Exact Analytic Solutions in Three-Body Problems

1. Exact Analytic Solutions in Three-Body Problems N.Takibayev Institute of Experimental and Theoretical Physics, Kazakh National University, Almaty teta@nursat.kz

2. 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. Takibayev N "Exact Analytic Solutions..."

3. 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: Takibayev N "Exact Analytic Solutions..."

4. 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 i, j, k = 1, 2, 3 where , and If pair amplitudes are separable we can introduce the P-matrix via connected part of T-matrix: Takibayev N "Exact Analytic Solutions..."

5. Then we come to the set of equations: where is the undiagonal matrix ; In the limit of we get simplifications - initial momentum of the light particle , , and , Takibayev N "Exact Analytic Solutions..."

6. 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” : . Here the effective potential between heavy particles is: Takibayev N "Exact Analytic Solutions..."

7. which can be determined with “electronic equation”: 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 Takibayev N "Exact Analytic Solutions..."

8. 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 And taking into account the conservation of the total momentum in the three-particle system we represent this potential in the integral form We label the heavy-particle variables at the exit from the interaction region with a prime; at the entrance, they carry no primes. Here Takibayev N "Exact Analytic Solutions..."

9. Then we introduce the Fourier transform of the solution and obtain 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 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. Takibayev N "Exact Analytic Solutions..."

10. Elements of diagonal matrix are And elements of matrix K are given as And we can write out the two modes of solution: where Takibayev N "Exact Analytic Solutions..."

11. 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: for the n=2,3. As , then The scattering amplitudeis Takibayev N "Exact Analytic Solutions..."

12. where - the wave function for the light particle. with In the limit , we come to an expression for the three particle scattering amplitudein the form: 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 Zeros of correspond to the poles of three-body amplitude in complex plane of energy. Takibayev N "Exact Analytic Solutions..."

13. In case of - the form-factor of S-wave pair potential known as Yamaguchi potential, - radius of pair forces, we can write We adduce expressions: Takibayev N "Exact Analytic Solutions..."

14. Zeros of the function D= (1−ηJ)correspondto the poles of the three-body amplitudeM(b; p0). Dependingon where these zeros lie in the complex planeof p0 —on the positive imaginary half-axis, onthe negative imaginary half-axis, or in the vicinityof the real axis—the singularities of the amplitudecorrespond to bound, virtual, or resonance states. Introducing the notation we can get relations for the wave numbers of bound, virtual and resonant three-particle states: Takibayev N "Exact Analytic Solutions..."

15. In the limit β →∞ (contactpair potentials) zeros of D give the values of which can be found from the algebraic equations: where and the pair bound energy is Three-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. Takibayev N "Exact Analytic Solutions..."

16. 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: The energy and width of resonance are determined with real and imaginary parts of resonance wave number: 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 Takibayev N "Exact Analytic Solutions..."

17. Then we obtain 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 Zeros of D are determined by equation and And we mark values of when Takibayev N "Exact Analytic Solutions..."

18. There are two sets of three-body resonances: 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. Takibayev N "Exact Analytic Solutions..."

19. 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: ; It is interesting that p,α-scattering amplitude has resonances in this P-wave too, but with parameters: 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. Takibayev N "Exact Analytic Solutions..."

20. Table 1. Quasi bound states of neutron and proton with subsystem of two -particles in model of fixed scatterer centers. 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. Takibayev N "Exact Analytic Solutions..."

21. Takibayev N "Exact Analytic Solutions..."

22. 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. Calculations give the points where Takibayev N "Exact Analytic Solutions..."

23. b p,α,α; 2,2 MeV 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]. n,α,α; 1,3 MeV p + e- → n + ν 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. Takibayev N "Exact Analytic Solutions..."

24. 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. Takibayev N "Exact Analytic Solutions..."

25. (p,α,α) – resonance state; ER ≈ 2.2 MeV p + e- → n + ν (n,α,α) – resonance state; ER ≈ 1.3 MeV Eν ≈ 0.16 MeV E < 2.1 MeV p' (p,α,α) → p' + (α,α); Ep'≈ 2.2 MeV – Eν -E 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. Takibayev N "Exact Analytic Solutions..."

26. 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. So, the process of stellar cooling may be the result of neutrinos radiation. 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. Takibayev N "Exact Analytic Solutions..."

27. 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. Takibayev N "Exact Analytic Solutions..."

28. The situation with neutrinos generation can be similar in case of the lattices of and particularly, and of more heavy nuclei... 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. Takibayev N "Exact Analytic Solutions..."

29. b p,α,α; 2,2 MeV n,α,α; 1,3 MeV 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 + e- → n + ν Thank you for your attention! Takibayev N "Exact Analytic Solutions..."