Quantum Mechanical Three-Body Systems and Its Application in Muon Catalyzed Fusion

A negative muon is a lepton of the second generation with mass number about times heavier than that of electrons, and has a finite lifetime of 6 2.197 10 sec      . This lifetime is amply long for most experiments. Muon catalyzed fusion (μCF) is a physical phenomenon in which the negative muon is able to cause fusion at room temperature and thereby eliminating the need for high temperature plasmas or powerful lasers (Owski, 2007; Imo et al., 2006; Filchenkov et al., 2005; Filipowicz et al., 2008; Pahlavani & Motevalli, 2008, 2009; Marshal, 2001; Bystritsky et al., 2006, Nagamine et al., 1987; Nagamine, 2001; Ponomarev, 2001). In comparison with (μCF), hot fusion schemes are made difficulty by the electrostatic (Coulomb) repulsion between positively charged nuclei. In the two conventional approaches to control fusion namely, Magnetic Confinement Fusion (MCF) and Inertial Confinement Fusion (ICF), barrier is partially surmounted by energetic collisions. The particle densities, n and confinement times,  in the plasma, ( 8 10 T K  ) are typically more than ten orders of magnitude difference for these two schemes but the product of these quantities required for d−t fusion is 14 3 10 sec/ n cm   . For the μCF, effectively 25 3 10 sec/ n cm   , but this criterion does not tell the whole truth because, in μCF the objective is to tunnel through the barrier without the benefit of kinetic energy. It is known that the d−t fusion by the usual magnetic or inertia confinement suffering a lot of difficulties and problems causing from tritium handling, neutron damage to materials and neutron-induced radioactivity, etc.


Introduction
A negative muon is a lepton of the second generation with mass number about times heavier than that of electrons, and has a finite lifetime of .This lifetime is amply long for most experiments.Muon catalyzed fusion (μCF) is a physical phenomenon in which the negative muon is able to cause fusion at room temperature and thereby eliminating the need for high temperature plasmas or powerful lasers (Owski, 2007;Imo et al., 2006;Filchenkov et al., 2005;Filipowicz et al., 2008;Pahlavani & Motevalli, 2008, 2009;Marshal, 2001;Bystritsky et al., 2006, Nagamine et al., 1987;Nagamine, 2001;Ponomarev, 2001).In comparison with (μCF), hot fusion schemes are made difficulty by the electrostatic (Coulomb) repulsion between positively charged nuclei.In the two conventional approaches to control fusion namely, Magnetic Confinement Fusion (MCF) and Inertial Confinement Fusion (ICF), barrier is partially surmounted by energetic collisions.The particle densities, n and confinement times,  in the plasma, ( 8 10 TK  ) are typically more than ten orders of magnitude difference for these two schemes but the product of these quantities required for d−t fusion is , but this criterion does not tell the whole truth because, in μCF the objective is to tunnel through the barrier without the benefit of kinetic energy.It is known that the d−t fusion by the usual magnetic or inertia confinement suffering a lot of difficulties and problems causing from tritium handling, neutron damage to materials and neutron-induced radioactivity, etc.
Study of the muon catalyzed fusion reactions is of great interest and carried out in many laboratories of the world recently (Ishida et al., 1999;Petitjean et al., 1992Petitjean et al., , 1993;;Bystritsky et al., 2005;Pahlavani & Motevalli, 2008, Bystritsky et al., 2000;Matsuzaki et al., 2001).Muons can be created by the decay of pion which is generated in the collision of intermediateenergy proton with target nuclei.In the muon catalyzed fusion process, after injection of muon in to deuterium and tritium mixture, either d or a t atom is formed, with a probability proportional to the relative concentrations of D and T in the mixture.These atoms are formed in exited states (Breunlich et al., 1989;Korenman, 1996) and then, due to cascade processes, de-excite to ground states.The following reactions illustrate direct formation of muonic dμ and tμ atoms where e  denotes an electron and d  and t  are the rate of reactions (1) and ( 2).The probability of formation of the d atom that will reach its 1s ground state is quantified by the parameter 1s q , which is a function of target density, φ and tritium concentration, t C .Also it is very sensitive to the d kinetic energy distribution (Menshikov & Ponomarev, 1984;Czaplinski et al., 1994).The difference between binding energies of t and d is about 48.1 eV (Bom et al., 2005).Therefore, the transfer of a muon from d to a triton is favorable for all temperatures in the given processes 48.1 ( ) with a rate of 8 2.8 10 dt    (Caffery et al., 1987;Jones et al., 1987;Bystritsky et al., 1980;Breunlich et al., 1987).The muon mass is about 206.77 times larger than the mass of electron.Consequently, the size of a muonic hydrogen atom is smaller than the one of the electronic hydrogen by the same rate approximately.These small muonic atoms can approach other hydrogen nuclei experiencing reduced Coulomb barrier and then induce d-t fusions.The process in which a muonic molecule is formed is the most important step in the μCF.The formation of muonic molecules of hydrogen isotopes and their nuclear reactions have been the subject of many experimental and theoretical studies (Caffery et al., 1987;Jones et al., 1987;Bystritsky et al., 1980).In collisions of t muonic atoms with D 2 and DT molecules, the muonic molecules dt are formed during a time interval 8 10 sec dt    (Jones et al., 1983;Eliezer & Henis, 1994) according to the following resonance reactions 6) in the excited rotational-vibrational (J ) state with quantum number J= =1, where C d and C t are concentrations of deuterium and tritium nuclei, respectively.A strong resonance effect appears due to degeneracy in the excited state of the dt and the electron molecule complex.
The rate of formation of the dt molecules has been found to depend strongly on temperature, density and on whether collision of the t atom occurs with a D 2 or a DT molecule (Bom et al., 2005;Faifman et al., 1996;Ackerbauer et al., 1999).
In fact, the radius of a muonic hydrogen ion (dt ) is much smaller (about ≈ 200 times) than a usual electron molecule, therefore the nuclei may tunnel the coulomb barrier with a high probability and fuse with a rate of ≈ 10 12 sec −1 (Bogdanova et al., 1982).Resonant formation of the dd molecule at very low temperatures was observed in solid and liquid D 2 targets (Bogdanova et al., 1982).
Developed methods in this field are based on detailed three-body equations which provide a correct description of the quantum mechanical three-body systems (Takahashi & Takatsuka, 2006;Kilic, Karr & Hilico, 2004;Nielsen et al., 2001;Pahlavani, 2010).Theoretical study of muonic three-body system comprises different theoretical methods, e.g.variational
The Born-Oppenheimer approach assumes the nuclei to be infinitely heavy with respect to the negatively charged particle.I t s h o u l d b e k e p t i n m i n d t h a t t h e f o l l o w i n g B o r n -Oppenheimer approach is the simplest solution to the three-body coulomb system.This approach is expected to be a poor approximation for calculations of muonic molecule eigenvalues.In this work, we calculate binding energies of the bound states of the dd muonic three-body system molecule using the adiabatic expansion method.

Adiabatic expansion approximation for the three-body system
The exact Hamiltonian that describes muonic three-body system can be shown by following relation: where 1 and 2 denote the two nuclei, their position is given by R 1 and R 2 , and the muon coordinate is r .The center of mass coordinate R CM is given by It is convenient to define Jacobi coordinate r and R as follow: where R is the internuclear coordinate and r is the muon coordinate to midpoint between the two nuclei.In these coordinates (R; r), the Hamiltonian denoted by equation ( 7) is change to the following operator: The three-body Hamiltonian (16) commutes with the total angular momentum operator for the three particle system, J, its projection on z-axis, J z , and the total parity operator, (, where ( 1) J p   is the eigenvalue of the parity operator: The functions presented in equation ( 23) (in bracket) are satisfying the following orthonormality condition: If m = 0, both the Wigner functions in ( 22) are reduced to the ordinary spherical function (,) JM Y  so that the dependence of ' disappears and the angular functions satisfying the conditions ( 24) and ( 25) are: ,0 (,) (,,) 2 In this case the parity is unambiguously specified by the quantum number J: p=+(-1) J .So, our basis functions have the following structure: () (, , ,,,)  ( , ,)  (,; ) The wave functions (,, ,,,) describing the muon motion around fixed nuclei separated by a distance R. E i (R) is the energy of a muon in the state i as a function of R.Here we show how to separate the variables through the use of the ellipsoidal (or, prolate spheroidal) coordinates Note that through the coordinate transformation (29-31), we have Writing the wave function as  and changing the variable to spheroidal coordinates, equation ( 28) can be separated into following three one-dimensional equations: Note that A and q are unknown parameters and should be obtained from ( 40) and ( 41) as eigenvalues of the coupled system.Once A and q are obtained, then E can be obtained from q 2 =-R 2 E/2.By substitution of expression ( 8) into the Schrödinger equation with Hamiltonian ( 16) and after averaging over spherical angles (,)   and the muon state, one obtains the radial equation where    is the collision energy and E is the total energy of the system and () is the ground state energy of muonic atom.
12 () is the potential corresponding to the Born-Oppenheimer approximation and 12 1 The adiabatic potential V Ad (R) for the (dd ) muonic three-body molecule is calculated in the adiabatic expansion method.The adiabatic potential curves and qualitatively similar for each of muonic molecules and are displayed for the (dd ) muonic molecule in Figure 1.
Results of the calculations of binding energies of the bound states   , J  of the (dd ) muonic molecule are compared with the results of the other methods used in (Korobov et al., 1992;Kilic, Karr & Hilico, 2004) and are given in Table 1.(Kilic, Karr & Hilico, 2004) (Korobov et al., 1992  The calculated binding energies are in good agreement with the previous calculations by other authors using different methods.

Charge-asymmetric three-body system in hyper-spherical elliptic coordinate system
Study of the nuclear synthesis reaction d− 3 He at low collision energies (below 1 keV) is of interest for its applications in nuclear and astrophysics (Belyaev et al.,1995).The relatively large energy gain as well as the lack of tritons in the initial and neutrons in the final channel makes this reaction a very attractive source of thermonuclear fusion energy.
The negatively charged energetic muons, after stopping in the D− 3 He mixture, fuse to d or 3 He in order to form the mesic atoms in excited states.After a sequence of cascade transitions lasting about 10 −11 sec at Liquid Hydrogen Density (LHD), mesic atoms are formed in the ground state (Ponomarev, 1991;Breunlich et al., 1989;Czaplinski et al., 1996).
The three-body molecules,   The muon is released after the fusion and can proceed to cause another fusion.Thus the muon works as a catalyst and this cycle can be repeated many times during its lifetime.
To test the stability of the mesic 3 He d system, we consider only Coulomb interaction between particles.As a starting point, we employ the aim of hyperspherical method to solve the multi-dimensional Schrödinger equation numerically for this three-body system.

() HT VE
    (52) The wave function,  , can be constructed explicitly by exploiting a specific representation, namely, the hyper-spherical adiabatic expansion method.Here, T is the kinetic energy in its enter-of-mass coordinate frame, V is the potential energy, and E is the total energy of the system.We briefly discuss the general structure of the method and formulate its basic equations for a three-body system in hyper-spherical elliptic coordinates.The Hamiltonian of this molecule in Jacobian coordinates (R, r) can be shown by the following equation (Gusev et al., 1990;Stuchi et al., 2000) 11 where i M and i m are reduced masses.It is convenient to define mass-scaled Jacobian vectors, ( i x  and i y  ), Therefore the kinetic energy of the system can be rewritten as, In this relation, It is convenient to calculate these sets of Jacobian coordinates ( i x  , i y  ) for mesic three-body, 3 He d system.These three sets should be used as coordinates in configuration space.Therefore this system contains six dimensions (d=6).In hyperspherical coordinates, (, )   , where s  denote a set of two angles defining the shape of the system and 0  refer to a set of three angles defining the overall orientation of the three-body system.The Hamiltonian, in this coordinate system, will take the following form: where 2  is regarded as the square of general angular momentum operator.Our aim is to solve the eigenvalue equation (, ) (, ) HE      in the adiabatic expansion method.The idea of adiabatic separability between the hyper-radius  and the hyper-angular variables  in three-body systems was first exploited by Macek (Macek, 1968) for studying doubly excited states of the Helium atom.The wave equation of the system in this method can be defined by: Here the quantum number, j characterizes a channel function, the radial functions, ( ) j F  satisfy the system of coupled ordinary differential equations and ( ; ) where CV   is the effective charge of the system.In the first step we try to solve the differential equation ( 62), which contains coordinate  as a parameter.The hyper-spherical elliptic coordinates (,)   on S (projection of the hyper-sphere .const   onto shape space) are induced by conical coordinates on its 3D image.The hyper-spherical elliptic coordinates (,)   are defined in the following intervals (Tolstikhin et al., 1995) 22     The definition of (,)   resembles the representation of plane elliptic coordinates.In order to rewrite Eq. ( 62) in a new set of coordinates, (,)   , it is necessary to define the square of general angular momentum operator, 2  in this set of coordinates, where m is azimuthal quantum number which is the projection of the general angular momentum along body-fixed axis.The potential energy, V, of the system is the sum of three inter-particle Coulombian interactions potential,   and V into Eq.( 62), we obtain a differential equation for adiabatic Hamiltonian that should be solve with appropriate boundary conditions.In the case, for infinite small values of  , the solutions of adiabatic Hamiltonian (62) can be constructed in the following form (Pahlavani & Motevalli, 2008): where N is the normalization parameter.With some mathematical simplification, one obtains the following set of ordinary differential equations: The resultant equations are subject to the regularity of boundary conditions and can be satisfied only for certain values of A and U.The method for solving these set of differential equations, are very similar to those equations which we presented in our previous work (Pahlavani & Motevalli, 2008), when we have studied the motion of muon in the two-center Coulomb problem in prolate spheroidal coordinate system for the symmetric mesic system dd .By solving these equations, one obtains the functions and   n  are quantum numbers correspond to number of zeros of the functions   X  and  Y  that appeared in Eqs. ( 69) and ( 70).These functions form a set of complete bases and satisfy the following normalization condition, where ds is the surface element that can be defined by, The results of the calculations are displayed graphically in Fig. 3.The normalization factor ()() nn N   , is a function of the rotational parameter, at different quantum numbers   n  and   n  .Calculated values of adiabatic potential   U  as a function of hyper-radius  have been shown in Fig. 4. By substituting Eq. ( 61) into Eq.( 52), one can obtain the following set of ordinary differential equations for radial functions ( ) The above equation is a set of differential equations coupled by the following nonadiabatic terms: where the brackets represent integration over the angular variables  .The hyperspherical adiabatic approximation amounts to retaining only one term in Eq. ( 61) (Macek, 1968).Then the radial function, ( ) This approximation turns out to be surprisingly accurate in the sense that in many situations the non adiabatic couplings in Eq. ( 75) are rather weak.Subject to this reality and considering appropriate boundary conditions, we obtain solutions of the differential equation ( 79) numerically.Finally, the calculated values of the binding energies of the bound states (, ) J  for the 3 He d system are compared with available data obtained using other methods in Table 2. (Pahlavani et  J  (the quantum numbers of rotational-vibrational state) for the 3 He d molecule.
The Born-Oppenheimer approach assumes the nuclei to be infinitely heavy with respect to the negatively charged particle.It should be kept in mind that Born-Oppenheimer approach is the simplest solution to the three-body Coulomb system.Usually, the most accurate results for the ground state energy levels of mesic three-body molecule were obtained from variational calculations.Comparison of our result for J = 0 with the ones obtained by the available variational calculation (Bogdanova et al., 1982) indicates difference that dose not exceed 0.2%.One can conclude that this fact supports the validity of adiabatic expansion in hyper-spherical elliptic coordinates method which have been used.

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Some Applications of Quantum Mechanics 124

Muon stripping in the muon catalyzed fusion
The sticking of muons to alpha particles after fusion is an unwanted process and eliminates muons from the chain of fusion reactions.This process is the main loss mechanism in the CF.The probability of forming a muonic helium ion is called initial sticking probability 0 ( 0.912%) S   (Hu, Hale & Cohen, 1994)


are the rates of populating and de-populating probability of state i, respectively.These rates can be given by the following relations: where Au , ra , de−ex , ex , Stark and strip are the Auger de-excitation, radiative, Coulomb deexcitation, Coulomb excitation, Stark mixing and striping rates, respectively.In general, is given by where N, v and are density of surrounded media, relative velocity and cross section for all processes under consideration, respectively.The time and velocity dependence in Eq. ( 80) are coupled through the energy-loss equation for muonic helium ion given by where S = −dE/dx is the stopping power of the surrounding media and m  is the mass of muonic helium ion.Ptat t .The intensity of X-ray transition in muonic helium ion is another quantity which can be measured experimentally and calculated along with reactivation coefficient (R).Muons in excited levels of the   may de-excite under X-ray emission.The X-ray spectrum depends not only on the initial sticking in the atomic levels and the reactivation of the muon but also on intra-atomic transitions due to inelastic collisions, internal and external Auger effect and Stark mixing.The photon intensity per sticking event is calculated using The number of X-ray photons emitted per fusion is the most useful quantity that can be measured experimentally.The X-ray yields for the nn   transition is given by The calculation for muon stripping probability from α + and the intensity of X-ray transitions have been done by solving a set of coupled differential equations numerically.The time-dependent population probabilities P i (t) for 1s, 2s, 2p, 3s, 3p, 3d are shown in Fig. 5 for a deuterium-tritium target at density φ=1.2 L.H.D (L.H.D≡ Liquid Hydrogen Density = 4.25 × 10 22 atoms/cm 3 ).The initial populations of all excited states are seen to drop to 0 during the stopping time, and only 1s orbital stays occupied.

Conclusions
The quantum-mechanical three-body problem plays an important role in modern physics by providing an appropriate description of three-particle systems in presence of Coulomb and nuclear forces.Developed methods in this field are based on detailed three-body equations which provide a correct description of the quantum mechanical three-body systems (Takahashi & Takatsuka, 2006;Kilic, Karr & Hilico, 2004;Nielsen et al., 2001;Pahlavani, 2010).Theoretical study of muonic three-body system comprises different theoretical methods, e.g.variational methods (Viviani et al. 1998;Frolov, 1993), Born-Oppenheimer approximation (Beckel et al., 1970;Kilic et al., 2004) and adiabatic expansion (Fano, 1981;Lin, 1995).In this investigation, we presented an appropriate method that enables us to study the solutions of Schrodinger equation for 3 He d system.The adiabatic expansion in hyper-spherical elliptic coordinates has shown a good approach for calculating the adiabatic potential.Fast convergent of this method led us to obtain precise results for the existence of the bound states in 3 He d three-body molecule.The obtained results for the adiabatic potential of this system are comparable with results gathered from other approximation methods.
The corresponding eigenvalue problem has been solved and the binding energy of this system is calculated.The obtained results agreed with the expected values of various theoretical methods.This approach can be applied for other three-body systems with variety of masses and charges.The obtained results are of significant importance for experimental and theoretical investigation of d− 3 He nuclear fusion especially at low collision energies.
In section 4, the obtained results show that the muon cycle coefficient increases almost slowly with the density of deuterium and tritium mixture.The energy required to produce a muon estimated to be about 5000 MeV.Since each deuterium and tritium fusion generates 17. 6 MeV, we see that the number of catalysis reactions by a muon should be about 285 to reach the scientific break-even (1/3 of the commercial break-even).The break-even point is reached when the fusion process generates as much energy as was initially put in (i.e., the energy output equals the energy input).The output energy of the number of catalysis reactions by a muon in it's lifetime ( = 2.197 sec), is much smaller than the input energy required to produce a muon.Therefore, a fusion energy system based on the muon catalyzed fusion in deuterium and tritium fuel seems to be viable at plasma conditions with fuel densities about 100 times of L.H.D.

Fig. 1 .
Fig. 1.Adiabatic potential curves, () Ad VR , corresponding to dd system (Pahlavani & any value of  , these functions form a set of complete orthogonal basis which satisfy the following relation:

Fig. 2 .
Fig. 2. Variation of effective charge C as a function of hyper-angular variables 22     Fig. 3. Variation of normalization parameter
Fig. 5.The population probabilities P i (t) as a function of time in a D-T target at density φ=1.2 L.H.D (Pahlavani & Motevalli, 2008).

Final
, the resultant Wigner functions would be different, and the angular functions consist both even and odd combinations.It is convenient to specify these combinations as follows: Note that the coordinates  ,  and  are orthogonal, and we have the first fundamental form 31)www.intechopen.comSome Applications of Quantum Mechanics 114

Table 1 .
Binding energies (eV ) of the states (, ) J  for the dd muonic molecule. www.intechopen.com Quantum Mechanical Three-Body Systems and Its Application in Muon Catalyzed Fusion 117The molecule dissociates quickly with a rate of about 10 12 sec −1 to the unbound ground state either by a well-known predissociation mechanism, via Auger transition or  -emission nucleus and a mesic helium atom.This mechanism leads to transfer rates of the order 10 8 sec −1 .The asymmetric-charged 3 He d molecule undergo nuclear fusion via two different channels, www.intechopen.com

Table 2 .
Binding energy E B (eV) of the bound states (, )   .The reactivation coefficient, R depends upon the stopping power of the media and several important cross sections.Stripping process can occur through several channels.Collisions of the (α ) 1s ions with the surrounding D 2 and DT molecules during the slowing down process can result in α charge transfer, ionization or excitation of the discrete α levels.Stripping (charge transfer plus ionization) can also happen from the α which is the results of the sticking or collisional excitation processes.The kinetic of reactivation is described by the various rates in a set of coupled differential equations.The fraction of stripped muonic helium ions in terms of population probabilities can be written as SS R