We introduce a new approach to a century old assumption which enhances not only planetary interior calculations but also high pressure material physics. We show that the polytropic index is the derivative of the bulk modulus with respect to pressure.
We then augment the traditional polytrope theory by including a variable polytrope index within the confines of the Lane-Emden differential equation. To investigate the possibilities of this method we create a high quality universal equation of state, transforming the traditional polytrope method to a tool with the potential for excellent predictive power. The theoretical foundation of our equation of state is the same elastic observable which we found equivalent to the polytrope index, the derivative of the bulk modulus with respect to pressure. We calculate the density-pressure of six common materials up to 10$^{18}$ Pa, mass-radius relationships for the same materials, and produce plausible density-radius models for the rocky planets of our solar system. We argue that the bulk modulus and its derivatives have been under utilized in previous planet formation methods. We constrain the material surface observables for the inner core, outer core, and mantle of planet Earth in a systematic way including pressure, bulk modulus, and the polytrope index in the analysis. We believe this variable polytrope method has the necessary apparatus to be extended further to gas giants and stars. As supplemental material we provide computer code to calculate multi-layered planets.
The differential cross section and the analyzing power are calculated for elastic scattering of $^6$He from a proton target using a microscopic folding optical potential, in which the $^6$He nucleus is described in terms of a $^4$He-core with two add
itional neutrons in the valence p-shell. In contrast to previous work of that nature, all contributions from the interaction of the valence neutrons with the target protons are taken into account.
Elastic scattering observables (differential cross section and analyzing power) are calculated for the reaction $^6$He(p,p)$^6$He at projectile energies starting at 71 MeV/nucleon. The optical potential needed to describe the reaction is derived desc
ribing $^6$He in terms of a $^4$He-core and two neutrons. The Watson first order multiple scattering ansatz is extended to accommodate the internal dynamics of a composite cluster model for the $^6$He nucleus scattering from a nucleon projectile. The calculations are compared with the recent experiments at the projectile energy of 71 MeV/nucleon. In addition, differential cross sections and analyzing powers are calculated at selected higher energies.
A global nucleon-nucleus optical potential for elastic scattering has been produced which replicates experimental data to high accuracy and compares well with other recently formulated potentials. The calculation that has been developed describes pro
ton and neutron scattering from target nuclei ranging from carbon to nickel and is applicable for projectile energies from 30 to 160 MeV. With these ranges it is suitable for calculations associated with experiments performed by exotic beam accelerators. The potential is also isospin dependent and has both real and imaginary isovector asymmetry terms to better describe the dynamics of chains of isotopes and mirror nuclei. An analysis of the validity and strength of the asymmetry term is included with connections established to other optical potentials and charge-exchange reaction data. An on-line observable calculator is available for this optical potential.
