aa r X i v : . [ nu c l - t h ] A p r Dirac oscillators and the relativistic R matrix J. Grineviciute and Dean Halderson
Department of Physics, Western Michigan University, Kalamazoo, MI 49008
The Dirac oscillators are shown to be an excellent expansion basis for solutions of the Diracequation by R -matrix techniques. The combination of the Dirac oscillator and the R -matrix ap-proach provides a convenient formalism for reactions as well as bound state problems. The utilityof the R -matrix approach is demonstrated in relativistic impulse approximation calculations whereexchange terms can be calculated exactly, and scattering waves made orthogonal to bound statewave functions. PACS numbers: 24.10.-i, 24.10.Jv, 25.40.Cm
I. INTRODUCTION
In a previous paper [1] a calculable form of the R -matrix procedure was derived for the scattering of twoparticles obeying a relative Dirac equation. The pro-cedure was demonstrated by calculations of 33.5 MeVneutrons from a Woods-Saxon well. An expansion ba-sis consisted of the set of free-particle Dirac solutionswhose upper components were zero at twice the R -matrixradius, a c . This basis did not provide the same sta-bility seen in nonrelativistic calculations [2] and henceother basis function were sought. Dirac oscillators giveone that stability. The combination of the R -matrix ap-proach with Dirac oscillators provides a convenient for-malism for coupled-channel reactions in which the bi-nary breakup channels satisfy relative Dirac equationsand also for bound state problems such as in relativisticmean field theory (RMFT).In this paper the procedure is applied to relativistic im-pulse approximation (RIA) calculations. Here the tech-nique allows one to calculate the exchange terms exactlyand also to make the scattered wave orthogonal to thebound states obtained from RMFT. Calculations are per-formed with the relativistic Love-Franey (RLF) model ofHorowitz [3] for the two-nucleon t matrix. Cross sections,analyzing power, and spin rotation matrices are calcu-lated for protons on O, Ca, and Zr. The RLF modelwas constructed to have a small exchange amplitude, buteven so, calculation of the exact exchange proved to besignificant, leading to a conclusion that the RLF t ma-trices with pseudoscalar coupling give better agreementwith experimental results than previous determined. Theprimary difference between exact calculation of the ex-change and the conventional approximation appears inthe matrix elements between negative energy states. II. DIRAC OSCILLATOR
Dirac oscillators are obtained by adding a three-vectorpotential, linear in the radial coordinate, to the free-particle Dirac equation. They were introduced by Ito etal. [4] and later revived by Moshinsky and Szczepaniak[5] where the name, Dirac oscillator, was applied due to the nonrelativistic reduction appearing as the ordinaryharmonic oscillator with a strong spin-orbit interaction.Their symmetry Lie algebra was discussed by Quesne andMoshinsky [6] and they were employed in calculations forwave packets in 3 + 1 dimensions by Rozmej and Arvieu[7].The Hamiltonian for the Dirac oscillator is given by i ~ ( ∂ψ/∂t ) = (cid:2) c α · ( p − imω r β ) + mc β (cid:3) ψ = Eψ. (1)Analytical solutions may be found by writing the wavefunction in two-component form ψ = (cid:18) [ F ( r ) /r ] Φ κm [ iG ( r ) /r ] Φ − κm (cid:19) , (2)where Φ κm = X m l m s C l / jm l m s m Y lm l ( θ, φ ) χ m s , (3) j = | κ | − /
2, and l = κ for κ >
0, but l = − ( κ + 1) for κ <
0. Insertion of Eq. (2) into Eq. (1) and projectingon the appropriate Φ κm yields two coupled equations for F and G ,( E − m ) F ( r ) = [ − dG ( r ) /dr + η ( j + 1 / G ( r ) + mωrG ( r )] (4)and( E + m ) G ( r ) = [ dF ( r ) /dr , + η ( j + 1 / F ( r ) + mωrF ( r )] , (5)where η = +1 for κ > η = − κ <
0. With ~ = c = 1 and α = ( mω ) / , Eqs. (4) and (5) havesolutions F ( r ) = e − x/ x ( l +1) / L l +1 / n ( x ) and G ( r ) = − αE + m e − x/ x ( l ′ +1 ) / L l ′ +3 / n ′ ( x ) (6)for κ < F ( r ) = 2 αE − m e − x/ x ( l +1) / L l +1 / n ( x ) and G ( r ) = e − x/ x ( l ′ +1 ) / L l ′ +1 / n ( x ) (7)for κ >
0, where x = mωr and L ab ( x ) is an associatedLaguerre polynomial. In Eq. (6) l ′ = l +1 and n ′ = n − l ′ = l − n ′ = n , with n starting at zero. Theenergies are given by E = m + 4 ( n + l + 1 / mw for κ > E = m + 4 nmw for κ < u nl ( r ). Eq. (6) becomes F ( r ) = u nl ( r ) / p A and G ( r ) = u n ′ l ′ ( r ) A/ p A , (8)where A = − αE + m √ n, (9)and Eq. (7) becomes F ( r ) = u nl ( r ) / p A and G ( r ) = u n ′ l ′ ( r ) A/ p A , (10)where A = ( E − m ) 12 α s n + l + 12 ( n + l ) + 1 , (11)A curiosity of the Dirac oscillators is that G ( r ) = 0 for κ < n = 0 and the negative energy state for κ < n = 0 does not exist. One needs bothpositive and negative energy solutions in the R -matrixexpansion. III. THE R MATRIX
The R-matrix for the Dirac equation was developed inRef. [1] by following the same steps that were followedin the nonrelativistic case. For the one-channel case thewave function is in the form of Eq. (2) and is expandedwithin the channel radius as Ψ = P λ A λ | λ i . The setof | λ i will be Dirac oscillators. The appropriate Blockoperator, whose purpose is described in Lane and Robson[8], is constructed to be L ( b ) = δ ( r − a c ) (cid:18) − b i ( σ · r ) /r (cid:19) (12)and added to the Hamiltonian. The natural choice ofthe boundary condition parameter, b = G ( a c ) /F ( a c ),requires that L ( b ) Ψ = 0 and, therefore, [ H + L ( b )] Ψ = E Ψ. Insertion of the expansion into this equation gives X λ ′ [ h λ | H − E | λ ′ i + γ λ ( b λ ′ − b ) γ λ ′ ] A λ ′ = 0 . (13)where b λ = G λ ( a c ) /F λ ( a c ) and γ λ = F λ ( a c ). One cansee that the lower component in the Dirac theory is tak-ing the place of the derivative of the wave function in thenonrelativistic theory. The theory is placed in calculable form in the methodof Philpott [2] in which one finds a transformation, T ,such that X λλ ′ T λµ [ h λ | H | λ ′ i + γ λ b λ ′ γ λ ′ ] T λ ′ µ ′ = E µ δ µµ ′ . (14)With this transformation, Eq. (13) becomes X µ ′ [( E µ − E ) δ µµ ′ − γ µ bγ µ ′ ] A µ ′ = 0 , (15)where γ µ = P λ γ λ T λµ and A µ = P λ T λµ A λ . Equation(15) is solved for the A µ , A µ = γ µ G ( a c ) / ( E µ − E ), andthese reinserted into Eq. (15) to give the R -matrix equa-tion (1 − bR ) γ = 0 or R = F ( a c ) /G ( a c ), where R = X µ γ µ / ( E µ − E ) . (16) IV. SCATTERING AMPLITUDES
The regular and irregular Dirac Coulomb functionsare generated as given by Young and Norrington [9]employing the code COULCC[10]. The upper (lower)component of the regular function will be specified by F F ( G F ), while the upper (lower) component of the ir-regular function will be specified by F G ( G G ). The func-tion F F ( r ) → sin ϕ ( r ) and F G ( r ) → cos ϕ ( r ), where ϕ ( r ) = kr + y ln 2 kr − lπ/ δ ′ κ , k is the momentum of theproton in the center-of-momentum system, y = Ze E/k , E = m p + k , δ ′ κ = Ψ − arg Γ ( γ + iy ) + π l + 1 − γ ) ,e i Ψ = ie Z/k − κγ + iy , and γ = (cid:0) κ − Z e (cid:1) . One constructs the incoming, F I = F G − iF F , and outgoing, F O = F G + iF F , solutionsand defines the collision matrix from F → F I − SF O .The relation R = F ( a c ) /G ( a c ) gives R = ( F I − SF O ) / ( G I − SG O ) , (17)and therefore the collision matrix is determined for each κ by S = [ F I ( a c ) /F O ( a c )] [(1 − L I R ) / (1 − L O R )] , (18)where L I = G I ( a c ) /F I ( a c ) and L O = G O ( a c ) /F O ( a c ).In practice one must chose an R -matrix radius andnumber of basis states for each κ . The choice is made bylooking at the phase shift as a function of a c . One findsa region where the phase shift is not changing. This isillustrated in Fig. 1 for protons on Ca at 181 MeV withthe phenomenological potential of Ref. [11]. The curvesare for a maximum value ( N ) of n set to 16 and 18. Onesees a range of about 2 fm where the real part of the phaseshift is accurate to three figures. In general the curveshave a wider flat range and the flat range moves to largerradii as the number of oscillators increases. Also, feweroscillators are required for larger values of l . Fixing theradii for each κ can usually be accomplished by finding aradius for a sufficient number of oscillators ( N ) for κ = − κ but changingthe number of oscillators according to N = N − . l . FIG. 1: The real part of the κ = − R -matrix radius for N = 16 and 18. The calculation is forthe phenomenological optical potential of Ref. [11]. The Coulomb scattering amplitudes are given by F C = 12 ip ∞ X l =0 h ( l + 1) (cid:16) e iδ ′− l − − (cid:17) + l (cid:16) e iδ ′ l − (cid:17)i × P l (cos θ ) , (19) G C = 12 p ∞ X l =0 h e iδ ′− l − − e iδ ′ l i P l (cos θ ) . The series is summed using the reduction method wherethe amplitudes are multiplied by a function which van-ishes at θ = 0 and then expanded in a series of Legendrepolynomials:(1 − cos θ ) m F c = X l a ( m ) l P l (cos θ ) = X l (cid:18) a ( m − l − l l − a ( m − l − − l + 12 l + 3 a ( m − l +1 (cid:19) P l (cos θ ) , (1 − cos θ ) m G c = X l a ( m ) l P l (cos θ ) = X l (cid:18) a ( m − l − l − l − a ( m − l − − l + 22 l + 3 a ( m − l +1 (cid:19) P l (cos θ ) . The nuclear scattering amplitudes are the obtained byreplacing the Coulomb t matrix, t C = i (cid:16) − e iδ ′ κ (cid:17) / e iδ ′ κ t N in Eq. (19), where t N = i (1 − S ) /
2. Thetotal scattering amplitude then becomes the sum nuclearplus Coulomb.
V. IMPULSE APPROXIMATION
As an example of combining the Dirac oscillators withthe R -matrix techniques, calculations have been madewith the RFL amplitudes of Horowitz. These amplitudeshave been employed in the previous RIA calculation ofMurdock and Horowitz [12]. The present calculation fol-lows the procedures of Ref. [12], but deviates by cal-culating the exchange term without the nuclear matterapproximation for the exchange density. As given in Ref.[12, 13], one can write the RIA optical potential, actingon state U , for L − S closed shells as h x | V opt | U i = − πikm p X L Z d x ′ ρ L ( x ′ , x ′ ) t LD ( | x ′ − x | ; E ) λ L U ( x ) − πikm p X L Z d x ′ ρ L ( x ′ , x ) t LX ( | x ′ − x | ; E ) λ L U ( x ′ ) , (20)where t LY ( | x | ; E ) ≡ Z d q (2 π ) t LY ( q , E ) e i q · x , (21) ρ L ( x ′ , x ) ≡ occ X α ¯ U α ( x ′ ) λ L U α ( x ) , (22) λ L = 1 or γ for the scalar or vector potential, and Y = D for direct or Y = X for exchange. The exchange potentialis, therefore, non-local, but can be made local by approx-imating ρ L ( x ′ , x ) in the method of Brieva and Rook [14]as was done in Refs. [12, 13].However, it is possible to calculate the exchange ex-actly in the R-matrix approach. Here one is calculatingthe matrix elements appearing in Eq. (13). They takethe form h γ ψ nlj | V exch | ψ n ′ l ′ j ′ i = − πikm p Z d x ′ d x ¯ ψ nlj ( x ) ρ L ( x ′ , x ) × t LX ( | x ′ − x | ; E ) λ L ψ n ′ l ′ j ′ ( x ′ ) , (23)where t LX ( | x ′ − x | ; E ) is replaced by the integral form inEq. (21). The exponential is then expanded as e − i q · ( x − x ′ ) = (4 π ) X LML ′ M ′ (cid:2) i L j L ( qx ′ ) Y ∗ LM (ˆ q ) Y LM (ˆ x ′ ) (cid:3) ( − i ) L ′ j L ′ ( qx ) Y ∗ L ′ M ′ (ˆ x ) Y L ′ M ′ (ˆ q ) . (24)The matrix element becomes h γ ψ nlj | V exch | ψ n ′ l ′ j ′ i = − ikm p (2 j + 1) X j α τ α L Z q dqt τ α X ( q ) (cid:0) R F Lnα h j α l α k Y L k jl i± R GLnα (cid:10) j α ¯ l α k Y L k j ¯ l (cid:11)(cid:1) (cid:0) R F Lαn ′ h j α l α k Y L k jl i± R GLαn ′ (cid:10) j α ¯ l α k Y L k j ¯ l (cid:11)(cid:1) , (25)where the plus (minus) signs are for the vector (scalar)potential, R F Lnα = R dxF nlj ( x ) j L ( x ) F α ( x ), R GLnα = R dxG n ¯ lj ( x ) j L ( x ) G α ( x ), ¯ l = l ± j = l ∓ /
2, andthe reduced matrix elements are as defined in de-Shalitamd Talmi [15].
VI. RESULTS
The cross sections, analyzing power, and spin rotationmatrix for elastic scattering of protons from Zr, O,and Ca are shown in Figs. 2-7, calculated with the RFL t matrices of Ref. [3]. Calculations were made for boththe pseudoscalar and pseudeovector π − N coupling. TheRFL parameters are those recommended in Ref. [13]. Nomedium corrections were applied to the t matrices. Thesolid lines in these figures correspond to calculations withthe exchange contribution approximated by the methodof Ref. [14]. This converts the non-local potential to alocal one which is generated by the program FOLDER[13]. The dashed lines in these figures correspond to cal-culations with the exchange contribution calculated withEq. (22), and hence, one has solved the non-local po-tential problem. Looking at Figs. 2, 4, and 6, one canconclude that calculating the exchange term exactly im-proves the agreements with data [16, 17], except, perhapsfor the O analyzing power. For pseudovector couplingin Figs. 3, 5, and 7, it is difficult to tell whether the ap-proximate or exact exchange give better agreement withdata. However, the point is that the approximate andexact exchange give different results. This is somewhatsurprising in that the Breiva and Rook approximationhas done reasonably well in nonrelativistic calculations.Unfortunately, it is not possible to compare approx-imate and exact exchange optical potentials. One can,however, compare the matrix elements of the Dirac os-cillators for both the approximate and exact exchange.The diagonal matrix elements of the exchange term for κ = − n = 0 to 21 positiveenergy states, and basis numbers 23 to 43 correspond to n = 1 to 21 negative energy states. The upper panel is forthe approximate exchange, the lower for the exact. Thesolid line corresponds to the real part, the dashed to theimaginary. The dotted line is also the approximate realvalues in the lower panel for comparison. For the posi-tive energy states the approximate matrix elements tendto be larger, perhaps an average of 1.7 larger. However, for the negative energy states, the approximate matrixelements are very much larger and have the wrong sign.The contributions to the pseudovector matrix elementsare broken down further in Fig. 10. The dashed line cor-responds to the contribution to the exact matrix elementsoriginating from the scalar component of the t matrix;the dotted line corresponds to the contribution to theexact matrix elements originating from the vector com-ponent of the t matrix; the solid line corresponds to thecontribution to the approximate matrix elements origi-nating from the scalar component of the t matrix; thedot-dashed line corresponds to the contribution to theapproximate matrix elements originating from the vec-tor component of the t matrix. One sees that the matrixelements between positive energy states are, on average,very similar when calculated with exact and approximateexchange. This would be consistent with the Brieva andRook approximations working well in nonrelativistic cal-culations. However, the approximate matrix elements be-tween negative energy states are far too large, and boththe scalar and vector contributions have the same sign.The result is a significant difference in the observables,even for the RFL amplitudes which do not rely on sensi-tive cancellation between direct and exchange terms.Another advantage of the R -matrix approach is theability to make the scattering states orthogonal to thebound states. This would be particularly importantif one were doing low energy capture reactions. Thisis demonstrated in Fig. 11 with the E p = 9 . κ = − p + Ca wave functions, generated with the po-tential from Ref. [11]. The dashed (solid) lines are theupper components of the wave function with (without)projecting the RMFT wave functions out of the expan-sion basis.The cross section is also affected by this projectionas shown in Fig. 12. The dashed line corresponds toprojecting out the bound states which makes the nucleuslook larger and more diffuse. The effect of projecting outthe bound states decreases as the energy increases, andby 50 MeV, it is not significant.
FIG. 2: Elastic scattering cross sections. Solid (dashed) lines are from calculations with approximate (exact) exchange andpseudoscalar coupling. Data are from Ref. [16] and [17] as given in [13].FIG. 3: Elastic scattering cross sections. Solid (dashed) lines are from calculations with approximate (exact) exchange andpseudovector coupling. Data are from Ref. [16] and [17] as given in [12].
FIG. 4: Analyzing power. Solid (dashed) lines are from calculations with approximate (exact) exchange and pseudoscalarcoupling. Data are from Ref. [16] and [17] as given in [12].FIG. 5: Analyzing power. Solid (dashed) lines are from calculations with approximate (exact) exchange and pseudovectorcoupling. Data are from Ref. [16] and [17] as given in [12].
FIG. 6: Spin rotation matrix. Solid (dashed) lines are from calculations with approximate (exact) exchange and pseudoscalarcoupling. Data are from Ref. [16] and [17] as given in [12].FIG. 7: Spin rotation matrix. Solid (dashed) lines are from calculations with approximate (exact) exchange and pseudovectorcoupling. Data are from Ref. [16] and [17] as given in [12].
FIG. 8: Diagonal matrix elements of the exchange potentialwith pseudoscalar coupling versus basis function number for κ = − n = 0 to 21 positive energy states, and basis numbers 23to 43 correspond to n = 1 to 21 negative energy states. Theupper panel is for the approximate exchange; the lower forthe exact exchange. The solid line corresponds to the realpart, the dashed to the imaginary. The dotted line repeatsthe approximate real values in the lower panel. FIG. 9: Same as Fig. 8 but for pseudovector coupling. FIG. 10: Diagonal matrix elements of the exchange potentialwith pseudovector coupling versus basis function number for κ = − t matrix;the solid (dot–dashed) line corresponds to the contribution tothe approximate matrix elements originating from the scalar(vector) component of the t matrix.FIG. 11: E p = 9 . κ = − p + Ca, generated with thepotential from Ref. [11]. The dashed (solid) lines are uppercomponents of the wave function with (without) projectingthe
RMF T wave functions out of expansion basis. FIG. 12: Cross section plot of the E p = 9 . p + Ca.The dashed (solid) lines is the cross section with (without)projecting the
RMF T wave functions out of expansion basis. VII. CONCLUSIONS
Simple expressions for the Dirac oscillators have beenpresented, and a review of the one-channel R -matrixapproach to the Dirac equation given. The formalismdescribed above is easily extended to the multi-channelcase, and as in the non-relativistic problem, many chan-nels can be included. The utility of employing Diracoscillators as an expansion basis in the R -matrix formal-ism was demonstrated by performing RIA calculationsfor calculated for protons on O, and Ca and Zr. Inthe R -matrix approach the exchange terms may be calcu-lated exactly. It was found that even for the t matrices ofRef.[3], which were deliberately constructed so as to nothave sensitive cancellations between exchange and directterms, a significant difference was found between observ- ables calculated with the exact exchange and the plane-wave approximation for the exchange density. Calcula-tions employing the exact exchange improved agreementwith data for pseudoscalar π − N coupling in the RFLamplitudes, but for pseudovector coupling the agreementwas similar to that of calculations with the approximateexchange. The differences between the exact and approx-imate exchange were traced back to the matrix elementsbetween negative energy states of the expansion basis,and are, therefore, relativistic in origin. Acknowledgments
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