Reaction Theory. Stefan Typel GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt Nuclear Astrophysics Virtual Institute

Transcription

1 Reaction Theory GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt Nuclear Astrophysics Virtual Institute Re-writing Nuclear Physics textbooks: 30 years of radioactive ion-beam physics Pisa, Italy July 20 July 24, 2015

2 Outline Introduction notation, interactions and reactions, scales, kinematics, conservation laws General Reaction Theory cross sections and theoretical methods stationary scattering theory partial-wave expansion operator formalism Applications astrophysics and indirect methods transfer reactions Summary Aim of Lecture introduction to basic methods in reaction theory with particularities for exotic nuclei no need to follow every step in derivations, only for completeness not a review of recent results or details of specific approaches/individual cases Reaction Theory - 1

3 Introduction

4 General Remarks definition reactions = all processes that occur if two (or more) particles collide within an interaction zone (depends on range of interaction) types of interactions electromagnetic interaction (long range) strong interaction (medium range) weak interaction (short range) possible competition or interference notation general form: X 1 +X 2 X 3 +X X n alternatively: X 2 (X 1,X 4,...,X n )X 3 with projectile X 1, target X 2, and ejectiles X 4,...,X n inverse kinematics, e.g. if X 2 is unstable nucleus X 1 (X 2,X 4,...,X n )X 3 with projectile X 2, target X 1, and ejectiles X 4,...,X n Reaction Theory - 2

5 Types of Reactions reactions with two particles in the initial state elastic scattering a+a A+a inelastic scattering a+a A +a rearrangement reactions a+a B +b with b a, B A radiative capture reactions a+a C +γ with photon in final state many-body reactions a+a B +b 1 +b example: p+ 7 3Li 7 3Li+p 7 3Li +p 7 4Be+n α+α 8 4Be+γ ( α+α+γ) α+t+p Reaction Theory - 3

6 Length and Time Scales radius of stable nucleus R = r 0 A 1/3 with r fm, mass number A corresponding area of circle S = πr 2 time for light to traverse nucleus t = 2R/c example 208 Pb: R 7.4 fm, S 172 fm 2 = 1.72 b, t 14.8 fm/c = s unit for area in reaction theory: barn (1 b = m 2, 1 fm 2 = 10 mb) time scales for reactions: fast, t s: direct reactions (few nucleons involved, one-step processes) intermediate, t > s: multi-step reactions (complicated) slow, t s: compound nucleus reactions (many nucleons involved, collective excitations, no memory of initial state, statistical features) Reaction Theory - 4

7 Energy Scales and Kinematics typical energies nuclear structure: binding and excitation energies a few MeV potentials: Coulomb barrier E C = Z az A e 2 R R C R a +R A e 2 = 1.44 MeV fm C reactions: very large range of energies, e.g. thermal neutrons 25 mev astrophysical reactions a few 10 or 100 kev direct nucleon transfer reactions a few MeV per nucleon Coulomb excitation reactions a few 10 MeV or 100 MeV per nucleon relativistic heavy-ion collisions a few GeV or TeV per nucleon Q value Q = (m a +m A m b m B )c 2 for reaction A(a,b)B Q > 0 exothermic reaction (release of kinetic energy due to larger binding) Q = 0 elastic reaction Q < 0 endothermic reaction (reaction not possible below threshold) Reaction Theory - 5

8 Conservation Laws reaction A(a, b)b conservation of energy E(a+A) = T a +T A +(m a +m A )c 2 = T b +T B +(m b +m B )c 2 = E(b+B) with kinetic energies T a, T A, T b, T B conservation of momentum P(a+A) = p a + p A = p b + p B = P(b+B) conservation of total angular momentum J(a+A) = J a + J A + J aa = J b + J B + J bb = J(b+B) conservation of parity P(a+A) = P a P A ( 1) l aa = P b P B ( 1) l bb = P(b+B) conservation of isospin T(a+A) = T a + T A = T b + T B = T(b+B) conservation of charge, baryon number and lepton number Reaction Theory - 6

9 General Reaction Theory

10 Theoretical Approaches large variety of reactions numerous theoretical approaches here only a selection, mainly for direct reactions many topics not covered: e.g. compound nucleus reactions, heavy-ion collisions theoretical description classical methods semiclassical methods quantal methods observables: cross sections general conditions in the following nonrelativistic kinematics no explicit consideration of antisymmetrization no weak interaction processes spins of particles mostly neglected Reaction Theory - 7

11 Definition of Cross Sections I quantifying the strength of reactions uniform beam of particles in z-direction hitting a thick target reduction of current J due to reactions: dj dz = σj(z)ρ J(z) = J(0)exp( σρz) proportial to current J(z) and density of target nuclei ρ proportionality constant = (interaction) cross section σ dimensions: [J] = L 2 T 1, [ρ] = L 3, [σ] = L 2 with length L and time T cross section σ depends on energy E of beam excitation function σ(e) more detailed information on reaction with dependence on scattering angle θ differential cross section dσ dω Reaction Theory - 8

12 Definition of Cross Sections II differential cross section initial state: uniform beam of particles in z direction with current J i scattering on a single target nucleus, detection of scattered particles at distance r from target in area ds = r 2 dω at scattering angle θ final state: current of particles J f in radial direction dσ fi = J fds J i in classical physics: or dσ fi dω = J fr 2 dσ dω = b sinθ J i dθ db 1 (elastic scattering i = f) with deflection function θ(b) depending on impact parameter b assuming azimuthal symmetry (φ independence) of scattering Reaction Theory - 9

13 Classical Description of Scattering determine trajectories of particles solve Newtonian equations of motion m i ri = F i (or use conservation laws for energy, momentum, angular momentum) with given initial conditions (position, velocity) ordinary time-dependent differential equations example: elastic Coulomb scattering of particle a with energy E and impact parameter b on target A F a = Z az A e 2 r 2 r r deflection function ( θ(b) = 2 arccot ) 2bE Z a Z A e 2 Rutherford cross section ( ) dσ R dω = Za Z A e E sin 4 ( 2) θ Reaction Theory - 10

14 Semiclassical Description combination of classical and quantal methods example: Coulomb excitation of nucleus a from ground state i to excited state f with excitation energy E = ω in time-dependent Coulomb potential V( x,t) = Z az A e 2 r(t)+ x of target nucleus A excitation cross section dσ fi dω = dσ R dω P fi Rutherford cross section dσ R dω excitation probability P fi = a fi 2 in first order time-dependent perturbation theory with amplitude a fi = 1 i dte iωt f V( x,t) i application to exotic nuclei: excitation to continuum states/breakup Reaction Theory - 11

15 Quantal Description determine scattering wavefunction solve Schrödinger equation with given boundary conditions time-dependent i tψ = Ĥψ time evolution of wave packet (not considered in the following) stationary Eψ = Ĥψ fixed energy E two formulations: partial differental equations integral equations boundary conditions: plane wave + outgoing (ingoing) spherical waves define channels c = i, f that characterize asymptotic states partition, e.g. A+a, B +b, C +γ,... additional quantum numbers, e.g. for particular states of nuclei A, a,... energy, momentum Reaction Theory - 12

16 General Reaction Theory Stationary Scattering Theory

17 Stationary Scattering Theory I theoretical formulation for reaction with nuclei a and A in initial channel total Hamiltonian Ĥ = Ĥa+ĤA+ ˆT aa + ˆV aa with Hamiltonians Ĥ a, Ĥ A of nuclei with wave functions φ a, φ A Ĥ a φ a = E a φ a Ĥ A φ A = E A φ A kinetic energy operator of relative motion ˆT aa = 2 2µ aa raa with reduced mass µ aa = m am A m a +m A and r aa = r a r A interaction potential ˆVaA Hamiltonian without aa interaction Ĥ (i) 0 = Ĥa+ĤA+ ˆT aa (i ) k i r i wave function Φ i = φ i exp ( r i = r aa k ki = µ a aa ma ) k A ma with φ i = φ a φ A, Ĥ (i) 0 Φ i = (E a +E A +E aa )Φ i and E aa = 2 k 2 i 2µ aa Reaction Theory - 13

18 Stationary Scattering Theory II full solution with total Hamiltonian ĤΨ (±) i = EΨ (±) i boundary condition: asymptotic form for large radii Ψ (±) i Φ i + φ f f (±) exp(±ik f r f ) fi with Φ i = φ i exp (i r ) k i r i f f and scattering amplitude f (±) fi in final channels f + solution: outgoing spherical waves solution: ingoing spherical waves differential cross section for reaction from initial channel i to final channel f dσ fi dω = J fr 2 J i with currents J i, J f of relative motion current in nonrelativistic quantum mechanics for wavefunction ψ for particle with mass m J = [ψ ( ) ψ ( ) ] ψ ψ 2mi Reaction Theory - 14

19 Stationary Scattering Theory III initial state: current for wave function of relative motion ψ = exp (i ) k i r i J i = k i = v i = v aa µ i final state: current for wave function of relative motion ψ = f (+) exp(±ik f r f ) f (+) fi 2 k f r f fi J f for r f r f µ f r f cross section for reaction from initial state i to final state f dσ fi dω = J frf 2 = k f f (+) fi 2 determine scattering amplitude f (+) fi J i k i methods to find f (+) fi : partial-wave expansion of wave function Ψ (+) i formulation with integral equation operator formalism r 2 f Reaction Theory - 15

20 General Reaction Theory Partial-Wave Expansion

21 Partial-Wave Expansion I elastic scattering A(a, a)a on a spherically symmetric short-range potential V aa (r) for nonrelativistic energies (single channel) expansion of full wave function Ψ (+) i = l l=0 m= l ϕ (+) l (r) Y lm (ˆr)φ a φ A r with radial wave functions ϕ (+) l ĤΨ (+) i = E i Ψ (+) i = (E a +E A +E aa )Ψ (+) i [ 2 d 2 2µ aa dr 2 + l(l+1) r 2 and spherical harmonics Y lm radial Schrödinger equation ] +V aa (r) ϕ (+) l (r) = E aa ϕ (+) l (r) with energy E aa = 2 k 2 2µ aa boundary conditions: r = 0 ϕ (+) l (r) = 0 r ϕ (+) l (r)?, but short-range potential V aa (r) 0 for r linear combination of regular and irregular spherical Bessel functions (modifications for Coulomb potentials) Reaction Theory - 16

22 Partial-Wave Expansion II wavefunction without potential Φ i = φ a φ A exp ( i ) k i r with k i = k i e z partial-wave expansion Φ i = 4π l,m i l j l (k i r)y lm (ˆr)Y lm(ˆk i )φ a φ A with spherical Bessel functions j l, spherical harmonics Y lm, ˆr = r/r, ˆk i = k i /k i use properties of spherical Bessel functions j 0 (z) = sinz = eiz e iz ( j l = z l 1 ) l d j 0 (z) = 1 [ ] u (+) l (z) u ( ) l (z) z 2iz zdz 2iz with in/outgoing spherical waves u (±) l (z) exp [ ±i ( z l π 2)] for z Φ i = 4π l,m i l 1 2ik i r [ ] u (+) l (k i r) u ( ) l (k i r) Y lm (ˆr)Y lm(ˆk i )φ a φ A Reaction Theory - 17

23 Partial-Wave Expansion III scattering can only affect outgoing spherical waves u (+) l introduce S-matrix elements S l (k i ) (complex numbers) asymptotics of solution of Schrödinger equation Ψ (+) i 4π l,m i l 1 2ik i r asymptotics of scattering part Ψ (+) i Φ i 4π l,m [ S l (k i )u (+) l (k i r) u ( ) l (k i r) ] Y lm (ˆr)Y lm(ˆk i )φ a φ A i l 1 2ik i r [S l(k i ) 1]u (+) l (k i r)y lm (ˆr)Y lm(ˆk i )φ a φ A elastic scattering amplitude f (+) ii (θ) = l 2l+1 2ik i [S l (k i ) 1]P l (cosθ) with Legendre polynomials P l, argument cosθ = ˆr ˆk i and P l (cosθ) = 4π m Y lm(ˆr)y lm (ˆk i ) Reaction Theory - 18

24 Partial-Wave Expansion IV differential elastic scattering cross section dσ ii dω = f (+) ii 2 2l+1 = [S l (k i ) 1]P l (cosθ) 2ik i l total elastic scattering cross section σ el = dω dσ 1 ii dω = 2π dcosθ f (+) ii (θ) 2 = π 1 ki 2 2 (2l+1) S l (k i ) 1 2 with orthogonality relation of Legendre polynomials 1 1 dz P l(z)p l (z) = 2 2l+1 δ ll knowledge of S-matrix elements S l sufficient to calculate cross sections l Reaction Theory - 19

25 Partial-Wave Expansion V parametrization S l = exp(2iδ l ) with scattering phase shifts δ l [0,π] asymptotics of radial wave functions [ ] u l = 1 2i S l u (+) l u ( ) l exp(iδ l )sin ( k i r +δ l l2) π for r radial wave function u 0 (k i r) s wave without potential with attractive potential δ 0 scattering phase shift δ l (E aa ) [deg] attractive potential repulsive potential resonance argument k i r energy E aa [a.u.] Reaction Theory - 20

26 Partial-Wave Expansion VI calculation of cross section with current of full wave function Ψ (+) i absorption cross section σ abs = π ki 2 (2l+1) [1 S l (k i ) 2] total reaction cross section σ tot = σ el +σ abs = 2π ki 2 (2l+1)Re[1 S l (k i )] l scattering phase shifts δ l real S l (k i ) = exp(2iδ l ) = 1 σ abs = 0 only elastic scattering l scattering phase shifts δ l complex with Im(δ l ) > 0 S l (k i ) = exp[ 2Im(δ l )] < 1 σ abs > 0 reactions with removal of flux from elastic scattering channel can be described phenomenologically by optical potential U = V +iw with real and imaginary contributions (V, W real) Reaction Theory - 21

27 Partial-Wave Expansion VII isolated narrow resonance in partial wave l resonance energy E r resonance width Γ E r different parametrisation of S-matrix element S l (E) = E E r i Γ 2 E E r +i Γ 2 Γ S l (E) = 1 S l (E) 1 = i E E r +i Γ 2 elastic scattering cross section σ el = π k 2 i (2l+1) Breit-Wigner form Γ 2 (E E r ) 2 + Γ2 4 general formulation with many resonances and many channels including Coulomb potential R-matrix theory with parameters: resonance energies and reduced widths Reaction Theory - 22

28 General Reaction Theory Operator Formalism

29 Operator Formalism I simplified notation: point-like nuclei without internal structure φ a = φ A = 1, E a = E A = 0, E = E i = E aa = 2 ki 2 2µ aa ) solve Schrödinger equation ĤΨ = (ˆT + ˆV Ψ = EΨ with ˆT = Ĥ 0 = 2 2µ aa Φ 0 ( k i ) = exp ( i ) k i r is solution of Schrödinger equation Ĥ 0 Φ 0 = EΦ 0 rewrite full Schrödinger equation (Ĥ0 + ˆV) Ψ = EΨ as ˆVΨ = ) (E Ĥ 0 Ψ and introduce operator G (±) 0 = ( ) 1 E Ĥ0±iǫ integral (Lippmann-Schwinger) equation Ψ (±) = Φ 0 ( k i )+G (±) 0 term with Φ 0 correct solution for ˆV = 0 (±) forms for different asymptotics ˆVΨ (±) Reaction Theory - 23

30 Operator Formalism II explicit form Ψ (±) ( r) = Φ 0 ( k i, r)+ d 3 r G (±) 0 ( r, r )ˆV( r )Ψ (±) ( r ) with Green s function G (±) 0 ( r, r ) = 2µ exp(±ik aa i r r ) 2 4π r r formal solution: Born series Ψ (±) = Φ 0 ( k i )+G (±) 0 ˆVΨ (±) = Φ 0 ( k i )+G (±) 0 ˆVΦ 0 +G (±) 0 with integral operator G (±) 0 [...] = d 3 r G (±) 0 ( r, r )[...] ˆVG (±) 0 integration range of coordinate r limited by extension potential V ˆVΦ for r r use approximation k i r r k i r k f r +... with r k f = k i r Ψ (±) ( r) = Φ 0 ( k i, r) 2µ aaexp(±ik i r) 2 4πr d 3 r exp ( i k f r ) ˆV( r )Ψ (±) ( r ) Reaction Theory - 24

31 Operator Formalism III scattering amplitude f (±) fi = µ aa d 3 r exp 2π 2 ( i k f r ) ˆV( r )Ψ (±) ( r ) = µ aa 2π 2 T fi with T-matrix element T fi = Φ 0 ( k f ) ˆV Ψ (+) ( k i ) knowledge of T-matrix elements T fi sufficient to calculate cross sections reformulation introduce potential U with known solutions χ (±) (distorted waves) of Schrödinger equation (Ĥ0 +Û) χ (±) = Eχ (±) use operator identity 1 A 1 B = 1 B (B A) 1 A two-potential formula T fi = Φ 0 ( k f ) Û χ(+) ( k i ) + χ ( ) ( k f ) ˆV Û Ψ(+) ( k i ) Reaction Theory - 25

32 Operator Formalism IV generalisation with Gell-Mann Goldberger relation for reaction A(a, b)b T fi = φ b φ B Φ 0 ( k f ) ÛaA φ a φ A χ (+) aa ( k i ) + φ b φ B χ ( ) bb ( k f ) ˆV bb ÛbB Ψ (+) aa ( k i ) potential U aa acting only on coordinates of relative motion, not internal coordinates φ b φ B Φ 0 ( k f ) Û aa φ a φ A χ (+) aa ( k i ) = 0 if aa bb for rearrangement reactions aa bb exact results: post form T fi = φ b φ B χ ( ) bb ( k f ) ˆV bb Û bb Ψ (+) aa ( k i ) prior form T fi = Ψ ( ) bb ( k f ) ˆV aa ÛaA φ a φ A χ ( ) aa ( k i ) exact scattering wave fuctions Ψ (+) aa or Ψ( ) bb still needed potentials ˆV aa and ˆV bb depend on coordinates of all nucleons in the nuclei, should be consistent with those used for microscopic description of nuclei itself challenge: combination of structure and reaction calculations Reaction Theory - 26

33 Cross Sections I general form for two-body reaction A(a,b)B in c.m. system energies E i = E a +E A + p2 aa 2µ aa, E f = E b +E B + p2 bb 2µ bb with spins averaging over initial states, summation over final states dσ(a+a b+b) = 2π µ aa 1 p aa (2J a +1)(2J A +1) m a,m A m b,m B d 3 p bb (2π ) 3 T bbaa 2 δ(e i E f +Q a+a B+b ) with d 3 p bb = p 2 bb dp bbdω bb, de f /dp bb = p bb /µ bb and integration over p bb dσ dω bb (a+a b+b) = µ aaµ bb (2π) 2 4 p bb p aa 1 (2J a +1)(2J A +1) similar expression for inverse reaction B(b, a)a m a,m A m b,m B T bbaa 2 result can be generalized to reactions with three or more particles in the final state Reaction Theory - 27

34 Cross Sections II cross sections for reactions A(a,b)B and B(b,a)A dσ dω bb (a+a b+b) = µ aaµ bb (2π) 2 4 p bb p aa 1 (2J a +1)(2J A +1) dσ dω aa (b+b a+a) = µ bbµ aa (2π) 2 4 p aa p bb 1 (2J b +1)(2J B +1) time-reversal symmetry T bbaa 2 = T aabb 2 theorem of detailed balance (for two-body reactions) (2J a +1)(2J A +1)p 2 aa = (2J b +1)(2J B +1)p 2 bb dσ (a+a b+b) dω dσ (b+b a+a) dω m a,m A m b,m B application: indirect methods (e.g. Coulomb dissociation method) m b,m B T bbaa 2 m a,m A T aabb 2 Reaction Theory - 28

35 Applications

36 Relevance of Reaction Theory applications direct interest in relevant reaction cross sections, e.g. prediction of production rates of exotic nuclei (not considered here) astrophysics: nucleosynthesis reactions to study of nuclear structure, e.g. gross properties: radii, density distributions,... elastic scattering (with electrons, protons, α-particles,... ), absorption reactions,... (not considered here) detailed structure: excitation of specific states with electromagnetic or nuclear interaction study of single-particle structure challenges with exotic nuclei correct treatment of continuum states combination of reaction theory with modern structure models choice of nuclear interaction Reaction Theory - 29

37 Applications Astrophysics and Indirect Methods

38 Reactions of Astrophysical Interest nuclear reactions rates input for astrophysical models various processes (pp-chain, CNO cycles, s-, r-, p-, rp-process) often unstable nuclei involved cross sections at low energies needed direct measurement practically impossible alternative: indirect methods nuclei in hot plasma Maxwellian-averaged reaction rate r aa = ρ aρ A 1+δ aa σv with σv = 8 πµ aa Z pp chains 4 2 CNO cycles (p,α) (p,γ) (α,γ) (α,p) (p,β + ), ( 3 He,2p) (β + ), (EC) (α) N ( de (kt) 3/2Eσ(E)exp E ) kt reactions with charged particles cross section needed in Gamov window around effective energy E eff = µ 1/3 aa (Z az Z T 9 ) 2/3 MeV with effective mass µ aa in amu and temperature T 9 = T/(10 9 K) Reaction Theory - 30

39 Reactions of Astrophysical Interest nuclear reactions rates input for astrophysical models various processes (pp-chain, CNO cycles, s-, r-, p-, rp-process) often unstable nuclei involved cross sections at low energies needed direct measurement practically impossible alternative: indirect methods nuclei in hot plasma Maxwellian-averaged reaction rate r aa = ρ aρ A 1+δ aa σv with σv = 8 πµ aa [a.u] E exp(-e/(kt)) σ E exp(-e/(kt)) Gamov window σ(e) E/E eff ( de (kt) 3/2Eσ(E)exp E ) kt reactions with charged particles cross section needed in Gamov window around effective energy E eff = µ 1/3 aa (Z az Z T 9 ) 2/3 MeV with effective mass µ aa in amu and temperature T 9 = T/(10 9 K) Reaction Theory - 30

40 Scattering with Coulomb Potential I modification with Coulomb potential V Coul = Z az A e 2 r in initial state replacement of plane wave in initial state with exact solution for Coulomb scattering (analytically known) asymptotics of radial wave functions with additional nuclear interaction u l exp(2iσ [ ] l) S l u (+) l u ( ) l for r 2i with Coulomb scattering phase shift σ l = argγ(l+1+iη) depending on the Sommerfeld parameter η = Z az A e 2 v aa, v aa = p aa µ aa [ ( and u (±) l (k i r) = exp( iσ l )[G l ±if l ] exp ±i k i r 2ηln(k i r)+σ l l π )] 2 with regular and irregular Coulomb wave functions F l and G l Reaction Theory - 31

41 Scattering with Coulomb Potential II elastic scattering amplitude f (+) ii (θ) = l 2l+1 2ik i [exp(2iσ l )S l (k i ) 1]P l (cosθ) = f (+) C (θ)+f(+) N (θ) with Coulomb scattering amplitude (analytically known) f (+) C (θ) = l 2l+1 2ik i [exp(2iσ l ) 1]P l (cosθ) and nuclear scattering amplitude f (+) N (θ) = l 2l+1 2ik i exp(2iσ l )[S l (k i ) 1]P l (cosθ) interference in cross sections! Reaction Theory - 32

42 Scattering with Coulomb Potential III effect of Coulomb barrier (and centrifugal barrier) reduced probability of finding the particles at small distance R introduce penetrability factor P l (R) = lim r u (±) l (η;kr) 2 1 = (η;kr) 2 F l 2(η,kR)+G2 l (η,kr) u (±) l example: p + 7 Be scattering solid lines: l = s-wave scattering (l = 0): lim P 0(R) = R 0 2πη exp(2πη) 1 penetrability P l define astrophysical S factor S(E) = σ(e)e exp(2πη) weak energy dependence E = 1.0 MeV E = 0.5 MeV E = 0.2 MeV E = 0.1 MeV E = 0.05 MeV E = 0.02 MeV distance R [fm] Reaction Theory - 33

43 Scattering with Coulomb Potential III effect of Coulomb barrier (and centrifugal barrier) reduced probability of finding the particles at small distance R introduce penetrability factor P l (R) = lim r u (±) l (η;kr) 2 1 = (η;kr) 2 F l 2(η,kR)+G2 l (η,kr) u (±) l example: p + 7 Be scattering solid lines: l = 0 dashed lines: l = s-wave scattering (l = 0): lim P 0(R) = R 0 2πη exp(2πη) 1 penetrability P l define astrophysical S factor S(E) = σ(e)e exp(2πη) weak energy dependence E = 1.0 MeV E = 0.5 MeV E = 0.2 MeV E = 0.1 MeV E = 0.05 MeV E = 0.02 MeV distance R [fm] Reaction Theory - 33

44 Scattering with Coulomb Potential III effect of Coulomb barrier (and centrifugal barrier) reduced probability of finding the particles at small distance R introduce penetrability factor P l (R) = lim r u (±) l (η;kr) 2 1 = (η;kr) 2 F l 2(η,kR)+G2 l (η,kr) u (±) l example: p + 7 Be scattering solid lines: l = 0 dashed lines: l = 1 dotted lines: l = s-wave scattering (l = 0): lim P 0(R) = R 0 2πη exp(2πη) 1 penetrability P l define astrophysical S factor S(E) = σ(e)e exp(2πη) weak energy dependence E = 1.0 MeV E = 0.5 MeV E = 0.2 MeV E = 0.1 MeV E = 0.05 MeV E = 0.02 MeV distance R [fm] Reaction Theory - 33

45 Indirect Methods I Coulomb dissociation ANC method Trojan-Horse method study inverse of radiative capture reaction b(x,γ)a a(γ,x)b use Coulomb field of target nucleus A as source of photons a(γ,x)b A(a,bx)A absolute S factors as a function of energy extract asymptotic normalization coefficient of ground state wave function of nucleus a from transfer reactions calculate matrix elements for radiative capture reaction b(x, γ)a S factor at zero energy study three-body reaction A + a C + c + b with Trojan horse a = b + x and spectator b extract cross section of two-body reaction A + x C + c energy dependence of S factor Reaction Theory - 34

46 Indirect Methods II Coulomb dissociation ANC method Trojan-Horse method A A A B=(A+x) A C x x x c a=(b+x) b a=(b+x) b a=(b+x) b photon exchange transfer of particle to bound state transfer of particle to continuum state similar reaction mechanisms: transfer of virtual particle final state with three particles (bound/continuum states) theoretical descripton with direct reaction theory Reaction Theory - 35

47 Indirect Methods III general characteristics: two-body reaction at low-energy is replaced by three-body reaction at high-energy with large cross section Coulomb dissociation b(x, γ)a A(a, bx)a ANC method b(x,γ)a A(a,B)b a = (b+x) B = (A+x) Trojan-horse method A(x, c)c A(a, Cc)b transfer of virtual particle (photon γ or nucleus x) relation of cross sections is found with the help of nuclear direct reaction theory theoretical approximations essential study of peripheral reactions asymptotics of wave functions relevant selection of suitable kinematical conditions important Reaction Theory - 36

48 Coulomb Dissociation Method I projectile a target A fragments b x radiative capture b(x, γ)a detailed balance photo absorption a(γ, x)b Coulomb dissociation A(a, bx)a equivalent photons in Coulomb field of target A electric field correspondence (Fermi 1924, Weizsäcker-Williams 1932) time-dependent electromagnetic field of highly-charged nucleus A during scattering of projectile a spectrum of (virtual, equivalent) photons stable nucleus breakup threshold exc exotic nucleus E ~ few MeV only ground state transitions! further application: study structure of nucleus a Reaction Theory - 37

49 Coulomb Dissociation Method II breakup cross section for reaction A(a, bx)a (in first-order approximation, with angular integration over relative momentum between fragments) d 2 σ = 1 de bx dω aa E γ πλ σ πλ (a+γ b+x) dn πλ dω aa π = E,M λ = 1,2,... photo absorption cross section σ πλ (a+γ b+x) virtual photon numbers dn πλ dω aa depend on kinematics: scattering angle ϑ aa or impact parameter b, projectile velocity v, excitation energy E γ = ω calculation in semiclassical approximation with trajectories with quantal methods using scattering wave functions in partial-wave expansion or eikonal approximation final state: usually three charged particles A, b, x higher-order effects = multi-step transitions/coulomb post-acceleration? better approximation of full scattering wave function needed Reaction Theory - 38

50 Coulomb Dissociation Method III Coulomb dissociation cross section d 2 σ = 1 σ πλ (a+γ b+x) dn πλ de bx dω Aa E γ dω Aa πλ theorem of detailed balance σ πλ (a+γ b+x) = (2J b+1)(2j x +1) 2(2J a +1) k 2 bx k 2 γ A a=(b+x) with photo absorpton and radiative capture cross sections σ πλ (b+x a+γ) A x b phase space factor k 2 bx k 2 γ = 2µ bxc 2 E bx (E bx +S bx ) 2 1 for not too small E bx virtual photon numbers large Coulomb dissociation cross sections dn πλ dω Aa 1 for large Z A and for not too high E bx Reaction Theory - 39

51 Applications Transfer Reactions

52 Transfer Reactions I application of transfer reactions indirect methods for astrophysics: Coulomb breakup, ANC method, Trojan Horse method study of nuclear structure: pickup reaction, e.g. (p,d), (d,t), (d, 3 He), (d, 6 Li),... stripping reaction, e.g. (d,p), (d,n), ( 3 He,p),... knockout/breakup reactions, e.g. (p,pn), (p,pα),... A x B=(A+x) a=(b+x) theoretical description information on reaction process in T-matrix elements A full scattering wave function Ψ (±) needed, in general complicated many-body wave function x choose appropriate approximations reactions with stable nuclei: two particles in final state, a=(b+x) transfer to bound states reaction with exotic nuclei: in most cases three (or more) particles in final state transfer to continuum states, study of correlations b C c b Reaction Theory - 40

53 Transfer Reactions II coordinates in two-body system a + A relative and center-of-mass coordinates r aa = r a r A p aa = µ aa ( pa m a p A m A R m a r a +m A r A = m a +m A ) P = p a + p A with reduced mass µ aa = m am A m a +m A coordinates in three-body system b + c + C = b + B Jacobi coordinates three possibilities, e.g. r cc = r c r C r b(cc) = r b r B R m b r b +m c r c +m C r C = m b +m c +m C ( pc p cc = µ cc p ) ( C pb p b(cc) = µ bb p ) B P = p b + p c + p C m c m C m b m B with r B = m c r c +m C r C m c +m C, p B = p c + p C, m B = m c +m C, µ bb = m bm B m b +m B Reaction Theory - 41

54 Transfer Reactions III general form of cross section for three-body reaction A(a, cc)b in c.m. system energies E i = E a +E A + p2 aa 2µ aa, E f = E b +E c +E C + p2 cc 2µ cc + p2 bb 2µ bb dσ(a+a b+c+c) = 2π µ aa p aa 1 (2J a +1)(2J A +1) m a,m A m b,m c,m C d 3 p bb (2π ) 3 d 3 p cc (2π ) 3 T (bcc)(aa) 2 δ(ei E f +Q a+a C+c+b ) with d 3 p bb = p 2 bb dp bbdω bb, d 3 p cc = p 2 cc dp ccdω cc, de f /dp bb = p bb /µ bb E cc = p 2 cc /(2µ cc), de cc /dp cc = p cc /µ cc and integration over p bb d 3 σ de cc dω cc dω cb (a+a b+c+c) = µ aaµ bb µ cc (2π) 5 7 p bb p cc p aa 1 (2J a +1)(2J A +1) m a,m A m b,m c,m C integration over unobserved quantities less detailed information T (bcc)(aa) 2 Reaction Theory - 42

55 Transfer Reactions IV T-matrix elements for transfer reactions A+a B +b with a = b+x, B = A+x introduce optical potentials U ij (ij = Aa,Bb) and distorted waves χ (±) ij with (T ij +U ij )χ (±) ij = E ij χ (±) ij post form: T (Bb)(Aa) = φ B φ b χ ( ) Bb ˆV Bb Û Bb Ψ (+) Aa exact! prior form: T (Bb)(Aa) = Ψ ( ) Bb ˆV Aa Û Aa φ A φ A χ (+) Aa exact! approximations for exact scattering wave functions: distorted-wave Born approximation Ψ (+) Aa φ aφ A χ (+) Aa better: Ψ (+) Aa other methods or Ψ( ) Bb from coupled-channel calculation or Ψ ( ) Bb φ Bφ b χ ( ) Bb Reaction Theory - 43

56 Transfer Reactions V introduction of spectroscopic amplitudes/factors introduce overlap functions ˆ= wave function of transferred particle A x B=(A+x) Φ a bx = φ b φ a Φ B Ax = φ A φ B a=(b+x) b (integration only over internal coordinates) approximation Φ a bx Aa bx ϕa bx ( r bx)φ x Φ B Ax AB Ax ϕb Ax ( r Ax)φ x with single-particle wave functions ϕ a bx, ϕa Bx generated from (standard) potentials (usually Woods-Saxon type) for bound states ϕ a bx ϕa bx = ϕa Bx ϕa Bx = 1 and spectroscopic amplitudes A a bx, AB Ax spectroscopic factors S a bx = Aa bx 2 model dependent quantities! S B Ax = A B Ax 2 Reaction Theory - 44

57 Transfer Reactions VI distorted-wave Born approximation (DWBA) and spectroscopic factors approximation for T-matrix elements (still expensive to calculate) post form: T (Bb)(Aa) A B Ax Aa bx ϕb Ax ( r Ax)χ ( ) Bb ˆV Bb ÛBb ϕ a bx ( r bx)χ (+) Aa prior form: T (Bb)(Aa) A B Ax Aa bx ϕb Ax ( r Ax)χ ( ) Bb ˆV Aa ÛAa ϕ a bx ( r bx)χ (+) Aa distorted waves χ (+) Aa, χ( ) Bb from full calculation in partial-wave expansion or eikonal approximation at high energies cross sections dσ T (Bb)(Aa) 2 dσ S a bx SB Ax dσ single particle experimental spectroscopic factors Sbx a, SB Ax from comparison of measured cross sections with single-particle cross sections microscopic nuclear structure models: S a bx = Φa bx Φa bx, SB Ax = ΦB Ax ΦB Ax choice of potentials ˆV aa, ˆV bb, Û aa, Û bb? often approximations interpretation if B = C +c is continuum state? Reaction Theory - 45

58 Applications Coupled-Channel Approach

59 Coupled-Channel Approach I explicit calculation of full scattering wave function expansion Ψ (+) aa = c ψ c with all channels c = a+a, b+b,... and correct asymptotics ψ c φ c f (+) exp(ik c r c ) c(aa) with φ c = φ a φ A, φ b φ B,... r c Hamiltonian Ĥ = Ĥ c + ˆT c + ˆV c with, e.g. Ĥ c = Ĥa+ĤA, ˆT c = ˆT Aa, ˆV c = ˆV aa for c = a+a stationary Schrödinger equation ĤΨ (+) aa = EΨ(+) aa projection on channel wave functions φ c coupled equations φ c ˆT c + ˆV c +E c E ψ c = c φ c Ĥ E ψ c with, e.g., E c = E a +E A for c = a+a in diagonal part Reaction Theory - 46

60 Coupled-Channel Approach II problems infinitely many channels (different excited states and partitions, partial waves) truncation needed choice of relevant channels asymptotic solution in channels with three particles different methods/approximations use hyperspherical coordinates, but no exact solution for three charges particles use product wave function φ bcc φ b Ψ (+) cc with two-body scattering wave function Ψ (+) cc continuum states depend on energy, not normalizable discretize continuum by introducing energy bins Emax φ bcc = φ b de w(e)ψ (+) E min cc (E) with appropriate weight functions w(e) φ bcc normalizable continuum-discretized coupled channels (CDCC) approach Reaction Theory - 47

61 Coupled-Channel Approach III CDCC approach single-particle states and collective states can be considered binning of continuum can be adapted to resonances numerical convergence can be tested computationable expensive, only for not too high energies optical potentials used in calculation of T-matrix elements imaginary part considers loss of flux to open channels consistency with explicit treatment of open channels? often systematic potentials used from fits of elastic scattering cross sections mostly available for scattering of nucleons and light nuclei (d, α,... ) usually not available for exotic nuclei other approaches: e.g. single-folding or double-folding potentials dispersive methods Reaction Theory - 48

62 Summary

63 Summary information on reactions is contained in cross sections depend only on asymptotics of scattering wave functions knowledge of S-matrix elements or T-matrix elements sufficient many different types of reactions and kinematical conditions many methods: partial-wave expansion, R-matrix theory, DWBA, CDCC,... reactions with exotic nuclei direct interest in reaction cross sections (e.g. astrophysics) reactions as tool to study nuclear structure major challenges: adaption of standard methods to specific conditions combination of reaction theory with modern nuclear structure models treatment of (many-body) continuum states choice and consistent application of potentials need for development and many exciting applications in the future Reaction Theory - 49