Remarks about weak-interaction processes

Transcription

1 Remarks about weak-interaction processes K. Langanke GSI Darmstadt and TUD, Darmstadt March 9, 2006 K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

2 Nuclear beta decay, energetics Q-value dened as the total kinetic energy released in the reaction β decay, Q β = M i M f + E i E f A(Z, N) A(Z + 1, N 1) + e + ν e β + decay, Q β + = M i M f + E i E f 2m e A(Z, N) A(Z 1, N + 1) + e + + ν e Electron capture, Q EC = M i M f + E i E f A(Z, N) + e A(Z 1, N + 1) + ν e. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

3 Transition rates for β decay Fermi's golden rule: λ = 2π M if 2 (2π ) 3 δ (4) (p f + p e + p ν p i ) d 3 p f (2π ) 3 d 3 p e (2π ) 3 d 3 p ν (2π ) 3 λ = 1 2π 7 M if 2 1 = 2J i + 1 M if 2 δ(m nuc f lepton spins M i,m f f H w i 2 + E e + E ν Mi nuc )pe 2 p2 ν dp dω e dω ν edp ν 4π 4π K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

4 Transition rates for β decay W = E e /(m e c 2 ); λ = m5 e c 4 G 2 V 2π 7 W 0 = Mnuc i Mf nuc = Q m e c 2 m e c W0 1 C(W )F (Z, W )(W 2 1) 1/2 W (W 0 W ) 2 dw C(W ) = 1 G 2 V M if 2 dω e 4π dω ν 4π F (Z, W ) Fermi function, accounts for distortion of the electron (positron) wave function due to Coulomb eects. We need to compute shape factor, C(W ) = 1 G 2 V 1 2J i + 1 between states: i = J i M i ; T i T zi ; lepton spins M i,m f f H w i 2 dω e 4π f = J f M f ; T f T zf ; e ; ν dω ν 4π K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

5 Weak Hamiltonian Current-Current interaction: H w = G V 2 d 3 rj µ (r)j µ (r) f H w i = G V 2 d 3 r J f M f ; T f T zf ; e, ν j µ J µ J i M i ; T i T zi Assuming plane waves for electron and neutrino: e; ν j µ 0 = e i(p +p e ν ) r ūγ µ (1 γ 5 )v f H w i = G V l µ d 3 re i q r J f M f ; T f T zf J µ J i M i ; T i T zi 2 l µ = ūγ µ (1 γ 5 )v K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

6 Non-relativistic reduction Assuming one nucleon participates in the decay and that we can use the free current (impulse approximation): f H w i = G V l µ 2 ( 1 ψ = σ p E+M f H w i = G V 2 Generalization to A particles: H w = G V 2 A k=1 d 3 re i q r ψf γ µ (1 + g A γ 5 )t ± ψ i ) ( φ φ 0 ) d 3 re i q r φ f (l g A l σ)t ± φ i e i q r k (l 0 1 k + g A l σ k )t k ± K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

7 Non-relativistic reduction H w = G V 2 A k=1 e i q r k (l 0 1 k + g A l σ k )t k ± e i q r = l 4π(2l + 1)( i) l j l (qr)y l0 (θ, ϕ) j l (qr) (qr)l (2l + 1)!! Zero order: Allowed transitions (Fermi, Gamow-Teller) Higher orders: Forbidden transitions. K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

8 ft-value λ = ln 2 K W0 1 C(W )F (Z, W )(W 2 1) 1/2 W (W 0 W ) 2 dw For allowed transitions: C(W ) = B(F ) + B(GT ), λ = ln 2 t 1/2 = ln 2 K [B(F ) + B(GT )]f (Z, W 0) ft 1/2 = K B(F ) + B(GT ), K = ± 1.6 s K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

9 Fermi Transitions B(F ) = 1 J f M f ; T f T zf 2J i + 1 M i,m f A t k ± J i M i ; T i T zi 2 k=1 B(F ) = [T i (T i + 1) T zi (T zi ± 1)]δ Ji,J f δ Ti,T f δ Tzf,T zi ±1 Energetics (Isobaric Analog State): E IAS = Q β + sign(t zi )[E C (Z + 1) E C (Z) (m n m H )] Selection rule: J = 0 T = 0 π i = π f Sum rule (sum over all the nal states): S(F ) = S (F ) S + (F ) = 2T zi = (N Z) K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

10 Gamow-Teller Transitions B(GT ) = Selection rule: g 2 A 2J i + 1 B(GT ) = m,m i,m f J f M f ; T f T zf g 2 A 2J i + 1 J f ; T f T zf A σm k tk ± J i M i ; T i T zi 2 k=1 A σ k t± J k i ; T i T zi 2 k=1 g A = ± J = 0, 1 (no J i = 0 J f = 0) T = 0, 1 π i = π f Ikeda sum rule: S(GT ) = S (GT ) S + (GT ) = 3(N Z) K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

11 Gamow-Teller transitions II β decay J i ; T, T Final state J f ; T 1, T 1 B(GT ) = 2g A 2 J f ; T 1 A k=1 σk t k J i ; T 2 2J i + 1 2T + 1 Final state J f ; T, T 1 B(GT ) = 2g A 2 J f ; T A k=1 σk t k J i ; T 2 2J i + 1 (2T + 1)(T + 1) Final state J f ; T + 1, T 1 B(GT ) = 2g 2 A 2J i + 1 J f ; T + 1 A k=1 σk t k J i ; T 2 (2T + 1)(T + 1)(2T + 3) K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

12 Forbidden Transitions Involve operators r l Y lm and r l [Y lm σ] K Selection rules Decay type J T π log ft Superallowed no Allowed 0,1 0,1 no First forbidden 0,1,2 0,1 yes 519 Second forbidden 1,2,3 0,1 no 1018 Third forbidden 2,3,4 0,1 yes 1722 Fourth forbidden 3,4,5 0,1 no 2224 K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

13 Electron Capture Fermi's golden rule: σ = 2π v e M if 2 (2π ) 3 δ (4) (p f + p ν p i p e ) d 3 p f d 3 p ν (2π ) 3 (2π ) 3 σ i,f (E e ) = G 2 V 2π 4 F (Z, E e)[b(f ) + B(GT )]p 2 ν K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

14 Neutrino cross sections Charged current: (Z, A) + ν e (Z + 1, A) + e σ i,f (E ν ) = G 2 V π p ee e F (Z + 1, E e )[B(F ) + B(GT )] Neutral current: (Z, A) + ν (Z, A) + ν σ i,f (E ν ) = G F π (E ν w) 2 B(GT 0 ) with w = E f E i In general, multipoles beyond allowed transitions are necessary. See Donelly and Peccei, Phys. Repts. 50, 1 (1979). K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

15 General considerations: Which multipoles? K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

16 Electron capture: energetics during collapse (MeV) 20 µ e 10 Q = µ n µ p 0 Q p ρ (g cm 3 ) capture rate becomes less dependent on details of GT distribution with increasing density (chem. potential); for ρ 11 > 1, it depends essentially only on total strength and centroid K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

17 Example: electron capture at high electron energy λec (s 1 ) ρ 11Y e=0.07, T=0.93 ρ 11Y e=0.62, T=1.32 ρ 11Y e=4.05, T= Q (MeV) Assumption: capture proceeds by a single transition (E f E i = const) with a constant strength K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

18 Remarks about response: Which models? K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

19 Shell Model. INTERACTING SHELL MODEL N S STATES INDEPENDENT SHELL MODEL SLATER DETERMINANT φ EXTERNAL SPACE (ALWAYS EMPTY) VALENCE SPACE i j RESIDUAL INTERACTION Heff + MOVES NUCLEON FROM STATE j TO STATE i O α = a i a j [O α, O β ] π 0 2 Ns SUCH OPERATORS CORE (ALWAYS OCCUPIED) HAMILTONIAN H = εo - _ 1 2VO 2 IF V = 0, H IS PURE I - BODY, SOLVABLE Ns x Ns MATRIX ELEMENTS Heff IF V π 0, φ {ALL POSSIBLE φ's} FULL COMBINATORIAL DIMENSION C K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

20 Diagonalization shell model. DIAGONALIZATION APPROACH "GIANT" MATRIX < H > N = Ns P N VAL Ns N N VAL DIMENSION N 10 9 FOR 60 Zn REDUCTION OF SIZE DUE TO SYMMETRIES MODERN ALGORITHM (STRASSBOURG - MADRID) ---- LANCZOS ALGORITHM (FEW LOWEST EIGENSTATES) ---- STORAGE OF ONLY Hpp, Hnn, Hpn ---- EFFICIENT ALGORITHM TO CONSTRUCT <... H... > FROM Hpp, Hnn, Hpn ALL EVEN-EVEN NUCLEI IN PF-SHELL C K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

21 Shell Model Monte Carlo. SHELL MODEL MONTE CARLO GOAL: DETERMINE NUCLEAR PROPERTIES, NOT ALL ~10 9 COMPONENTS OF W.F. CONSIDER THERMAL AVERAGE IN CANONICAL (FIXED NUMBER) EMSEMBLE < A > = Tr (e--βh A) Tr (e --βh ) β = T 1 HUBBARD - STRATONOVICH TRANSFORMATION 2-BODY MANY 1-BODY EVOLUTIONS IN FLUCTUATING EXTERNAL FIELDS SUPPOSE e --βh dσ 2π/βV e --βvo2 1 2 = e --βvσ2 1 2 e βσvo 1-BODY PROPAGATOR HOWEVER: MANY NON-COMMUTING O's e --βh = (e -- βh ) N β t ; β = TIME SLICE Nt SEPARATE σ-fields AT EACH TIME SLICE FOR EACH Ο MONTE-CARLO EVALUATION OF σ-integrals EMBARRASSINGLY PARALLEL C K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

22 Shell Model versus RPA. 100 σ (10 42 cm 2 ) O(ν e,e + )X CRPA SM 100 σ (10 42 cm 2 ) O(ν e,e )X CRPA SM Temperature T (MeV) The RPA considers only (1p-1h) correlations; it is well suited to describe the centroid of giant resonances K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

23 Inelastic electron scattering: RPA description of response K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

24 Beta-decay, electron capture, charged-current Electron capture 15 Beta decay 20 E (MeV) Finite T (Z,A) GT 5 F GT + 0 (Z+1,A) E (MeV) K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

25 Systematics of Fermi and Gamow-Teller centroids K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

26 How well do we know charged-current cross sections? RPA vs simple Gaussian: factor 2 (Surmann+Engel) K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

27 What about other multipoles? for neutrino spectrum with T = 4 MeV K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

28 Inuence of higher multipoles Spectrum for muon decay: µ + e + + ν e + ν µ ν/cm 2 /MeV ν e MeV K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

29 Inuence of higher multipoles 56 Fe(ν e, e ) 56 Co measured by KARMEN collaboration: σ exp = 2.56 ± 1.08(stat) ± 0.43(syst) cm 2 σ th = cm 2 σ (10 42 cm 2 ) σ (10 42 cm 2 ) Energy (MeV) Energy (MeV) K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

30 Typical supernova neutrino spectra K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

31 Multipole decomposition K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

32 Determining inelastic neutrino-nucleus cross sections INELASTIC NEUTRINO SCATTERING ON NUCLEI E E ν GT 0 resonance ν Finite T C N ν (Z,A) E* (v, v'), GT 0 (e, e'), M1 S S ISOVECTOR PIECE DOMINATES L T(GT 0) ~ Σt i z(i)s i T(M1)= 1 2 (L p - L n)+(g p - g n)σt z(i)s i µ N i M1 DATA YIELD GT 0 INFORMATION IF ORBITAL PART CAN BE REMOVED K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35

33 Neutrino cross sections from electron scattering Orbital 52 Cr B(M1) (µ N 2 ) Spin Total σ (10 42 cm 2 ) Only GS (SDalinac data) Only GS (Theory) T = 0.8 MeV 52 Cr Expt Excitation Energy (MeV) high-precision SDalinac data large-scale shell model Neutrino Energy (MeV) neutrino cross sections from (e, e ) data validation of shell model G.Martinez-Pinedo, P. v. Neumann-Cosel, A. Richter. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, / 35