A direct probe of the in-medium pn scattering cross section

Hard photon production from neutron-proton bremsstrahlung in intermediate energy heavy-ion reactions is examined as a probe of the in-medium pn scattering cross section within a transport model. Uncertainty of photon production probability pn-pngamma is cancelled out by using the ratio of hard photon spectra R_{12C+12C/p+n}(gamma) from two reactions. The in medium pn scattering cross section is constrained by using the ratio of hard photon production cross sections of proton-induced reactions p+12C and p+2H. A reduction factor sigma_pn^medium/sigma_pn^free of about 0.5 ~ 0.7 around saturation density is obtained by comparing with the existing experimental data.

the strong interactions in the final state. Ideally one expects more clean ways to study nucleon-nucleon scattering cross section. It is noted that the parityviolating electron scattering has been proposed to measure more precisely the size of the neutron-skin in 208 Pb [12]. Similarly to electrons, photons interact with nucleons only electromagnetically. Once produced they escape almost freely from the nuclear environment of nuclear reactions. Following the studies of using hard photon production to probe the symmetry energy [13], in this Letter, based on the Boltzmann-Uehling-Uhlenbeck (BUU) transport model, we report our results of using hard photons from neutron-proton bremsstrahlung in intermediate energy heavy-ion reactions as a direct probe of the in-medium pn scattering cross section.
Hard photon production in heavy-ion reactions at beam energies from about 10 to 200 MeV/nucleon had been extensively studied both experimentally and theoretically [14,15,16]. For instance, the TAPS collaboration carried out a series of comprehensive measurements at various experimental facilities (GSI, GANIL, KVI) studying in detail the properties of hard photons in a large variety of nucleus-nucleus systems in the range of energies spanning E lab ≈ 20−200 MeV/nucleon [17]. Theoretically, it was concluded that the neutron-proton bremsstrahlungs in the early stage of the reaction are the main source of high energy γ rays. Further, it was demonstrated clearly that the hard photons can be used to probe the reaction dynamics leading to the formation of dense matter [18,19,20,21,22]. Another favorable factor of using hard photons to probe the pn scattering cross section is that effects of the nuclear EoS on the hard photon production was found small [23]. One of the major uncertainties of hard proton studies is the input elementary pn → pnγ probability p γ which is still rather model dependent [24,25,26,27,28,16]. The recent systematic measurements of the pn → pnγ cross sections with neutron beams up to 700 MeV at Los Alamos and the subsequent state-of-the-art theoretical investigation may help to improve the above situation significantly in the near future [29,30].
Since the photon production probability is so small, i.e., only one photon is produced in roughly a thousand nucleon-nucleon collisions, a perturbative approach has been used in all dynamical calculations of photon production in heavy-ion reactions at intermediate energies [14,16]. In this approach, one calculates the photon production as a probability at each proton-neutron collision and then sum over all such collisions over the entire history of the reaction. As discussed in detail earlier in Ref. [16], the cross section for neutronproton bremsstrahlung in the long-wavelength limit separates into a product of the elastic pn scattering cross section and a γ-production probability. The probability is often taken from the semiclassical hard sphere collision model [14,15,16]. The single differential probability reads  Where ε γ is energy of emitting photon, β i and β f are the initial and final velocities of the proton in the proton-neutron center of mass frame. We notice that other expressions derived theoretically involving more quantum-mechanical effects exist in the literature, see, e.g., [24,25,26,27,28]. For a comparison we thus also use the prediction of the one boson exchange model by Gan et al. [27] where y = ε γ /E max , α = 0.7319 − 0.5898β i , and E max is the energy available in the center of mass of the colliding proton-neutron pairs. As noticed already in Ref. [27,13], the single differential probabilities p a γ and p b γ from the two models give quite similar but quantitatively different results especially near the kinematic limit where the photon production with p a γ is significantly higher than that with p b γ . As discussed in the paper of Gan et al. [27], compared with the result using the semiclassical expression, the quantum formula p b γ reduces proton production evidently near the kinematic limit. While we are not aiming at reproducing any data in this exploratory work, it is necessary to first gauge the model by comparing with the available data. Shown in Fig. 1 are the BUU calculations with both p a γ and p b γ and the experimental data for the inclusive cross sections of hard photon production in the reaction 12 C+ 12 C [13]. It is seen that both calculations are in reasonable agreements qualitatively with the experimental data, especially at higher beam energies.
The agreement is at about the same level as previous calculations by others in the literature [19,21,27]. It is noticed that the uncertainty in the elementary pn → pnγ probability leads to an appreciable effect on the inclusive γ-production in heavy-ion reactions. From Fig. 1, we can clearly see that the quantum formula p b γ seems more suitable for describing energetic photon production in intermediate energy heavy-ion reactions. It was noted that for nuclear bremsstrahlung, a strong suppression or coherence of the bremsstrahlung cross section were observed in comparison with predictions of transport models that include bremsstrahlung on the basis of quasi-free nucleon-nucleon collisions [31,32]. In this study, we use the BUU transport model [33] with an isospin-dependent in-medium reduced nucleon-nucleon (NN) cross section [4,6] to treat the in-medium bremsstrahlung semiclassically [34]. The energy and isospin dependent free-space NN cross sections σ f ree N N are taken from the experimental data. Another important input to the transport model is the momentum-and isospin-dependent single nucleon potential as given in Ref. [35]. The isoscalar part of the single nucleon potential was shown to be in good agreement with that of the variational many-body calculations and the results of the Brueckner-Hartree-Fock approach including three-body forces and the isovector potential is consistent with the experimental Lane potential [36].
Since the energetic photons are produced in pn collisions, it is quite obvious that pp or nn collisions do not influence the production of photons. We can understand this way: (1) The studied hard photon is produced via pn → pnγ, so it reflects pn scatterings directly. The nn or pp scatterings may affect pn scatterings, and then affect hard photon production. But they are both secondary effects. (2) The increased/decreased pp or nn cross sections may also increase/decrease the collision number of protons and neutrons, thus may increase/decrease the pn collision number. But, on the contrary, the larger/small pp or nn cross sections may reduce/enlarge the collision number of proton with neutron (or neutron with proton). The secondary effects are thus cancelled out each other. We in fact checked this deduction by BUU calculations and find that the hard photon production is really only sensitive to pn cross section, while not sensitive to pp or nn cross sections.
To reduce uncertainties of the input elementary pn → pnγ probability while using hard photon to probe in-medium pn cross section, we provide Fig. 2, the ratio of hard photon spectra in the reactions 12 C+ 12 C and p+n at a beam energy of 140 MeV/nucleon. The spectra ratio R12 C+ 12 C/p+n (γ) reads (3) It is seen that the R12 C+ 12 C/p+n (γ) is quite sensitive to the pn scattering cross section while not sensitive to theoretical formulas used. This ratio reduces the uncertainties of the theoretical elementary pn → pnγ probability maxi- mally. The change of the pn scattering cross section leads to R12 C+ 12 C/p+n (γ) a sensitivity of about 60%. Like in many experiments searching for minute but interesting effects, ratio of observables from two reactions can often reduce not only the systematic errors but also some unwanted effects. At least theoretically, the uncertainty of the γ-production probability gets almost completely cancelled out in the ratio of photon spectra here. From our BUU calculations, the defined quantity R12 C+ 12 C/p+n (γ) does not dependents on the input γproduction probability with a high credibility of 95%. Also in this definition we assume input γ-production probability in matter is the same as that in vacuum since photons interact with hadrons only electromagnetically.
The photon production in 12 C + 12 C is determined by pn colliding number and the input elementary pn → pnγ probability, the ratio R12 C+ 12 C/p+n (γ) thus only depends on the pn colliding number in the reaction 12 C + 12 C at certain energies. While the pn colliding number in a reaction depends on the pn cross section in matter. The spectra ratio R12 C+ 12 C/p+n (γ) is therefore a direct probe of the in-medium pn scattering cross section essentially free of the uncertainties associated with both the elementary photon production and the nn and pp scattering cross sections. Compared with other probes of nucleonnucleon scattering cross sections, such as nuclear flow and nuclear stopping, hard photon production directly affects the pn scattering cross section. Practically, besides cosmic-radiation background, one needs to consider photons from π 0 and fragment decays in the data analysis [37].
While comparing with the experimental data, we find that there are few existing data to use. As a rough comparison, we did simulations of proton induced reactions on 12 C and 2 H targets at a beam energy of 140 MeV/nucleon as shown in Fig. 3. Here σ p+ 12 C /σ p+ 2 H reads which in fact is the ratio of R p+ 12 C/p+ 2 H (γ) with different impact parameters.
In the calculations we use simple fermi-momentum as nucleonic momentum in deuteron and 12 C. And we find that photon production cross section σ p+ 2 H is not sensitive to the distribution of nucleonic momentum in deuteron [38,39,21]. We also assumed there is no medium effect on photon production from p+ 2 H. The reference reaction p+ 2 H thus plays roughly the same role as pn collision. We define the reduction factor Rd = σ medium pn /σ f ree pn . From Fig. 3 we can see that the experimental data are roughly within the range of Rd = 0.5 and 0.7 settings of our model. This reduction scale is somewhat larger than the Brueckner approach calculations [40,41,42]. Experimentally, heavy-ion collisions with N ∼ Z nuclei (to cancel the effect of symmetry energy) of symmetric system and p + n collision are more suitable to give constraints on the in-medium pn scattering cross section by using hard photon production.
In conclusions, we did an exploratory study about effect of the pn scattering cross section on the production of hard photons from intermediate energy heavy-ion reactions using a perturbative approach within the BUU transport model. The ratio of hard photon spectra R12 C+ 12 C/p+n (γ) is not only approximately independent of the uncertainties of nn, pp cross sections and the theoretical elementary pn → pnγ probability, but also quite sensitive to the pn scattering cross section. Compared with other probes of nucleon-nucleon scattering cross sections, hard photons are completely free of final state strong interactions, directly reflect the magnitude pn scattering cross section and are quite sensitive to the pn scattering cross section. Through comparing with existing experimental data, we obtain a reduction factor σ medium pn /σ f ree pn of about 0.5 ∼ 0.7 around saturation density. Heavy-ion collisions with N ∼ Z nuclei of symmetric system and p+n collision are needed to further constrain the in-medium pn scattering cross section at different densities and nucleonic momenta.