Orientation Dichroism Effect of Proton Scattering on Deformed Nuclei

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Li OU and Zhi-Gang XIAO. Orientation Dichroism Effect of Proton Scattering on Deformed Nuclei[J]. Chinese Physics C.
Li OU and Zhi-Gang XIAO. Orientation Dichroism Effect of Proton Scattering on Deformed Nuclei[J]. Chinese Physics C. shu
Received: 2020-03-09
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Orientation Dichroism Effect of Proton Scattering on Deformed Nuclei

  • 1. College of Physics and Technology and Guangxi Key Laboratory of Nuclear Physics and Technology, Guangxi Normal University, Guilin 541004, China
  • 2. Department of Physics, Tsinghua University, Beijing 100084, China

Abstract: Proton-induced scattering on 238U nuclei of spheroidal deformation at beam energies above 100 MeV has been simulated using an improved quantum molecular dynamics model. The angular distribution of the deflective protons depends sensitively on the orientation of the symmetrical long axis of the target nuclei with respect to the beam direction. As a result, in reverse kinematic reactions, the orientation dichroism effect is predicted that the absorbtion rate of the 238U beam on a proton target discerns the parallel orientation to the perpendicular orientation of the deformed 238U.


    1.   Introduction
    • The scattering of light or particles on an object of arbitrary shape is of significant importance in many fields in physics. In optics, anisotropic structure of microscopic particles causes birefringence which alters the polarization state of the passing light, leading to the improvements of polarization microscopy in extensive applications [14]. In astrophysics, the discrete dipole approximation (DDA) method has been developed to compute the extinction and scattering of the star light by the interstellar grains, which are unnecessarily in spherical shape [5, 6]. In nano sciences, large local-field enhancement and light-scattering efficiencies have been demonstrated in the nanorod compared to that in nanosphere metal particles, making the former interesting in optical applications [7].

      In nuclear physics, experiments of the scattering of protons and alpha particles on even-even nuclei were conducted in 1970s, leading to the discovery of the large multipole deformation of various even-even nuclei on the $ 0^+ $ ground state [1113]. For reactions induced by very exotic nuclei, it has been found that the deformation causes the extension of matter distribution and enhances the total reaction cross section in Glauber model analysis [14]. Very recently, the deformability of 238U has received increasing attention. In the synthesis of super heavy elements (SHE) via multi-nucleon transfer mechanism in $ ^{238} {\rm{U}}$+$ ^{238} {\rm{U}}$, the number of transferred nucleons and the survival probability of the giant system formed depends sensitively on the geometric configurations of the projectile and the target, according to the prediction by TDHF theory [15] and the improved quantum molecular dynamics (ImQMD) model [16]. Very recently, the significant orientation effect of the octupole deformed nuclei in fusion reactions has been demonstrated by calculations based on transport model [17]. In relativistic heavy ion collisions (HICs) at RHIC energies, the deformation and orientation effect of the colliding nuclei has been noticed as well. It has been shown that the tip-tip configuration of $ ^{238} {\rm{U}}$+238U collision is favorable in achieving the largest stopping, while the body-body collisions provide the way to study the initial geometry effect of the flow formation [1822]. The STAR collaboration has succeeded in selecting different overlap geometry configurations in $ ^{238} {\rm{U}}$+238U in extremely central collisions [23].

      On the other hand, the idea of making use of the deformed nuclei for producing highly compressed nuclear matter or enhancing the probability of synthesizing SHE will likely be handicapped unless the orientation of 238U can be determined before the collision occurs. For such even-even nucleus at ground state, however, the spin is zero and the conventional way to polarize the nuclei using magnetic field can hardly work. Thus, before one can take advantage of the deformed nuclei, more efforts are required to study the deformation and orientation effect of the reactions involving the deformed nuclei.

      In this letter, we revisit the scattering process involving the deformed nucleus. Instead of the reactions with both projectile and target being deformed, our motivation is to study the peripheral scattering of a light particle on a deformed nucleus to discern the deformation and orientation effect of the later in the scattering. It is expected that because of the small mass, the projectile is deflected easily and the deflection behavior depends sensitively on the deformation and orientation of the target nuclei. If such scenario is true, the deflected light projectile will carry clear information of the orientation of the deformed nucleus and expectedly bring novel implications in the usage of deformed nuclei. Contrast to the extremely central collisions, the peripheral scattering has an additional benefit that neither partners of the colliding system will be disintegrated.

      In order to circumvent the complication brought by the structure of the beam nucleus, the proton induced scattering on 238U is investigated. We use the conventional definition of the coordinator system. The incident direction of the projectile is defined as the z axis and the reaction plane is in the x-z plane in laboratory. Figure 1 depicts schematically three typical configurations in the scattering of protons on $ ^{238} {\rm{U}}$. The symmetrical long axis of 238U is parallel to x, y and z axis, marked by ${\rm{C}} _{{\rm{x}}} $, Cy and ${\rm{C}} _{{\rm{z}}} $, respectively. The shape of the 238U nucleus is represented by its density distribution projected to x-z plane with quadrupole deformation $ \beta_2 = 0.236 $ (see text later). It can be viewed that at a given impact parameter of peripheral scattering, the flight path length $ L_{\rm f} $ of the proton experiencing the field of 238U are different in the configurations of ${\rm{C}} _{{\rm{x}}} $, Cy and ${\rm{C}} _{{\rm{z}}} $. With the specific impact parameter $ b = 9 $ fm as shown in the figure, $ L_{\rm f} $ in Cx is moderate and the proton goes the deepest into the target, while $ L_{\rm f} $ in Cy is the shortest and the proton just grazes the target. The different path leads to different deflection of the projectile in these special cases, as shown later.

      Figure 1.  (Color online) Schematic view of three typical configurations, Cx, Cy and Cz in x-z plane in the scattering of p+238U. The contour depicts the density distribution of 238U projected onto x-z plane.

    2.   Theoretical Model Description
    • The ImQMD model is an extended version of the quantum molecular dynamics model with improvements that make it suitable for describing the transport process for light particle induced reactions as well as for HICs in wide energy range. The model has been applied in the nucleon-induced reactions [24, 25] and the deuteron-induced reactions [26, 27], leading to the prediction of the isovector orientation effect of polarized deuteron scattering. For the details of the ImQMD model, we refer to the literatures [2830]. Similar with the above studies, the calculations done in this letter only takes the beam energies above 100 MeV, at which the de Broglie wave length of the beam is much smaller than the radius of the target nucleus and hence the effect of the target granularity (including its size, shape and surface etc.) is exhibited. In addition, the validity of the transport model as a semi-classic approach is kept. The top energy treated here is 300 MeV, above which the inelastic channels of $ NN $ scattering are open and contribute increasingly.

      The initialization of the projectile and target is important to stabilize the properties of the two nuclei in simulating the reaction process. Particularly, the aim of this work is to study the orientation effect of the heavy deformed nuclei in proton-induced scattering, special attention should be paid to the shape and the orientation of the initial nuclei. For this purpose, the nuclei are sampled according to the density distribution with deformed Fermi form which reads

      $ \rho(r,\theta) = \rho_0\left[ 1+\exp\left( \frac{r-R(\theta)}{a}\right) \right]^{-1},$


      where $ a = 0.54 $ fm and

      $ R(\theta) = R_0 \left[ 1+\beta_2Y_{20}(\theta)+\beta_4Y_{40}(\theta) \right], $


      where $ Y(\theta) $ is the spherical harmonics. $ R_0 = 1.142A^{1/3}- 0.60 $ fm for proton distribution, and $ R_0 $ is multiplied by a factor $ 0.93(N/Z)^{1/3} $ for neutron distribution to get the reasonable neutron skin. Only the quadrupole deformation is considered in this work. The quadrupole deformation parameter is taken as $ \beta_2 = 0.236 $ for $ ^{238} {\rm{U}}$. With this scheme, the symmetrical long axis of the sampled nuclei are parallel to z axis initially. To start a scattering event, the nucleus can be rotated around its center of mass to make the symmetrical long axis parallel to x, y axis, or randomly directed. In figure 1, the initial density distributions of nucleons projected to x-z plane in 238U simulated by ImQMD model in ${\rm{C}} _{{\rm{x}}} $, Cy and Cz configurations are presented. The shape and the density distribution are examined until 200 fm/c, and it is found that they keep both stably well enough to satisfy the investigation request, because the incident proton already leaves the nuclear field of the target far enough after 200 fm/c in the peripheral scattering at 100 MeV incident energy.

    3.   Results and Discussions
    • To visualize the geometry effect of the 238U target nuclei, we first fix the impact parameter in the simulations. The incident proton is in X-Z plane with the target being fixed at the origin point and the impact parameter $ b = x $ when the proton crosses the X axis, while the orientation of the long axis of 238U can be varied to investigate the geometrical effect of the scattering. As the first step the orientation of $ ^{238} {\rm{U}}$ target is fixed in laboratory frame by hand. Figure 2 presents the angular distribution of the scattered proton at impact parameter $ b = 9 $ fm for the three geometrical configurations ${\rm{C}} _{{\rm{x}}} $, Cy and Cz in p+$ ^{238} {\rm{U}}$ at 200 MeV beam energy. Nuclear attraction dominates at this impact parameter since the scattering angle are negative as shown in Fig. 1). To assure that only the elastic or quasi-elastic scattering occurs, it is required that the total multiplicity $ M_{\rm t} = 2 $ and one of the outgoing particles must be proton with energy close to beam energy. For comparison, the case that the deformed target nucleus is randomly orientated is also calculated, as marked by $ rand $ in figure 2 and later. Generally, it is shown that the scattering angle decreases with the beam energy. More interestingly, it is commonly shown that at all beam energies the deflection angular distributions exhibit obvious difference in the ${\rm{C}} _{{\rm{x}}} $, Cy and Cz configurations. The smallest deflection is seen in ${\rm{C}} _{{\rm{y}}} $, i.e., the proton passes near the waist of the target with the symmetrical long axis being perpendicular to the beam direction, while the largest deflection occurs in Cx configuration where the proton traverses the tip of the target and experiences strongest nuclear attraction. The deflection angular distribution in Cz configuration is situating between Cx and ${\rm{C}} _{{\rm{y}}} $, and is similar with the $ rand $ case in which the direction of the symmetrical long axis of the target is randomly aligned in laboratory. The difference between the peak of angular distribution for Cx and Cy cases is larger than $ 10^\circ $ which is a large measurable quantity.

      Figure 2.  (Color online) The angular distribution of the deflective proton in p+238U with b = 9 fm at 100 (left), 200 (middle) and 300 (right) MeV beam energies for the three configurations Cx, Cy and Cz, respectively.

      Apparently the discrimination among the three configurations depends on the impact parameter. When the impact parameter decreases, the collision becomes violent and the competition between nuclear attraction and Coulomb repulsion takes effect and changes the deflective behavior of the proton. Figure 3 presents the deflective angular distribution at various impact parameters for p+$ ^{238} {\rm{U}}$ at 200 MeV beam energy. It is shown that the cross section of the elastic or quasi-elastic scattering events increases with impact parameter b. At $ b = 7 $ fm, the total events are much less because nucleons or clusters may be knocked out from the target leading to the opening of inelastic scattering channel. Compared to $ b = 9 $ fm shown in figure 2, the peaks of the angular distribution at $ b = 8 $ fm in Cy and Cz configurations move to right side and situates close to the Cx and $ rand $ cases. It implies that the length of interaction path becomes similar in these configurations and geometric effect is disappearing when the impact parameter is reduced. This trend is even more pronounced at small impact parameter $ b = 7 $ fm. On contrary, when the impact parameter increases to $ b = 10 $ fm where the nuclear interaction becomes weaker, the configuration of Cx clearly stands out with the largest deflection while other configurations show commonly much smaller deflection peaking at very forward angle. It indicates that at large impact parameter, the deflection depends mainly on the length of flight path in the field of the target.

      Figure 3.  (Color online) The angular distribution of the deflective proton in p+238U with b = 7 (a), 8 (b), and 10 (c) fm at 200 MeV beam energy for the three configurations Cx, Cy and Cz, respectively.

      It must be pointed out that in real experiment the azimuth can not be identified if only the inclusive proton is recorded. We further investigate the effect of the orientation of $ ^{238} {\rm{U}}$ in the transverse plane. Fig. 4(a) presents the angular distribution of the deflected proton by varying the azimuthal angle of the long axis of $ ^{238} {\rm{U}}$ in the x-y plane. Here $ \phi_{\rm U} $ indicates the angle of the long axis of $ ^{238} {\rm{U}}$ relative to x axis in x-y plane. It is seen that when $ \phi_{\rm U} $ increases from 0 to $ 90^\circ $, the peak of the $ \theta_{\rm{lab}} $ distributions moves to left. When $ \phi_{\rm U} $ continues to increases from $ 90^\circ $ to $ 180^\circ $, it moves back to the right side. More careful survey reveals that the width of the distribution also changes. At $ \phi_{\rm U} = 90^\circ $, the width of $ \theta_{\rm{lab}} $ distribution is smaller than that at $ \phi_{\rm U} = 0^\circ $. Fig. 4 (b) further presents the distribution of $ \theta_{\rm{lab}} $ by varying the orientation of 238U with respect to (w.r.t.) the beam direction in the x-z plane, characterized by $ \theta_{\rm U} $ as the angle of the long axis of 238U relative to z axis. With $ \theta_{\rm U} $ increasing, the peak of the $ \theta_{\rm{lab}} $ distributions first moves to right when $ \theta_{\rm U}<90^\circ $ and moves back towards left when $ \theta_{\rm U}>90^\circ $. The largest deflection occurs at $ \theta_{\rm U} = 90^\circ $.

      Figure 4.  (Color online) The $\theta_{\rm{lab}}$ distribution of the deflective proton in p+238U with $b = 9$ fm at $E_{\rm p} = 200$ MeV by rotating the long axis of 238U in x-y plane (a) and x-z plane (b), respectively. The meaning of $\phi^x_{xy}$ and $\phi^z_{xz}$ is clarified in text.

      The different behaviors of the $ \theta_{\rm{lab}} $ distribution reveals the deformation and orientation effect in p+238U scattering that the deflection of the incident nucleon depends on the orientation of the symmetrical long axis of the deformed target nuclei. Conbining all the effects of varying the orientation of 238U and the impact parameter, it is the quest of interest to see whether one can tag a certain orientation of the target nuclei in the scattering if measuring coincidentally the scattered proton at certain angles. It is usually the experimental motivation to select the polar orientation, i.e., the polar angle of the symmetric axis of 238U w.r.t. the beam, without counting the azimuth, one shall integrate over the azimuthal angle. Fig. 5 presents the percentage of two different configurations that are selected as a function of the deflective angle of the proton in p+238U summing over $ 7\leqslant b\leqslant 10 $ fm. At the initial state, the long symmetrical axis of 238U is randomly and isotropically oriented. Then after the scattering, the events corresponding to different configurations are counted and plotted as the ordinate in Fig. 5. Here C denotes the events with the symmetrical axis of 238U being approximately perpenticular to the beam direction with a cut $ \eta\leqslant 10^\circ $ (a) and $ \eta\leqslant 30^\circ $ (b), where $ \eta $ is the angle of the long axis of 238U w.r.t. the x-y plane, while Cz denotes the events that the long axis of 238U meets an angle of $ \eta\leqslant 10^\circ $ (a) and $ \eta\leqslant 30^\circ $ (b) with the beam axis. Thus C and ${\rm{C}} _{{\rm{z}}} $ can be viewed as the parallel and perpendicular orientation of $ ^{238} {\rm{U}}$, respectively. It is evident from the figure that the percentage of C exhibits an increasing trend as a function of $ \theta_{\rm{lab}} $, opposite against to the decreasing trend in Cz but with much larger scattering cross section. It suggests that if the scattered proton is measured in coincidence, the scattering angle can be used to select statistically, not event-by-event, the scattering events with different weight on the two polar orientations of C and ${\rm{C}} _{{\rm{z}}} $.

      Figure 5.  (Color online) The percentage of various geometric configurations selected as a function of the deflective angle of the proton in 200 MeV p+238U with b = 8-10 fm.

      The deformation and orientation effect of the heavy target leads to an orienttion dichroism effect in reverse kinetic scatterings using 238U as incident beam. After the primary beam 238U with intensity $ I_0(\Theta_s) $ passes through a proton target with thickness t, the remaining intensity $ I(\Theta_s,t) $ of the 238U in the original beam direction can be written model-independently as

      $ I(\Theta_s,t) = I_0(\Theta_s)\left[1-\sigma(\Theta_s)\rho_{\rm n}(t)\right] $


      where $ \Theta_s $ is the polar angle of the long symmetric axis of 238U with respect to the beam direction. $ \Theta_s\approx0^\circ $ ($ 90^{\circ} $) denotes the parallel (perpendicular) orientation corresponding to Cz (${\rm{C}} _{\perp} $) in Fig. 5, respectively. $ \sigma(\Theta_s) $ is the scattering cross section of p+238U and $ \rho_{\rm n}(t) $ is the areal number density of the target nuclei. Since the scattering cross section of the parallel orientation is much less than that of perpendicular orientation, as summarized in Table 1, the 238U nuclei in parallel orientation is relatively less absorbed than in perpendicular orientation, leading to an orientation dichroism effect. It is expected the ansatz (3) is model independent and the orientation dichroism effect is universal in the mdium energy scattering induced by deformed nuclei.

      $\eta$ $\sigma$[Cz] (mb) $\sigma$[C] (mb) $\sigma$[all] (mb)
      $\le 10^{\circ}$ 24.6(8.4) 259(112) 1527(609)
      $\le 30^{\circ}$ 212(73.6) 748(317)

      Table 1.  Elastic and quasi-elastic (inelastic) scattering cross section with $ 7\leq b\leq10 $ fm for different configurations.

      We note some cautions here. The nucleons are treated as Gaussian packages in ImQMD model. The spin of the incident proton is not taken into account in the calculation and the results do not contain the effect arising from the spin projection of the proton. In addition, the resonance scattering channels coupled to the excited states of 238U are not included. Therefore, it is requested to perform further theoretical calculation of the elastic or quasi-elastic scattering of proton on deformed target within the framework based on quantum mechanics in order to reproduce the novel effect predicted in our study, for instance by the approach developed in [31]. From the experimental point of view, it is feasible in principle to observe the deformation and orientation effect in p+238U scattering where the target nuclei of 238U are excited to the low lying states of the yrast rotational band for which the angular momentum is nonzero. The angular correlation of the cascade $ \gamma $-rays from the rotational band then carries the information of the orientation of $ ^{238} {\rm{U}}$. If the angular correlation of the $ \gamma $-rays shows expectedly dependence on the laboratory angle of the deflected proton, it will be a signal of the orientation effect in p+238U scattering.

    4.   Summary
    • In summary, the proton-induced scattering on deformed even-even nuclei 238U with spheroidal shape has been simulated with the ImQMD model. By surveying three special geometrical configurations ${\rm{C}} _{{\rm{x}}} $, Cy and ${\rm{C}} _{{\rm{z}}} $, it has been found that the angular distribution of the deflective proton exhibits clear dependence on the orientation of the deformed nucleus with various impact parameters. Summing over all impact parameters in peripheral scattering at the whole azimuth, the intensities of the deflected proton split between the parallel and perpendicular orientations of 238U as a function of scattering angle in laboratory. It is suggested that the polar orientation of the deformed nucleus in the scattering can be characterized by imposing an angular condition on the deflected proton. In reverse kinetics, the different scattering cross section between the parallel and perpendicular configurations causes an orientation dichroism effect, inferring a novel method to produce polarized secondary beam of deformed nuclei with nonzero spin, to which our method of calculation can be extended without changing the conclusion.

      We thank Prof. Hui Ma, Prof. Chunguang Du and Prof. Pengfei Zhuang from Tsinghua University, Prof. Zhuxia Li and Prof. Zhaochun Gao from CIAE for their valuable discussions.

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