1 Direct reactions methodologies for use at fragmentation beam energies TU Munich, February 14 th 2008 Jeff Tostevin, Department of Physics Faculty of Engineering and Physical Sciences University of Surrey, United Kingdom
The University of Surrey 2 Guildford, Surrey
Exotic nuclei presence of two Fermi surfaces 3 n 38 Si p Spherical Hartree Fock density (SkX)
Skyrme Hartree-Fock radii and densities 4 W.A. Richter and B.A. Brown, Phys. Rev. C67 (2003) 034317
5 Probing single particle (shell model) states One or two-nucleon removal at >100 MeV/nucleon J i 1 j 1 I,T j 2 2 9 Be J f c [fast] exotic projectile Experiments do not measure target final states. Final state of core c measured using decay gamma rays. P.G. Hansen and J.A. Tostevin, Ann Rev Nucl Part Sci 53 (2003) 219
Sampling the single-nucleon wave function 6 b R C + R T Interaction with the target probes wave functions at surface A target J i z
7 Single-neutron knockout momentum distributions =0,2 admixture =0,2 admixture pure =2 V. Maddalena et al. Phys. Rev. C 63 (2001) 024613
8 The force between nucleons is complicated Argonne v18 18 spin, angular momentum, ispin operators attractive
9 Implications for shell structure: asymmetric nuclei Attractive interaction between neutrons and protons occupying j > and j < levels, repulsive j> and j> levels Takaharu Otsuka et al, Phys. Rev. Lett. 87, 082502 (2001)
10 Magic numbers change with neutron richness 16 20 protons 8 10 12 neutrons 15 15 15
11 Migration of single particle levels for N=7 E*- Sn [MeV] 2 0-2 -4-6 -8 15 O 14 N 13 C 12 B 11 Be 10 Li 9 He removing protons 15 O r [fm] 0 10 1/2 + 8 p 11 Be r [fm] 0 10 20 n -10-12 -14 1/2 - r [fm] 10 Li 0.5 MeV 0 10 20 30 40 8 7 6 5 4 3 2 Atomic Number Z From: P.G. Hansen and J.A. Tostevin, Ann Rev Nucl Part Sci 53 (2003) 219
Otsuka: the np interaction tensor correlation 12 n 38 Si p Spherical Hartree Fock density (SkX)
A little scattering theory 13
Large r: The Asymptotic Normalisation Coefficient 14 Bound states but beyond the range of the nuclear forces, then Whittaker function ANC completely determines the wave function outside of the range of the nuclear potential only requirement if a reaction probes only these radii
Large r: The phase shift and partial wave S-matrix 15 Scattering states and beyond the range of the nuclear forces, then regular and irregular Coulomb functions
16 Ingoing and outgoing waves amplitudes 0
Eikonal approximation: point particles (1) 17 Approximate (semi-classical) scattering solution of assume small wavelength valid when high energy Key steps are: (1) the distorted wave function is written all effects due to U(r), modulation function (2) Substituting this product form in the Schrodinger Eq.
Eikonal approximation: point neutral particles (2) 18 The conditions imply that Slow spatial variation cf. k and choosing the z-axis in the beam direction phase that develops with z with solution b r 1D integral over a straight z line path through U at the impact parameter b
Eikonal approximation: point neutral particles (3) 19 So, after the interaction and as z Eikonal approximation to the S-matrix S(b) S(b) is amplitude of the forward going outgoing waves from the scattering at impact parameter b b r z Moreover, the structure of the theory generalises simply to few-body projectiles
Eikonal approximation: point particles 20 b z limit of range of finite ranged potential
Semi-classical models for the S-matrix - S(b) 21 b=impact parameter b k, for high energy/or large mass, semi-classical ideas are good kb, actually +1/2 1 absorption 1 b transmission
Point particle scattering cross sections 22 All cross sections, etc. can be computed from the S-matrix, in either the partial wave or the eikonal (impact parameter) representation, for example (spinless case): etc. and where (cylindrical coordinates) b z
Core-target effective interactions 23 Double folding V AB VAB (R) dr1 dr2 ρ A(r1 ) ρb (r2 ) t NN ( R + r2 r1 ) A ρ (r) A = R B ρ B (r) At higher energies for nucleus-nucleus or nucleon-nucleus systems first order term of multiple scattering expansion nucleon-nucleon cross section resulting in a COMPLEX nucleus-nucleus potential M.E. Brandan and G.R. Satchler, The Interaction between Light Heavy-ions and what it tells us, Phys. Rep. 285 (1997) 143-243.
Effective interactions Folding models 24 Double folding VAB Single folding VB VAB (R) dr1 dr2 ρ A(r1 ) ρb (r2 ) v NN ( R + r1 r2 ) A ρ (r) A = r 1 R R + r 1 r 2 R r 2 V (R) = B dr2 ρb (r2 ) vnn ( R r2 ) r 2 B ρ B (r) B R r 2 ρ B (r)
25 Sizes - Skyrme Hartree-Fock radii and densities B.A. Brown, S. Typel, and W.A. Richter, Phys. Rev. C65 (2002) 014612
Adiabatic (sudden) approximations in physics 26 Identify high energy/fast and low energy/slow degrees of freedom Fast neutron scattering from a rotational nucleus fast slow Ω E 3, 3 E 2, 2 E 1, 1 E 0, 0 E 0 Fix Ω, Ω, calculate scattering amplitude f(θ, Ω) Ω) for for each (fixed) Ω. Ω. moment of of inertia and rotational spectrum is is assumed degenerate Transition amplitudes ff αβ αβ (θ) = β f(θ, Ω) α Ω
27 Few-body projectiles the adiabatic model c r v slow R fast ε k φ k (r) Full spectrum of H p is assumed degenerate with the ground state ε 0 ε 0 Freeze internal co-ordinate r then scatter c+v from target and compute f(θ,r) for all required fixed values of r Physical amplitude for breakup to state φ k (r is then, f k (θ) = φ k f(θ, r) φ 0 r ) Achieved by by replacing H p ε 0 in in Schrödinger equation
28 Adiabatic approximation - time perspective The time-dependent equation is Ψ HΨ( r, R, t) = i t and can be written + Ψ( r, R, t) = Λ Φ( r(t), R), r(t) = Λ r Λ = exp{ i(hp + ε0)t/ } and where [TR + U( r(t), R) ε0] Φ( r(t), R) = i Adiabatic equation [TR + U( r, R)] Φ( r, R) = 0 Λ Φ t ( E + ε ) Φ( r, R) r c R v Adiabatic step assumes r(t) r(0)=r=fixed or or Λ=1 for for the collision time tt coll coll requires (H coll p + ε0)t / << 1
Adiabatic approximation: composite projectile 29 b 1 b 2 z Total interaction energy with composite systems: get products of the S-matrices
Few-body eikonal model amplitudes So, after the collision, as Z ω r, R) = S (b ) S (b ) Ψ Eik K ik R ( r, R) e Sc(bc) Sv(bv) φ0 ( c c v v ( r) 30 with S c and S v the eikonal approximations to the S-matrices for the independent scattering of c and v from the target - the dynamics c v at fixed r adiabatic b b c b v So, elastic amplitude (S-matrix) for the scattering of the projectile at an impact parameter b - i.e. The amplitude that it emerges in state is φ 0 ( r) Sp (b) = φ0 Sc(bc) Sv (bv ) φ0 r averaged over position probabilities of c and v amplitude that c,v survive interaction with b c and b v
31 Eikonal theory - dynamics and structure c Independent scattering information of c and v from target v dynamics φ α S c S v S αβ (b) = φβ Sc (bc) Sv (bv ) φα structure Use the best available few- or many-body wave functions More generally, S αβ (b) = ϕβ S1 (b1) S2(b2)... Sn (bn ) ϕα for any choice of 1,2,3,.. n clusters for which a most realistic wave function ϕ is available
Stripping of a nucleon 32 1 [ j 1 s] 1 A 2 2 σ strip = db φ0 SC (1 S1 ) φ0
Absorptive cross sections - target excitation 33 Since our effective interactions are complex all our S(b) include the effects of absorption due to inelastic channels 2 σabs = σr σdiff = db φ0 1 ScSv φ0 S S v c 2 2 (1 S c (1 S (1 S 2 c v 2 2 )(1 S ) ) v + + 2 ) v survives, c absorbed v absorbed, c survives v absorbed, c absorbed σstrip 0 2 2 = db φ0 Sc (1 Sv ) φ S(b) 2 1 stripping of v from projectile exciting the target. c scatters at most elastically with the target Related equations exist for the differential cross sections, etc.
34 Diffractive (breakup) removal of a nucleon 1 1 1 ] [ j s A { } S S S S d 2 0 v c 0 0 2 v c 0 diff = φ φ φ φ σ b
35 Diffractive dissociation of composite systems The total cross section for removal of the valence particle from the projectile due to the break-up is the sum over all final continuum states σ diff = dk db φk Sc(bc) Sv(bv) φ0 but, using completeness of the break-up states dk φk φk = 1 φ0 φ0 φ1 φ1... { 2 2 φ S S φ φ S S φ } σ = diff d b 0 c v 0 0 c v 0 2 If > 1 bound state can (for a weakly bound system with a single bound state) be expressed in terms of only the projectile ground state wave function as:
Bound states spectroscopic factors 36 In a potential model it is natural to define normalised bound state wave functions. The potential model wave function approximates the overlap function of the A and A 1 body wave functions (A and A n in the case of an n-body cluster) i.e. the overlap S( ) is the spectroscopic factor a structure calculation
Bound states microscopic overlaps 37 p Microscopic overlap from Argonne 9- and 8-body wave functions (Bob Wiringa et al.) Available for a few cases Normalised bound state in Woods-Saxon potential well x (0.23) 1/2 Spectroscopic factor
38 Bound states shell model overlaps USDA sd-shell model overlap from e.g. OXBASH (Alex Brown et al.). Provides spectroscopic factors but not the bound state radial wave function. p p
39 Bound states use mean field information But must make small correction as HF is a fixed centre calculation 17 17 O
We have the background we need to apply this in a serious context and confront experiment 40