Indirect Methods for Nuclear Astrophysics

Transcription

1 Indirect Methods for Nuclear strophysics Ecellence Cluster Universe Technische Universität München GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt 6th Rußach Workshop on Nuclear strophysics Rußach am Pass Gschütt, ustria 2-6 March 2009

2 Outline Motivation nuclear reactions of astrophysical interest, reaction rates and cross sections, charged-particle reactions, astrophysical S factor Indirect Methods overview, transfer reactions, T matri elements, DWB Coulom dissociation: theory, cross sections, characteristic parameters, eample spectroscopic factors, asymptotics of wave functions NC method: theory, continuum interaction Trojan-Horse method: theory, application, eamples, electron screening Summary Indirect Methods - 2

3 Nuclear Reactions of stophysical Interest nuclear astrophysics nuclear reaction rates are asic input in many astrophysical models (primordial nucleosynthesis, stellar evolution, novae, supernovae,... ) for various processes (pp chains, CNO cycles, s-, r-, p-, rp-process,... ) ideally: direct measurement of reaction cross sections at relevant energies ut in most cases practically impossile (small cross sections, often unstale nuclei) alternative approaches? depend on type of reaction Indirect Methods - 3

4 Types of Reactions radiative capture/ photo dissociation reactions: (n, γ), (p,γ), (α, γ),... / (γ, n), (γ,p), (γ,α), CNO cycles nuclear rearrangement reactions: (p, α), (α,p), ( 3 He,2p),... weak interaction reactions: β +, β, electron capture (EC)... Z 6 pp chains 4 (α,γ) (p,γ) (α,p) 2 (p,β + ), ( 3 He,2p) (p,α) (β + ), (EC) (α) N here: only charged-particle reactions only reactions with electromagnetic or strong interaction Indirect Methods - 4

5 Reaction Rates and Cross Sections astrophysical environment nuclei in hot plasma temperature-dependent distriution of relative velocities v for reaction + c... relevant quantity: Mawellian-averaged reaction rate r c = c 1 + δ c σv with densities, c and σv = s 8 πµ c Z 0 σ(e) E e E kt de (kt) 3/2 [a.u] E ep(-e/(kt)) σ E ep(-e/(kt)) Gamov window σ(e) E/E eff cross sections σ needed in Gamov window of width E around effective energy E eff Indirect Methods - 5

6 Gamov Window parameters effective energy E eff = µ 1/3 c (Z Z c T 9 ) 2/3 MeV width E = µ 1/6 c (Z Z c ) 1/3 T 5/6 9 MeV with temperature T 9 in 10 9 K and reduced mass µ c = m m c m +m c in amu [a.u] E ep(-e/(kt)) σ E ep(-e/(kt)) σ(e) Gamov window reaction E eff [kev] σ(e eff ) [p] 3 He( 3 He,2p) 4 He Be(p,γ) 8 B He(α,γ) 7 Be N(p,γ) 15 O E/E eff for T = K (center of the sun) Indirect Methods - 6

7 Charged-Particle Reactions Coulom arrier in reaction + c... with charged nuclei, c etremely small cross sections σ(e) with strong energy dependence astrophysical relevant energies (Gamov window) usually not accessile measurement at higher energies and etrapolation to low energies E with help of astrophysical S factor S(E) = σ(e)e ep(2πη) Sommerfeld parameter η = Z Z c e 2 /( v) Indirect Methods - 7

8 Charged-Particle Reactions Coulom arrier in reaction + c... with charged nuclei, c etremely small cross sections σ(e) with strong energy dependence astrophysical relevant energies (Gamov window) usually not accessile measurement at higher energies and etrapolation to low energies E with help of astrophysical S factor S(E) = σ(e)e ep(2πη) Sommerfeld parameter η = Z Z c e 2 /( v) direct measurement very difficult, often unstale nuclei involved indirect methods Indirect Methods - 7

9 Indirect Methods - Overview I Coulom dissociation G. Baur et al., NP 458 (1986) 188 study inverse of radiative capture reaction (, γ)a a(γ, ) use Coulom field of target nucleus as source of photons a(γ, ) (a, ) asolute S factors as a function of energy Indirect Methods - 8

10 Indirect Methods - Overview I Coulom dissociation G. Baur et al., NP 458 (1986) 188 NC method H. M. Xu et al., PRL 73 (1994) 2027 study inverse of radiative capture reaction (, γ)a a(γ, ) use Coulom field of target nucleus as source of photons a(γ, ) (a, ) asolute S factors as a function of energy etract asymptotic normalization coefficient of ground state wave function of nucleus a from transfer reactions calculate matri elements for radiative capture reaction (, γ)a S factor at zero energy Indirect Methods - 8

11 Indirect Methods - Overview I Coulom dissociation G. Baur et al., NP 458 (1986) 188 NC method H. M. Xu et al., PRL 73 (1994) 2027 Trojan-Horse method G. Baur, PLB 178 (1986) 35 study inverse of radiative capture reaction (, γ)a a(γ, ) use Coulom field of target nucleus as source of photons a(γ, ) (a, ) asolute S factors as a function of energy etract asymptotic normalization coefficient of ground state wave function of nucleus a from transfer reactions calculate matri elements for radiative capture reaction (, γ)a S factor at zero energy study three-ody reaction + a C + c + with Trojan horse a = + and spectator etract cross section of two-ody reaction + C + c energy dependence of S factor theoretical description? relation of methods? Indirect Methods - 8

12 Indirect Methods - Overview II Coulom dissociation NC method Trojan-Horse method B=(+) C c a=(+) a=(+) a=(+) photon echange transfer of particle to ound state transfer of particle to continuum state similar reaction mechanisms: transfer of virtual particle final state with three particles (ound/continuum states) theoretical descripton with direct reaction theory Indirect Methods - 9

13 Indirect Methods - Overview III general characteristics: two-ody reaction at low-energy is replaced y three-ody reaction at high-energy with large cross section Coulom dissociation (, γ)a (a, ) NC method (,γ)a (a, B) a = ( + ) B = ( + ) Trojan-horse method (, c)c (a, Cc) Indirect Methods - 10

14 Indirect Methods - Overview III general characteristics: two-ody reaction at low-energy is replaced y three-ody reaction at high-energy with large cross section Coulom dissociation (, γ)a (a, ) NC method (,γ)a (a, B) a = ( + ) B = ( + ) Trojan-horse method (, c)c (a, Cc) transfer of virtual particle (photon γ or nucleus ) relation of cross sections is found with the help of nuclear direct reaction theory theoretical approimations essential study of peripheral reactions asymptotics of wave functions relevant selection of suitale kinematical conditions important Indirect Methods - 10

15 Transfer Reactions - Cross Sections transfer reaction + a B + with a = + to ound state of B = + B=(+) cross section dσ = 2π µ a p a T fi 2 δ(e B + E E E a Q) d3 p B (2π ) 3 a=(+) Indirect Methods - 11

16 Transfer Reactions - Cross Sections transfer reaction + a B + with a = + to ound state of B = + B=(+) cross section dσ = 2π µ a p a T fi 2 δ(e B + E E E a Q) d3 p B (2π ) 3 a=(+) with T-matri in prior formulation T fi = Ψ ( ) (i) B V a ep(i k a r a )φ φ a Ψ (+) a /Ψ( ) B V (i) (f) a /V B or in post formulation T fi = ep(i k B r B )φ B φ V (f) B Ψ(+) a : eact scattering wave functions in initial/final state : full potentials in initial/final state T-matri element contains essential information on reaction process transferred particle is virtual, i.e. E p2 2m two poles in diagram factorization Indirect Methods - 12

17 T-Matri Elements - Transformations introduce optical potentials U ij (ij = a, B) and B=(+) distorted waves χ (±) ij with ( ˆT ij + U ij )χ (±) ij = E ij χ (±) ij apply Gell-Mann Golderger relation (Phys. Rev. 91 (1953) 398) a=(+) prior form T fi = Ψ ( ) (i) B V a U a χ (+) a φ φ a post form T fi = χ ( ) B φ Bφ V (f) B U B Ψ (+) a eact! eact! Indirect Methods - 13

18 T-Matri Elements - Transformations introduce optical potentials U ij (ij = a, B) and B=(+) distorted waves χ (±) ij with ( ˆT ij + U ij )χ (±) ij = E ij χ (±) ij apply Gell-Mann Golderger relation (Phys. Rev. 91 (1953) 398) a=(+) prior form T fi = Ψ ( ) (i) B V a U a χ (+) a φ φ a post form T fi = χ ( ) B φ Bφ V (f) B U B Ψ (+) a eact! eact! distorted-wave Born approimation (DWB): replace eact scattering wave functions Ψ (+) a χ(+) a φ φ a or Ψ ( ) B χ( ) B φ Bφ prior form T fi χ ( ) B φ Bφ V (i) a U a χ (+) a φ φ a post form T fi χ ( ) B φ Bφ V (f) B U B χ (+) a φ φ a Indirect Methods - 13

19 Coulom Dissociation - Idea projectile a target electric field fragments radiative capture detailed alance photo asorption equivalent photons in Coulom field of target Coulom dissociation (, γ)a a(γ,) (a,) (G. Baur, H. Reel, C. Bertulani, Nucl. Phys. 458 (1986) 188) correspondence (Fermi 1924, Weizsäcker-Williams 1932) time-dependent electromagnetic field of highly-charged nucleus during scattering of projectile a spectrum of (virtual, equivalent) photons stale nucleus reakup threshold eotic nucleus E ~ few MeV ec only ground state transitions! Indirect Methods - 14

20 Coulom Dissociation - Theory Coulom dissociation reaction: a with three-ody final state in the continuum only approimate theoretical treatment Indirect Methods - 15

21 Coulom Dissociation - Theory Coulom dissociation reaction: a with three-ody final state in the continuum only approimate theoretical treatment semiclassical methods classical description of projectile-target relative motion (valid for heavy targets if η a = Z Z a e 2 /( v) 1 with eam velocity v) time-dependent perturation V (t) of projectile system time-dependent perturation theory ecitation amplitude a fi Indirect Methods - 15

22 Coulom Dissociation - Theory Coulom dissociation reaction: a with three-ody final state in the continuum only approimate theoretical treatment semiclassical methods classical description of projectile-target relative motion (valid for heavy targets if η a = Z Z a e 2 /( v) 1 with eam velocity v) time-dependent perturation V (t) of projectile system time-dependent perturation theory ecitation amplitude a fi quantal methods valid for all projectile/target cominations and all eam energies time-independent scattering theory T-matri element T fi Indirect Methods - 15

23 Coulom Dissociation - Quantal Theory a=(+) prior-form distorted-wave Born approimation (DWB) T fi = χ ( ) () φ Ψ ( ) V a U a φ φ a χ (+) a neglection of nuclear interaction in V a and U a Indirect Methods - 16

24 Coulom Dissociation - Quantal Theory prior-form distorted-wave Born approimation (DWB) T fi = χ ( ) () φ Ψ ( ) V a U a φ φ a χ (+) a a=(+) neglection of nuclear interaction in V a and U a multipole epansion of Coulom potential in far-field approimation (r < r a ) V a U a = Z Z e 2 r r + Z Z e 2 r r Z Z a e 2 r a r 4πZ e λµ Z (λ) eff e 2λ+1 ( ) λ ( with effective charge numers Z (λ) eff = Z m m +m + Z and relative coordinates r = r r, r a = r r a r λ r λ+1 a Y λµ (ˆr )Y λµ (ˆr a) ) λ m m +m Indirect Methods - 16

25 Coulom Dissociation - Quantal Theory prior-form distorted-wave Born approimation (DWB) T fi = χ ( ) () φ Ψ ( ) V a U a φ φ a χ (+) a a=(+) neglection of nuclear interaction in V a and U a multipole epansion of Coulom potential in far-field approimation (r < r a ) V a U a = Z Z e 2 r r + Z Z e 2 r r Z Z a e 2 r a r 4πZ e λµ Z (λ) eff e 2λ+1 ( ) λ ( with effective charge numers Z (λ) eff = Z m m +m + Z and relative coordinates r = r r, r a = r r a factorization of T-matri element T fi Z e λµ 4π 2λ+1 Ψ( ) Z(λ) eff erλ Y λµ (ˆr ) }{{} φ a χ ( ) r λ r λ+1 a () r λ 1 Y λµ (ˆr )Y λµ (ˆr a) ) λ m m +m a a Y λµ (ˆr a) χ (+) M(Eλµ) electric multipole transition operator transition matri element quantal Coulom integral Indirect Methods - 16

26 Coulom Dissociation - Cross Section Coulom dissociation cross section (with angular integration over relative momentum etween fragments) d 2 σ = 1 de dω a E γ πλ σ πλ (a + γ + ) dn πλ dω a π = E, M λ = 1, 2,... photo asorption cross section σ πλ (a + γ + ) virtual photon numers dn πλ dω a Indirect Methods - 17

27 Coulom Dissociation - Cross Section Coulom dissociation cross section (with angular integration over relative momentum etween fragments) d 2 σ = 1 de dω a E γ πλ σ πλ (a + γ + ) dn πλ dω a π = E, M λ = 1, 2,... photo asorption cross section σ πλ (a + γ + ) virtual photon numers dn πλ dω a scattering angle ϑ a /impact parameter projectile velocity v ecitation energy E γ = ω that depend on kinematics: calculation in quantal approach/semiclassical approimation E2 enhancement dn E2 dω a / dne1 dω a 4 2 c 2 E 2 γ 2 M1 suppression dn M1 dω a / dne1 dω a v2 c 2 Indirect Methods - 17

28 Coulom Dissociation - Relation of Cross Sections a=(+) Coulom dissociation cross section d 2 σ = 1 σ πλ (a + γ + ) dn πλ de dω a E γ dω a πλ Indirect Methods - 18

29 Coulom Dissociation - Relation of Cross Sections a=(+) theorem of detailed alance σ πλ (a + γ + ) = (2J + 1)(2J + 1) 2(2J a + 1) Coulom dissociation cross section d 2 σ = 1 σ πλ (a + γ + ) dn πλ de dω a E γ dω a k 2 k 2 γ with photo asorption and radiative capture cross sections πλ σ πλ ( + a + γ) Indirect Methods - 18

30 Coulom Dissociation - Relation of Cross Sections a=(+) theorem of detailed alance σ πλ (a + γ + ) = (2J + 1)(2J + 1) 2(2J a + 1) Coulom dissociation cross section d 2 σ = 1 σ πλ (a + γ + ) dn πλ de dω a E γ dω a k 2 k 2 γ with photo asorption and radiative capture cross sections πλ σ πλ ( + a + γ) phase space factor k 2 k 2 γ = 2µ c 2 E (E + S ) 2 1 for not too small E virtual photon numers large Coulom dissociation cross sections dn πλ dω a 1 for large Z and for not too high E Indirect Methods - 18

31 Coulom Dissociation - Characteristic Parameters a=(+) virtual photon spectrum (E1) adiaaticity parameter ξ = ω γv = duration of scattering process ecitation period ξ = 0: sudden ecitation ξ 1: adiaatic ecitation ξ 1 Eec ma γv / Φ(ξ) Φ(ξ) = ξ 2 [K 0 2 (ξ)+k 1 2 (ξ)] ξ (Fermi 1924, Weizsäcker-Williams 1932) Indirect Methods - 19

32 Coulom Dissociation - Characteristic Parameters a=(+) virtual photon spectrum (E1) Φ(ξ) Φ(ξ) = ξ 2 [K 2 0 (ξ)+k 2 1 (ξ)] ξ adiaaticity parameter ξ = ω γv = duration of scattering process ecitation period ξ = 0: sudden ecitation ξ 1: adiaatic ecitation ξ 1 Eec ma γv / strength parameter χ = Z e f M(πλ) i v λ M(πλ) multipole operator Z e target charge χ small first-order perturation theory sufficient χ large higher-order effects (Fermi 1924, Weizsäcker-Williams 1932) Indirect Methods - 19

33 Coulom Dissociation - Eample: 8 B 8 B and solar neutrinos pp chains: main source of solar energy production flu of high-energy neutrinos proportional to synthesized 8 B precise knowledge of 7 Be(p,γ) 8 B S factor S 17 (E) in Gamov window (E 20 kev) required solar neutrino prolem solved with neutrino oscillation more precise direct capture data availale recently test case for Coulom dissociation method 1 + H(p,e νe ) H H(pe,νe ) 2 H ma E ν = 0.42 MeV E ν = 1.44 MeV 2 3 H(p,γ) He He( He,2p) He He(α,γ) Be 7 - Be(e,νe ) Li Be(p,γ) B 7 Li(p,α) 4 He E ν = 0.862/0.384 MeV 8 B -> 8 Be * + e + + νe E ν ma = 14 MeV 8 Be * -> 4 He + 4 He Indirect Methods - 20

34 Coulom Dissociation - Eample: 8 B 7 Be(p,γ) 8 B - direct eperiments R.W. Kavanagh, Nucl. Phys. 15 (1960) 411 P.D. Parker, Phys. Rev. 150 (1966) 851; strophys. J. 153 (1968) L85 R.W. Kavanagh et al., Bull. m. Phys. Soc. 14 (1969) 1209 F.J. Vaughn et al., Phys. Rev. C 2 (1970) 1657 C. Wiezoreck et al., Z. Phys. 282 (1977) 121 B.W. Filippone et al., Phys. Rev. Lett. 50 (1983) 412; Phys. Rev. C 28 (1983) 2222 M. Hass et al., Phys. Lett. B 462 (1999) 237 S 17 [ev ] Parker 66 Kanavagh et al. 69 Vaughn et al. 70 Filippone et al. 83 Hass et al. 99 Hammache et al. 01 Strieder et al. 01 Junghans et al. 03 Bay et al. 03 F. Hammache et al., Phys. Rev. Lett. 80 (1988) 928; Phys. Rev. Lett. 86 (2001) 3985 F. Strieder et al., Nucl. Phys. 696 (2001) 219.R. Junghans et al., Phys. Rev. Lett. 88 (2002) ; Phys. Rev. C 68 (2003) T. Bay et al., Phys. Rev. C 67 (2003) resonance 0,0 0,5 1,0 1,5 2,0 2,5 3,0 E [MeV] Indirect Methods - 21

35 Coulom Dissociation - Eample: 8 B 7 Be-p potential model depths of Woods-Saon potentials adjusted to inding energy of 2 + p-wave ground state ( MeV, halo system), 1 + and 3 + resonance energies, s-wave scattering lengths E1, E2 strength from model, M1 scaled to eperimental strength Coulom reakup in relativistic semiclassical approimation with quantal correction for diffraction nuclear reakup not investigated yet full triple differential cross section converted to event distriution, input for GENT simulation S 17 [ev ] dσ/de [m/mev] ,0 0,5 1,0 1,5 2,0 2,5 3,0 254 MeV E [MeV] E1 E2 M1 total E1 E2 M1 total ,0 0,5 1,0 1,5 2,0 2,5 3,0 E [MeV] Indirect Methods - 22

36 Coulom Dissociation - Eample: 8 B Coulom reakup eperiment GSI-2 high-energy 8 B eam: 254 MeV small higher-order effects small nuclear reakup (Re V pot 0), asorption important ( Im V opt ) kinematically complete with high statistics and precision various angular and momentum distriutions no sign for sustantial E2 contriution or higher-order effects M1 contriution suppressed as compared to capture reaction, ut oserved model predictions close to eperiment (F. Schümann et al., PRC 73 (2006) ) Intensity (ar. units) θ 17 (deg) counts order PT E1 only order PT E1+E θ 8 (deg) Indirect Methods - 23

37 Coulom Dissociation - Eample: 8 B dσ/de rel (ar.units) eperiment - simulation, E1+M1 - simulation, E1 - simulation, M1 difference eperiment - simulation in dσ de adjustment of model S 17 (E) etrapolation of S factor to zero energy with microscopic cluster model y P. Descouvement, PRC 70 (2004) normalization factor: ± S 17 (0) = 20.6 ± 0.8 (stat) ± 1.2 (syst) ev E rel (MeV) consistent with most precise direct results S 17 (ev ) this work (GSI-2) Iwasa et al. (GSI-1) Hammache et al. Junghans et al. Bay et al. Hammache et al. 01 Junghans et al. 03 Bay et al. 03 Schuemann et al Descouvemont E rel (MeV) S 17 (0) [ev ] Indirect Methods - 24

38 Spectroscopic Factors transfer reaction + a B + overlap functions ˆ= wave function of transferred particle Φ a = φ φ a Φ B = φ φ B a=(+) B=(+) Indirect Methods - 25

39 Spectroscopic Factors transfer reaction + a B + overlap functions ˆ= wave function of transferred particle Φ a = φ φ a Φ B = φ φ B B=(+) a=(+) approimation with spectroscopic amplitudes and single-particle wave functions Φ a a ϕa ( r )φ Φ B B ϕb ( r )φ ϕ a ϕa = ϕ B ϕ B = 1 spectroscopic factors S a = Φa Φa a 2 S B = ΦB ΦB B 2 Indirect Methods - 25

40 Spectroscopic Factors transfer reaction + a B + overlap functions ˆ= wave function of transferred particle Φ a = φ φ a Φ B = φ φ B B=(+) a=(+) approimation with spectroscopic amplitudes and single-particle wave functions Φ a a ϕa ( r )φ Φ B B ϕb ( r )φ ϕ a ϕa = ϕ B ϕ B = 1 spectroscopic factors S a = Φa Φa a 2 S B = ΦB ΦB B 2 T-matri elements in DWB post form: T (B)(a) = Φ B χ( ) B V B U B Φ a χ(+) a prior form: T (B)(a) = Φ B χ( ) B V a U a Φ a χ(+) a cross sections dσ T (B)(a) 2 dσ S a SB dσ sp factorization! Indirect Methods - 25

41 Wave Functions - General relative motion of two-ody system a = + (= c + y = d + z =...) many-ody wave function Ψ a is solution of Schrödinger equation HΨ a = (T + V a ) Ψ a = EΨ a with potential V a = V C + V N = V C cy + V N cy (Coulom + nuclear interaction) and oundary condition for ound/scattering state Indirect Methods - 26

42 Wave Functions - General relative motion of two-ody system a = + (= c + y = d + z =...) many-ody wave function Ψ a is solution of Schrödinger equation HΨ a = (T + V a ) Ψ a = EΨ a with potential V a = V C + V N = V C cy + V N cy (Coulom + nuclear interaction) and oundary condition for ound/scattering state for large distances r = r r, etc.: Coulom interaction remains, short-range nuclear interaction vanishes universal asymptotic form of Ψ a φ φ ψ (r ) +... for r,r cy,... with relative wave functions ψ, ψ cy,... eact solution: partial wave epansion Indirect Methods - 26

43 Wave Functions - Bound States general form of asymptotics (without particle spins, α = (), (cy),...) ψ α (m) 1 f αl (r α )Y lm (ˆr α ) for r α r α l with radial wave functions f αl (r α ) = C a α(l)w ηα,l+1/2(2q α r α ) and angular parts Y lm (ˆr α ) (spherical harmonic) Indirect Methods - 27

44 Wave Functions - Bound States general form of asymptotics (without particle spins, α = (), (cy),...) ψ α (m) 1 f αl (r α )Y lm (ˆr α ) for r α r α l with radial wave functions f αl (r α ) = C a α(l)w ηα,l+1/2(2q α r α ) and angular parts Y lm (ˆr α ) (spherical harmonic) Whittaker function W ηα,l+1/2(2q α r α ) ep( q α r α ) with Sommerfeld parameter η α, ound-state parameter q α e.g. η = Z Z e 2 µ 2 q q = 2µ S / and separation energy S of particle a into and asymptotic normalization coefficient (NC) C a α(l) Indirect Methods - 27

45 Wave Functions - Scattering States general form of asymptotics (without particle spins) Ψ (+) 4π 1 v φ α g (+) αl (r α )i l Y lm (ˆr α )Y k r α v lm(ˆk ) for r α α α=(),(cy),... lm with radial wave functions g (+) αl (r α ) = 1 2i [ S l α() u(+) l and angular parts Y lm (ˆr α ), Y lm (ˆk α ) (spherical harmonics) ] (η α, k α r α ) δ α() u ( ) l (η α,k α r α ) Indirect Methods - 28

46 Wave Functions - Scattering States general form of asymptotics (without particle spins) Ψ (+) 4π 1 v φ α g (+) αl (r α )i l Y lm (ˆr α )Y k r α v lm(ˆk ) for r α α α=(),(cy),... lm with radial wave functions g (+) αl (r α ) = 1 2i [ S l α() u(+) l and angular parts Y lm (ˆr α ), Y lm (ˆk α ) (spherical harmonics) ] (η α, k α r α ) δ α() u ( ) l (η α,k α r α ) Coulom wave functions u (±) l (η α, k α r α ) = e iσ l [G l ± if l ] ep { ±i [ ]} k α r α 2η α ln(k α r α ) lπ 2 with Sommerfeld parameter η α, momentum k α, energy E of relative motion e.g. η = Z Z e 2 µ 2 k k = 2µ E / µ = m m m +m S-matri elements S l α() e.g. elastic scattering S l αα = e 2i[σ l+δ l (α)] with Coulom and nuclear phase shifts Indirect Methods - 28

47 NC Method - Idea φ(r) [fm -1/2 ] a B + φ(r) C W -η,l+1/2 (2qr) NC 8 7 B Be + p r [fm] etract asymptotic normalization coefficient (NC) for reakup of nucleus B into + or nucleus a into + from cross section of transfer reaction + a B + with a = + and B = + calculate astrophysical S factor S(E) in the limit E 0 (H.M. Xu et al., Phys. Rev. Lett. 73 (1994) 2027) Indirect Methods - 29

48 NC Method - Theory I T-matri element in post-form DWB replace eact overlap functions y asymptotic form with asymptotic normalization coefficients (NCs) and Whittaker functions a=(+) B=(+) Indirect Methods - 30

49 NC Method - Theory I T-matri element in post-form DWB replace eact overlap functions y asymptotic form with asymptotic normalization coefficients (NCs) and Whittaker functions a=(+) overlap functions (ˆ= wave function of transferred particle, neglecting spins) φ φ a Ca (l a) r W η,l a +1/2(2q r )Y la m a (ˆr )φ φ φ B CB (l B) r W η,l B +1/2(2q r )Y lb m B (ˆr )φ B=(+) Indirect Methods - 30

50 NC Method - Theory I T-matri element in post-form DWB replace eact overlap functions y asymptotic form with asymptotic normalization coefficients (NCs) and Whittaker functions a=(+) overlap functions (ˆ= wave function of transferred particle, neglecting spins) φ φ a Ca (l a) r W η,l a +1/2(2q r )Y la m a (ˆr )φ φ φ B CB (l B) r W η,l B +1/2(2q r )Y lb m B (ˆr )φ B=(+) cross section of transfer reaction to ound state dσ = C dω a 2 C B 2 d σ with reduced DWB cross section B dω B d σ dω B Indirect Methods - 30

51 NC Method - Theory I T-matri element in post-form DWB replace eact overlap functions y asymptotic form with asymptotic normalization coefficients (NCs) and Whittaker functions a=(+) overlap functions (ˆ= wave function of transferred particle, neglecting spins) φ φ a Ca (l a) r W η,l a +1/2(2q r )Y la m a (ˆr )φ φ φ B CB (l B) r W η,l B +1/2(2q r )Y lb m B (ˆr )φ B=(+) cross section of transfer reaction to ound state dσ = C dω a 2 C B 2 d σ with reduced DWB cross section B dω B two NCs appear corresponding to two poles in diagram NC/Whittaker functions replace spectroscopic factors/full single-particle wave functions in conventional DWB d σ dω B Indirect Methods - 30

52 NC Method - Theory II approimations only valid for weakly ound states/ peripheral reactions precise optical potentials for + a and B + scattering required one additional NC needed a=(+) B=(+) Indirect Methods - 31

53 NC Method - Theory II approimations only valid for weakly ound states/ peripheral reactions precise optical potentials for + a and B + scattering required one additional NC needed a=(+) B=(+) calculate low-energy S factor of capture reaction (,γ)a numerically with etracted NC C a (l a) and asymptotic wave function C a 2 S(0) unique relation? effect of final-state interaction V? systematic model calculations Indirect Methods - 31

54 NC Method - Continuum Interaction effects of interaction in continuum states modification of shape of cross section, S factor (i.e. energy dependence) change of S(0) even though δ 0 calculation of zero-energy S factor S(0) in single-particle model with Woods-Saon potential with different depths V 0 Indirect Methods - 32

55 NC Method - Continuum Interaction effects of interaction in continuum states modification of shape of cross section, S factor (i.e. energy dependence) change of S(0) even though δ 0 calculation of zero-energy S factor S(0) in single-particle model with Woods-Saon potential with different depths V 0 eample: E1 s p wave capture for different nuclei with proton+core structure stronger variation of S(0) with V 0 with larger proton separation energy simple relation NC S(0) only correct for weakly ound nuclei S(0,V 0 )/S(0,0) Be(p,γ) 8 B (0.137 MeV) 11 C(p,γ) 12 N (0.601 MeV) 8 B(p,γ) 9 C (1.296 MeV) S. Typel and G. Baur, Nucl. Phys. 759 (2005) V 0 [MeV] Indirect Methods - 32

56 Trojan-Horse Method - Idea + C + c replace two-ody reaction + C + c y three-ody reaction + a C + c + + a C + c + with Trojan horse a = + and spectator small momentum transfer to spectator quasi-free scattering dominates large relative energy of system + a no suppression of cross section no electron screening small relative energies of system + accessile application to nuclear astrophysics... κǫκαλυµµǫνoι ιππo. Homer, Odyssey VIII, 503 (G. Baur, Phys. Lett. B 178 (1986) 35) Indirect Methods - 33

57 Trojan-Horse Method - Theory I T-matri element in post-form DWB with B = C + c T (B)(a) = φ B φ χ ( ) B V B U B φ a φ χ (+) a use asymptotic form of scattering wave function φ B = Ψ ( ) Cc in reaction channel C + c + ( surface approimation ) a=(+) C c Indirect Methods - 34

58 Trojan-Horse Method - Theory I T-matri element post-form DWB with B = C + c T (B)(a) = φ B φ χ ( ) B V B U B φ a φ χ (+) a use asymptotic form of scattering wave function φ B = Ψ ( ) Cc in reaction channel C + c + ( surface approimation ) a=(+) C c overlap function (ˆ= wave function of transferred particle, neglecting spins) φ Ψ ( ) Cc 4π vcc k Cc r v lm ξ l (r )i l Y lm (ˆr )Ylm (ˆk Cc )φ with ξ l (r ) = 1 2i [ S l Cc u(+) l ] (η ;k r ) δ Cc u ( ) l (η ;k r ) and S-matri element S l Cc of reaction C(c,) (S. Typel and G. Baur, nn. Phys. (N.Y.) 305 (2003) 228) Indirect Methods - 34

59 Trojan-Horse Method - Theory II post-form DWB T-matri element with surface approimation for a Cc factorization C c T (B)(a) lm Sl Cc a=(+) u(+) l (η ;k r ) k Cc r Y lm (ˆr )φ φ χ ( ) B V B U B φ a χ (+) a Indirect Methods - 35

60 Trojan-Horse Method - Theory II post-form DWB T-matri element with surface approimation for a Cc factorization C c T (B)(a) lm Sl Cc a=(+) u(+) l (η ;k r ) k Cc r Y lm (ˆr )φ φ χ ( ) B V B U B φ a χ (+) a cross section of transfer reaction to continuum (single channel, Cc) d 3 σ = SCc l dω B dω Cc de Cc 2 d3 σ l dω B dω Cc de Cc with reduced DWB cross section Indirect Methods - 35

61 Trojan-Horse Method - Theory II post-form DWB T-matri element with surface approimation for a Cc factorization C c T (B)(a) lm Sl Cc a=(+) u(+) l (η ;k r ) k Cc r Y lm (ˆr )φ φ χ ( ) B V B U B φ a χ (+) a cross section of transfer reaction to continuum (single channel, Cc) d 3 σ = SCc l dω B dω Cc de Cc 2 d3 σ l dω B dω Cc de Cc with reduced DWB cross section S-matri elements S l Cc dσ dω (C + c + ) = π k 2 determine cross section S l CcY l0 (ˆr ) l 2 Indirect Methods - 35

62 Trojan-Horse Method - Theory III additional approimations (not necessary in general, ut convenient) potential V B U B = V + V U B V plane waves for distorted waves χ ( ) B, χ(+) a a=(+) C c Indirect Methods - 36

63 Trojan-Horse Method - Theory III additional approimations (not necessary in general, ut convenient) potential V B U B = V + V U B V plane waves for distorted waves χ ( ) B, χ(+) a T-matri element in PWB with surface approimation T (B)(a) ) lm Sl Cc φ φ ep (i Q B r V φ a u(+) l with momentum transfers a=(+) ) (η ;k r ) k Cc r Y lm (ˆr ) ep (i Q a r C c Q a = k a µ m kb Q B = k B µ m ka factorization with three factors: two factors for two poles in diagram additional matri element, depends on kinematics (η!) Indirect Methods - 36

64 Trojan-Horse Method - Theory IV additional approimations (not necessary in general, ut convenient) potential V B U B V plane waves for distorted waves χ ( ) B, χ(+) a a=(+) cross section of transfer reaction to continuum (single channel) d 3 σ dω B dω Cc de Cc = K W( Q B ) dσ l dω ( Cc) T l(k ) with kinematic factor K momentum distriution W( Q B ) = Φ a ( Q B ) 2 depending on momentum transfer to spectator quasi-free scattering conditions cross section dσ l ( Cc) of two-ody reaction dω penetration factor T l (k ) k 3 ep(2πη ) C c Indirect Methods - 37

65 Trojan-Horse Method - Theory IV additional approimations (not necessary in general, ut convenient) potential V B U B V plane waves for distorted waves χ ( ) B, χ(+) a a=(+) cross section of transfer reaction to continuum (single channel) d 3 σ dω B dω Cc de Cc = K W( Q B ) dσ l dω ( Cc) T l(k ) with kinematic factor K momentum distriution W( Q B ) = Φ a ( Q B ) 2 depending on momentum transfer to spectator quasi-free scattering conditions cross section dσ l ( Cc) of two-ody reaction dω penetration factor T l (k ) k 3 ep(2πη ) cancels suppression of two-ody cross section y Coulom arrier for E 0 C c Indirect Methods - 37

66 Trojan-Horse Method - pplication selection of Trojan horse a = + (e.g. 2 H = n + p, 6 Li = α + d,... ) with inding energy ǫ a > 0 and well known ground state wave function momentum distriution W( Q B ) width of momentum distriution W Fermi motion of inside a momentum distriution W(Q B ) [a.u] momentum transfer Q B [a.u.] Q cut Q cut S factor S(E ) [a.u.] range of accessile energies qf E normalization scaled THM data asolute direct data energy E [a.u] Indirect Methods - 38

67 Trojan-Horse Method - pplication selection of Trojan horse a = + (e.g. 2 H = n + p, 6 Li = α + d,... ) with inding energy ǫ a > 0 and well known ground state wave function momentum distriution W( Q B ) width of momentum distriution W Fermi motion of inside a condition Q B = 0 defines quasi-free energy in + system E qf = E a ( 1 µ a µ B µ 2 m 2 ) ǫ a E a cutoff in Q B determines range of accessile energies E around E qf small momentum transfer dominance of quasi-free process normalization of cross section to direct data at higher E momentum distriution W(Q B ) [a.u] S factor S(E ) [a.u.] momentum transfer Q B [a.u.] qf E -Q cut Q cut range of accessile energies normalization scaled THM data asolute direct data energy E [a.u] Indirect Methods - 39

68 Trojan-Horse Method - Eample: 6 Li(p,α) 3 He direct reaction: 6 Li(p,α) 3 He eperimental data (J. Elwyn et al., Phys. Rev. C 20 (1979) 1084) differential cross section dσ/dω = l B lp l (cosθ) non-resonant s wave and resonant p wave contriution S matri from R-matri fit simulation of THM eperiment B l [m/sr] cross section [ar. units] B 0 B 1 B E p [MeV] E Li = 25 MeV E Li = 13.9 MeV E [MeV] E [MeV] Indirect Methods - 40

69 Trojan-Horse Method - Eample: 6 Li(p,α) 3 He direct reaction: 6 Li(p,α) 3 He eperimental data (J. Elwyn et al., Phys. Rev. C 20 (1979) 1084) differential cross section dσ/dω = l B lp l (cosθ) non-resonant s wave and resonant p wave contriution S matri from R-matri fit simulation of THM eperiment THM: 2 H( 6 Li,α 3 He)n eperiments with 13.9/25 MeV 6 Li eam (. Tumino et al., Phys. Rev. C 67 (2003) ) E qf = 0.24/1.35 MeV Q B < 30 MeV/c finite cross section at E = 0 MeV! B l [m/sr] cross section [ar. units] B 0 B 1 B E p [MeV] E Li = 25 MeV E [MeV] E Li = 13.9 MeV E [MeV] Indirect Methods - 40

70 Trojan-Horse Method - Electron Screening direct eperiments: reduction of Coulom arrier y electron cloud of target nucleus enhanced cross section at low energies are potential screened potential screening potential σ ep (E) = σ are (E)f(E) with f(e) = ep(πηu e /E) and electron screening potential energy U e discrepancy etween eperimental oservation and theoretical models, eplanation? V [kev] V [kev] stellar conditions: electron screening in plasma r [fm] r [fm] Indirect Methods - 41

71 Trojan-Horse Method - Eample: 2 H( 6 Li,α) 4 He direct reaction: 2 H( 6 Li,α) 4 He eperiment with gas target (S. Engstler et al., Z. Phys. 342 (1992) 471) S(0) = 17.4 MeV (corrected for electron screening) Indirect Methods - 42

72 Trojan-Horse Method - Eample: 2 H( 6 Li,α) 4 He direct reaction: 2 H( 6 Li,α) 4 He eperiment with gas target (S. Engstler et al., Z. Phys. 342 (1992) 471) S(0) = 17.4 MeV (corrected for electron screening) THM: 6 Li( 6 Li,αα) 4 He eperiment with 6 MeV 6 Li eam (C. Spitaleri et al., Phys. Rev. C 63 (2001) ;. Musumarra et al., Phys. Rev. C 64 (2001) ) E qf = 25 kev target and projectile reakup l = 0, Q B < 35 MeV/c normalization to direct data for E > 600 kev S(0) = (16.9 ± 0.5) MeV S factor S(E) [MeV ] direct data THM data fit to THM data electron screening normalization E [kev] electron screening potential: U e (direct) = (330 ± 120) ev U e (THM) = (320 ± 50) ev U e (theory) = 186 ev (adiaatic limit) Indirect Methods - 42

73 Summary Indirect methods (CD, NC, THM) provide complementary information for reactions of astrophysical interest peripheral reactions asymptotics of wave functions specific kinematical conditions description with direct reaction theory factorization of cross sections approimations great potential for future applications Indirect Methods - 43