Many students in upper-division physics courses struggle with the mathematically sophisticated tools and techniques that are required for advanced physics content. We have developed an analytical framework to assist instructors and researchers in characterizing students’ difficulties with specific mathematical tools when solving the long and complex problems that are characteristic of upper division. In this paper, we present this framework, including its motivation and development. We also describe an application of the framework to investigations of student difficulties with direct integration in electricity and magnetism (i.e., Coulomb’s law) and approximation methods in classical mechanics (i.e., Taylor series). These investigations provide examples of the types of difficulties encountered by advanced physics students, as well as the utility of the framework for both researchers and instructors.

Structural phase-change materials are of great importance for applications in information storage devices. Thermally driven structural phase transitions are employed in phase-change memory to achieve lower programming voltages and potentially lower energy consumption than mainstream nonvolatile memory technologies. However, the waste heat generated by such thermal mechanisms is often not optimized, and could present a limiting factor to widespread use. The potential for electrostatically driven structural phase transitions has recently been predicted and subsequently reported in some two-dimensional materials, providing an athermal mechanism to dynamically control properties of these materials in a nonvolatile fashion while achieving potentially lower energy consumption. In this work, we employ DFT-based calculations to make theoretical comparisons of the energy required to drive electrostatically-induced and thermally-induced phase transitions. Determining theoretical limits in monolayer MoTe₂and thin films of Ge₂Sb₂Te₅, we find that the energy consumption per unit volume of the electrostatically driven phase transition in monolayer MoTe₂at room temperature is 9% of the adiabatic lower limit of the thermally driven phase transition in Ge₂Sb₂Te₅. Furthermore, experimentally reported phase change energy consumption of Ge₂Sb₂Te₅is 100–10,000 times larger than the adiabatic lower limit due to waste heat flow out of the material, leaving the possibility for energy consumption in monolayer MoTe₂-based devices to be orders of magnitude smaller than Ge₂Sb₂Te₅-based devices.

We perform density functional theory (DFT) calculations of five solid phases and the liquid phase of tin. The calculations include cold curves of the five solid phases, phonon calculations in the quasiharmonic approximation over a range of volumes for each solid phase, and DFT-based molecular dynamics (DFT-MD) simulations of the liquid phase, including those of the melt curve using the $Z$ method. Using the DFT results, we construct a tabular multiphase sesame equation of state for tin, referred to as sesame 2162. Comparisons to experimental data are made and show a high level of agreement in isobaric data, isothermal data, shock data, and phase boundary measurements, including measurements of the melt curve. The 2162 EOS will be useful for hydrodynamics simulations and has been designed with an eye toward hydrodynamics simulations that incorporate materials strength models and allow for modeling of the kinetics of phase transitions.