systémes à petit nombre de corps (few-body systems) iphc strasbourg, france rimantas lazauskas,...

Systémes à petit nombre de Systémes à petit nombre de corpscorps

(Few-body systems)(Few-body systems)Rimantas Lazauskas, IPHC Strasbourg, FranceIPHC Strasbourg, France

http://iphc.in2p3.fr/index.php

http://www.cnrs.fr/

Rimantas Lazauskas, IPHC Strasbourg, France

Peculiarities of some simple few-body Peculiarities of some simple few-body systemssystemsPeculiarities of some simple few-body Peculiarities of some simple few-body systemssystems

15 Février, 2009

Few-body physics is…

Goal:Goal: Provide (numerically) exact solutions for QM systems with N2

Means:Means: General formalism for multidisciplinary applications

Motivation:Motivation: Systems with N>2 posses qualitative peculiarities, which even in principle can not be described as a one-particle system

No-adjustments!!

Raw interaction (Hamiltonian) Complete

description of QM systems:

• Bound states• Resonant state• Scattering process• EM & EW reactions



15 Février, 2009

In classical mechanics N>2 systems are non-In classical mechanics N>2 systems are non-integrable (chaotic) unlike quantum N>2 systems integrable (chaotic) unlike quantum N>2 systems

Thomas effect: N>2 boson system driven by the Thomas effect: N>2 boson system driven by the contact contact interactions collapse independently of C interactions collapse independently of C (Thomas, L. H. Phys. Rev. 47 (1935) 903 .)

Efimov states: binding energy of N=3 system can Efimov states: binding energy of N=3 system can decrease when two-particle interaction is made decrease when two-particle interaction is made more attractive. In addition, infinitely many weakly more attractive. In addition, infinitely many weakly bound three-boson states appear in the limit Bbound three-boson states appear in the limit B22 (Efimov, V. Phys. Lett. B. 33, (1970) 563., seen exp: T. Kraemer et al., Nature 440 (2006) 315)

( ) ( )V r C r r



15 Février, 2009

Bound states

Few-body physics is …

Wave function of finite size (bound to the box)

Variational method, multiple ways to discretize wave function and solve the Schrödinger eq.

Scattering

Wave functions extend to infinity.



15 Février, 2009

• Schrödinger eq. is not enough (can not provide an unique solution)

• Faddeev-Yakubovski equationsFaddeev-Yakubovski equations

…

The Formalism



15 Février, 2009

0ij ijF G V

12 12 23 13

23 23 12 23

13 13 23 12

12 0

23

3

1

0

13 0

( , ) ( , )ij ij iji j

F V F F

F V F F

F V F F

E V H

E

x y F

V

H

x y

H

E V

,

l

ij k ik jkij ij

kl

ij klij ij

G VK F F

G VH F

x13x23

y23 y13y12

x1212

3

12

3

12

3 4

y4

412,3K 34

12H1

2 3

1

2 3

z

yz

x x

12 6

, ,, ,

4 4

, , 1 , 1, ,( , , ) ( , ) ( , )l kll kll l kl kl

ij k ij k ij ijij k ijij k iji j k l i j k l

x y z y yx xK z H z

14 24

23 13

34 12

12 3

12

64

4 1 2 4412,30 12 23,1 23,4 13,4 13,2

2 10 12 34,1 34,2

12

12

E H V K V K K K K H H

E H V H V K K H

The Formalism

3-body Faddeev eq. 4-body Faddeev-Yakubovski eq.



15 Février, 2009

Numerical solution

Radial parts of wave function F (x,y,z) are developed in the basis of piecewise splines, converting

differential equations into linear algebra problem: (A-E·B)b=c

,( , )( , , ) ( ) ( ) ( )ˆ ˆ ˆx y z

x y z

x y z

l l l

LM

lll

lll

F x y zK x y z Y x Y y Y zx y z

Solution is searched by decomposing FY components in tripolar harmonic basis:

Boundary conditions:Boundary conditions:

3

ˆ ˆ( , , ) ( , ) ( ) ( ) ( )

( , , ) 0

x y z x y

z Z z

x y z

l l l l lHe l l lz

l l l

F x y z f x y h kz S k h kz scattering

F x y z bound states

4y

4

412,3K1

2 3

1

2 3

z

yz

x x3412H



15 Février, 2009

Interest

Fundamental quantum mechanics problemFundamental quantum mechanics problem

Application:Application: any non-relativistic quantum mechanical system

In particular:In particular:

Nuclear physics:Nuclear physics: to test nuclear interaction models« In the past quarter century physicists have devoted a huge amount of experimentation and mental labor to this problem –probably more man-hours than have been given to any other scientific question in the history of mankind. »

What Holds the Nucleus Together? by Hans A. Bethe

Scientific American, September 1953

Cold atom systemsCold atom systems…



15 Février, 2009

e-0 2 4 6 8

-0,2

-0,1

0,0

0,1

0,2

1Eg-E

H

3EU-E

H

E

r

6 8 10 12 14 16 18 20-0,0010

-0,0008

-0,0006

-0,0004

-0,0002

0,0000

0,0002

0,0004

0,0006

0,0008

0,0010

0 2 4 6 8-0,2

-0,1

0,0

0,1

0,2

1Eg-E

H

3EU-E

H

E

r

1Eg-E

H

3EU-E

H

E

r

Landau, Herringp+

e-

p+

20 nodes in E=0 p+-H singlet wave function, indicating presence of 20 vibrational bound states (in agreement with variational calculations) 1a=-29.3 (a. u.) 2 nodes in p+-H wave function at E=0, what does it mean? 3a= 750 +/- 5(a. u.)

p+-H scattering



15 Février, 2009

p+-H scatteringH2

+ 2pu excited state

0.0 2.0x10-7 4.0x10-7 6.0x10-7 8.0x10-7 1.0x10-6

-1.33x10-3

-1.32x10-3

-1.31x10-3

-1.30x10-3

-1.29x10-3

-1.28x10-3

-1.27x10-3

k ctg()

k ctg() = A + B1*k2 + B2*k4

Parameter Value Error------------------------------------------------------------A -0,00133 6,20158E-8B1 51,81769 0,26958------------------------------------------------------------

k0=i(0,001444 +/- 0,00004)

E0=-1,135E-9 +/- 0,035E-9

3-body results

k ct

g

k2 (u.a.)

• Big scattering lengths are due to the existence of nearthreshold negative energy pole (bound state)

• Position of this pole can be extracted by the analytical continuation of effective range formula:

2

0

1kctg bka

E=(-1.125E=(-1.125±0.03)±0.03)*10*10-9-9 (a. u.) (a. u.)

20 0

0

10bk k

a

EEvv=-1.08505*10=-1.08505*10-9-9 (a. u.) (a. u.)

kIm

kRe



15 Février, 2009

p+-H scatteringH2

+ 2pu excited state

-

+ -

+-

+ -

+

61r 7

1r

- ++

4r

2411

12 pm c r

c

r

3.285*10-12(a.u.)

• Casimir-Polder (retardation) effect

• Radiative corrections (Lamb shift)

1 10 100 1000

-0.02

-0.01

0.00

0.01

0.02

Ground state Excited state E=0

(r)

rp+-H

(a.u.)

J.C, R.L, D.Delande,L.Hilico, S. Kiliç: physics/0207007 (2002)

• Relativistic motion of the electron (Dirac Ham.)

- +- +

4.73*10-13(a.u.)

~10-15(a.u.)



15 Février, 2009

He atoms are

chemically the most

inert

(4He)2 – the weakest bound molecule in its ground state

(4He)3 excited excited

state is state is Efimov state

0 2 4 6-15

-10

-5

0

5

10

15

V (K

)

rHe-He

(A)

(4He)n have nearthreshold

excited and resonant states

He molecules

P. Bruhl et al. Phys. Rev. Lett 95 (2005) 063002

Superfluidity of Heat low T



15 Février, 2009

Direct bound state calculations are possible

Trimer excited state – Efimov state

B (mK) <T> (mK) xrms (Å) PDimer

He2 1.3035 99.42 35.54 1.0

He3 (g.s.) 126.37 1658 10.95 0.3301

He3 (e.s.) 2.268 122.1 104.3 0.7611

0 200 400

-1

0

1

2 4He trimer wave functions

(y)

y (Å)

He3

He*3

He-(He)2 at E=0

y

x

Resolved ambiguity in He-(He)2 scattering length

aa00=115.6 Å=115.6 Å

He trimers



15 Février, 2009

4He tetramer ground state He4 L=0 ground state binding energy B=558 mKB=558 mK and its properties are

calculated by solving FY equations. Tripolar harmonic basis contained

amplitudes up to max(lx,ly,lz) 8, which ensures three-digit accuracy.

0 10 20 300,00

0,04

0,08

0,12 One particle correlation function in He

4

(x 12

)

x12

(Å)

L=0

B (mK) <T> (mK) xrms (Å)

He4 557.7 4107 8.40

0 10 20 300,00

0,15

0,30

Different configuration overlap functionsin He

4 ground state

He-He3

He-He*3

He2-He

2

(z

)

z (Å)



15 Février, 2009

He-He3 elastic scattering

We predict a resonant He-He3 scattering length aa00=103.7 Å=103.7 Å

kIm

kRe

0 25 50 75 100-0.02

0.00

0.02

0.04

0.064He tetramer wave functions

(z)

z (Å)

He4

He*4

He-He3 at E=0

xy

z

0-energy scattering wave function contains two nodes in He-He3 separation

direction, indicating existence of He4 L=0 excited state close to He3 threshold.



15 Février, 2009

4He tetramer excited stateaa0 0 (Å)(Å) EEHe He (mK)(mK)

He-HeHe-He22 100.2 1.3035

He-HeHe-He22 115.6 0.965

He-HeHe-He33 103.7 1.09

E (mK) <T> (mK) rrms (Å)

He4 560.0 5388 8.40

He*4 127.5 1900 34.4

Using FY equations we can calculate rotational states with the same ease as L=0 ones

Possible existence of weakly bound He3 and He4 rotational states was explored:

• by enhancing He-He interaction• analyzing low energy L>0 scattering wave functions

He3 and He4 do not have rotational statesdo not have rotational states

0 25 50 75 100-0,02

0,00

0,02

0,04

0,064He tetramer wave functions

(z)

z (Å)

He4

He*4

He-He3 at E=0



15 Février, 2009

Multineutron

Controversial results in GANILControversial results in GANILF.M. Marqués et al: Phys. Rev. C 65 (2002) 044006 et arxiv:nucl-ex/0504009

14Be nn

nnn

p

pn

np

pn

nn

n

np

pn

np

pn

n10Be +n

n nn

E. Rich et al: proceedings Exxon conference 2004

nnn

n p

pn

n n

n p

pn +

nn n

np

n

p+

88HeHe 66LiLi22HH



15 Février, 2009

0 1 2 3 4 5-3

-2

-1

0

1

2

3

1S0 waves

Aziz (3He) MT I-III (nn) Av.18 (nn)

(m/h

2 )V (

MeV

*fm

2 )

[K

*A2 ]

r (fm) [A]

Quest for multineutrons

nn

nnn

p

pn

np

pn

nn

n

np

pn

np

pn

n +n

n nn

Recent experiment at GANIL suggested possible existence of a bound 4 neutron Recent experiment at GANIL suggested possible existence of a bound 4 neutron state (tetraneutron)state (tetraneutron)

1414BeBe 1010BeBe

F.M. Marqués et al: Phys. Rev. C 65 (2002) 044006.

Interaction nn not well defined in P-waves, but still <20%…

nn not bound, ann=-18.59 fm (it needs only Enn0.110 MeV)

If they were bosons… E(3n) 0.94 MeV, E(4n) 9.0MeV,…

But how to bind (nn)* ? nn S-waves untouchables!!!

Fermions sometimes consolidate… 3He atomcules



15 Février, 2009

Quest for multineutrons

• Effect of 3BF?… less than 1%!!!

• Enhancment of 3BF?

• Higher partial waves (P,D,F..)? … should be too strong,

Too strong charge dependency breaking, 3.52n becomes resonant in P waves E=5.2-2.5i MeV Violating nuclear properties: inversing order of nuclear states 4H becomes strongly bound B>40 MeV!!!

1S0

P,D,F,…

Violating nuclear properties Btriton 211 MeV

Abnormal nuclear matter density

… should be too strong,



15 Février, 2009

Basics of 2-body resonancesBasics of 2-body resonances

Even if not bound, they still should be somethere…

Re(k)

Im(k)2 2

2

kE

l=0

l>0

Resonance 2RE E i

S wave P wave Total

k

1S0: ann=-18.59 fm (virtual state

Enn0.110 MeV)

l=0l=0



15 Février, 2009

Basics of 2-body resonancesBasics of 2-body resonances


Resonance is complex eigenvalue of the Hamiltonian: their eigenfunctions are divergent!!!

-1.5

-1.0

-0.5

-1.0 -0.5 0.5

ERe

EIm

gˆ

2R iH E



15 Février, 2009

Multineutron resonances


Resonance is complex eigenvalue of the Hamiltonian: their eigenfunctions are divergent!!!

-1.5

-1.0

-0.5

-1.0 -0.5 0.5

ERe

EIm

g

But what is a resonance in multiparticle But what is a resonance in multiparticle system without any bound statessystem without any bound states

nnn

n p

pn

n n

n p

pn +

nn n

np

n

p+

88BeBe 66LiLi22HHD. Beaumel et al., (IPN Orsay)

ˆ2R iH E



15 Février, 2009

3n and 4n resonances are explored using realistic NN interaction Reid93 in conjunction with additional attractive force:

-6

-4

-2

1 2 3

0-

1-

2-

EIm

g (M

eV

)

ERe

(MeV)

-6

-4

-2

-1 1 2 3 E

Re (MeV)

EIm

g (M

eV)

0+

1+

2+

All the resonance trajectories end up in III-rd energy quadrant, with |E img|>6 MeV

Multineutron resonancesMultineutron resonances



15 Février, 2009

• Rigorous description of very cold p-H scattering

• Prediction of H2+ 2pu excited state, with proton separation energy only: EEvv=-1.08*10=-1.08*10-9-9 (a. u.) (a. u.)

• We predict, for the first time, 4He -(4He)3 scattering length aa00=103.7 Å=103.7 Å..

• Resonant value of this length, indicates existence of weakly bound (4He)4 excited state. We predict its binding energy (relative to He3 ground state) B=1.09 mK.B=1.09 mK.

• Non-existence of (4He)3 and (4He)4 rotational states have been shown.

• We have tested if some ‘brave’ experimental claims on the possible existence of bound or resonant tetraneutron can be supported with theoretical background

• We have demonstrated that theoretical nuclear interaction models exclude existence of both bound or physically observable resonant pure neutron systems with A≤4.

SummarySummary• I have presented Faddeev-Yakubovski equations, which is

rigorous and powerful method in describing QM few particle systems

• This method is very general and can be applied to study

very different physical systems.

systémes à petit nombre de corps (few-body systems) iphc strasbourg, france rimantas lazauskas,...

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