Charged pion contribution to the anomalous magnetic moment of the muon

The model dependence inherent in hadronic calculations is one of the dominant sources of uncertainty in the theoretical prediction of the anomalous magnetic moment of the muon. In this thesis, we focus on the charged pion contribution and turn a critical eye on the models employed in the few previous calculations of $a_\mu^{\pi^+\pi^-}$. Chiral perturbation theory provides a check on these models at low energies, and we therefore calculate the charged pion contribution to light-by-light (LBL) scattering to $\mathcal{O}(p^6)$. We show that the dominant corrections to the leading order (LO) result come from two low energy constants which show up in the form factors for the $\gamma\pi\pi$ and $\gamma\gamma\pi\pi$ vertices. Comparison with the existing models reveal a potentially significant omission - none include the pion polarizability corrections associated with the $\gamma\gamma\pi\pi$ vertex. We next consider alternative models where the pion polarizability is produced through exchange of the $a_1$ axial vector meson. These have poor UV behavior, however, making them unsuited for the $a_\mu^{\pi^+\pi^-}$ calculation. We turn to a simpler form factor modeling approach, generating two distinct models which reproduce the pion polarizability corrections at low energies, have the correct QCD scaling at high energies, and generate finite contributions to $a_\mu^{\pi^+\pi^-}$. With these two models, we calculate the charged pion contribution to the anomalous magnetic moment of the muon, finding values larger than those previously reported: $a_\mu^\mathrm{I} = -1.779(4)\times10^{-10}\,,\,a_\mu^\mathrm{II} = -4.892(3)\times10^{-10}$.

Optimal guidance of low-thrust interplanetary space vehicles

The low-thrust guidance problem is defined as the minimum terminal variance (MTV) control of a space vehicle subjected to random perturbations of its trajectory. To accomplish this control task, only bounded thrust level and thrust angle deviations are allowed, and these must be calculated based solely on the information gained from noisy, partial observations of the state. In order to establish the validity of various approximations, the problem is first investigated under the idealized conditions of perfect state information and negligible dynamic errors. To check each approximate model, an algorithm is developed to facilitate the computation of the open loop trajectories for the nonlinear bang-bang system. Using the results of this phase in conjunction with the Ornstein-Uhlenbeck process as a model for the random inputs to the system, the MTV guidance problem is reformulated as a stochastic, bang-bang, optimal control problem. Since a complete analytic solution seems to be unattainable, asymptotic solutions are developed by numerical methods. However, it is shown analytically that a Kalman filter in cascade with an appropriate nonlinear MTV controller is an optimal configuration. The resulting system is simulated using the Monte Carlo technique and is compared to other guidance schemes of current interest.

Dilution refrigeration and a calorimetric measurement of the quadruple coupling constant in rhenium metal

A dilution refrigerator has been constructed capable of producing steady state temperatures less than .075°K. The first part of this work is concerned with the design and construction of this machine. Enough theory is presented to allow one to understand the operation and critical design factors of a dilution refrigerator. The performance of our refrigerator is compared with the operating characteristics of three other dilution refrigerators appearing in the present literature.

The dilution refrigerator constructed was used to measure the nuclear contribution to the low temperature specific heat of a pure, single-crystalline sample of rhenium metal. Measurements were made in magnetic fields from 0 to 12.5 kOe for the temperature range .13°K - .52°K. The second part of this work discusses the results of these experiments. The expected nuclear contribution is not found when the sample is in the superconducting state. This is believed to be due to the long spin-lattice relaxation times in superconductors. In the normal state, for the temperature range studied, the nuclear contribution is given by A/T² where A = .061 ± .002 millijoules-K/mole. The value of A is found to increase to A = .077 ± .004 millijoules-K/mole when the sample is located in a magnetic field of 12.5 kOe.

From the measured value of A the splitting of the energy levels of the nuclear spin system due to the interaction of the internal crystalline electric field gradients with the nuclear quadrupole moments is calculated. A comparison is made between the predicted and measured magnetic dependence of the specific heat. Finally, predictions are made of future nuclear magnetic resonance experiments which may be performed to check the results obtained by calorimetery here and further, to investigate existing theories concerning the sources of electric field gradients in metals.

Time-dependent monoenergetic neutron transport in two adjacent semi-infinite media

An exact solution to the monoenergetic Boltzmann equation is obtained for the case of a plane isotropic burst of neutrons introduced at the interface separating two adjacent, dissimilar, semi-infinite media. The method of solution used is to remove the time dependence by a Laplace transformation, solve the transformed equation by the normal mode expansion method, and then invert to recover the time dependence.

The general result is expressed as a sum of definite, multiple integrals, one of which contains the uncollided wave of neutrons originating at the source plane. It is possible to obtain a simplified form for the solution at the interface, and certain numerical calculations are made there.

The interface flux in two adjacent moderators is calculated and plotted as a function of time for several moderator materials. For each case it is found that the flux decay curve has an asymptotic slope given accurately by diffusion theory. Furthermore, the interface current is observed to change directions when the scattering and absorption cross sections of the two moderator materials are related in a certain manner. More specifically, the reflection process in two adjacent moderators appears to depend initially on the scattering properties and for long times on the absorption properties of the media.

This analysis contains both the single infinite and semi-infinite medium problems as special cases. The results in these two special cases provide a check on the accuracy of the general solution since they agree with solutions of these problems obtained by separate analyses.

Experimental and theoretical studies of the electrothermal instability in a nonequilibrium plasma

Experimental and theoretical studies have been made of the electrothermal waves occurring in a nonequilibrium MHD plasma. These waves are caused by an instability that occurs when a plasma having a dependence of conductivity on current density is subjected to crossed electric and magnetic fields. Theoretically, these waves were studied by developing and solving the equations of a steady, one-dimensional nonuniformity in electron density. From these nonlinear equations, predictions of the maximum amplitude and of the half width of steady waves could be obtained. Experimentally, the waves were studied in a nonequilibrium discharge produced in a potassium-seeded argon plasma at 2000°K and 1 atm. pressure. The behavior of such a discharge with four different configurations of electrodes was determined from photographs, photomultiplier measurements, and voltage probes. These four configurations were chosen to produce steady waves, to check the stability of steady waves, and to observe the manifestation of the waves in a MHD generator or accelerator configuration.

Steady, one-dimensional waves were found to exist in a number of situations, and where they existed, their characteristics agreed with the predictions of the steady theory. Some extensions of this theory were necessary, however, to describe the transient phenomena occurring in the inlet region of a discharge transverse to the gas flow. It was also found that in a discharge away from the stabilizing effect of the electrodes, steady waves became unstable for large Hall parameters. Methods of prediction of the effective electrical conductivity and Hall parameter of a plasma with nonuniformities caused by the electrothermal waves were also studied. Using these methods and the values of amplitude predicted by the steady theory, it was found that the measured decrease in transverse conductivity of a MHD device, 50 per cent at a Hall parameter of 5, could be accounted for in terms of the electrothermal instability.

Wave induced oscillations in harbors of arbitrary shape

Theoretical and experimental studies were conducted to investigate the wave induced oscillations in an arbitrary shaped harbor with constant depth which is connected to the open-sea.

A theory termed the “arbitrary shaped harbor” theory is developed. The solution of the Helmholtz equation, ∇²f + k²f = 0, is formulated as an integral equation; an approximate method is employed to solve the integral equation by converting it to a matrix equation. The final solution is obtained by equating, at the harbor entrance, the wave amplitude and its normal derivative obtained from the solutions for the regions outside and inside the harbor.

Two special theories called the circular harbor theory and the rectangular harbor theory are also developed. The coordinates inside a circular and a rectangular harbor are separable; therefore, the solution for the region inside these harbors is obtained by the method of separation of variables. For the solution in the open-sea region, the same method is used as that employed for the arbitrary shaped harbor theory. The final solution is also obtained by a matching procedure similar to that used for the arbitrary shaped harbor theory. These two special theories provide a useful analytical check on the arbitrary shaped harbor theory.

Experiments were conducted to verify the theories in a wave basin 15 ft wide by 31 ft long with an effective system of wave energy dissipators mounted along the boundary to simulate the open-sea condition.

Four harbors were investigated theoretically and experimentally: circular harbors with a 10° opening and a 60° opening, a rectangular harbor, and a model of the East and West Basins of Long Beach Harbor located in Long Beach, California.

Theoretical solutions for these four harbors using the arbitrary shaped harbor theory were obtained. In addition, the theoretical solutions for the circular harbors and the rectangular harbor using the two special theories were also obtained. In each case, the theories have proven to agree well with the experimental data.

It is found that: (1) the resonant frequencies for a specific harbor are predicted correctly by the theory, although the amplification factors at resonance are somewhat larger than those found experimentally,(2) for the circular harbors, as the width of the harbor entrance increases, the amplification at resonance decreases, but the wave number bandwidth at resonance increases, (3) each peak in the curve of entrance velocity vs incident wave period corresponds to a distinct mode of resonant oscillation inside the harbor, thus the velocity at the harbor entrance appears to be a good indicator for resonance in harbors of complicated shape, (4) the results show that the present theory can be applied with confidence to prototype harbors with relatively uniform depth and reflective interior boundaries.