Effects of the interaction between the magnetic moments on the energy levels of a pionic atom
EEffects of the interaction between the magnetic moments on the energy levels of a pionic atom.
Voicu Dolocan
Faculty of Physics, University of Bucharest, Bucharest, Romania
We present the calculation of the binding energy of 2p pion for various central charges Z. We have taken into account the interaction between the magnetic moment of the nucleus and the orbital magnetic moment of the pion. The obtained results show that the interaction between the magnetic moments decreases the absolute value of the binding energy. This decrease is larger when the two magnetic moments add one another. Also, the binding energy increases with Z increasing.
Keywords: Klein-Gordon equation, pionic atom, interaction between magnetic moments.
1. Introduction
It has been of interest for physicists to study bound states of charged pions to nuclei, “pionic atoms”.The pion has the mass mc =139.577 MeV and spin 0. It obeys the Klein-Gordon equation [( E − V ( r )) − m c + ℏ c ∇ ] ψ( r )=
0 (1)The time independent field equation (1), in the static Coulomb field ,V(r) = -Ze /r , for the radial wave function =R(r)Y lm has the same form as the non-relativistic Schrödinger equation [ ℏ c E (− d d r − r ddr + l ( l + )− Z α r )− Z e r ] R = E −( m c ) E R (2)By comparing to the Schrödinger hydrogen-like equation [ ℏ μ (− d d r − r ddr + λ (λ + ) r )− Z e r ] R =ϵ R (3)we find μ= Ec λ= √ ( l + ) − Z α − ϵ= E −( m c ) E (4)Here is not an integer. As a result the bound state eigenvalues are given by ϵ= − Z α μ c ν where is the principal quantum number and takes values Next, −( m c ) E = − Z α E ν (5) = e /4 o ħc = 1/137.03602 is the fine structure constant. Solving for E one obtains E = m c √ + Z α / ν Expanding the above eigenvalue in a series of powers of Z yields E = m c ( − Z α ν + Z α ν + ... ) By expanding up to O(Z ) we can write λ= − Z α l + + ...and ν= n − Z α l + + ...Therefore [1], E = m c ( − Z α n − Z α ( l + ) n + Z α n + ... ) (5)where n = l +1, l +2, … The second term is the term we obtain in non-relativistic Schrödinger equation. The next two terms are the so-called “relativistic correction”, obtained by expanding the relativistic kinetic energy √ p c +( m c ) = m c + p m − ( p ) m c + ... (6)May be written p ψ= m ( Z e r − Z α m c n ) and therefore ⟨ Ψ | − ( p ) m c Ψ ⟩= − m c ⟨ Ψ | ( Z e r − Z α m c n ) ψ⟩ (7)Using Ψ | r Ψ ⟩= n a o ⟨ Ψ | r Ψ ⟩= ( l + ) n a o with a o = 4 o ħ /mZe , one obtains ⟨ Ψ | −( ⃗ p ) m c Ψ ⟩= m c ( − Z α ( l + ) n + Z α n + ... ) (8)Because the Klein-Gordon field has not spin, there is no spin-orbit coupling. Next, we calculate the energy levels of a pionic atom for n ≥ 2 and l ≥ 1, where play an important role the interaction between the magnetic moment of the nucleus and the orbital magnetic moment of the pion. We use a modified Coulomb potential due to the interaction between the magnetic moments as we have proceed to study ferromagnetism [2], high excitation energy levels of helium [3], deuteron energy states[4], Lamb shift in hydrogen atom [5] and Yukawa potential[6].
2. Effects of the interaction between the magnetic moments.
The magnetic moment of the nucleus is expressed in terms of the nuclear magneton N and nuclear spin I in the form μ=γ N μ N I where μ N = e ℏ M p c Experimentally values of N for some nuclei are presented in Table I [7,8] Table INuclear constants, I, N for some nucleiZ M, a.u. I N
10 20.183 ½ -1.8854220 40.08 3 1.602230 65.37 5/2 0.7690240 91.22 5/2 -1.3036250 118.69 2 0.04260 144.24 7/2 -1.06565 158.924 3/2 2.014 M p is the proton mass. The modified Coulomb potential due to the interaction between the magneticmoments is given by the expression [2,4] V ( r )=− Z e r [ + cos ( ar )] (9) = π e m c ( mM p γ N I − m l ) where m l is the magnetic quantum number of the pion. We have found [5] that the solution of the hydrogen atom Schrödinger equation in the constraint condition that the argument of the cosine and sine functions is a / n a o, is E =−ℏ m [ + ( a / n a o )+ ( a / n ) a o ] x [ √ n + mca π ℏ × ( cos ( a / n a o )− sin ( a / n a o ) N )− n ] ( ) a ( cos ( a / n a o )− sin ( a / n a o )) ;N = + ( a / n a o ) + [ cos ( a / n a o )+ sin ( a / n a o )] (10) which for a/ a o <<1 becomes E =−α m c n [ − α mcan ℏ ( − an a o ) + an a o ] (10 a )where =2×1.65013266/144 . For hydrogen-like atom is replaced by Z. By using the solution for the hydrogen-like atoms, we can write ⟨ ψ | [ p m − Z e r ( + ( a / r ))] ψ⟩= E (11)Further, Δ E =⟨ Ψ | −( p ) m c Ψ ⟩= − m c Z α ℏ c n a o l + − mc Z α m c n [ − Z α mcan ℏ ( − an a o ) + an a o ] + mc Z α ℏ cn a o Z α m c [ − Z α mcan ℏ ( − an a o ) + an a o ] (12)By using relation E − V ( r )= √ p c +( m c ) and relations (6) and (10-12) one obtains for the bound state energy E b = E – mc , the following xpression E b = E +Δ E (13)where E is given by Eq. (10 a ) and E is given by Eq. (12).The results are presented in Table II, E b (-) when the two magnetic moments substract, and E b (+) when the two magnetic moments add one another. Table IIBinding energies of 2 p pions for various central charges ZZ - E b (-), eV - E b (+), eV - E b (0), eV 10 92930.707320819 92930.6752206455 92930.97439493920 372587.68115116 372586.78741758 372581.5175170330 841565.39441219 841563.6760434 841584.4897771140 1504189.5656131 1504189.2593346 1504189.675664150 236506.2605144 2366505.9329577 2366512.990153160 3436318.6872157 3436295.066632 3436319.960413565 4051933.7664406 39360906.914632 4051935.2945534 The last column presents the binding energy when we neglect the interaction between magnetic moments, that is a = 0. For a → 0, relation (13) reduces to the well known expression[1] E b = m c [ − Z α n − Z α ( l + ) n + Z α n ] (14)where we have used relation a o = ħ/mcZ The number of pions per shell is (2 l + =3.30026532/144 . Note that we have attempted to find the binding energy of the pion by using equation (4) with E given by Eq. (10a ). Substituting = E /c , in this case, Eq. (4) becomes a quarten equation in E. We have solved this equation, but numerical results are far away from the acceptable results.
3. Conclusions
We have calculated the binding energies of 2p pions for various central charges Z. We have takeninto account the interaction between the magnetic moment of the nucleus and the orbital magnetic moment of the pion.The interaction between magnetic moments decreases the absolute value of thebinding energy. The derease is larger when the two magnetic moments add one another. The bindingenergy increases with Z increasing.
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