exact analytic solutions in three-body problems

Exact Analytic Exact Analytic Solutions in Three-Solutions in Three-

Body ProblemsBody ProblemsN.TakibayevN.Takibayev

Institute of Experimental and Institute of Experimental and Theoretical Physics, Kazakh Theoretical Physics, Kazakh National University, AlmatyNational University, Almaty

[email protected]@nursat.kz

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Takibayev N "Exact Analytic Solutions..."

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Exact analytic solutions are obtained in three-body problem for the scattering of light particle on the system of two fixed centers in case when pair potentials have a separable form.

By means of analytic solutions we can demonstrate how the new resonances appear in these three-body systems. It is remarkable that energies and widths of new three-body resonance states depend on distance between two fixed centers.

In the frame of this method the resonance scattering of neutrons on systems of two fixed nuclei is also considered when two-body neutron-nucleus amplitudes have the Breit-Wigner’s form.

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Analytic solutions to the problem of light-particle scattering on two heavy particles are found if two associated simplifications are acting in the system: a) the limit of , where - mass of the light particle and - mass of heavy particles, identical for simplicity,b) pair t-matrices have separable forms:

||V iiii ||t iiii

iiii G ||/1 0

1

0/ Mm 1Mm

32 MMM

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The mathematically rigorous solution to three-body problems was given by Faddeev. The set of Faddeev equations for the T-matrix elements can be written in the form

kjik0i

3

1kijiij T)Z(Gtt)Z(T

i, j, k = 1, 2, 3

ikik 1 ij

3

1j,i

TT

|| jjijiiijiij PtT

If pair amplitudes are separable we can introduce the P-matrix via connected part of T-matrix:

where , and

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jiZG jiij ,|)(| 0 0 ii

ljlill

ijij PP Then we come to the set of equations:

where is the undiagonal matrix

;

In the limit of we get simplifications 0/ Mm

mpmpZ iii

// 201

20 001 pp

, - initial momentum of

the light particle ,

and|)(|)( 12 pq

|)(|)( 13 pq ,

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)( 032 p

The enhancement factors in the t matrices for these pairs become functions only of the initial energy of the light particle - that is, they are functions of its initial momentum:

In common case, when the two-body interaction between heavy particles exists we can determine “the nuclear equation” :

'1''1''1''11''1

'11'11 PVVP .

Here the effective potential between heavy particles is:

'113,2,

'11 )( klklkllkl

MV

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which can be determined with “electronic equation”:

3,2,.,3,2

klMM kllklk

Above terms are taken from the well-known Born-Oppenheimer approximation.

This approximation is the assumption that the electronic motion and the nuclear motion in molecules can be separated. The electronic wave-function (i.e. the wave-function of light particle) depends upon the nuclear positions but not upon their velocities, i.e. the nuclear motion is so much slower than electron motion that they can be considered to be fixed.

See: Takibayev N.Zh. “Class of Model Problems in Three-Body Quantum Mechanics That Admit Exact Solutions”// Physics of Atomic Nuclei, V 71, No 3, p 460-468, 2008

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We label the heavy-particle variables at the exit from the interaction region with a prime; at the entrance, they carry no primes.

Here

In our case of fixed heavy centers we have to solve electronic equation, which gives total description of light particle scattering amplitude on these centers.

We can write );(0

)()(2 022

0

3223 ppf

ipp

ppm

And taking into account the conservation of the total momentum in the three-particle system we represent this potential in the integral form

)'exp();()exp()',( 302323223 priprJprirdpp

'32 ppp

);()exp();( 0023 ppfpripdprJ

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Then we introduce the Fourier transform of the solution

)''exp();',()exp(')',( 0 kkllklkllk pripppMpripdpdrrM

and obtain

)',()();()'();()',( 000 rrMpprJrrprJrrM kllklk

Since delta-functions remove the integration on the right-hand side, the equation for is reduced to an extremely simple form, and the solution of problem of light-particle scattering on two fixed centers can be represented in the analytical form

)'r,r(K)r(BI

1)'r,r(M ij

ii

ij

where i,j = 2,3 – numbers of heavy centers, r, r’ - radius-vectors of the initial and final fixed scattering centers in c.m., respectively.

)',( rrM lk

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Elements of diagonal matrix are

ijijijii )r(J)r(J)r(B

And elements of matrix K are given as

ij,)'rr(B)'r,r(K 1iiiii

ij,)'rr()r(J)'r,r(K ijij

And we can write out the two modes of solution:

)'rr()r(M)'rr()r(M)'r,r(M

)'()()(

1),( 0 rrrJ

rBIprM ij

ii

ij

where

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)'()()(

1),( 1

0 rrrBrBI

prM iii

ii

ii

In case when pair potentials are sums of separable terms, the expressions above have to be considered as matrix expressions with respect to additional indices.

We assume that heavy particles are strictly fixed in coordinate space, and we introduce their wave functions in the form:

2

2

2

)(exp)( n

n

RrCr

for the n=2,3. As 1| nn2/322 )( C, then

The scattering amplitude is fin T ||

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where || 321 in 1with - the wave function for the light particle.

In the limit 0 , we come to an expression for the three particle scattering amplitude in the form:

),'(||),( llklkk RrMRr

It should be emphasized that the positions of the heavy particles are specified in the c.m. frame of all three particles. This concerns their coordinates and momenta. It is then obvious that

2/2 bR

2/3 bR

Zeros of correspond to the poles of three-body amplitude in complex plane of energy.

)BI(detD )2/,( 2

00 mpErM

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- the form-factor of S-wave pair potential known as Yamaguchi potential, - radius of pair forces, we can write

In case of ,/1

|)(22

p

Npp

20

11 )/1( pi

We adduce expressions:

mN

42

D

J

J

JpbM

1),( 0

220

2220

0

/1

)2/exp(

)/1(

)2/exp()2/exp(4

p

b

pb

ibpbJ

1

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Zeros of the function D = (1−ηJ) correspond to the poles of the three-body amplitude M(b; p0).

Depending on where these zeros lie in the complex plane of p0 — on the positive imaginary half-axis, on the negative imaginary half-axis, or in the vicinity of the real axis — the singularities of the amplitude correspond to bound, virtual, or resonance states.

Introducing the notation IRresres ikkpk /,0 2/~ bb

we can get relations for the wave numbers of bound, virtual and resonant three-particle states:

)~

/()~

sin()~

exp()]~

exp(1[1

)1(2 22

bkbkbkbk

kkk

RRII

IRI

2222

22222

]1[)]~

cos()~

exp()~

)[exp(~

/2()~

exp(]1[

4]1[

IRIIR

IRIR

kbkbkbbbkk

kkkk

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In the limit β →∞ (contact pair potentials) zeros of D give the values of

2/0pbx )( IR xixx

)exp()cos( IRBI xxxx

)xexp()xsin(x IRR

where

which can be found from the algebraic equations:

2/rxB and the pair bound energy is mEB 2/2Three-body resonances depend on b. This dependence is an important feature of the three-body system. Note that bound, virtual and quasi-stationary states will be moving in complex plane with changing the distance between scattering centers.

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Now we consider the problem of neutron scattering on two fixed centers subsystem if the two-body scattering amplitudes have the Breit-Wigner resonant form:

2/

2/

)(

1

22.

2

iEEE

tR

i

The energy and width of resonance are determined with real and imaginary parts of resonance wave number:

mppE IRR 2/)( 22,

22,2, mpp IR /||2 2,2,2

Here, index 2 marks two-body parameters (below index 3 will mark three-body parameters).

The BW representation can be transformed to separable form if we determine a form-factor as

)(2/)( 2 Epi

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Then we obtain )2/( 22,1 iEE Ri

So, three-body exact solutions can be obtained in case of Breit-Wigner pair t-matrices, too.

Let we have an isolated pair resonance acting in S-wave. Then we can write the elements of matrix J as

rp

riprJJJ

Rjiij

2,

02 )exp(

2)(

Zeros of D are determined by equation

0)r(J2

iEE)r(J2

iEE 22,R

22,R

And we mark values of ),Re( 03, ppR 33, iEE Rres

when

and

0D

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There are two sets of three-body resonances:

rp

rpEE

R

RRR

2,

3,22,3,

)cos(

2

rp

rp

R

R

2,

3,23

)sin(1

rp

rpEE

R

RRR

2,

3,22,3,

)cos(

2

rp

rp

R

R

2,

3,23

)sin(1

Some of them are situated at energy scale above the energy of two-body resonance, and others - under this energy. Some of three-body resonances will have more narrowed width, others – more widened in comparison with the width of two-body resonance Note, there are no principal difficulties in including more complicated forms of two-body separable potentials and other partial components into the model.

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Example 1. The neutron resonance scattering on subsystem of two fixed α-particles.

Note that repulsive forces act between nucleon and α-particle in S-wave. It is known that n,α-scattering amplitude has resonances in P-waves. We take into account the resonance state in P-wave (J = 3/2) with parameters: ;MeVER 9.02, MeV6.02 It is interesting that p,α-scattering amplitude has resonances in this P-wave too, but with parameters: MeVER 9.12, MeV5.12

i.e. the position of p,α – resonance is above ~ nearly.

In framework of the method we obtained exact analytic solutions for “quasi-bound” states when the widths .

Near this points may cross zero and be even positive. Our calculations concerned a region of low energy only.

3,Ip

MeV1

03

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Table 1. Quasi bound states of neutron and proton with subsystem of two -particles in model of fixed scatterer centers.

3,REParticle and subsystem

(in MeV) b (in fm)

n + α,α0,88 18

1,37 31

p + α,α1,78 14

2,36 24

The estimations of resonant (p,α,α)-parameters have been performed without ordinary repulsive Coulomb force between the proton and α-particles.

The inclusion of repulsive Coulomb forces gives usually the results of widening of distance between centers and shifting of three-body resonance levels to higher energies.

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Example 2. The neutron resonance scattering on subsystem of two fixed nuclei. We take into account the lowest isolated two-body resonance state of neutron scattering on nucleus.

Quasi bound states of neutron with subsystem of two nuclei and in model of fixed scatterer centers.

O16

Mg24

2,RE 2 3,RE

On 16 )( 1616 OOn

Mgn 24 )( 2424 MgMgn

Two-body system

(in keV) (in keV)

Neutron and

subsystem (in keV)

b (in fm)

435 40231 10.53

286 12.179

84 1382 63.27

117 70.37

Calculations give the points where 03

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p,α,α;2,2 MeV

n,α,α;1,3 MeV

p + e- → n + ν

It had been found that a new mode of three-particle resonances appears in the process of light particle scattering on two fixed centers [1].

b

It should be noted that energies and widths of these resonances depend upon b - distance between centers. It means that the resonance energy and width are functions not only from parameters of two-body interactions but from lattice parameter – b, too. There are some values of parameter b (bres1, bres2, ..) when resonance widths become very small.

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For example, the system consisted of one free neutron and two fixed alpha-particles has the resonance energy ER ≈ 1.3 MeV with the width close to zero when b ≈ 30 fm.

Then, the analogous system consisted of proton and two fixed alpha-particles has the resonance too, with the energy ER ≈ 2.2 MeV and very small width.

It is remarkable that the resonance of p,α,α-system has the energy situated above the resonant energy of n,α,α-system

~ 1 MeV.In this case the transition between these resonance states becomes permissible in energetic aspect.

Protons can penetrate in crystal which have kinetic energies Ep >2.2 MeV and be captured in resonance states ER.

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Moreover, owing to reaction: p + e- → n + ν these protons can turn into the neutrons transiting from p,α,α-resonant states to n,α,α-resonant states situated below at energy scale. Neutrinos produced in the reactions will have the energy Eν ≈ 0.16 MeV, leave the lattice and then the star.

(p,α,α) → p' + (α,α) ; Ep' ≈ 2.2 MeV – Eν - E

(p,α,α) – resonance state; ER ≈ 2.2 MeV

p'

(n,α,α) – resonance state; ER ≈ 1.3 MeVp + e- → n + ν

epn

Eν ≈ 0.16 MeV

E < 2.1 MeV

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So, the process of stellar cooling may be the result of neutrinos radiation.

Above model of ideal crystalline lattice is considered where α-particles are fixed at nodes of the lattice. We supposed that distances between nodes of this perfect lattice become small as the result of very big pressure from outside.

This simple model may be interesting in astrophysics because the Helium crystal core can play the role of low energy neutrino generator.

The n,α,α-resonance state has a very big life-time when the lattice has b ≈ 31 fm. Moreover, the decay of neutron will be suppressed inside of crystal.

The process may be running without nuclear reactions – the calm neutrinos evaporation.

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}{

),,(),,(),(

nep

npp

}'{

),('),,( *

epn

epn

And then there may be a decay of neutrons because of the distortion of lattice

Otherwise, big number of neutrons can lead to nuclear explosion inside of star or generate neutrino flares.

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In case of distortions in the lattice the conditions for existence of neutron resonant state will not be supported and via β-decay neutron will turn into proton, producing electron and antineutrino. The distortions may be created periodically by satellites of star.

As a result the star can generate long time clouds of neutrinos spreading outside in space. Huge mass of neutrinos generated by star or galaxy center will be expanding and increasing far out to the galaxy frontiers in order to cripple the motion of satellites.

The situation with neutrinos generation can be similar in case of the lattices of and particularly, and of more heavy nuclei...

Mg24O16

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Thank you for your attention!

The analogous model of particle interactions with the quark lattice may also be interesting in astrophysical aspects. May be these models can be useful for problem of dark matter.

p,α,α;2,2 MeV

n,α,α;1,3 MeV

p + e- → n + ν

b

exact analytic solutions in three-body problems

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