several H atoms v-nh (ps) 180 experiment 150 ps Nb bulk number of H Hydrogen loading creation of vacancies surrounded for 4 hydrogen atoms

50 Hydrogen-induced vacancies well annealed Nb (1000 o C / 1h) hydrogen loading hydrogen induced defects 2 = ( ) ps hydrogen-induced volume expansion elastic process dislocations calculated lifetime for Nb vacancy (ATSUP): V = (222 1) ps vacancies surrounded by hydrogen shortening of positron lifetime c v at.% T 1850 o C (0.8 Tm) x x10-3 lifetime (ps s) I 2 (%) x10-3 C v-4h (at.% %) x x H x H

51 H-induced defects bulk Nb mechanism of formation effective medium theory: H v E b 0.50 ev vacancy formation energy: E f 2.32 ev P. Korzhavyi et al. PRB 59, (1999) vacancy 4H: H v E f 4E b 0.32 ev equilibrium concentration of vacancy 4H complexes: c p e S / k k e E f 4E H v b / kt 4.0x10-3 experiment 3.0x H in nearest neighbor positions required p at.% 4 99 ~ c H c 710.%) C v-4h (at. 20x10 2.0x10-3 theory 4 H in active volume V 0 3 V0 2.9 nm 80 unit cells of Nb 1.0x J. Čížek et al. PRB 69, (2004) x H

52 Hydrogen-induced vacancies 2 = 150 ps Loading unloading experiment loading: constant current 0.3 ma unloading: constant t voltage 0.8 V I 2 (%) PL intensity of trapped positrons 10 loading unloading x H unloading time (h) loading unloading (10-10 m) a ( XRD lattice constant Nb PDF x H unloading time (h)

53 Electron irradiated bulk Nb sample 1 bare Nb electron irradiated (10 MeV e -, F = m -2, T irr 100 o C) Pd cap sputtered after irradiation sample 2 Nb with Pd cap electron irradiated (10 MeV e -, F = m -2, T irr 100 o C) Sample 1 (ps) I 1 (%) 2 (ps) I 2 (%) bare Nb electron eecto irradiated adated. + Pd cap Nb electron irradiated with Pd cap vacancy-h complexes mixture of v-h ( v-h = 204 ps) and v-2h ( v-2h = 182 ps) complexes

54 Electron irradiated bulk Nb 3-component fit Sample 1 (ps) I 1 (%) 2 (ps) I 2 (%) 3 (ps) I 3 (%) bare Nb Fix Fix 25 3 electron irradiated. + Pd cap Fix Fix 29 4 Nb electron irradiated with Pd cap Fix Fix 11 3 v-2h complexes v-h complexes

55 Electron irradiated bulk Nb 14 application of 3-state trapping model (trapping coefficient: = at. s -1 ) 12 concentration n (10-5 at. -1 ) bare Nb irradiated Pd cap sputtered subsequently on the irradiated Nb Nb irradiated with Pd cap catalytic ti effect of Pd cap v-h transforms into v-2h 2 v-2h v-h 0

56 Doppler Broadening annihilation 1 2 E mec E 1 cp e - e + E L 2 annihilation peak 207 Bi E mec E 2 counts counts FWHM = 2.580(3) kev E (kev) FWHM = 1.103(1) kev E (kev)

57 Coincidence Doppler Broadening spectroscopy (CDB) 2 E mec E 1 E e cp L 2 p T HPGe detector t p p L HPGe detector CFD CFD Coincidence gate spectroscopy amplifier E 1 E 2 spectroscopy ADC amplifier

58 Coincidence Doppler Broadening spectroscopy (CDB) reference Fe 30 1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 E1+E E2-2 x 512 ((kev) E1- E2 (kev)

59 Coincidence Doppler Broadening spectroscopy (CDB) 1e+7 reference Fe 1e+6 1e+5 resolution function annihilation peak counts 1e+4 1e+3 1e+2 1e+1 1e E 1 -E 2 (kev)

60 Coincidence Doppler Broadening spectroscopy (CDB) CDB ratio curves Cu 1.4 ratio to Fe Al p (10-3 m 0 c) element sensitivity

61 Coincidence Doppler Broadening spectroscopy (CDB) Chemical environment of defects CDB measurements two HPGe detectors energy resolution 1.05 kev (FWHM, E = 512 kev) coincidence count rate 550 s counts in spectrum J. Čížek et al., Mat. Sci. Forum , 63 (2004)

62 Vacancy-H complexes calculated HMP s annihilation with core electrons effect of hydrogen 1s electrons to Nb ratio Nb vacancy Nb vacancy+h Nb vacancy+2h Nb vacancy+3h Nb vacancy+4h Nb vacancy+5h Nb vacancy+6h p (10-3 m 0 c)

63 Electron irradiated Nb experimental high momentum profiles effect of hydrogen 1s H electrons bare Nb irradiated ratio to Nb p (10-3 m 0 c)

64 Electron irradiated Nb experimental high momentum profiles bare Nb irradiated effect of hydrogen 1s H electrons Nb iradiated with Pd cap ratio to Nb p (10 m 0 c)

65 Electron irradiated Nb experimental high momentum profiles bare Nb irradiated effect of hydrogen 1s H electrons Nb iradiated with Pd cap hydrogen loaded Nb ratio to Nb p (10-3 m 0 c)

66 Other gas impurities calculated HMP s Nb vacancy Nb vacancy - Oxygen in center Nb vacancy - Nitrogen in center ratio to Nb p( (10-3 m 0 c)