Wolfram KORTEN 1 Euroschool Leuven – September 2009 Coulomb excitation with radioactive ion beams Motivation and introduction Theoretical aspects of Coulomb excitation Experimental considerations, set-ups and analysis techniques Recent highlights and future perspectives Lecture given at the Euroschool 2009 in Leuven Wolfram KORTEN CEA Saclay

Wolfram KORTEN 2 Euroschool Leuven – September 2009 Shell structure of atomic nuclei protons neutrons no 28 O ! stable double-magic nuclei 4 He, 16 O, 40 Ca, 48 Ca, 208 Pb radioactive: 56 Ni, 132 Sn, magic ? 48 Ni, 78 Ni, 100 Sn 70 Ca ? 48 Ca 40 Ca 42 Si 32 Mg

Wolfram KORTEN 3 Euroschool Leuven – September 2009 protons neutrons Changes in the nuclear shell structure 110 Zr Sn 82 “drip line” g 9/2 s 1/2 d 5/ h 11/2 d 3/2 g 7/2 f 7/2 N/Z g 9/2 s 1/2 d 5/2 82 stabil 50 f 7/2 h 11/2 d 3/2 g 7/2 p 1/2 Spin-orbit term of the nuclear int. ? Only observable in heavy nuciei: (Z,N) > 40

Wolfram KORTEN 4 Euroschool Leuven – September 2009 Changes in the nuclear shell structure Example: “Island of inversion” around 32 Mg  deformed ground state N E(2 + ) [MeV] 12 Mg 16 S 20 Ca N=20 N/Z 40 Ca 38 Ar 36 S 34 Si 32 Mg 30 Ne B(E2) [e 2 fm 4 ] N=20 sd sd+fp Masses and separation energies Observables: Properties of 2 + states

Wolfram KORTEN 5 Euroschool Leuven – September 2009 Shapes of atomic nuclei protons neutrons The vast majority of all nuclei shows a non-spherical mass distribution Z, N = magic numbers Closed shell = spherical shape Deformed Spherical sngle particle enegies elongation Nilssondiagramm Oblate Prolate

Wolfram KORTEN 6 Euroschool Leuven – September 2009 Quadrupole deformation of nuclei Coulomb excitation can, in principal, map the shape of all atomic nuclei M. Girod, CEA

Wolfram KORTEN 7 Euroschool Leuven – September 2009 Quadrupole deformation of nuclei M. Girod, CEA N=Z Oblate deformed nuclei are far less abundant than prolate nuclei Shape coexistence possible for certain regions of N & Z Prolate Oblate Pb & Bi N~Z Fission fragments

Wolfram KORTEN 8 Euroschool Leuven – September 2009 Shape variations and intruder orbitals single-particle energy (Woods-Saxon) quadrupole deformation ND 235 U SD 152 Dy Z=48 HD 108 Cd  i 13/2  (N+1) intruder  normal deformed, e.g. 235 U  (N+2) super-intruder  Superdeformation, e.g. 152 Dy, 80 Zr  (N+3) hyper-intruder  Hyperdeformation in 108 Cd, ? N+2 shell N+3 shell N shell N+1 shell Fermi level Energy Deformation

Wolfram KORTEN 9 Euroschool Leuven – September 2009 Exotic shapes and “deformation” parameters quadrupole octupole hexadecapole oblate non-collective prolate collective  2 : elongation  prolate non-collective Lund convention spherical oblate collective  : triaxiality Generic nuclear shapes can be described by a development of spherical harmonics    deformation parameters Tetrahedral Y 32 deformation Dynamic  vibration Static  rotation Triaxial Y 22 deformation

Wolfram KORTEN 10 Euroschool Leuven – September 2009 Coulomb excitation – an introduction

Wolfram KORTEN 11 Euroschool Leuven – September 2009 Rutherford scattering – some reminders Elastic scattering of charged particles (point-like  monopoles) under the influence of the Coulomb field F C = Z 1 Z 2 e 2 /r 2 with r(t) = |r 1 (t) – r 2 (t)|  hyperbolic relative motion of the reaction partners Rutherford cross section d  /d  Z 1 Z 2 e 2 /E cm 2 sin -4 (  cm /2) b projectile target r(t) a  beam energy b = impact parameter

Wolfram KORTEN 12 Euroschool Leuven – September 2009 Coulomb excitation – some basics Nuclear excitation by the electromagnetic interaction acting between two colliding nuclei. b projectile target Target and projectile excitation possible often heavy, magic nucleus (e.g. 208 Pb)  as projectile  target Coulomb excitation  as target  projectile Coulomb excitation (important technique for radioactive beams) small impact parameter  back scattering  close approach  strong EM field large impact parameter  forward scattering  large distance  weak EM field Coulomb trajectories only if the colliding nuclei do not reach the “Coulomb barrier”  purely electromagnetic process, no nuclear interaction, calculable with high precision

Wolfram KORTEN 13 Euroschool Leuven – September 2009 Reactions below the Coulomb barrier V C = Z 1 Z 2 e 2 /r b (1-a/r b ) r b = 1.07(A 1 1/3 +A 2 1/3 ) fm a=0.63 fm (surface diffuseness)  Beam energy per nucleon (E B /A) rather constant, e.g. for 208 Pb target 5.6 to 6 A.MeV  exp /  Ruth Pure Coulomb excitation requires a much larger distance between the nuclei  ”safe energy” requirement  Inelastic scattering (e.m/nucl.)  Transfer reactions  Fusion

Wolfram KORTEN 14 Euroschool Leuven – September 2009 „Safe“ energy requirement Rutherford scattering only if the distance of closest approach is large compared to nuclear radii + surfaces: „Simple“ approach using the liquid-drop model D min  r s = [1.25 (A 1 1/3 + A 2 1/3 ) + 5] fm More realistic approach using the half-density radius of a Fermi mass distribution of the nuclei : C i = R i (1-R i -2 ) with R = 1.28 A 1/ A -1/3  D min  r s = [ C 1 + C 2 + S ] fm b projectile target D min 

Wolfram KORTEN 15 Euroschool Leuven – September 2009 Empirical data on surface distance S as function of half-density radii C i require distance of closest approach S > fm  choose adequate beam energy (D > D min for all  ) low-energy Coulomb excitation  limit scattering angle, i.e. select impact parameter b > D min, high-energy Coulomb excitation „Safe“ energy requirement  exp /  Ruth  D min

Wolfram KORTEN 16 Euroschool Leuven – September 2009 He O Pb Ar Validity of classical Coulomb trajectories b=0 projectile target Sommerfeld parameter  >> 1 requirement for a semi classical treatment of equations of motion  measures the strength of the monopole-monopole interaction  equivalent to the number of exchanged photons needed to force the nuclei on a hyperbolic orbit D=2a

Wolfram KORTEN 17 Euroschool Leuven – September 2009 Coulomb trajectories – some more definitions  distance of closest approach (for w=0):  impact parameter:  angular momentum : Principal assumption  >>1  classical description of the relative motion of the center-of-mass of the two nuclei  hyperbolic trajectories r (w) = a (  sinh w + 1) t (w) = a/v  (  cosh w + w) a = Z p Z t e 2 E -1 b projectile target D(  ) 

Wolfram KORTEN 18 Euroschool Leuven – September 2009 Coulomb excitation – the principal process b projectile target vivi vfvf Inelastic scattering: kinetic energy is transformed into nuclear excitation energy e.g. rotation vibration Excitation probability (or cross section) is a measure of the collectivity of the nuclear state of interest  complementary to, e.g., transfer reactions

Wolfram KORTEN 19 Euroschool Leuven – September 2009 Coulomb excitation – “sudden impact” Excitation occurs only if nuclear time scale is long compared to the collision time: „sudden impact“ if  nucl >>  coll ~ a/v  10 fm / 0.1c  2-3  s  coll ~  nucl ~ ћ/  E  adiabatic limit for (single-step) excitations  : adiabacity paramater sometimes also  (  ) with D(  ) instead of a Limitation in the excitation energy  E for single-step excitations in particular for low-energy reactions (v<c)

Wolfram KORTEN 20 Euroschool Leuven – September 2009 Coulomb excitation – first conclusions Maximal transferable excitation energy and spin in heavy-ion collisions  =1

Wolfram KORTEN 21 Euroschool Leuven – September 2009 Coulomb excitation – the different energy regimes Low-energy regime (< 5 MeV/u) High-energy regime (>>5 MeV/u) Energy cut-off Spin cut-off:  L up to 30ћ  L~ n  with n~1

Wolfram KORTEN 22 Euroschool Leuven – September 2009 Summary Coulomb excitation is a purely electro-magnetic excitation process of nuclear states due to the Coulomb field of two colliding nuclei. Coulomb excitation is a very precise tool to measure the collectivity of nuclear excitations and in particular nuclear shapes. Coulomb excitation appears in all nuclear reactions (at least in the incoming channel) and can lead to doorway states for other excitations. Pure electro-magnetic interaction (which can be readily calculated without the knowledge of optical potentials etc.) requires “safe” distance between the partners at all times.