- Email: [email protected]
In-medium hadron–nucleon total cross sections inferred from quasifree scattering
Nuclear Physics A 740 (2004) 119–129 www.elsevier.com/locate/npe
In-medium hadron–nucleon total cross sections inferred from quasifree scattering R.J. Peterson Department of Physics, University of Colorado, Boulder, CO 80309-0390, USA Received 3 February 2004; received in revised form 1 April 2004; accepted 8 April 2004 Available online 20 April 2004
Abstract The target-mass dependence of hadron–nucleus quasifree scattering is used to select the hadron– nucleon total cross sections within the nuclear medium that yield the observed mass-dependence in a Glauber model. These in-medium cross sections are larger than found in free space for 705 MeV/c K+ , but smaller than in free space for protons and pions at several energies. 2004 Elsevier B.V. All rights reserved.
1. Introduction Much of what we have learned about the strong interaction has been determined by scattering of hadrons from free nucleons, with elastic scattering studies as the simplest. The most direct way to measure hadronic interactions within a strongly interacting medium would thus seem to be quasifree scattering, where the incident hadron has scattered incoherently and elastically once and only once with a bound nucleon. Alterations of hadron properties within complex nuclei are widely expected [1], and could affect quasifree scattering both by changing the differential hadron–nucleon cross sections and by changing the total hadron–nucleon cross sections, which limit the number of nucleons accessible once and only once. It has been found that quasifree scattering of electrons from complex nuclei shows no significant alterations of electric or magnetic form factors of the bound nucleons [2], but strong interactions are more difficult to investigate. Here we use experimental spectra for the non-charge-exchange continuum scattering (NCX) and single-charge-exchange (SCX) of protons, pions and positive K mesons from E-mail address: [email protected] (R.J. Peterson). 0375-9474/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2004.04.109
120
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
a range of target nuclei, under kinematic conditions that meet the conditions for quasifree scattering—that is, the incoherent scattering of the incident projectile from one and only one bound nucleon [3]. That the appropriate kinematic conditions have been met is demonstrated by reference to the y-scaling noted for quasifree scattering of high energy electrons [4]. The observed mass dependence of the proper portions of the hadron spectra is here summarized by a power law dependence across a wide range of nuclear target masses. The in-medium hadron–nucleon total cross sections within the several nuclear media are then adjusted within an eikonal or Glauber model [5] to match the exponent found in the power law analysis of the measured spectra. The validity of this model for hadron scattering and the factorization that places the mass dependence into a single power law term were demonstrated in Ref. [6]. These comparisons of data and model computations, summarized by the exponent of the power law, are carried out for several beam energies and momentum transfers. All relevant hadronic data that have been published are included in this study, being those cases with a suitable range of target masses and with beam energies (500 MeV or above, save for K+ ), energy losses, and momentum transfers suited to the quasifree conditions.
2. Methods Quasifree analyses have been shown to be somewhat successful for the continuum NCX spectra of 950 MeV/c π − scattered from 6 Li, C, Ca, Zr and 208 Pb [7] and for 750 MeV/c (π − , π 0 ) single charge exchange on C, Al and Cu [8]. Success is judged by a correct accounting of the spectral shape of the quasifree peak and by a successful y-scaling of the responses over some range of momentum transfers for each nuclear target. The spectral shape is determined largely by the internal momentum of the struck nucleons, and the magnitude of the y-scaling is determined by the number of nucleons struck once and only once (Aeff ) and by the differential cross sections at each angle observed. Free hadron– nucleon differential cross sections were used for those studies [9], without averaging to include internal momentum distributions of the nucleons. The number of nucleons with which a projectile interacts once and only once is computed as used in [10] from ∞ Aeff = 2π
T (b) exp −σt T (b) b db
(1)
0
with the profile function ∞ T (b) =
ρ(r) dz,
(2)
−∞
+ z2 . with Distributions ρ(r) of protons (not charge) were taken from the unfolding in [11] and neutron distribution parameters were taken to be the same as those for protons. Projectile– nucleon total cross sections σt in free space were taken from [9], or modified as described r2
= b2
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
121
below. Zeff or Neff for charge exchange were scaled from Aeff by Z/A or N/A of the nuclear sample. The responses for the π − NCX scattering were shown to follow the y-scaling hypothesis very well for light nuclei, with some deterioration in the agreement for the heaviest targets. Because of this mass-dependence, the concept of superscaling, as found in electron scattering [12], was not satisfied. In the present work it is assumed that the hadron responses must superscale, and that failure to do so is assigned to an incorrect computation of Aeff . Superscaling is to be restored by adjusting the hadron–nucleon total cross sections σt for the range of nuclear sizes investigated to bring agreement among the data at each angle for a wide range of nuclei. Superscaling is a concept guided by the Fermi gas model of nucleon momentum distributions, and uses a Fermi momentum kF derived from the observed width of the quasifree peak by use of the relativistic Fermi gas (RFG) model. Values of kF used here were taken from the observed hadron spectra, averaged over several angles if possible, and thus include instrumental influences, not only the internal momentum distributions. Since both features have the same effect of creating broader, lower, quasifree peaks, such use is appropriate. When possible from the data, comparisons of measured and computed power law exponents are carried out at several angles or momentum transfers, here chosen to be near 375 MeV/c (where ‘pionic enhancement’ of responses has been expected, but not sensed in quasifree (p, nX) charge exchange [13]), 500 and 625 MeV/c, where nuclear interactions can be expected to be small, permitting incoherent scattering. The nuclear absorption of the hadronic projectiles will give the sought after one-and-only-one scattering primarily in the nuclear surface, where the low density will impede collective effects [14], perhaps enabling a more reliable use of the Glauber model. The cases considered in this work are listed in Table 1. Only target nuclei of carbon and heavier were used, since lighter nuclei may exhibit irregular densities across small mass changes and computed values of Aeff are very sensitive to the details of the assumed matter distribution [6]. For comparison, the power law was also applied to continuum quasifree scattering of 2.02 GeV electrons at two angles [18]. The measured quasifree doubly differential cross sections at each lab energy loss ω at each angle for each nuclear sample were used to determine the exponent in d 2 σ/dω dΩ(ω) = σ0 Aα Zα
(3)
Nα
(or or for charge exchange, as appropriate). These exponents are plotted as functions of the scaling variable y, not ω, in order to compare results for the ranges of momentum transfer examined. The variable y is created from [4] y = (ω − SE)(ω − SE + 2M) − q, (4) with the separation energy SE = 20 MeV for reactions not changing the projectile charge, and 15 MeV for those that do. The three-momentum transfers q were taken to be those for free scattering at each energy loss for the data at fixed angles. The pion NCX data were collected at fixed lab frame momentum transfers. Figs. 1–3 show the resulting values of the exponents for the three cases of different momentum transfers for the inclusive spectra. In all three cases, the exponents drop
122
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
Table 1 Data samples used for the present study are summarized. Only nuclear targets of the mass of carbon and up were used. The lab frame free scattering momentum transfer q is cited for the angle of observation. Numbers in parentheses after each sample are the Fermi momenta determined from a relativistic Fermi gas fit to the widths of the measured spectra, in MeV/c Reaction
Energy (MeV)
(p, p X)
795
(π − , π − X)
820
(K+ , K+ X) (e, eX)
364 2002
(p, nX)
795
(π − , π 0 X)
500
(π − , π 0 )
623
Angle (deg)
Momentum transfer (MeV/c)
Nuclei
Reference
13 15 20 25 – – – 42 15 20 15
329 379 503 623 375 500 625 479 528 700 379
C(175), Al(197), Ca(195), V(191), Cu(209), 90 Zr(240), Pb(236)
[15]
C(183), Ca(199), Zr(214), 208 Pb(193)
[16]
C(198), Ca(234), Pb(247) C(220), Al(235), Fe(260), Au(265)
[17] [18] [19]
18 30 50 35 45 55
456 371 489 420 534 596
C(195), 13 C(195), Al(200), 58 Ni(239), Zr(243), Pb(210) C(195), Al(200), Pb(210) C(170), Al(175), Fe(180), Cu(180) Zr(185), Sn(190), Ta(190), Bi(190) C(235), Al(230), Cu(235)
[20] [8]
smoothly to a broad minimum near y = 0, corresponding to the peak of the quasifree spectra. The point y = 0 corresponds to scattering from a nucleon at rest, such that the recoil of the other nucleons does not contribute to the definition of y [12], and where arguments on the ‘optimum reference frame’ yield the best choice of the beam energy to evaluate in-medium cross sections as just the actual beam energy [21]. Fig. 4 shows a similar pattern for the smaller range of target masses available for the exclusive π − charge exchange data of Ref. [8], where the experiment enabled identification of reactions with only a single π 0 ejectile. In that work, a distinct minimum is found in the angular distribution, creating a sensitivity to the angle dependence not found for the other reactions used, where free and quasifree differential cross sections drop smoothly with angle. Coherent scattering of a projectile by a complex nucleus adds amplitudes, as for Rutherford scattering proportional to Z 2 , and scatters the projectile with an energy corresponding to the recoil of all the target nucleons, at low energy losses ω and very negative values of y. Incoherent quasifree scattering adds cross sections, and has a simplest yield proportional to A, with a recoil energy as for a single nucleon. Coherent scattering can thus be expected to have a mass dependence with a higher exponent α in Eq. (3). The extracted values of α shown in Figs. 1–3 do show this expected trend, with greater coherence for lower energy losses.
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
123
Fig. 1. Exponents derived from power law fits to the doubly differential cross sections for the cases listed in Table 1 are plotted for scattering angles that supply three-momentum transfers q near 375 MeV/c.
Fig. 2. As Fig. 1, but for scattering angles that yield q near 500 MeV/c.
Guided by the concepts of the Fermi gas and by the successes of superscaling for electron scattering, the Glauber model is used here to compute Aeff (Neff or Zeff for charge exchange, proportional to Aeff ) for beam hadron–nucleon total cross sections scaled from the free space values by a factor common to both neutrons and protons in the target nuclei β = σt (medium)/σt (free space),
(5)
with free space total cross sections on neutrons and protons from Ref. [9]. These computed Aeff are then used to compute a quantity proportional to the maximum d 2 σ/dω dΩ for each target nucleus and for a range of β by assuming the validity of the
124
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
Fig. 3. As Fig. 1, but for scattering angles that yield q near 625 MeV/c. The two curves show the y dependence expected from a simple representation of the nuclear Fermi momenta for each nuclear sample.
Fig. 4. As Fig. 1, but for 750 MeV/c (π − , π 0 ) exclusive spectra, on a smaller sample of target masses than those available for Figs. 1–3, shown for several reaction angles in the lab frame.
superscaling found for electron quasifree scattering [12]. Using the superscale variable Y = y/kF , the superscaling response in our case is d 2σ kF /Aeff dσ/dΩ|F , (6) dω dΩ where K is a kinematic constant K = q/ m2p + (q − y)2 , kF is the Fermi momentum, and dσ/dΩ|f is the free projectile–nucleon differential cross section (averaged over N and Z) [9]. In Ref. [12] the measured electron cross sections were used to determine f (Y ). f (Y ) = K
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
125
Here, we invert the usage to find a quantity proportional to an expected maximum quasifree hadron cross section at a fixed angle where dσ/dΩ|f is constant and requiring a constant, or superscaled, f (Y ); this would be Aeff /kF , computed for a range of medium modifications β. A power law in Aα was then fit to the computed quantities Aeff /kF , Neff /kF , or Zeff /kF , as in Eq. (3)—Aeff /kF ∼ Aα . Values of kF taken from the observed widths of quasifree peaks are listed in Table 1. These need not be the same for all studies, since experimental features are included in the way they were obtained. Comparison of arrays of the fitted exponents α for data and computed quantities will be used to infer enhancement factors β.
3. Results It is noted that many of the exponents in Figs. 1–4 show a shallow minimum near y = 0. This can be understood in view of the internal momentum distributions of the struck nucleons, where a larger Fermi momentum leads to a broader and lower quasifree peak for a fixed area. The mass dependence of the Fermi momenta observed in electron scattering [22] can be estimated from kF = 270 (1–2.5/A) MeV/c, and these values are used with an estimated parabolic quasifree peak shape for a range of nuclei to find the y-dependence of the power law exponents that match the changing quasifree peaks from this simple effect. These shapes are compared to several of the measured distributions of the exponents in Fig. 3, and found to account for the observed shape near y = 0. This observation tends to confirm the superscaling concept. Values of the exponents near y = 0 for each experimental case are listed in Table 2, with uncertainties estimated from the general trend of the points. The scatter among the points for the exponents is larger than the uncertainties arising from the fits to the observed cross sections and their stated uncertainties. This could arise from a number of experimental conditions not included in the published uncertainties for each nuclear target. Pion NCX data at 950 MeV/c at other momentum transfers q from 500 to 650 MeV/c in steps of 25 MeV/c give exponents α consistent with the values listed in Table 2 [16]. Larger exponents are found for data lacking sufficient momentum transfer to meet the quasifree conditions in that work. For these cases, coherence is still being sensed. Power law exponents derived from computed Aeff /kF (or Zeff /kF or Neff /kF ) for each of the experiments listed in Table 1 are shown in Fig. 5. Total cross sections in freespace were taken to be the average of those for the incoming beam energy and those for the outgoing beam energy for y = 0 at q near 500 MeV/c. Nuclear samples for these computations were taken to be just those of each experimental data set used for the exponents shown in Figs. 1–4. One then slides each curve along the measured exponents to find the value of β that yields the observed mass dependence, for each reaction. The curves in Fig. 5 are quite flat, limiting the accuracy of this method, especially towards large values of β. Those ratios of total in-medium to free-space hadron–nucleon total cross sections yielding the best agreement are listed in Table 2.
126
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
Table 2 The exponent α found near y = 0 for each of the cases listed in Table 1 is listed, with the ratio (β) of in-medium to free-space hadron–nucleon total cross sections that yields the mass dependence from application of the Glauber model reflected in that measured exponent Reaction
Energy (MeV)
(p, pX)
795
(π − , π − X)
820
(K+ , K+ X) (e, eX)
364 2020
(p, nX)
795
(π − , π 0 X)
500
(π − , π 0 )
623
q (MeV/c)
α
329 379 503 623 375 500 625 479 528 700 379 456 371 489 420 534 596
0.30(0.01) 0.347(0.012) 0.335(0.017) 0.380(0.009) 0.366(0.011) 0.440(0.007) 0.518(0.028) 0.461(0.056) 0.89 0.91 0.45(0.01) 0.48(0.02) 0.359(0.037) 0.406(0.039) 0.32(0.05) 0.42(0.04) 0.42(0.05)
β 1.1+0.5 −0.1 0.72+0.05 −0.01 0.79+0.10 −0.10 0.64+0.03 −0.03 0.97+0.02 −0.05 0.68+0.02 −0.04 0.52+0.06 −0.05 1.24+0.26 −0.22 <0.5 <0.5 1.00+0.15 −0.15 0.61+0.12 −0.10 >1.2 0.85+0.15 −0.15 0.80+0.20 −0.15
Fig. 5. Glauber model calculations with a range of in-medium enhancements β over free space hadron–nucleon total cross sections are combined with values of the Fermi momentum kF for the same ranges of target nuclei as used in Figs. 1–4 to find the power law exponents plotted. The Fermi momenta were obtained from the widths of the quasifree peaks seen in the experimental spectra.
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
127
Fig. 6. As Fig. 1, but for the 795 MeV (p, nX) inclusive spectra of Ref. [19] at two angles.
Figs. 2 and 3 show exponents for the mass dependence of quasifree electron scattering that are not simply unity. If the same Glauber model is used for these data as for the hadron results, an electron–nucleon total cross section of 2.5 mb is inferred. In Fig. 6 are shown the exponents resulting from the power law analysis of the inclusive (p, nX) spectra of Prout [19] at two angles. The 15 degree data cover a good range of target masses, but the momentum transfer is too low to allow the incoherent model to be valid. Indeed, this angle of 15 degrees was chosen to maximize the expected role of longrange correlations. The 18 degree data include only three nuclear targets. The exponents for both are in agreement with one another, but not with those observed in the 795 MeV (p, pX) data seen in Figs. 1 and 2 at the same beam energy. If only in-medium total cross sections were to determine these mass dependences, the results should be the same for the arrays of exponents. It could be inferred that the charge exchange data continue to show coherent, collective, effects due to the long-range of the residual interactions in the spin/isospin channel driving the (p, n) reaction, as suggested in the figures of Ref. [23].
4. Conclusions It is seen in Table 2 that K+ -nucleon total cross sections within complex nuclei as inferred in this work are enhanced above their free space values. This has been noted in several other reactions of K+ with nuclei [24], and is confirmed by the present quasifree method. All other quasifree cases other than K+ treated here, for proton and pion quasifree scattering at sufficient q, are accounted for by enhancement factors β less than one, showing smaller in-medium hadron–nucleon total cross sections than found in free space. The results from q near 375 MeV/c (Fig. 1) may not be free of nuclear coherence and Pauli blocking effects, which may enhance or diminish the quasifree cross sections from
128
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
free-space expectations. Conclusions are thus most reliable from data with q near 500 or 625 MeV/c, which are to be compared to several theoretical expectations. Recent calculations of in-medium proton–nucleon total cross sections are shown in Ref. [25], where σt is found to reach asymptotic values of 15–20 mb at higher beam energies. These values correspond to ratios β to the free space values of about 0.55 and 0.41 at proton beam energies of 500 and 800 MeV. The average nucleon–nucleon value of β found from Table 2 above for q near or above q = 500 MeV/c is 0.68. The smaller inmedium total cross sections are indeed found in this analysis of quasifree data. Similarly, the transparencies computed by Kelly [26] for 650 MeV protons underestimate the data systematically for heavier nuclei, as another indication of smaller in-medium nucleon– nucleon total cross sections, in an energy range similar to that examined here. In-medium pion–nucleon total cross sections seem not to have been addressed yet, but are shown to be damped from free space values in Table 2. It has also been inferred from high energy proton collisions with Be, Cu and Au that antiproton absorption cross sections within nuclei are smaller than found in free space [27]. Quasifree pion scattering at lower beam energies, but with large scattering angles to achieve the same range of q considered here at energies above 500 MeV, have been considered in Ref. [28]. There, a model similar to that used here was used to predict quasifree cross sections using free-space pion–nucleon cross sections. The effective number of nucleons expected to have one and only one elastic scattering at 120 degrees are lower than observed by 30% at 245 MeV and 50% at 315 MeV. The model overestimated pion attenuation, as would be found for smaller in medium pion–nucleon total cross sections. This conclusion agrees with the present work. Quasifree electron scattering has been used to infer the momentum distributions of nucleons bound within complex nuclei. If such scattering is not fully quasifree, coherences among nucleons can alter the experimental spectra. Here we find power law exponents for hadron scattering that are not constant across a wide range of ω or y, indicating that the spectra are influenced by features other than just the momentum distributions. Evidently, hadron quasifree scattering is not suited to measure nucleon momentum distributions. The present consideration of the mass dependence of the magnitudes of quasifree hadron scattering has found evidence of a decrease in the hadron–nucleon total cross sections for pions and nucleons within complex nuclei by a method not previously used; Glauber calculations can match the power-law summary dependence of the quasifree cross sections only with smaller values of σt . Although much of the analysis here relies on both the Glauber model and the Fermi gas model, the shapes of the quasifree peaks in the original data and the trends of the exponents α around y = 0 indicate that these models are sufficient to account for the observations. Within the limits of the models and approximations made here, it is concluded that observation of incoherent one-on-one quasifree scattering of hadrons from bound nuclei can serve as a measure of hadron–nucleon total cross sections within nuclei, and finds lower pion–nucleon and nucleon–nucleon total cross sections than those observed in free space.
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
129
Acknowledgements This work was supported in part by the US Department of Energy, and by sabbatical leave granted by the University of Colorado.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
[21] [22] [23] [24] [25] [26] [27] [28]
G.E. Brown, M. Rho, Phys. Rev. Lett. 66 (1991) 2720. K.S. Kim, L.E. Wright, Phys. Rev. C 68 (2003) 027601. M.L. Goldberger, K.M. Watson, Collision Theory, Wiley, New York, 1964, Chapter 11. D.B. Day, J.S. McCarthy, T.W. Donnelly, I. Sick, Annu. Rev. Nucl. Part. Sci. 40 (1990) 357. R.J. Glauber, in: W.E. Brittin, L.G. Dunham (Eds.), Lectures in Theoretical Physics, Interscience, New York, 1959. G.F. Bertsch, O. Scholten, Phys. Rev. C 25 (1982) 804. R.J. Peterson, et al., Phys. Rev. C 65 (2002) 054601. R.J. Peterson, et al., Phys. Rev. C, submitted for publication. R.A. Arndt, et al., Scattering Analysis Interactive Dialin (SAID), solution SM95, http://gwdac.phys. gwu.edu/. J. Ouyang, S. Hoibraten, R.J. Peterson, Phys. Rev. C 47 (1993) 2809. J.D. Patterson, R.J. Peterson, Nucl. Phys. A 717 (2003) 235. T.W. Donnelly, I. Sick, Phys. Rev. C 60 (1999) 065502. X.Y. Chen, et al., Phys. Rev. C 47 (1993) 2159. W.M. Alberico, A. DePace, M. Ericson, M.B. Johnson, A. Molinari, Phys. Rev. C 38 (1988) 109. R.E. Chrien, et al., Phys. Rev. C 21 (1980) 1014. Y. Fujii, et al., Phys. Rev. C 64 (2001) 034608. C.M. Kormanyos, et al., Phys. Rev. C 51 (1995) 669. D.B. Day, et al., Phys. Rev. C 48 (1993) 1853. D.L. Prout, et al., Phys. Rev. C 52 (1995) 228; D.L. Prout, Ph.D. Thesis, University of Colorado, 1992. J. Ouyang, Ph.D. Thesis, University of Colorado; J. Ouyang, Los Alamos National Laboratory Report LA-12457-T; R.J. Peterson, et al., Phys. Lett. B 297 (1992) 238. S.A. Gurvitz, Phys. Rev. C 33 (1986) 422. E.J. Moniz, Phys. Rev. Lett. 26 (1971) 445. T. Shigehara, K. Shimizu, A. Arima, Nucl. Phys. A 492 (1989) 388. P.B. Siegel, W.B. Kaufman, W.R. Gibbs, Phys. Rev. C 31 (1985) 2184; P.B. Jiang, D.J. Ernst, C.M. Chen, Phys. Rev. C 51 (1995) 857. C. Fuchs, A. Faessler, M. El-Shabsiry, Phys. Rev. C 64 (2001) 024003. J.J. Kelly, Phys. Rev. C 54 (1996) 2547. I. Chemakin, et al., Phys. Rev. C 64 (2001) 064908. E. Piasetzky, R.R. Silbar, M.M. Sternheim, Phys. Rev. C 25 (1982) 2830.
In-medium hadron–nucleon total cross sections inferred from quasifree scattering R.J. Peterson Department of Physics, University of Colorado, Boulder, CO 80309-0390, USA Received 3 February 2004; received in revised form 1 April 2004; accepted 8 April 2004 Available online 20 April 2004
Abstract The target-mass dependence of hadron–nucleus quasifree scattering is used to select the hadron– nucleon total cross sections within the nuclear medium that yield the observed mass-dependence in a Glauber model. These in-medium cross sections are larger than found in free space for 705 MeV/c K+ , but smaller than in free space for protons and pions at several energies. 2004 Elsevier B.V. All rights reserved.
1. Introduction Much of what we have learned about the strong interaction has been determined by scattering of hadrons from free nucleons, with elastic scattering studies as the simplest. The most direct way to measure hadronic interactions within a strongly interacting medium would thus seem to be quasifree scattering, where the incident hadron has scattered incoherently and elastically once and only once with a bound nucleon. Alterations of hadron properties within complex nuclei are widely expected [1], and could affect quasifree scattering both by changing the differential hadron–nucleon cross sections and by changing the total hadron–nucleon cross sections, which limit the number of nucleons accessible once and only once. It has been found that quasifree scattering of electrons from complex nuclei shows no significant alterations of electric or magnetic form factors of the bound nucleons [2], but strong interactions are more difficult to investigate. Here we use experimental spectra for the non-charge-exchange continuum scattering (NCX) and single-charge-exchange (SCX) of protons, pions and positive K mesons from E-mail address: [email protected] (R.J. Peterson). 0375-9474/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2004.04.109
120
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
a range of target nuclei, under kinematic conditions that meet the conditions for quasifree scattering—that is, the incoherent scattering of the incident projectile from one and only one bound nucleon [3]. That the appropriate kinematic conditions have been met is demonstrated by reference to the y-scaling noted for quasifree scattering of high energy electrons [4]. The observed mass dependence of the proper portions of the hadron spectra is here summarized by a power law dependence across a wide range of nuclear target masses. The in-medium hadron–nucleon total cross sections within the several nuclear media are then adjusted within an eikonal or Glauber model [5] to match the exponent found in the power law analysis of the measured spectra. The validity of this model for hadron scattering and the factorization that places the mass dependence into a single power law term were demonstrated in Ref. [6]. These comparisons of data and model computations, summarized by the exponent of the power law, are carried out for several beam energies and momentum transfers. All relevant hadronic data that have been published are included in this study, being those cases with a suitable range of target masses and with beam energies (500 MeV or above, save for K+ ), energy losses, and momentum transfers suited to the quasifree conditions.
2. Methods Quasifree analyses have been shown to be somewhat successful for the continuum NCX spectra of 950 MeV/c π − scattered from 6 Li, C, Ca, Zr and 208 Pb [7] and for 750 MeV/c (π − , π 0 ) single charge exchange on C, Al and Cu [8]. Success is judged by a correct accounting of the spectral shape of the quasifree peak and by a successful y-scaling of the responses over some range of momentum transfers for each nuclear target. The spectral shape is determined largely by the internal momentum of the struck nucleons, and the magnitude of the y-scaling is determined by the number of nucleons struck once and only once (Aeff ) and by the differential cross sections at each angle observed. Free hadron– nucleon differential cross sections were used for those studies [9], without averaging to include internal momentum distributions of the nucleons. The number of nucleons with which a projectile interacts once and only once is computed as used in [10] from ∞ Aeff = 2π
T (b) exp −σt T (b) b db
(1)
0
with the profile function ∞ T (b) =
ρ(r) dz,
(2)
−∞
+ z2 . with Distributions ρ(r) of protons (not charge) were taken from the unfolding in [11] and neutron distribution parameters were taken to be the same as those for protons. Projectile– nucleon total cross sections σt in free space were taken from [9], or modified as described r2
= b2
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
121
below. Zeff or Neff for charge exchange were scaled from Aeff by Z/A or N/A of the nuclear sample. The responses for the π − NCX scattering were shown to follow the y-scaling hypothesis very well for light nuclei, with some deterioration in the agreement for the heaviest targets. Because of this mass-dependence, the concept of superscaling, as found in electron scattering [12], was not satisfied. In the present work it is assumed that the hadron responses must superscale, and that failure to do so is assigned to an incorrect computation of Aeff . Superscaling is to be restored by adjusting the hadron–nucleon total cross sections σt for the range of nuclear sizes investigated to bring agreement among the data at each angle for a wide range of nuclei. Superscaling is a concept guided by the Fermi gas model of nucleon momentum distributions, and uses a Fermi momentum kF derived from the observed width of the quasifree peak by use of the relativistic Fermi gas (RFG) model. Values of kF used here were taken from the observed hadron spectra, averaged over several angles if possible, and thus include instrumental influences, not only the internal momentum distributions. Since both features have the same effect of creating broader, lower, quasifree peaks, such use is appropriate. When possible from the data, comparisons of measured and computed power law exponents are carried out at several angles or momentum transfers, here chosen to be near 375 MeV/c (where ‘pionic enhancement’ of responses has been expected, but not sensed in quasifree (p, nX) charge exchange [13]), 500 and 625 MeV/c, where nuclear interactions can be expected to be small, permitting incoherent scattering. The nuclear absorption of the hadronic projectiles will give the sought after one-and-only-one scattering primarily in the nuclear surface, where the low density will impede collective effects [14], perhaps enabling a more reliable use of the Glauber model. The cases considered in this work are listed in Table 1. Only target nuclei of carbon and heavier were used, since lighter nuclei may exhibit irregular densities across small mass changes and computed values of Aeff are very sensitive to the details of the assumed matter distribution [6]. For comparison, the power law was also applied to continuum quasifree scattering of 2.02 GeV electrons at two angles [18]. The measured quasifree doubly differential cross sections at each lab energy loss ω at each angle for each nuclear sample were used to determine the exponent in d 2 σ/dω dΩ(ω) = σ0 Aα Zα
(3)
Nα
(or or for charge exchange, as appropriate). These exponents are plotted as functions of the scaling variable y, not ω, in order to compare results for the ranges of momentum transfer examined. The variable y is created from [4] y = (ω − SE)(ω − SE + 2M) − q, (4) with the separation energy SE = 20 MeV for reactions not changing the projectile charge, and 15 MeV for those that do. The three-momentum transfers q were taken to be those for free scattering at each energy loss for the data at fixed angles. The pion NCX data were collected at fixed lab frame momentum transfers. Figs. 1–3 show the resulting values of the exponents for the three cases of different momentum transfers for the inclusive spectra. In all three cases, the exponents drop
122
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
Table 1 Data samples used for the present study are summarized. Only nuclear targets of the mass of carbon and up were used. The lab frame free scattering momentum transfer q is cited for the angle of observation. Numbers in parentheses after each sample are the Fermi momenta determined from a relativistic Fermi gas fit to the widths of the measured spectra, in MeV/c Reaction
Energy (MeV)
(p, p X)
795
(π − , π − X)
820
(K+ , K+ X) (e, eX)
364 2002
(p, nX)
795
(π − , π 0 X)
500
(π − , π 0 )
623
Angle (deg)
Momentum transfer (MeV/c)
Nuclei
Reference
13 15 20 25 – – – 42 15 20 15
329 379 503 623 375 500 625 479 528 700 379
C(175), Al(197), Ca(195), V(191), Cu(209), 90 Zr(240), Pb(236)
[15]
C(183), Ca(199), Zr(214), 208 Pb(193)
[16]
C(198), Ca(234), Pb(247) C(220), Al(235), Fe(260), Au(265)
[17] [18] [19]
18 30 50 35 45 55
456 371 489 420 534 596
C(195), 13 C(195), Al(200), 58 Ni(239), Zr(243), Pb(210) C(195), Al(200), Pb(210) C(170), Al(175), Fe(180), Cu(180) Zr(185), Sn(190), Ta(190), Bi(190) C(235), Al(230), Cu(235)
[20] [8]
smoothly to a broad minimum near y = 0, corresponding to the peak of the quasifree spectra. The point y = 0 corresponds to scattering from a nucleon at rest, such that the recoil of the other nucleons does not contribute to the definition of y [12], and where arguments on the ‘optimum reference frame’ yield the best choice of the beam energy to evaluate in-medium cross sections as just the actual beam energy [21]. Fig. 4 shows a similar pattern for the smaller range of target masses available for the exclusive π − charge exchange data of Ref. [8], where the experiment enabled identification of reactions with only a single π 0 ejectile. In that work, a distinct minimum is found in the angular distribution, creating a sensitivity to the angle dependence not found for the other reactions used, where free and quasifree differential cross sections drop smoothly with angle. Coherent scattering of a projectile by a complex nucleus adds amplitudes, as for Rutherford scattering proportional to Z 2 , and scatters the projectile with an energy corresponding to the recoil of all the target nucleons, at low energy losses ω and very negative values of y. Incoherent quasifree scattering adds cross sections, and has a simplest yield proportional to A, with a recoil energy as for a single nucleon. Coherent scattering can thus be expected to have a mass dependence with a higher exponent α in Eq. (3). The extracted values of α shown in Figs. 1–3 do show this expected trend, with greater coherence for lower energy losses.
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
123
Fig. 1. Exponents derived from power law fits to the doubly differential cross sections for the cases listed in Table 1 are plotted for scattering angles that supply three-momentum transfers q near 375 MeV/c.
Fig. 2. As Fig. 1, but for scattering angles that yield q near 500 MeV/c.
Guided by the concepts of the Fermi gas and by the successes of superscaling for electron scattering, the Glauber model is used here to compute Aeff (Neff or Zeff for charge exchange, proportional to Aeff ) for beam hadron–nucleon total cross sections scaled from the free space values by a factor common to both neutrons and protons in the target nuclei β = σt (medium)/σt (free space),
(5)
with free space total cross sections on neutrons and protons from Ref. [9]. These computed Aeff are then used to compute a quantity proportional to the maximum d 2 σ/dω dΩ for each target nucleus and for a range of β by assuming the validity of the
124
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
Fig. 3. As Fig. 1, but for scattering angles that yield q near 625 MeV/c. The two curves show the y dependence expected from a simple representation of the nuclear Fermi momenta for each nuclear sample.
Fig. 4. As Fig. 1, but for 750 MeV/c (π − , π 0 ) exclusive spectra, on a smaller sample of target masses than those available for Figs. 1–3, shown for several reaction angles in the lab frame.
superscaling found for electron quasifree scattering [12]. Using the superscale variable Y = y/kF , the superscaling response in our case is d 2σ kF /Aeff dσ/dΩ|F , (6) dω dΩ where K is a kinematic constant K = q/ m2p + (q − y)2 , kF is the Fermi momentum, and dσ/dΩ|f is the free projectile–nucleon differential cross section (averaged over N and Z) [9]. In Ref. [12] the measured electron cross sections were used to determine f (Y ). f (Y ) = K
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
125
Here, we invert the usage to find a quantity proportional to an expected maximum quasifree hadron cross section at a fixed angle where dσ/dΩ|f is constant and requiring a constant, or superscaled, f (Y ); this would be Aeff /kF , computed for a range of medium modifications β. A power law in Aα was then fit to the computed quantities Aeff /kF , Neff /kF , or Zeff /kF , as in Eq. (3)—Aeff /kF ∼ Aα . Values of kF taken from the observed widths of quasifree peaks are listed in Table 1. These need not be the same for all studies, since experimental features are included in the way they were obtained. Comparison of arrays of the fitted exponents α for data and computed quantities will be used to infer enhancement factors β.
3. Results It is noted that many of the exponents in Figs. 1–4 show a shallow minimum near y = 0. This can be understood in view of the internal momentum distributions of the struck nucleons, where a larger Fermi momentum leads to a broader and lower quasifree peak for a fixed area. The mass dependence of the Fermi momenta observed in electron scattering [22] can be estimated from kF = 270 (1–2.5/A) MeV/c, and these values are used with an estimated parabolic quasifree peak shape for a range of nuclei to find the y-dependence of the power law exponents that match the changing quasifree peaks from this simple effect. These shapes are compared to several of the measured distributions of the exponents in Fig. 3, and found to account for the observed shape near y = 0. This observation tends to confirm the superscaling concept. Values of the exponents near y = 0 for each experimental case are listed in Table 2, with uncertainties estimated from the general trend of the points. The scatter among the points for the exponents is larger than the uncertainties arising from the fits to the observed cross sections and their stated uncertainties. This could arise from a number of experimental conditions not included in the published uncertainties for each nuclear target. Pion NCX data at 950 MeV/c at other momentum transfers q from 500 to 650 MeV/c in steps of 25 MeV/c give exponents α consistent with the values listed in Table 2 [16]. Larger exponents are found for data lacking sufficient momentum transfer to meet the quasifree conditions in that work. For these cases, coherence is still being sensed. Power law exponents derived from computed Aeff /kF (or Zeff /kF or Neff /kF ) for each of the experiments listed in Table 1 are shown in Fig. 5. Total cross sections in freespace were taken to be the average of those for the incoming beam energy and those for the outgoing beam energy for y = 0 at q near 500 MeV/c. Nuclear samples for these computations were taken to be just those of each experimental data set used for the exponents shown in Figs. 1–4. One then slides each curve along the measured exponents to find the value of β that yields the observed mass dependence, for each reaction. The curves in Fig. 5 are quite flat, limiting the accuracy of this method, especially towards large values of β. Those ratios of total in-medium to free-space hadron–nucleon total cross sections yielding the best agreement are listed in Table 2.
126
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
Table 2 The exponent α found near y = 0 for each of the cases listed in Table 1 is listed, with the ratio (β) of in-medium to free-space hadron–nucleon total cross sections that yields the mass dependence from application of the Glauber model reflected in that measured exponent Reaction
Energy (MeV)
(p, pX)
795
(π − , π − X)
820
(K+ , K+ X) (e, eX)
364 2020
(p, nX)
795
(π − , π 0 X)
500
(π − , π 0 )
623
q (MeV/c)
α
329 379 503 623 375 500 625 479 528 700 379 456 371 489 420 534 596
0.30(0.01) 0.347(0.012) 0.335(0.017) 0.380(0.009) 0.366(0.011) 0.440(0.007) 0.518(0.028) 0.461(0.056) 0.89 0.91 0.45(0.01) 0.48(0.02) 0.359(0.037) 0.406(0.039) 0.32(0.05) 0.42(0.04) 0.42(0.05)
β 1.1+0.5 −0.1 0.72+0.05 −0.01 0.79+0.10 −0.10 0.64+0.03 −0.03 0.97+0.02 −0.05 0.68+0.02 −0.04 0.52+0.06 −0.05 1.24+0.26 −0.22 <0.5 <0.5 1.00+0.15 −0.15 0.61+0.12 −0.10 >1.2 0.85+0.15 −0.15 0.80+0.20 −0.15
Fig. 5. Glauber model calculations with a range of in-medium enhancements β over free space hadron–nucleon total cross sections are combined with values of the Fermi momentum kF for the same ranges of target nuclei as used in Figs. 1–4 to find the power law exponents plotted. The Fermi momenta were obtained from the widths of the quasifree peaks seen in the experimental spectra.
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
127
Fig. 6. As Fig. 1, but for the 795 MeV (p, nX) inclusive spectra of Ref. [19] at two angles.
Figs. 2 and 3 show exponents for the mass dependence of quasifree electron scattering that are not simply unity. If the same Glauber model is used for these data as for the hadron results, an electron–nucleon total cross section of 2.5 mb is inferred. In Fig. 6 are shown the exponents resulting from the power law analysis of the inclusive (p, nX) spectra of Prout [19] at two angles. The 15 degree data cover a good range of target masses, but the momentum transfer is too low to allow the incoherent model to be valid. Indeed, this angle of 15 degrees was chosen to maximize the expected role of longrange correlations. The 18 degree data include only three nuclear targets. The exponents for both are in agreement with one another, but not with those observed in the 795 MeV (p, pX) data seen in Figs. 1 and 2 at the same beam energy. If only in-medium total cross sections were to determine these mass dependences, the results should be the same for the arrays of exponents. It could be inferred that the charge exchange data continue to show coherent, collective, effects due to the long-range of the residual interactions in the spin/isospin channel driving the (p, n) reaction, as suggested in the figures of Ref. [23].
4. Conclusions It is seen in Table 2 that K+ -nucleon total cross sections within complex nuclei as inferred in this work are enhanced above their free space values. This has been noted in several other reactions of K+ with nuclei [24], and is confirmed by the present quasifree method. All other quasifree cases other than K+ treated here, for proton and pion quasifree scattering at sufficient q, are accounted for by enhancement factors β less than one, showing smaller in-medium hadron–nucleon total cross sections than found in free space. The results from q near 375 MeV/c (Fig. 1) may not be free of nuclear coherence and Pauli blocking effects, which may enhance or diminish the quasifree cross sections from
128
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
free-space expectations. Conclusions are thus most reliable from data with q near 500 or 625 MeV/c, which are to be compared to several theoretical expectations. Recent calculations of in-medium proton–nucleon total cross sections are shown in Ref. [25], where σt is found to reach asymptotic values of 15–20 mb at higher beam energies. These values correspond to ratios β to the free space values of about 0.55 and 0.41 at proton beam energies of 500 and 800 MeV. The average nucleon–nucleon value of β found from Table 2 above for q near or above q = 500 MeV/c is 0.68. The smaller inmedium total cross sections are indeed found in this analysis of quasifree data. Similarly, the transparencies computed by Kelly [26] for 650 MeV protons underestimate the data systematically for heavier nuclei, as another indication of smaller in-medium nucleon– nucleon total cross sections, in an energy range similar to that examined here. In-medium pion–nucleon total cross sections seem not to have been addressed yet, but are shown to be damped from free space values in Table 2. It has also been inferred from high energy proton collisions with Be, Cu and Au that antiproton absorption cross sections within nuclei are smaller than found in free space [27]. Quasifree pion scattering at lower beam energies, but with large scattering angles to achieve the same range of q considered here at energies above 500 MeV, have been considered in Ref. [28]. There, a model similar to that used here was used to predict quasifree cross sections using free-space pion–nucleon cross sections. The effective number of nucleons expected to have one and only one elastic scattering at 120 degrees are lower than observed by 30% at 245 MeV and 50% at 315 MeV. The model overestimated pion attenuation, as would be found for smaller in medium pion–nucleon total cross sections. This conclusion agrees with the present work. Quasifree electron scattering has been used to infer the momentum distributions of nucleons bound within complex nuclei. If such scattering is not fully quasifree, coherences among nucleons can alter the experimental spectra. Here we find power law exponents for hadron scattering that are not constant across a wide range of ω or y, indicating that the spectra are influenced by features other than just the momentum distributions. Evidently, hadron quasifree scattering is not suited to measure nucleon momentum distributions. The present consideration of the mass dependence of the magnitudes of quasifree hadron scattering has found evidence of a decrease in the hadron–nucleon total cross sections for pions and nucleons within complex nuclei by a method not previously used; Glauber calculations can match the power-law summary dependence of the quasifree cross sections only with smaller values of σt . Although much of the analysis here relies on both the Glauber model and the Fermi gas model, the shapes of the quasifree peaks in the original data and the trends of the exponents α around y = 0 indicate that these models are sufficient to account for the observations. Within the limits of the models and approximations made here, it is concluded that observation of incoherent one-on-one quasifree scattering of hadrons from bound nuclei can serve as a measure of hadron–nucleon total cross sections within nuclei, and finds lower pion–nucleon and nucleon–nucleon total cross sections than those observed in free space.
R.J. Peterson / Nuclear Physics A 740 (2004) 119–129
129
Acknowledgements This work was supported in part by the US Department of Energy, and by sabbatical leave granted by the University of Colorado.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
[21] [22] [23] [24] [25] [26] [27] [28]
G.E. Brown, M. Rho, Phys. Rev. Lett. 66 (1991) 2720. K.S. Kim, L.E. Wright, Phys. Rev. C 68 (2003) 027601. M.L. Goldberger, K.M. Watson, Collision Theory, Wiley, New York, 1964, Chapter 11. D.B. Day, J.S. McCarthy, T.W. Donnelly, I. Sick, Annu. Rev. Nucl. Part. Sci. 40 (1990) 357. R.J. Glauber, in: W.E. Brittin, L.G. Dunham (Eds.), Lectures in Theoretical Physics, Interscience, New York, 1959. G.F. Bertsch, O. Scholten, Phys. Rev. C 25 (1982) 804. R.J. Peterson, et al., Phys. Rev. C 65 (2002) 054601. R.J. Peterson, et al., Phys. Rev. C, submitted for publication. R.A. Arndt, et al., Scattering Analysis Interactive Dialin (SAID), solution SM95, http://gwdac.phys. gwu.edu/. J. Ouyang, S. Hoibraten, R.J. Peterson, Phys. Rev. C 47 (1993) 2809. J.D. Patterson, R.J. Peterson, Nucl. Phys. A 717 (2003) 235. T.W. Donnelly, I. Sick, Phys. Rev. C 60 (1999) 065502. X.Y. Chen, et al., Phys. Rev. C 47 (1993) 2159. W.M. Alberico, A. DePace, M. Ericson, M.B. Johnson, A. Molinari, Phys. Rev. C 38 (1988) 109. R.E. Chrien, et al., Phys. Rev. C 21 (1980) 1014. Y. Fujii, et al., Phys. Rev. C 64 (2001) 034608. C.M. Kormanyos, et al., Phys. Rev. C 51 (1995) 669. D.B. Day, et al., Phys. Rev. C 48 (1993) 1853. D.L. Prout, et al., Phys. Rev. C 52 (1995) 228; D.L. Prout, Ph.D. Thesis, University of Colorado, 1992. J. Ouyang, Ph.D. Thesis, University of Colorado; J. Ouyang, Los Alamos National Laboratory Report LA-12457-T; R.J. Peterson, et al., Phys. Lett. B 297 (1992) 238. S.A. Gurvitz, Phys. Rev. C 33 (1986) 422. E.J. Moniz, Phys. Rev. Lett. 26 (1971) 445. T. Shigehara, K. Shimizu, A. Arima, Nucl. Phys. A 492 (1989) 388. P.B. Siegel, W.B. Kaufman, W.R. Gibbs, Phys. Rev. C 31 (1985) 2184; P.B. Jiang, D.J. Ernst, C.M. Chen, Phys. Rev. C 51 (1995) 857. C. Fuchs, A. Faessler, M. El-Shabsiry, Phys. Rev. C 64 (2001) 024003. J.J. Kelly, Phys. Rev. C 54 (1996) 2547. I. Chemakin, et al., Phys. Rev. C 64 (2001) 064908. E. Piasetzky, R.R. Silbar, M.M. Sternheim, Phys. Rev. C 25 (1982) 2830.