Comparison of Electron and Neutron Compton Scattering from Entangled Protons in a Solid Polymer

We present, for the first time, a direct comparison between electron (ECS) and neutron (NCS) Compton scattering results from protons of a solid polymer. The momentum distributions of hydrogen obtained from ECS and NCS are in excellent agreement. In both experiments, a strong “anomalous” shortfall in the scattering intensity of protons {first detected in liquid water with NCS [C. A. Chatzidimitriou-Dreismann et al., Phys. Rev. Lett. 79, 2839 (1997)]} is found ranging from about $20\%$ up to $50\%$, depending on the momentum transfer applied. The characteristic times of electron- and neutron-proton collisions lie in the subfemtosecond range. The presented ECS and NCS results provide further direct evidence for this striking effect, which has been ascribed to attosecond quantum entanglement of the protons.

Figure 1

ECS spectrum of a formvar film taken using 25 keV electrons, thus effectuating a momentum transfer $\hslash q$ with $q=61.8{\AA }^{-1}$. The main peak at $\epsilon =0.6\mathrm{e}\mathrm{V}$ energy loss is due to electrons scattered from C and O. The small peak near 8 eV is due to electrons scattered from protons. Inset: The fit of the peak related to proton recoils (open symbols) with a Gaussian and a quadratic background (broken line) is shown by the full line.

Figure 2

Distributions $J(y)$ derived from the ECS and NCS measurements, with $\hslash y$ being the H-momentum component (before collision) along the direction of momentum transfer $\hslash \mathbf{q}$. The abscissa is given in ${\AA }^{-1}$ and in atomic units (a.u., on the top). The scatter in the four ECS measurements (energies as indicated, individual symbols) is an indication of the statistical errors. The graph given by the full line represents $J(y)$ derived with NCS at scattering angle $\theta =66°$. NCS results at different angles are similar. For comparison, all distributions are normalized to have the maximum about 1. Note that all distributions derived by ECS and NCS have similar widths and shapes.

Figure 3

Example of a measured TOF spectrum by NCS of formvar (self-supporting foil, 0.1 mm thick), with the associated fit (line). The spectrum is taken with the detector at scattering angle $\theta =51.27°$, which corresponds to a mean momentum transfer (for the neutron-proton collision) with $q\approx 60.7{\AA }^{-1}$. The broad H peak is well resolved from the narrow recoil peak from C and O. The ratio of peak areas $R_{\mathrm{e}\mathrm{x}\mathrm{p}}$ is about $40\%$ lower than the conventionally expected value $R_{ia}$; cf. Equations (1, 2).

Figure 4

Anomalous reduction of scattering intensity from H of formvar, as a function of applied momentum transfer $\hslash q$ (with $q=|\mathbf{q}|$); cf. Equations (1, 2). The $q$ range shown corresponds to scattering times ${\tau }_{sc}\approx 300–1000\times {10}^{-18}\mathrm{s}$. Squares, circles: Values of $R_{\mathrm{e}\mathrm{x}\mathrm{p}}$ of NCS spectrum areas (for 32 detectors in the angular range about $32°–68°$), relative to $R_{ia}$, Eq. (2). Full squares represent results for $t=0.1\mathrm{m}\mathrm{m}$ formvar foils, open circles those for $t=0.2\mathrm{m}\mathrm{m}$. They are equal within experimental errors, indicating that multiple scattering effects are insignificant. Large open triangles: $R_{\mathrm{e}\mathrm{x}\mathrm{p}}/R_{ia}$ as measured by ECS from formvar films of $50–100\AA$ thickness, for electrons with kinetic energies 15–30 keV. (The overall errors of these results are estimated to be about $10\%$, see the text.) The results demonstrate a strong ($20–50\%$) shortfall of scattering intensity from H, which is $q$ dependent. Note that the ECS value at $q=47.6{\AA }^{-1}$ may be larger than the corresponding NCS value. ECS also provides a new confirmation (independent from NCS) of the considered QE effect [4, 17, 18].