09/03/2017 1 r. lazauskas · 2017-03-10 · • observables extracted using integral relations....
TRANSCRIPT
09/03/2017 2
Contents
R. Lazauskas09/03/2017 2
R. Lazauskas
• Solution of the 3-5 body Faddeev-Yakubovsky equationsfor nuclear systems
• Application of the complex-scaling method to solvecomplicated scattering problems using trivial boundaryconditions
09/03/2017 3
Introduction
R. Lazauskas09/03/2017 3
R. Lazauskas
• In configuration space wave functions extend to infinity!
• Increasingly complex asymptotic behaviour for A>2 systems!!
Collisions
…
How to take care of the boundary condition?
Conceptual difficulties to uncouple different particle channel, to constrain assymptotes of the solutions in all directions and thus get unique (physical) solution to the Schrodinger eq.
• It is ok, as long as there is single particle channel (elastic plus target excitations)
• Mathematically Ill-conditioned problem when several particle channels are open
Faddeev-Yakubovsky equations efficiently separates asymptotes of the binary channels
L. D. Faddeev, Zh. Eksp. Teor. Fiz. 39, 1459 (1960). [Sov. Phys. JETP 12, 1014(1961)].O. A. Yakubovsky, Sov. J. Nucl. Phys. 5, 937 (1967).
09/03/2017 4
Faddeev-Yakubovsky eq
R. Lazauskas09/03/2017 4
R. Lazauskas
ij12
lk3
3-body (Faddeev eq.)
4-body (Faddeev-Yakubovsky eq.)
𝜙12 = 𝐺0𝑉12Ψ
23
1
𝜙23 = 𝐺0𝑉23Ψ
31
2
𝜙31 = 𝐺0𝑉31Ψ
Ψ = 𝜙12 + 𝜙23 + 𝜙31
Equations for short-ranged pairwise interactions
k
l
ij
K-type(12 components)
H-type(6 components)
𝜙𝑗𝑘 = 𝐺0𝑉𝑗𝑘Ψ
𝐾𝑖𝑗,𝑘𝑙 = 𝐺𝑖𝑗𝑉𝑖𝑗 𝜙𝑗𝑘 + 𝜙𝑖𝑘 ; 𝐻𝑖𝑗
𝑘𝑙 = 𝐺𝑖𝑗𝑉𝑖𝑗𝜙𝑘𝑙
Ψ =
𝑖<𝑗
𝜙𝑖𝑗 =
𝑖<𝑗
(𝐾𝑖𝑗,𝑘𝑙 +𝐾𝑖𝑗,𝑙
𝑘 + 𝐻𝑖𝑗𝑘𝑙)
09/03/2017 5
5-body Faddeev-Yakubovski eq
R. Lazauskas09/03/2017 5
R. Lazauskas
12
5
4 3
1212 12124 3
4 33
4
5 5
3
45 5
09/03/2017 6
Faddeev-Yakubovsky eq
R. Lazauskas09/03/2017 6
R. Lazauskas
12
5
4 3
1212 12124 3
4 33
4
5 5
3
45 5
Problem Number eq.(identical particles)
Number eq. (different particles)
A=2 1 1
A=3 1 3
A=4 2 18
A=5 5 180
A=6 15 2700
A=N 𝑁! 𝑁 − 1 !
2𝑁−1 J.W. Waterhouse : « Pandora »
Merits: Handling of symmetries Boundary conditions for binary channels Easy reduction to subsystemsPrice Overcomplexity
09/03/2017 7
5-body Faddeev-Yakubovski eq
R. Lazauskas09/03/2017 7
R. Lazauskas
12
5
4 3
1212 12124 3
4 33
4
5 5
3
45 5
NUMERICAL SOLUTION*R.L., PhD Thesis, Université Joseph Fourier, Grenoble (2003).• PW decomposition of the components K,H,T,S,F• Radial parts expanded using Lagrange-mesh methodD. Baye, Physics Reports 565 (2015) 1• Resulting linear algebra problem solved using iterative methods• Observables extracted using integral relations
09/03/2017 8
Numerical costs
R. Lazauskas09/03/2017 8
R. Lazauskas
12
5
4 3
1212 12124 3
4 33
4
5 5
3
45 5
Problem Number eq.
(ident particles)
Number eq.
(diff. particles)
PW basis. Radial
disc.
2N 1 1 2 ~N
3N 1 3 ~100 ~N2
4N 2 18 ~104 ~N3
5N 5 180 ~106 ~N4NUMERICAL SOLUTION*R.L., PhD Thesis, Université Joseph Fourier, Grenoble (2003).• PW decomposition of the components K,H,T,S,F• Radial parts expanded using Lagrange-mesh methodD. Baye, Physics Reports 565 (2015) 1• Resulting linear algebra problem solved using iterative methods• Observables extracted using integral relations
09/03/2017 9
Short overview of nuclear problems by FY eq’s
R. Lazauskas09/03/2017 9
R. Lazauskas
1st solution: A. Laverne and C. Gignoux: Nucl. Phys. A 203 (1973) 597 G. Gignoux, A. Laverne, and S. P. Merkuriev Phys. Rev. Lett. 33 (1974) 1350Review: W. Glockle et al., Physics Reports 274 ( 1996) 107-285
3N-problem(Faddeev eq.)
4N-problem (Faddeev eq.)
1st solution: S. P. Merkuriev, S.L. Yakovlev, C. Gignoux, Nucl. Phys. A 431 (1984) 125.Benchmarks: A. Nogga, et al., Phys. Rev. C 65 (2002) 054003 (bound state)R. Lazauskas et al.,Phys. Rev. C 71 (2005) 034004 (n3H scattering)M. Viviani et al., Phys. Rev. C 84 (2011) 054010 (p3He scattering)M. Viviani et al., arXiv:1610.09140 (p3H,n3He scattering)
𝑝 +3 𝐻 ↔ 𝑛 +3 𝐻𝑒
09/03/2017 10
5N problem: n-4He scattering
R. Lazauskas09/03/2017 10
R. Lazauskas
0 4 8 12-90
-60
-30
0
30
60
90
120
2P
3/2
2P
1/2
No Coulomb
+ Coulomb
Ecm
(MeV)
2S
1/2
MT I-III S-wave potential
09/03/2017 11
R. Lazauskas09/03/2017 11
R. Lazauskas
K.M. Nollett et. al., Phys.Rev.Lett.99:022502,2007P. Navratil, INT, Nuclear Physics from Lattice QCD, March 21 - May 27, 2016
Idaho N3LO pot.
n-4He scattering
0 4 8-90
-60
-30
0
30
60
90
120
2P
3/2
2P
1/2
N3LO Idaho
AV18
INOY
Ecm
(MeV)
2S
1/2
09/03/2017 12
n-3H total cross section
R. Lazauskas09/03/2017 12
R. Lazauskas
0.01 0.1 10.0
0.5
1.0
1.5
2.0
2.5
MT I-III
Av.18
Av.18+UIX
INOY
(
b)
E (MeV)
0.1 11.0
1.5
2.0
2.5 I-N3LO+3BF(N2LO)
NLO
Av.18+UIX
INOY
(
b)
E (MeV)
I-N3LO
09/03/2017 13
Contents
R. Lazauskas09/03/2017 13
R. Lazauskas
• Solution of the 3-5 body Faddeev-Yakubovsky equationsfor nuclear systems
• Application of the complex-scaling method to solvecomplicated scattering problems using trivial boundaryconditions
09/03/2017 14
Introduction
R. Lazauskas09/03/2017 14
R. Lazauskas
• In configuration space wave functions extend to infinity!
• Increasingly complex assymptoticbehavior for A>2 systems!!
Collisions
…
What to do?
Take care of the boundary condition
Faddeev-Yakubovsky equations efficiently separates assymptotes of the binary channels
BUT
Number of the reaction channels increases rapidly with A
Even for 3-body systems there may exist infinite number of binary channels
e+He+H(n=1,2,…,)
Complex behavior of the breakup
assymptotes
FIND SOME TRICKS TO AVOID PROBLEMS AT BOUNDARY!
09/03/2017
R. Lazauskas09/03/2017
R. Lazauskas
A dream: to solve scattering problems with a similar ease as bound state one (avoiding complex singularities or boundary conditions)
A. Deltuva, R.L. et al.,PPNP 74 (2014)
• “Calculable” R-matrix E.P. Wigner, Phys. Rev. 70 (1946) 15; P. Descouvemont, D. Baye, RPP 73 (2010) 036301.
• Lorentz integral transform V . D. Efros, W. Leidemann, and G. Orlandini, Phys. Rev. Lett. 78, 4015 (1997).
• Complex energy method F. A. McDonald and J. Nuttal, PRL 23 ,361 (1969) (config. space); E. Uzu, H. Kamada, and Y. Koike, Phys. Rev. C 68, 061001 (2003).; A. Deltuva and A. C. Fonseca, Phys. Rev. C 86, 011001(2012).
• Momentum lattice technique O. A. Rubtsova, V. N. Pomerantsev, and V. I. Kukulin, Phys. Rev. C 79, 064602 (2009).
• Continuum discretization A. Kievsky, M. Viviani, and L. E. Marcucci, Phys. Rev. C 85, 014001 (2012).
• Complex scaling method B. Giraud, K.Kato and A. Ohnishi, J. of Phys. A37 , 11575 (2004) (passing via spectral function);Y. Suzuki, W.Horiuchi, D.Baye, PTP,123 (2010) (passing via spectral function); A. T. Kruppa et al., Phys. Rev. C 75 (2007); R.L. & J. Carbonell, Phys. Rev. C 84, 034002 (2011).
09/03/2017
Complex scaling: resonances, reactions & bound states in an unified formalism
R. Lazauskas09/03/2017
In Quantum-Mechanics problem of N-particle dynamics may be often formulated in time-independent formalism and for the wave functions whose asymptotes contain only nontrivial outgoing waves
• Bound state problem: 𝑆 𝒓𝒊 ≡ 𝟎
• Resonances: 𝑆 𝒓𝒊 ≡ 𝟎
• Reactions due to external probe: 𝑆 𝜌 → ∞ = 𝟎
• Collisions: 𝑆 𝒓𝒊 ≡ 𝟎
R. Lazauskas
Resonances
𝑘𝑟𝑒𝑠 = 𝑘𝑅 − 𝑖𝑘𝐼= |𝑘𝑟𝑒𝑠|𝑒−𝑖𝜂𝑟𝑒𝑠
𝑘𝑅
𝑘𝐼
𝜓(𝜌 → ∞) ∝ 𝑒𝑖𝑘𝑥𝑟𝑥
𝐸𝑠𝑐 − 𝐻0 −
𝑖>𝑗
𝑉𝑖𝑗(𝑟𝑖𝑗) 𝜓 𝒓𝒊 = 𝑆 𝒓𝒊
𝜓𝑠𝑐(𝜌 → ∞) ∝ 𝜓𝑖𝑛 𝒓𝒊 +
𝑐
𝐴𝑐( 𝑘𝑐) 𝑒𝑖|𝑘𝑐|𝑟𝑐
Scattering
B. States
09/03/2017
R. Lazauskas09/03/2017
Complex scaling kills outgoing waves in the assymptote
R. Lazauskas
irer
Complex scaling: resonances, reactions & bound states in an unified formalism
𝜓 𝜌 → ∞ ∝ 𝑒𝑖𝑘𝑥𝑟𝑥𝑒𝑖𝜃= 𝑒𝑖𝑘𝑥𝑟𝑥𝑐𝑜𝑠𝜃𝑒−𝑘𝑥𝑟𝑥𝑠𝑖𝑛𝜃
R. Hartree , J.G. L. Michel, P. Nicolson (1946)J. Nuttal and H. L. Cohen, Phys. Rev. 188 (1969) 1542
09/03/2017
Complex scaling: resonances, reactions & bound states in an unified formalism
R. Lazauskas09/03/2017
In Quantum-Mechanics problem of N-particle dynamics may be often formulated in time-independent formalism and for the wave functions whose asymptotes contain only nontrivial outgoing waves
• Bound state problem: 𝑆 𝒓𝒊 ≡ 𝟎
• Resonances: 𝑆 𝒓𝒊 ≡ 𝟎
• Reactions due to external probe: 𝑆 𝜌 → ∞ = 𝟎
• Collisions: 𝑆 𝒓𝒊 ≡ 𝟎
R. Lazauskas
𝜓(𝜌 → ∞) ∝ 𝑒𝑖𝑘𝑥𝑟𝑥
𝐸𝑠𝑐 − 𝐻0 −
𝑖>𝑗
𝑉𝑖𝑗(𝑟𝑖𝑗) 𝜓 𝒓𝒊 = 𝑆 𝒓𝒊
𝜓𝑠𝑐(𝜌 → ∞) ∝ 𝜓𝑖𝑛 𝒓𝒊 +
𝑐
𝐴𝑐( 𝑘𝑐) 𝑒𝑖|𝑘𝑐|𝑟𝑐
Diverges
09/03/2017
19
R. Lazauskas09/03/2017
19
Complex scaling kills outgoing waves in the assymptote
R. Lazauskas
irer
Complex scaling: resonances, reactions & bound states in an unified formalism
𝜓 𝜌 → ∞ ∝ 𝑒𝑖𝑘𝑥𝑟𝑥𝑒𝑖𝜃= 𝑒𝑖𝑘𝑥𝑟𝑥𝑐𝑜𝑠𝜃𝑒−𝑘𝑥𝑟𝑥𝑠𝑖𝑛𝜃
R. Hartree , J.G. L. Michel, P. Nicolson (1946)J. Nuttal and H. L. Cohen, Phys. Rev. 188 (1969) 1542
• Schrödinger equation in a driven form:
exp. bound for the shortrange pot. term Vs
exp. bound if 0<<p
𝑟Ψ 𝑟 = 𝐹𝑙𝑖𝑛
𝑟 + 𝐹𝑙𝑠𝑐(𝑟)
ℏ2
2𝜇−
𝑑2
𝑑𝑟2+
𝑙 𝑙+1
𝑟2+ 𝑉𝑙 𝑟 − 𝑘2 𝐹𝑙
𝑠𝑐𝑟 =-𝑉𝑙 𝑟 𝐹𝑙
𝑖𝑛𝑟
irer Complex scaling
ℏ2
2𝜇−
𝑑2
𝑒2𝑖𝜃𝑑𝑟2+
𝑙 𝑙+1
𝑒2𝑖𝜃𝑟2+ 𝑉𝑙 𝑟𝑒
𝑖𝜃 − 𝑘2 𝐹𝑙𝑠𝑐
𝑟 =-𝑉𝑙 𝑟𝑒𝑖𝜃 𝐹𝑙
𝑖𝑛𝑟𝑒𝑖𝜃
ℏ2
2𝜇−
𝑑2
𝑑𝑟2+𝑙 𝑙 + 1
𝑟2− 𝑘2 𝐹𝑙
𝑖𝑛𝑟 = 0
Complex scaling: guide for pedestrians
09/03/2017
R. Lazauskas
2 22
2 2 2 2
( 1( )
)( (() ) )
2 l
s i in i
l
i
i
s
l
c
li
d l lV re k
e drV r e
re r
er
• Solution (using standard bound state techniques)
1. Expand c.s. outgoing wave in your favorite basis
2. Convert c.s. Schrödinger equation into linear algebra problem3.
4. Solve linear algebra problem to determine coefficients 𝑐𝑖i. Spectral expansion (Y.Suzuki, W.Horiuchi, K. Kato): determine all
eigenvalues/eigenvectors limit 105 basis st.
ii. (Iterative) linear algebra methods upto 1010 basis st.
5. Extract scattering observables from obtained solution(s) : multiple choice, most efficient way via integral expressions
Ψ𝑠𝑐 𝑟 ≈
𝑖=1
𝑁
𝑐𝑖 𝜓𝑖(𝑟)
square integrable basis functions
Complex coefficients
𝜓𝑗 𝑟 𝑑𝑟 ⋮
𝑗 = 1, . . 𝑁
𝐻𝑗,𝑖𝑐𝑖 = 𝑏𝑗
09/03/2017 21
Review of the method*
R. Lazauskas
• Complex scaling method is very functional, adaptable to almost any b.s. technique. Already successfully applied using: Spline basis Laguerre, HO, Gaussian, correlated Gaussian, Sturmian basis functions Lagrange mesh method
T. Myo, Y. Kikuchi, and K. Katō, Phys. Rev. C 85, 034338 (2012) , Attila Csótó, Phys. Rev. C 49, 2244 (1994), B. Gyarmati and A. T. Kruppa, Phys. Rev. C 34, 95 (1986)
• External probes 2-body, 3-body,4-body systems, including repulsive Coulomb
T.M, K. Kato, S. Aoyama and K. Ikeda PRC63(2001)054313, T.Myo, K. Kato, H. Toki, K. Ikeda, PRC76(2007) 024305
• Collisions, demonstrated to work for:
2-body collisions including Coulomb interaction, Optical potentials, 1
𝑟𝑛potentials with n≥4
3-body scattering including the break-up 3-body scattering including non-local, Optical potential, attractive/repulsive Coulomb interactions 3-body break-up amplitude for n-d & p-d scattering 4-body scattering in 4N systems, including Coulomb
…R.L. & J. Carbonell, Phys. Rev. C 84, 034002 (2011); A. Deltuva, R.L., A.C. Fonseca, “Clusters in Nuclei Vol.3 –LNP 875, 1 (2013); A. Deltuva, R.L. et al.,PPNP 74 (2014) ; R.L., Phys. Rev. C 91, 041001(R) (2015),..
09/03/2017 22
R. Lazauskas
Ref.[23] J. L. Friar et al.:, Phys. Rev. C 51 (1995) 2356.
-
-
n-d scattering for MT I-III potential
09/03/2017 23
R. Lazauskas
2H+12C↔p+13C scattering within 3-body model
12C
p
n
AV18
Optical CH89 pot.
pnC
pC
dC
pC
12
13
12
13
pnC
pC
dC
dC
12
13
12
12
A. Deltuva, A.C. Fonseca and R.L., “Clusters in nuclei vol.3”, (2014) 1.
p+13C d+12C
Ecm (MeV)
Ep (MeV)
0 2.7 28.4
30.6
p+n+12C
4.9
09/03/2017 24
R. Lazauskas
p-3He scattering/realistic NN interactions
p+3Hep+p+2H
p+p+n+p
Ecm (MeV)
Ep (MeV)
0 5.5 22.5
30.07.7
0 30 60 90 120 150 180
1
10
100
d/d
(m
b/s
r)
c.m.
Ep=30 MeV
AV18
Deltuva
This Work
INOY
Deltuva
This Work
*A. Deltuva and A.C. Fonseca, Phys. Rev. C87 (2013) 054002**Exp: B.T. Murdoch et al., Phys. Rev. C21(1984) 2001; J. Birchall, Phys. Rev. C29(1984) 2009.
30 60 90 120 150 180
-0.4
0.0
0.4
0.8
Ep=30 MeV
c.m.
Ay
09/03/2017 25
Price to pay
R. Lazauskas
• C.S. outgoing waves asymtoticaly converges as 𝑒−|𝑘𝑐𝑝|𝑟𝑠𝑖𝑛(𝜃)
Slow convergence for small 𝑘𝑐𝑝, difficulties in close-threshold region
Convergence is faster for large
• One should work with c.s. potential or c.s. basis functions Imposes upper limit for the q to be used, since
𝑉(𝑟)~exp(−𝜇𝑟𝑛) → exp(−𝜇𝑟𝑛𝑒𝑖𝑛𝜃)
Starts to diverge for >p/(2n)
09/03/2017 26
Conclusion
R. Lazauskas
• FY eq. formalism remains reference in few-body scattering calculations. Four-nucleon scattering problem approaches fixed status. The first solution of 5-body FY equations is presented.
•At the same time several promising methods emerge to solve scattering problemsbased on trivial boundary conditions.
•In particular, complex-scaling method has been revived in nuclear physics. This method enables to solve few-body scattering problem employing standard bound state techniques (feasible by almost any config. space bound state technique and requires very limited effort to be implemented)
• Simple extension of the formalism to many-body scattering case
• Very accurate results are already obtained for 3-body and 4-body elastic and breakup scatteringAcknowledgements: The numerical calculations have been performed at IDRIS (CNRS, France). We thank the staff members of the IDRIS computer center for their constant help.