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Pierre Descouvemont Université Libre de Bruxelles, Brussels, Belgium

R-matrix theory of nuclear reactions. Pierre Descouvemont Université Libre de Bruxelles, Brussels, Belgium. Introduction General collision theory: elastic scattering Phase-shift method R-matrix theory (  theory) Phenomenological R matrix (  experiment) Conclusions.

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Pierre Descouvemont Université Libre de Bruxelles, Brussels, Belgium

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  1. R-matrix theory of nuclear reactions Pierre DescouvemontUniversité Libre de Bruxelles, Brussels, Belgium • Introduction • General collision theory: elastic scattering • Phase-shift method • R-matrix theory ( theory) • Phenomenological R matrix ( experiment) • Conclusions

  2. General context: two-body systems • Low energies (E ≲ Coulomb barrier), few open channels (one) • Low masses (A ≲ 15-20) • Low level densities (≲ a few levels/MeV) • Reactions with neutrons AND charged particles Introduction ℓ>0 V E Only a few partial waves contribute r ℓ=0

  3. Low-energy reactions • Transmission (“tunnelling”) decreases with ℓ  few partial waves  semi-classic theories not valid • astrophysics • thermal neutrons • Wavelengthλ=ℏc/Eλ(fm)~197/E(MeV) example: 1 MeV: λ ~200 fm radius~2-4 fm quantum effects important • Some references: • C. Joachain: Quantum Collision Theory, North Holland 1987 • L. Rodberg, R. Thaler: Introduction to the quantum theory of scattering, Academic Press 1967 • J. Taylor : Scattering Theory: the quantum theory on nonrelativistic collisions, John Wiley 1972

  4. Different types of reactions 1. Elastic collision : entrance channel=exit channel A+B  A+B: Q=0 2. Inelastic collision (Q≠0) A+B  A*+B (A*=excited state)A+B  A+B*A+B  A*+B* etc.. 3. Transfer reactions A+B  C+DA+B  C+D+EA+B  C*+D etc… 4. radiativecapture reactions A+B C + g covered here NOT covered here

  5. 2. Collision theory: elastic scattering Assumptions: elastic scattering no internal structure no Coulomb spins zero 2 q 1 2 1 Before collision After collision Center-of-mass system

  6. Scattering wave functions Schrödinger equation: V(r)=interaction potential Assumption: the potential is real and decreases faster than 1/r A large distances : V(r) 0 2 independent solutions : Incoming plane wave Outgoing spherical wave where: k=wave number: k2=2mE/2 A=amplitude (scattering wave function is notnormalized) f(q) =scattering amplitude (length)

  7. Cross sections q solid angle dW Cross section • Cross section obtained from the asymptotic part of the wave function • “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,

  8. How to solve the Schrödinger equation for E>0? • With (z along the beam axis) • Several methods: • Formal theory: Lippman-Schwinger equation  approximations • Eikonal approximation • Born approximation • Phase shift method: well adapted to low energies • Etc…

  9. Lippman-Schwinger equation: is solution of the Schrödinger equation With =Green function r is supposed to be large (r >> r') • Y(r) must be known • Y(r) only needed where V(r)≠0  approximations are possible • Born approximation : • Eikonal approximation: Valid at high energies:V(r) ≪ E

  10. 3. Phase-shift method • Definitions, cross sections Simple conditions: neutral systems spins 0 single-channel • Extension to charged systems • Extension to multichannel problems • Low energy properties • General calculation • Optical model

  11. The wave function is expanded as 3.a Definition, cross section When inserted in the Schrödinger equation To be solved for any potential (real) For r∞ V(r) =0 =Bessel equation  ul(r)= jl(kr), nl(kr)

  12. For small x For large x Examples: At large distances: ul(r) is a linear combination of jl(kr) and nl(kr): With dl = phase shift (information about the potential)If V=0  d=0 Cross section: Provide the cross section

  13.  factorization of the dependences in E and q • General properties of the phase shifts • Expansion useful at low energies: small number of ℓ values • The phase shift (and all derivatives) are continuous functions of E • The phase shift is known within np: exp(2id)=exp(2i(d+np)) • Levinson theorem • d(E=0) - d(E=)= Np , where N is the number of bound states • Example: p+n, ℓ=0: d(E=0) - d(E=)=p (bound deuteron)

  14. Interpretation of the phase shift from the wave function V=0 u(r)sin(kr-lp/2)dl=0 u(r) r V≠0 u(r)sin(kr-lp/2+dl) dl<0 V repulsive dl>0 V attractive

  15. d(E) p 3p/4 p/2 p/4 E ER ER-G/2 ER+G/2 Resonances: =Breit-Wigner approximation ER=resonance energyG=resonance width • Narrow resonance: G small • Broad resonance: G large Example: 12C+p With resonance: ER=0.42 MeV, G=32 keV lifetime: t=}/G~2x10-20 s Without resonance: interaction range d~10 fm interaction time t=d/v ~1.1x10-21 s

  16. 3.b Generalization to the Coulomb potential The asymtotic behaviour Y(r) exp(ikz) + f(q)*exp(ikr)/r Becomes: Y(r) exp(ikz+h log(kr)) + f(q)*exp(ikr-h log(2kr))/r With h=Z1Z2e2/ℏv =Sommerfeld parameter, v=velocity Bessel equation: Coulomb equation Solutions: Fl(h,kr): regular, Gl(h,kr): irregular Ingoing, outgoing functions: Il= Gl-iFl, Ol=Gl+iFl

  17. Example: hard sphere V(r) V(r)= ∞ for r<a =Z1Z2e2/r for r>a a r = Hard-Sphere phase shift

  18. Special case: Neutrons, with l=0: F0(x)=sin(x), G0(x)=cos(x)  d=-ka example : a+n phase shift l=0  hard sphere is a good approximation

  19. Elastic cross section with Coulomb: Still valid, but converges very slowly. Coulomb: exact Nuclear = Coulomb cross section: diverges for q=0 increases at low energies

  20. fC(q): Coulomb part: exact • fN(q): nuclear part: converges rapidly • The total Coulomb cross section is not defined (diverges) • Coulomb is dominant at small anglesused to normalize data • Increases at low energies • Minimum at q=180o  nuclear effect maximum

  21. 3.c Extension to multichannel problems • One channel: phase shift d U=exp(2id) • Multichannel: collision matrix Uij, (symmetric , unitary)with i,j=channels • Good quantum numbers J=total spin p=total parity • Channel i characterized by I=I1 I2 =channel spin J=Ill =angular momentum • Selection rules: |I1- I2| ≤ I ≤ I1+ I2 |l - I| ≤J ≤l+ I p=p1*p2* (-1)l Example of quantum numbers a+3He a=0+, 3He=1/2+ p+7Be 7Be=3/2-, p=1/2+ X X X X X X X X X X X X

  22. Cross sections One channel: Multichannel With: K1,K2=spin orientations in the entrance channel K’1,K’2=spin orientations in the exit channel • Collision matrix • generalization of d: Uij=hijexp(2idij) • determines the cross section

  23. 3.d Low-energy properties of the phase shifts One defines =logarithmic derivative at r=a (a large) Then, for very low E (neutral system): For ℓ=0: d l=0, a<0 l>0 strong difference between l=0 and l>0 E l=0, a>0 Generalization a=scattering lengthre=effective range

  24. Cross section at low E: In general (neutrons): Thermal neutrons (T=300K, E=25 meV): only l=0 contributes example : a+n phase shift l=0 At low E: d~-ka with a>0  replusive

  25. 3.e General calculation For some potentials: analytic solution of the Schrödinger equation In general: no analytic solution  numerical approach with: • Numerical solution : discretization N points, with mesh size h • ul(0)=0 • ul(h)=1 (or any constant) • ul(2h) is determined numerically from ul(0) and ul(h) (Numerov algorithm) • ul(3h),… ul(Nh) • for large r: matching to the asymptotic behaviour  phase shift • Bound states: same idea

  26. Experimental phase shifts Example: a+a Experimental spectrum of 8Be 4+E~11 MeVG~3.5 MeV Potential: VN(r)=-122.3*exp(-(r/2.13)2) 2+E~3 MeVG~1.5 MeV VB~1 MeV a+a 0+E=0.09 MeVG=6 eV

  27. 3.f Optical model Goal: to simulate absorption channels • High energies: • many open channels • strong absorption • potential model extended to complex potentials (« optical ») d+18F n+19Ne p+19F Phase shift is complex: d=dR+idIcollision matrix: U=exp(2id)=hexp(2idR) whereh=exp(-2dI)<1 Elastic cross section Reaction cross section: a+16O 20Ne

  28. 4. The R-matrix Method • Goals: • To solve the Schrödinger equation (E>0 or E<0) • Potential model • 3-body scattering • Microscopic models • Many applications in nuclear and atomic physics • To fit cross sections • Elastic, inelastic  spectroscopic information on resonances • Capture, transfer  astrophysics • References: • A.M. Lane and R.G. Thomas, Rev. Mod. Phys. 30 (1958) 257 • F.C. Barker, many papers

  29. Principles of the R-matrix theory • Standard variational calculations • Hamiltonian • Set of N basis functions ui(r) with • Calculation of Hij=<ui|H|uj> over the full space Nij=<ui|uj> • (example: gaussians: ui(r)=exp(-(r/ai)2)) • Eigenvalue problem :  upper bound on the energy • But: Functions ui(r) tend to zero  not directly adapted to scattering states

  30. Principles of the R-matrix theory Extension to the R-matrix: includes boundary conditions a. Hamiltonian b. Wave functionsSet of N basis functions ui(r)  correct asymptotic behaviour (E>0 and E<0)

  31. Spectroscopy: short distances onlyCollisions : short and large distances • R matrix: deals with collisions • Main idea: to divide the space into 2 regions (radius a) • Internal: r ≤ a : Nuclear + coulomb interactions • External: r > a : Coulomb only Exit channels 12C(2+)+a Entrance channel 12C+a Internal region 16O 12C+a 15N+p, 15O+n Nuclear+Coulomb:R-matrix parameters Coulomb Coulomb

  32. Here: elastic cross sections Cross sections measurements Phase shifts R matrix models Microscopic models Etc,…. Potentialmodel

  33. Plan of the lecture: • The R-matrix method: two applications • Solving the Schrödinger equation (essentially E>0) • Phenomenological R-matrix (fit of data) • Applications • Other reactions: capture, transfer, etc.

  34. c. R matrix • N basis functions ui(r) are valid in a limited range (r ≤ a) in the internal region: (ci=coefficients) • At large distances the wave function is Coulomb in the external region: • Idea: to solve • with defined in the internal region • But T is not hermitian over a finite domain,

  35. Bloch-Schrödinger equation: • With L(L0)=Bloch operator (surface operator) – constant L0 arbitrary • Now we have T+L(L0) hermitian: • Since the Bloch operator acts at r=a only: • Here L0=0 (boundary-condition parameter)

  36. Summary: • Using (2) in (1) and the continuity equation: Provides ci(inversion of D) Dij(E) Continuity at r=a

  37. With the R-matrix defined as If the potential is real: R real, |U|=1, d real Procedure (for a given ℓ): • Compute matrix elements • Invert matrix D • Compute R matrix • Compute the collision matrix U

  38. S(E)=shift factorP(E)=penetration factor Charged system: Neutral system: a + 3He a + n Pl Sl E (MeV)

  39. Interpretation of the R matrix From (1): Then: =inverse of the logarithmic derivative at a

  40. Example Gaussians with different widths l=0 Can be done exactly(incomplete g function) Matrix elements of H can be calculated analytically for gaussian potentials Other potentials: numerical integration

  41. Input data • Potential • Set of n basis functions (here gaussians with different widths) • Channel radius a • Requirements • a large enough : VN(a)~0 • n large enough (to reproduce the internal wave functions) • n as small as possible (computer time) compromise • Tests • Stability of the phase shift with the channel radius a • Continuity of the derivative of the wave function

  42. Results for a+a • potential : V(r)=-126*exp(-(r/2.13)2) (Buck potential) • Basis functions: ui(r)=rl*exp(-(r/ai)2) • with ai=x0*a0(i-1) (geometric progression) typically x0=0.6 fm, a0=1.4 potential VN(a)=0  a > 6 fm

  43. Elastic phase shifts • a=10 fm too large (needs too many basis functions) • a=4 fm too small (nuclear not negligible) • a=7 fm is a good compromise

  44. Wave functions at 5 MeV, a= 7 fm 0 0 5 10 15 20 25 -50 del exact n=15 -100 n=10 -150 n=8 n=7 -200 -250 -300 a=7 fm -350 -400

  45. Other example : sine functions • Matrix elements very simple • Derivative ui’(a)=0  Not a good basis (no flexibility)

  46. Example of resonant reaction: 12C+p Resonance l=0E=0.42 MeVG=32 keV • potential : V=-70.5*exp(-(r/2.70)2) • Basis functions: ui(r)=rl*exp(-(r/ai)2)

  47. Wave functions Phase shifts a=7 fm

  48. Resonance energies with Hard-sphere phase shift R-matrix phase shift • Resonance energy Er defined by dR=90o • In general: must be solved numerically • Resonance width G defined by

  49. Application of the R-matrix to bound states Positive energies: Coulomb functions Fl, Gl Negative energies: Whittaker functions Asymptotic behaviour: Starting point of the R matrix:

  50. R matrix equations With C=ANC (Asymptotic Normalization Constant): important in “external” processes • Using (2) in (1) and the continuity equation: Dij(E) • But: L depends on the energy, which is not known iterative procedure

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