Few-body model approach to the lowest bound S-state of non-symmetric exotic few-body systems
FFew-body model approach to the lowest bound S-state ofnon-symmetric exotic few-body systems
Md. A. Khan* and M. Hasan
Department of Physics, Aliah University,IIA/27, Newtown, Kolkata-700156, India
Email: drakhan.rsm.phys @ gmail.com ; drakhan.phys @ aliah.ac.in Abstract
Lowest bound S-state energy of Coulomb three-body systems ( N Z + µ − e − ) hav-ing a positively charged nucleus of charge number Z ( N Z + ), a negatively chargedmuon ( µ − ) and an electron ( e − ), is investigated in the framework of hyperspheri-cal harmonics expansion method. A Yukawa type Coulomb potential with an ad-justable screening parameter ( λ ) is chosen for the 2-body subsystems. In the result-ing Schrödinger equation (SE), the three-body relative wave function is expanded inthe complete set of hyperspherical harmonics (HH). Thereafter use of orthonormal-ity of HH in the SE, led to a set of coupled differential equations which are solvednumerically to get the energy (E) of the systems investigated. Keywords:
Hyperspherical Harmonics (HH), Raynal-Revai Coefficient (RRC),Renormalized Numerov Method (RNM), Exotic Ions.
PACS:
I Introduction
Helium like exotic systems may those can be formed by replacing an orbital electron in atoms (or ions)by an exotic particle of the same electric charge can in general be represented by N Z + x − e − , where Nrepresents the atomic nucleus having Z number of protons and x represents exotic particles like muon,kaon, taon, baryon or their anti-matters. Among the many possible exotic particles, the one which isthe most fascinating is the muon, which is sometimes called as big cousin of electron. Muons are widelyused as an important probe to the physics beyond standard model. This is because this particle has fairchances of interacting with virtual particles as inferred by the anomalous value of its measured magneticmoment over theoretical predictions [1]. literature survey reveals the fact, studies of muon decay mayyield very important information on the overall strength and chiral structure of weak interactions inaddition to charged-lepton-flavor-violating processes [2]. Spectroscopy of muonium and muonic atomsgives unmatched determinations of fundamental quantities including the magnetic moment ratio µ µ /µ p ,lepton mass ratio m µ /m e , and proton charge radius r p [2]. Moreover, muon capture experiments areexploring elusive features of weak interactions involving nucleons and nuclei. For production of muon onemay refer to decay of pions produced during bombardment of target by fast moving proton beams apart a r X i v : . [ phy s i c s . a t m - c l u s ] M a y rom its cosmic origin. Out of several types of exotic few-body systems, the one, which contains at leastone muon are the most exciting. These exciting muonic atoms/ions can be formed in exact analogy withelectronic atoms/ions. It is interesting to note that muon being nearly 200 times heavier than an electronrevolves round the nucleus in orbit much smaller than that of an orbital electron. In fact, a 1S muonspends about one half of its periodic life inside the nucleus in muonic atoms/ions having nuclear chargenumber Z ∼ [3]. It is worth mentioning here that the spectra of muonic atom (for Z ∼
82) lies in theX-ray region. Further, muonic atoms/ions ( N Z + µ − e − ) are the rare kind of atomic three-body systemshaving no restriction due to Pauli principle for the muon and electron being non-identical fermions.These atoms/ions are generally formed as by-products of the process of muon catalyzed fusion, henceare useful to understand the fusion reactions in a proper way [4-5]. Although, hyperfine structure of theseatoms are studied by several groups to interpret the nature of electromagnetic interaction between theelectron and negatively charged muon [6-7], less attention have been given towards ab-initio calculationfor the bound state observables of such exotic atoms/ions in the few-body model approach.As exotic particles are mostly unstable, their parent atoms (or ions) are also very short lived, and canbe formed by trapping the accelerated exotic particle(s) inside matter at the cost of same number ofelectron(s). The trapped exotic particle revolves round the nucleus of the target atom in orbit of radiusequal to that of the electron before its ejection from the atom. Which subsequently cascades down theladder of resulting exotic atomic states by the emission of X-rays and Auger transitions before being loston its way to the atomic nucleus. If the absorbed exotic particle is a negatively charged muon, it passesthrough various intermediate environments undergoing scattering from atom to atom (in a way similarto that for a free electron) resulting in a gradual loss of its energy until it is captured into an atomicorbit. In the lowest energy level (1S), it feels only Coulomb interaction with nuclear protons and alsoexperiences weak interaction with the rest of the nucleons.In this work, we considered only those atoms/ions in which the positively charged nucleus is being orbitedby one electron and one negatively charged muon. We adopt hyperspherical harmonics expansion (HHE)approach for a systematic study of the ground state of atoms/ ions of the form N Z + µ − e − assuming athree-body model for each of them. In our model, we assume that the electromagnetic interaction ofthe valence particles with the nucleus is weak enough to influence the internal structure of the nucleus.Again, the fact that the muon is much lighter than the nucleus allows us to regard the nucleus to remaina static source of electrostatic potential. However, a hydrogen-like two-body model, consisting a quasi-nucleus ( µ − N ( Z − , formed by the muon and the nucleus plus an orbital electron will also be testeddue to significantly smaller size of muonic orbital compared to an electronic orbital.In HHE approach to a general three-body system consisting three unequal mass particles the choice ofJacobi coordinates, corresponds to three different partitions and in the i th partition, particle labeled by ‘ i ’ acts as a spectator while the remaining two labeled ‘ j ’ and ‘ k ’ form the interaction pair. So the totalpotential is the sum of three two-body potential terms (i.e. V = V jk ( r jk ) + V ki ( r ki ) + V ij ( r ij ) ). For thecomputation of matrix element of V( r ij ), potential of the ( ij ) pair, the chosen HH is expanded in theset of HH corresponding to the partition in which (cid:126)r ij is proportional to the first Jacobi vector [8] by theuse of Raynal-Revai coefficients (RRC) [9]. The binding energies obtained for the lowest bound s-stateare compared with the ones of the literature wherever available .In Section II, we briefly introduce the hyperspherical coordinates and the scheme of the transformationof HH belonging to two different partitions. Results of calculation and discussions will be presented inSection III and finally we shall draw our conclusion in section IV. Figure 1
1 3 2 1 2 3 1 2 3 𝜂 𝜉 𝜂 𝜉 𝜉 𝜂 Figure 1: Choice of Jacobi coordinates in different partitions of a three-body system.
II HHE Method
The choice of Jacobi coordinates for systems of three particles of mass m i , m j , m k is shown in Figure1. The Jacobi coordinates [10] in the i th partition can be defined as: (cid:126)η i = C ( (cid:126)r j − (cid:126)r k ) (cid:126)ζ i = C − (cid:16) (cid:126)r i − m j (cid:126)r j + m k (cid:126)r k m j + m k (cid:17) (cid:126)R = ( m i (cid:126)r i + m j (cid:126)r j + m k (cid:126)r k ) /M (1)where C = (cid:104) m j m k Mm i ( m j + m k ) (cid:105) , M = m i + m j + m k and the sign of (cid:126)η i is determined by the condition that( i, j, k ) should form a cyclic permutation of (1, 2, 3).The Jacobi coordinates are connected to the hyperspherical coordinates [11] as η i = ρ cos φ i ; ζ i = ρ sin φ i ρ = (cid:112) η i + ζ i ; φ i = tan − ( ζ i /η i ) (cid:41) (2)The relative three-body Schrődinger’s equation in hyperspherical coordinates can be written as (cid:20) − ¯ h µ (cid:26) ∂ ∂ρ + 5 ρ ∂∂ρ + ˆ K (Ω i ) ρ (cid:27) + V ( ρ, Ω i ) − E (cid:21) Ψ( ρ, Ω i ) = 0 (3)where Ω i → { φ i , θ η i , φ η i , θ ζ i , φ ζ i } , effective mass µ = (cid:2) m i m j m k M (cid:3) , potential V ( ρ, Ω i ) = V jk + V ki + V ij .The square of hyper angular momentum operator ˆ K (Ω i ) satisfies the eigenvalue equation [11] ˆ K (Ω i ) Y Kα i (Ω i ) = K ( K + 4) Y Kα i (Ω i ) (4)where the eigen function Y Kα i (Ω i ) is the hyperspherical harmonics (HH). An explicit expression for theHH with specified grand orbital angular momentum L (= | (cid:126)l η i + (cid:126)l ζ i | ) and its projection M is given by Y Kα i (Ω i ) ≡ Y Kl ηi l ζi LM ( φ i , θ η i , φ η i , θ ζ i , φ ζ i ) ≡ (2) P l ηi l ζi K ( φ i ) (cid:104) Y m ηi l ζi ( θ η i , φ η i ) Y m ζi l ζi ( θ ζ i , φ ζ i ) (cid:105) LM (5)with α i ≡ { l η i , l ζ i , L, M } and [] LM denoting angular momentum coupling. The hyper-angular momentumquantum number K ( = 2 n i + l η i + l ζ i ; n i → a non-negative integer) is not a conserved quantity for thehree-body system. In a chosen partition (say partition “ i "), the wave-function Ψ( ρ, Ω i ) is expanded inthe complete set of HH Ψ( ρ, Ω i ) = (cid:88) Kα i ρ − / U Kα i ( ρ ) Y Kα i (Ω i ) (6)Substitution of Eq. (6) in Eq. (3) and use of orthonormality of HH, leads to a set of coupled differentialequations (CDE) in ρ (cid:104) − ¯ h µ d dρ + ( K +3 / K +5 / h µρ − E (cid:105) U Kα i ( ρ )+ (cid:80) K (cid:48) α (cid:48) i < Kα i | V ( ρ, Ω i ) | K (cid:48) α (cid:48) i > U K (cid:48) α (cid:48) i ( ρ ) = 0 . (7)where < Kα i | V ( ρ, Ω i ) | K (cid:48) , α (cid:48) i > = (cid:90) Y ∗ Kα i (Ω i ) V ( ρ, Ω i ) Y K (cid:48) α (cid:48) i (Ω i ) d Ω i (8) III Results and discussions
For the present calculation, we assign the label ‘ i ’ to the nucleus of mass m i = M N = Am N ( m N → nucleon mass, A → mass number), the label ‘ j ’ to the negatively charged muon of mass m j = m µ (andcharge -e) and the label ‘ k ’ to the electron of mass m k = m e (and charge -e). Hence, for this particularchoice of masses, Jacobi coordinates of Eq. (1) in the partition “ i " become (cid:126)η i = C ( (cid:126)r j − (cid:126)r k ) (cid:126)ζ i = C − ( (cid:126)r i − m µ (cid:126)r j + m e (cid:126)r k m µ + m e ) (cid:41) (9)and the corresponding Schrödinger equation (Eq. (7)) is (cid:104) − ¯ h µ (cid:110) d dρ − ( K +3 / K +5 / ρ (cid:111) − E (cid:105) U Kα i ( ρ )+ (cid:80) K (cid:48) α (cid:48) i < Kα i | C exp( − λρ cos φ i ) ρcosφ i − Z exp( − λρ | C sin φ i ˆ ζ i − C cos φ i ˆ η i | ) ρ | C sin φ i ˆ ζ i − C cos φ i ˆ η i |− Z exp( − λρ | C sin φ i ˆ ζ i + C cos φ i ˆ η i | ) ρ | C sin φ i ˆ ζ i + C cos φ i ˆ η i | | K (cid:48) α (cid:48) i > U K (cid:48) α (cid:48) i ( ρ ) = 0 (10)where C = (cid:104) m µ m e ( M N + m µ + m e ) M N ( m µ + m e ) (cid:105) , µ = (cid:16) M N m µ m e M N + m µ + m e (cid:17) is the effective mass of the system and λ ( ≥ )is an adjustable screening parameter. In atomic units we take ¯ h = m e = m = e = 1 . Masses of theparticles involved in this work are partly taken from [11-14].In the ground state (1 S µ S e ) of N Z + µ − e − three-body system, the total orbital angular momentum, | (cid:126)L | = | (cid:126)l η i + (cid:126)l ζ i | = 0 and there is no restriction (on l η i ) due to Pauli requirement as electron and muonare non-identical fermions. Since L = 0 , l η i = l ζ i , and the set of quantum numbers α i is { l η i , l η i , , } .Hence, the set of quantum numbers { Kα i } can be represented by { Kl η i } only. Eq. (10) is solvedfollowing the method described in our previous work [11] to get the ground state energy E.One of the major drawbacks of HH expansion technique is its slow convergence for Coulomb-type longrange potentials, unlike for the Yukawa-type short-range potentials for which the convergence is rea-sonably fast [10,15]. So, to achieve the desired degree of convergence, sufficiently large K m value hasto be included in the calculation. But, if all K values up to a maximum of K m are included in theHH expansion then the number of the basis states and hence the number of CDEs involved equals (1 + K m / K m / . Which shows a rapid increase in the number of basis states and hence thesize of coupled differential equations (CDE) (Eq. (7)) with increase in K m . For the available computer Binding energy, B (au) K m a x e - m - H e e - m - L i e - m - B e
F ig u r e 2 .
Relative energy difference, c K m a x e - m - H e
F ig u r e 3 . C a lc u la te d d a ta E s tim a te d d a ta [E q .( 1 3 ) ] F itte d d a ta
Binding energy, B (a.u.)
N u c le a r c h a r g e , ZF ig u r e 4 .
Figure 2: Pattern of dependence of the ground-state energy (B= -E) of muonic atom/ionson the increase in K max .Figure 3: Pattern of dependence of the ground-state relative energy difference χ = B Km +4 − B Km B Km +4 of muonic helium ( ∞ He µ − e − ) on the increase in K max .Figure 4: Pattern of dependence of the ground-state energy (B) of muonic atom/ions onthe increase in nuclear charge Z. facilities, we are allowed to solve up to K m = 28 only, reliably. The calculated ground state energies( B K m ) with increasing K m for some representative cases are presented in columns 2, 4 and 6 of TableI. Energies for a number of muonic atom/ions of different atomic number (Z) at K m = 28 are presentedin column 4 of Table II. The calculated energies at K m = ∞ of column 5 in Table II are obtained byextrapolating those obtained at K m ≤ . The pattern of convergence of the energy of the lowest boundS-state with respect to increasing K m can be checked by gradually increasing K m values in suitable steps( dK ) and comparing the corresponding relative energy difference χ = B K + dK − B K B K + dK with that found inthe previous step. From Table I, it can be seen that at K m = 28 , the energy of the lowest bound S-stateof e − µ − ∞ He converges up to 4th decimal places and similar convergence trends are observed in theremaining cases.The pattern of increase in binding energy, B (=-E) with respect to increasing K max is shown in Figure2 for few representative cases. In Figure 3 the relative energy difference χ is plotted against K max todemonstrate the relative convergence trend in energy. The calculated ground state energies for muonicthree-body systems of infinite nuclear mass but of different nuclear charge Z, have been plotted againstZ as shown in Figure 4 to study the Z-dependence of the binding energies. The curve of Figure 4 showsa gradual increase in energy with increasing Z approximately following the formula B ( Z ) = 0 . − . Z + 103 . Z − . × − Z + 8 . × − Z (11)Eq. (11) may be used to estimate the ground state energy of muonic atom/ions of given Z and ofinfinite nuclear mass. Finally, in Table II, energies of the lowest bound S-state of several muonic three-body systems obtained by numerical solution of the coupled differential equations by the renormalizedNumerov method [16] have been compared with the ones of the literature wherever available. Sincereference values are not available for systems having nuclear charge Z > , we made a crude estimationf the ground-state ( S µ S e ) energies following the relation B Best = Am N Z (cid:20)
11 + Am N + m µ m µ + Am N (cid:21) (12)where m N = 1836 , muon mass m µ = 206 . (in atomic unit) and A is mass number of the nucleus.Here we assumed two hydrogen-like subsystems for the muonic atom/ions. This estimate can furtherbe improved by assuming a compact ( A ( Z ) , µ − ) S positive muonic ion and an electron in the 1S state,"feeling" a (Z-1) charge and an Am N + m µ mass. For this case the estimation formula (Eq. (12)) canbe replaced by B Best = 12 (cid:20) Am N + m µ Am N + m µ ( Z − + Am N m µ m µ + Am N Z (cid:21) (13)The energies estimated by Eq. (13)) is shown in column 3 of Table II. IV Conclusion
A comparison among the different results tabulated in columns 2 to 4 of Table II reveals that theestimation formula given by Eq. (13) gives better result in comparison with the highly accurate varia-tional results listed in bold in column 4 of Table II. This indicates the fact that the three-body systems( A ( Z ) , µ − , e − ) form weakly bound three-body states very close to the ( A ( Z ) , µ − ) S + e − threshold.Again, the convergence trend of the three-body wave function and energy for the muonic systems for thelong range Coulomb potential is poor even at K max = 28 . It can be seen from Table II, that, between theestimated energies B Best in column 2 and B Best in column 3, the latter are closer to the calculated energiesin columns 4 and 5. This indicates that the muonic three-body systems have more possibility of havingsuper heavy-hydrogen-like two-body structure over three helium-like three-body structure. Finally, itcan be noted that a further improvement in the calculations could have been done by incorporating theKato’s cusp conditions[17] in the limit r jk → appropriately.The authors acknowledge Aliah University for providing computer facilities and one of the authorsM. Hasan gladly acknowledges Aliah University for granting a research assistantship. V ReferencesReferences [1] Gibney Elizabeth, 2017
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EPJ Web of Conferences , 02009. VI Figure Caption
Fig. 1. Choice of Jacobi coordinates in different partitions of a three-body system.Fig. 2. Pattern of dependence of the ground-state energy, B (=-E) of muonic atom/ions on theincrease in K max .Fig. 3. Pattern dependence of the ground-state relative energy difference χ = B Km +4 − B Km B Km +4 ofmuonic helium ( ∞ He µ − e − ) on the increase in K max .Fig. 4. Pattern of dependence of the ground-state energy (B) of muonic atom/ions on the increasein nuclear charge Z. VII Tables able I. Convergence trend in the lowest s-state energy (-E) with increasing K max inrepresentative cases of muonic three-body systems. Energy (-E) and the corresponding relative energy difference ( χ )System → ∞ He µ − e − ∞ Li µ − e − ∞ Be µ − e − K max − E χ − E χ − E χ able II. Energy (B) of the lowest bound S-state of electron-muon-nucleus three-body systems.
System Binding energies expressed in atomic unit (a.u.)Estimated Calculated Screening B Best [ Eq. (12)] B Best [ Eq. (13)] B calcK m =28 B calcK m = ∞ parameter, λe − µ − He a , 399.043 b e − µ − He c , 402.641 d e − µ −∞ He e , 414.037 f e − µ − Li e , 915.231 g e − µ − Li e , 917.650 g e − µ −∞ Li e e − µ − Be e − µ −∞ Be e − µ − B e − µ −∞ B e − µ − C e − µ −∞ C e − µ − O e − µ −∞ O e − µ − Ne e − µ −∞ Ne e − µ − Mg e − µ −∞ Mg e − µ − Si e − µ −∞ Si e − µ − S e − µ −∞ S e − µ − Ar e − µ −∞ Ar e − µ − Ca e − µ −∞ Ca a Ref[2,18-20,26], b Ref[21], c Ref[2,18-19,21-22], d Ref[1,23-24], e Ref[2], f Ref[19], gg