Understanding Nuclear Deformation and Shape Determination
Dive into the intricate world of nuclear physics as we explore the challenges and techniques involved in determining prolate or oblate shape through low-energy inelastic scattering. Discover the significance of nuclear deformation and the complexities of deciphering the sign of deformation. Unravel the reorientation effect and distorted-wave Born series, shedding light on the self-coupling effects of excited states. Join us in this journey through the fascinating realm of nuclear structure analysis.
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Determination of Prolate or Oblate Shape via Low-energy Inelastic Scattering S. WatanabeA,B, Y. SuzukiC, M. KimuraB, K. OgataD,C NIT, Gifu Coll.A, RIKENB, RCNPC, Kyushu Univ.D ? ?2< 0 ?2> 0 2025/05/29 The 29thInternational Nuclear Physics Conference DCC, Deajeon, Korea
Nuclear Deformation and its Determination + 41 The magnitude of ?2has been deduced from many experiments. The systematic determination of ?2is an important subject in nuclear physics. + 21 ?2 + 0gs Examples: B(E2): S. Raman et al., Atomic Data and Nuclear Data Tables 78, 1 (2001). (inel): T. Motobayashi, Phys. Lett. B 346, 9 (1995). (reac): M. Takechi et al., Phys. Rev. C 90, 061305 (2014). Oblate Prolate ? Despite the numerous studies on ?2, the determination of its sign is still a challenging task. ?2< 0 ?2> 0
Difficulty in determining the sign of deformation: Case of ????? Deformed potential between and T T ?: Relative coordinate between and T ?: Internal coordinate of T ? ?,? = ? ? + ? ?,? ? ? Example: Deformed Woods-Saxon Potential (1storder) ?? ? ?? ?0+?0 1+exp? ?0 ?20? ? ?,? = ?2?0 ? ? = , ? cos? = ? ? Radial part of ? Assumption Only the quadrupole deformation ?2 Prolate deformation ?2= ?+> 0 Oblate deformation ?2= ? < 0 Same magnitude ?+= ? Only the 0+ to 2+transtion MeV ? ?2?0?? ? ?? ?+ ? fm
Difficulty in determining the sign of deformation: Case of ????? Deformed potential between and T T ?: Relative coordinate between and T ?: Internal coordinate of T ? ?,? = ? ? + ? ?,? ? ? Example: Deformed Woods-Saxon Potential (1storder) ?? ? ?? ?0+?0 1+exp? ?0 ?20? ? ?,? = ?2?0 ? ? = , ? cos? = ? ? DWBA: Distorted wave Born Approximation Inelastic T-matrix elements for ? in DWBA ??? ? 2+ ?? ?0+ + 0+ 2 ?? = ?2+ = ? ?? =???+ ? ??? ? Opposite phase Identical Identical ???+ ? 2 2+ ??+?0+ + 0+ = ? ??+ ??+= ?2+ Impossible to distinguish between ?+and ? in DWBA.
Reorientation Effect and Distorted-Wave Born Series Distorted-Wave Born Series (Exact inelastic T matrix) Reorientation effect (RE): Self-coupling effect of excited states 1 2 2+ ????2 = ??? ?2 + ??? ?2 1 1 1 ??? ?? +?? ??? ? = ??? ?+ ??? ?2 = ?? Reorientation 2 2 2 ??? ?+? ?? +?? 2= +??? 2 2= ?? ??? ? ?+ ??? ?2 2+ 1: Distorted-wave Green s function ?+= ? ? ? ?+ ?? ??= 2, 1,0,1,2 Independent of ?2 0+ ??= 0 1and ??? 2can The coherent sum of ??? carry the information on the sign of ??.
Purpose and Method Purpose We propose a method for determining the sign of quadrupole deformation using inelastic scattering and demonstrate its effectiveness. Method Standard coupled-channel method based on the macroscopic model inelastic scattering data Well established theoretically and experimentally 154Sm Example: +154Sm at 50 MeV ? ? Elastic and inelastic scattering data are available 154Sm is often considered to be a prolate nucleus ?2> 0
Coupled-channel method based on the macroscopic model Schr dinger equation 154Sm ? ??+ ? ?,? + ? ? ???,? = 0 ? Total wave function ? = 0 or 2 ?0? ?0? ? ??+?? 2 ?0?,? =??0 ?? ?? ? 2? excited state ???? ? 0? ground state ?? ??+ ? Couple-channel equation Channel: ? = ?,? ?0= ?0,? = 0 (Initial channel) 2 2? ?2 ??2+ 2 ? ? + 1 ?2 ?0? = ?0(?) + ???? ???? ??? ? ?? 2? Coupling potential: ??? ? = ???? ?? ?,? ?? ?? ? ?,? ? ? Boundary condition ???0: S matrix Elastic and Inelastic cross sections ?0? ?? + ?? ??0? ???0 ??0/?????0?? ???
Determining the sign of deformation from experimental data
Elastic cross section and the determination of the potential Elastic scattering cross section for +154Sm at 50 MeV First, we fix the potential parameters (?0,?0,?0,?) by fitting to the elastic scattering cross section. ?0= 65.9 MeV,?0= 27.3 MeV, ?0= 1.44 154 Exp: D. L. Hendrie et al., Phys. Lett. B 26, 127 (1968). 1 3fm,? = 0.70 fm The result with ?2= 0reproduces the experimental data except for the strong oscillation at backward angles. This oscillation diminishes when the 2+ coupling is considered.
Determining the Sign of ?2 from the Experimental Data Next, we vary ?2 from negative to positive values to determine the optimized deformation parameters: ? optand ?+ Inelastic scattering cross section for +154Sm at 50 MeV opt. opt= +0.25 is in good optcan The result with ?+ agreement with the experimental data from the forward to the backward angles. The result with ? optdeviates from the data even at the forward angle. Slope of the cross section Position of the diffraction minimum Only ?+ reproduce the data The sign of deformation can be determined by analyzing the RE in low-energy inelastic scattering.
Summary We have proposed a method for determining the sign of quadrupole deformation (?2) using low-energy inelastic scattering data and demonstrated its effectiveness. Our approach is the standard coupled-channel method based on the macroscopic model. We utilize the nuclear reorientation effect as a probe sensitive to the sign of deformation. We apply this method to the realistic case ( +154Sm at 50 MeV) and numerically confirm its effectiveness. The broad applicability of inelastic scattering will make this approach a powerful tool to study shapes of nuclei, especially unstable nuclei. S. Watanabe, Y. Suzuki, M. Kimura, and K. Ogata, Phys. Rev. C 110, 034618 (2024).
Numerical demonstration of the reorientation effect
Reorientation Effect: One- and Two-Step Calculations We will clarify the main part of the RE on the cross section. Inelastic scattering cross section for +154Sm at 50 MeV Distorted-Wave Born Series 1 2 2+ ????2 = ??? ?2 + ??? ?2 No difference is observed for ? in the 1step calculation (DWBA). Clear difference is seen in the 2step calculation. 2acts destructively for ?+and constructively for ? ??? 2is the primary term of ??? the reorientation effect.
Channel-Coupling Effect and Reorientation Inelastic scattering cross section for +154Sm at 50 MeV Distorted-Wave Born Series 1 2 2+ ????2 = ??? ?2 + ??? ?2 The CC effects merely smoothen the cross section. Without the RE, ? yield exactly the same cross section. Even-order perturbations vanish. ????+ = ???? is realized. Therefore, ?? ?+ ? . ? = ?? ? With the RE, even-order terms interfere differently for ? , and we can determine the sign.
