Tunnelling and barrier interaction times of neutrons were previously measured. Here we show that the neutron interaction time with barriers corresponds to the universal tunnelling time of wave mechanics which was formerly observed with elastic, electromagnetic and electron waves. The universal tunnelling time seems to hold for neutrons also. Such an adequate general wave mechanical behaviour was conjectured by Brillouin. Remarkably, wave mechanical effects, and even virtual particles, hold from the microcosm to the macrocosm.
Neutron tunnelling and neutron interaction times were recently studied , , . As described by Brillouin, in his textbook Wave Propagation in Periodic Structures, waves of all fields should act in a similar way . Here, we evidence this speculation in the case of tunnelling.
Photonic tunnelling time was studied in various barrier systems and at different electromagnetic wave frequencies. A tunnelling barrier example, i.e. the double prism is displayed in Figure 1. This tunnelling process is based on frustrated internal reflection as shown in Figure 2. The comparison of the available experimental data obtained on different barriers revealed a universal tunnelling time , , . The tunnelling time was found to be approximately equal to the reciprocal photon or phonon frequency
where τ is the measured group tunnelling time, ν the radiation frequency, h the Planck constant and E the wave packet energy. This universal tunnelling time is seen only in the case of opaque barriers, which means the penetration depth of the beam is smaller than 1/e, where e is the Euler’s number. As shown in Table 1, it was observed that besides photons, phonons and electrons also exhibit such an approximate universal tunnelling time.
Esposito , and somewhat later Olkhovsky et al. , delivered a theoretical description of this universal tunnelling time for rectangular barriers. Their theoretical data fit the experimental photonic values to the first-order approximation.
Recently, Matiwane et al. measured the reflectivity of tunnelling neutrons, i.e. of Schrödinger waves . The experiments were carried out in grazing angle geometry, i.e. in the angle range of frustrated total internal reflection as this reflection is called in optics. The double prism is the quantum mechanical analogue of the tunnelling process. Total reflection becomes frustrated if the second prism approaches the first one and light is partially transmitted to the second prism . In this neutron experiment, a sandwich nanostructure of 58Ni62Ni58Ni layers represents a Fabry-Perot resonator. The two Ni isotopes have a different neutron scattering potential. Seven tunnelling resonances of this resonator were detected in the reflectivity spectrum . The so-called thickness or Kiessig fringes are seen at larger angles and shorter wavelengths. The period of these fringes gives an approximate value of the resonator thickness.
|Tunneling barriers||τ||T = 1/ν|
|Frustrated total reflection||117 ps||120 ps|
|Double prisms||87 ps||100 ps|
|Photonic lattice||2.13 fs||2.34 fs|
|Photonic lattice||2.7 fs||2.7 fs|
|Undersized waveguide||130 ps||115 ps|
|Electron field-emission tunnelling||7 fs||6 fs|
|Electron ionization tunnelling||≤6 as||0 as|
|Acoustic (phonon) tunnelling||0.8 μs||1 μs|
|Acoustic (phonon) tunnelling||0.9 ms||1 ms|
|Neutron (resonant level)||0.217 μs||0.236 μsa|
|Neutron (non-resonant)||>19 ns||33 nsb|
In the interaction experiments of Frank et al. , the Bragg reflection time was studied with neutrons of ≈2 nm wavelength. The barrier structures were built from 30 pairs of a periodic thin film structure of 13.0 nm thick Ni-V alloy and 7.0 nm thick Ti layers based on a Si substrate. Grazing incident neutron beams were applied. An average interaction time of the order of 0.2 μs was measured over the measured grazing angle range. Assuming the double Bragg reflection is due to the Ni-V alloy Ti layers, a single peak time was 0.156 μs. This time corresponds with (1) to a reflected energy of 26.6 neV, assuming the Bragg interaction as due to a periodic lattice.
The energy of a resonant level of a sandwich structure of Ni-Ti/Zr alloy-Ni layers was estimated to be ≈127 neV. The two Ni layers present a double barrier separated by the Ti-Zr alloy. From (1), a tunnelling time of 33 ns is obtained for non-resonant tunnelling. However, a resonant level tunnelling time is always much longer as it represents a cavity . Olkhovsky et al.  have calculated the value given in the table for the interaction time with the resonant level. A different structure was studied by Maaza et al. . They observed five levels and estimated the lifetimes of the resonant neutron waves as being between 0.1 and 1 μs.
The above value of 33 ns approximately equals the measured value near resonance and was assumed to represent the travel time through the Si substrate.
Summing up, elastic, electromagnetic and Schrödinger waves reveal an approximate and universal collision duration in the case of opaque barriers. The effect is observed in the tunnelling process and in the Bragg diffraction time. It is described by the Wigner phase time and Helmholtz and Schrödinger equations, respectively. The universal tunnelling time agrees with the theoretical study of Hartman in 1962 . At that time he was motivated by the first tunnelling experiment in solid state physics by Esaki’s tunnelling diode and by Giaever’s tunnelling between two superconductors separated by a thin metal oxide layer. The collision and tunnelling time arise at the barrier front. The non-local propagation inside a barrier is described by evanescent modes and virtual wave packets , , .
This remarkable result should be further studied with respect to the Eisenbud and Wigner energy derivation of the scattering phase .
We are very grateful to Prof. M. Maaza for his support with Figure 2.
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