Nuclear Momentum Distribution Research Articles

Direct Measurements of Quantum Kinetic Energy Tensor in Stable and Metastable Water near the Triple Point: An Experimental Benchmark.

This study presents the first direct and quantitative measurement of the nuclear momentum distribution anisotropy and the quantum kinetic energy tensor in stable and metastable (supercooled) water near its triple point, using deep inelastic neutron scattering (DINS). From the experimental spectra, accurate line shapes of the hydrogen momentum distributions are derived using an anisotropic Gaussian and a model-independent framework. The experimental results, benchmarked with those obtained for the solid phase, provide the state of the art directional values of the hydrogen mean kinetic energy in metastable water. The determinations of the direction kinetic energies in the supercooled phase, provide accurate and quantitative measurements of these dynamical observables in metastable and stable phases, that is, key insight in the physical mechanisms of the hydrogen quantum state in both disordered and polycrystalline systems. The remarkable findings of this study establish novel insight into further expand the capacity and accuracy of DINS investigations of the nuclear quantum effects in water and represent reference experimental values for theoretical investigations.

Generalized nuclear contacts and momentum distributions

The general nuclear contact matrices are defined, taking into consideration all partial waves and finite-range interactions, extending Tan's work for the zero range model. The properties of these matrices are discussed and the relations between the contacts and the one-nucleon and two-nucleon momentum distributions are derived. Using these relations, a new asymptotic connection between the one-nucleon and two-nucleon momentum distributions, describing the two-body short-range correlations in nuclei, is obtained. Using available numerical data, we extract few connections between the different contacts and verify their relations to the momentum distributions. The numerical data also allows us to identify the main nucleon momentum range affected by two-body short-range correlations. Utilizing these relations and the numerical data, we also verify a previous independent prediction connecting between the Levinger constant and the contacts. This work provides an important indication for the relevance of the contact formalism to nuclear systems, and should open the path for revealing more useful relations between the contacts and interesting quantities of nuclei and nuclear matter.

Correlated fermions in nuclei and ultracold atomic gases

Background: The high momentum distribution of atoms in two spin-state ultra-cold atomic gases with strong short-range interactions between atoms with different spins, which can be described using Tan's contact, are dominated by short range pairs of different fermions and decreases as $k^{-4}$. In atomic nuclei the momentum distribution of nucleons above the Fermi momentum ($k>k_F \approx 250$ Mev/c) is also dominated by short rangecorrelated different-fermion (neutron-proton) pairs. Purpose: Compare high-momentum unlike-fermion momentum distributions in atomic and nuclear systems. Methods: We show that, for $k>k_F$ MeV/c, nuclear momentum distributions are proportional to that of the deuteron. We then examine the deuteron momentum distributions derived from a wide variety of modern nucleon-nucleon potentials that are consistent with $NN$-scattering data. Results: The high momentum tail of the deuteron momentum distribution, and hence of the nuclear momentum distributions appears to decrease as $k^{-4}$. This behavior is shown to arise from the effects of the tensor part of the nucleon-nucleon potential. In addition, when the dimensionless interaction strength for the atomic system is chosen to be similar to that of atomic nuclei, the probability for finding a short range different-fermion pair in both systems is the same. Conclusions: Although nuclei do not satisfy all of the conditions for Tan's contact, the observed similarity of the magnitude and $k^{-4}$ shape of nuclear and atomic momentum distributions is remarkable because these systems differ by about $20$ orders of magnitude in density. This similarity may lead to a greater understanding of nuclei and the density dependence of nuclear systems.

Mass-selective Neutron Spectroscopy Beyond the Proton

We discuss ongoing methodological developments underpinning the determination of nuclear-momentum distributions from mass-resolved neutron Compton data of lightweight materials. To this end, two systems are considered in detail, namely, lithium hydride (including its deuterated counterpart) and squaric acid, an organic antiferroelectric material containing hydrogen, carbon, and oxygen. Beyond the usual case of the proton, our approach enables direct access to detailed line shape information associated with the underlying nuclear-momentum distributions of both deuterium and lithium. For oxygen and carbon, mean kinetic energies can also be obtained directly from the neutron data, as demonstrated by a detailed analysis of mass- resolved data from squaric acid. From an instrumentation point of view, this work provides a suitable platform for a detailed assessment of existing capabilities and future developments in mass-selective neutron spectroscopy on the VESUVIO spectrometer at ISIS.

Mass dependence of short-range correlations in nuclei and the EMC effect

We sketch an approximate method to quantify the number of correlated pairs in any nucleus $A$. It is based on counting independent-particle model (IPM) nucleon-nucleon pairs in a relative $S$-state with no radial excitation. We show that IPM pairs with those quantum numbers are most prone to short-range correlations and are at the origin of the high-momentum tail of the nuclear momentum distributions. Our method allows to compute the $a_2$ ratios extracted from inclusive electron scattering. Furthermore, our results reproduce the observed linear correlation between the number of correlated pairs and the magnitude of the EMC effect. We show that the width of the pair center-of-mass distribution in exclusive two-nucleon knockout yields information on the quantum numbers of the pairs.

Mass-selective neutron spectroscopy of lithium hydride and deuteride: Experimental assessment of the harmonic and impulse approximations

Neutron Compton scattering experiments on crystalline lithium hydride and deuteride are presented and compared with existing experimental data and first-principles predictions. With currently available instrumentation, these measurements demonstrate sufficient mass selectivity for studies of nuclear-momentum distributions of protons, deuterons, and lithium using forward and backscattering geometries. In both materials, spectral discrimination of lithium-recoil features is highest in backscattering geometry, although mass isolation is also possible in the forward-scattering direction. These results evince the possibility of performing simultaneous mass-selective neutron spectroscopic studies for nuclei with $m>4$ amu. We also provide an in-depth analysis and assessment of departures from the harmonic and impulse approximations as described by the celebrated Sears expansion of the dynamic structure factor, as well as how these can manifest themselves in the experimental data. We close by outlining the potential of our experimental strategy for mass-selective spectroscopic studies of materials containing protons and other light nuclides.

Time-dependent importance sampling in semi-classical initial value representation calculations for time correlation functions. III. A state-resolved implementation to electronically non-adiabatic dynamics

The recently developed time-dependent (TD) Monte Carlo (MC) importance sampling method [Tao and Miller; JCP 135, 024104 (2011)] provides an efficient implementation of the semi-classical (SC) initial value representation (IVR) methodology for the evaluation of time correlation functions. The key idea in this TD-SC-IVR method is to perform importance sampling of trajectories for the MC averages in the SC calculations with a sampling function that includes information about the final (time-evolved) values of the coordinates and momenta of trajectories in addition to (the usual) information of their initial values. This paper shows how this approach deals with electronically non-adiabatic dynamics, i.e. dynamics that involves coupled multiple electronic states. We suggest that a state-resolved sampling function may facilitate the efficient implementation of TD-SC-IVR for such processes. This is illustrated by application to the calculation of the nuclear momentum distribution function (i.e. the nuclear energy-loss spectrum) for a benchmark non-adiabatic scattering problem.

High-momentum tails from low-momentum effective theories

In a recent work \cite{Anderson:2010aq}, Anderson \emph{et al.} used the renormalization group (RG) evolution of the momentum distribution to show that, under appropriate conditions, operator expectation values exhibit factorization in the two-nucleon system. Factorization is useful because it provides a clean separation of long- and short-distance physics, and suggests a possible interpretation of the universal high-momentum dependence and scaling behavior found in nuclear momentum distributions. In the present work, we use simple decoupling and scale-separation arguments to extend the results of Ref. \cite{Anderson:2010aq} to arbitrary low-energy $A$-body states. Using methods that are reminiscent of the operator product expansion (OPE) in quantum field theory, we find that the high-momentum tails of momentum distributions and static structure factors factorize into the product of a universal function of momentum that is fixed by two-body physics, and a state-dependent matrix element that is the same for both and is sensitive only to low-momentum structure of the many-body state. As a check, we apply our factorization relations to two well-studied systems, the unitary Fermi gas and the electron gas, and reproduce known expressions for the high-momentum tails of each.

Effect of Short-Range and Tensor Force Correlations on High-Density Behavior of Symmetry Energy

The effect of short-range and tensor correlations on nuclear symmetry energy is investigated by using realistic nuclear momentum distribution n(k). For finite nuclei, the n(k) is discussed in detail within an analytical model that allows inclusion of both short-range and tensor correlation effects in a physically very transparent way. The equation of state of symmetric nuclear matter is calculated by using realistic n(k) parameterization. The tensor-force induced short-range correlations are shown to have a significant impact on the behavior of symmetry energy.

Efficient First-Principles Calculation of the Quantum Kinetic Energy and Momentum Distribution of Nuclei

Light nuclei at room temperature and below exhibit a kinetic energy which significantly deviates from the predictions of classical statistical mechanics. This quantum kinetic energy is responsible for a wide variety of isotope effects of interest in fields ranging from chemistry to climatology. It also furnishes the second moment of the nuclear momentum distribution, which contains subtle information about the chemical environment and has recently become accessible to deep inelastic neutron scattering experiments. Here, we show how, by combining imaginary time path integral dynamics with a carefully designed generalized Langevin equation, it is possible to dramatically reduce the expense of computing the quantum kinetic energy. We also introduce a transient anisotropic Gaussian approximation to the nuclear momentum distribution which can be calculated with negligible additional effort. As an example, we evaluate the structural properties, the quantum kinetic energy, and the nuclear momentum distribution for a first-principles simulation of liquid water.

Fragment momentum distributions obtained from coupled electron-nuclear dynamics

We theoretically investigate fragmentation processes induced by femtosecond laser pulses within a model which incorporates electronic and nuclear motion. Single-pulse excitation leads to diffraction patterns in the electron momentum distribution which depend on the nature of the electronic state and also on the nuclear charge distribution. Additional structures appear in the nuclear momentum distribution if two time-delayed pulses produce fragments in the same dissociation channel. It is shown that these functions are modified by the electronic degree-of-freedom. A simultaneous excitation of two different electronic states results in further interferences which are related to electronic wave-packet dynamics on the attosecond time-scale.

Nuclear and electronic momentum distributions from pulse induced photodissociation

We study fragmentation processes induced by femtosecond laser pulses. Within an adiabatic treatment of the nuclear motion, the interaction with two identical but time-delayed pulses leads to interference structures in the momentum distribution which are related to the pulse delay. These structures are modified if an electronic degree of freedom is included in the wave-packet propagation. Additional time-dependent patterns are found in the electron momentum distributions which are also present for a single-pulse excitation. They are related to the delocalization of the electron density over the different charge centers.

Ab initionuclear momentum distributions in lithium hydride: Assessing nonadiabatic effects

Theoretical nuclear momentum distributions for solid lithium hydride and lithium deuteride are presented. Electronic-structure calculations were performed within the framework of plane-wave density functional theory, followed by the computation of phonon-dispersion relations and vibrational densities of states. The generalized-gradient-approximation functional of Perdew, Burke, and Ernzerhof was used in these first-principles calculations. Our computational results are compared with existing neutron Compton scattering and inelastic neutron scattering experiments on solid LiH. We find an excellent agreement between theory and experiment within the harmonic Born-Oppenheimer approximation. On the basis of the above, we estimate an upper conservative bound of $~$2 to 3% for the effects of nonadiabatic dynamics on the second moment and Laplacian of the atomic momentum distributions in this benchmark system. We close by discussing the implications of this study on future theoretical studies of atomic momentum distributions from isolated molecules and extended condensed-matter systems.

Neutron Compton scattering investigation of sodium hydride: From bulk material to encapsulated nanoparticulates in amorphous silica gel

In this study we utilize neutron Compton scattering (NCS) to determine differences in nuclear momentum distributions in NaH, both as bulk material and encapsulated as nanoscale particles (from 20 to 50 nm in diameter) within an amorphous silica-gel matrix (SiGNaH). In addition, elemental Na dispersed in such a matrix is also studied (SiGNa). Data treatment and fitting of experimental spectra yields comparison of the nuclear Compton profiles and radial momentum distributions for the proton in both bulk NaH and nanoscale SiGNaH, with resultant proton kinetic energies being in agreement with previous inelastic neutron studies of bulk NaH. Slight differences in proton radial momentum distributions for bulk and nanoscale systems are witnessed and discussed. The technique of stoichiometric-fixing is applied to the backscattering spectra of each system in order to examine changes in the Na profile width, and NCS is shown to be sensitive to the chemical environment change of this heavier nucleus. Examination of the Si and O profile widths in the gel samples also supports this method.

Nuclear momentum distribution in solid and liquid HF from ab initio calculation

A calculation of nuclear momentum distribution of liquid and solid hydrogen fluoride was performed. In both systems, density functional theory generalized gradient approximation functional of Perdew, Burke, and Ernzerhof was used for the calculation: for liquid hydrogen fluoride, using an atom centered basis set for an isolated molecule with optimized geometry, and for solid hydrogen fluoride using plane-wave basis sets on optimized orthorhombic crystal cell. For liquid hydrogen fluoride, a semiclassical approach was adopted with the vibrational contribution to momentum distribution obtained from the density functional theory calculation and translational and rotational contributions calculated classically. Nuclear momentum distribution in the solid hydrogen fluoride was calculated entirely quantum mechanically using phonon dispersion and vibrational density of states calculated in the framework of plane-wave density functional theory. Theoretical results were contrasted with recently obtained results of Compton (deep inelastic) neutron scattering on liquid and solid hydrogen fluoride. In case of liquid hydrogen fluoride, almost a perfect agreement between theory and experiment was achieved within the harmonic Born-Oppenheimer approximation. For the solid system under investigation, the harmonic approximation leads to small (4%) overestimation of the square root of the second moment indicating that neutron Compton scattering technique is sensitive to proton delocalization due to hydrogen bonding in solid hydrogen fluoride.

Superscaling analysis of inclusive electron scattering and its extension to charge-changing neutrino-nucleus cross sections beyond the relativistic Fermi gas approach

Superscaling analyses of inclusive electron scattering from nuclei are extended from the quasielastic processes to the delta excitation region. The calculations of $(e,{e}^{'})$ cross sections for the target nucleus $^{12}\mathrm{C}$ at various incident electron energies are performed using scaling functions $f({\ensuremath{\psi}}^{'})$ obtained in approaches going beyond the mean-field approximation, such as the coherent density fluctuation model (CDFM) and the one based on the light-front dynamics method. The results are compared with those obtained using the relativistic Fermi gas (RFG) model and the extended RFG model (ERFG). Our method utilizes in an equivalent way both basic nuclear quantities, density and momentum distributions, showing their role for the scaling and superscaling phenomena. The approach is extended to consider scaling function for medium and heavy nuclei with $Z\ensuremath{\ne}N$ for which the proton and neutron densities are not similar. The asymmetry of the CDFM quasielastic scaling function is introduced, simulating in a phenomenological way the effects that violate the symmetry for ${\ensuremath{\psi}}^{\ensuremath{'}}\ensuremath{\geqslant}0$, including the role of the final-state interaction. The superscaling properties of the electron scattering are used to predict charge-changing neutrino-nucleus cross sections at energies from 1 to 2 GeV. A comparison with the results of the ERFG model is made. The analyses make it possible to gain information about the nucleon correlation effects on both local density and nucleon momentum distributions.

Anomalous neutron Compton scattering: comparison of the convolution approximation with a model-free approach

A shortfall of the scattering intensity from protons has been observed in liquids (water, benzene, etc.) and solids (metal hydrogen systems, organic polymers, etc.) using neutron Compton scattering (NCS). The conventional NCS data reduction scheme, used to obtain the scattering intensities, is based on the convolution approximation (CA) of the nuclear momentum distribution with the instrument resolution function. This scheme has been severely criticized. It was claimed that it is the way the energy resolution function is incorporated in the CA data processing that leads to an underestimation of the scattering intensity. In the present work, the CA data reduction scheme is compared to a model-free approach newly proposed by Dorner. This new scheme is independent of the form of the momentum distribution and the resolution function. Specifically, the ratios of the scattering cross section densities of H to C are presented for a measurement on a polyethylene (PE) sample. The new data reduction scheme is shown...

Anomalous neutron Compton scattering: Comparison of the convolution approximation with a model-free approach

A shortfall of the scattering intensity from protons has been observed in liquids (water, benzene, etc.) and solids (metal hydrogen systems, organic polymers, etc.) using neutron Compton scattering (NCS). However, the suitability of the conventional NCS data reduction scheme, used to obtain the scattering intensities, based on the convolution approximation (CA) of the nuclear momentum distribution with the instrument resolution function has been questioned. In the present work the commonly used data reduction scheme is compared to a model-free approach [B. Dorner, J. Neutron Res. (to be published)] that is independent of the form of the momentum distribution and the resolution function. Specifically, the ratios of the scattering cross section density of H to C are presented for polyethylene. The model-free approach is shown to lead to the same results as for experimental data published earlier and treated with the CA thus addressing a great part of the previous comments.

Formalism for obtaining nuclear momentum distributions by the deep inelastic neutron scattering technique

We present a new formalism to obtain momentum distributions in condensed matter from Neutron Compton Profiles measured by the Deep Inelastic Neutron Scattering technique. The formalism describes exactly the Neutron Compton Profiles as an integral in the momentum variable $y$. As a result we obtain a Volterra equation of the first kind that relates the experimentally measured magnitude with the momentum distributions of the nuclei in the sample. The integration kernel is related with the incident neutron spectrum, the total cross section of the filter analyzer and the detectors efficiency function. A comparison of the present formalism with the customarily employed approximation based on a convolution of the momentum distribution with a resolution function is presented. We describe the inaccuracies that the use of this approximation produces, and propose a new data treatment procedure based on the present formalism.

Momentum distribution of relativistic nuclei with Hartree-Fock mesonic correlations

The impact of Hartree-Fock correlations on the nuclear momentum distribution is studied in a fully relativistic one boson exchange model. Hartree-Fock equations are exactly solved to first order in the coupling constants. The renormalization of the Dirac spinors in the medium is shown to affect the momentum distribution, as opposed to what happens in the non-relativistic case. The unitarity of the model is shown to be preserved by the present renormalization procedure.