Corrections to Eikonal Approximation for Nuclear Scattering at Medium Energies
NNT@UW-14-12
Corrections to Eikonal Approximation for Nuclear Scattering atMedium Energies
Micah Buuck and Gerald A. Miller
Department of Physics, University of Washington, Seattle, WA 98195-1560 (Dated: July 21, 2018)
Abstract
The upcoming Facility for Rare Isotope Beams (FRIB) at the National Superconducting CyclotronLaboratory (NSCL) at Michigan State University has reemphasized the importance of accurate mod-eling of low energy nucleus-nucleus scattering. Such calculations have been simplified by using theeikonal approximation. As a high energy approximation, however, its accuracy suffers for the mediumenergy beams that are of current experimental interest. A prescription developed by Wallace [1, 2]that obtains the scattering propagator as an expansion around the eikonal propagator (Glauber ap-proach) has the potential to extend the range of validity of the approximation to lower energies.Here we examine the properties of this expansion, and calculate the first-, second-, and third-ordercorrections for the scattering of a spinless particle off of a Ca nucleus, and for nuclear breakupreactions involving Be. We find that, including these corrections extends the lower bound of therange of validity of the down to energies of 40 MeV. At that energy the corrections provide as muchas a 15% correction to certain processes.
PACS numbers:Keywords: a r X i v : . [ nu c l - t h ] J un . INTRODUCTION Ongoing and planned experiments using rare isotopes promise to further our understandingof nuclei and their role in astrophysics [3]. Nuclear reaction theory is needed both to interpretthe data and to determine the necessary experiments [4–6]. Use of the eikonal approximation(also known as Glauber theory [7]) has long been known as appealing procedure to simplify thecalculations, for medium and low energies see e.g. [8–12]. This technique has often been usedto analyze experiments, see e.g [13–15] performed at energies less than 100 MeV per nucleon.A computer program using the eikonal approximation, described as being appropriate forknockout reactions for energies between 30 and 2000 MeV per nucleon, has been published [16].However, as stated in the orignal article [7] the Glauber theory rests on the approximationthat the product of the wave number k and the range of the relevant potential a satisfy k a (cid:29) V be very small compared to the scatteringenergy, E , so that V /E (cid:28) . (2)For a nucleon of energy 100 MeV and nucleus of radius ≈ k a ≈
6, and
V /E ≈ / . It is far from obvious that the conditions for the accuracy of the Glauber approximation aresatisfied. Moreover, it is not clear if the relevant distance appearing in the term ka should bethe nuclear radius or the nuclear diffuseness. If the latter, the beam energy must be higherfor the eikonal approximation to be valid. It is therefore of interest to assess the accuracyof Glauber theory and the lower limits on energy for which it may be applied [17]. In thefollowing we treat the terms eikonal approximation and Glauber theory as synonymous.The conclusions of Ref. [17] have been summarized [4] as showing that the eikonal approx-imation is accurate to within a few percent for energies as low as 20 MeV/nucleon. Thisconclusion is based on a comparison between the results of using the eikonal approximationand a time-dependent Schr¨odinger equation. The incoming projectile is treated as a boundstate of a nucleon and a core. The time-dependent Schr¨odinger equation that includes thedynamics of the interaction of the nucleon with the core as well as the nucleon-target in-teraction was solved. We do not believe the conclusion that the eikonal approximation isvalid at 20 MeV [4] to be a valid summary of the work of Ref. [17]. This is because thetime-dependent equation (their Eqs.(5,6)) treats the motion of the core of the projectile asfollowing the linear trajectory R = b + v t . In other words, the eikonal approximation is usedin the time-dependent Schroedinger equation. Thus the work contains no actual test of theeikonal approximation. However, Ref. [17] does have the very useful result that the interactionbetween the nucleon and the core that occurs during the nuclear reaction can be neglectedfor energies as low as 20 MeV/nucleon. Thus the so-called sudden approximation is justified,at least for one particular state. However, the use of the eikonal approximation has not beenjustified and the range of its validity has not been fully determined. Thus the present paperis devoted to studying the corrections to the eikonal approximation.In this paper we assess the validity of the eikonal approximation by computing the cor-rections Sect. II to this approximation for potential scattering Sect. III, and for reactions2nvolving halo nuclei Sect. IV. The principal tool is the expansion developed by Wallace [1, 2]in which the complete Green’s function is expanded about the Glauber approximation to thecomplete Green’s function. Our results and directions for further research are summarized ina final Sect. V. II. CORRECTIONS TO THE EIKONAL THEORY
We first apply the corrections to the eikonal approximation described by Wallace [1, 2] forscattering of a spin-zero particle off a generic potential. This exercise is useful because we cancalculate the scattering amplitude exactly using a partial wave expansion and compare it withsuccessive corrections in the eikonal expansion. We will give a quick review of the correctionshere using the same notation as [1, 2] before showing the results of our calculations.The T matrix for scattering at a center of mass energy E = K / M is given by T ( E ) = V + V G ( E ) T = V + V G ( E ) T ( E ) , (3)where G − ( E ) = E − P / M − V + i(cid:15) is the particle propagator and V is the interactionpotential.The Wallace eikonal expansion consists of expanding the momentum operator P about aparticular vector k and dropping all terms quadratic in P − k . The choice k = K (cid:98) k with (cid:98) k as the average of the projectile initial and final direction ( k = ( k i + k f ) /
2) gives the Glauberapproximation and the propagator: g − = v · ( k − P ) − V + i(cid:15). (4)The difference between the full propagator G and the reduced eikonal propagator g is givenby g − − G − = N (5) N = (1 − cos( θ/ g − + V ) + [( P − k f ) · ( P − k i )] / M, (6)where θ is the scattering angle.It is then possible to solve for the T matrix as a perturbation series: T = ( V + V gV ) +
V gN gV + V gN gN gV + V gN gN gN gV + . . . . (7)The Glauber approximation consists of keeping only the terms in parentheses, and Wal-lace showed how to systematically calculate higher order correction terms. The result is anexpansion in powers of the interaction energy over the kinetic energy with corrections due tothe spatial non-uniformity of the potential. He explicitly calculates the first three correctionterms, and first with the conjecture of some advantageous cancellations [1, 2], and later [18]in an explicit calculation obtained the following expressions: T (0) ( b ) = e iχ ( | b | ) − T (1) ( b ) = e i ( χ ( | b | )+ τ ( | b | )) − T (2) ( b ) = e i ( χ ( | b | )+ τ ( | b | )+ τ ( | b | )) e − ω ( | b | ) − T (3) ( b ) = e i ( χ ( | b | )+ τ ( | b | )+ τ ( | b | )+ τ ( | b | )+ φ ( | b | )) e − ω ( | b | ) − ω ( | b | ) − . (11)3ere b is the impact parameter, T (0) is the Glauber approximation, and the phases are definedbelow, with z ⊥ b , r = b + z , U ( r ) = V ( r ) /V (0), (cid:98) β n ≡ b n ∂ n /∂b n , and (cid:15) = V (0) / E : χ ( b ) = − K(cid:15) (cid:90) ∞ d z U ( r ) (12) τ ( b ) = − K(cid:15) (1 + (cid:98) β ) (cid:90) ∞ d z U ( r ) (13) τ ( b ) = − K(cid:15) (1 + 53 (cid:98) β + 13 (cid:98) β ) (cid:90) ∞ d z U ( r ) − b [ χ (cid:48) ( b )] K (14) ω ( b ) = bχ (cid:48) ( b ) ∇ χ ( b )8 K (15) τ ( b ) = − K(cid:15) ( 54 + 114 (cid:98) β + (cid:98) β + 112 (cid:98) β ) (cid:90) ∞ d z U ( r ) − bτ (cid:48) ( b )[ χ (cid:48) ( b )] K (16) φ ( b ) = − K(cid:15) (1 + 53 (cid:98) β + 13 (cid:98) β ) (cid:90) ∞ d z (cid:20) ∂U ( r ) /∂r K (cid:21) (17) ω ( b ) = bχ (cid:48) ( b ) ∇ τ ( b ) + bτ (cid:48) ( b ) ∇ χ ( b )8 K (18)We see that the corrections related to (cid:98) β n involve the derivatives of the nuclear potential whichare large in the region of the nuclear surface. This indicates that the product of the wavenumber and the nuclear diffuseness parameter, needs to be large compared to unity for theeikonal approximation to be valid. This condition is more stringent than the one involvingthe product of the wave number and the nuclear radius.The scattering amplitude is then simply: f ( q ) = − iK/ (cid:90) d b e i q · b T ( n ) ( b ) . (19) III. EIKONAL EXPANSION
VS.
EXACT PARTIAL WAVE RESULTS
Our focus is on reactions at FRIB energies. We therefore evaluate the scattering amplitudefor protons scattering off of Ca using the potential described by Varner et. al. [19] forincident center-of-mass kinetic energy between 16 and 98 MeV. We neglect the spin-orbit andthe Coulomb interaction because such terms are neglected in Ref. [12].This potential is then given by V ( r, E ) = − V r ( E ) f ws ( r, R , a ) − iW v ( E ) f ws ( r, R w , a w ) − i W s ( E ) ddr f ws ( r, R w , a w ) , (20)with f ws ( r, R, a ) = 11 + exp[( r − R ) /a ] (21) V r ( E ) = V + V e ( E − E c ) ± V t ε (22) W v ( E ) = W v0 f ws ( W ve0 , ( E − E c ) , W vew ) (23) W s ( E ) = ( W s0 ± W st ε ) f ws (( E − E c ) , W se0 , W sew ) , (24)4here the ± indicates + for proton projectiles and − for neutron projectiles. Parameters inthe model can be found in Table I and in the text. Parameter Value Uncertainty ε ( N − Z ) /A – V ± . V t ± . V e -0.299 ± . r ± . r (0)0 -0.225 fm ± . R r A / + r (0)0 – a ± . r c r (0)c R c r c A / + r (0)c fm – E c Ze / R c MeV – W v0 ± . W ve0
35 MeV ± W vew
16 MeV ± W s0 ± . W st
18 MeV ± W se0
36 MeV ± W sew
37 MeV ± r w ± . r (0)w -0.42 fm ± . R w r w A / + r (0)w fm – a w ± . The approximate eikonal solution to potential scattering can be compared with an exactsolution obtained by using a partial wave technique. Phase shifts for arbitrary values of (cid:96) areobtained by numerically solving the radial Schr¨odinger equation, d u (cid:96) dr + (cid:18) k − mV − (cid:96) ( (cid:96) + 1) r (cid:19) u (cid:96) = 0 , matching u (cid:96) ( R ) = Re iδ (cid:96) [cos( δ (cid:96) ) j (cid:96) ( kR ) − sin( δ (cid:96) ) n (cid:96) ( kR )] for R large enough such that V ( R ) ≈ δ (cid:96) . Here, j (cid:96) and n (cid:96) are spherical Bessel functions of the first and second kind,respectively. The scattering amplitude is then: f ( θ ) = ∞ (cid:88) (cid:96) =0 (2 (cid:96) + 1) e iδ (cid:96) sin δ (cid:96) P (cid:96) (cos θ ) . Σ ! b a r n s " ! MeV " FIG. 1: (Color online) Total nuclear cross-section σ for a proton incident on Ca as a function ofbeam energy. The exact partial wave result is the thick blue line, the zeroth-order eikonalapproximation is the thin magenta line, the first-order eikonal approximation is the beige dashedline, the second-order eikonal approximation is the green dot-dashed line, and the third-ordereikonal approximation is the red dotted line.
A. Results of Calculations
Our main results for these calculations are presented in figures 1-4. Figure 1 shows the totalelastic nuclear cross-section of a spinless proton incident on a Ca nuclear potential in theexact calculation, and in successive orders in the eikonal expansion. The zeroth order eikonalapproximation has an error of at least 5% up to 100 MeV, while including the correction termsreduces the error to <
1% above about 45 MeV. The relative degree of agreement between6he zeroth order approximation and the exact calculation at energies below 20 MeV is likelya coincidence. d Σ (cid:144) d W H b a r n s L - - Θ H Radians L FIG. 2: (Color online) Differential elastic cross-section dσ/d
Ω for p + Ca at a beam energy of 40MeV in log scale. The angle θ is the scattering angle from the forward direction. The designationsfor the lines are the same as in Fig. 1. Figures 2 and 3 show the differential elastic cross-section for the same reaction at a beamenergy of 40 MeV. Even for forward scattering, the zeroth-order eikonal approximation sev-erly underestimates the exact value, and successive corrections monotonically improve theestimate. The corrections also successively improve the range in the polar angle θ over whichthe approximation is accurate.Figure 4 gives the real and imaginary parts of the T -matrix elements T ( n ) ( b ) for successiveorders n in the expansion as a function of the impact parameter b at a variety of beamenergies. The rapid oscillations and drastic changes in T with each correction at 20 MeVimply that the expansion is not appropriate there. This is because the interaction potentialis energy dependent. In this case, it is the imaginary part of the potential that is important.It has both a surface and a volume term which have magnitudes that behave oppositely as afunction of beam energy, as shown in fig. 5a. At low energies, the surface term dominates andthe derivative operators (cid:98) β n in τ ( b ) are large, negative, and imaginary, (see Fig. 5b) whichgenerate large oscillations in T (3) ( b ). (The frequencies of such oscillations are given by the realpart of τ ( b ).) This behavior also occurs in τ ( b ) and τ ( b ), but at lower energies. The point7 Σ (cid:144) d W H b a r n s L Θ H Radians L FIG. 3: (Color online) Same as Fig. 2, but zoomed in on the forward scattering region, and with alinear scale. at which this breakdown occurs provides a lower bound on the effectiveness of the expansionthat can be computed order-by-order. For example, in fig. 6a, which was calculated at 25MeV, the third-order correction has a real part of about 1, which is already an amplitude ofoscillations in T (3) ( b ) of about 2.7 at b ≈ .
75. The second-order correction is about to enterpositive territory in fig. 6a at b ≈ .
25, and will start to generate similar rapid oscillationsin T (2) (3 .
25) at lower eneries. This can be seen in fig. 4a, which was calculated at 20 MeV.Thus, empirically, the second-order correction is effective to about 25 MeV for this potential.Using the same method, we find the third-order correction to be effective to about 30 MeV.Since the convergence of the expansion improves at higher energies, calculating only thefirst-order correction should be sufficient at some sufficiently high beam energy. From fig. 1this appears to happen for this potential at a beam energy of about 60 MeV. Above this value,the fractional error in the second- and third-order corrections is only marginally lower thanthe fractional error in the first-order correction.With these calculations, it is apparent that for at least some interactions, these correctionsto the eikonal approximation are meaningful over a range of energies. It is therefore worthwhileto apply the corrections to a more interesting interaction to further evaluate their effectiveness.8 e @ T D - - -
101 b H fm L (a) Beam Energy at 20 MeV I m @ T D - - H fm L (b) Beam Energy at 20 MeV R e @ T D - - - - - - H fm L (c) Beam Energy at 40 MeV I m @ T D - - H fm L (d) Beam Energy at 40 MeV R e @ T D - - - - - - H fm L (e) Beam Energy at 98 MeV I m @ T D H fm L (f) Beam Energy at 98 MeVFIG. 4: (Color online) Real and imaginary parts of the transition matrix elements T ( n ) ( b ) forsuccessive orders n in the eikonal expansion. The designations for the lines are the same as infigs. 1. IV. BREAKUP REACTIONS OF HALO NUCLEI BE We now apply these calculations to the study of scattering of Be off of various targets,using the reaction theory of Hencken, Bertsch & Esbensen [12]. They computed the diffractive,9 m a g i n a r y P a r t s o f P o t e n ti a l
20 40 60 80 10051015 E (cid:72)
MeV (cid:76) (a) Imaginary surface (solid blue) W s ( E )and volume (dashed magenta) W v ( E ) termsof Varner potential as a function of beamenergy. Τ (cid:45) (cid:45) (cid:45) (cid:45) (b) Real (solid blue) and imaginary (dashedmagenta) parts of τ ( b ) at a beam energy of20 MeV.FIG. 5: (Color online) R e (cid:64) l n (cid:72) T (cid:76) (cid:68) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45)
101 b (a) I m (cid:64) l n (cid:72) T (cid:76) (cid:68) (b)FIG. 6: (Color online) Real and imaginary parts of ln( T ( n ) ( b )) for n = 0 (thin magenta), n = 1(dashed beige), n = 2 (dot-dashed green), and n = 3 (dotted red) calculated at a beam energy of 25MeV. neutron stripping, core stripping, and total absorption cross sections for Be scattered offtargets with mass number ranging from 9-208 using the Glauber eikonal approximation at anenergy of 40 MeV/nucleon. They used the Varner potential [19] as the model for nucleon-nucleon scattering. Given that we see a significant improvement in the performance of theeikonal approximation at that energy when the Wallace corrections are included for the Varnerpotential, it is fruitful to investigate whether or not the cross-sections evaluated by Henckenand Bertsch also experience similar improvement.The relevant formulae of Ref. [12] are displayed next. The reaction considered is H + T → c + X , where the projectile halo nucleus H is treated in a single particle model as c + n with c corresponding to a specific final state of the core. The halo nuclear ground state is10escribed by a wave function φ LM ( r ) which depends on the relative coordinate (cid:126)r between thenucleon and the core, see Fig. 7. The function is generally specified by φ LM ( r ) = R L ( r ) Y LM ( (cid:98) r )where Y LM ( (cid:98) r ) are spherical harmonics. Here we take R L ( r ) to be the solution to the radialSchr¨odinger equation in an L = 0 state with the appropriate binding energy of 0.503 MeV. c T n r R n R c R FIG. 7: Coordinates used in this calculation. R is the coordinate of the center of mass of the halonucleus, and b c and b n denote the components of R c and R n that are transverse to the beamdirection. The scattering wave function of the halo nucleus has the form,Ψ( r , R ) = S n ( b n ) S c ( b c ) φ LM ( r ) , (25)in its rest frame, where (Fig. 7) R is the coordinate of the center of mass of the halo nucleus,and b c and b n are the impact parameters of the core and the nucleon with respect to thetarget nucleus, i. e. b n = R ⊥ + r ⊥ A c / ( A c + 1) and b c = R ⊥ − r ⊥ / ( A c + 1), where A c is themass number of the core and the designation ⊥ refers to components transverse to R n and R c . The two profile functions, S n ( b n ) for the nucleon and S c ( b c ) for the core, are generatedby interactions with the target nucleus. In the eikonal approximation, they are defined by thelongitudinal integrals over the corresponding potentials: S ( b ) = exp (cid:20) − i (cid:126) v (cid:90) dzV ( b + z (cid:98) z ) (cid:21) , (26)where v is the beam velocity and potential V is the optical potential. The relation between S ( b ) and the quantities denoted as T ( b ) of Sect. II) is given by S ( b ) = T ( b ) + 1 . (27)We compute the order n corrections by replacing S ( b ) from equation 26 with T ( n ) ( b ) + 1 fromequations 18.The scattering wave function is the difference between eq. (25) and the wave function ofthe undisturbed beam, Ψ scat = ( S n S c − φ LM . (28)11ith the shorthand notation S n = S n ( b n ) and S c = S c ( b c ).Scattering cross sections are calculated by taking overlaps of Ψ scat with different final states.For diffractive breakup the final state depends on the relative momentum (cid:126)k of nucleon andcore in their center-of-mass frame as well as on the transverse momentum (cid:126)K ⊥ of the center ofmass. Writing the continuum nucleon-core wave function as φ k ( (cid:126)r ) (normalized asymptoticallyto a plane wave: φ k ∼ exp( i k · r )) the diffractive breakup cross section is given by dσ diff. ( d K ⊥ d k ) = 1(2 π ) L + 1 (cid:88) M (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d rd R ⊥ e − i K ⊥ · R ⊥ φ ∗ k ( r ) S c S n φ LM ( r ) (cid:12)(cid:12)(cid:12)(cid:12) . (29)To obtain the relative momentum distribution in (cid:126)k , integrate over K ⊥ to get dσ diff. d k = 1(2 π ) L + 1 (cid:88) M (cid:90) d R ⊥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d rφ ∗ k ( r ) S c S n φ L,M ( r ) (cid:12)(cid:12)(cid:12)(cid:12) . (30)A convenient expression for the total diffractive cross section can be derived using com-pleteness if φ LM is the only bound state of the system. The result is σ diff. = 12 L + 1 (cid:88) M (cid:90) d R ⊥ (cid:34) (cid:90) d (cid:126)rφ L,M ( r ) ∗ | S c S n | φ L,M ( r ) − (cid:88) M (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d rφ ,M (cid:48) ( r ) ∗ S c S n φ L,M ( r ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:35) . (31)Other contributions to the total cross section come from absorption, present when theeikonal S -factors have moduli less than 1. There are three of these so-called stripping pro-cesses. The nucleon-absorption cross section, differential in the momentum of the core, isgiven by dσ n-str. d k c = 1(2 π ) L + 1 (cid:88) M (cid:90) d b n (cid:104) − (cid:12)(cid:12) S n ( b n ) (cid:12)(cid:12) (cid:105) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) d r e − i k c · r S c ( b c ) φ L,M ( r ) (cid:12)(cid:12)(cid:12)(cid:12) . (32)The corresponding total cross section for stripping of the nucleon is σ n-str. = 12 L + 1 (cid:88) M (cid:90) d b n (cid:104) − (cid:12)(cid:12) S n ( b n ) (cid:12)(cid:12) (cid:105) (cid:90) d r φ L,M ( r ) ∗ (cid:12)(cid:12) S c ( b c ) (cid:12)(cid:12) φ L,M ( r ) . (33)The stripping of the core is expressed in a similar way, interchanging subscripts n and c .The expression for absorption of both nucleon and core is given by σ abs. = 12 L + 1 (cid:88) M (cid:90) d b c (cid:104) − (cid:12)(cid:12) S c ( b c ) (cid:12)(cid:12) (cid:105) × (cid:90) d (cid:126)rφ ∗ L,M ( r ) (cid:104) − (cid:12)(cid:12) S n ( b n ) (cid:12)(cid:12) (cid:105) φ L,M ( r ) . (34)12 . The potential for the Nucleon-Target and Core-Target Interaction Evaluation of the profile functions requires a potential model for the interaction betweenthe target nucleus and the constituents of the halo nucleus. At low energies, extending up toabout 100 MeV/ n , one can find optical potentials that are fit to nucleon-nucleus scattering.We use the optical potential, V op of ref. [19], which was fit to scattering data in the rangeof 10 to 60 MeV. The potential has the usual Woods-Saxon form, with volume and surfaceimaginary terms, but we neglect the spin-orbit and Coulomb interactions as does [12]. Thispotential represents the target-nucleon interaction. The core-target interaction potential isobtained by folding V op with the core density distribution, V c ( r ) = (cid:90) d xρ c ( x ) V op ( | r − x | ) . (35)For the core density we use a harmonic oscillator density with parameters taken from thecharge distribution of the core nucleus [20] ( a =2.5 fm and α =0.61). B. Results of Eikonal Expansion Calculations Σ (cid:72) b a r n s (cid:76) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236)(cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242)(cid:231) (cid:231) (cid:231) (cid:231) (cid:231) (cid:231)(cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225)(cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237)(cid:243) (cid:243) (cid:243) (cid:243) (cid:243) (cid:243)
10 1005020 2003015 150700.101.000.500.202.000.303.000.151.500.70 A
FIG. 8: (Color online) Comparison of first-order corrections with standard (zeroth-order) eikonalapproximation with a beam energy of 40 MeV/nucleon. The zeroth-order terms are shown withsolid markers, and the first-order terms are with outlined markers. The solid (blue) line with circlesis diffractive scattering, the dashed (magenta) line with squares is core stripping, the dotted (beige)line with diamonds is neutron stripping, and the dash-dotted (green) line with triangles is totalabsorption of the core and neutron.
In this subsection we present the results of applying the Wallace corrections to the totalcross-sections described above (eqns. 31, 33, and 34). Our primary results for these calculationsare summarized by figs. 8-10. 13 (cid:72) b a r n s (cid:76) (cid:198) (cid:198) (cid:198)(cid:196) (cid:196) (cid:196) (cid:230) (cid:230) (cid:230)
10 1005020 2003015 150700.20.51.02.0 A
FIG. 9: (Color online) Comparison of corrections to scattering data at 41 MeV/nucleon from [21].The designations for the lines are the same as in fig. 4, with the Coulomb breakup cross-section indashed black. Because we did not calculate corrections to the Coulomb cross-section, thecorrections appear smaller on this plot at high energies where the Coulomb term is larger.
Figure 8 shows the effect of the first order corrections for scattering at 40 MeV/nucleon.These corrections are generally not negligible for any value of A Figure 9 compares our results to scattering data collected at 41 MeV/nucleon by Anneet. al. [21]. The data were collected by detecting the Be core, so the processes that con-tribute are diffractive scattering, neutron stripping, and Coulomb breakup, which we did notconsider. The Coulomb cross-section was taken from [21] and added to our calculations.Although the corrections have a noticeable effect when compared to the zeroth-order calcu-lations, it is unclear from this data whether the effect is actually significant since the errorin the measurements is so large. With more precise experimental measurements the utility ofthese corrections will become clearer.Figure 10 gives the fractional correction at each order, which more clearly illustrates theeffects of the corrections. The corrections to neutron stripping and total absorption are onlysignificant at first-order, and appear to be independent of A . The corrections to diffractivescattering are significant at large values of A all the way through third-order, but are lesssignificant at low values of A . We have also performed the same calculations at the higher14 r ac ti on a l C o rr ec ti on æ æ æ æ æ æà à à à à àì ì ì ì ì ìò ò ò ò ò ò
10 1005020 2003015 15070 - - (a) Effect of first-order corrections at beamenergy of 40 MeV. F r ac ti on a l C o rr ec ti on æ æ æ æ æ æà à à à à àì ì ì ì ì ìò ò ò ò ò ò
10 1005020 2003015 15070 - (b) Effect of second-order corrections atbeam energy of 40 MeV. F r ac ti on a l C o rr ec ti on æ æ æ æ æ æà à à à à àì ì ì ì ì ìò ò ò ò ò ò
10 1005020 2003015 15070 - - (c) Effect of third-order corrections at beamenergy of 40 MeV. F r ac ti on a l C o rr ec ti on æ æ æ æ æ æà à à à à àì ì ì ì ì ìò ò ò ò ò ò
10 1005020 2003015 15070 - - (d) Effect of first-order corrections at abeam energy of 100 MeV/nucleon at beamenergy of 40 MeV.FIG. 10: (Color online) Fractional corrections at various orders and beam energies. Thedesignations for the lines are the same as in fig. 8. energy of 100 MeV (see figure 10d). As expected, the corrections are smaller at first order(less than 10%), and are less than 1% at higher orders.Diffractive scattering is primarily a surface effect, (see Fig. 11) which is why the diffractivecorrections have a markedly different behavior as a function of A than the other types ofscattering. As A changes, the radius of the target nucleus changes as well. The correctionsare larger for surface effects than for volume effects (especially at low energies) because of thederivative operators (cid:98) β n that arise. Since diffractive scattering is the only type of scatteringstudied here that is almost entirely a surface effect, changes in the radius of the target affectit more than the other types of scattering we studied.15 (cid:222) FIG. 11: (Color online) The integrand of eq. 31 (solid blue) for a
Pb target, and the second termin the same integrand, which is the elastic scattering for the system (dashed magenta). Both aregiven as functions of R ⊥ (see Fig. 7) with all other variables integrated out. V. SUMMARY AND DISCUSSION
We have calculated corrections to the eikonal approximation to nuclear scattering in aneikonal expansion framework for many different processes. We find that for the case of simplepotential scattering it is clear that application of these corrections improves the accuracy ofthe eikonal approximation at beam energies between 30 and 100 MeV. It is reasonable toexpect that the first-order correction would be significant at even higher beam energies.We also see from application to the interactions of Be with nuclei at 40 MeV that thesecorrections can be as high as 15% for neutron stripping and diffractive scattering. As expected,the corrections decrease as the beam energy increases.We compare our theory with the data of Anne et al. [21] and find that the corrections aresubstantial, although not as large as the experimental uncertainties.It is interesting to note that the diffractive corrections have a strikingly different behaviorfrom the corrections to stripping and absorption. We attribute this to surface effects thathave a stronger influence on diffractive scattering than on the other processes.The first-order corrected cross-sections do not require much more computational effort tocalculate than the zeroth-order calculations. We performed our calculations on an 8 core nodeof the Hyak scientific computing cluster at the University of Washington, and saw less thana factor of 2 increase in computation time after including the first-order corrections. Evenadding in the second- and third-order corrections usually resulted in less than a factor of 2increase in computation time, although the calculation of the T -matrix elements does increasein complexity (eqns. 8-18).Thus we believe that our proposed framework of using the eikonal approximation as im-proved by the corrections of Wallace would be a useful way to analyze data produced at FRIB.Future work will focus on specific reactions of experimental interest.16 cknowledgements The authors would like to thank George Bertsch for sharing some of his data, providingadvice on some calculations and commenting on the manuscript. This work has been par-tially supported by U.S. D. O. E. Grant No. DE-FG02-97ER-41014 and by the University ofWashington eScience Institute. [1] S. J. Wallace, Phys. Rev. Lett. , 622 (1971).[2] S. J. Wallace, Annals Phys. , 219 (2003).[5] A. Gade and T. Glasmacher, Prog. Part. Nucl. Phys. 60, 161 (2008)[6] C. A. Bertulani and A. Gade, Phys. Rept. , 195 (2010)[7] R. J. Glauber, “High Energy Collision Theory”,p. 315 in “Lectures in Theoretical Physics”Ed. by W. E. Brittain and L. G. Dunham, Vol. I, Interscience, New York, 1959 R. J. Glauber,“Theory of high energy hadron-nucleus collisions,” p. 207, In “High-Energy Physics And NuclearStructure”, ed. by S. Devons, Plenum Press, New York 1970[8] G. Bertsch, H. Esbensen and A. Sustich, Phys. Rev. C , 758 (1990).[9] Y. Ogawa, K. Yabana and Y. Suzuki, Nucl. Phys. A , 722 (1992).[10] J. S. Al-Khalili, J. A. Tostevin and I. J. Thompson, Phys. Rev. C , 1843 (1996).[11] T. Aumann, A. Navin, D. P. Balamuth, D. Bazin, B. Blank, B. A. Brown, J. E. Bush andJ. A. Caggiano et al. , Phys. Rev. Lett. , 35 (2000).[12] K. Hencken, G. Bertsch and H. Esbensen, Phys. Rev. C , 3043 (1996)[13] Y. . L. Parfenova, M. V. Zhukov and J. S. Vaagen, Phys. Rev. C , 044602 (2000).[14] I. Licot, N. Added, N. Carlin, G. M. Crawley, S. Danczyk, J. Finck, D. Hirata and H. Laurent et al. , Phys. Rev. C , 250 (1997).[15] E. Sauvan, F. Carstoiu, N. A. Orr, J. C. Angelique, W. N. Catford, N. M. Clarke, M. MacCormick and N. Curtis et al. , Phys. Lett. B , 1 (2000)[16] C. A. Bertulani and A. Gade, Comput. Phys. Commun. , 372 (2006)[17] H. Esbensen and G. F. Bertsch, Phys. Rev. C , 014608 (2001).[18] S. J. Wallace, Phys. Rev. D , 1846 (1973).[19] R. L. Varner, W. J. Thompson, T. L. McAbee, E. J. Ludwig and T. B. Clegg, Phys. Rept. ,57 (1991).[20] H. de Vries, C. W. de Jager, and C. de Vries, At. Data Nucl. Data Tables , 495 (1987).[21] R. Anne, et al. , Nuc. Phys. A575 , 125 (1994)., 125 (1994).