Nuclear reactions in astrophysics

Transcription

1 Nuclear reactions in astrophysics Pierre Descouvemont Université Libre de Bruxelles Brussels, Belgium 1

2 Content of the lectures 1. Introduction 1. Reaction networks 2. Needs for astrophysics 3. Specificities of nuclear astrophysics 2. Low-energy cross sections 1. Definitions 2. General properties 3. S-factor 3. Reaction rates 1. Definitions 2. Gamow peak 3. Resonant and non-resonant rates 4. General scattering theory (simple case: spins 0, no charge, single channel) 1. Different models 2. Optical model 3. Scattering amplitude and cross sections 4. Phase-shift method 5. Resonances 6. Generalizations: Coulomb interaction, absorption, non-zero spins 2

3 5. Models used in nuclear astrophysics 1. Brief overview 2. The potential model : Radiative-capture reactions 3. The R-matrix method 4. Microscopic models 3

4 1. Introduction 4

5 1. Introduction Goal of nuclear astrophysics: understand the abundances of the elements Nuclear astrophysics Masses Cross sections b lifetimes Fission barriers H, 4 He most abundant (~75%, ~25%) «Gap» between A=4 and A=12: no stable element with A=5 and 8 Iron peak (very stable) astrophysicist Stellar model Etc 5

6 1. Introduction Years ~ : Hoyle, Gamow Role of nuclear reactions in stars Energy production Nucleosynthesis (Hoyle state in 12 C) 1957: B 2 FH: Burbidge, Burbidge, Fowler, Hoyle (Rev. Mod. Phys. 29 (1957) 547) Wikipedia site: Cycles: pp chain: converts 4p 4 He CNO cycle: converts 4p 4 He (via 12 C) s (slow) process: (n,g) capture followed by b decay r (rapid) process: several (n,g) captures p (proton) process: (p,g) capture Nucleosynthesis: Primordial (Bigbang): 3 first minutes of the Universe Stellar: star evolution, energy production Essentially two (experimental) problems in nuclear astrophysics Low energies very low cross sections (Coulomb barrier) Need for radioactive beams in most cases a theoretical support is necessary (data extrapolation) 6

7 1. Introduction Reaction networks: set of equations with abundances of nucleus m: Y m Destruction of m by b decay: l m =1/t m Production of m by b decay from elements k Destruction of m by reaction with k Production of m by reaction k+l m with [kl] (m) ~ <sv>, <sv>=reaction rate (strongly depends on temperature) In practice: Many reactions are involved (no systematics) s must be known at very low energies very low cross sections Reactions with radioactive elements are needed At high temperatures: high level densities properties of many resonances needed 7

8 1. Introduction Specificities of nuclear astrophysics low energies (far below the Coulomb barrier) small cross sections (in general not accessible in laboratories at stellar energies) low angular momenta (selection of resonances) radioactive nuclei need for radioactive beams ( 7 Be, 13 N, 18 F, ) different types of reactions: transfer (a,n), (a,p), (p,a), etc radiative-capture: (p,g), (a,g), (n,g), etc weak processes: p+p d+ e + + n fusion: 12 C+ 12 C, 16 O+ 16 O, etc. different situations capture, transfer resonant, non resonant low level density (light nuclei), high level density (heavy nuclei) peripheral, internal processes different approaches, for theory and for experiment 8

9 1. Introduction Some key reactions d(a,g) 6 Li, 3 He(a,g) 7 Be: Big-Bang Triple a, 12 C(a,g) 16 O: He burning 7 Be(p,g) 8 B: solar neutrino problems 18 F(p,a) 15 O: nova nucleosynthesis Etc 9

10 2. Low-energy cross sections Definitions General properties S-factor 10

11 2. Low-energy cross sections Types of reactions: general definitions valid for all models Type Example Origin Transfer 3 He( 3 He,2p)a Strong Radiative capture 2 H(p,g) 3 He Electromagnetic Weak capture p+p d+ e + + n Weak cross section decreases 11

12 2. Low-energy cross sections Transfer: A+B C+D (s t, strong interaction, example: 3 He(d,p) 4 He) σ t,c c (E) = π 2J + 1 k 2 2I I U cc E 2 Jπ U cc E =collision (scattering) matrix (obtained from scattering theory various models) c, c =entrance and exit channels Transfer reaction: Nucleons are transfered Scattering energy E A+B threshold, ex: 3 He+d C+D threshold, ex: 4 He+p Compound nucleus, ex: 5 Li 12

13 2. Low-energy cross sections Radiative capture : A+B C+g (s C, electromagnetic interaction, example: 12 C(p,g) 13 N) σ C J f π f E λ k 2λ+1 γ < Ψ J fπ f M λ Ψ J iπ i E > 2 J i π i J f π f =final state of the compound nucleus C Ψ J iπ i E =initial scattering state of the system (A+B) M λμ =electromagnetic operator (electric or magnetic): M λμ e r λ Y μ λ (Ω r ) Capture reaction: A photon is emitted g Long wavelength approximation: Wave number k γ = E γ /ħc, wavelength: λ γ = 2π/k γ Typical value: E γ = 1 MeV, λ γ 1200 fm >> typical dimensions of the system (R) k γ R 1= Long wavelength approximation 13

14 2. Low-energy cross sections initial state E>0, contains all J i π i, A+B threshold, ex: 12 C+p final states E f <0, specific J f π f σ C J f π f E J i π i k γ = E E f /ħc = photon wave number In practice λ k γ 2λ+1 < Ψ J fπ f M λ Ψ J iπ i E > 2 o Summation over λ limited to 1 term (often E1, or E2/M1 if E1 is forbidden) E2 E1 k γr 2 1 (from the long wavelength approximation) o Summation over J i π i limited by selection rules J i J f λ J i + J f π i π f = 1 λ for electric, π i π f = 1 λ+1 for magnetic 14

15 2. Low-energy cross sections Example 1: 8 Be(a,g) 12 C Initial partial wave J i = 0 + (includes the Hoyle state. E2 dominant (E1 forbidden in N=Z) essentially the J f = 2 + state is populated. 15

16 2. Low-energy cross sections Example 2: 14 N(p,g) 15 O 14 N+p E (all J i values) Spin of 14 N: I 1 =1 +, proton I 2 =1/2 + Channel spin I: I 1 I 2 I I 1 + I 2 I = 1/2, 3/2 Orbital momentum l I l J i I + l At low energies, l = 0 is dominant J i = 1/2 +, 3/2 + multipolarity E1 transitions to J f = 1/2, 3/2, 5/2 Resonance 1/2 + determines the cross section 16

17 2. Low-energy cross sections Example 3: 12 C(a,g) 16 O, 15 N(p,g) 16 O, 15 N(p,a) 12 C 15 N+p threshold 15 N(p,g) 16 O and 15 N(p,a) 12 C are open 15 N(p,g) 16 O negligible 12 C+a threshold only possibility: 12 C(a,g) 16 O 12 C(a,g) 16 O (very) important 17

18 2. Low-energy cross sections Weak capture (p+p d+n+e - ): tiny cross section no measurement (only calculations) J σ f π f W E < Ψ Jfπf O β Ψ Jiπi E > 2 J i π i o Calculations similar to radiative capture o O β = Fermi ( i t i± ) and Gamow-Teller ( i t i± σ i ) operators o 3 He+p 4 He+n+e - : produces high-energy neutrinos (more than tiny!) 18

19 2. Low-energy cross sections Fusion: similar to transfer, but with many output channels statistical treatment optical potentials E many states experimental cross section Satkowiak et al. PRC 26 (1982) C+ 12 C 19

20 2. Low-energy cross sections General properties (common to all reactions) V(r) l > 0 E Scattering energy E: wave function Ψ i E common to all processes Reaction threshold Cross sections dominated by Coulomb effects Sommerfeld parameter η = Z 1 Z 2 e 2 /ħv Coulomb functions at low energies F l η, x exp πη F l x, G l η, x exp πη G l x, l = 0 r E astro Coulomb effect: strong E dependence : exp 2πη neutrons: σ E 1/v Strong l dependence l l+1 r 2 Centrifugal term: ħ2 2μ stronger for nucleons (μ 1) than for a (μ 4) 20

21 2. Low-energy cross sections General properties: specificities of the entrance channel common to all reactions All cross sections (capture, transfer) involve a summation over l: σ E = l σ l (E) The partial cross sections σ l (E) are proportional to the penetration factor P l 1.E+01 1.E+00 1.E-01 1.E-02 1.E-03 P l E = l=0 ka F l ka 2 +G l ka 2 l=2 l= E (MeV) d+d (a =typical radius) Consequences l > 0 are often negligible at low energies l = l min is dominant (often l min = 0) For l = 0, P 0 E exp 2πη Astrophysical S factor: S(E) = σ(e)eexp(2πη) (Units: E*L 2 : MeV-barn) removes the coulomb dependence only nuclear effects weakly depends on energy σ E S 0 exp 2πη /E (any reaction at low E) 21

22 S (kev-b) s (barn) 2. Low-energy cross sections 10-5 non resonant:s E = σ(e)eexp(2πη) He(a,g) 7 Be Parker et al. Krawinkel et al. Di Leva et al Example: 3 He(a,g) 7 Be reaction Cross section s(e) Strongly depends on energy Logarithmic scale S factor Coulomb effects removed Weak energy dependence Linear scale 0.2 Parker et al. Krawinkel et al. 0 Di Leva et al

23 2. Low-energy cross sections Resonant cross sections: Breit-Wigner form σ R E π k 2 2J R + 1 Γ 1 (E)Γ 2 (E) E R E 2 + Γ 2 /4 2I I J R, E R =spin, energy of the resonance Valid for any process (capture, transfer) Valid for a single resonance several resonances need to be added (if necessary) Γ 1 =Partial width in the entrance channel (strongly depends on E, l) Γ 1 E = 2γ 2 1 P l (E) with γ 2 1 =reduced width (does not depend on E) P l E exp 2πη A resonance at low energies is always narrow (role of P l E ) Γ 2 =Partial width in the exit channel (weakly depends on E, l) o Transfer: Γ 2 E = 2γ 2 2 P lf (E + Q) (in general Q E P lf (E + Q) almost constant) o Capture: Γ 2 E E E f 2λ+1 B(Eλ) weak energy dependence S factor near a resonance S(E) = σ(e)eexp(2πη) γ S R E 2 1 Γ 2 P E R E 2 +Γ 2 /4 l E exp 2πη Almost constant Simple estimate at low E (at the Breit-Wigner approximation) 23

24 2. Low-energy cross sections S-factor (kev-b) C+p 13 N 12 C(p,g) 13 N 1/2-5/2+ 3/2-1/ S R E γ 2 1 Γ 2 E R E 2 + Γ 2 /4 P l E exp 2πη γ 2 1 Γ 2 E R E 2 + Γ 2 /4 For l = 0 : P 0 E exp 2πη constant For l > 0, P l E P 0 E l > 0 resonances are suppressed In 12 C(p,g) 13 N: Resonance 1/2 + : l = 0 Resonances 3/2 -, 5/2 + l = 1,2 negligible Note: BW is an approximation Neglects background, external capture Assumes an isolated resonance Is more accurate near the resonance energy 24

25 2. Low-energy cross sections 3 He(d,p) 4 He: isolated resonance in a transfer reaction 3 He+d 3/2+ 20 a+p Q=18.4 MeV 1/2-3/2- S (MeV-b) He(d,p) 4 He Zhichang 77 Krauss 87 Geist E cm (MeV) 3/2+ resonance: Entrance channel: spin S=1/2,3/2, parity + l = 0,2 Exit channel: spin S=1/2, parity + l = 1 25

26 2. Low-energy cross sections Breit Wigner approximation S (MeV-b) σ dp E π k 2 2J R + 1 2I I Γ d (E)Γ p (E) E R E 2 + Γ 2 /4 Width at half maximum=total width G Amplitude: Γ d Γ p /Γ 2 5 Zhichang 77 Krauss He(d,p) 4 He Geist E cm (MeV) S(E) constant for l = 0 26

27 2. Low-energy cross sections Two comments: 1. Selection of the main resonances 2. Going beyond the Breit-Wigner approximation 1. Selection of the main resonances 11 C(p,g) 12 N (spin 11 C=3/2 - ) Resonance 2 - : l = 0, E1 Resonance 2 + : l = 1, E2/M1 negligible 18 F(p,a) 15 O (spin 18 F=1 + ) Many resonances Only l = 0 resonances are important J = 1/2 +, 3/2 + only In general a small number of resonances play a role 27

28 2. Low-energy cross sections 2. Going beyond the Breit-Wigner approximation How to go beyond the BW approximation? Problem of vocabulary Direct capture External capture Non-resonant capture = «direct» capture confusion! External capture σ E = M int + M ext 2 With σ BW E = M int 2 M ext C, with C=Asymptotic Normalization Constant (ANC) is needed Non resonant capture : σ E = l σ l (E) = σ R E + l l R σ l (E) scanning the resonance is necessary 28

29 2. Low-energy cross sections Many different situations Transfer cross sections (strong interaction) Non resonant: 6 Li(p,a) 3 He Resonant, with l R =l min : 3 He(d,p)a Resonant, with l R >l min : 11 B(p,a) 8 Be Multiresonance: 22 Ne(a,n) 25 Mg Capture cross sections (electromagnetic interaction) Non resonant: 6 Li(p,g) 7 Be Resonant, with l R =l min : 12 C(p,g) 13 N Resonant, with l R >l min : 7 Be(p,g) 8 B Multiresonance: 22 Ne(a,g) Mg Subthreshold state: 12 C(a,g) 16 O Weak capture cross sections (weak interaction) Non resonant p(p,e n) 2 H He(p,e + n) 4 He S-factor (MeV-b) Li(p,a) 3 He 6 Li(p,g) 7 Be Cross sections E (MeV) 29

30 S-factor (ev-b) S-factor (ev-b) Li(p,g) 7 Be Be(p,g) 8 B Li+p a+ 3 He 7 Be+p 7 Be 8 B 7/2-5/2-5/2-1/2-3/ Non resonant Resonant l min =0, E1 l R =1, M1 S-factor (kev-b) C(p,g) 13 N C+p 13 N 1/2-5/2+ 3/2-1/2+ Resonant l min =0 l R =0 30

31 2. Low-energy cross sections S-factor (kev-b) S-factor (MeV-b) C(a,g) 16 O E (MeV) 22 Ne(a,n) 25 Mg a+ 12 C 20 states 125 states 16 O n+ 25 Mg a+ 22 Ne Subthreshlod states 2 +, 1 - Multiresonant General situation for heavy nuclei E (MeV) 0 26 Mg 0+ 31

32 3. Reaction rates 1. Definitions 2. Gamow peak 3. Non-resonant rates 4. Resonant rates 32

33 3. Reaction rates 1. Definition Quantity used in astrophysics: reaction rate (integarl over the energy E) N A < σv > = N A σ E vn E, T de Definition valid for resonant and non resonant reactions N A =Avogadro number T=temperature, v = velocity, k B = Boltzmann constant (k B MeV/109 K) N E, T = 1 8E πμm N k B T 3 =typical reaction time N A <σv> 1/2 exp E k B T = Maxwell Boltzmann distribution 2 approaches numerical analytical: non-resonant and resonant reactions treated separately essentially two energy dependences: exp E : decreases with E k B T exp 2πη : increases with E 33

34 3. Reaction rates 2. The Gamow peak Defines the energy range relevant for the reaction rate (non-resonant reactions) s Logarithmic scale exp(-e/kt) s.v.exp(-e/kt) Ecm Linear scale s.v.exp(-e/kt) E 0 DE 0 Ecm Gamow peak (~ Gaussian, depends on T) s Gamow peak : E 0 = 0.122μ 1/3 Z 1 Z 2 T 9 2/3 MeV: lower than the Coulomb barrier increases with T ΔE 0 = μ 1/6 Z 1 Z 2 1/3 T 9 5/6 MeV =Energy range where σ(e) must be known (T 9 = T in 10 9 K) 34

35 3. Reaction rates Examples Reaction T (10 9 K) E 0 (MeV) DE 0 (MeV) E coul (MeV) s(e 0 )/s(e coul ) d + p He + 3 He a + 12 C C + 12 C E 0 /E coul 0.3 T 9 2/3 (p and a) V(r) At low T 9, E 0 E coul (coulomb barrier) E coul Very low cross sections at stellar temperatures (different for neutrons: no barrier) l = 0 r E 0 35

36 3. Reaction rates 3. Non-resonant reaction rates Approximation: Taylor expansion about the minimum E = E 0 : 2πη + E/k B T c 0 + E E 0 2ΔE 0 2 Then < σv > 8 1/2 E πμm N k B T exp 3 0 S E exp E E k B T 2ΔE 0 S(E) is assumed constant (= S(E 0 )) in the Gamow peak < σv > S E 0 exp 3 E 0 k B T de /T2/3, with E 0 = 0.122μ 1/3 Z 1 Z 2 T 9 2/3 MeV Very fast variation with the temperature 3 He(a,g) 7 Be 36 36

37 3. Reaction rates 4. Resonant reaction rates General definition: N A < σv > = N A σ E vn E, T de here σ E This provides is given by the Breit-Wigner approximation σ E π 2J R + 1 Γ 1 (E)Γ 2 (E) k 2 2I I E R E 2 + Γ 2 /4 wg=resonance «strength» 2π < σv > R = μm N k B T 2J R + 1 ωγ = 2I I /2 ħ 2 ωγ exp Γ 1 Γ 2 Γ 1 +Γ 2 G 1,G 2 =partial widths in the entrance and exit channels E R k B T For a reaction (p,g) : G g <<G p wg~g g Valid for capture and transfer Rate strongly depends on the resonance enrgy In general: competition between resonant and non-resonant contributions 37

38 3. Reaction rates Tail contribution: for a given resonance For a resonance: < σv > S E exp 2πη E/k B T de Non resonant: S E S 0 : 1 maximum at E = E 0 Resonant: S E =BW : 2 maxima at E = E R does not depend on T E = E 0 : depends on T 2 contributions to the rate : N A < σv > N A < σv > R +N A < σv > T «resonant» «tail» 2π 3/2 < σv > R = μm N k B T ħ 2 ωγ exp E R k B T < σv > T S E 0 exp 3 E 0 k B T /T2/3, with S E 0 Γ 1(E 0 )Γ 2 (E 0 ) E R E 2 0 +Γ 2 /4 o Both contributions depend on temperature: in most cases one term is dominant o «Critical temperature»: when E 0 = E R separation not valid 38

39 3. Reaction rates Example 12 C(p,g) 13 N: E R = 0.42 MeV Integrant S E exp 2πη E/k B T E 0 E R 12 C(p,g) 13 N Above T 9 0.3: Below T 9 0.2: «resonant» contribution is dominant requires E R, ωγ only (no individual partial widths) strongly depends on E R : exp k B T E R E 0 E R : «tail» contribution is dominant requires both widths weakly depends on E R : 1/( E R E Γ 2 /4) 39

40 4. General scattering theory 1. Different models 2. Potential/optical model 3. Scattering amplitude and cross section (elastic scattering) 40

41 4. General scattering theory Scheme of the collision (elastic scattering) 1 Before collision 2 1 After collision 2 q Center-of-mass system 41

42 4. General scattering theory 1. Different models Schrödinger equation: HΨ r 1, r 2, r A = EΨ r 1, r 2, r A with E > 0: scattering states A-body equation (microscopic models) H = i t i i,j v ij ( r i r j ) r v ij =nucleon-nucleon interaction Optical model: internal structure of the nuclei is neglected the particles interact by a nucleus-nucleus potential absorption simulated by the imaginary part = optical potential HΨ(r) = ħ2 Δ + V r 2μ Ψ(r) = EΨ(r) r Additional assumptions: elastic scattering no Coulomb interaction spins zero 42

43 4. General scattering theory V (MeV) 2. Potential/Optical model Two contributions to the nucleus-nucleus potential: nuclear V N (r) and Coulomb V C (r) Typical nuclear potential: V N (r) (short range, attractive) examples: Gaussian V N r = V 0 exp r/r 0 2 Real at low energies Woods-Saxon: V N r = parameters are fitted to experiment no analytical solution of the Schrödinger equation V 0 1+exp r r 0 a Woods-Saxon potential r 0 =range (~sum of the radii) a= diffuseness (~0.5 fm) -40 Figure: V 0 =50 MeV, r 0 =5 fm, a = 0.5 fm r (fm) 43

44 4. General scattering theory Coulomb potential: long range, repulsive «point-point» potential : V C r = Z 1Z 2 e 2 r «point-sphere» potential : (radius R C ) V C r = Z 1Z 2 e 2 for r R r C V C r = Z 1Z 2 e 2 3 r 2R C R C 2 for r R C Total potential : V r = V N r + V C r : presents a maxium at the Coulomb barrier radius r = R B height V R B = E B 44

45 4. General scattering theory 3. Scattering amplitude and cross section HΨ(r) = ħ2 Δ + V r 2μ Ψ(r) = EΨ(r) At large distances : Ψ(r) A e ik r + f θ eikr r (with z along the beam axis) where: k=wave number: k 2 = 2μE/ħ 2 A =amplitude (scattering wave function is not normalized to unity) f θ =scattering amplitude (length) Incoming plane wave Outgoing spherical wave 45

46 4. General scattering theory Cross section: dσ dω = f θ 2 q, σ = dσ dω dω solid angle dw Cross section obtained from the asymptotic part of the wave function General problem for scattering states: the wave function must be known up to large distances Direct problem: determine s from the potential Inverse problem : determine the potential V from s Angular distribution: E fixed, q variable Excitation function: q variable, E fixed, 46

47 4. General scattering theory Main issue: determining the scattering amplitude f θ (and wave function Ψ r ) At low energies: partial wave expansion: Ψ r = lm Ψ l r Y m l θ, φ o Scattering wave function necessary to compute cross sections o Must be determined for each partial wave l o Main interest: few partial waves at low energies l > 0 V E l = 0 r centrifugal term : ħ2 l l+1 2μ r 2 Partial-wave expansion Only a few partial waves contribute Effect more important for nucleonnucleus: μ 1 Strongest for neutron: no barrier for l = 0. 47

48 4. General scattering theory 4. Phase-shift method Goal: solving the Schrodinger equation with a partial-wave expansion ħ2 Δ + V r 2μ Ψ r = l,m u l r Simplifying assumtions neutral systems (no Coulomb interaction) spins zero single-channel calculations elastic scattering r Ψ(r) = EΨ(r) Y l m (Ω r ) Y l m (Ω k ) 48

49 4. General scattering theory The wave function is expanded as u l r Ψ r = r l,m Y l m (Ω r ) Y l m (Ω k ) This provides the Schrödinger equation for each partial wave (Ω k = 0 m = 0) ħ2 2μ d 2 dr 2 l l + 1 r 2 u l + V r u l = Eu l Large distances : r, V r 0 u l l l + 1 r 2 u l + k 2 u l = 0 Bessel equation u l r = rj l kr, rn l kr Remarks must be solved for all l values at low energies: few partial waves in the expansion at small r: u l r r l+1 49

50 4. General scattering theory For small x: j l x xl n l x 2l+1!! 2l 1!! x l+1 For large x: j l x 1 sin(x lπ/2) x n l x 1 cos(x lπ/2) x Examples: j 0 x = sin x, n cos x x 0 x = x j0(x) n0(x) At large distances: u l (r) is a linear combination of rj l kr and rn l kr u l r C l r j l kr tan δ l n l kr With δ l = phase shift (provides information about the potential): If V=0 δ l = 0 50

51 4. General scattering theory Derivation of the elastic cross section Identify the asymptotic behaviours Ψ(r) A e ik r + f θ eikr r Ψ r l C l j l kr tan δ l n l kr Y l 0 (Ω r ) 2l+1 Provides coefficients C l and scattering amplitude f θ (elastic scattering) f θ, E = 1 dσ(θ,e) dω 2ik l=0 = f θ, E 2 4π (2l + 1)(exp(2iδ l (E)) 1) P l (cos θ) Integrated cross section (neutral systems only) σ = π k 2 l=0 2l + 1 sin 2 δ l In practice, the summation over l is limited to some l max 51

52 4. General scattering theory dσ(θ, E) dω = f θ, E 2 with f θ, E = 1 2ik l=0 (2l + 1)(exp(2iδ l (E)) 1) P l (cos θ) factorization of the dependences in E and q low energies: small number of l values (δ l 0 when l increases) high energies: large number ( alternative methods) General properties of the phase shifts 1. The phase shift (and all derivatives) are continuous functions of E 2. The phase shift is known within np: exp 2iδ = exp(2i δ + nπ ) 3. Levinson theorem δ l (E = 0) is arbitrary δ l 0 δ l = Nπ, where N is the number of bound states in partial wave l Example: p+n, l = 0: δ 0 0 δ 0 = π (bound deuteron) l = 1: δ 1 0 δ 1 = 0 (no bound state for l = 1) 52

53 4. General scattering theory Example: hard sphere (radius a) continuity at r = a j l ka tan δ l n l ka = 0 tan δ l = j l ka n l ka δ 0 = ka V(r) a r At low energies: δ l (E) ka 2l+1 2l+1!! 2l 1!!, in general: δ l(e) k 2l+1 Strong difference between l = 0 (no barrier) et l 0 (centrifugal barrier) (typical to neutron-induced reactions) 53

54 4. General scattering theory example : a+n phase shift l = 0 consistent with the hard sphere (a 2.2 fm) 54

55 4. General scattering theory 5. Resonances Resonances: δ R E atan Γ 2 E R E = Breit-Wigner approximation E R =resonance energy G=resonance width: related to the lifetime Γτ = ħ d(e) p 3p/4 p/2 Narrow resonance: G small, τ large Broad resonance: G large, τ small p/4 Bound states: Γ = 0, E R < 0 E R -G/2 E R E R +G/2 E 55

56 4. General scattering theory Cross section (for neutrons) σ(e) = π k 2 l 2l + 1 exp 2iδ l 1 2 maximum for δ = π 2 Near the resonance: σ E s 4π k 2 2l R + 1 G Γ 2 /4 E R E 2 +Γ 2 /4, where l R=resonant partial wave E E R In practice: Peak not symmetric (G depends on E) «Background» neglected (other l values) Differences with respect to Breit-Wigner 56

57 4. General scattering theory Example: n+ 12 C E cm =1.92 MeV G= 6 kev Comparison of 2 typical times: 197 a. Lifetime of the resonance: τ R = ħ/γ s b. Interaction time without resonance: τ NR = d/v s t NR << t R 57

58 4. General scattering theory Narrow resonances Small particle width long lifetime can be approximetly treated by neglecting the asymptotic behaviour of the wave function proton width=32 kev can be described in a bound-state approximation 58

59 4. General scattering theory Broad resonances Large particle width Short lifetime asymptotic behaviour of the wave function is important rigorous scattering theory bound-state model complemented by other tools (complex scaling, etc.) very broad resonance: G=1990 kev ground state unstable: G=120 kev 59

60 5. Generalizations Extension to charged systems Numerical calculation Optical model (high energies absorption) Extension to multichannel problems 60

61 5. Generalizations Generalization 1: charged systems Scattering energy E E E B : weak coulomb effects (V negligible with respect to E) E < E B : strong coulomb effects (ex: nuclear astrophysics) 61

62 4. Generalizations A. Asymptotic behaviour Neutral systems Charged systems ħ2 2μ Δ + V N(r) E Ψ r = 0 Ψ r exp(ik r) + f θ exp ikr r ħ2 2μ Δ + V N(r) + Z 1Z 2 e 2 r Ψ r exp ik r + iη ln(k r kr) exp i(kr η ln 2kr ) + f θ r η = Z 1Z 2 e 2 ħv Sommerfeld parameter E Ψ r = 0 «measurement» of coulomb effects Increases at low energies Decreases at high energies 62

63 5. Generalizations B. Phase shifts with the coulomb potential d Neutral system: 2 l l+1 + k 2 R dr 2 r 2 l = 0 Bessel equation : solutions j l kr, n l (kr) Charged system: F 0 (η,x) d 2 dr 2 l l+1 r 2 2 ηk r + k2 R l = 0: Coulomb equation: solutions F l η, kr, G l η, kr eta=0 x eta=1 eta=5 eta= G 0 (η,x) eta=0 eta=1 eta=5 eta= x 63

64 5. Generalizations Incoming and outgoing functions (complex) I l η, x = G l η, x if l η, x e i(x lπ 2 η ln 2x+σ l) : incoming wave O l η, x = G l η, x + if l η, x e i(x lπ 2 η ln 2x+σ l) : outgoing wave Phase-shift definition o neutral systems R l r o charged systems: R l r ra j l kr tan δ l n l kr A F l η, kr + tan δ l G l η, kr B(cos δ l F l η, kr + sin δ l G l η, kr C I l η, kr U l O l η, kr 3 equivalent definitions (amplitude is different) Collision matrix (=scattering matrix) U l = e 2iδ l : modulus U l = 1 64

65 5. Generalizations Example: hard-sphere potential V a r V r = Z 1Z 2 e 2 for r > a r for r < a phase shift: tan δ l = F l η,ka G l η,ka 0 k (fm 1 ) η= a=4 fm η=0 l=2 l=1 l=0 65

66 5. Generalizations C. Rutherford cross section For a Coulomb potential (V N = 0): scattering amplitude : f c θ = η 2k sin 2 θ/2 e2i(σ 0 η ln sin θ/2) Coulomb phase shift for l = 0: σ 0 = arg Γ 1 + iη We get the Rutherford cross section: dσ C dω = f c θ 2 = Increases at low energies Diverges at θ = 0 no integrated cross section Z 1 Z 2 e 2 4E sin 2 θ/2 2 66

67 5. Generalizations D. Cross sections with nuclear and Coulomb potentials The general defintions are still valid f θ = 1 dσ dω 2ik l=0 = f θ 2 (2l + 1)(exp(2iδ l ) 1) P l (cos θ) Problem very slow convergence with l separation of the nuclear and coulomb phase shifts δ l = δ l N + σ l σ l = arg Γ 1 + l + iη Scattering amplitude f θ written as f θ = f C θ + f N θ f C θ = 1 (2l + 1)(exp(2iσ l ) 1) P l (cos θ) = analytical 2ik l=0 η 2k sin 2 θ/2 e2i(σ 0 η ln sin θ/2) f N θ = 1 2ik l=0 converges rapidly (2l + 1) exp(2iσ l ) (exp(2iδ l N ) 1) P l (cos θ) 67

68 5. Generalizations Total cross section: dσ dω = f θ 2 = f C θ + f N θ 2 Nuclear term dominant at 180 Coulomb term coulombien dominant at small angles used to normalize experiments Coulomb amplitude strongly depends on the angle dσ dω dσ C dω Integrated cross section dσ dω is not defined dω System 6 Li+ 58 Ni E cm = E lab Coulomb barrier E B 17 MeV 7 Below the barrier: σ σ C Above E B : σ is different from σ C 68

69 5. Generalizations Generalization 2: numerical calculation For some potentials: analytic solution of the Schrödinger equation In general: no analytical solution numerical approach with: Numerical solution : discretization N points, with mesh size h u l 0 = 0 u l h = 1 (or any constant) u l 2h is determined numerically from u l 0 and u l h (Numerov algorithm) u l 3h, u l Nh for large r: matching to the asymptotic behaviour phase shift Bound states: same idea (but energy is unknown) 69

70 5. Generalizations Example: a+a Experimental spectrum of 8 Be Experimental phase shifts l=0 l=2 l=4 a+a 4 + E~11 MeV G~3.5 MeV 2 + E~3 MeV G~1.5 MeV 0 + E=0.09 MeV G=6 ev delta (degres) V (MeV) Ecm (MeV) Potential: V N (r)=-122.3*exp(-(r/2.13) 2 ) nuclear Coulomb -10 Total -15 V B ~1 MeV -20 r (fm) 70

71 5. Generalizations a+a wave function for l = 0 2 a-a potential E=1 MeV, l variable ell=0 ell=2 ell=4 71

72 5. Generalizations Generalization 3: complex potentials V = V R + iw Goal: to simulate absorption channels d+ 18 F n+ 19 Ne p+ 19 F a+ 16 O High energies: many open channels strong absorption potential model extended to complex potentials («optical») Phase shift is complex: δ = δ R + iδ I collision matrix: U = exp 2iδ = η exp 2iδ R where η = exp 2δ I < 1 Elastic cross section Reaction cross section: 20 Ne 72

73 5. Generalizations In astrophysics, optical potentials are used to compute fusion cross sections Fusion cross section: includes many channels Example: 12 C+ 12 C: Essentially 20 Ne+a, 23 Na+p, 23 Mg+n channels absorption simulated by a complex potential V = V R + iw E 12 C+ 12 C many states 73

74 5. Generalizations Typical potentials A. Real part Woods-Saxon: V R r = V 0 1+exp r r 0 a Folding V R r = λ dr 1 dr 2 v NN r r 1 + r 2 ρ 1 r 1 ρ 2 (r 2 ) r 1 r 2 Nucleus 1 density ρ 1 (r 1 ) r Nucleus 2 density ρ 2 (r 2 ) with parameters V 0, r 0, a adjusted to experiment v NN =nucleon-nucleon interaction λ =amplitude (~1), adjustable parameter ρ 1, ρ 2 =nuclear densities (in general known experimentally) Main advantage: only one parameter λ 74

75 5. Generalizations B. Imaginary part Woods-Saxon: Volume: W r = W 0 f r = Surface W r Folding W(r) = N I V R r = W 0 df r dr W 0 1+exp r r 0 a Surface Volume 75

76 5. Generalizations Example: a+ 144 Sm P. Mohr et al., Phys. Rev. C55 (1997) 1523 Measurement of elastic scattering optical potential used for astrophysics Elastic cross section at E lab =20 MeV (E cm =9.5 MeV) a+ 144 Sm potential (folding)

77 6. Models used for nuclear reactions in astrophysics 80

78 6. Models used in nuclear astrophysics (for reactions) Theoretical methods: Many different cases no unique model! Model Applicable to Comments Potential/optical model R-matrix Capture Fusion Capture Transfer Internal structure neglected Antisymetrization approximated No explicit wave functions Physics simulated by some parameters DWBA Transfer Perturbation method Wave functions in the entrance and exit channels Light systems Low level densities Microscopic models Capture Transfer Based on a nucleon-nucleon interaction A-nucleon problems Predictive power Hauser-Feshbach Shell model Capture Transfer Capture Statistical model Only gamma widths Heavy systems 81

79 7. Radiative capture in the potential model 82

80 7. Radiative capture in the potential model Potential model: two structureless particles (=optical model, without imaginary part) o Calculations are simple o Physics of the problem is identical in other methods o Spins are neglected R cm =center of mass, r =relative coordinate r r 1 r 2 r 1 = R cm A 2 A r r 2 = R cm + A 1 A r Initial wave function: Ψ l im i r = 1 r u l i r Y li m i Ω, energy E l i =scattering energy E Final wave function: Ψ l fm f r = 1 r u l f r Y lf m f Ω, energy E l f The radial wave functions are given by: ħ2 2μ d 2 dr 2 l l + 1 r 2 u l + V r u l = E l u l 83

81 7. Radiative capture in the potential model Schrödinger equation: ħ2 2μ d 2 l l+1 dr 2 r 2 u l + V r u l = E l u l Typical potentials: coulomb =point-sphere nuclear: Woods-Saxon, Gaussian parameters adjusted on important properties (bound-state energy, phase shifts, etc.) Potentials can be different in the initial and final states Wave functions computed numerically (Numerov algorithm) Limitations initial (scattering state): must reproduce resonances (if any) final (bound) state: must have a A+B structure initial state E>0 final state E <0 84

82 7. Radiative capture in the potential model Some typical examples a+ 3 H structure n+ 7 Li structure n+ 16 O structure 7 Li Problem more and more important when the level density increases in practice: limited to low-level densities (light nuclei or nuclei close to the drip lines) 85

83 7. Radiative capture in the potential model Electric operator for two particles: M μ Eλ = e Z 1 r 1 R cm λ Y λ μ Ω r1 R cm + Z 2 r 2 R cm λ Y λ μ Ω r2 R cm which provides M μ Eλ = e Z 1 A 2 A λ + Z 2 A 1 A λ r λ Y λ μ Ω r Matrix elements needed for electromagnetic transitions < Ψ J fm f M μ Eλ Ψ J im i > = ez eff < Y Jf m f Y λ μ Y Ji m i > = ez eff r λ Y λ μ Ω r 0 u Ji r u Jf r r λ dr Reduced matrix elements: < Ψ J f M Eλ Ψ J i > = ez eff < J f 0λ0 J i 0 > 2J i+1 2λ+1 4π 2J f +1 simple one-dimensional integrals 1/2 0 uji r u Jf r r λ dr 86

84 7. Radiative capture in the potential model Assumptions: spins zero: l i = J i, l f = J f given values of J i, J f, λ initial state E: all J i are possible wave functions u Ji (r, E) photon with multipolarity λ Integrated cross section with σ λ E = 8π e 2 k 2 ħc Z eff 2 k 2λ+1 γ F λ, J i, J f Z eff = Z 1 A 2 A F λ, J i, J f =< J i λ0 0 J f 0 > 2J i + 1 k γ = E E f ħc λ + Z2 A 1 A λ λ+1 2λ+1 λ 2λ+1!! 2 0 u Ji (r, E) u Jf Final state E f, J f wave function: u Jf r r λ dr 2 r Normalization final state (bound): normalized to unity u J r CW 2k B r Cexp( k B r) initial state (continuum): u J r F J kr cos δ J + G J kr sin δ J 87

85 7. Radiative capture in the potential model Integrated vs differential cross sections Total (integrated) cross section: σ E = no interference between the multipolarities Differential cross section: λ σ λ E dσ dθ = a λ E P λ (θ) λ P λ (θ)=legendre polynomial a λ E are complex, σ λ E a λ E 2 interference effects angular distributions are necessary to separate the multipolarities in general one multipolarity is dominant (not in 12 C(a,g) 16 O: E1 and E2) 2 88

86 7. Radiative capture in the potential model Example: 12 C(p,g) 13 N First reaction of the CNO cycle Well known experimentally Presents a low energy resonance (l = 0 J = 1/2 + ) S (MeV-b) 1.00E E E E E (MeV) Potential : V= -55.3*exp(-(r/2.70) 2 ) (final state) -70.5*exp(-(r/2.70) 2 ) (initial state) 89

87 7. Radiative capture in the potential model Final state: J f = 1/2 Initial state: l i = 0 J i = 1/2 + E1 transition 1/2 + 1/2-1.60E E E E E E E-03 Initial wave function: Ei=100 kev 2.00E-03 E 13 N,J=1/2-12 C+p 0.00E E E E E E E E E E-03 Integrant Ei=100 kev -2.00E E E E E E E-01 Final state Ef= MeV 1.00E E E-01 Integral Ei=100 kev 3.00E E E E E E E E-01 90

88 7. Radiative capture in the potential model The calculation is repeated at all energies 1.00E+00 S (MeV-b) 1.00E E E E E (MeV) S 1 S 0.45 Necessity of a spectroscopic factor S Assumption of the potential model: 13 N= 12 C+p In reality 13 N= 12 C+p 12 C*+p 9 Be+a. to simulate the missing channels: u f (r) is replaced by S 1/2 u f (r) S=spectroscopic factor Other applications: 7 Be(p,g) 8 B, 3 He(a,g) 7 Be, etc 91

89 8. The R-matrix method General presentation Single resonance system Applications to elastic scattering 12 C+p Application to 12 C(p,g) 13 N and 12 C(a,g) 16 O 99

90 2. MODELS: the R-matrix method 8. The R-matrix method Introduced by Wigner (1937) to parametrize resonances (nuclear physics) In nuclear astrophysics: used to fit data Provides scattering properties at all energies (not only at resonances) Based on the existence of 2 regions (radius a): Internal: coulomb+nuclear external: coulomb Entrance channel 12 C+a Internal region 16 O 12 C(2 + )+a 12 C+a Exit channels Coulomb Nuclear+Coulomb: R-matrix parameters 15 N+p, 15 O+n Coulomb 100

91 S E2 (kev b) 8. The R-matrix method Main Goal: fit of experimental data Ne+p elastic scattering resonance properties Nuclear astrophysics: 12 C(a,g) 16 O Extrapolation to low energies

92 8. The R-matrix method Internal region: The R matrix is given by a set of resonance parameters E i, γ i 2 R E = i 2 γ i = a Ψ a E i E Ψ a i=3, E 3, γ 3 2 i=2, E 2, γ 2 2 External region: Coulomb behaviour of the wave function Ψ r = I r UO r the collision matrix U is deduced from the R-matrix (repeated for each spin/parity Jπ) Two types of applications: phenomenological R matrix: γ i 2 and E i are fitted to the data (astrophysics) calculable R matrix: γ i 2 and E i are computed from basis functions (scattering theory) i=1, E 1, γ 1 2 R-matrix radius a is not a parameter: the cross sections must be insensitive to a Can be extended to multichannel calculations (transfer), capture, etc. Well adapted to nuclear astrophysics: low energies, low level densities 102

93 8. The R-matrix method A simple case: elastic scattering with a single isolated resonance E resonance: energy E R, width Γ (or γ 2 ) = observed parameters calculated parameters: E 0, γ 0 2 threshold From the total width Γ reduced width Γ = 2γ 2 P l (E R ) P l (E R )=penetration factor Link between (E R, γ 2 ) (E 0, γ 0 2 ) Calculation of the R-matrix R E = γ 0 2 E 0 E Calculation of the scattering matrix: U E = I ka O ka 1 L R E 1 LR E Calculation of the cross section E 0 and/or γ 0 2 can be fitted (must be done for each l) 103

94 8. The R-matrix method Example: 12 C+p: E R =0.42 MeV Resonance l = 0 13 N In the considered energy range: resonance J=1/2+ (l = 0) Phase shift for l = 0 is treated by the R matrix Other phase shifts l > 0 are given by the hard-sphere approximation 104

95 s (mb) s (mb) 8. The R-matrix method First example: Elastic scattering 12 C+p Data from H.O. Meyer et al., Z. Phys. A279 (1976) E (MeV) q=146.9 R matrix fits for different channel radii 2500 q= E (MeV) a E R G E 0 g 0 2 c E R, Γ very stable with a global fit independent of a 105

96 8. The R-matrix method Extension to transfer, example: 18 F(p,a) 15 O E resonance: energy E R, partial widths Γ 1, Γ 2 (or γ 2 1, γ 2 2 ) = observed parameters calculated parameters: E 0, γ , γ 02 threshold 1 (p+ 18 F) threshold 2 (a+ 15 O) Link between (E R, γ 2 1, γ 2 2 ) E 0, γ 2 01, γ 2 02 more complicated R-matrix: 2x2 matrix R ii E = γ 01 2 E 0 E 2 R ff E = γ 02 E 0 E associated with the entrance channel associated with the exit channel R if E = γ 01γ 02 associated with the transfer E 0 E Scattering matrix: 2x2: U 11, U 22 elastic cross sections U 12, transfer cross section More parameters, but some are common to elastic scattering (E 0, γ 2 01 ) constraints with elastic scattering 106

97 8. The R-matrix method Recent application to 18 F(p,p) 18 F and 18 F(p,a) 15 O D. Mountford et al, Phys. Rev. C 85 (2012) simultaneous fit of both cross sections angle: 176 o for each resonance: Jπ, E R, Γ p, Γ α 8 resonances 24 parameters 107

98 8. The R-matrix method Extension to radiative capture E resonance: energy E R, reduced γ 2, gamma width Γ γ = observed parameters calculated parameters: E 0, γ 0 2, Γ 0γ threshold Final state: ANC C f Capture reaction= transition between an initial state at energy E to bound states Cross section σ C E < Ψ f H γ Ψ i E > 2 Additional pole parameter: gamma width Γ γi < Ψ f H γ Ψ i E > = < Ψ f H γ Ψ i E > int + < Ψ f H γ Ψ i E > ext internal part: < Ψ f H γ Ψ i E > int N γi Γ γi i=1 E i E external part: < Ψ f H γ Ψ i E > ext C f a W 2kf r r λ I i (kr) UO i kr dr 108

99 8. The R-matrix method External part: < Ψ f H γ Ψ i E > ext C f a W 2kf r r λ I i (kr) UO i kr dr Essentially depends on k f E E f = ħ2 2μ k f 2 Witthaker function W 2k f r exp( k f r) k f large: fast decrease example 12 C(a,g) 16 O, E f =7.16 MeV, μ = 3 external term negligible insensitive to C f k f small: slow decrease example: 7 Be(p,g) 8 B, E f =0.137 MeV, μ = 7/8 external term dominant mainly given by C f Contribution of internal/external terms depends on energy (external larger at low energies) 109

100 8. The R-matrix method S-factor (MeV-b) Example 1: 12 C(p,g) 13 N: R-matrix calculation with a single pole 12 C+p a=6 fm a=5 fm a=6 fm a=5 fm 13 N E (MeV) Experiment: E R = 0.42 MeV, Γ p = 31 kev, Γ γ = 0.4 ev Red line: internal contribution, pure Breit-Wigner approximation Green lines: external contribution: important at low energies, sensitive to the ANC 110

101 8. The R-matrix method Example 2: 12 C(a,g) 16 O General presentation of 12 C(a,g) 16 O Determines the 12 C/ 16 O ratio Cross section needed near E cm =300 kev (barrier ~2.5 MeV) cannot be measured in the Gamow peak 1 - and 2 + subthreshold states extrapolation difficult E1 and E2 important (E1 forbidden when T=0) 12 C+a Interferences between 1-1,1-2 and between 2 + 1, Capture to gs dominant but also cascade transitions 16 O 111

102 8. The R-matrix method Many experiments Direct 12 C(a,g) 16 O (angular distributions are necessary: E1 and E2) Indirect: spectroscopy of 1-1 and subthreshold states Constraints a+ 12 C phase shifts (1 - E1, 2 + E2) E1: 16 N beta decay (Azuma et al, Phys. Rev. C50 (1994) 1194) probes J=1 - E1 E2:??? 112

103 8. The R-matrix method Current situation 100 S (kev b) E E cm (MeV) SE2 (kev-b) Dyer Redder Ouellet Rotters Kunz Orsay 2 par E2 0.1 Orsay 3 par E cm (MeV) 113

104 8. The R-matrix method S(300 kev): current situation for E1 SE1 (kev-barn) Weisser 74 Dyer 74 Koonin 74 Humblet C(a,g) 16 O E1 Descouvemont 87 Redder 87 Filippone year Kremer 88 Ouellet 92 Humblet 93 Azuma 94 Ouellet 96 Trautvetter 97 Gialanella 01 Fey N data available 114

105 8. The R-matrix method S E2 (kev-barn) S(300 kev): current situation for E C(a,g) 16 O E Kettner 82 Barker 91 Angulo 00 Langanke 85 Redder 87 Plaga 87 Buchmann 96 Descouvemont 93 Fey 03 Descouvemont 84 Redder 87 Descouvemont 87 Redder 87 Barker 87 Ouellet 96 Tichhauser 02 Filippone 89 Brune 99 Dufour 08 year 115

106 9. Microscopic models 116

107 9. Microscopic models o Goal: solution of the Schrödinger equation HΨ = EΨ o Hamiltonian: H = i T i + j>i V ij T i = kinetic energy of nucleon i V ij = nucleon-nucleon interaction o Cluster approximation Ψ = Aφ 1 φ 2 g(ρ) with φ 1, φ 2 = internal wave functions (input, shell-model) g(ρ) =relative wave function (output) A = antisymmetrization operator r o Generator Coordinate Method (GCM): the radial function is expanded in Gaussians Slater determinants (well adapted to numerical calculations) o Microscopic R-matrix: extension of the standard R-matrix reactions 117

108 9. Microscopic models Many applications: not only nuclear astrophysics spectroscopy, exotic nuclei, elastic and inelastic scattering, etc. Extensions: Multicluster calculations: deformed nuclei (example: 7 Be+p) R 2 R 1 Multichannel calculations: Ψ = Aφ 1 φ 2 g ρ + Aφ 1 φ 2 g ρ + better wave functions inelastic scattering, transfer Ab initio calculations: no cluster approximation very large computer times limited to light nuclei difficult for scattering (essentially limited to nucleon-nucleus) 118

109 9. Microscopic models Example: 7 Be(p,g) 8 B Important for the solar-neutrino problem Since 1995, many experiments: Direct (proton beam on a 7 Be target) Indirect (Coulomb break-up) Extrapolation to zero energy needs a theoretical model (energy dependence) From E. Adelberger et al., Rev. Mod. Phys.83 (2011) 196 astrophysical energies 119

110 9. Microscopic models Example: 7 Be(p,g) 8 B Microscopic cluster calculations: 3-cluster calculations P. D. and D. Baye, Nucl. Phys. A567 (1994) 341 P.D., Phys. Rev. C 70, (2004) Includes the deformation of 7 Be: cluster structure a+ 3 He Includes rearrangement channels 5 Li+ 3 He Can be applied to 8 B/ 8 Li spectroscopy Can be applied to 7 Be(p,g) 8 B and 7 Li(n,g) 8 Li p 3 He 7 Be=a+ 3 He 5 Li=a+p 120

111 9. Microscopic models Spectroscopy of 8 B experiment Volkov Minnesota m (2 + ) (m N ) Q(2 + ) (e.fm 2 ) 6.83 ± B(M1, ) (W.u.) 5.1 ± Channel components in the 8 B ground state 7 Be(3/2 - )+p 47% 7 Be(1/2 - )+p 9% 5 Li(3/2 - )+ 3 He 34% Important role of the 5+3 configuration 5 Li(1/2 - )+ 3 He 3% 121

112 9. Microscopic models 7 Be(p,g) 8 B S factor S-factor (ev b) Be(p,g) 8 B V2 DB94 MN Hammache et al. Hass et al. Strieder at al. Baby et al. Junghans et al. Schumann et al. Davids et al E c.m. (MeV) Low energies (E<100 kev): energy dependence given by the Coulomb functions 2 NN interactions (MN, V2): the sensitivity can be evaluated Overestimation: due to the 8 B ground state (cluster approximation) 122

113 9. Microscopic models Cluster models In general a good approximation, but do not allow the use of realistic NN interactions Example: a particle described by 4 0s orbitals intrinsic spin =0 no spin-orbit, no tensor force, no 3-body force these terms are simulated by (central) NN interactions Ab initio models No cluster approximation Use of realistic NN interactions (fitted on deuteron, NN phase shifts, etc.) Application: d+d systems 2 H(d,g) 4 He, 2 H(d,p) 3 H, 2 H(d,n) 3 He two physics issues Analysis of the d+d S factors (Big-Bang nucleosynthesis) Role of the tensor force in 2 H(d,g) 4 He 123

114 9. Microscopic models S factor (kev-b) 2 H(d,g) 4 He S factor Ground state of 4 He=0+ E1 forbidden main multipole is E2 2 + to 0 + transition d wave as initial state Experiment shows a plateau below 0.1 MeV: typical of an s wave Interpretation : the 4 He ground state contains an admixture of d wave final 0 + state: Ψ 0+ = Ψ 0+ L = 0, S = 0 + Ψ 0+ L = 2, S = 2 = 0 +, 0 > + 0 +, 2 > initial 2 + state: Ψ 2+ = Ψ 2+ L = 2, S = 0 + Ψ 2+ L = 0, S = 2 = 2 +, 0 > + 2 +, 2 > 1.E-02 1.E-03 1.E-04 2 H(d,g) 4 He d s E2 matrix element < Ψ 0+ E2 Ψ 2+ > < 0 +, 0 E2 2 +, 0 >: d s, dominant E>100 kev +< 0 +, 2 E2 2 +, 0 >: s d, tensor (E<100 kev) 1.E-05 1.E-06 s d E cm (kev) direct effect of the tensor force 124

115 9. Microscopic models Application: d+d systems Collaboration Niigata (K. Arai, S. Aoyama, Y. Suzuki)-Brussels (D. Baye, P.D.) Phys. Rev. Lett. 107 (2011) Mixing of d+d, 3 H+p, 3 He+n configurations r x 1 x 2 x 2 d+d 3H+p, 3He+n x 1 r The total wave function is written as an expansion over a gaussian basis Superposition of several angular momenta 4-body problem (in the cluster approximation we would have: x 1 =x 2 =0) 125

116 9. Microscopic models We use 3 NN interactions: Realistic: Argonne AV8, G3RS Effective: Minnesota MN No parameter MN does not reproduce the plateau (no tensor force) D wave component in 4 He: 13.8% (AV8 ) 11.2% (G3RS) 126

117 9. Microscopic models Transfer reactions 2 H(d,p) 3 H, 2 H(d,n) 3 He 127

118 10. Conclusions Needs for nuclear astrophysics: low energy cross sections resonance parameters Experiment: direct and indirect approaches Theory: various techniques fitting procedures (R matrix) extrapolation non-microscopic models: potential, DWBA, etc. microscopic models: cluster: developed since 1960 s, applied to NA since 1980 s ab initio: problems with scattering states, resonances limited at the moment Current challenges: new data on 3 He(a,g) 7 Be, triple a process, 12 C(a,g) 16 O, etc. D(d,g) 4 He : 4 nucleons 4 clusters 128