Search Results (26 total)

Although much is known about the differences between expert and novice problem solvers, knowledge of those differences typically does not provide enough detail to help instructors understand why some students seem to learn physics while solving problems and others do not. A critical issue is how students access the knowledge they have in the context of solving a particular problem. In this paper, we discuss our observations of students solving physics problems in authentic situations in an algebra-based physics class at the University of Maryland. We find that when these students are working together and interacting effectively, they often use a limited set of locally coherent resources for blocks of time of a few minutes or more. This coherence appears to provide the student with guidance as to what knowledge and procedures to access and what to ignore. Often, this leads to the students failing to apply relevant knowledge they later show they possess. In this paper, we outline a theoretical phenomenology for describing these local coherences and identify six organizational structures that we refer to as epistemic games. The hypothesis that students tend to function within the narrow confines of a fairly limited set of games provides a good description of our observations. We demonstrate how students use these games in two case studies and discuss the implications for instruction.

Decades of education research have shown that students can simultaneously possess alternate knowledge frameworks and that the development and use of such knowledge are context dependent. As a result of extensive qualitative research, standardized multiple-choice tests such as Force Concept Inventory and Force-Motion Concept Evaluation tests provide instructors tools to probe their students’ conceptual knowledge of physics. However, many existing quantitative analysis methods often focus on a binary question of whether a student answers a question correctly or not. This greatly limits the capacity of using the standardized multiple-choice tests in assessing students’ alternative knowledge. In addition, the context dependence issue, which suggests that a student may apply the correct knowledge in some situations and revert to use alternative types of knowledge in others, is often treated as random noise in current analyses. In this paper, we present a model analysis, which applies qualitative research to establish a quantitative representation framework. With this method, students’ alternative knowledge and the probabilities for students to use such knowledge in a range of equivalent contexts can be quantitatively assessed. This provides a way to analyze research-based multiple choice questions, which can generate much richer information than what is available from score-based analysis.

We study the isospin dependence of nucleon-nucleon potentials by examining solutions to the Schrödinger equation in paronic (Pauli-forbidden) states. We find a model dependence to the isovector content and discuss its origin. We observe a qualitative difference between meson-exchange-based models and those that are more phenomenological. This has consequences for a number of issues of current interest.

The optical potential for elastic scattering of protons from ¹⁶O at incident beam energies between 100 and 500 MeV is calculated from a full-folding integral of a simple s-p-shell representation of the target density matrix together with fully off-shell nucleon-nucleon t matrices derived from two different Bonn meson exchange models. Elastic scattering observables calculated from this full-folding optical potential are compared to those obtained from ‘‘optimum factorized’’ as well as on-shell local (‘‘tρ’’) approximations. The optimum factorization is found to provide a good approximation to elastic scattering observables obtained from the full-folding optical potential, although the potentials differ in the structure of their nonlocality. A perturbative treatment of the nonlocality is used to extract approximate localized potentials and associated Perey damping factors for the interior wave functions. The results indicate that the interior wave function from the optimum factorization potential and the full-folding potential are very similar.

We solve the Bethe-Goldstone equation in nuclear matter for the scattering of a nucleon of a few hundred MeV. The angular momentum coupling caused by the nonspherical part of the Pauli blocking operator is treated exactly. It is found that standard approximation of angle averaging the Pauli operator is quite accurate for bulk properties of the reaction matrix even at energies as high as 300 MeV. Our result justifies one of the common approximations previously untested in the microscopic calculation of the optical model potential.

The partial wave decomposition of the Franey-Love effective nucleon-nucleon amplitude is used to show that it satisfies on-shell unitarity to better than 5% in all partial waves, but its off-shell continuations are strikingly nonunitary.

The behavior of the fully-off-the-energy-shell T matrices for the Reid-soft-core and Paris-80 potentials are studied for a variety of partial waves, both uncoupled and coupled. Although the potentials have qualitatively different structures in both coordinate space and momentum space, the resulting off-energy-shell behavior is very similar in the range of energy and momentum relevant to low and medium energy nuclear physics.

New prescriptions are presented for the single scattering approximations for the optical potential and distorted-wave impulse approximation transition amplitudes. They are based on the application of systematics observed in the behavior of off-shell T matrices calculated from realistic potential models.

The distorted wave Born series for inelastic scattering can be formulated as arising from the iteration of an integral equation. The small parameter determining the convergence of the series can then be identified as the spectral radius of the integral kernel. Such a formulation allows a detailed investigation of what controls the accuracy of the distorted-wave Born approximation. Several important questions can then be answered, such as the following: what are the relevant dimensionless parameters, what physics is important, and what is the best distortion potential to use. We consider these questions in a simple, analytically solvable two-channel model.

The accuracy of the distorted-wave approximation for inelastic scattering is studied in a solvable model. By formulating the scattering problem as an integral equation, the inhomogeneity can be identified as the distorted-wave approximation, and the norm of the kernel as the convergence parameter of the multistep series. The convergence parameter is studied as a function of dimensionless control parameters in order to extract what governs the convergence of the series.

The 800-MeV (p, p^′) inclusive proton spectra at forward angles are considered. A plane-wave impulse approximation is used to calculate quasifree nucleon knockout and quasifree isobar production. When the quasifree peaks are normalized to data, it is found that the integrated cross section for the two processes can account for the total reaction cross section.

The connected kernel equations of Bencze, Redish, and Sloan for the scattering operators in the N-body problem are derived by a new and more direct method. It is seen that the equations can be obtained by distributing the final state's residual interaction over all partitions in a particular way. A different resolvent equation is then applied to each term of the sum. For the specific distribution chosen the disconnected parts of the kernel vanish exactly. The validity of the off-shell transformation which simplifies the inhomogeneous term is demonstrated.

NUCLEAR REACTIONS N-body problem, scattering theory, Bencze-Redish-Sloan equations, Kouri-Levin method, Faddeev equations.

The reaction (K⁺, K⁺p) is considered as a probe of the deep hole states in nuclei. Distorted-wave impulse-approximation calculations are presented for the knockout of a 1s_{1 / 2} proton from ⁴⁰Ca. We conclude that the experiment should be valuable.

We investigate the single-scattering optical potential in the multiple scattering approaches of Watson and of Kerman, McManus, and Thaler. Since the kinematics of single scattering is three body in nature, we build a three-body model of this term. This approach can include the proper kinematics for the struck nucleon, the identity of the target nucleons, and the binding interaction of the struck nucleon. Integral equations of the Faddeev type are derived for both the Watson and Kerman-McManus-Thaler single-scattering optical potentials. Unitarity relations are investigated and we observe that these relations can be expanded in order to identify the intermediate states responsible for the absorptive parts. The transition amplitudes to the inelastic states implicit in the model are extracted and evaluated. This permits one to understand the physical meaning of the imaginary part in precise terms. The same procedure is applied to the closure and impulse approximations for the single-scattering term and their implicit inelastic states and reaction amplitudes are identified. These approximations are evaluated by analyzing the inelastic data. We conclude that the impulse approximation to the Watson single-scattering term should provide the best two-body approximation to a single-scattering optical potential.

NUCLEAR REACTIONS Study optical potential in Watson and KMT multiple-scattering theory, three-body model introduced, Faddeev equations derived, unitarity studied; applied to intermediate energy nucleon-nucleus scattering.

Proposed forms for nuclear momentum distributions are investigated. Calculations of (p ,p^′) reactions using those forms are done in a plane-wave impulse approximation at angles where the quasielastic peak is seen and also at back angles. The parameters used are derived from (e ,e^′) data, where the nuclear momenta probed overlap with those of the low angle (p ,p^′) experiment. Although there is reasonable agreement for the (p ,p^′) data at 180°, the inclusion of distortion necessitates a different parameter set to obtain agreement for the quasifree process. We conclude that the (p ,p^′) reaction cannot be readily understood with a simple momentum distribution.

NUCLEAR STRUCTURE ¹²C; low and high momentum behavior of single particle momentum distributions, compared for PWIA calculations of (p ,p^′).

A unitarity relation is derived for the nucleon-nucleus scattering amplitude which displays the reaction mechanisms associated with the absorptive part of the single-scattering optical potential. The reactions implicit in the absorption are nucleon pickup and knockout. The transition amplitudes for these processes are extracted from the unitarity relation and their accuracy is discussed.

We have studied the reaction mechanisms of the (p,2p) reaction on ⁴He at energies of 65 and 100 MeV. The dynamics of the reaction is taken to be that of a three-particle system, the ⁴He target being viewed as a proton bound to an inert triton. The particles interact via separable potentials chosen to fit bound state and scattering data of the two-particle subsystems. The protons are treated as identical bosons interacting with each other only in the relative S state while the proton-triton potential is limited to relative L≤2. The ⁴He(p,2p)³H doubly differential cross section can be calculated exactly within this model by solving the Faddeev equations numerically using the method of deformed contours and Padé approximants. This can be compared with the distorted-wave impulse approximation (DWIA), calculated in this model by summing the appropriate subset of the multiple scattering series. This is done in momentum space and treats off-shell and finite range effects of the nucleon-nucleon T matrix as well as recoil effects exactly. We find that the DWIA is an adequate approximation at 100 MeV but is almost double the exact result at 65 MeV. Its use in the extraction of spectroscopic information below 100 MeV is therefore suspect. The shapes of the exact, distorted, and plane wave cross sections agree quite well with each other (and with the experimental shape), differing only in magnitude.

NUCLEAR REACTIONS ⁴He(p,2p)³H, E=65,100 MeV; calculated σ. Threebody model, reaction mechanisms.

The newly developed connected kernel equations for N-body scattering are applied to the study of rearrangement reactions. An integral equation for the transition operator is obtained which has a connected kernel and whose inhomogeneous terms contain the standard distorted-wave Born approximation (DWBA) and a two-step part. The two-step part is similar to the expression usually used for the calculation of two-step reactions but with a non-standard prescription for the driving potentials. The difference comes about because of the off-shell transformations used to simplify the original connected kernel equations. A multi-step DWBA series is generated and the conditions for its convergence are discussed.

The separation method of Moszkowski and Scott has recently been used to generate an effective interaction for inelastic scattering. We investigate this approximation for the two-nucleon ¹S₀ state in a realistic model. We find that the method only yields reliable off-shell matrix elements for low energies and momenta near the on-shell point.

The numerical accuracy of an approximation used in the authors' wave-function model of the two-nucleon interaction is discussed. In addition, we consider the energy dependence of the upper and lower bounds on the parameter used to mock up the off-shell uncertainty.

The distorted-wave impulse approximation for knockout reactions, when written following a Faddeev prescription, contains a fully off-shell T matrix whose arguments are coupled to those of the distorted waves. We show how the off-shell T matrix may be expanded around its half-shell point and an approximate factorized distorted-wave impulse approximation obtained with the major off-shell effect included. The off-shell ambiguity present in the standard treatments is thereby eliminated.

The p-p cross sections relevant to a factorized-impulse-approximation treatment of (p,2p) reactions from 150 to 350 MeV are calculated. Several on-shell prescriptions are used and compared with the half-off-shell prescription suggested by the Faddeev-Watson multiplescattering series. Four phenomenological potentials, Hamada-Johnston, Bryan-Scott III, Reid hard core, and Reid soft core, are investigated as well as three potentials which are phase-shift equivalent to the Reid soft-core potential. We find significant differences between the various prescriptions for all of the potentials. We also observe a wide range in the predictions on the various phenomenological potentials for each prescription. However, the ratio of the half-shell cross section to the on-shell cross section is remarkably insensitive to the choice of potential, especially where the half-shell prescription is needed. This suggests a simple method for extrapolating from the elastic scattering data to the appropriate half-shell cross section.

Current descriptions of (p,2p) reactions, nucleon-nucleon bremsstrahlung, and low-energy pion production use the half-off-shell two-nucleon T matrix in the neighborhood of the on-shell point. We present a method for expanding the amplitude t₀(p,k,;k²) in powers of p-k. After explicitly extracting the contributions arising from the long-range part of the interaction, we obtain a power series with an infinite radius of convergence. The coefficients of this series are expressed in terms of weighted integrals of the difference function, thus permitting us to determine precisely how variations of the interior wave function affect the near off-shell behavior of the half-shell T matrix. This allows treatment of a large class of models for the two-nucleon interaction, including wave-function models as well as local, nonlocal, and energy-dependent potentials. As an example, we apply this method to a wave-function model of the two-nucleon ¹S₀ T matrix constructed by the authors in a previous paper. We find that the expansion converges rapidly and that the inclusion of more terms extends the representation farther off the energy shell. The expansion coefficients are smooth, rapidly decreasing functions of energy. The even coefficients and the odd coefficients each form a family of similar functions which fall off rapidly with index. The T matrix arising from this model is very similar to those obtained with realistic potentials. Since the coefficient functions can be tabulated or parametrized quite simply, this expansion should provide a compact, useful method of expressing and comparing the near off-shell behavior of the two-nucleon half-shell T matrices arising from different models while permitting one to maintain both a fixed phase shift and the correct long-range behavior of the potential.

The T matrix half off the energy shell can be expressed in terms of the on-shell T matrix and the difference between the full scattering wave function and the phase-shifted free wave function. This representation allows one to investigate variations in the half-shell T matrix by means of a parametrization of the scattering wave function in the interaction region. Most of our knowledge of the physical constraints on that wave function may then be included. In particular we have studied the arbitrariness remaining in the half-shell T matrix given the following constraints: (1) The on-shell T matrix is given; (2) a specific local potential acts beyond a certain radius; and (3) the wave function is suppressed at short distances. We construct models of the ¹S₀ two-nucleon half-shell T matrix. In a simple one-parameter model satisfying these constraints, the range of variation of the near off-shell behavior is displayed.