Relativistic Studies of the Charmonium and Bottomonium Systems Using the Sucher Equation

In this dissertation, bound states of quarks and anti-quarks (mesons) are studied with a relativistic equation known as the Sucher equation. Prior to the work in this dissertation, the Sucher equation had never been used for meson mass spectra. Furthermore, a full angular momentum analysis of the Sucher equation has never been studied. The Sucher equation is a relativistic equation with positive energy projectors imposed on the interaction. Since spin is inherent to the equation, the Sucher equation is equivalent to a relativistic Schrödinger equation with a spin-dependent effective potential. Through a complete general angular momentum analysis of the equation, we found that different angular momenta can couple through the effective potential without explicitly using tensor interaction. Next we expanded the wave functions in a complete set of basis functions and converted the Sucher equation into a matrix eigenvalue equation. As a practical application, we fit to the low lying states of the bottomonium and charmonium systems with the minimal number of input parameters, and we were able to predict the remaining spectra. We find that the the Sucher equation may be used for charmonium and bottomonium spectra. However, the spin dependent interactions inherent to the Sucher equation do not produce adequate energy level splitting between singlet and triplet states

SHIELD and HZETRN Comparisons of Pion Production Cross Sections

A program of comparing American (NASA) and Russian (ROSCOSMOS) space radiation transport codes has recently begun, and the rst paper directly comparing the NASA and ROSCOSMOS space radiation transport codes, HZETRN and SHIELD respectively has recently appeared. The present work represents the second time that NASA and ROSCOSMOS calculations have been directly compared, and the focus here is on models of pion production cross sections used in the two transport codes mentioned above. It was found that these models are in overall moderate agreement with each other and with experimental data. Disagreements that were found are discussed

Elastic Differential Cross Sections for Space Radiation Applications

The eikonal, partial wave (PW) Lippmann-Schwinger, and three-dimensional Lippmann- Schwinger (LS3D) methods are compared for nuclear reactions that are relevant for space radiation applications. Numerical convergence of the eikonal method is readily achieved when exact formulas of the optical potential are used for light nuclei (A $\le$ 16), and the momentum-space representation of the optical potential is used for heavier nuclei. The PW solution method is known to be numerically unstable for systems that require a large number of partial waves, and, as a result, the LS3D method is employed. The effect of relativistic kinematics is studied with the PW and LS3D methods and is compared to eikonal results. It is recommended that the LS3D method be used for high energy nucleon-nucleus reactions and nucleus-nucleus reactions at all energies because of its rapid numerical convergence and stability

Relativistic Three-Dimensional Lippman-Schwinger Cross Sections for Space Radiation Applications

Radiation transport codes require accurate nuclear cross sections to compute particle fluences inside shielding materials. The Tripathi semi-empirical reaction cross section, which includes over 60 parameters tuned to nucleon-nucleus (NA) and nucleus-nucleus (AA) data, has been used in many of the world’s best-known transport codes. Although this parameterization fits well to reaction cross section data, the predictive capability of any parameterization is questionable when it is used beyond the range of the data to which it was tuned. Using uncertainty analysis, it is shown that a relativistic three-dimensional Lippmann-Schwinger (LS3D) equation model based on Multiple Scattering Theory (MST) that uses 5 parameterizations—3 fundamental parameterizations to nucleon-nucleon (NN) data and 2 nuclear charge density parameterizations—predicts NA and AA reaction cross sections as well as the Tripathi cross section parameterization for reactions in which the kinetic energy of the projectile in the laboratory frame (TLab) is greater than 220 MeV/n. The relativistic LS3D model has the additional advantage of being able to predict highly accurate total and elastic cross sections. Consequently, it is recommended that the relativistic LS3D model be used for space radiation applications in which TLab \u3e 220MeV/n

Numerical Gram-Schmidt Orthonormalization

A numerical Gram-Schmidt orthonormalization procedure is presented for constructing an orthonormal basis function set from a non-orthonormal set, when the number of basis functions is large. This method will provide a pedagogical illustration of the Gram-Schmidt procedure and can be presented in classes on numerical methods or computational physics

Elastic differential cross sections for space radiation applications

The eikonal, partial wave (PW) Lippmann-Schwinger, and three-dimensional Lippmann-Schwinger (LS3D) methods are compared for nuclear reactions that are relevant for space radiation applications. Numerical convergence of the eikonal method is readily achieved when exact formulas of the optical potential are used for light nuclei (\u3eA≤\u3e16), and the momentum-space representation of the optical potential is used for heavier nuclei. The PW solution method is known to be numerically unstable for systems that require a large number of partial waves, and, as a result, the LS3D method is employed. The effect of relativistic kinematics is studied with the PW and LS3D methods and is compared to eikonal results. It is recommended that the LS3D method be used for high-energy nucleon-nucleus reactions and nucleus-nucleus reactions at all energies because of its rapid numerical convergence and stability

Elastic Differential Cross Sections

The eikonal, partial wave (PW) Lippmann-Schwinger, and three-dimensional Lippmann-Schwinger (LS3D) methods are compared for nuclear reactions that are relevant for space radiation applications. Numerical convergence of the eikonal method is readily achieved when exact formulas of the optical potential are used for light nuclei (A less than or equal to 16) and the momentum-space optical potential is used for heavier nuclei. The PW solution method is known to be numerically unstable for systems that require a large number of partial waves, and, as a result, the LS3D method is employed. The effect of relativistic kinematics is studied with the PW and LS3D methods and is compared to eikonal results. It is recommended that the LS3D method be used for high energy nucleon- nucleus reactions and nucleus-nucleus reactions at all energies because of its rapid numerical convergence and stability

Correlated Uncertainties in Radiation Shielding Effectiveness

The space radiation environment is composed of energetic particles which can deliver harmful doses of radiation that may lead to acute radiation sickness, cancer, and even death for insufficiently shielded crew members. Spacecraft shielding must provide structural integrity and minimize the risk associated with radiation exposure. The risk of radiation exposure induced death (REID) is a measure of the risk of dying from cancer induced by radiation exposure. Uncertainties in the risk projection model, quality factor, and spectral fluence are folded into the calculation of the REID by sampling from probability distribution functions. Consequently, determining optimal shielding materials that reduce the REID in a statistically significant manner has been found to be difficult. In this work, the difference of the REID distributions for different materials is used to study the effect of composition on shielding effectiveness. It is shown that the use of correlated uncertainties allows for the determination of statistically significant differences between materials despite the large uncertainties in the quality factor. This is in contrast to previous methods where uncertainties have been generally treated as uncorrelated. It is concluded that the use of correlated quality factor uncertainties greatly reduces the uncertainty in the assessment of shielding effectiveness for the mitigation of radiation exposure

Evidence Report: Risk of Radiation Carcinogenesis

As noted by Durante and Cucinotta (2008), cancer risk caused by exposure to space radiation is now generally considered a main hindrance to interplanetary travel for the following reasons: large uncertainties are associated with the projected cancer risk estimates; no simple and effective countermeasures are available, and significant uncertainties prevent scientists from determining the effectiveness of countermeasures. Optimizing operational parameters such as the length of space missions, crew selection for age and sex, or applying mitigation measures such as radiation shielding or use of biological countermeasures can be used to reduce risk, but these procedures have inherent limitations and are clouded by uncertainties. Space radiation is comprised of high energy protons, neutrons and high charge (Z) and energy (E) nuclei (HZE). The ionization patterns and resulting biological insults of these particles in molecules, cells, and tissues are distinct from typical terrestrial radiation, which is largely X-rays and gamma-rays, and generally characterized as low linear energy transfer (LET) radiation. Galactic cosmic rays (GCR) are comprised mostly of highly energetic protons with a small component of high charge and energy (HZE) nuclei. Prominent HZE nuclei include He, C, O, Ne, Mg, Si, and Fe. GCR ions have median energies near 1 GeV/n, and energies as high as 10 GeV/n make important contributions to the total exposure. Ionizing radiation is a well known carcinogen on Earth (BEIR 2006). The risks of cancer from X-rays and gamma-rays have been established at doses above 50 mSv (5 rem), although there are important uncertainties and on-going scientific debate about cancer risk at lower doses and at low dose rates (<50 mSv/h). The relationship between the early biological effects of HZE nuclei and the probability of cancer in humans is poorly understood, and it is this missing knowledge that leads to significant uncertainties in projecting cancer risks during space exploration (Cucinotta and Durante 2006; Durante and Cucinotta 2008)

Nucleus-Nucleus Relativistic Multiple Scattering Theory with Delta Degrees of Freedom

It is well known that multiple scattering theories are very useful in the study of nucleon-nucleus and nucleus-nucleus scattering processes. The derivation of a nonrelativistic multiple scattering theory (NRMST) is well-established and clear. A key component to the formulation of an NRMST is the ability to separate the unperturbed Hamiltonian from the residual interaction. For the relativistic problem, it is not clear how to perform this separation starting from a field theoretical Lagrangian. Instead, one starts from an infinite set of Feynman diagrams, which play the role of the kernel in the Bethe-Salpeter equation for nucleus-nucleus scattering. Once the kernel is defined, it is straightforward to develop a relativistic multiple scattering theory (RMST). To be more complete than previous studies, delta degrees of freedom are included, which is a minimum requirement to explain pion production. It is demonstrated that an RMST can be formulated by expressing the kernel in a form that is similar to the residual interaction in the NRMST