nuclei: many body problembrown/uio-msu-ornl-ut-2011/...three-body force for oxygen isotopes winter...

nuclei: many body problem

winter school, jan 2011

effective interactions

Solve the problem in a part of the Hilbert space (P) where P+Q=1


Winning the (nobel) prize was not half as exciting as doing the work!

Maria Meyer

structure models

Ab-initio models and moreVariational monte carlo (VMC) Green’s function monte carlo (GFMC) no core shell model (NCSM) shell model coupled clusters

cluster models


coupled clusterscluster models resonating group model generator coordinate model few body models

Mean field models hartree-fock hartree-fock bogoliubov density functional methods

Collective models

orbitals and magic numbers away from stability?


magic numbers

Doubly magic nuclei

208Pb(d,p)209Pb


132Sn(d,p)133Sn

SF

[K. Jones et [K. Jones et al, Nature 465 (2010) 454al, Nature 465 (2010) 454]]

studying double magic nuclei away from stability

d(132Sn,133Sn)p@5 MeV/u


132Sn(n,γγγγ)133Sn

[K. Jones et [K. Jones et al, Nature 465 (2010) 454al, Nature 465 (2010) 454]]

connection to r-process

where is the oxygen dripline?

winter school, jan 2011 Slide from INT talk, A. Schwenk

three-body force for Oxygen isotopes

winter school, jan 2011 Otsuka, Suzuki, Holt, Schwenk, Akaishi, PRL (2009)

doubly magic at the dripline?


halo nuclei


Very large spatial extension:correct asymptotic behaviour needed

finite range effects crucial

the latest on halos


why bother with reactions?

a) nuclei of interest are beams


b) offers much more than energy levels

example of deuteron and Vnp

any simple central interaction can give correct binding

only the large body of reaction exp could


only the large body of reaction exp could provide the detailed structure of the deuteronand the tensor interaction

Pieper and Wiringa, ANL

and then there is the whole universe…


indirect methods: nuclear reactions

• direct measurement 7Be(p,γγγγ)8B

• Coulomb dissociation

• transfer reaction



p

208Pb

7Be

γγγγ

low relative energy

8B

14N(7Be, 8B)13C

direct versus indirect measurements?

Coulomb dissociation S17(0)=20.6(0.8 stat)(1.2 sys) eV b

Direct measurement S17(0)=22.1(0.6 exp)(0.6 th) eV b

7Be(p,γγγγ)8B


Transfer reactionS17(0)=17.3(1.8) eV b

Solar Fusion II, Rev. Mod. Phys. (2011)

heavy element: r-process in the chart


indirect methods: nuclear reactions

• direct measurement 14C(n,γγγγ)15C


• transfer reaction



n

208Pb

14

Cγγγγ

low relative energy

15C

14C(d,p)15C

19

breakup reactions and (n,γ)

Nakamura

208Pb(15C,14C+n)208Pb@68 MeV/u

winter school, jan 2011Nakamura et al, NPA722(2003)301c

Reifarth et al, PRC77,015804 (2008)

• Reifarth

14C(n,γγγγ)15C

20

putting in perspective reaction theory


Compare theory to data: structure=data/reaction



Compare theory to data: cross section(theory)=cross section(exp) ?

If yes: structure assumptions correctIf no: try again!

need absolute confidence in reaction model

multiple channel reaction theory


Reaction theories need to keep track of multiple channels

from the many body problem to few body reactions


Reaction theories need to map onto the many-body problem!

meeting point between reactions and structure

overlap function A

B=A+n

I


spectroscopic factor (Snlj):norm of overlap function

I

microscopic one nucleon overlap functions

winter school, jan 2011 [[WiringaWiringa et al., ANL 2007]et al., ANL 2007]

microscopic 2n overlap functions

6He(p,t)4He @ 25 MeV

6He 2n overlap functions

winter school, jan 2011[[BridaBrida, PhD thesis, MSU 2009], PhD thesis, MSU 2009]

[Brida and Nunes, [Brida and Nunes, NPA]NPA]

single particle approximation

A0)()( =++ rVT nljnAr ϕε


nucleons feels mean field generated by core nucleons VnA

• specific n,l,j and separation energy • assumptions about single particle parameters

single particle approximation

A2

22

nljb

CAS lj

nljBnlj ==


Same radial dependence at large distances:

)()( rArI nljnljAB ϕ=

)()( rihibr lnljRr

nljN κκϕ → >)()( rihiCrI llj

RrAB

N κκ → >

Extend that assumption within the range of the interaction:

Overlap function, SF and ANC

S (spectroscopic factor) – volume property


C (asymptotic normalization coefficient) – asymptotic property

Some references

• Reviews: Baye and Capel arxiv:1011.6427Nunes and Al-Khalili, JPG (2003)

• Notes: JINA school on direct reactions, April 2007• Book: Nuclear reactions for astrophysics, Thompson and Nunes


• Book: Nuclear reactions for astrophysics, Thompson and Nunes

direct reactions

nuclear reactions

Important concepts:

• projectile (A)• target (B)• residual nuclei (C+D)• q-value of a reaction


A+B C+D

B(A,C)D

Notations for the reaction

34

forces present in reactions

• nuclear (strong)• weak• electromagnetic• gravitational


classification of reactions

Resonance reactions

Direct reactionstransfer momentum is small compared to initial momentumtypically peripheralshort timescale (10-22 s) E>10 MeVmostly one stepfinal states keep memory of initial states


Resonance reactionsreactions that go through a resonance (peak in the cross section)intermediate step in the reactionlonger time scale

Compound reactionslonger timescalemany steps in the reactionall nucleons share the beam energyloss of memory from the initial statelow energy reactions

36

resonant reaction

a + A C*


Breit-Wigner shape

( ) ( ) 412

22

2

2 Γ+−Γ+=

R

scattEEk

ℓπσ

energyresonantE

FWHM

R ==Γ

37

compound reactions

the decay of the compound state does not depend on the initial state.


energy distribution: compound vs direct

pure compound nucleus formation can be seen at

backward angles


The more forward you go, the more pronounced direct

components will appear

39

angular distribution: compound vs direct

Direct reactions (ID):Forward peaked (large b)


Compound reactions (NC):Distribution is generally

isotropic (except for heavy ion collision where L large)

40

direct reactions


why do reactions? elastic

traditionally used to extract optical potentials,


extract optical potentials, rms radii, density distributions.

[[LapouxLapoux et al, PRC 66 (02) 034608]et al, PRC 66 (02) 034608]

43

why do reactions? inelastic

winter school, jan 2011 [Summers et al, PLB 650 (2007) 124][Summers et al, PLB 650 (2007) 124]

traditionally used to extract electromagnetic transitions or nuclear

deformations

44

why do reactions? transfer

6

8

10

12

14

/dΩ

(mb/

sr)

ground state1

st Ex = 462 keV

l = 2 DWBAl = 0 DWBA

example:84Se(d,p)85Se traditionally used to extract spin,parity and spectroscopic factors


g.s. SF = 0.33 ± 0.09 (d5/2)

1st SF = 0.30 ± 0.08 (s1/2)

0 5 10 15 20 25 30 35 40θc.m. (deg.)

0

2

4

6

dσ/d

Ω

[J. Thomas et al, PRC 76, 044302 (2007)][J. Thomas et al, PRC 76, 044302 (2007)]

45

why do reactions? transfer


11Li(p,t)9Li@ 3 A MeV

measured both ground state and excited state 9Li

[[TanihataTanihata et al, PRL 100, 192502 (2008)]et al, PRL 100, 192502 (2008)]

traditionally used to study two nucleon correlations

and pairing

46

why do reactions? breakup

two nucleon

correlation function

14Be n+n+12Be

Pb


23O(Pb,Pb)22O+n+[[NociforoNociforo et al, PLB 605 (2005) 79]et al, PLB 605 (2005) 79]

n

23O + γ n + 22O208Pb

22O

γγγγ23O

γγγγ

[Marques et al, PRC 64 (2001) 061301][Marques et al, PRC 64 (2001) 061301]

C

47

nuclear knockout


Includes elastic and inelastic breakup as well as transfer

Maddalena et al., PRC63(01)024613

transfer versus knockout

winter school, jan 2011 [Jenny Lee et al, PRL 2009][Jenny Lee et al, PRL 2009]49

interaction


bound states


potential shape


single particle ordering


binding potential parameters


nodes in single nucleon bound states


nodes in cluster states


spectroscopic factor


unbound states at low energy


completeness and orthogonality


distorted waves


absorption


Schrodinger equation


schrodinger equation


asymptotic normalization coefficients


S-matrix


phase shifts


resonances


phase shifts and resonances


at what energy is the resonance?

• peak in the cross section• phase shift goes through 90 deg• peak of derivative of phase shift


• peak of derivative of phase shift• pole of the S-matrix

at what energy is the resonance?

• peak in the cross section• phase shift goes through 90 deg• peak of derivative of phase shift


• peak of derivative of phase shift• pole of the S-matrix

Breit-Wigner shape

( ) ( ) 412

22

2

2 Γ+−Γ+=

R

scattEEk

ℓπσ

energy of resonances: peak of cross section


energy of resonances: δ=900


Breit-Wigner resonances: pole of S-matrix


virtual states and scattering length


continuum bins


optical potential: conventions


S-matrix and absorption


amplitudes of ingoing and outgoing waves


barrier penetration and S-matrix


nuclei: many body problembrown/uio-msu-ornl-ut-2011/...three-body force for oxygen isotopes winter...

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