Mixing of isoscalar and isovector characteristics in the low-energy dipole mode

We investigated isospin splitting in low-energy dipole (LED) states of spherical nuclei such as 40 Ca, 90 Zr, 132 Sn, 208 Pb, and several <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>50</mml:mn></mml:mrow></mml:math> isotones using self-consistent Hartree–Fock plus random phase approximation calculations. Our analysis of isovector dipole (IVD) and isoscalar dipole (ISD) strengths, along with transition densities, reveals a clear energy-dependent relationship between IS and IV modes in 40 Ca and 90 Zr. For 208 Pb and 132 Sn, LED states show mixed IS + IV characteristics due to different neutron and proton shell structures. In <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:mrow><mml:mi>N</mml:mi><mml:mo>=</mml:mo><mml:mn>50</mml:mn></mml:mrow></mml:math> isotones, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"><mml:mrow><mml:mi>E</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:math> modes exhibit varying IS and IV properties with smooth transitions as neutron excess increases. Our results suggest that compressional ISD strengths could provide valuable insights into the slope parameter of the nuclear equation of state. The observed dependence on nuclear shell structures and neutron–proton correlations highlights the need for precise measurements and further research in nuclear physics.