Fundamentals in Nuclear Physics

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1 Fundamentals in Nuclear Physics Kenichi Ishikawa () 1

2 Nuclear decays and fundamental interactions 2

3 Four fundamental interactions interaction gravity exchanged particle (gauge boson) graviton decay weak W ±, Z 0 beta decay electromagnetic photon gamma decay strong nuclear force gluon pion and other hadrons alpha decay tunnel effect 3

4 Decay rate, natural width probability to decay in an interval dt dp = dt = number of unstable nuclei dt decay rate mean life time N(t) =N(t = 0)e t/ half life t 1/2 = (ln 2) = Li (7.459 MeV) n 6 Li, 3 H 4 He τ = sec 76 Ge 76 Se 2e 2 e t 1/2 = yr > (age of universe)! An unstable particle has an energy uncertainty or natural width = = = MeV sec 4

5 Branching ratio Often, an unstable state (nucleus, isotope) has more than one decay channels Energy (MeV) n + 6 Li Li H + He 8 10 channel k branching ratio k Bk B k =1 partial decay rate partial width k = B k k k = k = B k k k = 5

6 Decay diagram half life branching ratio 6

7 Measurement of half life τ > 10 8 yr (α decay, double β decay) isotopically enriched (98.4%) scintillators 160 helium gas e 100 Mo foil 127g e scintillators still present on Earth can be chemically and isotopically isolated in macroscopic quantity detected decays, quantity lifetime events/0.1 MeV events (background subtracted) during 6140 h electron energy sum (MeV) Mo 100 Ru 2e 2 e double β decay half-life: (0.95±0.11) yr 7

8 10 min < τ < 10 8 yr (α decay, β decay) no longer present on Earth and must be produced in nuclear reactions purify chemically or isotopically detect decays and derive τ s < τ < 10 3 s (α decay, β decay, γ decay) chemical and isotopic purification impossible particles produced in nuclear reactions, slowed down, and stopped detect decays and derive τ τ < s (γ decay, dissociation) standard timing techniques not applicable a variety of ingenious techniques: Doppler-shift attenuation method, Mössbauer spectroscopy 8

9 Formula for decay rates T interaction decay a b 1 + b b N rest mass M energy E = Mc 2 f state of final particles decay rate probability per unit time that a decays into f a f a f = 2 f T a 2 Mc 2 j transition matrix element E j energy conservation Fermi s golden rule 9

10 Gamma decay 10

11 Energetics gamma decay A A+ gamma ray spontaneous emission unstable high-energy state (stable) low-energy state m A >m A m A m A m A momentum conservation p = E c energy conservation E + p2 2m A =(m A m A ) c 2 recoil energy (energy loss) E R = E2 2m A c 2 m A c 2 A MeV E R E E (m A m A ) c 2 but E R > in general Emitted gamma rays are not resonantly re-absorbed by other nuclei in gases 11

Electric-dipole transitions Classical image radiation from an oscillating electric dipole Quantum mechanically i f = 4 q 2 E 3 rate 3 e 2 3 f r i 2 c2 fine-structure constant Atomic transition = e2 4

12 Electric-dipole transitions Classical image radiation from an oscillating electric dipole Quantum mechanically i f = 4 q 2 E 3 rate 3 e 2 3 f r i 2 c2 fine-structure constant Atomic transition = e2 4 0 c f r i = d 3 r f (r)r i (r) ev r m s = / 10 7 ev E R = E 2 /(2m A c 2 ) Nuclear transition r A 1/ m (E1) E MeV s 10 ev E E 3 A 1/3 fm c 10 9 ev 2 12

13 Higher multi-pole transitions Often, ( electric-dipole ) ((E1) decay ) is forbidden. f r i =0 may still decay radiatively by higher-order and slower processes Table 4.1. Selection rules for radiative transitions angular type symbol momentum parity change J change τ (sec) 6 10 E5 E4 M4 electric dipole E1 1 yes magnetic dipole M1 1 no electric quadrupole E2 2 no magnetic quadrupole M2 2 yes electric octopole E3 3 yes magnetic octopole M3 3 no electric 16-pole E4 4 no magnetic 16-pole M4 4 yes E3 M3 M E2 E1 M1 Lifetime of excited nuclear states as a function of E γ for various multipoles E(MeV) Lifetimes of excited nuclear states as a function of E(MeV) for various electric 13

14 Internal conversion An excited nucleus can interact with an electron in one of the lower atomic orbitals, causing the electron to be emitted (ejected) from the atom. s-electrons have finite probability density at the nuclear position. s for a hydrogen atom 1s The electron may couple to the excited state of the nucleus and take the energy of the nuclear transition directly, without an intermediate gamma ray. probability density s 2p 3s interaction Energy of the conversion electron followed by characteristic x-ray emission Auger effect E ce (m A m A ) c 2 E b E E b r (atomic unit) binding energy of the electron 14

15 + 7/2 137 Cs _ 11/2 + 3/2 137 Ba γ (90%) internal conversion (10%) 137 internal conversion beta most internal conversion electrons from the K shell K E ce (m A m A ) c 2 E b ejection from higher orbitals generally less probable L, M electron momentum 15

16 Mössbauer effect recoil energy (energy loss) Emitted gamma rays are not resonantly re-absorbed by other nuclei in gases. but... E R = Inverse transition (resonant re-absorption) possible when nuclear recoil is suppressed in a crystal ( very very large ma ) Mössbauer effect (discovered in 1957) the excited nucleus decays in flight with the Doppler effect compensating the nuclear recoil E2 2m A c 2 16

17 Mössbauer spectroscopy 191 Os Ir 191 Os source γ 191 Ir γ absorber detector γ v v v(cm/sec) % absorption E (µ ev)

18 Doppler-shift attenuation method germanium photon beam θ detectors 70 MeV 19 F 58Ni target counts per channel θ=24 θ=52 at rest in flight θ=156 θ= (kev) E γ Fig Measurement of radiative-decay lifetimes by the Doppler-shift attenu- 74Br 1068 kev gamma-ray 0.25 ps lifetime 18

Mössbauer effect + Doppler shift Test of Albert Einstein's theory of general relativity Gravitational red shift of light Clocks run differently at different places in a gravitational field

19 Mössbauer effect + Doppler shift Test of Albert Einstein's theory of general relativity Gravitational red shift of light Clocks run differently at different places in a gravitational field Gravitational shift h(f r f e )=mgh hf e = mc 2 f r =1+ gh f e c 2 Doppler shift s f r 1 v/c = f e 1+v/c 1 v = gh c v c = m/s gamma ray (14.4 kev) 57 Fe f e f r blue shift by falling H = 22.5 m 57Fe v by Pound and Rebka, 1959 Jefferson laboratory (Harvard University) 19

20 Weak interaction and beta decay 20

21 Four fundamental interactions interaction gravity exchanged particle (gauge boson) graviton decay weak W ±, Z 0 beta decay electromagnetic photon gamma decay strong nuclear force gluon pion and other hadrons alpha decay tunnel effect 21

22 beta decay decay + decay A ZN A ZN A Z+1N +e A Z + e 1N +e + + e half life = 5730 years dating 22

23 Emitted electron (positron) energy has a broad distribution 64Cu _ + β β 64Cu p (MeV/c) p (MeV/c) β β β 23

24 beta decay decay + decay A ZN A ZN A Z+1N +e A Z + e 1N +e + + e half life = 5730 years dating The existence of the neutrino was predicted by Wolfgang Pauli in 1930 to explain how beta decay could conserve energy, momentum, and angular momentum. Pauli 24

fundamental processes n pe e p ne + e mp = 938.

only in nuclei Feynman diagram p n weak boson W

25 fundamental processes n pe e p ne + e mp = MeV/c 2 < mn = MeV/c 2 mean life = ± 1.5 s - free proton does NOT decay - takes place only in nuclei Feynman diagram p n weak boson W mw = GeV/c 2 e ν e cf. mpion = MeV/c 2 (±), MeV/c 2 (neutral) 25

26 Fermi theory of beta decay Decay rate w = 2 ~ h p e H n i 2 dn de Fermi s golden rule density of state weak interaction is a short-range force H (r 2 r 1 ) G (r 2 r 1 ) G Electron energy distribution dominated by density of state 放出される電のエネルギー分布は状態密度で決まる 26

27 Density of state 状態密度 assuming plane waves dn / p 2 dpq 2 dq electron neutrino p : electron momentum q : neutrino momentum energy E = cq E e = p m 2 ec 4 + p 2 c 2 de = de = cdq Cu Q = kev dn de / p2 q 2 dp / (Q statistical factor 統計因 Experiment Experiment Theory E e ) 2 p 2 dp n (p) (arb. unit) 3 2 Coulomb repulsion p /m e c

28 Electron capture (EC) a) b) (A,Z) k l m c) (A,Z 1) (A,Z 1) l m νe 40 Ar 18 γ followed by characteristic x-ray emission Auger effect 40 K % MeV EC γ a radiation from the human body % MeV β 40 20Ca 0+ A ZN + e A Z 1N + e fundamental process: pe n e neutrino energy: E = M(A, Z)c 2 M(A, Z 1)c 2 atomic mass (not nuclear mass) 28

29 β + decay and electron capture + decay A ZN A Z 1N +e + + e M N (A, Z)c 2 >M N (A, Z 1)c 2 + m e c 2 nuclear mass electron capture A ZN + e A Z 1N + e M N (A, Z)c 2 >M N (A, Z 1)c 2 m e c 2 Both may not always be energetically possible! 29

30 By transforming the Feynman diagram... p n W e n pe e p ne + e ν e betabeta+ pe e p n e e + n electron capture (EC) neutrino detection 30

31 Parity violation... but before that... β Symmetry and conservation law 31

no change under a transformation Any symmetry of a physical law has a corresponding conservation law Noether s theorem symmetry temporal translation spatial translation rotation reflection r -r (P)

32 no change under a transformation Any symmetry of a physical law has a corresponding conservation law Noether s theorem symmetry temporal translation spatial translation rotation reflection r -r (P) time reversal (T) charge conjugation (C) gauge invariance conserved quantity energy momentum angular momentum parity T-parity C-parity electric charge Example: Coulomb force V (r) = q 1q r 2 or V (r 1, r 2 )= q 1 q r 1 r

33 example in the classical mechanics Hamilton equations q i = H p i ṗ i = H q i If the Hamiltonian does not explicitly depend on qi (invariant under the spatial translation) gauge invariance ṗ i =0 ゲージ不変性 B = A, E = invariant under the gauge transformation p i = const Conservation of momentum A t 運動量保存 A A = A +, = Invariance of the Action S t Conservation of the electric charge t + j =0 33

34 Parity reflection ˆ (r) = ( r) parity operator ˆ2 (r) = (r) Eigenvalues ± 1 If the physical law is invariant under the reflection (gravitational, electromagnetic, and strong interaction) i tˆ = H ˆ i tˆ =ˆH ˆH = H ˆ [ˆ, H] =0 Heisenberg s equation of motion i dˆ dt =[ˆ, H] =0 Conservation of parity 34

Wu in 1957 4BAPBPF 7PK ^ ^ M b by d c e M c Y d n A o F 76?

35 parity violation nonconservation of parity in the weak interaction Prediction by T.-D. Lee and C. N. Yang in 1956 Experimental verification by C.S. Wu in BAPBPF 7PK ^ ^ M b by d c e M c Y d n A o F 76? anisotropy 7PPP real world in the mirror ol l 2 i m ph M go =FPIFK= ^ ^ M b T cy d c _ c M h _ P asymmetry {a n Low-temperature cryostat Cerium magnesium nitrate crystal PPPPPPPPP)PPPP 8BDDBE 35

36 parity violation nonconservation of parity in the weak interaction Prediction by T.-D. Lee and C. N. Yang in 1956 Experimental verification by C.S. Wu in BAPBPF 7PK ^ ^ M b by d c e M c Y d n A o F 76? anisotropy 7PPP real world inverted world θ e - ol l 2 i m ph M go =FPIFK= ^ ^ M b T cy d c _ c M h _ P asymmetry {a θ 60 Co 60 Co n Low-temperature cryostat Cerium magnesium nitrate crystal PPPPPPPPP)PPPP 8BDDBE e- 36

37 37

38 Lee Yang Wu Nobel prize in physics (1957) 38

39 CP violation Makoto Kobayashi Toshihide Maskawa Nobel prize in physics (2008) 39

40 CPT theorem CPT Preservation of CPT symmetry by all physical phenomena Any Lorentz invariant local quantum field theory with a Hermitian Hamiltonian must have CPT symmetry 40

Fundamentals in Nuclear Physics

2014/ Fundamentals in Nuclear Physics Kenichi Ishikawa () http://ishiken.free.fr/english/lecture.html ishiken@atto.t.u-tokyo.ac.jp 1 Nuclear decays and fundamental interactions (II) Weak interaction and