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, 2006 1 / 35
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, 2006 2 / 35
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, 2006 3 / 35
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 + 1 2 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, 2006 4 / 35
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, 2006 5 / 35
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 0 1 + 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, 2006 6 / 35
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, 2006 7 / 35
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 = 6144.4 ± 1.6 s K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, 2006 8 / 35
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, 2006 9 / 35
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 = 1.2720 ± 0.0018 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, 2006 10 / 35
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, 2006 11 / 35
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 0 + 0 + 0 no 3.13.6 Allowed 0,1 0,1 no 2.910 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, 2006 12 / 35
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, 2006 13 / 35
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, 2006 14 / 35
General considerations: Which multipoles? K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, 2006 15 / 35
Electron capture: energetics during collapse 40 30 (MeV) 20 µ e 10 Q = µ n µ p 0 Q p 10 10 10 11 10 12 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, 2006 16 / 35
Example: electron capture at high electron energy 10 5 10 4 λec (s 1 ) 10 3 10 2 10 1 10 0 10 1 10 2 ρ 11Y e=0.07, T=0.93 ρ 11Y e=0.62, T=1.32 ρ 11Y e=4.05, T=2.08 15 10 5 0 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, 2006 17 / 35
Remarks about response: Which models? K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, 2006 18 / 35
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-195-1 K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, 2006 19 / 35
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-195-2 K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, 2006 20 / 35
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-195-3 K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, 2006 21 / 35
Shell Model versus RPA. 100 σ (10 42 cm 2 ) 10 1 0.1 16 O(ν e,e + )X CRPA SM 100 σ (10 42 cm 2 ) 10 1 0.1 16 O(ν e,e )X CRPA SM 0.01 2 4 6 8 10 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, 2006 22 / 35
Inelastic electron scattering: RPA description of response K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, 2006 23 / 35
Beta-decay, electron capture, charged-current Electron capture 15 Beta decay 20 E (MeV) 10 5 0 Finite T (Z,A) 15 10 GT 5 F GT + 0 (Z+1,A) E (MeV) K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, 2006 24 / 35
Systematics of Fermi and Gamow-Teller centroids K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, 2006 25 / 35
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, 2006 26 / 35
What about other multipoles? for neutrino spectrum with T = 4 MeV K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, 2006 27 / 35
Inuence of higher multipoles Spectrum for muon decay: µ + e + + ν e + ν µ 10 12 ν/cm 2 /MeV 5 4.5 4 3.5 ν e 3 2.5 2 1.5 1 0.5 0 0 10 20 30 40 50 60 MeV K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, 2006 28 / 35
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) 10 40 cm 2 σ th = 2.38 10 40 cm 2 σ (10 42 cm 2 ) 120 100 80 60 40 σ (10 42 cm 2 ) 20 15 10 5 1 2 0 5 10 15 20 25 30 Energy (MeV) 20 0 0 5 10 15 20 Energy (MeV) K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, 2006 29 / 35
Typical supernova neutrino spectra K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, 2006 30 / 35
Multipole decomposition K. Langanke (GSI and TUD, Darmstadt)Remarks about weak-interaction processes March 9, 2006 31 / 35
Determining inelastic neutrino-nucleus cross sections INELASTIC NEUTRINO SCATTERING ON NUCLEI E E ν GT 0 resonance ν Finite T C-195-14 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, 2006 32 / 35
Neutrino cross sections from electron scattering 0.1 0.08 Orbital 52 Cr B(M1) (µ N 2 ) 0.06 0.04 0.02 0 1.5 1.0 0.5 0.0 1.5 Spin Total σ (10 42 cm 2 ) 10 3 10 2 10 1 10 0 10 1 10 2 10 3 Only GS (SDalinac data) Only GS (Theory) T = 0.8 MeV 52 Cr 1.0 0.5 0.0 1.5 1.0 0.5 0.0 Expt. 6 8 10 12 14 Excitation Energy (MeV) high-precision SDalinac data large-scale shell model 10 4 0 5 10 15 20 25 30 35 40 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, 2006 33 / 35