Upload
others
View
0
Download
0
Embed Size (px)
Citation preview
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
winter school, jan 2011
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
winter school, jan 2011
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
winter school, jan 2011
orbitals and magic numbers away from stability?
winter school, jan 2011
magic numbers
Doubly magic nuclei
208Pb(d,p)209Pb
winter school, jan 2011
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
winter school, jan 2011
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?
winter school, jan 2011
halo nuclei
winter school, jan 201111
Very large spatial extension:correct asymptotic behaviour needed
finite range effects crucial
the latest on halos
winter school, jan 201112
why bother with reactions?
a) nuclei of interest are beams
winter school, jan 2011
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
winter school, jan 2011
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…
winter school, jan 2011
indirect methods: nuclear reactions
• direct measurement 7Be(p,γγγγ)8B
• Coulomb dissociation
• transfer reaction
winter school, jan 2011
• Coulomb dissociation
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
winter school, jan 2011
Transfer reactionS17(0)=17.3(1.8) eV b
Solar Fusion II, Rev. Mod. Phys. (2011)
heavy element: r-process in the chart
winter school, jan 201118
indirect methods: nuclear reactions
• direct measurement 14C(n,γγγγ)15C
• Coulomb dissociation
• transfer reaction
winter school, jan 2011
• Coulomb dissociation
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
winter school, jan 2011
Compare theory to data: structure=data/reaction
putting in perspective reaction theory
winter school, jan 2011
Compare theory to data: cross section(theory)=cross section(exp) ?
If yes: structure assumptions correctIf no: try again!
need absolute confidence in reaction model
putting in perspective reaction theory
winter school, jan 2011
multiple channel reaction theory
winter school, jan 2011
Reaction theories need to keep track of multiple channels
from the many body problem to few body reactions
winter school, jan 2011
Reaction theories need to map onto the many-body problem!
meeting point between reactions and structure
overlap function A
B=A+n
I
winter school, jan 2011
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
winter school, jan 2011
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 ==
winter school, jan 2011
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
winter school, jan 2011
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
winter school, jan 2011
• 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
winter school, jan 2011
A+B C+D
B(A,C)D
Notations for the reaction
34
forces present in reactions
• nuclear (strong)• weak• electromagnetic• gravitational
winter school, jan 201135
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
winter school, jan 2011
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*
winter school, jan 2011
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.
winter school, jan 201138
energy distribution: compound vs direct
pure compound nucleus formation can be seen at
backward angles
winter school, jan 2011
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)
winter school, jan 2011
Compound reactions (NC):Distribution is generally
isotropic (except for heavy ion collision where L large)
40
direct reactions
winter school, jan 2011
direct reactions
winter school, jan 2011
why do reactions? elastic
traditionally used to extract optical potentials,
winter school, jan 2011
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
winter school, jan 2011
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
winter school, jan 2011
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
winter school, jan 2011
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
winter school, jan 2011
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
winter school, jan 2011
bound states
winter school, jan 2011
potential shape
winter school, jan 2011
single particle ordering
winter school, jan 2011
binding potential parameters
winter school, jan 2011
binding potential parameters
winter school, jan 2011
nodes in single nucleon bound states
winter school, jan 2011
nodes in cluster states
winter school, jan 2011
nodes in cluster states
winter school, jan 2011
spectroscopic factor
winter school, jan 2011
unbound states at low energy
winter school, jan 2011
completeness and orthogonality
winter school, jan 2011
distorted waves
winter school, jan 2011
absorption
winter school, jan 2011
Schrodinger equation
winter school, jan 2011
schrodinger equation
winter school, jan 2011
asymptotic normalization coefficients
winter school, jan 2011
S-matrix
winter school, jan 2011
phase shifts
winter school, jan 2011
resonances
winter school, jan 2011
phase shifts and resonances
winter school, jan 2011
at what energy is the resonance?
• peak in the cross section• phase shift goes through 90 deg• peak of derivative of phase shift
winter school, jan 2011
• 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
winter school, jan 2011
• 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
winter school, jan 2011
energy of resonances: peak of cross section
winter school, jan 2011
energy of resonances: δ=900
winter school, jan 2011
Breit-Wigner resonances: pole of S-matrix
winter school, jan 2011
virtual states and scattering length
winter school, jan 2011
continuum bins
winter school, jan 2011
optical potential: conventions
winter school, jan 2011
S-matrix and absorption
winter school, jan 2011
amplitudes of ingoing and outgoing waves
winter school, jan 2011
barrier penetration and S-matrix
winter school, jan 2011