A Formula of Particles Oscillating in Electromagnetic and Weak Fields Giving Energy and Decay of Particles, Beta Decay and Energy Levels of Hydrogen Atom

Abstract

A Formula of Particles Oscillating in Electromagnetic and Weak Fields Giving Energy and Decay of Particles, Beta Decay and Energy Levels of Hydrogen Atom Kh. M. Mannan, Physics Department, Curzon Hall, Dhaka University, Dhaka, Bangladesh. Phone: (880)-(2)-9614468; Mob.: 88-01715327301. Mailing address: House#59, Road#3/A, Dhanmondi R/A, Dhaka, Bangladesh. Corresponding Author: "mailto:mafmannan@yahoo.com" Received on April 08, 2015; revised on June 14, 2015. ABSTRACT: A formula for the mass-energy and the decay of particles is obtained from a model of particles oscillating in the electromagnetic and in the weak fields. For some selected displacements of the particle, important results are obtained. From this formula it is suggested that particles acquire their masses while massless gauge bosons (W±) get their masses by Higgs mechanism. Mass-energies and decay of particles are obtained from the decay of a W± boson coupled to the electromagnetic or the weak fields in which the particles oscillate. In the centre of mass (CM) frame, the spin of a particle is shown as localized circular motion of the centre of charge around the CM confined within the reduced Compton wavelength for the particle. Decay constants of pseudo-mesoens are obtained from zero point energies of the oscillators. The beta decay and kinetic energies of beta particles are obtained from this formula. Dimension (d) of a three-dimensional harmonic oscillator explains the build up of charge(Q) by the oscillation of quarks constituting a particle. Particle appearance quantum number (N) gives parity (P) of particles. Discrete energy levels for the hydrogen atom are obtained from the same formula with appropriate conditions. 1. INTRODUCTION In the Standard Model (SM) [1] Higgs mechanism [2,3] gives masses to the W± and Z0 bosons by spontaneous symmetry breaking. In the current picture of Higgs mechanism W± and Z0 bosons and quark masses are generated by interactions with the vacuum field [4,5]. In the for the flavour changing transition probability between one quark to another [6] and, meson mass eigenstates are obtained by mixing quark flavour eigenstate [7]. In this work a formula for the mass-energies of hadrons and leptons (excluding neutrinos) are obtained. It is suggested that particles when oscillate in the electromagnetic or in the weak fields get their masses while massless gauge bosons (W±) acquire their masses by Higgs mechanism. This formula also gives the decay of hadrons. Spins of particles are shown to arise from the centre of charge of the rest mass-energy circulating around the centre of mass confined within the reduced Compton wavelength. This formula is applicable in calculating masses of particles arising from strong interactions via π0, ρ0, or ω0 coupled to the strong field by strong coupling constant αs. This formula explains the energetics of beta (β±) decay, and the maximum kinetic energy KE(β±)max of the beta particles. Dimension (d) of a 3-dimensional harmonic oscillator shows how charge (Q) in a particle builds up from the oscillation of the quarks constituting a particle. The quantum number N for the distance at which a particle appears gives the parity (P). The decay constants of pseudoscalar mesons are obtained from the zero point energy of the harmonic oscillators. Mass-energies of particles and their decay, beta decay and the discrete energy levels for the hydrogen atom obtained from one formula shows the importance of the work. For the first time important properties of particles, beta decay and decay of particles are explained in terms of a 3D quantum mechanical harmonic oscillator. 2. OSCILLATION OF PARTICLES AND HGGS MECHANISM, SPIN AND PARITY 2.1. OSCILLATION OF PARTICLES AND HIGGS MECHANISM From uncertainty principle the reduced Compton wavelength of a particle is thought of as the fundamental limitation on measuring the position of a particle i.e. this is the minimum deviation away from the eqilibrium point at which a particle appears. Let us consider a particle at a distance Xp ≥ ƛpc when, ƛpc=(ħ)/(m0pc) (1) where, ƛpc is the reduced Compton wavelength of the particle, ћ the reduced Planck constant, m0p rest mass of the particle and c velocity of light. The Einstein-de-Broglie equation gives ħω0p=m0p.c2 (2) when de Broglie suggested that ω0p is the natural circular frequency of a particle. The particle property arises from the localized vibrational energy (ħω0p) divided by c2 so that the inertial mass m0p is given by [8,9] m0p=(ħω0p)/c2 (3) and, for the corresponding Compton wavelength, (ƛpc.ωop)= [(ħ ω0p) / (m0p.c)]=c (4) In the celebrated Schrodinger equation for wave number k=(1/ƛpc), Einstein-de-Broglie equation (2) is retained as m0p.c2 (1+ 2α(0))=2ħω0p (5) when, Coulomb potential energy V(1/k)=α(0)m0p.c2 and α(0) is the fine structure constant. From relativity E= ±mop.c2 where, ± means a paticle and an antiparticle respectively. When a W± boson is displaced by a distance Xdisp=D.ƛw, mass-energy state (±mop.c2) of a particle appears at a distance Xp=N.ƛpc where, N and D’s are integers, and ƛw the reduced Compton wavelength of the boson. These particles are attracted by the respective Coulomb force (-/+Fp) between the particle and the boson. For neutrons a W+ boson attracts a d-quark. Particles/antiparticles are put into orbital motion by the emission of a W+ or a W- boson which provides the centripetal force and for orbital motion centripetal force equals the Coulomb force. The charged W boson is connected with an oppositely charged or a neutral particle as if by a spring and this is put into oscillation due to the emission of a W± boson from the particle. Let us consider that the mass-energy state for a particle appears at a distance Xp=N.ƛpc away from the point of equilibrium when a W± boson pops up at a distance Xdisp=D.ƛw.The particle is put into oscillation in a Coulomb field by the emission of a W± boson from the particle. For simple harmonic oscillation, the spring constant is K given by K= m0p(ω0p)2 = - (Fp/Xdisp). Let the displacement of the particle be Xdisp=D.ƛw and this is in the opposite direction of the Coulomb restoring force (Fp). For spring constant K and from equation (4) as ƛpc.ω0p=c, we get, mop(ω0p)2 = [{(q2)/(4πε0cɦ)}.(cɦ)]/[(N.ƛpc)2.(D.ƛw)] m0p.c2= [{α(0)} / (N2D)]. (Mw) (6) when, Mw=mw.c2=(cћ)/(ƛw), N the particle quantum number, D boson displacement quantum number, α(0)=[(q2)/(4πεocћ)]=[1/(137.03599)], the fine structure constant [10] , q charge, and ε0 permittivity of free space. Equation (6) tells that the rest mass-energy (mop.c2) of a particle is proportional to (α/(N2.D)), the constant of proportionality being mass-energy Mw (=±mw.c2) of the charged boson W±. The constant of proportionality imparts the properties of a gauge boson (Mw) to the rest mass-energy of a particle. It is suggested that while a massless gauge (W±) boson acquires mass by Higgs mechanism [2,3], then any particle as well acquires mass in accordance with equation (6). 2.2. COUPLING CONSTANT The charged current electroweak interaction is caused by the emission and absorption of W± bosons. In the beta decay n(udd)---˃( p(uud)+ W- ---à(p(udd) + e- + (anti) עe, Coulomb force between the quark d(Q= -1/3) and W(Q= -1) boson pushes the neutron into oscillation. The coupling constants α(0) and αw are used for the electroweak forces. In the strong interaction exchange of π-meson between nucleons takes the place of W boson. At low energy when neutral current is zero, a gauge coupling constant αw for weak charged-current interaction is obtained by Kushtan [11] as: αw=(GF Mw 2)/(4π.21/2) (7) where, GF=Fermi’s constant=1.166 GeV2 , Mw=80.399 GeV [10], so that αw=4.244x10-3. 2.3. SPIN OF PARTICLES A particle subjected to a restoring force undergoes a time varying displacement; when Xdisp= T..ƛpc instead of Xdisp=D.ƛw in equation (6), we get m0p.c2= [(α(0)Mw)/(T2.D)]=[(±Fp).(T.ƛpc)2]/ (T2.ƛpc) =(Fp).[(ƛpc)/T] (8) In the orbitting charge model for particles [Rivas,12], it is considered that all charges are concentrated in a point known as centre of charge (CC) which is spinning around the centre of mass (CM) at the constant speed of light (c) while for CM observer CM is at rest. The appearance of the CC is expressed in terms of an angle of rotation T.(2π) after which the wavefunction of the particle returns to itself when T is the number of rotations by an angle of 2π. Also, T=(1/Sz) where Sz is the z-component of the spin. From equations (2) and (8) as T.mop.c2= T.Iop.ωop2 = ɦωop, when (Iop.ωop) is the angular momentum for rotation around the centre of charge, we get the Z-component of spin as Iop.ωop=(1/T)ɦ= Sz.ɦ (9) Thus, T=2, means Sz=(1/2), the angle of rotation being 2.( 2π); T=0 means Sz=0 for rotation invariance; T=1 means Sz=1, the angle of rotation being (2π); T=1/2 means Sz= 2 for rotation being (π). In the CM frame, from equations (8) and (9) when the centripetal force [(mop.c2)/RCC] balances the Coulomb force Fp, the CC rotates in a non-radiating circlular orbit of radius RCC =ƛpc/T with velocity of light. For spin ½ particles Rcc=(ƛpc/2). This local microscopic circulatory motion of the CC confined within the reduced Compton wavelength ƛpc is presumed to be the basis of spin and magnetic moment for particles. In the inertial reference frame the CC moves around the CM in a helical path with constant velocity v=c, while the CM moves with velocity v<c perpendicular to the plane of rotation along the axis of the helix [12]. 2.4. PARITY The angular momentum quantum number [L] of a particle is given by L=N-1 as in equation (6). Particle Data Group [13] gives parity (P) as For mesons: P=(-1)L+1 For baryons: P=(-1)L (10) 3. ZERO POINT ENERGY AND DECAY CONSTANTS FOR PSEUDOSCALAR MESONS In Quantum Field Theory (QFT), each point in a field is a quantum harmonic oscillator. In QFT a field arises from the vibration of inter-connected springs and balls filling the vacuum space [14]. Excitations of the field gives elementary particles. The energy eigenvalues (En) of a 3-D isotropic quantum mechanical harmonic oscillator at energy level n is given by [15]: En =ħω0p (n+ d/2) (11) when, the position for the 3-dimensional oscillator is given by x1, x2, and x3. Mass-energy and kinetic energy of particles is obtained by transition to lower energy states(m) for m<n. From equations (6) and (11) the mass-energy and decay formula for a particle is given as : En=(m0p.c2)n=[(α.Mw)]/[ (N2.D) ].( n+d/2) (12) where, ħω0p =(αMw)/(N2.D), Mw= mass-energy of the W± boson, n the energy level of the particle of mass-energy=(m0p.c2) , and α the coupling constant for the strong, weak or the electromagnetic fields in which the particle oscillates. The zero point energy (ZPE) from equation (12) is E0 =[(αMw)/(N2.D)].(d/2) when d is the dimension of the oscillator. According to QFT, there is ZPE for strong, weak and electromagnetic interactions. ZPE drives the Compton oscillation of the wavepacket representing a particle at n=0 [16,17]. The charged pion decay amplitude <M2> is given by [18] <M2>=(1/16)[(gw2. fπ2)/(Mw2)]2.ml2(mπ2-ml2)│Vi j│2 (13) where, gw is the weak charge, fπ charged pion decay constant, ml mass of lepton, mπ mass of pion, and Vi j KM matrix. In the SM the decay constant is determined from experiment. For quark-antiquark coupled to the W± boson, the lepton weak vertex factor is (-igw/(8)1/2.γµ(1-γ5) , when, γµ gives a vector current (V) and, (γµ.γ5) gives the axial vector current (A) . For the πW interaction the pure vertex factor is given by [(-igw/(8)1/2]).fπ].pπµ when, pπ is the momentum of π [19]. Taking zero point energy equal to the energy of the weak interaction as obtained from the vertex factor, from equationss (12) and (13) we get, [(gw)/(8)1/2].fπ=(α.Mw)/(N2.D) (14) 4. DIMENSION (d) AND OSCILLATION OF QUARKS The values for dimension (d) are obtained considering that quarks constituting a particle also oscillate and this oscillation builds up charge Q in a particle. It was found that the charge Q is explained by the number of quarks (Nq) constituting a particle and dimension (d) of the oscillator by the relation as given below: Q=±(Nq-d) (15) where, the + sign is for the particle and the – sign for the anti-particle. For neutral particles as ƩQquark=0, we get Nq=d. These values of d are used in calculating the hadron masses from equation (12). The significance of dimension(d) in building up charge in a particle in accordance with equation (15) is outlined below: (a) For the mesons, when d=1 the oscillating quarks are coupled as x1=x2 giving Q=±1. For baryons d=1 gives x1=x2=x3 i.e. all the quarks are coupled giving Q=±2. (b) For mesons, when d=2, x1≠x2 i.e. the quarks are not coupled when oscillation giving Q=0. For baryons when d=2, (x1=x2)≠x3 gives Q=±1 where two of the oscillating quarks are coupled and the the third quark oscillates being uncoupled to the two coupled quarks. (c) For baryons d=3 means x1≠x2≠x3 i.e. all the quarks oscillate randomly as they are uncoupled giving Q=0. From equation (15) changes in Q, Nq and d are given as Δ(Q)=±(Δ(Nq)-Δ(d)) (16) For flavour changing weak charged-current interactions when Δ(Nq)=0, we get Δd=-ΔQ i.e. dimension of the QMHO increases with decrease in Q. Flavour change produces change in charge and dimension of a particle. When ΔNq=0 and Δd=0, we get ΔQ=0. From equation(15) and Gell-Mann Nishijima formula, isospin I3 in this model is given as I3 =±((Nq-d)-Y/2) (17) and hypercharge Y is given in the SM as: Y=S+C+B/+T+B (18) when, strangeness(S=-1), charm(C=1), bottomness(B/=1), topness (T=1), and baryon number(B=1) are the quantum numbers in the Standard Model. Weak hypercharge Yw=2(Q-I3) plays the same role in weak interactions as the charge Q in electromagnetic interaction. 5. RESULTS AND DISCUSSIONS 5.1. PARTICLES AND THEIR DECAY From equation (12) we get mass-energy formula for particles as En=[(αMw)/(N2.D)].(n+d/2) (19) The decay process of a particle is explained by the transition to the lower mass-energy state(m) for m<n. From these transitions the kinetic energy (KE) of a neutrino is inferred. 5.1.1. MASS-ENERGY OF (π± ) MESONS AND THEIR DECAY Mass-energy The π± decays by weak interaction into a lepton-neutrino pair [20,21] as shown below: π+ -->μ+ + ﬠμ (muon neutrino) (20) For weak interaction α=αw=4.244x10-3, and for N=1, D=11, d=1, n=4, the mass-energy of π± is 139.58(139.57)MeV. Decay For E4-->3 decay, as Δd=0 and ΔNq=0, we get ΔQ=0, so that the emiited particle arising from this transition is uncharged e.g. a neutrino, and neglecting mass-energy of the neutrino, the KE(ﬠμ)= E4-E3≈ 31.0 (29.8)MeV [22]. The partcle arising from E3−−>X=∞ has Q=+1 and, KE(μ+)=[(E3–mμ.c2] =(108.56-105.66)MeV=2.90(4.12)MeV [22]. From equations(15) to (18), I3=±1(±1), and Q=±1(±1), and P=-1(-1) when the SM values are given in brackets. Fig.1 depicts the energy level diagram for the decay of π+ meson. N D n En MeV ΔE MeV --------------------- 1 11 4 139.58 --------------------- 1 11 3 108.56 31.00 -------------------- 1 11 x 0(Xp=∞) 108.56 Fig.1. Energy level diagram for the decay of π+ meson into μ+ and νμ . 5.1.2. MASS ENERGY OF K± Charged kaons decay by weak interaction as shown in the Feynman diagrams [20,21]. For α=αw, and N=1, D=10, d=1, and (n+d/2)=14.5, mass-energy of K± is obtained as 494.74 (493.70)MeV. Present Model values of isospin(I3), charge(Q), and Parity(P) are given with SM values in the bracket as I3=±1/2(±1/2), Q=±1(±1), and P=-1(-1). 5.1.3. NEUTRAL KAON AND STRONG COUPLING CONSTANT (αs) Neutral kaons are produced in strong interaction given by π+(ud) +p(uud)----˃ K+(us)+K0(sd)+p(uud) and they decay via weak interactions [20,21]. These suggest that mass-energies of K0 obtained from equation (19) for both electroweak and strong interactions are the same so that αs=[(m0p.c2)em.(Ds)]/[(Ms).(ns+d/2)] (21) when, subscripts s and em stand for strong and electroweak interactions respectively, Ms mass-energy of the exchange particle for strong interaction (π0=135MeV). For electroweakweak interaction, α(0)emMw=586.7MeV, Dem=13, (nem+d/2)=11 gives (m0p.c2)em= 496.44 MeV so that from equation (20) for Ds=1, (ns+d/2)=31, we get, αs=0.1186(0.1184) [10]. 5.1.4. MASS-ENERGY AND DECAY OF ∑0 AND Λ0 BARYONS Decays of Σ0 and Λ0 The decays ∑0 -----˃ Λ0+ﬠ and Ʌ0----˃ p + π- are explained by weak interactions [20,21]. For N=1, D=9, d=3, (n+d/2)=(30+1.5), and α=αw= 4.24x10-3, the mass-energy of ∑0 is 1193.68 (1192.64)MeV, and, ∑0 decays to Λ0 of mass-energy 1115.68(1115.68)MeV when KE(Λ0)max= 2.21MeV for (n+3/2)=29.5 from the transition E30--˃28. From equation (15) as ΔNq=0 and Δd=0, the emitted mass-energy does not carry any charge. It is suggested that a gamma ray(ﬠ) of energy 75.79MeV is emitted. The ∑0 is an excited state of Λ0 as there is no change in flavour when Δd=0 in Ʃ0 -à Ʌ0 decay and strangeness quantum number (S) is conserved in this decay. Decay of Ʌo For αw=4.24x10-3 , N=1, D=9, d=1, and (n+1/2) = 4.5, E4 =170.525MeV giving (E28 – E4 )=(1115.68-170.525)= 945.155 MeV. From equation (16), ΔQ=(Δ(Nq)-Δd)=(-1+2)=+1 giving Q=+1 for the emitted particle. The KE(p+)max= (945.155-938.27) MeV=6.89MeV, and for the transition to Xp=∞ gives KE(π-)max=(170.525-139.57) MeV=30.96MeV when ΔQ=-1 for conservation of charge. The predicted values for the KEmax’s of the decay products need experimental confirmation. 5.2. BETA ( β±) DECAY The beta (β±) decay process is explained as weak interaction from equation (12) with the help of Feynman diagrams [20,21]. In the weak charged current interaction quarks change flavour by emitting or by absorbing a virtuall W± boson [23]. For example a neutron becomes a proton by emitting a virtual W– boson which decays into β- particle and an anti-neutrino (ﬠe) [24,25]. The processes of β- decay are shown below: n——›p + W- —— ˃ p +e- +ﬠe(antineutrino) (22) The flavour change at quark level is given by d——› u + W-——˃ u + e- + ﬠe (23) The β+ decay is given by p ——> n + e+ + ﬠe (neutrino) (24) At quark level this is explained as u ——> d + W+ ——> d + e + +ﬠe (25) The emitted positron annihilates with the nearest shell electron giving rise to two photons as 2mec2= 2hγ. The mass-energy difference (ΔMo.c2)=[m(Z,A)-m(Z±1,A)].c2 between the parent and daughter atoms gives the released energy (Q) as Q(β-) =[m(Z,A)-m(Z+1,A)].c2; Q(β+)=[m(Z,A)-m(Z-1,A)-2me].c2 (26) In equation (25), Q(β±) gives KE’s of the β± particles and gamma energy (Ɛgam) from excited nuclei neglecting the energy of neutrinos. Equation (15) gives d=1 for a point like beta particle. From equation (12) for beta decay the atomic mass-energy difference [ΔMo.c2] between the parent and daughter atoms considering weak interaction (α=αw) and for the Xdisp=(D.αw-1ƛw), we get, Ɛn=[ΔMo.c2]n=[(αw2 .Mw)/(N2.D)](n+1/2) (27) The KEmax for β-decays arise from [Ɛn-Ɛ(X=∞)] transitions and for β-decays arising from collisions with excited daughter nuclei. Excited nuclei come to lower energy states by emitting gamma rays of energy Ɛγ=(Ɛn-Ɛm) where n>m. The proposed model accurately explains β± decay energies. 5.2.1. SINGLE β- DECAY TO GROUND STATE A neutron outside the nucleus is not stable and decays as 1n0 --- -> 1p1 + β- + ﬠe (anti neutrino) (28) As (mn-mp).c2 =1.292 MeV [10], maximum calculated kinetic energy of β- is KE(β-)max =0.782MeV. As mass of a neutrino is very small, it is neglected. From equation (27), for N=1, D=12, (n+1/2)=6.5, as ΔƐ=Ɛ6-Ɛ(Xp=∞)= KE (β-)max =0.784(0.781)MeV when the calculated value of KE is given in bracket. 5.2.2. SINGLE β- DECAY TO AN EXCITED STATE For this process: E=β-+KE(β-)max+Ɛgam (29) Ɛgam is the energy of gamma ray. For example the process as Au198 ----˃ Hg198 + β- + KE(β-)max + Ɛgam (30) (i) For N=1, D=9 and (n+1/2)= 8.5 gives KE(β-)max=1.37 (1.37)MeV. (ii) For N=1, D=9, (n+1/2)=6.5, Ɛ6 =Ɛgam=1.05(1.09)MeV;KE(β-)max =(Ɛ8 – Ɛ6)=0.32(0.285)MeV. (iii) For N=1, D=9, (n+1/2)=2.5: Ɛ2=Ɛgam=0.40(0.41)MeV; KE(β-)max= (ε8 -Ɛ2)=0.97(0.96); Ɛgam=(Ɛ4-Ɛ2)=0.63(0.65)MeV. All observed values are given in brackets. 5.2.3. POSITRON (β+) EMISSION This is given as: p –--> n + β+ + KE(β+)max + Ɛgam (31) For the β+ emission as shown below [26] : O15 --- ˃ N15 + β++KE(β+)max (32) For N=1, D=5, (n+1/2)=9.5: KE(β+)max= 1.73(1.735)MeV. 5.3. LEPTONS OTHER THAN NEUTRINOS Electron: From equation(19) for N=10, D=10, d=1, (n+d/2)=1.5 and αwMw= 341.2MeV giving for electron mec2=0.512 (0.511)MeV, and KE plus mass-energy of ﬠe =0.8KeV. At n=0, we get the vacuum expecation value for the electron as 0.17MeV i.e. one-third of its mass-energy. Muon: For muon, N=1, D=8, d=1, n=2, and α=αw, E2= 106.625 (105.66)MeV, and E1=63.975MeV. As charge must be conserved, one of the muon decay products must be an electron so that the muon decay is as given below: µ- -----˃ e- + ﬠe + ﬠµ (33) From equation(15) as ΔQ=0 for Δd=0, charges for mass-energy states 1 and 2 are the same, and the particle arising from E2−−>1 transition is thus neutral. We assign E1-E2=42.65MeV as the mass-energy plus KE of muon neutrino(ﬠμ). The KE of the electron comes from the transition E1--->X(p)=∞ = 63.98(50.0)MeV when calculated value is given in bracket. The charge at n=1 is Q1=(Nq-d1)=(0-1)=-1 as Nq=0 considering leptons as poinlike particles. Taun: For the taun, N=1, D=7, d=1, n=36, E36=1778.3(1776.8)MeV, K.E. of ﬠτ< 1.4MeV. 5.4. THE HYDROGEN ATOM For neutral hydrogen atom (Qp+Qe)=0 gives d=Nq=2 and the factor (n̷+d/2)=(n̷+1)=n. Replacing MW by me.c2, Xdisp= -(2n).(α-1 ƛe) and N=n, from equation (6) the energy of electron in the hydrogen atom(E)Hn is obtained for α=α(0) as (E)Hn =[(α(0)2.me.c2)].[(n̷ +1)/(N2.D)] =-[(e4me)/(8ϵ02h2)][1/n2] (34) Equation (34) means that when an electron oscillating in an electromagnetic field, say, due to a proton, is coupled to the electromagnetic field by coupling constant α(0) and when it is displaced by Xdisp= -(2n)(α-1ƛe), then discrete energy levels same as those in the Bohr”s model appear. From equation (16) as Δd=0 and ΔNq=0, then ΔQ=0 for any change in n, so that the energy absorbed or released is due to an uncharged particle and conservation of spin means this particle is a photon. The radius of the n-th stable Bohr orbit is given by Rn=n. ƛDB (35) where, ƛDB is the reduced de Broglie wavelength. Applying condition (34), radii of stable atomic orbitals are obtained. Also we get, ƛDB =α-1. ƛe (36) From equations (35) and (36), we get, R1=0.529x10-10m(0.529x10-10m) when the value obtained from Bohr’s theory is given in bracket. 5.5. DECAY CONSTANTS OF PSEUDOSCALAR MESONS Quark-antiquark annihilates via a virtual W+ boson to the final l+ע state for π+, k+, D+ and Ds+ pseudoscalar mesons are reported by the Particle Data Group (PDG) from experimental data of various sources with radiative corrections [27] . There are considerable differences in the radiative corrections from different groups in these results. Values of decay constants for these pseudoscalar mesons obtained from equation (14) with PDG values in the bracket are given below: For π+ decay as N=1, D=11, we get fπ=133.3(131.1) MeV; similarly, for K+ decay, as N=1 and D=10, fk=146.4(156.1) MeV; For D+ decay, as N=1, D=7, fD=209.2(206.7±11) MeV; For Ds+ decay, as N=1, D=6, fD(s)+=244.1(257.6) MeV. 7. CONCLUSIONS (1) The present model of particles oscillating in different fields due to Coulomb force, explains mass-energies of particles and their decay, beta decay and kinetic energies of beta particles, mass-energies of leptons excluding neutrinos and other properties of hadrons from a single formula. (2) Dimension (d) of the quantum mechanical harmonic oscillator explains bulid up of charges due to oscillation of the coupled and uncoupled quarks constituting a particle. 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