Results for “Phase correlation” — Page 39

DOE OSTI2025

Statistically elucidated responses from low-signal contrast mechanisms in ultrafast electron microscopy

The emergence of ultrafast electron microscopy (UEM) has enabled the discovery of strongly correlated dynamic mechanisms, including electron–phonon coupling, structural phase transitions, thermal transport, and electromagnetic deflection. Most UEM systems operate stroboscopically, meaning that the technique is susceptible to artifacts, mistakes, and misinterpretation of the data due to extensive experimental effort. In contrast to the ultrafast designation, data acquisition is extraordinarily slow because the electron beam has significantly reduced signal compared to traditional transmission electron microscopy due to pulsing the electron beam. Consequently, the sample may drift, tilt, or undergo irreversible structural changes that are independent of the time-resolved dynamics throughout the experimental time frame. Furthermore, these datasets require significant user interpretation that can be problematic when proper controls are not implemented thoroughly. Here, we demonstrate a new algorithm designed to separate ultrafast structural dynamics from long-term artifacts using a LiNbO 3 sample experiencing electrically driven surface acoustic wave propagation. Additionally, we provide examples of the impact of user bias when analyzing the data and provide a methodology, which enables the extraction of time-resolved responses when the image signal is extraordinarily low. Overall, the goal of this publication is to provide methods that validate the experimental results and reduce researcher biases during UEM data interpretation.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2020

NWChem: Past, present, and future

Specialized computational chemistry packages have permanently reshaped the landscape of chemical sciences by providing tools to support and guide the experimental effort and for prediction of chemical and materials properties. In this regard, a special role has been played by electronic structure packages where complex chemical and materials processes can be modeled using first-principle-driven methodologies. Over the last few decades, the rapid development of computing technologies and a tremendous increase in computational power has offered a unique chance to study complex chemical transformations using sophisticated and predictive many-body techniques to describe correlated behavior of electrons in molecular and condensed phase systems at different levels of theory. In enabling these simulations, a critical role has been played by novel parallel algorithms capable of taking advantage of computational resources to address polynomial scaling of electronic structure methods. NWChem was among the first electronic structure codes that focused on delivering scalable performance for electronic structure simulations. Herein, we briefly review the NWChem suite of computational codes including its history, design principles, parallel tools, current capabilities, outreach and outlook.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2022

Approximate bounds and temperature dependence of adiabatic connection integrands for the uniform electron gas

Thermal density functional theory is commonly used in simulations of warm dense matter, a highly energetic phase characterized by substantial thermal effects and by correlated electrons demanding quantum mechanical treatment. Methods that account for temperature dependence, such as Mermin–Kohn–Sham finite-temperature density functional theory and free energy density functional theory, are now employed with more regularity and available in many standard code packages. However, approximations from zero-temperature density functional theory are still often used in temperature-dependent simulations using thermally weighted electronic densities as an input to exchange–correlation functional approximations, a practice known to miss temperature-dependent effects in the exchange–correlation free energy of these systems. Here, in this work, the temperature-dependent adiabatic connection is demonstrated and analyzed using a well-known parameterization of the uniform electron gas free energy. Useful tools based on this formalism for analyzing and constraining approximations of the exchange–correlation at zero temperature are leveraged for the finite-temperature case. Inspired by the Lieb–Oxford inequality, which provides a lower bound for the ground-state exchange–correlation energy, bounds for the exchange–correlation at finite temperatures are approximated for various degrees of electronic correlation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

Effective description of ajar systems with a 𝑈⁡(1) symmetry

We introduce the concept of “ajar systems” as an intermediate case between closed and open systems, where the time scale for charge exchange with the environment is parametrically larger than all other characteristic time scales. The Schwinger-Keldysh effective action for such finite-temperature systems exhibits a symmetry group G 1 × G 2 weakly broken to its diagonal subgroup G diag , which we systematically describe using spurion techniques. For G = U(1), we calculate leading-order corrections to correlation functions in both diffusive and spontaneously broken phases. Unlike systems with approximate symmetries where both real and imaginary parts of dispersion relations receive corrections, ajar systems modify only the imaginary part—preserving gapless modes while adding finite damping.

Effective field theory↗

DOE OSTI2024

Observation of paramagnetic spin-degeneracy lifting in EuZn 2 Sb 2

Taken together, time-reversal and spatial inversion symmetries impose a twofold spin degeneracy of the electronic states in crystals. In centrosymmetric materials, this degeneracy can be lifted by introducing magnetism, either via an externally applied field or through internal magnetization. However, a correlated alignment of spins, even in the paramagnetic phase, can lift the spin degeneracy of electronic states. Here, we report an in-depth study of the electronic band structure of the Eu-ternary pnictide EuZn 2 Sb 2 through a combination of high-resolution angle-resolved photoemission spectroscopy measurements and first-principles calculations. An analysis of the photoemission line shapes over a range of incident photon energies and sample temperatures is shown to reveal the presence of band spin-degeneracy lifting in the paramagnetic phase. Our angle-resolved photoemission spectroscopy results are in good agreement with theoretical ferromagnetic-phase calculations, which indicates the importance of ferromagnetic fluctuations in the system. Through our calculations, we predict that spin-polarized bands in EuZn 2 Sb 2 generate a single pair of Weyl nodes. Our observation of band splittingin EuZn 2 Sb 2 provides a key step toward realizing time-reversal symmetry breaking physics in the absence of long-range magnetic order.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2021

Modeling quasielastic interactions of monoenergetic kaon decay-at-rest neutrinos

Monoenergetic muon neutrinos at 236 MeV are readily produced in intense medium-energy proton facilities (≳2–3GeV) when a positive kaon decays at rest (KDAR; K + →μ + ν μ ). These neutrinos provide a unique opportunity to both study the neutrino interaction and probe the nucleus with a monoenergetic weak-interaction-only tool. We present cross-section calculations for quasielastic scattering of these 236-MeV neutrinos off 12 C and 40 Ar, paying special attention to low-energy aspects of the scattering process. Our model takes the description of the nucleus in a mean-field approach as the starting point, where we solve Hartree-Fock equations using a Skyrme type nucleon-nucleon interaction. Thereby, we introduce long-range nuclear correlations by means of a continuum random-phase approximation (CRPA) framework where we solve the CRPA equations using a Green's function method. The model successfully describes (e,e') data on 12 C and 40 Ca in the kinematic region that overlaps with the KDAR ν μ phase space. In addition to these results, we present future prospects for precision KDAR cross-section measurements and applications of our calculations in current and future experiments that will utilize these neutrinos.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

DOE OSTI2021

Many-body forces and nucleon clustering near the QCD critical point

It has been proposed that one can look for the QCD critical point (CP) by the Beam Energy Scan accurately monitoring event-by-event fluctuations. This experimental program is under way at the BNL RHIC collider. Separately, it has been studied how clustering of nucleons at freeze out affects proton multiplicity distribution and light nuclei production. It was found that even a minor increase of the range of nuclear forces dramatically increases clustering, while large correlation length ξ near CP makes attraction due to binary forces unrealistically large. Here, in this paper, we show that repulsive many-body forces near CP should overcome the binary ones and effectively suppress clustering. We also discuss current experimental data and point out locations at which a certain drop in clustering may already be observed.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2025

Dihadron azimuthal correlations in deep-inelastic scattering off nuclear targets

We measured the nuclear dependence of the di-pion azimuthal correlation function in deepinelastic scattering (DIS) using the CEBAF Large Acceptance Spectrometer (CLAS) and a 5 GeV electron beam. As the nuclear-target size increases, transitioning from deuterium to carbon, iron, and lead, the correlation function broadens monotonically. Its shape exhibits a significant dependence on kinematics, including the transverse momentum of the pions and the difference in their rapidity. None of the various Monte-Carlo event generators we evaluated could fully replicate the observed correlation functions and nuclear effects throughout the entire phase space. As the first study of its kind in DIS experiments, this research provides an important baseline for enhancing our understanding of the interplay between the nuclear medium and the hadronization process in these reactions.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2022

Binary neutron star mergers as a probe of quark-hadron crossover equations of state

It is anticipated that the gravitational radiation detected in future gravitational wave (GW) detectors from binary neutron star (NS) mergers can probe the high-density equation of state (EOS). In this work, we perform the first simulations of binary NS mergers which adopt various parametrizations of the quark-hadron crossover (QHC) EOS. These are constructed from combinations of a hadronic EOS ($n_b$<2 $n_0$) and a quark-matter EOS ($n_b$>5 $n_0$), where $n_b$ and $n_0$ are the baryon number density and the nuclear saturation density, respectively. At the crossover densities (2 $n_0$ < $n_b$ < 5 $n_0$) the QHC EOSs continuously soften, while remaining stiffer than hadronic and first-order phase transition EOSs, achieving the stiffness of strongly correlated quark matter. This enhanced stiffness leads to significantly longer lifetimes of the postmerger NS than that for a pure hadronic EOS. We find a dual nature of these EOSs such that their maximum chirp GW frequencies $f_{\text{max}}$ fall into the category of a soft EOS while the dominant peak frequencies ($f_{\text{peak}}$) of the postmerger stage fall in between that of a soft and stiff hadronic EOS. An observation of this kind of dual nature in the characteristic GW frequencies will provide crucial evidence for the existence of strongly interacting quark matter at the crossover densities for QCD.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2023

Spin alignment of vector mesons by glasma fields

We explain how spin alignment of vector mesons can be induced by background fields, such as electromagnetic fields or soft gluon fields. Our study is based on the quantum kinetic theory of spinning quarks and antiquarks and incorporates the relaxation of the dynamically generated spin polarization. The spin density matrix of vector mesons is obtained by quark coalescence via the Wigner function and kinetic equation. Our approach predicts a local spin correlation that is distinct from the nonlocal expressions previously obtained in phenomenological derivations. We estimate the magnitude of such local correlations in the glasma model of the preequilibrium phase of relativistic heavy ion collisions. It is found that the resulting spin alignment could be greatly enhanced and may be comparable to the experimental measurement in order of magnitude. We further propose new phenomenological scenarios to qualitatively explain the transverse-momentum and centrality dependence of spin alignment in a self-consistent framework.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2026

Dichotomy of flat bands in the van der Waals ferromagnet Fe 5 ⁢GeTe 2

Quantum materials with bands of narrow bandwidth near the Fermi level represent a promising platform for exploring a diverse range of fascinating physical phenomena, as the high density of states within the small energy window often enables the emergence of many-body physics. On one hand, flat bands can arise from strong Coulomb interactions that localize atomic orbitals. On the other hand, quantum destructive interference can quench the electronic kinetic energy. Although both have a narrow bandwidth, the two types of flat bands should exhibit very distinct spectral properties arising from their distinctive origins. So far, the two types of flat bands have only been realized in very different material settings and chemical environments, preventing a direct comparison. Here, in this study, we report the observation of the two types of flat bands within the same material system—an above-room-temperature van der Waals ferromagnet, Fe 5−x GeTe 2 , distinguishable by a switchable iron site order. The contrasting nature of the flat bands is also identified by the remarkably distinctive temperature evolution of the spectral features, indicating that one arises from electron correlations in the Fe(1) site-disordered phase, while the other geometrical frustration in the Fe(1) site-ordered phase. Our results therefore provide a direct juxtaposition of the distinct formation mechanism of flat bands in quantum materials and an avenue for understanding the distinctive roles flat bands play in the presence of magnetism, topology, and lattice geometrical frustration, utilizing sublattice ordering as a key control parameter.

Wu, Han [Rice Univ., Houston, TX (United States)] ↗

DOE OSTI2022

Experimental Investigation of Interactions Between Two Closely Spaced Azimuthal Modes in a Multinozzle Can Combustor

Thermoacoustic instabilities in annular or circular combustors are often coupled with azimuthal modes. These modes can be characterized by a spin ratio (SR), which quantifies the dominant mode between two counter-rotating waves, and a phase difference between them, which is directly related to the orientation of the antinodal line. This study investigates the instability amplitudes, SR, and phase difference of two closely spaced (3% of their mean frequency), yet distinct azimuthal modes; one is the first azimuthal (1A) mode, and the other is a combination of first azimuthal and first longitudinal (1A1L) mode. Each mode itself consists of two peaks that are spaced even more closely in frequency (0.8%). Furthermore, distinct harmonics at 2× and 3× of these frequencies, presumably associated with nonlinearities, are also evident in the spectra. Each mode is bandpass filtered in spectrum to analyze them separately. For the 1A mode, the SR and phase difference exhibit a variety of behaviors—including quasi-periodic standing waves, spinning waves, and intermittency—depending on operating conditions such as thermal power and azimuthal fuel staging. Similar trends are observed for the 1A1 L mode. Moreover, there is clear coupling between the 1A and 1A1 L modes, as their SRs are almost synchronized during the quasi-periodic standing wave. This synchronization is observed in phase differences as well, but not in the instability amplitude. For spinning dominant wave conditions, the SRs of each mode have similar average values, but they fluctuate in a seemingly random fashion. For the phase difference, both average and fluctuation are not correlated. In contrast, the instability amplitudes are strongly correlated, with modulation of the 1A mode leading to that of the 1A1 L mode. Furthermore, these results clearly indicate that complex coupling occurs across closely spaced frequencies under instability conditions, coupling that must be understood in order to capture limit cycle dynamics.

33 ADVANCED PROPULSION SYSTEMS↗

DOE OSTI2024

KCl-UCl 3 molten salts investigated by Ab Initio Molecular Dynamics (AIMD) simulations: A comparative study with three dispersion models

Ab Initio Molecular Dynamics (AIMD) simulations are performed on molten KCl-UCl 3 salt mixtures to determine energies, heat capacities, and densities. The density-dependent energy correction (DFT-dDsC), Grimme et al.’s DFT-D3, and Langreth & Lundqvist (vdW-cx) models are used for dispersion forces and combined with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation potential with a Hubbard U param eter for the 5$f$ electrons of uranium. After validating predictions for the end-member systems to literature data, KCl-UCl 3 mixtures are studied at select temperatures. Densities and energies both deviate from ideal solution behavior, with the maximum deviation occurring around 36% UCl 3 for mixing energies and slightly lower (29% UCl 3 ) for densities. Compared to the NaCl-UCl 3 system, which was previously investigated using the same simulation methodologies, the KCl-UCl 3 density and mixing energy deviations from ideal solution behavior are larger by almost a factor of two. No deviation from ideal solution behavior for heat capacity was observed. The AIMD predictions for mixing energies and densities agree qualitatively with experimental data, though the spread in data obtained from the various dispersion force models utilized, measurements, and empirical estimates makes strong conclusions difficult. The dependence of thermodynamic and thermophysical properties on composition is correlated with the local chemistry of the solution phase, in particular, the tendency of UCl 3 to form network structures.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

DOE OSTI2025

Transformation of TiN to TiNO Films via In-Situ Temperature-Dependent Oxygen Diffusion Process and Their Electrochemical Behavior

Titanium oxynitride (TiNO) thin films represent a multifaceted material system applicable in diverse fields, including energy storage, solar cells, sensors, protective coatings, and electrocatalysis. This study reports the synthesis of TiNO thin films grown at different substrate temperatures using pulsed laser deposition. A comprehensive structural investigation was conducted by X-ray diffraction (XRD), X-ray photoelectron spectroscopy (XPS), Non-Rutherford backscattering spectrometry (N-RBS), and X-ray absorption spectroscopy (XAS), which facilitated a detailed analysis that determined the phase, composition, and crystallinity of the films. Structural control was achieved via temperature-dependent oxygen in-diffusion, nitrogen out-diffusion, and the nucleation growth process related to adatom mobility. The XPS analysis indicates that the TiNO films consist of heterogeneous mixtures of TiN, TiNO, and TiO2 phases with temperature-dependent relative abundances. The correlation between the structure and electrochemical behavior of the thin films was examined. The TiNO films with relatively higher N/O ratio, meaning less oxidized, were more electrochemically active than the films with lower N/O ratio, i.e., more oxidized films. Films with higher oxidation levels demonstrated enhanced crystallinity and greater stability under electrochemical polarization. These findings demonstrate the importance of substrate temperature control in tailoring the properties of TiNO film, which is a fundamental part of designing and optimizing an efficient electrode material.

Cherono, Sheilah↗

DOE OSTI2021

xsacrcfrspccopol (a1)

ARM's scanning cloud radars are fully coherent dual-frequency, dual-polarization Doppler radars mounted on a common scanning pedestal. Each pedestal includes a Ka-band radar (2kW peak power) and the deployment location determines whether the second radar is a W-band (WSACR; 1.7 kW peak power) or X-band (XSACR; 20 kW peak power). At ARM's tropical sites, X-band radars are paired with the Ka-band because they are better suited for the atmospheric attenuation in this region. Beamwidth for the X-band is approximately 1 degree, and the Ka-band beamwidth is roughly 0.3 degrees. Due to the narrow antenna beamwidth, ARM’s scanning cloud radars use scanning strategies unlike typical weather radars. Rather than focusing on plan position indicator, or PPI, scans, the XSACR uses range height indicator, or RHI, scans at numerous azimuths to obtain cloud volume data. For the second ARM Mobile Facility (AMF2), the XSACR can be dismounted from the pedestal it shares with the KASACR and mounted on a separate pedestal. This allows the XSACR to operate more like a weather (precipitation) radar in deployments where such local coverage is lacking. Measurements collected with the XSACR are copolar and cross-polar radar reflectivity, Doppler velocity, spectra width and spectra when not scanning, differential Reflectivity (Zdr), correlation coefficient (rho-hv), and specific differential phase (phi-dp). XSACR data from the 2018–2019 Cloud, Aerosol, and Complex Terrain Interactions (CACTI) field campaign in Argentina are now available as b1-level products. Building on the original CACTI operational data, the b1-level products feature improved data quality resulting from extensive analyses and corrections. The data are cross-calibrated to a common point, datastreams are corrected for operational issues that occurred during the campaign, and several data quality masks and basic derived products are incorporated. For more information, read the CACTI radar b1-level processing report.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

xsacrcfrspcxpol (a1)

ARM's scanning cloud radars are fully coherent dual-frequency, dual-polarization Doppler radars mounted on a common scanning pedestal. Each pedestal includes a Ka-band radar (2kW peak power) and the deployment location determines whether the second radar is a W-band (WSACR; 1.7 kW peak power) or X-band (XSACR; 20 kW peak power). At ARM's tropical sites, X-band radars are paired with the Ka-band because they are better suited for the atmospheric attenuation in this region. Beamwidth for the X-band is approximately 1 degree, and the Ka-band beamwidth is roughly 0.3 degrees. Due to the narrow antenna beamwidth, ARM’s scanning cloud radars use scanning strategies unlike typical weather radars. Rather than focusing on plan position indicator, or PPI, scans, the XSACR uses range height indicator, or RHI, scans at numerous azimuths to obtain cloud volume data. For the second ARM Mobile Facility (AMF2), the XSACR can be dismounted from the pedestal it shares with the KASACR and mounted on a separate pedestal. This allows the XSACR to operate more like a weather (precipitation) radar in deployments where such local coverage is lacking. Measurements collected with the XSACR are copolar and cross-polar radar reflectivity, Doppler velocity, spectra width and spectra when not scanning, differential Reflectivity (Zdr), correlation coefficient (rho-hv), and specific differential phase (phi-dp). XSACR data from the 2018–2019 Cloud, Aerosol, and Complex Terrain Interactions (CACTI) field campaign in Argentina are now available as b1-level products. Building on the original CACTI operational data, the b1-level products feature improved data quality resulting from extensive analyses and corrections. The data are cross-calibrated to a common point, datastreams are corrected for operational issues that occurred during the campaign, and several data quality masks and basic derived products are incorporated. For more information, read the CACTI radar b1-level processing report.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2021

xsacrcfrspccross (a1)

ARM's scanning cloud radars are fully coherent dual-frequency, dual-polarization Doppler radars mounted on a common scanning pedestal. Each pedestal includes a Ka-band radar (2kW peak power) and the deployment location determines whether the second radar is a W-band (WSACR; 1.7 kW peak power) or X-band (XSACR; 20 kW peak power). At ARM's tropical sites, X-band radars are paired with the Ka-band because they are better suited for the atmospheric attenuation in this region. Beamwidth for the X-band is approximately 1 degree, and the Ka-band beamwidth is roughly 0.3 degrees. Due to the narrow antenna beamwidth, ARM’s scanning cloud radars use scanning strategies unlike typical weather radars. Rather than focusing on plan position indicator, or PPI, scans, the XSACR uses range height indicator, or RHI, scans at numerous azimuths to obtain cloud volume data. For the second ARM Mobile Facility (AMF2), the XSACR can be dismounted from the pedestal it shares with the KASACR and mounted on a separate pedestal. This allows the XSACR to operate more like a weather (precipitation) radar in deployments where such local coverage is lacking. Measurements collected with the XSACR are copolar and cross-polar radar reflectivity, Doppler velocity, spectra width and spectra when not scanning, differential Reflectivity (Zdr), correlation coefficient (rho-hv), and specific differential phase (phi-dp). XSACR data from the 2018–2019 Cloud, Aerosol, and Complex Terrain Interactions (CACTI) field campaign in Argentina are now available as b1-level products. Building on the original CACTI operational data, the b1-level products feature improved data quality resulting from extensive analyses and corrections. The data are cross-calibrated to a common point, datastreams are corrected for operational issues that occurred during the campaign, and several data quality masks and basic derived products are incorporated. For more information, read the CACTI radar b1-level processing report.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2023

Beam dynamics corrections in the measurement of the anomalous precession frequency at the Muon $g-2$ experiment at Fermilab

The Muon $g-2$ experiment at Fermilab (E989) aims to measure the anomalous magnetic moment of the muon with an accuracy of 140 ppb (parts per billions). This accuracy, obtained by adding in quadrature a statistical and a systematic contribution of comparable value (100 ppb), will allow to reduce the experimental uncertainty (from the previous E821 experiment at BNL \cite{bnl}) of a factor of 4, and represents one of the most precise tests of the Standard Model (SM) theory of elementary particles. The first result on the Run-1 dataset \cite{prl} was released on April 7, 2021, showing a very good agreement with the previous result from BNL experiment, with a slightly better uncertainty. The corresponding experimental average increases the significance of the discrepancy between the measured and Standard Model prediction of 4.2$\sigma$ \cite{white_paper}.\\ \noindent The measured quantity is the muon magnetic anomaly a$_\mu$=$\frac{g_\mu-2}{2}$ where g$_\mu$ is the gyromagnetic factor of the muon. Dirac's equation predicts g$_\mu$ = 2, while radiative corrections, dominated by the QED contribution due to an exchange of a virtual photon, causes a per-mille correction on this quantity. By including all the SM contributions, a$_\mu$ is known at 370 ppb. The E989 experiment measures $a_\mu$ injecting positive muons with momentum of 3.1 GeV/$c$ polarized longitudinally in bunches (called $fills$) with an average rate of 12 Hz, in a storage ring of 14 meters diameter. Due to the parity violation in the weak muon decay, high energy positrons produced are emitted preferably in the muon's spin direction. By counting the number of positrons with energy greater then 1.7 GeV in function of the time, the frequency precession of the muon spin is measured, that together with the measurement of the magnetic field, allows to extract $a_\mu$. The positrons are detected with 24 electromagnetic calorimeters, that measure the energy and the arrival time of the positrons, each made of 54 crystals of lead fluoride (PbF$_2$) read by silicon photomultipliers (SiPM). Together with calorimeters, two tracking detectors are used to make non-destructive measurements of the muon beam characteristic by reconstructing the muon decay position extrapolating backward the decay positrons. The knowledge of the beam motion inside the ring plays a fundamental role in the analysis of $a_{\mu}$, where the measured anomalous precession frequency $\omega_a$ must be corrected for four main beam dynamics effects. A first correction is associated with the presence of an electric field responsible of the vertical focusing on the storage ring, where vertical direction is orthogonal to the orbit and horizontal direction is along the storage ring radius. Due to the oscillations in the vertical direction, the so called vertical betatron oscillations, a second correction is necessary to account for an average angle associated with the muon motion off the ideal orbit. A third correction is caused by lost muons in the ring which have a different spin phase at the injection respect to the decay ones. Finally, due to the correlation between the vertical and horizontal beam motion of the muons and the acceptance of the calorimeter, a correction (called ``phase-acceptance") arises. In Run1, due to the presence of two damaged resistors in one electrostatic quadrupole, this correction had a prominent role. The replacement of the damaged resistors before Run2 decreased this effect by one order of magnitude, and a further reduction in Run-3 was provided by an improved orbit.\\ \noindent The work of this Thesis focuses on the beam dynamics corrections on $\omega_a$. Due to the prominent role in Run-1 a special attention has been put to the phase-acceptance correction\footnote{The effect of this correction on the E821 BNL measurement of the $g-2$ has been evaluated to be within the quoted systematic error.}. Chapter \ref{ch:anomalous_magnetic_moment} introduces the anomalous magnetic moment of the muon. Chapter \ref{ch:early_experiment} describes the history of the Muon $g-2$ experiments. Chapter \ref{ch:theory} discusses the Standard Model prediction and possible new physics scenario. Chapter \ref{ch:muon_g2_experiment} describes the E989 experiment, whose experimental technique and the apparatus are discussed focusing on the improvements needed to reach the final goal on $a_\mu$ measurement. The original work of the Thesis is discussed in the last 5 chapters. Chapter \ref{ch:experiment_technique} presents the analysis technique to extract $\omega_a$, Chapter \ref{ch:beam_dynamics} describes the beam dynamics in the E989 experiment, Chapter \ref{ch:beam_dynamics_correction} discusses the beam dynamics correction to $\omega_a$, and Chapter \ref{sec:pa_corr} focuses on the phase acceptance correction. Finally Chapter \ref{ch:run23_analysis_improvements} contains the Run-2/3 improvements which are expected to allow for an increased precision ons the beam dynamics co...

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗