09/03/2017 1 r. lazauskas · 2017-03-10 · • observables extracted using integral relations....

09/03/2017 1

R. Lazauskas

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. 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).

http://www.krys-delanne.com/photos/tele3.jpg


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Faddeev-Yakubovsky eq


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)

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5-body Faddeev-Yakubovski eq


R. Lazauskas

12

5

4 3

1212 12124 3

4 33

4

5 5

3

45 5

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Faddeev-Yakubovsky eq


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

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5-body Faddeev-Yakubovski eq


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

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Numerical costs


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

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Short overview of nuclear problems by FY eq’s


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

https://arxiv.org/abs/1610.09140

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5N problem: n-4He scattering


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. 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

http://www.int.washington.edu/PROGRAMS/16-1/

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n-3H total cross section


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. 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. 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


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

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Complex scaling kills outgoing waves in the assymptote

R. Lazauskas

irer


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



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

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19


19

Complex scaling kills outgoing waves in the assymptote

R. Lazauskas

irer


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),..

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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.

09/03/2017 1 r. lazauskas · 2017-03-10 · • observables extracted using integral relations....

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