Helium Trimer in the Framework Helium Trimer in the Framework of Faddeev Approachof Faddeev Approach

Elena KolganovaElena KolganovaBLTP JINR, Dubna, BLTP JINR, Dubna,

RussiaRussia

In collaboration with

A.K.Motovilov (JINR Dubna)

W.Sandhas (PI Bonn)

1Few-Body Conference, August 31, 2009 Elena Kolganova (JINR, Dubna)

Few-Body Conference, August 31, 2009 Elena Kolganova (JINR, Dubna)2

First observation by Luo et al. (1993) and Schöllkopf, Toennies (1994)

First measurement of the bond length by Grisenti et al.(2000)

Estimation of the binding energy and scattering length

Two-body and three-body, experiment4He - 4He

o0.3 80.2 181.1 mK 104 Ad scl

-71 mK 10 eV

o

52 4AR

4He - 4He - 4He

Experiment – Toennies et al. JCP 104, 1155 (1996), JCP 117, 1544 (2002)0.4

0.51.1 nmd

Trimer pair distance

Theory – variational, hyperspherical, Faddeev equations (integral, differential)

126 mK 2.28 mKgs exE E

Few-Body Conference, August 31, 2009 Elena Kolganova (JINR, Dubna)3

Two-body, theory

Potential models: Aziz et al. – HFD-B (1987), LM2M2 (1991),Tang et al. – TTY (1995)

4He2

( ), the dimer wave functiond r

4He – 4He potential (LM2M2)

( ), the dimer wave functiond r

where

Few-Body Conference, August 31, 2009 Elena Kolganova (JINR, Dubna)4

Potential models: Aziz et al. – HFD-B (1987), LM2M2 (1991)

Two-body, theory4He - 4He

Tang et al. – TTY (1995)

Few-Body Conference, August 31, 2009 Elena Kolganova (JINR, Dubna)5

Three-body, theory formalismFaddeev differential equations for hard-core potentials

Few-Body Conference, August 31, 2009 Elena Kolganova (JINR, Dubna)6

Three-body, theory formalism

Faddeev differential equations for hard-core potentials

Boundary condition model is mathematically rigorous ! Merkuriev, Motovilov, LMP 7 (1983) 497; Motovilov, VLU 22 (1983) 76; Merkuriev, Motovilov, Yakovlev, TMP 94(1993)306 JPB 31(1998) 1279

E.Kolganova (Dubna)E.Kolganova (Dubna) 77

When the total angular momentum L of the system is fixed, the three-body dynamics is constrained onto three-dimensional internal space [5], which can be parametrized by coordinates

ˆ ˆ| |, | |, cos ( , )x y z x y x y

[5] - V.V.Kostrykin,A.A.Kvitsinsky,S.P.Merkuriev, Few-Body Syst. 6 (1989) 97

2 2 2 2

2 2 2 2

2 2 2 2

2

2

( ) ( )

x c x s y c s x y z

y s x c y c s x y z

x y z c s x y z c s x y

0( ) ( , , ) ( , , )H V E F x y z V F x y z

2 2

1/ 20 2 2 2 2

1 1( ) (1 )H z

x y x y z z

For zero angular momentum the Faddeev equations in internal space are given by the set of three coupled three-dimensional equations

0( ) ( , , ) ( , , )H V E V

or in hyperspherical coordinates 2 2 ˆ ˆ, tan / , ( , )x y y x x y

1/ 20

exp( )( , , ) ( ) exp( ) ( ; ) ( , ; )d

i Ex y x ipy a E A E

7Few-Body Conference, August 31, 2009 Elena Kolganova (JINR, Dubna)

E.Kolganova (Dubna)E.Kolganova (Dubna) 88

For computational purposes, one can reduce the dimension by expanding the Faddeev components into an auxiliary basis, at the expense of dealing with an infinite number of partial equations. Expanding the function F in a series of bispherical harmonics

2 2

0 2 2 2 2

( 1) ( 1)l lH

x y x y

( )

,

( , )( , , ) | 0l

l

x yF x y l

xy

One can obtain the partial equation

' '

1( ) ( ) ( )

0 ' '' ' 1

( ) ( , ) ( , , ) ( , )l ll l

l

H V E x y V d h x y x y

2 2 2 2

2 2 2 2

2

2

x c x s y c s x y

y s x c y c s x y

The asymptotic boundary conditions for the partial-wave Faddeev components of the 2’2,3 scattering wave function for and/or reads ’ and/or y’ reads

( ) ( )' ' ' ' ' ' ' ' ' '

exp( )( , ) ( ) ( ) ( ) ( ) ( ) ( , )l ll l l l l

i Ex y x j py x h py a p A p

Where p is the relative moment conjugate to Jacoby variable y, E is the scattering energy,

stand for the spherical Bessel and Hankel functions2

and' ',E p j h

8Few-Body Conference, August 31, 2009 Elena Kolganova (JINR, Dubna)

Bogolyubov Conference, August 24, 2009 Elena Kolganova (JINR, Dubna)9

Three-body, theorybound states and scattering4He3 and 3He 4He2

9Few-Body Conference, August 31, 2009 Elena Kolganova (JINR, Dubna)

Bogolyubov Conference, August 24, 2009 Elena Kolganova (JINR, Dubna)10

Three-body, theory bound states

4He3

V. Kolontsov

10Few-Body Conference, August 31, 2009 Elena Kolganova (JINR, Dubna)

• My collaborators - Prof. A.K.Motovilov and Prof. W.Sandhas

•Diploma student – V.Kolontsov

•Alexander von Humboldt foundation

•Heisenberg-Landau Program

Few-Body Conference, August 31, 2009 Elena Kolganova (JINR, Dubna)11

Thank you!

Acknowledgements

Outgoing results and future directions

• Scattering calculations, especially, above three-body threshold

• Resonances calculations,

to see the influence of applying 3D eq. to their values and positionsKolganova E.A. and Motovilov A.K. Mechanism of the Emergence of Efimov States in the 4He Trimer Yad. Fiz. 1999. V. 62. P. 1253–1267 [Phys. At. Nucl. 62, 1179 (1999)].