# CRYSTALLIZATION OF HELIUM-3 AND NEUTRON MATTER FERMIONS IN CONDENSED MATTER PHYSICS

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Thesis
In this Thesis crystallization of a hard-sphere assembly of fermions with densities ranging from low to high values has been theoretically investigated, the fermions are very close to each other, and the interaction between a pair of fermions is assumed to be a hard-sphere one. The hard-sphere system is a useful first approximation of a many-body system interacting via a potential containing a short-ranged repulsive part, which happens to be even better at very low densities at which the particles experience weakly attractive potential tail surrounding the repulsion, or at very high densities where the repulsion is predominant. However, at intermediate densities, the attractive potential is assumed to play a significant role. Since fermions (degenerate Fermi gas or liquid) such as a gas or liquid 𝐻𝑒3 or neutrons satisfy Pauli’s exclusion principle which leads to repulsion between two fermions when they try to approach each other to occupy the same quantum (energy) state. A hard-sphere assembly of fermions of densities ranging from very low to very high values was considered to obtain an expression for the energy per particle, 𝐸𝑁⁄. For an N-identical fermion hard-sphere system with 𝜈 intrinsic degrees of freedom for each fermion, the total energy, E in terms of other parameters, a generalized London equation was applied to obtain the total energy of the system, and to obtain the saturation density, 𝜌𝑠, leading to crystallization of the system. Particle number densities, 𝜌𝑠, for low and high densities were 7.11x1027 particles/𝑐𝑚3 and 1.502x1029 particles/𝑐𝑚3 respectively, at which the fermions close pack or crystallize. Transition Temperature 𝑇𝑐 at crystallization of fermions was calculated whose value was 19.26K. These results are consistent with experimental and computer-simulated results whose value is 20.3K. Next, similar techniques were applied to neutron stars which were considered as reservoirs of high density fermions. The energy per neutron in a neutron star was calculated for low and high density neutron stars whose findings demonstrated that the energy per neutron for both low and high density of the neutron star increases for a given value of the scattering length, in agreement with experimental results, since increasing the density leads to strong interactions which in turn increase energy of the assembly. This also confirms that under high pressure, the system attains large density and huge amount of energy on crystallization.

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