Ferromagnetic Resonance Force Microscopy

A new approach to magnetic resonance was introduced in 1992 based upon detection of spin-induced forces by J. Sidles [1]. This technique, now called magnetic resonance force microscopy (MRFM), was first demonstrated that same year via electron paramagnetic resonance (EPR) by D. Rugar et al. [2]. This new method combines principles of magnetic resonance with those of scanned probe technology to detect spin resonance through mechanical, rather than inductive, means. In this thesis the development and use of ferromagnetic resonance force microscopy (FMRFM) is described. This variant of MRFM, which allows investigation of ferromagnetic samples, was first demonstrated in 1996 by Z. Zhang et al. [3]. FMRFM enables characterization of (a) the dynamic magnetic properties of microscale magnetic devices, and (b) the spatial dependence of ferromagnetic resonance within a sample. Both are impossible with conventional ferromagnetic resonance techniques.

Ferromagnetically coupled systems, however, pose unique challenges for force detection. In this thesis the attainable spatial resolution - and the underlying physical mechanisms that determine it - are established. We analyze the dependence of the magnetostatic modes upon sample dimensions using a series of microscale yttrium iron garnet (YIG) samples. Mapping of mode amplitudes within these sample is attained with an unprecedented spatial resolution of 15μm. The modes, never before analyzed on this scale, fit simple models developed in this thesis for samples of micron dimensions. The application of stronger gradient fields induces localized perturbation of the ferromagnetic resonance modes. The first demonstrations of this effect are presented in this study, and a simple theoretical model is developed to explain our observations. The results indicate that the characteristics of the locally-detected ferromagnetic modes are still largely determined by the external fields and dimensions of the entire sample, rather than by the localized interaction volume (i.e., the locale most strongly affected by the local gradient field). Establishing this is a crucial first step toward understanding FMRFM in the high gradient field limit where the dispersion relations become locally determined. In this high gradient field regime, FMRFM imaging becomes analogous with that of EPR MRFM.

FMRFM has also been employed to characterize magnetic multilayers, similar to those utilized in giant magnetoresistance (GMR) devices, on a lateral scale 40 x 40μm. This is orders of magnitude smaller than possible via conventional methods. Anisotropy energies, thickness, and interface qualities of individual layers have been resolved.

This initial work clearly demonstrates the immense and unique potential that FMRFM offers for characterizing advanced magnetic nanostructures and magnetic devices.

Exact solutions and transformation properties of nonlinear partial differential equations from general relativity

Various families of exact solutions to the Einstein and Einstein-Maxwell field equations of General Relativity are treated for situations of sufficient symmetry that only two independent variables arise. The mathematical problem then reduces to consideration of sets of two coupled nonlinear differential equations.

The physical situations in which such equations arise include: a) the external gravitational field of an axisymmetric, uncharged steadily rotating body, b) cylindrical gravitational waves with two degrees of freedom, c) colliding plane gravitational waves, d) the external gravitational and electromagnetic fields of a static, charged axisymmetric body, and e) colliding plane electromagnetic and gravitational waves. Through the introduction of suitable potentials and coordinate transformations, a formalism is presented which treats all these problems simultaneously. These transformations and potentials may be used to generate new solutions to the Einstein-Maxwell equations from solutions to the vacuum Einstein equations, and vice-versa.

The calculus of differential forms is used as a tool for generation of similarity solutions and generalized similarity solutions. It is further used to find the invariance group of the equations; this in turn leads to various finite transformations that give new, physically distinct solutions from old. Some of the above results are then generalized to the case of three independent variables.

Synthetic molecular machines for active self-assembly : prototype algorithms, designs, and experimental study

Computer science and electrical engineering have been the great success story of the twentieth century. The neat modularity and mapping of a language onto circuits has led to robots on Mars, desktop computers and smartphones. But these devices are not yet able to do some of the things that life takes for granted: repair a scratch, reproduce, regenerate, or grow exponentially fast–all while remaining functional.

This thesis explores and develops algorithms, molecular implementations, and theoretical proofs in the context of “active self-assembly” of molecular systems. The long-term vision of active self-assembly is the theoretical and physical implementation of materials that are composed of reconfigurable units with the programmability and adaptability of biology’s numerous molecular machines. En route to this goal, we must first find a way to overcome the memory limitations of molecular systems, and to discover the limits of complexity that can be achieved with individual molecules.

One of the main thrusts in molecular programming is to use computer science as a tool for figuring out what can be achieved. While molecular systems that are Turing-complete have been demonstrated [Winfree, 1996], these systems still cannot achieve some of the feats biology has achieved.

One might think that because a system is Turing-complete, capable of computing “anything,” that it can do any arbitrary task. But while it can simulate any digital computational problem, there are many behaviors that are not “computations” in a classical sense, and cannot be directly implemented. Examples include exponential growth and molecular motion relative to a surface.

Passive self-assembly systems cannot implement these behaviors because (a) molecular motion relative to a surface requires a source of fuel that is external to the system, and (b) passive systems are too slow to assemble exponentially-fast-growing structures. We call these behaviors “energetically incomplete” programmable behaviors. This class of behaviors includes any behavior where a passive physical system simply does not have enough physical energy to perform the specified tasks in the requisite amount of time.

As we will demonstrate and prove, a sufficiently expressive implementation of an “active” molecular self-assembly approach can achieve these behaviors. Using an external source of fuel solves part of the the problem, so the system is not “energetically incomplete.” But the programmable system also needs to have sufficient expressive power to achieve the specified behaviors. Perhaps surprisingly, some of these systems do not even require Turing completeness to be sufficiently expressive.

Building on a large variety of work by other scientists in the fields of DNA nanotechnology, chemistry and reconfigurable robotics, this thesis introduces several research contributions in the context of active self-assembly.

We show that simple primitives such as insertion and deletion are able to generate complex and interesting results such as the growth of a linear polymer in logarithmic time and the ability of a linear polymer to treadmill. To this end we developed a formal model for active-self assembly that is directly implementable with DNA molecules. We show that this model is computationally equivalent to a machine capable of producing strings that are stronger than regular languages and, at most, as strong as context-free grammars. This is a great advance in the theory of active self- assembly as prior models were either entirely theoretical or only implementable in the context of macro-scale robotics.

We developed a chain reaction method for the autonomous exponential growth of a linear DNA polymer. Our method is based on the insertion of molecules into the assembly, which generates two new insertion sites for every initial one employed. The building of a line in logarithmic time is a first step toward building a shape in logarithmic time. We demonstrate the first construction of a synthetic linear polymer that grows exponentially fast via insertion. We show that monomer molecules are converted into the polymer in logarithmic time via spectrofluorimetry and gel electrophoresis experiments. We also demonstrate the division of these polymers via the addition of a single DNA complex that competes with the insertion mechanism. This shows the growth of a population of polymers in logarithmic time. We characterize the DNA insertion mechanism that we utilize in Chapter 4. We experimentally demonstrate that we can control the kinetics of this re- action over at least seven orders of magnitude, by programming the sequences of DNA that initiate the reaction.

In addition, we review co-authored work on programming molecular robots using prescriptive landscapes of DNA origami; this was the first microscopic demonstration of programming a molec- ular robot to walk on a 2-dimensional surface. We developed a snapshot method for imaging these random walking molecular robots and a CAPTCHA-like analysis method for difficult-to-interpret imaging data.

Arithmetic of cyclotomic fields and Fermat-type equations

Let l be any odd prime, and ζ a primitive l-th root of unity. Let C_l be the l-Sylow subgroup of the ideal class group of Q(ζ). The Teichmüller character w : Z_l → Z^*_l is given by w(x) = x (mod l), where w(x) is a p-1-st root of unity, and x ∈ Z_l. Under the action of this character, C_l decomposes as a direct sum of C^((i))_l, where C^((i))_l is the eigenspace corresponding to w^i. Let the order of C^((3))_l be l^h_3). The main result of this thesis is the following: For every n ≥ max( 1, h_3 ), the equation x^(ln) + y^(ln) + z^(ln) = 0 has no integral solutions (x,y,z) with l ≠ xyz. The same result is also proven with n ≥ max(1,h_5), under the assumption that C_l^((5)) is a cyclic group of order l^h_5. Applications of the methods used to prove the above results to the second case of Fermat's last theorem and to a Fermat-like equation in four variables are given.

The proof uses a series of ideas of H.S. Vandiver ([Vl],[V2]) along with a theorem of M. Kurihara [Ku] and some consequences of the proof of lwasawa's main conjecture for cyclotomic fields by B. Mazur and A. Wiles [MW]. In [V1] Vandiver claimed that the first case of Fermat's Last Theorem held for l if l did not divide the class number h^+ of the maximal real subfield of Q(e^(2πi/i)). The crucial gap in Vandiver's attempted proof that has been known to experts is explained, and complete proofs of all the results used from his papers are given.

Did galaxies reionize the universe?

The epoch of reionization remains one of the last uncharted eras of cosmic history, yet this time is of crucial importance, encompassing the formation of both the first galaxies and the first metals in the universe. In this thesis, I present four related projects that both characterize the abundance and properties of these first galaxies and uses follow-up observations of these galaxies to achieve one of the first observations of the neutral fraction of the intergalactic medium during the heart of the reionization era.

First, we present the results of a spectroscopic survey using the Keck telescopes targeting 6.3 < z < 8.8 star-forming galaxies. We secured observations of 19 candidates, initially selected by applying the Lyman break technique to infrared imaging data from the Wide Field Camera 3 (WFC3) onboard the Hubble Space Telescope (HST). This survey builds upon earlier work from Stark et al. (2010, 2011), which showed that star-forming galaxies at 3 < z < 6, when the universe was highly ionized, displayed a significant increase in strong Lyman alpha emission with redshift. Our work uses the LRIS and NIRSPEC instruments to search for Lyman alpha emission in candidates at a greater redshift in the observed near-infrared, in order to discern if this evolution continues, or is quenched by an increase in the neutral fraction of the intergalactic medium. Our spectroscopic observations typically reach a 5-sigma limiting sensitivity of < 50 AA. Despite expecting to detect Lyman alpha at 5-sigma in 7-8 galaxies based on our Monte Carlo simulations, we only achieve secure detections in two of 19 sources. Combining these results with a similar sample of 7 galaxies from Fontana et al. (2010), we determine that these few detections would only occur in < 1% of simulations if the intrinsic distribution was the same as that at z ~ 6. We consider other explanations for this decline, but find the most convincing explanation to be an increase in the neutral fraction of the intergalactic medium. Using theoretical models, we infer a neutral fraction of X_HI ~ 0.44 at z = 7.

Second, we characterize the abundance of star-forming galaxies at z > 6.5 again using WFC3 onboard the HST. This project conducted a detailed search for candidates both in the Hubble Ultra Deep Field as well as a number of additional wider Hubble Space Telescope surveys to construct luminosity functions at both z ~ 7 and 8, reaching 0.65 and 0.25 mag fainter than any previous surveys, respectively. With this increased depth, we achieve some of the most robust constraints on the Schechter function faint end slopes at these redshifts, finding very steep values of alpha_{z~7} = -1.87 +/- 0.18 and alpha_{z~8} = -1.94 +/- 0.23. We discuss these results in the context of cosmic reionization, and show that given reasonable assumptions about the ionizing spectra and escape fraction of ionizing photons, only half the photons needed to maintain reionization are provided by currently observable galaxies at z ~ 7-8. We show that an extension of the luminosity function down to M_{UV} = -13.0, coupled with a low level of star-formation out to higher redshift, can fit all available constraints on the ionization history of the universe.

Third, we investigate the strength of nebular emission in 3 < z < 5 star-forming galaxies. We begin by using the Infrared Array Camera (IRAC) onboard the Spitzer Space Telescope to investigate the strength of H alpha emission in a sample of 3.8 < z < 5.0 spectroscopically confirmed galaxies. We then conduct near-infrared observations of star-forming galaxies at 3 < z < 3.8 to investigate the strength of the [OIII] 4959/5007 and H beta emission lines from the ground using MOSFIRE. In both cases, we uncover near-ubiquitous strong nebular emission, and find excellent agreement between the fluxes derived using the separate methods. For a subset of 9 objects in our MOSFIRE sample that have secure Spitzer IRAC detections, we compare the emission line flux derived from the excess in the K_s band photometry to that derived from direct spectroscopy and find 7 to agree within a factor of 1.6, with only one catastrophic outlier. Finally, for a different subset for which we also have DEIMOS rest-UV spectroscopy, we compare the relative velocities of Lyman alpha and the rest-optical nebular lines which should trace the cites of star-formation. We find a median velocity offset of only v_{Ly alpha} = 149 km/s, significantly less than the 400 km/s observed for star-forming galaxies with weaker Lyman alpha emission at z = 2-3 (Steidel et al. 2010), and show that this decrease can be explained by a decrease in the neutral hydrogen column density covering the galaxy. We discuss how this will imply a lower neutral fraction for a given observed extinction of Lyman alpha when its visibility is used to probe the ionization state of the intergalactic medium.

Finally, we utilize the recent CANDELS wide-field, infra-red photometry over the GOODS-N and S fields to re-analyze the use of Lyman alpha emission to evaluate the neutrality of the intergalactic medium. With this new data, we derive accurate ultraviolet spectral slopes for a sample of 468 3 < z < 6 star-forming galaxies, already observed in the rest-UV with the Keck spectroscopic survey (Stark et al. 2010). We use a Bayesian fitting method which accurately accounts for contamination and obscuration by skylines to derive a relationship between the UV-slope of a galaxy and its intrinsic Lyman alpha equivalent width probability distribution. We then apply this data to spectroscopic surveys during the reionization era, including our own, to accurately interpret the drop in observed Lyman alpha emission. From our most recent such MOSFIRE survey, we also present evidence for the most distant galaxy confirmed through emission line spectroscopy at z = 7.62, as well as a first detection of the CIII]1907/1909 doublet at z > 7.

We conclude the thesis by exploring future prospects and summarizing the results of Robertson et al. (2013). This work synthesizes many of the measurements in this thesis, along with external constraints, to create a model of reionization that fits nearly all available constraints.

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.

A kinetic theory description for external spherical flows with arbitrary Knudsen number by a moment method

The Maxwell integral equations of transfer are applied to a series of problems involving flows of arbitrary density gases about spheres. As suggested by Lees a two sided Maxwellian-like weighting function containing a number of free parameters is utilized and a sufficient number of partial differential moment equations is used to determine these parameters. Maxwell's inverse fifth-power force law is used to simplify the evaluation of the collision integrals appearing in the moment equations. All flow quantities are then determined by integration of the weighting function which results from the solution of the differential moment system. Three problems are treated: the heat-flux from a slightly heated sphere at rest in an infinite gas; the velocity field and drag of a slowly moving sphere in an unbounded space; the velocity field and drag torque on a slowly rotating sphere. Solutions to the third problem are found to both first and second-order in surface Mach number with the secondary centrifugal fan motion being of particular interest. Singular aspects of the moment method are encountered in the last two problems and an asymptotic study of these difficulties leads to a formal criterion for a "well posed" moment system. The previously unanswered question of just how many moments must be used in a specific problem is now clarified to a great extent.

Surface impedance theory for superconductors in large static magnetic fields

A theory of electromagnetic absorption is presented to explain the changes in surface impedance for Pippard superconductors (ξ_o ≫λ) due to large static magnetic fields. The static magnetic field penetrating the metal near the surface induces a momentum dependent potential in Bogolubov's equations. Such a potential modifies a quasiparticle's wavefunction and excitation spectrum. These changes affect the behavior of the surface impedance in a way that in large measure agrees with available observations.

Microwave dynamics of quasiparticles and critical fields in superconducting films

The microwave response of the superconducting state in equilibrium and non-equilibrium configurations was examined experimentally and analytically. Thin film superconductors were mostly studied in order to explore spatial effects. The response parameter measured was the surface impedance.

For small microwave intensity the surface impedance at 10 GHz was measured for a variety of samples (mostly Sn) over a wide range of sample thickness and temperature. A detailed analysis based on the BCS theory was developed for calculating the surface impedance for general thickness and other experimental parameters. Experiment and theory agreed with each other to within the experimental accuracy. Thus it was established that the samples, thin films as well as bulk, were well characterised at low microwave powers (near equilibrium).

Thin films were perturbed by a small dc supercurrent and the effect on the superconducting order parameter and the quasiparticle response determined by measuring changes in the surface resistance (still at low microwave intensity and independent of it) due to the induced current. The use of fully superconducting resonators enabled the measurement of very small changes in the surface resistance (< 10^-9 Ω/sq.). These experiments yield information regarding the dynamics of the order parameter and quasiparticle systems. For all the films studied the results could be described at temperatures near T_c by the thermodynamic depression of the order parameter due to the static current leading to a quadratic increase of the surface resistance with current.

For the thinnest films the low temperature results were surprising in that the surface resistance decreased with increasing current. An explanation is proposed according to which this decrease occurs due to an additional high frequency quasiparticle current caused by the combined presence of both static and high frequency fields. For frequencies larger than the inverse of the quasiparticle relaxation time this additional current is out of phase (by π) with the microwave electric field and is observed as a decrease of surface resistance. Calculations agree quantitatively with experimental results. This is the first observation and explanation of this non-equilibrium quasiparticle effect.

For thicker films of Sn, the low temperature surface resistance was found to increase with applied static current. It is proposed that due to the spatial non-uniformity of the induced current distribution across the thicker films, the above purely temporal analysis of the local quasiparticle response needs to be generalised to include space and time non-equilibrium effects.

The nonlinear interaction of microwaves arid superconducting films was also examined in a third set of experiments. The surface impedance of thin films was measured as a function of the incident microwave magnetic field. The experiments exploit the ability to measure the absorbed microwave power and applied microwave magnetic field absolutely. It was found that the applied surface microwave field could not be raised above a certain threshold level at which the absorption increased abruptly. This critical field level represents a dynamic critical field and was found to be associated with the penetration of the app1ied field into the film at values well below the thermodynamic critical field for the configuration of a field applied to one side of the film. The penetration occurs despite the thermal stability of the film which was unequivocally demonstrated by experiment. A new mechanism for such penetration via the formation of a vortex-antivortex pair is proposed. The experimental results for the thinnest films agreed with the calculated values of this pair generation field. The observations of increased transmission at the critical field level and suppression of the process by a metallic ground plane further support the proposed model.