Three- and four-particle scattering A. Deltuva Centro de Física Nuclear da Universidade de Lisboa
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Three- and four-particle scattering
A. Deltuva
Centro de Física Nuclear da Universidade de Lisboa
Few-particle scattering
Three-particle scattering equations
Three-nucleon system
Three-body direct nuclear reactions
Four-particle scattering equations
Four-nucleon system
Four-boson universal physics
Three-particle system
Hamiltonian H0+∑α
vα
• v1
•
v2 •
v3
1
2
3
Three-particle system
Hamiltonian H0+∑α
vα
• v1
•
v2 •
v3
1
2
3
Faddeev equations
(E−H0−vα)|ψα〉 = vα ∑σ
δασ|ψσ〉
|Ψ〉 = ∑α|ψα〉
Alt, Grassberger, and Sandhas equations
Uβα = δβαG−10 +∑
σδβσTσG0Uσα
U0α = G−10 +∑
σTσG0Uσα
Tσ = vσ +vσG0Tσ
G0 = (E + i0−H0)−1
channel states (E−H0−vα)|φα〉 = 0
...
AGS equations: numerical solution
Uβα = δβαG−10 +∑
σδβσTσG0Uσα
3 sets of Jacobi momenta
• pα
•
•
αqα
momentum-space partial wave basis
set of coupled 2-variable integral equations
integrable singularities in kernel
Coulomb interaction: screening and renormalization
Proton-deuteron elastic scattering at Ep = 135 MeV
1
10
0 60 120 180
dσ/d
Ω (
mb/
sr)
Θc.m. (deg)
AV18AV18 + UIXCD BonnCD Bonn + ∆
-0.4
0.0
0.4
0 60 120 180A
y (N
)
Θc.m. (deg)
-0.4
0.0
0.4
0 60 120 180
Ay
(d)
Θc.m. (deg)
0.0
0.4
0.8
0 60 120 180
Ayy
Θc.m. (deg)
-0.4
0.0
0.4
0 60 120 180
Axx
Θc.m. (deg)
0.0
0.4
0.8
0 60 120 180
Axz
Θc.m. (deg)
[PRC 80, 064002]
Coulomb vs 3NF: 1H(d,pp)n at Ed = 130 MeV
0.0
0.2
(15o,15o,160o)0.0
0.2
(20o,15o,160o)0.0
0.2
(25o,15o,160o)
0.0
0.2
0.4
d5 σ/dS
dΩ
1 dΩ
2 (
mb
MeV
-1sr
-2)
(30o,15o,160o)0.0
0.1
(20o,20o,160o)0.0
0.1
(25o,20o,160o)
0.0
0.1
50 100 150
S (MeV)
(30o,20o,160o)
AV18(nd)AV18(pd)AV18+UIX(pd)
0.0
0.1
50 100 150
S (MeV)
(25o,25o,160o)0.0
8.0
50 100 150
S (MeV)
(13o,13o,20o)
Application to 3-body nuclear reactions
p+(nA)
d+A
→
n+(pA)
p+(nA)
d+Ap+n+A
with A = 4He, 10Be, 12C, 14C, 16O, 28Si, 40Ca, 48Ca, 58Ni, . . .
Validity test of approximate nuclear reaction methods:DWBA, ADWA, CDCC, . . .
Novel dynamic input: nonlocal potentials, . . .
CDCC test: 12C(d, pn)12C & 58Ni(d,d)58Ni
-60 -40 -20 0 20 40 60θp (deg)
100
101
102
103
d4 σ/dΩ
ndΩp (
mb/
sr2 )
Matsuoka (1982)FaddeevCDCC-BU
12C(d,pn)
12C @ Ed=56 MeV
θn=15o
0 30 60 90 120θc.m. (deg)
10-2
10-1
100
(dσ/
dΩ)/
(dσ R
/dΩ
)
Exp. (80.0 MeV)Exp. (79.0 MeV)FaddeevCDCC-BU
CDCC: A. M. Moro & F. M. Nunes
[PRC 76, 064602 (2007)]
CDCC test: 12C(d, p)13C and12C(d, pn)12C
0
20
40
60
8012 MeV
56 MeV
CDCCAGSADWA
0 20 40 60 80θ (degrees)
0
5
10
15
dσ /
dΩ (m
b/sr
)
0
20
40
6012 MeV
20 40 60 80 100θ (deg)
0
100
200
dσ /
dΩ (m
b/sr
)
56 MeVCDCCAGS
0 4 8Epn (MeV)
0
5
10
dσ /
dE
0 20 40Epn (MeV)
0
2
4
6
dσ /
dE
CDCC/ADWA: F. M. Nunes, N. Upadhyay [PRC 85, 054621]
Comparison with r-space FM results
1.0
10.0
(dσ/
dΩ)/
(dσ R
/dΩ
)
p+13C → p+13C
AGSFM
0.0
0.5
0 60 120 180
Ay
Θc.m. (deg)
0.1
1.0
10.0
(dσ/
dΩ)/
(dσ R
/dΩ
)
d+12C → d+12C
AGSFM
0.1
1.0
10.0
0 60 120 180
dσ/d
Ω
(mb/
sr)
Θc.m. (deg)
d+12C → p+13C(1/2-)
FM: R. Lazauskas [arXiv:1201.4979]
Nonlocal OP: transfer reactions
1.0
10.0
dσ/d
Ω (
mb/
sr)
d+16O → p+17O(5/2+)
local OPnonlocal OP
0.1
1.0
10.0
0 50 100 150
dσ/d
Ω (
mb/
sr)
Θc.m. (deg)
Ed = 36 MeV
d+16O → p+17O(1/2+)
1
10
100
dσ/d
Ω (
mb/
sr)
d+14C → p+15C(1/2+)
Ed = 14 MeV
local OPnonlocal OP
1
10
100
0 50 100 150
dσ/d
Ω (
mb/
sr)
Θc.m. (deg)
d+14C → p+15C(5/2+)
[PRC 79, 021602, PRC 79, 054603]
Four-particle scattering
Hamiltonian H0+∑i> j
vi j
• •
•
•
v12
1
2
3
4
Wave function:Schrödinger equation
Wave function components:Faddeev-Yakubovsky equations
Transition operators:Alt-Grassberger-Sandhas equations
Symmetrized AGS equations
t = v+vG0t
G0 = (E + iε−H0)−1
u j = PjG−10 +Pj tG0u j
3+1 : P1 = P12P23+P13P23
2+2 : P2 = P13P24
U11 = (G0 tG0)−1ζP34+ζP34u1G0 tG0U11+u2G0 tG0U21
U21 = (G0 tG0)−1(1+ζP34)+(1+ζP34)u1G0 tG0U11
U12 = (G0 tG0)−1 +ζP34u1G0 tG0U12+u2G0 tG0U22
U22 = (1+ζP34)u1G0 tG0U12
ζ = −1 (+1) for fermions (bosons)
basis states partially symmetrized
Scattering amplitudes: E + iε → E + i0
2-cluster reactions:
Tf i = sf i〈φ f |U f i|φi〉
|φ j〉 = G0tPj |φ j〉
|Φ j〉 = (1+Pj)|φ j〉
3-cluster breakup/recombination:
T3i = s3i〈φ3|[(1+ζP34)u1G0 tG0U1i +u2G0 tG0U2i]|φi〉
4-cluster breakup/recombination:
T4i = s4i〈φ4|[1+(1+P1)ζP34](1+P1)tG0u1G0 tG0U1i|φi〉
+ 〈φ4|(1+P1)(1+P2)tG0u2G0 tG0U2i|φi〉
Wave function
|Ψi〉 = si[1+(1+P1)ζP34](1+P1)|ψ1,i〉+(1+P1)(1+P2)|ψ2,i〉
with Faddeev-Yakubovsky components
|ψ j,i〉 = δ ji |φi〉+G0 tG0u jG0 tG0U ji |φi〉
Solution of 4N AGS equations
U11|φ1〉 = −G−10 P34P1|φ1〉−P34u1G0 tG0U11|φ1〉+u2G0 tG0U21|φ1〉
1
l l lx l lx
l
y
z
z y
2
momentum-space partial-wave basis|kxkykz[lz(ly[(lxSx) jxsy]SyJysz)Sz]JM, [(Txty)Tytz]TMT〉1
|kxkykz[lz(lxSx) jx[ly(sysz)Sy] jySz]JM, [Tx(tytz)Ty]TMT〉2
large system (up to 30000) of coupled 3-variableintegral equations with integrable singularities
Coulomb interaction: screening and renormalization[PRC 75, 014005, PRL 98, 162502]
Singularities of 4N AGS equations
3H, 3He, or d+d bound state poles
G0u jG0 →Pj |φ j〉sj j 〈φ j |Pj
E + iε−Ebj −k2
z/2µj
treated by subtraction below 3-cluster threshold
Z q
pk2
z dkzF(kz)
k20−k2
z + i0
= PZ q
pk2
z dkzF(kz)
k20−k2
z−
12
iπk0F(k0)
=Z q
pdkz
k2zF(kz)−k2
0F(k0)
k20−k2
z
−12
k0F(k0)[
iπ+ ln(k0 + p)(q−k0)
(k0− p)(k0 +q)
]
p-3Hescattering
0 60 1200
100
200
300
400
500dσ
/dΩ
[mb/
sr] Famularo 1954
Fisher 2006I-N3LOAV18low-k
2.25 MeV
0 60 1200
0.2
0.4
Ay0
Fisher 2006George 2001
0 60 120θc.m. [deg]
0
0.1
A0y
Daniels 2010
0 60 120
Mcdonald 1964Fisher 2006
4.05 MeV
0 60 120
Fisher 2006
0 60 120θc.m. [deg]
Daniels 2010
0 60 120 180
Mcdonald 1964
5.54 MeV
0 60 120 180
Alley 1993
0 60 120 180θc.m. [deg]
Alley 1993Daniels 2010
AGS/HH/FY (Lisbon/Pisa/Strasbourg, PRC 84, 054010)
∆-isobar excitation: effective 3N and 4N forces
Fujita-Miyazawa higher order 3N force
4N force
[PLB 660, 471]
n-3Heelastic scattering
0
200
400
dσ/d
Ω (
mb/
sr)
En = 1 MeV
AV18N3LO
CD BonnCD Bonn + ∆
INOY04
En = 3.5 MeV
0.0
0.4
0 50 100 150
Ay
Θc.m. (deg)
En = 1 MeV
0 50 100 150Θc.m. (deg)
En = 3.7 MeV
[PRC 76, 021001]
p-3H elastic scattering
Ep = 4.15 MeV
0
200
400
0 50 100 150
dσ/d
Ω (
mb/
sr)
Θc.m. (deg)
N3LOINOY04
AV18CD Bonn
0.0
0.2
0.4
0.6
0 50 100 150
Ay
Θc.m. (deg)
0
0.2
0.4
0.6
3 4 5 6
max
(Ay)
Ep (MeV)
p-3He
p-3H
CD Bonn
Charge exchange reaction 3H(p,n)3He
0
40
80
dσ/d
Ω (
mb/
sr)
Ep = 2.48 MeVAV18N3LO
CD BonnCD Bonn + ∆
INOY04
Ep = 6 MeV
0.0
0.4
0 50 100 150
Ay
Θc.m. (deg)0 50 100 150
Θc.m. (deg)
[PRC 76, 021001]
d-d elastic scattering at Ed = 3 MeV
102
103
0 50 100 150
dσ
/dΩ
(m
b/s
r)
Θc.m. (deg)
CD BonnCD Bonn + ∆
0.000
0.002
0 50 100 150
T2
2
Θc.m. (deg)
[PLB 660, 471]
2H(d,p)3H and 2H(d,n)3He
0
20dσ
/dΩ
(m
b/sr
) d+d → p+3H d+d → n+3He
Ed = 1.5 MeV
0
20
40
dσ/d
Ω (
mb/
sr) AV18
N3LOCD BonnCD Bonn + ∆INOY04 Ed = 3 MeV
0
20
0 50 100 150
dσ/d
Ω (
mb/
sr)
Θc.m. (deg)0 50 100 150
Θc.m. (deg)
Ed = 4 MeV
[PRC 81, 054002]
2H(d,p)3H and 2H(d,n)3He
0.0
0.2
iT11
d+d → p+3H d+d → n+3He
Ed = 1.5 MeV
0.0
0.2
iT11
INOY04CD Bonn + ∆CD BonnN3LOAV18
Ed = 3 MeV
0.0
0.2
0 50 100 150
iT11
Θc.m. (deg)0 50 100 150
Θc.m. (deg)
Ed = 4 MeV
0.0
0.2
T21
d+d → p+3H d+d → n+3He
Ed = 1.5 MeV
0.0
0.2
T21
AV18N3LOCD BonnCD Bonn + ∆INOY04
Ed = 3 MeV
0.0
0.2
0 50 100 150T
21Θc.m. (deg)
0 50 100 150Θc.m. (deg)
Ed = 4 MeV
Above breakup: additional singularities in AGS equations
deuteron bound state poles
t →v|φd〉〈φd|v
E + iε−ed −k2y/2µy
j −k2z/2µj
free resolvent
G0 →1
E + iε−k2x/2µx
j −k2y/2µy
j −k2z/2µj
Above breakup: additional singularities in AGS equations
deuteron bound state poles
t →v|φd〉〈φd|v
E + iε−ed −k2y/2µy
j −k2z/2µj
free resolvent
G0 →1
E + iε−k2x/2µx
j −k2y/2µy
j −k2z/2µj
treated by complex-energy method:
1. solve for U f i(E + iε) with finite ε = ε1, ...,εn
2. extrapolate to ε → 0 for physical amplitudes U f i(E + i0)
[H. Kamada et al, Prog. Theor. Phys. 109, 869L (2003)]
Integration with special weights
accuracy & efficiency of the complex-energy method isgreatly improved by a special integration
Z b
a
f (x)xn
0 + iy0−xndx≈
N
∑j=1
f (x j)w j(n,x0,y0,a,b)
where the quasi-singular factor is absorbed into specialweights
w j(n,x0,y0,a,b) =Z b
a
Sj(x)xn
0 + iy0−xndx
that may be calculated using spline functions Sj(x) forstandard Gaussian grid x j [PRC 86, 011001]
Extrapolation ε → 0: n+3H at 22.1 MeV
10
100
dσ/d
Ω (
mb/
sr)
[εmin,εmax]/MeV:
[1.0, 2.0]
[1.2, 2.0]
[1.4, 2.0]
[1.2, 1.8]
ε = 1.4 MeV
-0.4
0.0
0.4
0.8
0 60 120 180
Ay
Θc.m. (deg)
Extrapolation ε → 0: n+3H at 22.1 MeV
[εmin,εmax] δ(1S0) η(1S0) δ(3P0) η(3P0) δ(3P2) η(3P2)
[1.0,2.0] 62.63 0.990 43.03 0.959 65.27 0.950[1.2,2.0] 62.60 0.991 43.04 0.959 65.29 0.951[1.4,2.0] 62.67 0.991 43.03 0.958 65.27 0.950[1.2,1.8] 62.65 0.992 43.03 0.959 65.28 0.9501.4 73.37 0.916 44.77 0.840 67.38 0.933
[PRC 86, 011001]
n+3H elastic scattering
10
100
dσ/d
Ω (
mb/
sr)
En = 14.1 MeV
INOY04CD BonnFrenjeDebertin
10
100
dσ/d
Ω (
mb/
sr)
En = 18.0 MeV
DebertinSeagrave
1
10
100
0 60 120 180
dσ/d
Ω (
mb/
sr)
Θc.m. (deg)
En = 22.1 MeV
Seagrave, 21 MeVSeagrave, 23 MeV -0.4
0.0
0.4
0.8
0 60 120 180
Ay
Θc.m. (deg)
En = 22.1 MeV
INOY04CD BonnSeagraveINOY04, 14.1 MeVINOY04, 18.0 MeV
n+3H total and breakup cross sections
1.5
2.0
2.5
0 2 4 6
σ t (
b)
En (MeV)
AV18N3LOCD BonnCD Bonn + ∆INOY04
0
1
2
0 5 10 15 20
σ t (
b)
En (MeV)
PhillipsBattatINOY04CD Bonn
0
20
40
60
10 15 20
σ b (
mb)
En (MeV)
INOY04
Recombination reaction 2H+n+n → n+3H
1 × 10-5
2 × 10-5
3 × 10-5
0 1 2 3 4
NA2 K
3 (
cm6 s
-1 m
ol-2
)
E3 (MeV)
dρt
dt= Kγ
2ρdρn +K3ρdρ2n + . . .
p+3He elastic scattering
10
100dσ
/dΩ
(m
b/sr
) Ep = 21.3 MeV
INOY04
Murdoch
-0.4
0.0
0.4
0.8
0 60 120 180
Ay
Θc.m. (deg)
Birchall
4-boson system with resonant interactions
T(n−1,1)
T(n−1,2)
a+t(n−1)
a+t(n)
IVS T(n,2)
Energy
d+d
d+a+a
1/a
T(n,1)
T(n,0)d
Four-atom recombination at threshold
100
103
106
109
1012
0.2 0.4 0.6 0.8 1.0
κ n
a/an0
n = 1
n = 2
n = 3
n = 4
[PRA 85, 012708]
T(n−1,1)
T(n−1,2)
a+t(n−1)
a+t(n)
IVS T(n,2)
Energy
d+d
d+a+a
1/a
T(n,1)
T(n,0)d
a0n : bn = 0
a0n,k : Bn,k = 0
a0n,1/a0
n = 0.4254a0
n,2/a0n = 0.9125
Atom-trimer scattering length
-200
0
200
0.1 1.0
An/
a ndd
andd/a
Re An/andd
Im An/andd
[EPL 95, 43002, PRA 85, 042705]
T(n−1,1)
T(n−1,2)
a+t(n−1)
a+t(n)
IVS T(n,2)
Energy
d+d
d+a+a
1/a
T(n,1)
T(n,0)d
addn : bn = 2bd
Summary
3/4-body Faddev/AGS equations in momentum space
complex-energy methodwith special integration weights
Summary
3/4-body Faddev/AGS equations in momentum space
complex-energy methodwith special integration weights
3N scattering
3-body nuclear reactions
4N scattering
universal 4-boson physics
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