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.

Part I. Approximate Hartree-Fock wavefunctions one-electron properties and electronic structure of the water molecule. Part II. Perturbation-variational calculation of the nuclear spin-spin isotope coupling constant in HD

Part I

Several approximate Hartree-Fock SCF wavefunctions for the ground electronic state of the water molecule have been obtained using an increasing number of multicenter s, p, and d Slater-type atomic orbitals as basis sets. The predicted charge distribution has been extensively tested at each stage by calculating the electric dipole moment, molecular quadrupole moment, diamagnetic shielding, Hellmann-Feynman forces, and electric field gradients at both the hydrogen and the oxygen nuclei. It was found that a carefully optimized minimal basis set suffices to describe the electronic charge distribution adequately except in the vicinity of the oxygen nucleus. Our calculations indicate, for example, that the correct prediction of the field gradient at this nucleus requires a more flexible linear combination of p-orbitals centered on this nucleus than that in the minimal basis set. Theoretical values for the molecular octopole moment components are also reported.

Part II

The perturbation-variational theory of R. M. Pitzer for nuclear spin-spin coupling constants is applied to the HD molecule. The zero-order molecular orbital is described in terms of a single 1s Slater-type basis function centered on each nucleus. The first-order molecular orbital is expressed in terms of these two functions plus one singular basis function each of the types e^-r/r and e^-r ln r centered on one of the nuclei. The new kinds of molecular integrals were evaluated to high accuracy using numerical and analytical means. The value of the HD spin-spin coupling constant calculated with this near-minimal set of basis functions is J_HD = +96.6 cps. This represents an improvement over the previous calculated value of +120 cps obtained without using the logarithmic basis function but is still considerably off in magnitude compared with the experimental measurement of J_HD = +43 0 ± 0.5 cps.

π+ photoproduction at angles from 6° to 90° C.M. and energies from 589 to 1269 MeV

The differential cross section for the reaction γp → π⁺n was measured at 32 laboratory photon energies between 589 and 1269 MeV at the Caltech Synchrotron. At each energy, data have been obtained at typically fifteen π⁺ c.m. angles between 6° and 90°. A magnetic spectrometer was used to detect the π⁺ photo-produced in a liquid hydrogen target. Two Cherenkov counters were used to reject the background of positrons and protons. The data clearly show the presence of a pole in the production amplitude due to the one pion exchange. Moravcsik fits to the 32 angular distributions, including data from another experiment, are presented. The extrapolation of these fits to the pole gives a value for the pion-nucleon coupling constant of 14.5 which is consistent with the accepted value. The second and third pion-nucleon resonances are evident as peaks in the total cross section and as changes in the shape of the angular distributions. At the third resonance there is evidence for both a D5/2 and an F5/2 amplitude. The absence of large variations in the 0° and 180° cross sections implies that the second and third resonances are mostly produced from an initial state with helicity ± 3/2.

A perturbation procedure for nonlinear oscillations (The dynamics of two oscillators with weak nonlinear coupling)

This thesis considers in detail the dynamics of two oscillators with weak nonlinear coupling. There are three classes of such problems: non-resonant, where the Poincaré procedure is valid to the order considered; weakly resonant, where the Poincaré procedure breaks down because small divisors appear (but do not affect the O(1) term) and strongly resonant, where small divisors appear and lead to O(1) corrections. A perturbation method based on Cole's two-timing procedure is introduced. It avoids the small divisor problem in a straightforward manner, gives accurate answers which are valid for long times, and appears capable of handling all three types of problems with no change in the basic approach.

One example of each type is studied with the aid of this procedure: for the nonresonant case the answer is equivalent to the Poincaré result; for the weakly resonant case the analytic form of the answer is found to depend (smoothly) on the difference between the initial energies of the two oscillators; for the strongly resonant case we find that the amplitudes of the two oscillators vary slowly with time as elliptic functions of ϵ t, where ϵ is the (small) coupling parameter.

Our results suggest that, as one might expect, the dynamical behavior of such systems varies smoothly with changes in the ratio of the fundamental frequencies of the two oscillators. Thus the pathological behavior of Whittaker's adelphic integrals as the frequency ratio is varied appears to be due to the fact that Whittaker ignored the small divisor problem. The energy sharing properties of these systems appear to depend strongly on the initial conditions, so that the systems not ergodic.

The perturbation procedure appears to be applicable to a wide variety of other problems in addition to those considered here.

Electrical and optical interconnects for high-performance computing

Technology scaling has enabled drastic growth in the computational and storage capacity of integrated circuits (ICs). This constant growth drives an increasing demand for high-bandwidth communication between and within ICs. In this dissertation we focus on low-power solutions that address this demand. We divide communication links into three subcategories depending on the communication distance. Each category has a different set of challenges and requirements and is affected by CMOS technology scaling in a different manner. We start with short-range chip-to-chip links for board-level communication. Next we will discuss board-to-board links, which demand a longer communication range. Finally on-chip links with communication ranges of a few millimeters are discussed.

Electrical signaling is a natural choice for chip-to-chip communication due to efficient integration and low cost. IO data rates have increased to the point where electrical signaling is now limited by the channel bandwidth. In order to achieve multi-Gb/s data rates, complex designs that equalize the channel are necessary. In addition, a high level of parallelism is central to sustaining bandwidth growth. Decision feedback equalization (DFE) is one of the most commonly employed techniques to overcome the limited bandwidth problem of the electrical channels. A linear and low-power summer is the central block of a DFE. Conventional approaches employ current-mode techniques to implement the summer, which require high power consumption. In order to achieve low-power operation we propose performing the summation in the charge domain. This approach enables a low-power and compact realization of the DFE as well as crosstalk cancellation. A prototype receiver was fabricated in 45nm SOI CMOS to validate the functionality of the proposed technique and was tested over channels with different levels of loss and coupling. Measurement results show that the receiver can equalize channels with maximum 21dB loss while consuming about 7.5mW from a 1.2V supply. We also introduce a compact, low-power transmitter employing passive equalization. The efficacy of the proposed technique is demonstrated through implementation of a prototype in 65nm CMOS. The design achieves up to 20Gb/s data rate while consuming less than 10mW.

An alternative to electrical signaling is to employ optical signaling for chip-to-chip interconnections, which offers low channel loss and cross-talk while providing high communication bandwidth. In this work we demonstrate the possibility of building compact and low-power optical receivers. A novel RC front-end is proposed that combines dynamic offset modulation and double-sampling techniques to eliminate the need for a short time constant at the input of the receiver. Unlike conventional designs, this receiver does not require a high-gain stage that runs at the data rate, making it suitable for low-power implementations. In addition, it allows time-division multiplexing to support very high data rates. A prototype was implemented in 65nm CMOS and achieved up to 24Gb/s with less than 0.4pJ/b power efficiency per channel. As the proposed design mainly employs digital blocks, it benefits greatly from technology scaling in terms of power and area saving.

As the technology scales, the number of transistors on the chip grows. This necessitates a corresponding increase in the bandwidth of the on-chip wires. In this dissertation, we take a close look at wire scaling and investigate its effect on wire performance metrics. We explore a novel on-chip communication link based on a double-sampling architecture and dynamic offset modulation technique that enables low power consumption and high data rates while achieving high bandwidth density in 28nm CMOS technology. The functionality of the link is demonstrated using different length minimum-pitch on-chip wires. Measurement results show that the link achieves up to 20Gb/s of data rate (12.5Gb/s/$\mu$m) with better than 136fJ/b of power efficiency.

Nonlinear optics and wavelength translation via cavity-optomechanics

The field of cavity-optomechanics explores the interaction of light with sound in an ever increasing array of devices. This interaction allows the mechanical system to be both sensed and controlled by the optical system, opening up a wide variety of experiments including the cooling of the mechanical resonator to its quantum mechanical ground state and the squeezing of the optical field upon interaction with the mechanical resonator, to name two.

In this work we explore two very different systems with different types of optomechanical coupling. The first system consists of two microdisk optical resonators stacked on top of each other and separated by a very small slot. The interaction of the disks causes their optical resonance frequencies to be extremely sensitive to the gap between the disks. By careful control of the gap between the disks, the optomechanical coupling can be made to be quadratic to first order which is uncommon in optomechanical systems. With this quadratic coupling the light field is now sensitive to the energy of the mechanical resonator and can directly control the potential energy trapping the mechanical motion. This ability to directly control the spring constant without modifying the energy of the mechanical system, unlike in linear optomechanical coupling, is explored.

Next, the bulk of this thesis deals with a high mechanical frequency optomechanical crystal which is used to coherently convert photons between different frequencies. This is accomplished via the engineered linear optomechanical coupling in these devices. Both classical and quantum systems utilize the interaction of light and matter across a wide range of energies. These systems are often not naturally compatible with one another and require a means of converting photons of dissimilar wavelengths to combine and exploit their different strengths. Here we theoretically propose and experimentally demonstrate coherent wavelength conversion of optical photons using photon-phonon translation in a cavity-optomechanical system. For an engineered silicon optomechanical crystal nanocavity supporting a 4 GHz localized phonon mode, optical signals in a 1.5 MHz bandwidth are coherently converted over a 11.2 THz frequency span between one cavity mode at wavelength 1460 nm and a second cavity mode at 1545 nm with a 93% internal (2% external) peak efficiency. The thermal and quantum limiting noise involved in the conversion process is also analyzed and, in terms of an equivalent photon number signal level, are found to correspond to an internal noise level of only 6 and 4 times 10x^-3 quanta, respectively.

We begin by developing the requisite theoretical background to describe the system. A significant amount of time is then spent describing the fabrication of these silicon nanobeams, with an emphasis on understanding the specifics and motivation. The experimental demonstration of wavelength conversion is then described and analyzed. It is determined that the method of getting photons into the cavity and collected from the cavity is a fundamental limiting factor in the overall efficiency. Finally, a new coupling scheme is designed, fabricated, and tested that provides a means of coupling greater than 90% of photons into and out of the cavity, addressing one of the largest obstacles with the initial wavelength conversion experiment.

Synthetic, kinetic and mechanistic studies in early transition metal systems. I. α- and β-hydrogen elimination reactions in derivatives of permethyltitanocene, -zirconocene, and -hafnocene. II. The reactions of aluminum alkyls with derivatives of permethylniobocene

The thermal decomposition of Cp*Ti(CH_3)_2 (Cp*≡ ƞ^5-C_5Me_5) toluene solution follows cleanly first-order kinetics and produces a single titanium product Cp*(C_5Me_4CH_2)Ti(CH_3) concurrent with the evolution of one equivalent of methane. Labeling studies using Cp*_2Ti- (CD_3)_2 and (Cp*-d_(15))_2Ti(CH_3)_2 show the decomposition to be intramolecular and the methane to be produced by the coupling of a methyl group with a hydrogen from the other TiCH_3 group. Activation parameters, ΔH^‡ and ΔS^‡, and kinetic deuterium isotope effects have been measured. The alternative decomposition pathways of α-hydrogen abstraction and a-hydrogen elimination, both leading to a titanium-methylidene intermediate, are discussed.

The insertion of unactivated acetylenes into the metal-hydride bonds of Cp*_2MH_2 (M = Zr, Hf) proceeds rapidly at low temperature to form monoand/ or bisinsertion products, dependent upon the steric bulk of the acetylene substituents. Cp*_2M(H)(C(Me)=CHMe), Cp*_2M(H)(CH=CHCMe_3), Cp*_2M(H)-(CH=CHPh), Cp*_2M(CH=CHPh)_2, Cp*_2M(CH=CHCH_3)_2 and Cp*_2Zr- (CH=CHCH_2CH_3)_2 have been isolated and characterized. To extend the study of unsaturated-carbon ligands, Cp*_2M(C≡CCH_3)_2 have been prepared by treating Cp*_2MCl_2 with LiC≡CCH_3. The reactivity of many of these complexes with carbon monoxide and dihydrogen is surveyed. The mono(2- butenyl) complexes Cp*_2M(H)(C(Me)=CHMe) rearrange at room temperature, forming the crotyl-hydride species Cp*_2M(H)(ƞ^3-C_4H_7). The bis(propenyl) and bis(l-butenyl) zirconium complexes Cp*_2Zr(CH=CHR)_2 (R = CH_3, CH_2CH_3) also rearrange, forming zirconacyclopentenes. Labeling studies, reaction chemistry, and kinetic measurements, including deuterium isotope effects, demonstrate that the unusual 6-hydrogen elimination from an sp^2-hybridized carbon is the first step in these latter rearrangements but is not observed in the former. Details of these mechanisms and the differences in reactivity of the zirconium and hafnium complexes are discussed.

The reactions of hydride- and alkyl-carbonyl derivatives of permethylniobocene with equimolar amounts of trialkylaluminum reagents occur rapidly producing the carbonyl adducts Cp*_2Nb(R)(COAlR'_3) (R = H, CH_3, CH_2CH_3, CH_2CH_2Ph, C(Me)=CHMe; R' = Me, Et). The hydride adduct Cp*_2NbH_3•AlEt_3 has also been formed. In solution, each of these compounds exists in equilibrium with the uncomplexed species. The formation constants for Cp*_2Nb(H)(COA1R'_R) have been measured. They indicate the steric bulk of the Cp* ligands plays a deciding factor in the isolation of the first example of an aluminum Lewis acid bound to a carbonyl-oxygen in preference to a metalhydride. Reactions of Cp*_2Nb(H)CO with other Lewis acids and of the one:one adducts with H_2, CO and C_2H_4 are also discussed.

Cp*_2Nb(H)(C_2H_4) also reacts with equimolar amounts of trialkylaluminum reagents, forming a one:one complex that ^1H NMR spectroscopy indicates contains a Nb-CH_2CH_2-Al bridge. This adduct also exists in equilibrium with the uncomplexed species in solution. The formation constant for Cp*_2N+/b(H)(CH_2CH_2ĀlEt_3) has been measured. Reactions of Cp*_2Nb(H)(C_2H_4) with other Lewis acids and the reactions of Cp*_2N+b(H)- (CH_2CH_2ĀlEt_3) with CO and C_2H_4 are described, as are the reactions of Cp_*2Nb(H)(CH_2=CHR) (R = Me, Ph), Cp*_2Nb(H)(CH_3C≡CCH_3) and Cp*_2Ti-(C_2H_4) with AlEt_3.

Theory of ozone isotopic effects and various electron transfer reactions

Three separate topics, each stimulated by experiments, are treated theoretically in this dessertation: isotopic effects of ozone, electron transfer at interfaces, and intramolecular directional electron transfer in a supramolecular system.

The strange mass-independent isotope effect for the enrichment of ozone, which has been a puzzle in the literature for some 20 years, and the equally puzzling unconventional strong mass-dependent effect of individual reaction rate constants are studied as different aspects of a symmetry-driven behavior. A statistical (RRKM-based) theory with a hindered-rotor transition state is used. The individual rate constant ratios of recombination reactions at low pressures are calculated using the theory involving (1) small deviation from the statistical density of states for symmetric isotopomers, and (2) weak collisions for deactivation of the vibrationally excited ozone molecules. The weak collision and partitioning among exit channels play major roles in producing the large unconventional isotope effect in "unscrambled" systems. The enrichment studies reflect instead the non-statistical effect in "scrambled" systems. The theoretical results of low-pressure ozone enrichments and individual rate constant ratios obtained from these calculations are consistent with the corresponding experimental results. The isotopic exchange rate constant for the reaction ^(16)O + ^(18)O ^(18)O→+ ^(16)O ^(18)O + ^(18)O provides information on the nature of a variationally determined hindered-rotor transition state using experimental data at 130 K and 300 K. Pressure effects on the recombination rate constant, on the individual rate constant ratios and on the enrichments are also investigated. The theoretical results are consistent with the experimental data. The temperature dependence of the enrichment and rate constant ratios is also discussed, and experimental tests are suggested. The desirability of a more accurate potential energy surface for ozone in the transition state region is also noted.

Electron transfer reactions at semiconductor /liquid interfaces are studied using a tight-binding model for the semiconductors. The slab method and a z-transform method are employed in obtaining the tight-binding electronic structures of semiconductors having surfaces. The maximum electron transfer rate constants at Si/viologen^(2-/+) and InP /Me_(2)Fc^(+/O) interfaces are computed using the tight-binding type calculations for the solid and the extended-Huckel for the coupling to the redox agent at the interface. These electron transfer reactions are also studied using a free electron model for the semiconductor and the redox molecule, where Bardeen's method is adapted to calculate the coupling matrix element between the molecular and semiconductor electronic states. The calculated results for maximum rate constant of the electron transfer from the semiconductor bulk states are compared with the experimentally measured values of Lewis and coworkers, and are in reasonable agreement, without adjusting parameters. In the case of InP /liquid interface, the unusual current vs applied potential behavior is additionally interpreted, in part, by the presence of surface states.

Photoinduced electron transfer reactions in small supramolecular systems, such as 4-aminonaphthalimide compounds, are interesting in that there are, in principle, two alternative pathways (directions) for the electron transfer. The electron transfer, however, is unidirectional, as deduced from pH-dependent fluorescence quenching studies on different compounds. The role of electronic coupling matrix element and the charges in protonation are considered to explain the directionality of the electron transfer and other various results. A related mechanism is proposed to interpret the fluorescence behavior of similar molecules as fluorescent sensors of metal ions.

Investigations of charge-carrier dynamics at semiconductor/liquid interfaces

Theoretical and experimental investigations of charge-carrier dynamics at semiconductor/liquid interfaces, specifically with respect to interfacial electron transfer and surface recombination, are presented.

Fermi's golden rule has been used to formulate rate expressions for charge transfer of delocalized carriers in a nondegenerately doped semiconducting electrode to localized, outer-sphere redox acceptors in an electrolyte phase. The treatment allows comparison between charge-transfer kinetic data at metallic, semimetallic, and semiconducting electrodes in terms of parameters such as the electronic coupling to the electrode, the attenuation of coupling with distance into the electrolyte, and the reorganization energy of the charge-transfer event. Within this framework, rate constant values expected at representative semiconducting electrodes have been determined from experimental data for charge transfer at metallic electrodes. The maximum rate constant (i.e., at optimal exoergicity) for outer-sphere processes at semiconducting electrodes is computed to be in the range 10^-17-10^-16 cm⁴ s^-1, which is in excellent agreement with prior theoretical models and experimental results for charge-transfer kinetics at semiconductor/liquid interfaces.

Double-layer corrections have been evaluated for semiconductor electrodes in both depletion and accumulation conditions. In conjuction with the Gouy-Chapman-Stern model, a finite difference approach has been used to calculate potential drops at a representative solid/liquid interface. Under all conditions that were simulated, the correction to the driving force used to evaluate the interfacial rate constant was determined to be less than 2% of the uncorrected interfacial rate constant.

Photoconductivity decay lifetimes have been obtained for Si(111) in contact with solutions of CH₃OH or tetrahydrofuran containing one-electron oxidants. Silicon surfaces in contact with electrolyte solutions having Nernstian redox potentials > 0 V vs. SCE exhibited low effective surface recombination velocities regardless of the different surface chemistries. The formation of an inversion layer, and not a reduced density of electrical trap sites on the surface, is shown to be responsible for the long charge-carrier lifetimes observed for these systems. In addition, a method for preparing an air-stable, low surface recombination velocity Si surface through a two-step, chlorination/alkylation reaction is described.

Design strategies for ultra-high efficiency photovoltaics

While concentrator photovoltaic cells have shown significant improvements in efficiency in the past ten years, once these cells are integrated into concentrating optics, connected to a power conditioning system and deployed in the field, the overall module efficiency drops to only 34 to 36%. This efficiency is impressive compared to conventional flat plate modules, but it is far short of the theoretical limits for solar energy conversion. Designing a system capable of achieving ultra high efficiency of 50% or greater cannot be achieved by refinement and iteration of current design approaches.

This thesis takes a systems approach to designing a photovoltaic system capable of 50% efficient performance using conventional diode-based solar cells. The effort began with an exploration of the limiting efficiency of spectrum splitting ensembles with 2 to 20 sub cells in different electrical configurations. Incorporating realistic non-ideal performance with the computationally simple detailed balance approach resulted in practical limits that are useful to identify specific cell performance requirements. This effort quantified the relative benefit of additional cells and concentration for system efficiency, which will help in designing practical optical systems.

Efforts to improve the quality of the solar cells themselves focused on the development of tunable lattice constant epitaxial templates. Initially intended to enable lattice matched multijunction solar cells, these templates would enable increased flexibility in band gap selection for spectrum splitting ensembles and enhanced radiative quality relative to metamorphic growth. The III-V material family is commonly used for multijunction solar cells both for its high radiative quality and for the ease of integrating multiple band gaps into one monolithic growth. The band gap flexibility is limited by the lattice constant of available growth templates. The virtual substrate consists of a thin III-V film with the desired lattice constant. The film is grown strained on an available wafer substrate, but the thickness is below the dislocation nucleation threshold. By removing the film from the growth substrate, allowing the strain to relax elastically, and bonding it to a supportive handle, a template with the desired lattice constant is formed. Experimental efforts towards this structure and initial proof of concept are presented.

Cells with high radiative quality present the opportunity to recover a large amount of their radiative losses if they are incorporated in an ensemble that couples emission from one cell to another. This effect is well known, but has been explored previously in the context of sub cells that independently operate at their maximum power point. This analysis explicitly accounts for the system interaction and identifies ways to enhance overall performance by operating some cells in an ensemble at voltages that reduce the power converted in the individual cell. Series connected multijunctions, which by their nature facilitate strong optical coupling between sub-cells, are reoptimized with substantial performance benefit.

Photovoltaic efficiency is usually measured relative to a standard incident spectrum to allow comparison between systems. Deployed in the field systems may differ in energy production due to sensitivity to changes in the spectrum. The series connection constraint in particular causes system efficiency to decrease as the incident spectrum deviates from the standard spectral composition. This thesis performs a case study comparing performance of systems over a year at a particular location to identify the energy production penalty caused by series connection relative to independent electrical connection.

The heat capacity of superfluid ^4He in the presence of a constant heat flux near Tλ

We present the first experimental evidence that the heat capacity of superfluid ⁴He, at temperatures very close to the lambda transition temperature, T_λ,is enhanced by a constant heat flux, Q. The heat capacity at constant Q, C_Q,is predicted to diverge at a temperature T_c(Q) < T_λ at which superflow becomes unstable. In agreement with previous measurements, we find that dissipation enters our cell at a temperature, T_DAS(Q),below the theoretical value, T_c(Q). Our measurements of C_Q were taken using the discrete pulse method at fourteen different heat flux values in the range 1µW/cm² ≤ Q≤ 4µW /cm². The excess heat capacity ∆C_Q we measure has the predicted scaling behavior as a function of T and Q:∆C_Q • t^α ∝ (Q/Q_c)², where Q_cT) ~ t^2ν is the critical heat current that results from the inversion of the equation for T_c(Q). We find that if the theoretical value of T_c( Q) is correct, then ∆C_Q is considerably larger than anticipated. On the other hand,if T_c(Q)≈ T_DAS(Q),then ∆C_Q is the same magnitude as the theoretically predicted enhancement.

Acceleration Sensing, Feedback Cooling, and Nonlinear Dynamics with Nanoscale Cavity-Optomechanical Devices

Light has long been used for the precise measurement of moving bodies, but the burgeoning field of optomechanics is concerned with the interaction of light and matter in a regime where the typically weak radiation pressure force of light is able to push back on the moving object. This field began with the realization in the late 1960's that the momentum imparted by a recoiling photon on a mirror would place fundamental limits on the smallest measurable displacement of that mirror. This coupling between the frequency of light and the motion of a mechanical object does much more than simply add noise, however. It has been used to cool objects to their quantum ground state, demonstrate electromagnetically-induced-transparency, and modify the damping and spring constant of the resonator. Amazingly, these radiation pressure effects have now been demonstrated in systems ranging 18 orders of magnitude in mass (kg to fg).

In this work we will focus on three diverse experiments in three different optomechanical devices which span the fields of inertial sensors, closed-loop feedback, and nonlinear dynamics. The mechanical elements presented cover 6 orders of magnitude in mass (ng to fg), but they all employ nano-scale photonic crystals to trap light and resonantly enhance the light-matter interaction. In the first experiment we take advantage of the sub-femtometer displacement resolution of our photonic crystals to demonstrate a sensitive chip-scale optical accelerometer with a kHz-frequency mechanical resonator. This sensor has a noise density of approximately 10 micro-g/rt-Hz over a useable bandwidth of approximately 20 kHz and we demonstrate at least 50 dB of linear dynamic sensor range. We also discuss methods to further improve performance of this device by a factor of 10.

In the second experiment, we used a closed-loop measurement and feedback system to damp and cool a room-temperature MHz-frequency mechanical oscillator from a phonon occupation of 6.5 million down to just 66. At the time of the experiment, this represented a world-record result for the laser cooling of a macroscopic mechanical element without the aid of cryogenic pre-cooling. Furthermore, this closed-loop damping yields a high-resolution force sensor with a practical bandwidth of 200 kHZ and the method has applications to other optomechanical sensors.

The final experiment contains results from a GHz-frequency mechanical resonator in a regime where the nonlinearity of the radiation-pressure interaction dominates the system dynamics. In this device we show self-oscillations of the mechanical element that are driven by multi-photon-phonon scattering. Control of the system allows us to initialize the mechanical oscillator into a stable high-amplitude attractor which would otherwise be inaccessible. To provide context, we begin this work by first presenting an intuitive overview of optomechanical systems and then providing an extended discussion of the principles underlying the design and fabrication of our optomechanical devices.

Multipole moments in general relativity and dynamical perturbations of black-hole magnetospheres

This thesis consists of two parts. In Part I, we develop a multipole moment formalism in general relativity and use it to analyze the motion and precession of compact bodies. More specifically, the generic, vacuum, dynamical gravitational field of the exterior universe in the vicinity of a freely moving body is expanded in positive powers of the distance r away from the body's spatial origin (i.e., in the distance r from its timelike-geodesic world line). The expansion coefficients, called "external multipole moments,'' are defined covariantly in terms of the Riemann curvature tensor and its spatial derivatives evaluated on the body's central world line. In a carefully chosen class of de Donder coordinates, the expansion of the external field involves only integral powers of r ; no logarithmic terms occur. The expansion is used to derive higher-order corrections to previously known laws of motion and precession for black holes and other bodies. The resulting laws of motion and precession are expressed in terms of couplings of the time derivatives of the body's quadrupole and octopole moments to the external moments, i.e., to the external curvature and its gradient.

In part II, we study the interaction of magnetohydrodynamic (MHD) waves in a black-hole magnetosphere with the "dragging of inertial frames" effect of the hole's rotation - i.e., with the hole's "gravitomagnetic field." More specifically: we first rewrite the laws of perfect general relativistic magnetohydrodynamics (GRMHD) in 3+1 language in a general spacetime, in terms of quantities (magnetic field, flow velocity, ...) that would be measured by the ''fiducial observers” whose world lines are orthogonal to (arbitrarily chosen) hypersurfaces of constant time. We then specialize to a stationary spacetime and MHD flow with one arbitrary spatial symmetry (e.g., the stationary magnetosphere of a Kerr black hole); and for this spacetime we reduce the GRMHD equations to a set of algebraic equations. The general features of the resulting stationary, symmetric GRMHD magnetospheric solutions are discussed, including the Blandford-Znajek effect in which the gravitomagnetic field interacts with the magnetosphere to produce an outflowing jet. Then in a specific model spacetime with two spatial symmetries, which captures the key features of the Kerr geometry, we derive the GRMHD equations which govern weak, linealized perturbations of a stationary magnetosphere with outflowing jet. These perturbation equations are then Fourier analyzed in time t and in the symmetry coordinate x, and subsequently solved numerically. The numerical solutions describe the interaction of MHD waves with the gravitomagnetic field. It is found that, among other features, when an oscillatory external force is applied to the region of the magnetosphere where plasma (e⁺e^-) is being created, the magnetosphere responds especially strongly at a particular, resonant, driving frequency. The resonant frequency is that for which the perturbations appear to be stationary (time independent) in the common rest frame of the freshly created plasma and the rotating magnetic field lines. The magnetosphere of a rotating black hole, when buffeted by nonaxisymmetric magnetic fields anchored in a surrounding accretion disk, might exhibit an analogous resonance. If so then the hole's outflowing jet might be modulated at resonant frequencies ω=(m/2) Ω_H where m is an integer and Ω_H is the hole's angular velocity.

I. Electron spin relaxation studies of manganese(II) complexes in acetonitrile. II. Nuclear spin relaxation studies of bromine in tetrabromomanganese(II) in acetonitrile

Magnetic resonance techniques have given us a powerful means for investigating dynamical processes in gases, liquids and solids. Dynamical effects manifest themselves in both resonance line shifts and linewidths, and, accordingly, require detailed analyses to extract desired information. The success of a magnetic resonance experiment depends critically on relaxation mechanisms to maintain thermal equilibrium between spin states. Consequently, there must be an interaction between the excited spin states and their immediate molecular environment which promote changes in spin orientation while excess magnetic energy is coupled into other degrees of freedom by non-radiative processes. This is well known as spin-lattice relaxation. Certain dynamical processes cause fluctuations in the spin state energy levels leading to spin-spin relaxation and, here again, the environment at the molecular level plays a significant role in the magnitude of interaction. Relatively few electron spin relaxation studies of solutions have been conducted and the present work is addressed toward the extension of our knowledge in this area and the retrieval of dynamical information from line shape analyses on a time scale comparable to diffusion controlled phenomena.

Specifically, the electron spin relaxation of three Mn⁺²3d⁵ complexes, Mn(CH₃CN)₆⁺², MnCl₄^-2 in acetonitrile has been studied in considerable detail. The effective spin Hamiltonian constants were carefully evaluated under a wide range of experimental conditions. Resonance widths of these Mn⁺² complexes were studied in the presence of various excess ligand ions and as a function of concentration, viscosity, temperature and frequency (X-band, ~9.5 Ԍ Hz and K-band, ~35 Ԍ Hz).

A number of interesting conclusions were drawn from these studies. For the Et₄NCl-₄^-2 system several relaxation mechanisms leading to resonance broadening were observed. One source appears to arise through spin-orbit interactions caused by modulation of the ligand field resulting from transient distortions of the complex imparted by solvent fluctuations in the immediate surroundings of the paramagnetic ion. An additional spin relaxation was assigned to the formation of ion pairs [Et₄N⁺…MnCl₄^-2] and it was possible to estimate the dissociation constant for this specie in acetonitrile.

The Bu₄NBr-MnBr₄^-2 study was considerably more interesting. As in the former case, solvent fluctuations and ion-pairing of the paramagnetic complex [Bu₄N⁺…MnBr₄^-2] provide significant relaxation for the electronic spin system. Most interesting, without doubt, is the onset of a new relaxation mechanism leading to resonance broadening which is best interpreted as chemical exchange. Thus, assuming that resonance widths were simply governed by electron spin state lifetimes, we were able to extract dynamical information from an interaction in which the initial and final states are the same

MnBr₄^-2 + Br^- = MnBr₄^-2 + Br^-.

The bimolecular rate constants were obtained at six different temperatures and their magnitudes suggested that the exchange is probably diffusion controlled with essentially a zero energy of activation. The most important source of spin relaxation in this system stems directly from dipolar interactions between the manganese 3d⁵ electrons. Moreover, the dipolar broadening is strongly frequency dependent indicating a deviation between the transverse and longitudinal relaxation times. We are led to the conclusion that the 3d⁵ spin states of ion-paired MnBr₄^-2 are significantly correlated so that dynamical processes are also entering the picture. It was possible to estimate the correlation time, ^Td, characterizing this dynamical process.

In Part II we study nuclear magnetic relaxation of bromine ions in the MnBr4-2-Bu4NBr-acetonitrile system. Essentially we monitor the 79Br and 81Br linewidths in response to the [MnBr4-2]/[Br-] ratio with the express purpose of supporting our contention that exchange is occurring between "free" bromine ions in the solvent and bromine in the first coordination sphere of the paramagnetic anion. The complexity of the system elicited a two-part study: (1) the linewidth behavior of Bu4NBr in anhydrous CH3CN in the absence of MnBr4-2 and (2) in the presence of MnBr4-2. It was concluded in study (1) that dynamical association, Bu4NBr k1= Bu4N+ + Br-, was modulating field-gradient interactions at frequencies high enough to provide an estimation of the unimolecular rate constant, k1. A comparison of the two isotopic bromine linewidth-mole fraction results led to the conclusion that quadrupole interactions provided the dominant relaxation mechanism. In study (2) the "residual" bromine linewidths for both 79Br and 81Br are clearly controlled by quadrupole interactions which appear to be modulated by very rapid dynamical processes other than molecular reorientation. We conclude that the "residual" linewidth has its origin in chemical exchange and that bromine nuclei exchange rapidly between a "free" solvated ion and the paramagnetic complex, MnBr4-2.

I. Nuclear spin-internal-rotation-coupling. II. Fluorine spin-rotation interaction and magnetic shielding in fluorobenzene

Part I.

The interaction of a nuclear magnetic moment situated on an internal top with the magnetic fields produced by the internal as well as overall molecular rotation has been derived following the method of Van Vleck for the spin-rotation interaction in rigid molecules. It is shown that the Hamiltonian for this problem may be written

H_SR = Ῑ · M · Ĵ + Ῑ · M” · Ĵ”

Where the first term is the ordinary spin-rotation interaction and the second term arises from the spin-internal-rotation coupling.

The F¹⁹ nuclear spin-lattice relaxation time (T₁) of benzotrifluoride and several chemically substituted benzotrifluorides, have been measured both neat and in solution, at room temperature by pulsed nuclear magnetic resonance. From these experimental results it is concluded that in benzotrifluoride the internal rotation is crucial to the spin relaxation of the fluorines and that the dominant relaxation mechanism is the fluctuating spin-internal-rotation interaction.

Part II.

The radiofrequency spectrum corresponding to the reorientation of the F¹⁹ nuclear moment in flurobenzene has been studied by the molecular beam magnetic resonance method. A molecular beam apparatus with an electron bombardment detector was used in the experiments. The F¹⁹ resonance is a composite spectrum with contributions from many rotational states and is not resolved. A detailed analysis of the resonance line shape and width by the method of moments led to the following diagonal components of the fluorine spin-rotational tensor in the principal inertial axis system of the molecule:

F/Caa = -1.0 ± 0.5 kHz

F/Cbb = -2.7 ± 0.2 kHz

F/Ccc = -1.9 ± 0.1 kHz

From these interaction constants, the paramagnetic contribution to the F¹⁹ nuclear shielding in C₆H₅F was determined to be -284 ± ppm. It was further concluded that the F¹⁹ nucleus in this molecule is more shielded when the applied magnetic field is directed along the C-F bond axis. The anisotropy of the magnetic shielding tensor, σ_” - σ_⊥, is +160 ± 30 ppm.