Abstract
1 Introduction
The Weibel instability, which is mainly characterized as being able to generate strong magnetic fields, is of great significance for a range of scenarios in plasma physics, and has been studied for many decades since it was first proposed[
The classical Weibel instability, which is referred as the thermal anisotropy-driven Weibel instability, arises from electron thermal anisotropy and can increase when a plasma slab is expanding into the vacuum[
Experimentally, when space charge effects between the current filaments have a negligible impact on deflection of the probe proton, which has been proved using the difference in deflection between the electric field and the magnetic field, the magnetic field generated by the Weibel instability can be transversely imaged by proton radiography[
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However, the strength and spatial wavelength of such an isotropic and stochastic magnetic field still cannot be directly inferred from the experiments, as far as we are aware of. The reason for this is given below. The toroidal magnetic fields have an almost constant amplitude along the longitudinal direction; their transversal distribution, however, is two-dimensionally isotropic and stochastic[
In proton radiography of three-dimensionally isotropic and stochastic magnetic field turbulence, the mean strength and the main spatial wavelength of the magnetic turbulence can be inferred by extracting the energy spectrum of the magnetic energy
In this paper, by theoretically analyzing proton radiography of a two-dimensionally isotropic and stochastic magnetic field, a relationship is built between the proton flux density perturbation on the detection plane and the energy spectrum of the magnetic field. We demonstrate for the first time that the mean strength and main spatial wavelength of the Weibel-instability-generated magnetic field can be inferred by means of proton radiography. Numerical calculations based on ray tracing methods are also conducted to verify our theory.
2 Proton radiography
The Weibel-instability-generated magnetic field has been investigated previously by many particle-in-cell simulations and experiments[
In proton radiography of a magnetic field or an electric field, the proton probe will be deflected by the field and acquire a deflection velocity after it passes. The spatial distribution of the deflection velocity on exiting the field region is an important parameter. It creates a relationship between the proton flux density obtained on the detection plane and the field to be probed. Furthermore, it has also been proved to be helpful in field reconstruction from the proton flux density in previous studies[
Assuming that the deflection distance of the proton probe inside the plasma is small and negligible, when a parallel proton beam has passed through the magnetic field region along the
After leaving the field region and traversing a distance
It is necessary to mention that the impact of the electric field on the probe proton, which is mainly due to the space charge effect between the current filaments, has been experimentally proved to be negligible when compared with that of the magnetic field[
3 Energy spectrum of the magnetic field and deflection velocity
In fluid physics, the velocity autocorrelation tensor is a commonly used tool to deduce the energy spectrum of three-dimensional fluid turbulence[
Similar to the definition of
From
4 Strength and wavelength of the magnetic field
By combining Equations (
The mean strength of the magnetic field is then acquired through
5 Numerical verification
In order to validate the above theory, numerical calculations based on ray tracing methods are conducted to simulate radiography of the Weibel-instability-generated two-dimensionally isotropic and stochastic magnetic field[
The three-dimensional magnetic field to be radiographed is pre-set as follows. Two hundred tube-like magnetic field structures are distributed in a cross-section of
The cross-section of the magnetic field strength at
The energy spectrum of the pre-set magnetic field in Figure
In this paper, our interest is limited to the case in which the Weibel instability is collisionless. In proton radiography of the Weibel-instability-generated magnetic field, the spatial density perturbation of the proton beam, which is introduced from the magnetic field, is small enough to avoid the probe trajectory crossing or overlapping. Also, space charge effects between the probe protons are neglected. This allows the ray tracing method to be used to simulate the process of proton radiography of the magnetic field. In this paper, the kinetic energy of the mono-energy probe proton is 10 MeV, emitted parallel with the
Numerical results show that after traversing the field region, the maximum proton deflection velocity in the
Figure
From Figure
We can see from Figure
In the extracted spectrum, peaks 1 and 2 have close wavelengths and evidently higher amplitudes when compared to the other peaks in the blue line. Therefore, the extracted main wavelength for the magnetic energy is taken from the average wavelength of peaks 1 and 2. The result shows that
It is necessary to mention that our theoretical analyses predict that the energy spectra are strictly equal for an ideal stochastic magnetic field; the difference between the two energy spectra in Figure
Comprehensively examining the consistency between the pre-set values and the extracted results, we conclude that the method demonstrated above is applicable for extracting the main wavelength and mean strength of the Weibel-instability-generated magnetic field with proton radiography.
6 Discussion
In the above demonstration, some ideal model simplifications are assumed. The situations in experiment, however, could be much more complicated.
In this paper, all the probe protons have the same kinetic energy and are emitted from the source at the same time. Experimentally, the proton probes used in the radiography are mostly generated through the target normal sheath acceleration mechanism. The difference in the proton emission time and energy spread can broaden the probe duration. Generally, the probe pulse duration can reach about one picosecond[
At the same time, as has been discussed in Section
7 Conclusion
A method to infer the strength and wavelength of the Weibel-instability-generated magnetic field from the proton radiography has been demonstrated in our paper. With theoretical analyses, it is found that in the proton radiography of a Weibel-instability-induced magnetic field, which is usually two-dimensionally isotropic and stochastic, the proton flux density perturbation on the detection plane can be related to the energy spectrum of the magnetic field. It further allows us to obtain the mean strength and main wavelength of the magnetic field. Using ray tracing methods, a proton beam has been emitted numerically to simulate radiography of a two-dimensionally isotropic and stochastic magnetic field with the pre-set main wavelength equal to approximately
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