Download presentation

Presentation is loading. Please wait.

Published byCyrus Womble Modified about 1 year ago

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

2
Wolfram KORTEN 2 Euroschool Leuven – September 2009 Coulomb excitation theory - the general approach Solving the time-dependent Schrödinger equation: iħ d (t)/dt = [H P + H T + V (r(t))] (t) with H P/T being the free Hamiltonian of the projectile/target nucleus and V(t) being the time-dependent electromagnetic interaction (remark: often only target or projectil excitation are treated) Expanding (t) = n a n (t) n with n as the eigenstates of H P/T leads to a set of coupled equations for the time-dependent excitation amplitudes a n (t) iħ da n (t)/dt = m n |V(t)| m exp[i/ħ (E n -E m ) t] a m (t) The transition amplitude b nm are calculated by the (action) integral b nm = i ħ -1 a n n |V(t)| a m m exp[i/ħ (E n -E m ) t] dt Finally leading to the excitation probability P(I n I m ) = (2I n +1) -1 b nm 2 b projectile r(t) target r (w) = a ( sinh w + 1) t (w) = a/v ( cosh w + w) a = Z p Z t e 2 E -1

3
Wolfram KORTEN 3 Euroschool Leuven – September 2009 Coulomb excitation theory - the general approach The coupled equations for a n (t) are usually solved by a multipole expansion of the electromagnetic interaction V(r(t)) V P-T (r) = Z T Z P e 2 /rmonopole-monopole (Rutherford) term + V P ( ) electric multipole-monopole target excitation, + V T ( ) electric multipole-monopole project. excitation, + V P ( ) magnetic multipole project./target excitation + V T ( )(but small at low v/c) + O( ’ ’>0)higher order multipole-multipole terms (small) V P/T ( ) = Z T/P e /(2 r –( +1) Y ( , ) · M P/T ( V P/T ( ) = Z T/P e /(2 +1) i/c r –( +1) dr/dtLY ( , ) · M P/T ( electric multipole moment: M(E (r‘) r‘ Y (r‘) d 3 r‘ magnetic multipole moment: M(M -i/c( j(r‘) r‘ l (ir )Y (r‘) d 3 r‘ Coulomb excitation cross section is sensitive to electric multipole moments of all orders, while angular correlations give also access to magnetic moments

4
Wolfram KORTEN 4 Euroschool Leuven – September 2009 Nuclear shapes and electric multipole moments Electric multipole moments can be linked to Deformation parameters of the nuclear mass distribution For axially symmetric shapes ( and a homogenous density distribution the quadrupole, octupole and hexadecupole moments (Q 2,Q 3,Q 4 ) become:

5
Wolfram KORTEN 5 Euroschool Leuven – September 2009 Transition rates in the Coulomb excitation process 1 st order perturbation theory applicable if only one state is excited, e.g. 0 + 2 + excitation, and for small excitation probability (e.g. semi-magic nuclei) 1 st order transition probability for multipolarity Orbital integrals Adiabacity parameter Strength parameter

6
Wolfram KORTEN 6 Euroschool Leuven – September 2009 Strength parameter E2 as function of (Z p,Z T ) 132 Sn 70 Se 46 Ar ZTZT | | 2 /B(E2) · e 2 b 2 78 Ni

7
Wolfram KORTEN 7 Euroschool Leuven – September 2009 Orbital integrals R(E2) as function of and R 2 (E2;, ) =0.0 =0.2 =0.4 =0.8 Ar+Sn 0.4 MeV 0.8 MeV 1.6 MeV

8
Wolfram KORTEN 8 Euroschool Leuven – September 2009 Cross section for Coulomb excitation Differential and total cross sections Rutherford P( )

9
Wolfram KORTEN 9 Euroschool Leuven – September 2009 Angular distribution functions for different multipolarities df ( )

10
Wolfram KORTEN 10 Euroschool Leuven – September 2009 Total cross sections for different multipolarities B( ) values for single particle like transitions (W.u.): B sp ( ) = (2 +1) 9e 2 /4 R 2 x 10(ħc/M p R 0 ) 2 B( ) [e 2 b ] 208 Pb E1: A 2/ E2: A 4/ E3: A E4: A 8/ M1: 1.79 M2: A 2/ f E ( )f M ( )

11
Wolfram KORTEN 11 Euroschool Leuven – September 2009 Second order perturbation theory becomes necessary if several states can be excited from the ground state or when multiple excitations are possible i.e. for larger excitation probabilities 2 nd order transition probability for multipolarity Transition rates in the Coulomb excitation process IfIf IiIi b (1) b (2) 2 nd order: InIn InIn IfIf IiIi b (1) 1 st order:

12
Wolfram KORTEN 12 Euroschool Leuven – September 2009 Second order perturbation theory (cont.) IfIf IiIi b (1) b (2) 2 nd order: InIn InIn IfIf IiIi b (1) 1 st order: P (22) often negligible unless direct excitation through i f small/forbidden

13
Wolfram KORTEN 13 Euroschool Leuven – September 2009 Shape coexistence and excited 0 + states oblateprolate Shape isomer, E0 transition Configuration mixing: | = | o + | p 74 Kr IfIf IiIi InIn E0 E2 0202 2020 0 + states can only be excited via an intermediate 2 + state ( if (E0) = 0)

14
Wolfram KORTEN 14 Euroschool Leuven – September 2009 Examples of double-step E2 excitations 0 + states can only be excited via an intermediate 2 + state ( if (E0) = 0) IfIf IiIi InIn E0 E2 0202 2020

15
Wolfram KORTEN 15 Euroschool Leuven – September 2009 Examples of double-step E2 excitations 4 + states can be excited through a double-step E2 or a direct E4 excitation InIn IiIi IfIf E2 E4 Double E2Direct E4 quadrupolehexadecapole +

16
Wolfram KORTEN 16 Euroschool Leuven – September 2009 Double-step E2 vs. E4 excitation of 4 + states and d functions for different scattering angles and 1 - 2 ratios

17
Wolfram KORTEN 17 Euroschool Leuven – September 2009 Application to double-step (E2) excitations Double-step excitations are important if if P (12) 0 + states can only be excited via an intermediate 2 + state ( if = 0) P (2) = | 0 2 | 2 | 2 0 | 2 ,s, ) with ,s, ) = 25/4 (|R 20 | 2 +|G 20 | 2 ) with = 1 + 2 and s= 1 /( 1 + 2 ) P (2) ( 1 = 2 0) 5/4 | 0 2 | 2 | 2 0 | 2 4 + states are usually excited through a double-step E2 since the direct E4 excitation is small P (2) = | 0 2 | 2 | 2 4 | 2 ,s, ) with ,s, ) = 25/4 (|R 24 | 2 +|G 24 | 2 ) P (2) ( 1 = 2 0) 5/14 | 0 2 | 2 | 2 4 | 2 IfIf IiIi InIn E0 E2 InIn IiIi IfIf E4

18
Wolfram KORTEN 18 Euroschool Leuven – September 2009 The reorientation effect Specific case of second order perturbation theory where the „intermediate“ states are the m substates of the state of interest 2 nd order excitation probability for 2 + state IfIf IiIi a (1) a (2) 2 nd order: IjIj IjIj reorientation effect: IfIf IiIi a (2) MfMf

19
Wolfram KORTEN 19 Euroschool Leuven – September 2009 Strength of the reorientation effect (2 + ) [b] CM 76 Kr on 208 Pb sensitive to diagonal matrix elements intrinsic properties of final state: quadrupole moment including sign

20
Wolfram KORTEN 20 Euroschool Leuven – September 2009 Quadrupole deformation of nuclear ground states Coulomb excitation can, in principal, map the shape of all atomic nuclei: Quadrupole (and higher-order multipole moments) of I>½ states M. Girod, CEA

21
Wolfram KORTEN 21 Euroschool Leuven – September 2009 Quadrupole deformation and sum rules Model-independent method to determine charge distribution parameters (Q, from a (full) set of E2 matrix elements ground state shape can be determined by a full set of E2 matrix elements i.e. linking the ground state to all collective 2 + states ~ Q 2 ~ Q 3 cos3

22
Wolfram KORTEN 22 Euroschool Leuven – September 2009 Multi-step Coulomb excitation q = 45/16 0 2 Possible if >> 1 (no perturbative treatment) Example : Rotational band in a strongly deformed nucleus: I I+2 I+6 I+4 I+8 I+10 I+14 I+12 EE ħ [keV] J (2) [ħ 2 MeV -1 ] superdeformed 152 Dy

23
Wolfram KORTEN 23 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 max :up to 30ћmainly single-step excitations Cross section: d /d ~ I i |M( )|I f ~ (Z p e 2 / ħc) 2 B( , 0→ ) differential integral Luminosity: low mg/cm 2 targetshigh g/cm 2 targets Beam intensity:high >10 3 ppslow a few pps Comprehensive study of low-lying exitations First exploration of excited states in very “exotic” nuclei

24
Wolfram KORTEN 24 Euroschool Leuven – September 2009 Summary Coulomb excitation probability P(I ) increases with increasing strength parameter ( ), i.e. Z P/T, ), 1/D, cm decreasing adiabacity parameter ( ),i.e. E, a/v Differential cross sections d /d show varying maxima depending on multipolarity and adiabacity parameter allows to distinguish different multipolarities (E2/M1, E2/E4 etc.) Total cross section tot decreases with increasing adiabacity parameter and multipolarity is generally smaller for magnetic than for electric transitions Second and higher order effects lead to “virtual” excitations influencing the real excitation probabilities allow to excite 0 + states and to measure static moments lead to multi-step excitations

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google