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y-scaling in proton continuum scattering
Nuclear Physics A 769 (2006) 95–114
y-scaling in proton continuum scattering R.J. Peterson ∗ Department of Physics, UCB 390, University of Colorado, Boulder, CO 80309-0390, USA Received 3 November 2005; received in revised form 18 January 2006; accepted 10 February 2006 Available online 3 March 2006
Abstract Data for inclusive proton scattering spectra at kinematics suited to incoherent quasifree scattering from single nucleons bound within complex nuclei are analysed in the y-scaling format found to unify many electron scattering spectra into universal nuclear responses. Proton continuum spectra are analyzed for beam energies from 392 MeV to 19.2 GeV. Scaling is somewhat successful for proton scattering on light nuclei or at fixed angles, with differences from electron scattering that indicate modifications of proton scattering on nucleons within nuclei from scattering in free space. Comparisons to inclusive continuum meson scattering in the same y-scaling format are made, and a new hadronic scaling relation is demonstrated in light nuclei. 2006 Elsevier B.V. All rights reserved. PACS: 25.40.-h; 25.80.-e Keywords: Hadron; Continuum scattering; Quasifree; Scaling
1. Introduction The scattering of a projectile from an array of individual entities bound within a complex system can be treated as quasifree, that is incoherent and following the impulse approximation, when the projectile and ejectile wavelengths are sufficiently short and the momentum transfers sufficiently large. These kinematic conditions, as presented in Chapter 11 of Ref. [1], are met for protons scattered from complex nuclei with beam energies above about 400 MeV, with momentum transfers above about q = 450 MeV/c. Such scattering can be recognized by the peaks in the continuum spectra following single-nucleon kinematics. These quasifree ideas have been pursued by electron scattering, with the simplest kinematic tool being the y-scaling assumption. * Tel.: +1 303 492 1686; fax: +1 303 492 3352.
E-mail address: [email protected] (R.J. Peterson). 0375-9474/$ – see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2006.02.003
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Guided by the conditions for quasifree scattering, a kinematic transformation allows comparison of a wide range of spectra, seeking universal relations among them, even beyond the spectral regions expected to exhibit only single-nucleon quasifree scattering [2]. These electron data have been an important means to understand the momentum distributions and other dynamic quantities of nucleons within nuclei, and have served to check that the charge of 40 Ca indeed resides in twenty protons [3]. The details of the continuum responses have been interpreted as seeing no modifications of the electron–nucleon cross sections in carbon as different from observations on free nucleons [4]. Here, the same y-scaling ideas are applied to previously published data for proton inelastic spectra, at beam energies of 392 MeV [5,6], 400 MeV [7], 558 MeV [8], 795 MeV [9], 1014 MeV [10], 9.5 GeV/c [11] and 19.2 GeV/c [12] for a range of scattering angles and nuclear targets. This wide range for the more complex hadron spectra will allow strong tests of scaling, with deviations from scaling or differences from electron scaling to be examined for the role of hadronic continuum scattering reaction mechanisms. Both the proton–nucleon total and differential cross sections enter this analysis, as described below. If y-scaling is found in hadron continuum cross sections, the data would enable a quite direct measure of hadron–nucleon interactions within nuclei, similar to scattering of hadrons from free nucleons. With a wide range of hadron energies and nuclear samples, the present work examines the validity of such mediumdependent scattering. Momentum transfers considered are those suited to probing the length scale of individual nucleons and their interactions within nuclei. An upper limit to the validity of yscaling based on single nucleons might be set near q = 1000 MeV/c. Beyond this, other scaling phenomena are noted, due to features other than simple single-nucleon scattering [13,14]. Similar y-scaling analyses of pion and K + scattering [15,16] and pion charge exchange [17] have been presented, and the scattering results will be compared below to the results from the present proton analysis. Differences of hadron scaling responses from those obtained by electron scattering are to be expected, and it is these differences that the present work is designed to make evident through the common application of the same scaling methods under conditions where incoherent scattering is expected. The present work does not emphasize the reasons for the differences to be presented below. 2. Methods The present definitions of the scaling variables are taken to be those found in the major electron scattering review [2] and a recent presentation [18]. The laboratory frame energy loss ω and three-momentum transfer q at each data point are transformed into a scaling variable y, which can be considered to be the projection of the nucleon internal momentum along q, for large values of q. Without nuclear recoil this variable is (1) y∞ = (ω − SE)(ω − SE + 2mN ) − qeff . With recoil of the residual nucleus, this becomes [19] m2N + (q + y∞ )2 y∞ , y = y∞ 1 − 2(A − 1)mN q + y∞
(2)
with mN the free nucleon mass. Here, SE is the positive separation energy, to represent the average binding energy of the removed nucleon. Values for low momentum cases are taken to be those used in electron scattering scaling analyses [19]; these are listed in Table 1. The effective
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Table 1 Parameters entering the y-scale analyses for continuum proton and meson scattering on complex nuclei are listed. The free projectile-nucleon total cross sections listed in mb from Ref. [22] enter the computation of Aeff . At higher energies total cross sections are constant at 40 mb [25], and a value of kF = 400 MeV/c is used to match the observed quasifree peak widths. Separation energies SE and Fermi momenta kF are as used for related analyses of electron scattering [19] and are near those obtained by fits to electron scattering spectra in [20] A
SE (MeV)
kF (MeV/c)
392 MeV Aeff
795 MeV Aeff
1014 MeV Aeff
500 MeV π + Aeff
705 MeV/c K + Aeff
6 Li C Al Ca V Zr Pb σt (p) σt (n)
10. 15. 20. 20. 20. 22. 25.
147 220 230 232 235 238 240
– 3.42 – 6.80 – – 11.68 26.73 33.32
2.26 2.40 3.85 4.45 4.02 6.01 7.24 46.13 38.33
– 2.38 – 4.40 – – – 46.66 38.50
– 3.95 – 8.18 – 10.69 13.14 19.15 32.26
– 6.13 – 14.87 – – 33.31 12.36 16.27
momentum transfer qeff is used to consider the nuclear Coulomb effect on the incident proton. No mean field potential for the protons is considered, since optical model analyses of 800 MeV proton scattering find only small repulsive real potentials [21]. The roles of proton–nucleon elastic cross sections and nuclear distortion effects in quasifree scattering have been modeled by factorization for some years [7]. Here, the measured doubly differential cross sections are transformed to form a factored scaling response by F (y) =
d 2 σ/dω dΩ qeff , Aeff dσ/dΩ (free) m2 + (q + y)2 eff N
(3)
with qeff evaluated for each energy loss for spectra taken at fixed angles. These scaling variables y and F (y) are derived from a quasifree scattering model, but will be used here, as in electron scattering, across the wide continuum spectra. The single-scattering differential cross section dσ/dΩ is taken from free-space values, averaged over the neutrons and protons of each target. Below 1.6 GeV these are from Ref. [22]. The off-shell nature of the scattering away from free kinematics at y = 0 (xBj = 1) will be considered below. The number of nucleons that the incident proton may see once and only once is evaluated in the eikonal Glauber model [23], as described in [24]. Proton–nucleon total cross sections σt are first taken from free scattering [22,25] and listed in Table 1; modifications within the medium will be discussed below. Distributions ρ(r) for protons were taken from Ref. [26], with the same geometrical parameters used for neutrons. This usage of the Glauber model may be tested for our cases by examination of scaling of the second kind, as described in [18,19]. The width and maximum of the scaling responses F (y) are influenced by the internal momentum distributions, here modeled as a relativistic Fermi gas with Fermi momentum kF . This usage accounts for the dominant differences expected in the continuum responses of a wide range of nuclei, as discussed in [19]. Values of kF , as used in the electron scattering analysis of Ref. [19], and Aeff used here are listed in Table 1. Most of the proton scattering experiments considered here below 2 GeV had good energy and angle resolution that would not significantly alter the spectral shapes. At higher momenta the experimental resolution gave peak widths requiring kF = 400 MeV/c for all cases. Examples of the sensitivity of the hadron scaling results to this kF parameter will be shown below.
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The scaling variables of the second kind are then taken to be Y = y/kF
and f (Y ) = kF F (y).
(4)
3. Scaling of the first kind Using Aeff as obtained from the Glauber model with free hadron–nucleon total cross sections, scaling of the first kind [18] may be examined, sensitive for hadron beams to the in-medium projectile-nucleon differential cross sections, and to the assumptions of the incoherent reaction model. Free (on-shell) projectile-nucleon differential cross sections are used in this section. Data for a wide range of beam energies at comparable momentum transfers will form a strong test of y-scaling, since the hadronic scattering mechanisms for proton and pion beams are not so simple as for electrons. 3.1. 392 MeV A beam energy of 392 MeV is somewhat low to meet the kinematic conditions for quasifree scattering, especially at larger angles where the outgoing proton momentum is low. Nonetheless, Fig. 1 shows data for three targets at a range of angles [5,6]. Scaling of the first kind would be found if each individual nucleus gave the same response F (y) at all angles.
Fig. 1. Data for inclusive 392 MeV proton scattering [5,6] are compared for scaling of the first kind, where each nuclear system should give the same scaling response F (y) over a range of angles. These responses rise rapidly at large angles and positive y, indicating the role of multinucleon, pion production or other inelastic processes, especially for the heavy nucleus. The second set of data points for carbon show the effect of using the optimum reference frame to compute proton–nucleus differential cross sections to apply to the scaling calculation. Effects are small for negative y, where scaling is most expected to apply. The irregularity in the 40◦ carbon points is found in the original data. Momentum transfers at these angles are 400, 473, 609, 670, 725 and 819 MeV/c for free scattering.
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The peak near y = 0 is found out to 40◦ , with a rise at larger angles for positive y, especially for lead. These latter events correspond to inelastic or multiple scattering processes, and do not meet the ‘billiard ball’ scattering assumption used here. An extended scaling analysis of electron data has been applied for cases with y > 0 [20]. At 25◦ , with q = 400 MeV/c, the incoherent conditions are marginally met at 392 MeV, and the full scaling response seems not to be obtained. At angles larger than 60◦ , not shown here, the scaling peak is submerged among more complex contributions to the spectra and a strong rise is seen for the response at positive values of y. Only for carbon at moderate angles might some success be claimed for scaling of the first kind at 392 MeV. The second data points for carbon at 25◦ and 40◦ show the results when the optimum reference frame [27] is used to determine the beam energies and scattering angles to evaluate proton–nucleon differential cross sections in order to minimize the effects of scattering from bound nucleons not at rest. Effects are small in the regions of y where scaling is to be sought. 3.2. 558 MeV Tables of doubly-differential cross sections from 558 MeV proton scattering are found in Ref. [8], without good energy resolution. Transformations to F (y) for carbon and copper are shown in Fig. 2, with uncertainties in y determined from the stated energy resolutions. Good scaling of the first kind is noted for carbon at the smaller angles, with the larger angles showing the strong role of other processes. These trends away from y-scaling are stronger for copper and other heavy nuclei, not shown.
Fig. 2. Proton continuum spectra at 558 MeV [8] are transformed to y-scaling functions for carbon and copper. Free scattering momentum transfers are 400, 587, 756, 900, and 1015 MeV/c at these angles. The second data points for carbon use the optimum reference frame for singly-differential cross sections.
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3.3. 795 MeV
The 795 MeV proton data of Ref. [9] were normalized to hydrogen cross sections known at that time. These cross sections have been renormalized here to match the proton-hydrogen cross sections of Ref. [22]. At this higher energy the kinematic conditions are more completely met, and a wide range of target nuclei has been studied. Scaling responses of the first kind are shown in Figs. 3 and 4. As was found for pion scattering [16], 6 Li exhibits a very good scaling response beyond 15◦ , where the conditions are not yet met and the scattering can still be partially coherent. For heavier nuclei, the magnitudes of the maxima are increasing with angle, with at least some of this increase arising from a continuum background most evident at positive y. Transverse electron scattering scaling responses also show the strong continuum for positive y, with the rise stronger for larger values of the momentum transfer and for heavier nuclei, as in [19,20]. The proton data shown in Figs. 3 and 4 extend to larger q and heavier nuclei than shown in Fig. 11 of Ref. [19]. The comparison between charge and transverse electron scattering continuum responses leads one to expect that pion production is an important source of the rise for positve y, and the same is likely for the proton data. An extensive analysis of scaling for electrons in this region is found in [20].
Fig. 3. The test of scaling of the first kind is applied to 795 MeV proton inclusive spectra [9]. At 15◦ the momentum transfer is too low to expect the conditions for quasifree scattering to apply. The second set of data points for carbon shows the result of using the optimum reference frame for free proton–nucleon elastic differential cross sections. Free scattering momentum transfers are 379, 503, 623 and 739 MeV/c at these angles.
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Fig. 4. As Fig. 3, extended to zirconium and lead [9]. Scaling for these heavy nuclei is not found at 795 MeV.
3.4. 1014 MeV Clear quasifree peaks are seen for carbon at this energy [10], as seen in Fig. 5, with the magnitudes and shapes in good agreement. This figure shows a very good example of scaling of the first kind. The calcium sample used in [10] yielded uncertain magnitudes of the cross sections, but consistency was assumed for this analysis. The higher beam energy seems to have provided a better scaling of the first kind than is seen at 795 MeV, perhaps because the kinematic conditions for quasifree scattering are more securely met at 1014 MeV. A recent report of 1 GeV proton scattering with quasifree kinematics suffers from poorly known resolution [28], and is not used here. The 6 Li data shown in that work at 25◦ , transformed to the scaling variables of the first kind, do nearly match the data shown here for carbon at 1014 MeV, with a maximum of 7.1 GeV−1 . 3.5. 500 MeV (π + , π + ) Doubly-differential cross sections for inclusive π + scattering from carbon at a beam energy of 500 MeV are transformed into y-scaling responses and shown in Fig. 6. Data are from Ref. [15], and scaling of the first kind is found for the angles shown. That pion scattering experiment found π − scattering cross sections on carbon to be equal to those for π + . Since pion–nucleon scattering near this energy is dominated by a number of resonances, the effects of internal nucleon motion may be important to treat the in-medium differential cross sections correctly. The second set of points for 50◦ uses the optimum reference frame, with little change. Scaled data from 950 MeV/c π − scattering from a number of nuclei were shown in Ref. [16]. Scaling of the first kind was documented well for 6 Li, with increasing deviations from scaling for heavier nuclei, as noted here in the proton scattering analyses.
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Fig. 5. Doubly-differential cross sections for 1014 MeV proton spectra [10] have been transformed into the y-scaling format. The text of Ref. [10] warns of an uncertain overall magnitude for the calcium data. Free-scattering momentum transfers for the data shown are 402, 450, 512 and 599 MeV/c. Good scaling of the first kind is noted using free proton–nucleon differential cross sections. Second data points use the optimum frame prescription for free proton–nucleon differential cross sections.
Fig. 6. Inclusive π + spectra at 500 MeV (624 MeV/c) from [15] are shown in the y-scaling format. Free momentum transfers are 406 and 488 MeV/c. The second points for 50◦ show the role of the optimum reference frame.
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3.6. The optimum reference frame Projectile-nucleon scattering in nuclei at values of y other than zero is off-shell, relative to free scattering, since a moving nucleon was struck. It has been shown [27] that a different reference frame (the optimum frame) can minimize the differences between free and quasifree scattering. Proton–nucleon differential cross sections evaluated within the standard phase shift solution [22] in these reference frames are used to form proton y-scaling responses at 392, 558, 795 and 1014 MeV and π + responses at 500 MeV, with the results shown in Figs. 1, 2, 3, 5 and 6 for carbon. Little change is seen at negative values of y, with the effect of the optimum frame prescription being a slight decrease in the magnitude of the scaling response. The scaling response at positive y increases with use of the optimum frame, but this region could also include nonelastic processes and is not the crucial region to consider for scaling. We conclude that the off-shell correction, as modeled with the optimum frame free cross sections, has little effect on the important regions of y-scaling responses for proton cross sections. 3.7. Comparisons of scaling of the first kind It is the hard collisions (y < 0) that have been of greatest interest in continuum electron scattering at high momentum transfers, since this region can yield information on nuclear dynamics [18]. It is an advantage of inclusive electron scattering that questions of the reaction mechanism have a small effect, whereas issues of both nuclear dynamics and reaction mechanism occur together for hadron scattering in the same kinematic region. The present analysis seeks to test the consistency of the simple scattering hypothesis for hadrons, to determine the conditions when the simple ideas work, and to examine the systematics of hadron deviations from y-scaling. The most extensive angular range is for 392 MeV [5,6]. The scaling response at this energy increases with angle or momentum transfer, but not smoothly. Scaling is maintained for 40◦ (q = 609 MeV/c) and 60◦ (q = 819 MeV/c) at this beam energy, as noted in Fig. 1. A similar trend is noted for carbon at 558 MeV (with poor energy resolution) and 795 MeV. The average of hadron–nucleon total cross sections for the incident beam and the emerging hadron could be used to compute Aeff , thus building in a means for the scaling responses to change with angle. For all the cases presented here, no more than a 6% change of the magnitude of F (y = 0) would result, save for a 16% increase of F (y = 0) from 30◦ to 45◦ at a proton beam energy of 392 MeV. The proton continuum spectra fail to show a scaling response of the first kind across any wide range of angles, especially for very negative y. In contrast to electron scattering, one cannot extend hadron y-scaling to large momentum transfers. The more moderate ranges of momentum transfer and y used for Figs. 1–6 are evidently the only ranges where y-scaling can be approached for hadrons. At higher energies, larger angles, and heavier nuclei there are strong deviations from scaling. A mechanism demostrated to cause these deviations at 19.2 GeV/c will be presented below. 4. Scaling of the second kind Since scaling of the first kind is not consistently observed for y < 0, it is appropriate to examine mainly the magnitudes of the maxima of the scaling responses, near y = 0, for mass dependences. Scaling of the second kind [18] will be examined in this section, using data near
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Fig. 7. Scaling functions of the second kind are shown for three nuclear species at 392 MeV, with data from Refs. [5,6]. Momentum transfers for free scattering are 406 and 609 MeV/c at these angles. Scaling is noted, with use of free-space proton–nucleon total cross sections to evaluate Aeff .
the maxima for a wide range of nuclei at the same scattering angle. The scaling variables have been normalized to Fermi momenta kF for better comparisons, as in Ref. [19]. 4.1. 392 MeV Fig. 7 shows 392 MeV proton spectra for three nuclei [5,6], with free scattering momentum transfers q = 406 and 609 MeV/c. The simplest Fermi gas would give a parabola ranging from Y = −1 to +1, with a maximum f (Y = 0) = 0.75. Few conclusions can be reached for this limited range of nuclei, although scaling of the second kind is approximately followed at both angles. Free proton–nucleon total cross sections are used to evaluate Aeff for this analysis, using an average value of 30.0 mb [22]. Responses f (Y = 0) would be about 4% higher if the fitted value of kF from [20] were used. 4.2. 400 MeV Doubly-differential cross sections for three nuclei are shown in the Y -scaling format in Fig. 8, as traced from data shown in Ref. [7]. Scaling of the second kind is observed to within about 10% at 30◦ , near the accuracy of the data. Data for 58 Ni at 25◦ are also seen to scale, as a limited test of scaling of the first kind. The thrust of Ref. [7] was the comparison of predictions from a surface RPA model to the data shown. The RPA model curves, transformed to scaling functions of the second kind, show very good Y -scaling consistency, with a maximum f (Y = 0) = 1.01.
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Fig. 8. Cross sections for 400 MeV proton inclusive spectra, traced from Ref. [7], yield scaling responses of the second kind as shown. The momentum transfers for free scattering are 405 MeV/c at 25◦ and 479 MeV/c at 30◦ .
4.3. 558 MeV The 558 MeV continuum spectra did not have good energy resolution, but covered a wide range of target masses [8]. The upper panel of Fig. 9 shows the resulting scaling responses f (Y ), using free-space p–N total cross sections [22] to evaluate Aeff in the Glauber model. The neutron/proton average is 35.3 mb for carbon. A strong scatter is noted, not showing scaling of the second kind for free scattering with q = 587 MeV/c over a more complete range of nuclear masses than was available at the lower beam energies. These mass-dependent data at 558 MeV were not considered in Ref. [29], where scaling of the second kind was forced by altering the in-medium p–N total cross sections. With the ratio β = σt (in-medium)/σt (free) of total proton–nucleon total cross sections taken to be 0.71, the larger values of Aeff provide the data points in the lower panel of Fig. 9. Although points for carbon and lead lie above those for the other nuclei, quite good scaling of the second kind is found. The lower in-medium total cross section is consistent with the conclusions from Ref. [29]. 4.4. 795 MeV The high resolution and accurate data of Ref. [9] are transformed into scaling functions f (Y ) in the upper panel of Fig. 10 using free-space p–N total cross sections, averaging 42.2 mb [22]. In the lower panel a smaller in-medium σt yields a closer match to the expectations of scaling of the second kind for a wide range of nuclei. This case was one of those considered in [29]. The smaller in-medium cross sections provide close scaling of the second kind for a wide range of nuclei at each angle. The fits to electron scattering spectra in [20] found slightly smaller values of kF than used here, with the most significant difference being for lithium. If the fit value (kF = 165 MeV/c)
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Fig. 9. Double-differential proton spectra at 558 MeV [8] are transformed to scaling functions of the second kind with free-space proton–nucleon total cross sections in the top panel, at an angle giving a free-space momentum transfer q = 587 MeV/c. In the lower panel the in-medium proton–nucleon total cross sections are reduced by a factor β = 0.71, giving larger values of Aeff . This usage provides closer agreement with scaling of the second kind.
were used for 6 Li in Fig. 9, the data points would be a factor of 1.12 higher than those shown, nearer the trend of the other samples. 4.5. 9.5 and 19.2 GeV/c These higher beam momenta, with short wavelengths and very forward scattering, should be well-suited to the conditions for the incoherent quasifree scattering process. Free differential cross sections are not available from [22], but are taken here from the same experiment as for the complex nuclei at 9.5 GeV/c [11] and from compiled p–p elastic data at 19.2 GeV/c and t = −0.328 (GeV/c)2 [30]. The differential cross sections used are 65 mb/sr at 9.5 GeV/c and 62 mrad and 213 mb/sr at 19.2 GeV/c and 35 mrad. The effective number of nucleons is computed using free p–N total cross sections of 40 mb for both 9.5 and 19.2 GeV/c [25]. Fits to the broad peaks in the low resolution spectra at both momenta require effective values of the Fermi momentum to be 400 MeV/c. Fig. 11 shows 9.5 GeV/c continuum inclusive proton spectra at 62 mrad, for a free momentum transfer of 611 MeV/c [11]. Free total cross sections give the Y -scaling responses in the upper panel, while smaller in-medium total cross sections give the responses in the lower panel. Only this one scattering angle was used in Ref. [11], and the stated uncertainty in the magnitudes of the cross sections was 20%. The scaling maxima do not line up at Y = 0 as expected for scaling of the second kind at 9.5 GeV/c, although use of smaller in-medium total cross sections does reduce the scatter among
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Fig. 10. Above are the 20◦ doubly differential inclusive proton scattering cross sections at 795 MeV [9], transformed to f (Y ) as described in the text, using free p–N total cross sections. These in-medium total cross sections are reduced by a factor of 0.79 to compute larger values of Aeff for the lower panel. The free-space momentum transfer is 503 MeV/c at this angle.
the magnitudes near Y = 0. The fitted power law dσ/dt ∼ Aα with α = 0.44 [11] also yields a smaller in-medium total cross section, with β = 0.7, much as found in Ref. [29]. Only at the smallest scattering angle of 35 mrad does the quasifree peak occur at the expected location of the 19.2 GeV/c spectra of [12], with a free scattering momentum transfer of 710 MeV/c. Inclusive spectra for three nuclei at this angle are shown in the upper panel of Fig. 12, using free-space total cross sections of 40 mb. The severe scatter is much reduced in the lower panel, using an in-medium σt only 70% of that in free space. The maxima do not line up as would be expected for scaling of the second kind. The 19.2 GeV/c experiment also measured spectra at larger angles, but the maxima are not near the expected quasifree single nucleon energy loss, that for free p–N scattering, and no peak is to be discerned at the proper quasifree energy loss. The explanation given in [14] is that the sharply forward angle peaked p–N elastic differential cross sections enable a given energy loss to be obtained at larger angles by a series of small angle scatterings on several nucleons. This model is able to explain the features of the larger angle spectra at 19.2 GeV/c in [12], and goes beyond the single scattering usage in the present work. Since this multiple scattering becomes more likely for heavier nuclei, it is also likely to be the cause of the shift of f (Y ) towards more negative Y (smaller energy losses) as noted in both Figs. 11 and 12, even at moderate momentum transfers. At the lower beam momenta used for scaling analyses of the first kind the proton–nucleon differential cross sections are much less forward peaked, and the explanation of Ref. [14] is less applicable.
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Fig. 11. Doubly-differential cross sections for 9.5 GeV/c protons at a scattering angle of 62 mrad are transformed into scaling functions of the second kind from tables of Ref. [11]. Free scattering in this case would give a momentum transfer q = 611 MeV/c. Free-space total proton–nucleon cross sections of 40 mb are used to compute Aeff for the top panel, while lower in-medium values are used for the lower panel, giving much closer agreement among this range of nuclei. The shift of the maxima for heavier nuclei is discussed in the text.
4.6. Proton comparisons Fig. 13 shows a collection of continuum proton scattering spectra with a beam kinetic energy range of a factor of 50, from 392 MeV to 19.2 GeV, all with similar three-momentum transfers q suited to the conditions for quasifree scattering. Free proton–nucleon total cross sections are used to compute Aeff in the upper panel, with target nuclei ranging from lithium to lead. These Y -scaling responses show a scatter of about a factor of two. One adjustment of the in-medium proton–nucleon total cross section is sufficient to bring about the much tighter scaling responses in the lower panel of Fig. 13. The scale factor β ranges from 0.70 to 0.75, depending upon the magnitude of the free space total cross sections, which differ only slightly among this range of proton beam energies. Many other arrays of data could be added to Fig. 13, as shown in previous sections of this work. It is to be concluded that the in-medium proton–nucleon total cross sections must be lower at all beam energies to achieve this level of consistency. The curve in Fig. 13b is taken from the ‘universal’ curve used to represent all longitudinal electron scattering [20]. Parameters used to transform to the present usage are kF = 230 MeV/c and q = 500 MeV/c. This curve is seen to match the magnitudes of the lower of the proton scattering scaling functions, with smaller in-medium total cross sections, but the average proton response, even with smaller total cross sections, lies above the curve.
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Fig. 12. As for Fig. 11, but with the 19.2 GeV/c data from Ref. [12], at a scattering angle of 35 mrad for a free momentum transfer q = 709 MeV/c.
Fig. 13. A sample of the many proton continuum spectra treated in this work is shown in the upper panel, in the format for scaling of the second kind. Free proton–nucleon total cross sections are used to compute Aeff , with a resulting scatter in magnitudes. Smaller in-medium total cross sections, with larger values of Aeff , yield the data in the lower panel. The curve is from the ‘universal’ fit to longitudinal electron scattering [20].
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5. Scaling of the third kind The y-scaling format permits comparison of a wide range of proton continuum spectra, as shown in Sections 3 and 4. Here, the comparison is extended to include meson inclusive continuum spectra. Values of Aeff computed from the Glauber model with in-medium hadron–nucleon total cross sections are used to compute the scaling responses for carbon shown in Fig. 14, using only results with fairly good energy resolution. The optimum reference frame is not used here; this makes no difference at the maxima of the responses, and, as shown above, has little effect for negative y. Where a wide range of nuclei was studied with a given beam, the method of Ref. [29] is used to find the ratios β. For cases with too few nuclei to carry this through, values of β are estimated by interpolation among the relations between total cross sections and β needed to restore scaling of the second kind [29]. Values of β used for Fig. 15 are: K + (1.24), 850 MeV/c π − (0.88), 950 MeV/c π − (0.68), 1000 MeV/c π − (0.64), 392 MeV proton (0.76), 795 MeV proton (0.79), and 1014 MeV proton (0.79). The broader peaks, due to lower resolution, at 9.5 GeV/c and 19.2 GeV/c lie near 3 GeV−1 (using β = 0.7); the poorer resolution cross sections at 558 MeV (β = 0.71) are also seen to lie near the other cases. A 500 MeV π + (β = 1) spectrum with q = 488 MeV/c gives a maximum at 3.9 GeV−1 [15]. All these responses for carbon lie within 20% of one another, when treated in the consistent scaling fashion with altered in-medium hadron–nucleon total cross sections, and show their maxima at very nearly the same value of y, near −40 MeV/c. This might then be called scaling of the third kind, with near consistency for a wide range of hadron species. These hadron scaling
Fig. 14. A wide range of hadron–carbon spectra is shown, using in-medium hadron–nucleon total cross sections as described in the text. The scaling responses for all these strongly interacting probes nearly coincide, but are stronger than observed when a longitudinal (charge) electron scattering spectrum from Ref. [31] is subjected to the same analysis; these points are denoted as EEL. The optimum reference frame has not been used for these plots. These hadron response spectra exhibit a new scaling of the third kind.
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Fig. 15. Maxima of the hadron non-charge exchange (NCX) response functions F (y) from experiments with good energy resolution are fit to determine their maxima, as described in the text. These maxima are not found to lie at y = 0, and the differences from zero are plotted, with a negative shift indicating a maximum at y < 0. A systematic trend is noted for the four different beam species. The energy shift parameter fit to electron spectra [20] would force a shift of zero in a corresponding plot.
responses lie above those for longitudinal (charge) electron scattering, for carbon, as shown in Fig. 14 using the formulations of the present work with the data of Ref. [31] at q = 550 MeV/c. The apparent discrepancy between the hadron and electron comparisons in Figs. 13 and 14 is due to several small effects, with the data points shown in Fig. 14 [31] only one of the many examples analyzed for the universal fL curve in [20]. The larger in-medium total cross sections for the K + , with its long mean free path in nuclei, have been widely noted [32], while the other hadrons demonstrate smaller in-medium total cross sections. 6. Comparisons and medium effects The several hadronic probes at the energies and momentum transfers used in Fig. 14 couple to bound nucleons with a range of isospin sensitivities. These can be compared by the magnitudes of the fractional differences between hadron–proton and hadron–neutron differential cross sections and their sums at the angles corresponding to q near 575 MeV/c, as used in Fig. 14. These are 0.33, 0.27, 0.13, 0.14, 0.17, 0.17 and 0.21 for 392 MeV protons, 558 MeV protons, 795 MeV protons, 1014 MeV protons, 705 MeV/c K + , 950 MeV/c π − and 500 MeV π + , respectively [22]. No pattern determined by these isospin couplings is to be discerned in Fig. 14. As noted for a wider range of meson scattering responses in Ref. [16], all these hadron responses lie somewhat above the charge scattering responses for electron scattering, which have a unit difference-over-sum ratio for neutrons and protons. The hadron scattering probes are predominantly isoscalar in their coupling to nucleons for q near 575 MeV/c, and have small spin transfer cross sections [22].
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The maxima of the present hadron scattering responses do not line up on y = 0, while electron scaling responses do maximize very near y (or Y or ψ ) = 0 [19,20]. The separation energies SE used for the present analysis are taken to be the same as used for electrons, as listed in Table 1. Fig. 15 shows the differences between the observed maxima and y = 0 for values of q near 500 MeV/c. These maxima are determined by forcing a parabolic Fermi gas shape to the F (y) responses near the maxima, using values of kF as listed in Table 1. Although uncertainties are large, the shift towards more negative values of y does increase smoothly with target mass. Note that the separation energies used (Table I) change with mass, as also found by fits to electron spectra in [20], but this does not account for the shifts observed for hadrons. Note that the fraction of Aeff to A is such that larger nuclei are probed a lower densities than lighter samples. If Aeff is used to count nucleons beyond some radius, a typical total cross section of 42 mb as used at 795 MeV gives access to densities (relative to central densities) of 0.23, 0.135, 0.096 and 0.087 for A = 12, 40, 90 and 208, respectively. The multistep scattering used for the 19.2 GeV/c analysis [14] is a means for the nominal single-scattering quasifree peak to occur at lower energy losses for heavier nuclei, as noted in Fig. 15. Another means to move the maximum of F (y) to zero would be to apply an effective mass for the struck nucleons greater than that in free space, as might be expected in the nuclear surface [33] where the quasifree scattering is localized. The magnitudes of the scaling responses of hadrons on nuclei are influenced in two ways by their interactions with bound nucleons, within the quasifree scaling context. Differential in-medium proton–nucleon cross sections were considered by Ray [34], as applied to coherent proton–nucleus elastic scattering. Especially for the S = T = 0 amplitudes, very small changes in the nuclear medium are indicated near q = 500 MeV/c for beam energies of 500 MeV or above. This is confirmed in the present study of scaling of the first kind, as noted for light nuclei across a range of scattering angles. The agreement between meson and nucleon scattering responses further indicates that the roles of exchanged mesons do not change within nuclei; the spin-zero meson probes may not exchange single pions to interact with nucleons, and by q = 500 MeV/c shorter range exchanges are expected to dominate. Proton–nucleon in-medium total cross sections enter the scaling analysis in a manner that influences most directly the scaling of the second kind, since Aeff does not change uniformly with nuclear mass as the in-medium total cross sections change. This mass dependence of scaling was examined for 795 MeV proton scattering in Ref. [29], with the conclusion that scaling of the second kind can be obtained with proton–nucleon total cross sections within nuclei smaller than found in free space by a factor β = 0.79 (0.10). This is consistent with an extrapolation of the theoretical curves of Fuchs et al. [35], where such total cross sections are reduced by 10–20% at half nuclear density. The discrepancy with measured electron charge scattering responses is diminished by use of smaller in-medium total cross sections as well. The lower panels of Figs. 9 and 10 show scaling responses of the second kind at 558 and 795 MeV with Aeff computed by use of in-medium total cross sections taken to be about 75% of the free-space values. The scatter among the array of proton data from 6 Li to lead is much smaller than found in the upper panels using free-space total cross sections, for closer scaling of the second kind. Similar effects are noted at 9.5 and 19.2 GeV/c. 7. Conclusions Three kinds of scaling have been examined using these analyses of a wide range of hadron data. Scaling of the first kind is that in which a given proton energy and nuclear sample give
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the same response F (y) for a range of angles (or momentum transfers). This tests the conditions for scaling and the use of free-space dσ/dΩ across this range of angles. The hadron data fail to scale in general in this first fashion, save for high beam energies and light nuclei. The extreme case of 19.2 GeV/c [14] suggests that this failure is due to second scattering effects, notably at larger angles and for heavier nuclei. Scaling of the second kind examines responses f (Y ) at a given beam energy and scattering angle for a range of nuclear samples, thus testing the validity of the Aeff approximation and the total cross sections that determine this parameter. This scaling can be obtained, in concord with that noted with electron charge scattering, with altered in-medium hadron–nucleon total cross sections. If both first and second kinds of scalings were to be obtained with hadrons, this would be equivalent to the ‘superscaling’ found for electrons [19]. Comparison of all hadronic scatterings at similar momentum transfers with altered in-medium total cross sections gives a new common scaling response of the third kind, with no analogy in electron scattering. Only for carbon is a suitably broad range of data with hadronic probes usable for this demonstration. With β near 0.75, hadron NCX responses are larger than charge electron scattering responses by a factor near 1.5. Continuum electron scattering experiments have demonstrated that a simple and weak scattering probe finds simple and universal scaling responses from complex nuclei, with no indications of nucleons within nuclei being different from those in free space [4]. In the same spirit, this work has examined a large body of proton and meson continuum scattering spectra, seeking similar scaling relations when the interactions of the projectiles are not simple and not weak, to determine if such quasifree scattering can be a useful measure of in-medium hadron–nucleon cross sections. Only a fraction of the nucleons within a nucleus will scatter the hadron beam once and only once, as modeled here in a Glauber calculation. Deviations from scaling of the first kind are observed for all but the lightest nuclei; previous analysis of the highest energy case indicates that multinucleon processes are strong there even for negative y. The present work also indicates this explanation for lower energies with smaller deviations from scaling. Scaling of the second kind for the maxima of hadron quasifree spectra can only be obtained with smaller in-medium hadron–nucleon total cross sections (save for the known K + case), consistent with theoretical expectations in general [35]. In summary, the simple concept of direct observation of hadron–nucleon elastic scattering on nucleons within complex nuclei has been tested through scaling analyses of scattering data. Some cases do exhibit such a simple process, while other examples document systematic violations of scaling requiring further study. The following paper [36] will extend these y-scaling analyses to hadron continuum spectra with charge exchange, using very similar methods and another wide range of data. Acknowledgements The author wishes to thank M. Nakano for permission to cite unpublished data, G.F. Steyn for tables of cross sections and S.G. Mashnik for a most useful critical reading. This work was supported in part by the USDOE. References [1] [2] [3] [4]
M.L. Goldberger, K.M. Watson, Collision Theory, Wiley, New York, 1964. D. Day, J.S. McCarthy, T.W. Donnelly, I. Sick, Annu. Rev. Nucl. Part. Sci. 40 (1990) 357. C.F. Williamson, et al., Phys. Rev. C 56 (1997) 3152. K.S. Kim, L.E. Wright, Phys. Rev. C 68 (2003) 027601.
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[5] A.A. Cowley, et al., Phys. Rev. C 62 (2000) 064604. [6] T. Kin et al., Research Center for Nuclear Physics, Osaka University, 2001, Annual Report and private communication; T. Kin et al., Research Center for Nuclear Physics, Osaka University, 2002, Annual Report. [7] H. Esbensen, G.F. Bertsch, Phys. Rev. C 34 (1986) 1419. [8] S.M. Beck, C.A. Powell, NASA Technical Note D-8119, Washington, DC, 1976. [9] R.E. Chrien, T.J. Krieger, R.J. Sutter, M. May, H. Palevsky, R.L. Stearns, T. Kozlowski, T. Bauer, Phys. Rev. C 21 (1980) 1014. [10] D.M. Corley, et al., Nucl. Phys. A 184 (1972) 437. [11] L.Z. Barabash, et al., Sov. J. Nucl. Phys. 36 (1982) 90. [12] A. Klovning, O. Kofoed-Hansen, K. Schlupmann, Nucl. Phys. B 54 (1973) 29. [13] L.L. Frankfurt, M.I. Strikman, Phys. Rev. 76 (1981) 214. [14] O. Kofoed-Hansen, Nucl. Phys. A 402 (1973) 515. [15] J.E. Wise, et al., Phys. Rev. C 48 (1993) 1840. [16] R.J. Peterson, et al., Phys. Rev. C 65 (2002) 054601. [17] R.J. Peterson, et al., Phys. Rev. C 69 (2004) 064612. [18] A.N. Antonov, et al., Phys. Rev. C 69 (2004) 044321. [19] T.W. Donnelly, I. Sick, Phys. Rev. C 60 (1999) 065502. [20] C. Maieron, T.W. Donnelly, I. Sick, Phys. Rev. C 65 (2002) 025502. [21] N.J. DiGiacomo, et al., Phys. Rev. C 19 (1979) 1132. [22] R.A. Arndt, L.D. Roper, program SAID (unpublished); R.A. Arndt, J.M. Ford, L.D. Roper, Phys. Rev. D 31 (1985) 1085, solution SM95. [23] R.J. Glauber, in: W.E. Brittin, L.G. Dunham (Eds.), Lectures in Theoretical Physics, Interscience, New York, 1959. [24] J. Ouyang, S. Hoibraten, R.J. Peterson, Phys. Rev. C 47 (1993) 2809. [25] Particle Data Group, Phys. Rev. D 66 (2002) 010001. [26] J.D. Patterson, R.J. Peterson, Nucl. Phys. A 717 (2003) 235. [27] S.A. Gurvitz, Phys. Rev. C 33 (1986) 422. [28] V.N. Baturin, et al., Nucl. Phys. A 736 (2004) 283. [29] R.J. Peterson, Nucl. Phys. A 740 (2004) 119. [30] J.V. Allaby, et al., Phys. Lett. B 28 (1968) 67. [31] P. Barreau, et al., Nucl. Phys. A 402 (1983) 515. [32] A. Gal, Nucl. Phys. A 639 (1998) 485c. [33] Z.Y. Ma, J. Wambach, Nucl. Phys. A 402 (1983) 275. [34] L. Ray, Phys. Rev. C 41 (1990) 2816. [35] C. Fuchs, A. Faessler, M. El-Shabsiry, Phys. Rev. C 64 (2001) 024003. [36] R.J. Peterson, Nucl. Phys. A, following paper.
y-scaling in proton continuum scattering R.J. Peterson ∗ Department of Physics, UCB 390, University of Colorado, Boulder, CO 80309-0390, USA Received 3 November 2005; received in revised form 18 January 2006; accepted 10 February 2006 Available online 3 March 2006
Abstract Data for inclusive proton scattering spectra at kinematics suited to incoherent quasifree scattering from single nucleons bound within complex nuclei are analysed in the y-scaling format found to unify many electron scattering spectra into universal nuclear responses. Proton continuum spectra are analyzed for beam energies from 392 MeV to 19.2 GeV. Scaling is somewhat successful for proton scattering on light nuclei or at fixed angles, with differences from electron scattering that indicate modifications of proton scattering on nucleons within nuclei from scattering in free space. Comparisons to inclusive continuum meson scattering in the same y-scaling format are made, and a new hadronic scaling relation is demonstrated in light nuclei. 2006 Elsevier B.V. All rights reserved. PACS: 25.40.-h; 25.80.-e Keywords: Hadron; Continuum scattering; Quasifree; Scaling
1. Introduction The scattering of a projectile from an array of individual entities bound within a complex system can be treated as quasifree, that is incoherent and following the impulse approximation, when the projectile and ejectile wavelengths are sufficiently short and the momentum transfers sufficiently large. These kinematic conditions, as presented in Chapter 11 of Ref. [1], are met for protons scattered from complex nuclei with beam energies above about 400 MeV, with momentum transfers above about q = 450 MeV/c. Such scattering can be recognized by the peaks in the continuum spectra following single-nucleon kinematics. These quasifree ideas have been pursued by electron scattering, with the simplest kinematic tool being the y-scaling assumption. * Tel.: +1 303 492 1686; fax: +1 303 492 3352.
E-mail address: [email protected] (R.J. Peterson). 0375-9474/$ – see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysa.2006.02.003
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Guided by the conditions for quasifree scattering, a kinematic transformation allows comparison of a wide range of spectra, seeking universal relations among them, even beyond the spectral regions expected to exhibit only single-nucleon quasifree scattering [2]. These electron data have been an important means to understand the momentum distributions and other dynamic quantities of nucleons within nuclei, and have served to check that the charge of 40 Ca indeed resides in twenty protons [3]. The details of the continuum responses have been interpreted as seeing no modifications of the electron–nucleon cross sections in carbon as different from observations on free nucleons [4]. Here, the same y-scaling ideas are applied to previously published data for proton inelastic spectra, at beam energies of 392 MeV [5,6], 400 MeV [7], 558 MeV [8], 795 MeV [9], 1014 MeV [10], 9.5 GeV/c [11] and 19.2 GeV/c [12] for a range of scattering angles and nuclear targets. This wide range for the more complex hadron spectra will allow strong tests of scaling, with deviations from scaling or differences from electron scaling to be examined for the role of hadronic continuum scattering reaction mechanisms. Both the proton–nucleon total and differential cross sections enter this analysis, as described below. If y-scaling is found in hadron continuum cross sections, the data would enable a quite direct measure of hadron–nucleon interactions within nuclei, similar to scattering of hadrons from free nucleons. With a wide range of hadron energies and nuclear samples, the present work examines the validity of such mediumdependent scattering. Momentum transfers considered are those suited to probing the length scale of individual nucleons and their interactions within nuclei. An upper limit to the validity of yscaling based on single nucleons might be set near q = 1000 MeV/c. Beyond this, other scaling phenomena are noted, due to features other than simple single-nucleon scattering [13,14]. Similar y-scaling analyses of pion and K + scattering [15,16] and pion charge exchange [17] have been presented, and the scattering results will be compared below to the results from the present proton analysis. Differences of hadron scaling responses from those obtained by electron scattering are to be expected, and it is these differences that the present work is designed to make evident through the common application of the same scaling methods under conditions where incoherent scattering is expected. The present work does not emphasize the reasons for the differences to be presented below. 2. Methods The present definitions of the scaling variables are taken to be those found in the major electron scattering review [2] and a recent presentation [18]. The laboratory frame energy loss ω and three-momentum transfer q at each data point are transformed into a scaling variable y, which can be considered to be the projection of the nucleon internal momentum along q, for large values of q. Without nuclear recoil this variable is (1) y∞ = (ω − SE)(ω − SE + 2mN ) − qeff . With recoil of the residual nucleus, this becomes [19] m2N + (q + y∞ )2 y∞ , y = y∞ 1 − 2(A − 1)mN q + y∞
(2)
with mN the free nucleon mass. Here, SE is the positive separation energy, to represent the average binding energy of the removed nucleon. Values for low momentum cases are taken to be those used in electron scattering scaling analyses [19]; these are listed in Table 1. The effective
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Table 1 Parameters entering the y-scale analyses for continuum proton and meson scattering on complex nuclei are listed. The free projectile-nucleon total cross sections listed in mb from Ref. [22] enter the computation of Aeff . At higher energies total cross sections are constant at 40 mb [25], and a value of kF = 400 MeV/c is used to match the observed quasifree peak widths. Separation energies SE and Fermi momenta kF are as used for related analyses of electron scattering [19] and are near those obtained by fits to electron scattering spectra in [20] A
SE (MeV)
kF (MeV/c)
392 MeV Aeff
795 MeV Aeff
1014 MeV Aeff
500 MeV π + Aeff
705 MeV/c K + Aeff
6 Li C Al Ca V Zr Pb σt (p) σt (n)
10. 15. 20. 20. 20. 22. 25.
147 220 230 232 235 238 240
– 3.42 – 6.80 – – 11.68 26.73 33.32
2.26 2.40 3.85 4.45 4.02 6.01 7.24 46.13 38.33
– 2.38 – 4.40 – – – 46.66 38.50
– 3.95 – 8.18 – 10.69 13.14 19.15 32.26
– 6.13 – 14.87 – – 33.31 12.36 16.27
momentum transfer qeff is used to consider the nuclear Coulomb effect on the incident proton. No mean field potential for the protons is considered, since optical model analyses of 800 MeV proton scattering find only small repulsive real potentials [21]. The roles of proton–nucleon elastic cross sections and nuclear distortion effects in quasifree scattering have been modeled by factorization for some years [7]. Here, the measured doubly differential cross sections are transformed to form a factored scaling response by F (y) =
d 2 σ/dω dΩ qeff , Aeff dσ/dΩ (free) m2 + (q + y)2 eff N
(3)
with qeff evaluated for each energy loss for spectra taken at fixed angles. These scaling variables y and F (y) are derived from a quasifree scattering model, but will be used here, as in electron scattering, across the wide continuum spectra. The single-scattering differential cross section dσ/dΩ is taken from free-space values, averaged over the neutrons and protons of each target. Below 1.6 GeV these are from Ref. [22]. The off-shell nature of the scattering away from free kinematics at y = 0 (xBj = 1) will be considered below. The number of nucleons that the incident proton may see once and only once is evaluated in the eikonal Glauber model [23], as described in [24]. Proton–nucleon total cross sections σt are first taken from free scattering [22,25] and listed in Table 1; modifications within the medium will be discussed below. Distributions ρ(r) for protons were taken from Ref. [26], with the same geometrical parameters used for neutrons. This usage of the Glauber model may be tested for our cases by examination of scaling of the second kind, as described in [18,19]. The width and maximum of the scaling responses F (y) are influenced by the internal momentum distributions, here modeled as a relativistic Fermi gas with Fermi momentum kF . This usage accounts for the dominant differences expected in the continuum responses of a wide range of nuclei, as discussed in [19]. Values of kF , as used in the electron scattering analysis of Ref. [19], and Aeff used here are listed in Table 1. Most of the proton scattering experiments considered here below 2 GeV had good energy and angle resolution that would not significantly alter the spectral shapes. At higher momenta the experimental resolution gave peak widths requiring kF = 400 MeV/c for all cases. Examples of the sensitivity of the hadron scaling results to this kF parameter will be shown below.
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The scaling variables of the second kind are then taken to be Y = y/kF
and f (Y ) = kF F (y).
(4)
3. Scaling of the first kind Using Aeff as obtained from the Glauber model with free hadron–nucleon total cross sections, scaling of the first kind [18] may be examined, sensitive for hadron beams to the in-medium projectile-nucleon differential cross sections, and to the assumptions of the incoherent reaction model. Free (on-shell) projectile-nucleon differential cross sections are used in this section. Data for a wide range of beam energies at comparable momentum transfers will form a strong test of y-scaling, since the hadronic scattering mechanisms for proton and pion beams are not so simple as for electrons. 3.1. 392 MeV A beam energy of 392 MeV is somewhat low to meet the kinematic conditions for quasifree scattering, especially at larger angles where the outgoing proton momentum is low. Nonetheless, Fig. 1 shows data for three targets at a range of angles [5,6]. Scaling of the first kind would be found if each individual nucleus gave the same response F (y) at all angles.
Fig. 1. Data for inclusive 392 MeV proton scattering [5,6] are compared for scaling of the first kind, where each nuclear system should give the same scaling response F (y) over a range of angles. These responses rise rapidly at large angles and positive y, indicating the role of multinucleon, pion production or other inelastic processes, especially for the heavy nucleus. The second set of data points for carbon show the effect of using the optimum reference frame to compute proton–nucleus differential cross sections to apply to the scaling calculation. Effects are small for negative y, where scaling is most expected to apply. The irregularity in the 40◦ carbon points is found in the original data. Momentum transfers at these angles are 400, 473, 609, 670, 725 and 819 MeV/c for free scattering.
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The peak near y = 0 is found out to 40◦ , with a rise at larger angles for positive y, especially for lead. These latter events correspond to inelastic or multiple scattering processes, and do not meet the ‘billiard ball’ scattering assumption used here. An extended scaling analysis of electron data has been applied for cases with y > 0 [20]. At 25◦ , with q = 400 MeV/c, the incoherent conditions are marginally met at 392 MeV, and the full scaling response seems not to be obtained. At angles larger than 60◦ , not shown here, the scaling peak is submerged among more complex contributions to the spectra and a strong rise is seen for the response at positive values of y. Only for carbon at moderate angles might some success be claimed for scaling of the first kind at 392 MeV. The second data points for carbon at 25◦ and 40◦ show the results when the optimum reference frame [27] is used to determine the beam energies and scattering angles to evaluate proton–nucleon differential cross sections in order to minimize the effects of scattering from bound nucleons not at rest. Effects are small in the regions of y where scaling is to be sought. 3.2. 558 MeV Tables of doubly-differential cross sections from 558 MeV proton scattering are found in Ref. [8], without good energy resolution. Transformations to F (y) for carbon and copper are shown in Fig. 2, with uncertainties in y determined from the stated energy resolutions. Good scaling of the first kind is noted for carbon at the smaller angles, with the larger angles showing the strong role of other processes. These trends away from y-scaling are stronger for copper and other heavy nuclei, not shown.
Fig. 2. Proton continuum spectra at 558 MeV [8] are transformed to y-scaling functions for carbon and copper. Free scattering momentum transfers are 400, 587, 756, 900, and 1015 MeV/c at these angles. The second data points for carbon use the optimum reference frame for singly-differential cross sections.
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3.3. 795 MeV
The 795 MeV proton data of Ref. [9] were normalized to hydrogen cross sections known at that time. These cross sections have been renormalized here to match the proton-hydrogen cross sections of Ref. [22]. At this higher energy the kinematic conditions are more completely met, and a wide range of target nuclei has been studied. Scaling responses of the first kind are shown in Figs. 3 and 4. As was found for pion scattering [16], 6 Li exhibits a very good scaling response beyond 15◦ , where the conditions are not yet met and the scattering can still be partially coherent. For heavier nuclei, the magnitudes of the maxima are increasing with angle, with at least some of this increase arising from a continuum background most evident at positive y. Transverse electron scattering scaling responses also show the strong continuum for positive y, with the rise stronger for larger values of the momentum transfer and for heavier nuclei, as in [19,20]. The proton data shown in Figs. 3 and 4 extend to larger q and heavier nuclei than shown in Fig. 11 of Ref. [19]. The comparison between charge and transverse electron scattering continuum responses leads one to expect that pion production is an important source of the rise for positve y, and the same is likely for the proton data. An extensive analysis of scaling for electrons in this region is found in [20].
Fig. 3. The test of scaling of the first kind is applied to 795 MeV proton inclusive spectra [9]. At 15◦ the momentum transfer is too low to expect the conditions for quasifree scattering to apply. The second set of data points for carbon shows the result of using the optimum reference frame for free proton–nucleon elastic differential cross sections. Free scattering momentum transfers are 379, 503, 623 and 739 MeV/c at these angles.
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Fig. 4. As Fig. 3, extended to zirconium and lead [9]. Scaling for these heavy nuclei is not found at 795 MeV.
3.4. 1014 MeV Clear quasifree peaks are seen for carbon at this energy [10], as seen in Fig. 5, with the magnitudes and shapes in good agreement. This figure shows a very good example of scaling of the first kind. The calcium sample used in [10] yielded uncertain magnitudes of the cross sections, but consistency was assumed for this analysis. The higher beam energy seems to have provided a better scaling of the first kind than is seen at 795 MeV, perhaps because the kinematic conditions for quasifree scattering are more securely met at 1014 MeV. A recent report of 1 GeV proton scattering with quasifree kinematics suffers from poorly known resolution [28], and is not used here. The 6 Li data shown in that work at 25◦ , transformed to the scaling variables of the first kind, do nearly match the data shown here for carbon at 1014 MeV, with a maximum of 7.1 GeV−1 . 3.5. 500 MeV (π + , π + ) Doubly-differential cross sections for inclusive π + scattering from carbon at a beam energy of 500 MeV are transformed into y-scaling responses and shown in Fig. 6. Data are from Ref. [15], and scaling of the first kind is found for the angles shown. That pion scattering experiment found π − scattering cross sections on carbon to be equal to those for π + . Since pion–nucleon scattering near this energy is dominated by a number of resonances, the effects of internal nucleon motion may be important to treat the in-medium differential cross sections correctly. The second set of points for 50◦ uses the optimum reference frame, with little change. Scaled data from 950 MeV/c π − scattering from a number of nuclei were shown in Ref. [16]. Scaling of the first kind was documented well for 6 Li, with increasing deviations from scaling for heavier nuclei, as noted here in the proton scattering analyses.
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Fig. 5. Doubly-differential cross sections for 1014 MeV proton spectra [10] have been transformed into the y-scaling format. The text of Ref. [10] warns of an uncertain overall magnitude for the calcium data. Free-scattering momentum transfers for the data shown are 402, 450, 512 and 599 MeV/c. Good scaling of the first kind is noted using free proton–nucleon differential cross sections. Second data points use the optimum frame prescription for free proton–nucleon differential cross sections.
Fig. 6. Inclusive π + spectra at 500 MeV (624 MeV/c) from [15] are shown in the y-scaling format. Free momentum transfers are 406 and 488 MeV/c. The second points for 50◦ show the role of the optimum reference frame.
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3.6. The optimum reference frame Projectile-nucleon scattering in nuclei at values of y other than zero is off-shell, relative to free scattering, since a moving nucleon was struck. It has been shown [27] that a different reference frame (the optimum frame) can minimize the differences between free and quasifree scattering. Proton–nucleon differential cross sections evaluated within the standard phase shift solution [22] in these reference frames are used to form proton y-scaling responses at 392, 558, 795 and 1014 MeV and π + responses at 500 MeV, with the results shown in Figs. 1, 2, 3, 5 and 6 for carbon. Little change is seen at negative values of y, with the effect of the optimum frame prescription being a slight decrease in the magnitude of the scaling response. The scaling response at positive y increases with use of the optimum frame, but this region could also include nonelastic processes and is not the crucial region to consider for scaling. We conclude that the off-shell correction, as modeled with the optimum frame free cross sections, has little effect on the important regions of y-scaling responses for proton cross sections. 3.7. Comparisons of scaling of the first kind It is the hard collisions (y < 0) that have been of greatest interest in continuum electron scattering at high momentum transfers, since this region can yield information on nuclear dynamics [18]. It is an advantage of inclusive electron scattering that questions of the reaction mechanism have a small effect, whereas issues of both nuclear dynamics and reaction mechanism occur together for hadron scattering in the same kinematic region. The present analysis seeks to test the consistency of the simple scattering hypothesis for hadrons, to determine the conditions when the simple ideas work, and to examine the systematics of hadron deviations from y-scaling. The most extensive angular range is for 392 MeV [5,6]. The scaling response at this energy increases with angle or momentum transfer, but not smoothly. Scaling is maintained for 40◦ (q = 609 MeV/c) and 60◦ (q = 819 MeV/c) at this beam energy, as noted in Fig. 1. A similar trend is noted for carbon at 558 MeV (with poor energy resolution) and 795 MeV. The average of hadron–nucleon total cross sections for the incident beam and the emerging hadron could be used to compute Aeff , thus building in a means for the scaling responses to change with angle. For all the cases presented here, no more than a 6% change of the magnitude of F (y = 0) would result, save for a 16% increase of F (y = 0) from 30◦ to 45◦ at a proton beam energy of 392 MeV. The proton continuum spectra fail to show a scaling response of the first kind across any wide range of angles, especially for very negative y. In contrast to electron scattering, one cannot extend hadron y-scaling to large momentum transfers. The more moderate ranges of momentum transfer and y used for Figs. 1–6 are evidently the only ranges where y-scaling can be approached for hadrons. At higher energies, larger angles, and heavier nuclei there are strong deviations from scaling. A mechanism demostrated to cause these deviations at 19.2 GeV/c will be presented below. 4. Scaling of the second kind Since scaling of the first kind is not consistently observed for y < 0, it is appropriate to examine mainly the magnitudes of the maxima of the scaling responses, near y = 0, for mass dependences. Scaling of the second kind [18] will be examined in this section, using data near
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Fig. 7. Scaling functions of the second kind are shown for three nuclear species at 392 MeV, with data from Refs. [5,6]. Momentum transfers for free scattering are 406 and 609 MeV/c at these angles. Scaling is noted, with use of free-space proton–nucleon total cross sections to evaluate Aeff .
the maxima for a wide range of nuclei at the same scattering angle. The scaling variables have been normalized to Fermi momenta kF for better comparisons, as in Ref. [19]. 4.1. 392 MeV Fig. 7 shows 392 MeV proton spectra for three nuclei [5,6], with free scattering momentum transfers q = 406 and 609 MeV/c. The simplest Fermi gas would give a parabola ranging from Y = −1 to +1, with a maximum f (Y = 0) = 0.75. Few conclusions can be reached for this limited range of nuclei, although scaling of the second kind is approximately followed at both angles. Free proton–nucleon total cross sections are used to evaluate Aeff for this analysis, using an average value of 30.0 mb [22]. Responses f (Y = 0) would be about 4% higher if the fitted value of kF from [20] were used. 4.2. 400 MeV Doubly-differential cross sections for three nuclei are shown in the Y -scaling format in Fig. 8, as traced from data shown in Ref. [7]. Scaling of the second kind is observed to within about 10% at 30◦ , near the accuracy of the data. Data for 58 Ni at 25◦ are also seen to scale, as a limited test of scaling of the first kind. The thrust of Ref. [7] was the comparison of predictions from a surface RPA model to the data shown. The RPA model curves, transformed to scaling functions of the second kind, show very good Y -scaling consistency, with a maximum f (Y = 0) = 1.01.
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Fig. 8. Cross sections for 400 MeV proton inclusive spectra, traced from Ref. [7], yield scaling responses of the second kind as shown. The momentum transfers for free scattering are 405 MeV/c at 25◦ and 479 MeV/c at 30◦ .
4.3. 558 MeV The 558 MeV continuum spectra did not have good energy resolution, but covered a wide range of target masses [8]. The upper panel of Fig. 9 shows the resulting scaling responses f (Y ), using free-space p–N total cross sections [22] to evaluate Aeff in the Glauber model. The neutron/proton average is 35.3 mb for carbon. A strong scatter is noted, not showing scaling of the second kind for free scattering with q = 587 MeV/c over a more complete range of nuclear masses than was available at the lower beam energies. These mass-dependent data at 558 MeV were not considered in Ref. [29], where scaling of the second kind was forced by altering the in-medium p–N total cross sections. With the ratio β = σt (in-medium)/σt (free) of total proton–nucleon total cross sections taken to be 0.71, the larger values of Aeff provide the data points in the lower panel of Fig. 9. Although points for carbon and lead lie above those for the other nuclei, quite good scaling of the second kind is found. The lower in-medium total cross section is consistent with the conclusions from Ref. [29]. 4.4. 795 MeV The high resolution and accurate data of Ref. [9] are transformed into scaling functions f (Y ) in the upper panel of Fig. 10 using free-space p–N total cross sections, averaging 42.2 mb [22]. In the lower panel a smaller in-medium σt yields a closer match to the expectations of scaling of the second kind for a wide range of nuclei. This case was one of those considered in [29]. The smaller in-medium cross sections provide close scaling of the second kind for a wide range of nuclei at each angle. The fits to electron scattering spectra in [20] found slightly smaller values of kF than used here, with the most significant difference being for lithium. If the fit value (kF = 165 MeV/c)
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Fig. 9. Double-differential proton spectra at 558 MeV [8] are transformed to scaling functions of the second kind with free-space proton–nucleon total cross sections in the top panel, at an angle giving a free-space momentum transfer q = 587 MeV/c. In the lower panel the in-medium proton–nucleon total cross sections are reduced by a factor β = 0.71, giving larger values of Aeff . This usage provides closer agreement with scaling of the second kind.
were used for 6 Li in Fig. 9, the data points would be a factor of 1.12 higher than those shown, nearer the trend of the other samples. 4.5. 9.5 and 19.2 GeV/c These higher beam momenta, with short wavelengths and very forward scattering, should be well-suited to the conditions for the incoherent quasifree scattering process. Free differential cross sections are not available from [22], but are taken here from the same experiment as for the complex nuclei at 9.5 GeV/c [11] and from compiled p–p elastic data at 19.2 GeV/c and t = −0.328 (GeV/c)2 [30]. The differential cross sections used are 65 mb/sr at 9.5 GeV/c and 62 mrad and 213 mb/sr at 19.2 GeV/c and 35 mrad. The effective number of nucleons is computed using free p–N total cross sections of 40 mb for both 9.5 and 19.2 GeV/c [25]. Fits to the broad peaks in the low resolution spectra at both momenta require effective values of the Fermi momentum to be 400 MeV/c. Fig. 11 shows 9.5 GeV/c continuum inclusive proton spectra at 62 mrad, for a free momentum transfer of 611 MeV/c [11]. Free total cross sections give the Y -scaling responses in the upper panel, while smaller in-medium total cross sections give the responses in the lower panel. Only this one scattering angle was used in Ref. [11], and the stated uncertainty in the magnitudes of the cross sections was 20%. The scaling maxima do not line up at Y = 0 as expected for scaling of the second kind at 9.5 GeV/c, although use of smaller in-medium total cross sections does reduce the scatter among
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Fig. 10. Above are the 20◦ doubly differential inclusive proton scattering cross sections at 795 MeV [9], transformed to f (Y ) as described in the text, using free p–N total cross sections. These in-medium total cross sections are reduced by a factor of 0.79 to compute larger values of Aeff for the lower panel. The free-space momentum transfer is 503 MeV/c at this angle.
the magnitudes near Y = 0. The fitted power law dσ/dt ∼ Aα with α = 0.44 [11] also yields a smaller in-medium total cross section, with β = 0.7, much as found in Ref. [29]. Only at the smallest scattering angle of 35 mrad does the quasifree peak occur at the expected location of the 19.2 GeV/c spectra of [12], with a free scattering momentum transfer of 710 MeV/c. Inclusive spectra for three nuclei at this angle are shown in the upper panel of Fig. 12, using free-space total cross sections of 40 mb. The severe scatter is much reduced in the lower panel, using an in-medium σt only 70% of that in free space. The maxima do not line up as would be expected for scaling of the second kind. The 19.2 GeV/c experiment also measured spectra at larger angles, but the maxima are not near the expected quasifree single nucleon energy loss, that for free p–N scattering, and no peak is to be discerned at the proper quasifree energy loss. The explanation given in [14] is that the sharply forward angle peaked p–N elastic differential cross sections enable a given energy loss to be obtained at larger angles by a series of small angle scatterings on several nucleons. This model is able to explain the features of the larger angle spectra at 19.2 GeV/c in [12], and goes beyond the single scattering usage in the present work. Since this multiple scattering becomes more likely for heavier nuclei, it is also likely to be the cause of the shift of f (Y ) towards more negative Y (smaller energy losses) as noted in both Figs. 11 and 12, even at moderate momentum transfers. At the lower beam momenta used for scaling analyses of the first kind the proton–nucleon differential cross sections are much less forward peaked, and the explanation of Ref. [14] is less applicable.
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Fig. 11. Doubly-differential cross sections for 9.5 GeV/c protons at a scattering angle of 62 mrad are transformed into scaling functions of the second kind from tables of Ref. [11]. Free scattering in this case would give a momentum transfer q = 611 MeV/c. Free-space total proton–nucleon cross sections of 40 mb are used to compute Aeff for the top panel, while lower in-medium values are used for the lower panel, giving much closer agreement among this range of nuclei. The shift of the maxima for heavier nuclei is discussed in the text.
4.6. Proton comparisons Fig. 13 shows a collection of continuum proton scattering spectra with a beam kinetic energy range of a factor of 50, from 392 MeV to 19.2 GeV, all with similar three-momentum transfers q suited to the conditions for quasifree scattering. Free proton–nucleon total cross sections are used to compute Aeff in the upper panel, with target nuclei ranging from lithium to lead. These Y -scaling responses show a scatter of about a factor of two. One adjustment of the in-medium proton–nucleon total cross section is sufficient to bring about the much tighter scaling responses in the lower panel of Fig. 13. The scale factor β ranges from 0.70 to 0.75, depending upon the magnitude of the free space total cross sections, which differ only slightly among this range of proton beam energies. Many other arrays of data could be added to Fig. 13, as shown in previous sections of this work. It is to be concluded that the in-medium proton–nucleon total cross sections must be lower at all beam energies to achieve this level of consistency. The curve in Fig. 13b is taken from the ‘universal’ curve used to represent all longitudinal electron scattering [20]. Parameters used to transform to the present usage are kF = 230 MeV/c and q = 500 MeV/c. This curve is seen to match the magnitudes of the lower of the proton scattering scaling functions, with smaller in-medium total cross sections, but the average proton response, even with smaller total cross sections, lies above the curve.
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Fig. 12. As for Fig. 11, but with the 19.2 GeV/c data from Ref. [12], at a scattering angle of 35 mrad for a free momentum transfer q = 709 MeV/c.
Fig. 13. A sample of the many proton continuum spectra treated in this work is shown in the upper panel, in the format for scaling of the second kind. Free proton–nucleon total cross sections are used to compute Aeff , with a resulting scatter in magnitudes. Smaller in-medium total cross sections, with larger values of Aeff , yield the data in the lower panel. The curve is from the ‘universal’ fit to longitudinal electron scattering [20].
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5. Scaling of the third kind The y-scaling format permits comparison of a wide range of proton continuum spectra, as shown in Sections 3 and 4. Here, the comparison is extended to include meson inclusive continuum spectra. Values of Aeff computed from the Glauber model with in-medium hadron–nucleon total cross sections are used to compute the scaling responses for carbon shown in Fig. 14, using only results with fairly good energy resolution. The optimum reference frame is not used here; this makes no difference at the maxima of the responses, and, as shown above, has little effect for negative y. Where a wide range of nuclei was studied with a given beam, the method of Ref. [29] is used to find the ratios β. For cases with too few nuclei to carry this through, values of β are estimated by interpolation among the relations between total cross sections and β needed to restore scaling of the second kind [29]. Values of β used for Fig. 15 are: K + (1.24), 850 MeV/c π − (0.88), 950 MeV/c π − (0.68), 1000 MeV/c π − (0.64), 392 MeV proton (0.76), 795 MeV proton (0.79), and 1014 MeV proton (0.79). The broader peaks, due to lower resolution, at 9.5 GeV/c and 19.2 GeV/c lie near 3 GeV−1 (using β = 0.7); the poorer resolution cross sections at 558 MeV (β = 0.71) are also seen to lie near the other cases. A 500 MeV π + (β = 1) spectrum with q = 488 MeV/c gives a maximum at 3.9 GeV−1 [15]. All these responses for carbon lie within 20% of one another, when treated in the consistent scaling fashion with altered in-medium hadron–nucleon total cross sections, and show their maxima at very nearly the same value of y, near −40 MeV/c. This might then be called scaling of the third kind, with near consistency for a wide range of hadron species. These hadron scaling
Fig. 14. A wide range of hadron–carbon spectra is shown, using in-medium hadron–nucleon total cross sections as described in the text. The scaling responses for all these strongly interacting probes nearly coincide, but are stronger than observed when a longitudinal (charge) electron scattering spectrum from Ref. [31] is subjected to the same analysis; these points are denoted as EEL. The optimum reference frame has not been used for these plots. These hadron response spectra exhibit a new scaling of the third kind.
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Fig. 15. Maxima of the hadron non-charge exchange (NCX) response functions F (y) from experiments with good energy resolution are fit to determine their maxima, as described in the text. These maxima are not found to lie at y = 0, and the differences from zero are plotted, with a negative shift indicating a maximum at y < 0. A systematic trend is noted for the four different beam species. The energy shift parameter fit to electron spectra [20] would force a shift of zero in a corresponding plot.
responses lie above those for longitudinal (charge) electron scattering, for carbon, as shown in Fig. 14 using the formulations of the present work with the data of Ref. [31] at q = 550 MeV/c. The apparent discrepancy between the hadron and electron comparisons in Figs. 13 and 14 is due to several small effects, with the data points shown in Fig. 14 [31] only one of the many examples analyzed for the universal fL curve in [20]. The larger in-medium total cross sections for the K + , with its long mean free path in nuclei, have been widely noted [32], while the other hadrons demonstrate smaller in-medium total cross sections. 6. Comparisons and medium effects The several hadronic probes at the energies and momentum transfers used in Fig. 14 couple to bound nucleons with a range of isospin sensitivities. These can be compared by the magnitudes of the fractional differences between hadron–proton and hadron–neutron differential cross sections and their sums at the angles corresponding to q near 575 MeV/c, as used in Fig. 14. These are 0.33, 0.27, 0.13, 0.14, 0.17, 0.17 and 0.21 for 392 MeV protons, 558 MeV protons, 795 MeV protons, 1014 MeV protons, 705 MeV/c K + , 950 MeV/c π − and 500 MeV π + , respectively [22]. No pattern determined by these isospin couplings is to be discerned in Fig. 14. As noted for a wider range of meson scattering responses in Ref. [16], all these hadron responses lie somewhat above the charge scattering responses for electron scattering, which have a unit difference-over-sum ratio for neutrons and protons. The hadron scattering probes are predominantly isoscalar in their coupling to nucleons for q near 575 MeV/c, and have small spin transfer cross sections [22].
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The maxima of the present hadron scattering responses do not line up on y = 0, while electron scaling responses do maximize very near y (or Y or ψ ) = 0 [19,20]. The separation energies SE used for the present analysis are taken to be the same as used for electrons, as listed in Table 1. Fig. 15 shows the differences between the observed maxima and y = 0 for values of q near 500 MeV/c. These maxima are determined by forcing a parabolic Fermi gas shape to the F (y) responses near the maxima, using values of kF as listed in Table 1. Although uncertainties are large, the shift towards more negative values of y does increase smoothly with target mass. Note that the separation energies used (Table I) change with mass, as also found by fits to electron spectra in [20], but this does not account for the shifts observed for hadrons. Note that the fraction of Aeff to A is such that larger nuclei are probed a lower densities than lighter samples. If Aeff is used to count nucleons beyond some radius, a typical total cross section of 42 mb as used at 795 MeV gives access to densities (relative to central densities) of 0.23, 0.135, 0.096 and 0.087 for A = 12, 40, 90 and 208, respectively. The multistep scattering used for the 19.2 GeV/c analysis [14] is a means for the nominal single-scattering quasifree peak to occur at lower energy losses for heavier nuclei, as noted in Fig. 15. Another means to move the maximum of F (y) to zero would be to apply an effective mass for the struck nucleons greater than that in free space, as might be expected in the nuclear surface [33] where the quasifree scattering is localized. The magnitudes of the scaling responses of hadrons on nuclei are influenced in two ways by their interactions with bound nucleons, within the quasifree scaling context. Differential in-medium proton–nucleon cross sections were considered by Ray [34], as applied to coherent proton–nucleus elastic scattering. Especially for the S = T = 0 amplitudes, very small changes in the nuclear medium are indicated near q = 500 MeV/c for beam energies of 500 MeV or above. This is confirmed in the present study of scaling of the first kind, as noted for light nuclei across a range of scattering angles. The agreement between meson and nucleon scattering responses further indicates that the roles of exchanged mesons do not change within nuclei; the spin-zero meson probes may not exchange single pions to interact with nucleons, and by q = 500 MeV/c shorter range exchanges are expected to dominate. Proton–nucleon in-medium total cross sections enter the scaling analysis in a manner that influences most directly the scaling of the second kind, since Aeff does not change uniformly with nuclear mass as the in-medium total cross sections change. This mass dependence of scaling was examined for 795 MeV proton scattering in Ref. [29], with the conclusion that scaling of the second kind can be obtained with proton–nucleon total cross sections within nuclei smaller than found in free space by a factor β = 0.79 (0.10). This is consistent with an extrapolation of the theoretical curves of Fuchs et al. [35], where such total cross sections are reduced by 10–20% at half nuclear density. The discrepancy with measured electron charge scattering responses is diminished by use of smaller in-medium total cross sections as well. The lower panels of Figs. 9 and 10 show scaling responses of the second kind at 558 and 795 MeV with Aeff computed by use of in-medium total cross sections taken to be about 75% of the free-space values. The scatter among the array of proton data from 6 Li to lead is much smaller than found in the upper panels using free-space total cross sections, for closer scaling of the second kind. Similar effects are noted at 9.5 and 19.2 GeV/c. 7. Conclusions Three kinds of scaling have been examined using these analyses of a wide range of hadron data. Scaling of the first kind is that in which a given proton energy and nuclear sample give
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the same response F (y) for a range of angles (or momentum transfers). This tests the conditions for scaling and the use of free-space dσ/dΩ across this range of angles. The hadron data fail to scale in general in this first fashion, save for high beam energies and light nuclei. The extreme case of 19.2 GeV/c [14] suggests that this failure is due to second scattering effects, notably at larger angles and for heavier nuclei. Scaling of the second kind examines responses f (Y ) at a given beam energy and scattering angle for a range of nuclear samples, thus testing the validity of the Aeff approximation and the total cross sections that determine this parameter. This scaling can be obtained, in concord with that noted with electron charge scattering, with altered in-medium hadron–nucleon total cross sections. If both first and second kinds of scalings were to be obtained with hadrons, this would be equivalent to the ‘superscaling’ found for electrons [19]. Comparison of all hadronic scatterings at similar momentum transfers with altered in-medium total cross sections gives a new common scaling response of the third kind, with no analogy in electron scattering. Only for carbon is a suitably broad range of data with hadronic probes usable for this demonstration. With β near 0.75, hadron NCX responses are larger than charge electron scattering responses by a factor near 1.5. Continuum electron scattering experiments have demonstrated that a simple and weak scattering probe finds simple and universal scaling responses from complex nuclei, with no indications of nucleons within nuclei being different from those in free space [4]. In the same spirit, this work has examined a large body of proton and meson continuum scattering spectra, seeking similar scaling relations when the interactions of the projectiles are not simple and not weak, to determine if such quasifree scattering can be a useful measure of in-medium hadron–nucleon cross sections. Only a fraction of the nucleons within a nucleus will scatter the hadron beam once and only once, as modeled here in a Glauber calculation. Deviations from scaling of the first kind are observed for all but the lightest nuclei; previous analysis of the highest energy case indicates that multinucleon processes are strong there even for negative y. The present work also indicates this explanation for lower energies with smaller deviations from scaling. Scaling of the second kind for the maxima of hadron quasifree spectra can only be obtained with smaller in-medium hadron–nucleon total cross sections (save for the known K + case), consistent with theoretical expectations in general [35]. In summary, the simple concept of direct observation of hadron–nucleon elastic scattering on nucleons within complex nuclei has been tested through scaling analyses of scattering data. Some cases do exhibit such a simple process, while other examples document systematic violations of scaling requiring further study. The following paper [36] will extend these y-scaling analyses to hadron continuum spectra with charge exchange, using very similar methods and another wide range of data. Acknowledgements The author wishes to thank M. Nakano for permission to cite unpublished data, G.F. Steyn for tables of cross sections and S.G. Mashnik for a most useful critical reading. This work was supported in part by the USDOE. References [1] [2] [3] [4]
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