12 November 2003 Fusion 03 1

Modelling Effects of Halo Breakup on Fusion.Synopsis and a new proposal

Ian ThompsonUniversity of Surrey,Guildford, EnglandwithAlexis Diaz-Torres (Frankfurt)

12 November 2003 Fusion 03 2

Topics to Discuss

Irreversibility: in fusion (yes!), in breakup (?)

Fusion defined: complete & incomplete Fate of neutrons

Three-body dynamics where to put imaginary potentials?

in time-dependent or CDCC calculations.

Modelled: 11Be + 208Pb, 6,7Li + 59Co,209Pb‘Optical Decoherence Model’: a proposal

12 November 2003 Fusion 03 3

Quantum Irreversibility?

Imaginary potentials in the optical model describe an irreversible step. Loss of phase coherence between channels, Also gives loss of flux.

Decoherence: In QM: from coupling to the environment Fusion channels definitely like an environment! Are breakup channels like an environment?

12 November 2003 Fusion 03 4

Definition of ‘Fusion’

‘Theory’: Complete fusion: capture all proj. fragments Incomplete: capture only some of fragments

‘Experiment’ Complete fusion: capture all projectile charge Incomplete: capture only some of charge

6Li, 7Li: Definitions agree (mostly)Halos: What happens to the neutrons?!?

12 November 2003 Fusion 03 5

Halo & clusters as testing grounds for fusion theories

Consider halo nuclei, and others e.g. 6,7Li with strong clusters,

Good test-benches for theories of breakup, theories of incomplete & complete fusion

because the breakup and partial-fusion modes are clearly defined, (assuming experimentalists will fix neutrons soon!)

12 November 2003 Fusion 03 6

Fusion of Halo nuclei

The two 6He examplesSignorini, “Fusion at the barrier with light Radioactive Ion Beams” 2001

12 November 2003 Fusion 03 7

Imaginary Potentials (0)

Try: no imaginary potentials, or short-range potentials in only the coordinate of the core relative to target, ‘complete fusion’ not calculated ‘incomplete core fusion’ = absorption from

this core imaginary potentialneed very many partial waves to describe motion

of neutron on target: including transfer channels. suggested by K. Yabana, M. Ueda, T. Nakatsukasa, Nucl. Phys.

A722, 261c (2003); also this conference

12 November 2003 Fusion 03 8

Imaginary Potentials (1)

Try: short-range potentials in only the coordinate of valence nucleon relative to target, ‘complete fusion’ not calculated ‘incomplete valence fusion’ =

absorption from this valence imaginary potential

suggested by H. Esbensen & G. Bertsch, Phys. Rev. C 59, 3240 (1999)

12 November 2003 Fusion 03 9

Imaginary Potentials (2)

Try: short-range potentials in the coordinate of whole projectile-c.m. relative to target. Then guess that

‘complete fusion’ = absorption from g.s.‘incomplete fusion’ = absorption from breakup

states suggested by K. Hagino, A. Vitturi, C.H. Dasso and S.M. Lenzi,

Phys Rev C 61, 037602 (2000), used also in A Diaz-Torres & I Thompson, Phys Rev C65, 024606

(2002)

12 November 2003 Fusion 03 10

no couplings

complete fusion including couplings to and from the g.s.

total

no couplings

neglecting excited state of 11Be

Effect of continuum couplings

[A Diaz-Torres and I Thompson, [A Diaz-Torres and I Thompson, Phys Rev C65 (2002) 024606]Phys Rev C65 (2002) 024606]

Above the barrier:both total and complete fusion are suppressedBelow the barrier:strong enhancement

CDCC 11Be + 208Pb at Coulomb barrier.

12 November 2003 Fusion 03 11

Imaginary Potentials (3)

Try: short-range potentials in the coordinate of each projectile fragment relative to target.

‘total fusion’ = complete+incomplete fusion, from any absorption by an imaginary part

capture of the c.m. of the projectile is not necessarily connected to the capture of the large charged fragments

suggested in A. Diaz-Torres, I. Thompson, C. Beck, Phys Rev C68, 044607 (2003)

12 November 2003 Fusion 03 12

Fragment-target imaginary fusion potentials

Used for 6,7Li on targets of 59Co, 209Pb

See that there are many events where one of the fragments of 6Li is captured, but the c.m. of the projectile does not reach the absorption (fusion) region

from A. Diaz-Torres, I. Thompson, C. Beck, Phys Rev C68, 044607 (2003)

12 November 2003 Fusion 03 13

Complete vs Incomplete Fusion

Need to theoretically distinguish complete from incomplete fusion.

Need correlations after breakup need time-dependent or CDCC calculations

Need, after absorption of one fragment, to follow evolution of remaining parts.

12 November 2003 Fusion 03 14

Classical Trajectories

Three-body classical trajectory model of 6Li and 7Li breakup on 209Bi. find whether none, one or two fragments are

‘captured’. identify these with elastic breakup,

incomplete and complete fusion, respectively.

“CDCC would give a more realistic picture”

from M. Dasgupta et al, Phys. Rev. C 89, 41602 (2002)

12 November 2003 Fusion 03 15

Theory of Incomplete Fusion

Integral expressions for incomplete fusion Survey by M. Ichimura, Saha conference (Jan 1989):

Three-body model, Austern et al Post-form DWBA, Ichimura-Austern-Vincent Integral of Imaginary Potl, Hussein-McVoy Elastic-breakup-fusion, Udagawa-Tamura

More recent series of papers: L.F. Canto et al, Phys Rev C 58, 1107 (1998) M. Hussein et al, ISPUN02 conference (Nov 2002)

12 November 2003 Fusion 03 16

New theoretical model?

But: do we need more information than is present in a ‘full CDCC’ calculation of dynamics of elastic scattering & breakup?

We need, after absorption of one fragment, to follow evolution of remaining parts. This is not in CDCC wfn!

I will now outline a dynamical approach with density-matrix semigroups.

12 November 2003 Fusion 03 17

Density Matrix Evolution

Schrödinger time evolution: ρ/t = - i[H,ρ] = - i[H0,ρ] – W ρ – ρ W when H=H0–iW

Semigroup time evolution ρ' = – i[H,ρ] + Σm(2LmρLm

+–Lm+Lmρ– ρLm

+Lm)/2 - general trace-preserving master equation Lm are any operators: ‘Lindblads’ from G. Lindblad, Commun Math Phys 48, 119 (1976):

12 November 2003 Fusion 03 18

Unitary density evolution

Schrödinger time evolution:If initially

ρ(x,x',0) = |Ψ(x,0)> < Ψ(x',0)|

then the density matrix stays pure: ρ(x,x',t) = |Ψ(x,t)> < Ψ(x',t)| where ih Ψ(x,t)/t = H Ψ(x,t) describes evolution

according to the Schrödinger equation.

12 November 2003 Fusion 03 19

Optical-Model Decoherence

The decoherence effect of an imaginary optical potential is:

If H0 = 0 in a toy model, then ρ(x,x',t)/t = –W(x) ρ(x,x',t) – ρ(x,x',t)W(x'), solution: ρ(x,x',t) = e–W(x)t ρ(x,x',0) e–W(x')t

off-diagonal terms reduced, but diagonal also!

12 November 2003 Fusion 03 20

Choose Lindblad form to give same decoherence effect as an imaginary optical potential: L1 = [2W(x)]

If H0=0, then ρ/t = 2 WρW – Wρ – ρW so ρ(x,x',t) = ρ(x,x',0) exp(– [W(x') –W(x)]2 t) = e–W(x)t ρ(x,x',0) e–W(x')t e+2W(x)W(x')t

Optical Decoherence Model

Schrödinger density evolution Semigroup evolution

12 November 2003 Fusion 03 21

Semigroup Decoherence

The toy model H0=0 (still no unitary evolution):

ρ(x,x',t) = ρ(x,x',0) exp(– [W(x') –W(x)]2 t)

If x=x' or W(x) = W(x') then ρ(x,x',t) = ρ(x,x',0):

no diagonal decoherence, no decoherence where W(x) is constant

If W(x) W(x') then ρ(x,x',t) 0:

off-diagonal decoherence where W(x) varies.

12 November 2003 Fusion 03 22

Optical Decoherence Model

Solve time evolution of composite projectile with semigroup evolution eqn: ρ/t = – i[H0,ρ] + 2WρW – Wρ – ρW

The imaginary potential –iW(x)= –iW1(x1) –iW2(x2), for each fragment-

target interaction, gives decoherence, but NO loss of flux

After one fragment fused, can still follow time evolution & flux of other fragments.

12 November 2003 Fusion 03 23

ODM Agenda

Rederive optical-model scattering theory, by evolving the density matrix, not the wf.

Include second semigroup term in ρ/t = – i[H0 – iW,ρ] + 2WρW

in eg model examples of reduced dimensionsDetermine practicality of evolving three-

body-model density matrices (number of spatial dimensions are squared!)

Exercises for the reader

12 November 2003 Fusion 03 24

Conclusions

Halo and cluster nuclei, with well-defined breakup & fusion modes, are good test-benches for theories of breakup & fusion.

Still theoretical uncertainties & varieties in modelling incomplete/complete fusion

(Optical Decoherence Model) ODM Proposal: Semigroup evolution of density matrix;

Evolution even after absorption!

12 November 2003 Fusion 03 25

Stationary-state scattering

If ρ(x,x',t) = |Ψ(x,t)> < Ψ(x',t)| and Ψ(x,t) = u(x) exp(ihE/t) then ρ(x,x',t) = |u(x)> < u(x')| - independent of time. ħ

Evolution eqn. for time-independent ρ(x,x'): (H0 – iW)xρx,x' – ρx,x'(H0 – iW)x' + 2iWxρx,x'Wx' = 0

Two-coordinate x,x' second-order p.d.e. Energy E only appears in the boundary conditions.

ħħhđħħħħ