Radiative-capture reactions P. Descouvemont Physique Nucléaire Théorique et Physique Mathématique, CP229, Université Libre de Bruxelles, B1050 Bruxelles - Belgium 1. Introduction, definitions 2. Electromagnetic transitions 3. Radiative-capture cross sections 4. Radiative capture in nuclear astrophysics 5. Application to the potential model 6. Conclusions 1
1. Introduction - definitions Capture reaction = Electromagnetic transition from a scattering state to a bound state A E i B g E γ E i > 0 A+B Initial scattering state A+B E i >0 C, E f Final bound state: E f <0 In general: several states C E f < 0 Energy conservation: E γ = E i E f (recoil energy of nucleus C is negligible) Examples : (p,g) reactions: 7 Be(p,g) 8 B, 12 C(p,g) 13 N, 16 O(p,g) 17 F (a,g) reactions: 3 He(a,g) 7 Be, 12 C(a,g) 16 O Typical values: up to a few MeV Photon wavelength for E γ = 1 MeV: λ γ = 2π = 2πħc 1200 fm k γ E γ λ γ typical dimension R k γ R 1: long wavelength approximation 2
1. Introduction - definitions Importance in astrophysics: 1. Hydrogen burning: CNO cycle, pp chain CNO cycle(s) Capture reactions Transfer reactions Beta decays 2. Helium burning: 8 Be(a,g) 12 C, 12 C(a,g) 16 O, 16 O(a,g) 20 Ne, 3. Neutron capture reactions: 7 Li(n,g) 8 Li, many others 3
1. Introduction - definitions Difference fusion / radiative capture Radiative capture: g channel only (electromagnetic) Fusion: all channels with mass higher than the projectile and target In general, many channels statistical approach Example: 12 C+ 12 C fusion experimental cross section Satkowiak et al. PRC 26 (1982) 2027 E many states 12 C+ 12 C 4
1. Introduction - definitions Other example: 12 C( 12 C,g) 24 Mg radiative capture Existence of molecular states in 24 Mg? Transitions to 3 states of 24 Mg Cross section much smaller than fusion Rad. cap: ~100 nb *4p ~1 mb (electromagnetic) Fusion: ~200 mb (nuclear) 12 C+ 12 C 24 Mg 5
2. Electromagnetic transitions 6
2. Electromagnetic transitions Transition probability Initial state: energy E i Final state: energy E f g system of charged particles: H = H 0 Wave function Ψ i system of charged particles + photon: H = H 0 + H γ (H γ small) Wave function Ψ f Energy conservation: E i = E f + E γ Perturbation theory (H γ small) Fermi golden rule Transition probability from an initial state to a final state w i f < Ψ f H γ Ψ i > 2 : very general definition 7
2. Electromagnetic transitions Three types of electromagnetic transitions Nucleus C=A+B A B C E i > 0 E i > 0 E = 0 A+B A+B E f > 0 A+B E i < 0 E f < 0 E f < 0 C C C Bound bound E i < 0, E f < 0 scattering bound E i > 0, E f < 0 =radiative capture scattering scattering E i > 0, E f > 0 =bremsstrahlung In all cases w i f < Ψ f H γ Ψ i > 2 But: different types of wave functions Bremsstralhung: transition from continuum to continuum strong convergence problems 8
2. Electromagnetic transitions Bound bound: Initial and final states characterized by spins J i π i and J f π f Transition probability: w i f < Ψ J fπ f H γ Ψ J iπ i > 2 w i f provides the g width Γ γ = ħw, lifetime τ = ħ/γ γ Scattering to bound state: radiative capture Final state: characterized by spin J f π f Initial state : Ψ(E) = scattering state (expansion in partial waves, all J i π i ) Transition probability: w i f (E) < Ψ J fπ f H γ Ψ(E) > 2 w i f provides the capture cross section General property : H γ small electromagnetic processes have a low probability Gamma widths small compared to particle widths Capture cross sections small compared to elastic or transfer cross sections (nuclear origin) g decay: G g =0.5 ev, T=10-14 s proton decay: G p =32 kev, T=2x10-20 s G g /G p <<1 9
2. Electromagnetic transitions S-factor (MeV-b) Comparaison transfer capture reactions: 6 Li(p,a) 3 He and 6 Li(p,g) 7 Be 10 1 6 Li(p,a) 3 He S- Factor S E = σ E Eexp 2πη 0.1 0.01 0.001 factor 10 4 typical of nucl/elec. g channel negligible compared to a channel 6 Li(p,g) 7 Be 0.0001 0.00001 0.01 0.1 1 10 E (MeV) S factor proportional to s Same energy dependence (identical entrance channel) s t (transfer)>> s C (capture) : general property 10
2. Electromagnetic transitions Electromagnetic operator H γ Depends on nuclear coordinates (nuclei or nucleons, according to the model) photon properties (energy E γ, emission angle Ω γ ) Deduced from the Maxwell equations (+quantification) H γ λμσ k γ λ M μ σλ (r 1, r A )D λ μq (Ω γ ) nucleons photon σ = E or M (electric or magnetic) λ=order of the multipole (from 1 to, in practice essentially λ = 1,2) m is between l and +l D λ μq (Ω γ )=Wigner fonction, Ω γ =photon-emission angle M μ σλ = multipole operators (depend on nucleon coordinates r i ) 11
2. Electromagnetic transitions Electric operators M Eλ μ = e ( 1 t 2 iz )r λ i Y μ i λ Ω ri for a many-nucleon system (sensitive to protons only) with t iz =isospin= +1/2 (neutron) -1/2 (proton) r i = r i, Ω i = nucleon space coordinate Magnetic operators with M μ Mλ = μ N ħ [ (r λ Y λ μ Ω r ] r=ri i S i =spin of nucleon i L i =orbital momentum of nucleon i ( 2g li λ+1 L i+g si S i ) Transition probability (integral over Ω γ ) w i f (E) < Ψ J fπ f H γ Ψ J iπ i > 2 H γ is expanded in multipoles W Ji π i J f π f k 2λ+1 γ < Ψ J fπ f M σλ Ψ J iπ i > 2 λσ photon nucleus 12
2. Electromagnetic transitions Reduced transition probability B σλ, J i π i J f π f = 2J f + 1 2J i + 1 < ΨJ fπ f M σλ Ψ J iπ i > 2 Units: electric (s=e): e 2 fm 2λ magnetic (s=m): μ N 2 fm 2λ 2 Gamma width: Γ γ (J i π i J f π f ) = ħw Ji π i J f π f Total gamma width : sum over final states example: 15 O from a given state: several transitions are possible in practice: selection rules factor k γ 2λ+1 favors large E γ 13
2. Electromagnetic transitions Important properties: Hierarchy between the multipoles: w σ,λ+1 w σ,λ E1 E2 M1 E3 M2, only a few multipoles contribute (one) ~ k γ R 2 1 (long wavelength approximation) Selection rules: < Ψ J fπ f M σλ Ψ J iπ i > angular momentum J i J f λ J i + J f parity: π i π f = 1 λ for σ = E π i π f = 1 λ+1 for σ = M E1 forbidden in N=Z nuclei (T=0) M E1 ( 1 2 t iz)(r i R cm ) 3 No transition with λ = 0 Examples: transition 2 + 0 + : E2 transition 1 0 + : E1 transition 2 + 1 + : E2, M1, M3 transition 3 2 + : E1, E3, E5, M2, M4 + 0 2 2 + 0 + E2 E2 E3 14
2. Electromagnetic transitions Capture cross section (for a given final state J f ): σ(j f, E) k γ 2λ+1 < Ψ J fπ f M σλ Ψ(E) > 2 λσ with Ψ(E) =initial state, expanded in partial waves Ψ E = Ψ J iπ i J i (E) E1 (or E2) dominant picks up a few partial waves examples: 12 C(p,g) 13 N, 12 C(a,g) 16 O 12 C(p,g) 13 N A single final state: J f = 1/2 (ground state) Main multipolarity: E1 Initial states: J i = 1/2 +, 3/2 + Resonance for J i = 1/2 + enhances the cross section 15
2. Electromagnetic transitions 12 C+a E 12 C(a,g) 16 O Several final states: (J f = 0 + dominant) E1 multipolarity: J i = 1 E2 multipolarity: J i = 2 + Cascade transitions small (factor k γ 2λ+1 ) N=Z=6 E1 forbidden (T=0, long wavelength approx.) Exp: E1 E2 (T=0 not strict) 16
3. Cross sections for radiative capture 17
3. Radiative-capture cross sections We assume a spin zero for the colliding nuclei σ(j f, E) k γ 2λ+1 < Ψ J fπ f M σλ Ψ(E) > 2 λσ General definition, valid for any model. Potential model (structure neglected) Microscopic models R-matrix, etc. Spins zero for the colliding nuclei: Ψ E = Ψ J (E) J Essentially astrophysics applications low energies low J values At low scattering energies J i = 0 + is dominant (centrifugal barrier) J f = 1 E1, J f = 2 + E2 are expected to be dominant J > 0 J = 0 E small penetrability decreases with J 18
3. Radiative-capture 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. 19
3. Radiative-capture 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 20
Radiative capture in astrophysics 21
4. Radiative capture in astrophysics Elastic scattering is always possible, but does not affect the nucleosynthesis Essentially two types of reactions Transfer Capture capture is important only if transfer channels are closed 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 22
4. Radiative capture in astrophysics exp(-2*pi*eta) Low energies cross sections dominated by coulomb effects Coulomb functions at low energies (h=sommerfeld parameter=z 1 Z 2 e 2 / v) F(h,x) exp(-ph)f(x), G(h,x) exp(ph)g(x) 2 coulomb effects: strong E dependence : factor exp(-2ph) strong l dependence V(r) E (MeV) 0.1 0 0.5 1 1.5 2 2.5 0.001 1E-05 r E astro 1E-07 1E-09 a+ 3 He 1E-11 Astrophysical S factor: S(E)=s(E)*E*exp(2ph) (Units: E*L 2 : MeV-barn) removes the coulomb dependence only nuclear effects weakly depends on energy s(e) S 0 exp(-2ph)/e 23
4. Radiative capture in astrophysics Penetration factor P l for d+d (2 different radii, a=5 fm and a=6 fm) Typical Coulomb effect depends on the radius strongly depends on energy E, and on angular momentum l P l (E) exp 2πη for l = 0 1.E+01 1.E+00 l=0 l=2 P l 1.E-01 l=4 1.E-02 d+d 1.E-03 0 2 4 6 8 10 12 E (MeV) 24
4. Radiative capture in astrophysics S (kev-b) s (barn) Non resonant: S(E)=s(E)*E*exp(2ph) approximately constant 10-5 10-6 10-7 10-8 10-9 0.8 0.6 10-10 Parker et al. 0 1 2 3 4 6 Li+p a+ 3 He 3 He(a,g) 7 Be 5/2-5/2-7/2-1/2-3/2- No resonance with low J 0.4 7 Be 0.2 0 Parker et al. 0 1 2 3 4 E (MeV)
4. Radiative capture in astrophysics Resonances: the Breit-Wigner approximation General properties of a resonance: spin J R Energy E R Entrance width (here: particle width) Output width (here: gamma width) J R = 1/2 + E R = 0.42 MeV Γ p = 32 kev Γ γ = 0.5 ev Breit-Wigner approximation near a resonance: E E R : σ E π k 2 2J Γ γ E Γ p E R + 1 E E 2 R + Γ E 2 /4 Valid for the resonant partial wave J R possible contribution of other partial waves Two typical examples: 12 C(p,g) 13 N: a resonance is present in the dominant partial wave l = 0 7 Be(p,g) 8 B: resonance in the l = 1 partial wave superposition of resonant and non-resonant contributions 26
4. Radiative capture in astrophysics S-factor (kev-b) Example 1: 12 C(p,g) 13 N J f =1/2 - E1 J i =1/2 +, l i =0 dominant J i =3/2 +, l i =2 Resonance for J i =1/2 + enhancement Two effects make J i =1/2 + dominant 1000 100 12 C(p,g) 13 N 12 C+p 5/2+ 3/2-1/2+ 10 1 0 0.2 0.4 0.6 13 N 1/2-27
S-factor (ev-b) 4. Radiative capture in astrophysics Example 2: 7 Be(p,g) 8 B J f =2 + E1 J i =1 -, l i =0,2 1 -,2 - dominant J i =2 -, l i =0,2,4 J i =3 -, l i =2,4 Resonance for J i =1 + only E2 or M1 are possible l i =1,3 limited effect of the 1 + resonance Both contributions must be treated separately 200 150 7 Be(p,g) 8 B 3+ 100 50 0 0 1 2 3 7 Be+p 8 B 1+ 2+ 28
5. Radiative capture in the potential model 29
5. 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 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 30
5. 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 with parameters adjusted on important properties (bound-state energy, phase shifts, etc.) Potentials can be different in the initial and final states Numerical method: Numerov (finite differences, based on a step h and N points) u l 0 = 0 u l h = 1 u l 2h is determined from u l 0 and u l h u l Nh u l r is adjusted on u l r = F J kr cos δ J + G J kr sin δ J at large distances phase shift the wave function is available in a numericall format any matrix element can be computed 31
5. 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 = ez eff 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 > u Ji r 0 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 u Ji r u Jf r r λ dr 32
5. 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 u Ji (r, E) 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 0 λ+1 2λ+1 λ 2λ+1!! 2 Final state E f, J f wave function: u Jf u Jf r r λ dr 2 r Normalization final state (bound): normalized to unity u J r Cexp ( k B r) initial state (continuum): u J r F J kr cos δ J + G J kr sin δ J Test: λ = 0 provides σ λ E = 0 (orthogonality of the wave functions) 33
5. Radiative capture in the potential model Integrated vs differential cross sections Total (integrated) cross section: σ E = σ λ E λ no interference between the multipolarities Differential cross section: dσ dθ = λ a λ E P λ (θ) P λ (θ)=legendre polynomial interference effects angular distributions are necessary to separate the multipolarities (in general one multipolarity is dominant) 2 34
5. Radiative capture in the potential model S (MeV-b) 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+) 1.00E+00 1.00E-01 1.00E-02 1.00E-03 0 0.1 0.2 0.3 0.4 0.5 0.6 E (MeV) Potential : V= -55.3*exp(-(r/2.70) 2 ) (final state) -70.5*exp(-(r/2.70) 2 ) (initial state) 35
5. Radiative capture in the potential model Final state: Jf=1/2 - Initial state: li=0 Ji=1/2 + E1 transition 1/2 + 1/2-1.60E-02 1.40E-02 1.20E-02 1.00E-02 8.00E-03 6.00E-03 4.00E-03 2.00E-03 Initial wave function: Ei=100 kev 0.00E+00 0 5 10 15 20 25 30-2.00E-03-4.00E-03 5.00E-03 E 12C+p 4.00E-03 3.00E-03 2.00E-03 Integrant Ei=100 kev 1.00E-03 13 N,J=1/2-0.00E+00 0 5 10 15 20 25 30-1.00E-03-2.00E-03 7.00E-01 1.40E+00 6.00E-01 5.00E-01 4.00E-01 Final state Ef=- 1.94 MeV 1.20E+00 1.00E+00 8.00E-01 6.00E-01 Integral Ei=100 kev 3.00E-01 4.00E-01 2.00E-01 2.00E-01 1.00E-01 0.00E+00 0 5 10 15 20 25 30 0.00E+00 0 5 10 15 20 25 30-2.00E-01 36
5. Radiative capture in the potential model S (MeV-b) The calculation is repeated at all energies 1.00E+00 1.00E-01 1.00E-02 S 1 1.00E-03 S 0.45 1.00E-04 0 0.2 0.4 0.6 0.8 1 E (MeV) 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 37