Abstract
We analytically solve the relativistic equation of motion for an electron in ion plasma channels and calculate the corresponding trajectory as well as the synchrotron radiation. The relativistic effect on a trajectory is strong, i.e., many highorder harmonic terms in the trajectory, when the ratio of the initial transverse velocity to the longitudinal velocity of the electron injected to ion plasma channels is high. Interestingly, these highorder harmonic terms result in a quite broad and intense radiation spectrum, especially at an oblique angle, in contrast to an earlier understanding. As the initial velocity ratio decreases, the relativistic effect becomes weak; only the first and second harmonic terms remain in the transverse and longitudinal trajectories, respectively, which coincides with the result of Esarey et al. [Phys. Rev. E65, 056505 (2002)]. Our formalism also allows the description of electron’s trajectory in the presence of an applied magnetic field. Critical magnetic fields for cyclotron motions are figured out and compared with semiclassical results. The cyclotron motion leads to more highorder harmonic terms than the trajectory without magnetic fields and causes an immensely broad spectrum with vastly large radiation amplitude for high initial velocity ratios . The radiation from hard xray to gammaray regions can be generated with a broad radiation angle, thus available for applications.
This work was financially supported by a project of the National Science Council, Taiwan, under Contract Nos. NSC 972811M009055 and NSC 972112M009002MY3.
I. INTRODUCTION
II. THEORY
A. Trajectory
B. Radiation
III. RESULTS AND DISCUSSION
A. Trajectory
B. Radiation
IV. CONCLUSION
Key Topics
 Magnetic fields
 51.0
 Semiclassical theories
 28.0
 Betatrons
 24.0
 Gamma rays
 10.0
 Equations of motion
 8.0
Figures
Schematic diagram for an electron in an ion plasma channel subject to an electric field and a magnetic field.
Schematic diagram for an electron in an ion plasma channel subject to an electric field and a magnetic field.
(a) Transverse trajectory and (b) longitudinal trajectory as a function of time calculated by the relativistic (abbreviated as R) and semiclassical (abbreviated as S) theories at incident energies (E) of 1 GeV (solid line for R, hollow triangle for S), 500 MeV (dashed line for R, hollow circle for S), and 100 MeV (dasheddotted line for R, cross for S): and . The inset plots show the corresponding velocities and trajectories in the xz plane for relativistic results only.
(a) Transverse trajectory and (b) longitudinal trajectory as a function of time calculated by the relativistic (abbreviated as R) and semiclassical (abbreviated as S) theories at incident energies (E) of 1 GeV (solid line for R, hollow triangle for S), 500 MeV (dashed line for R, hollow circle for S), and 100 MeV (dasheddotted line for R, cross for S): and . The inset plots show the corresponding velocities and trajectories in the xz plane for relativistic results only.
(a) Transverse trajectory and (b) longitudinal trajectory as a function of time calculated by the relativistic theory at initial velocity ratios of (solid line), 5:5 (dashed line), and 3:7 (dasheddotted line): and . The inset plots show the corresponding velocities and trajectory in the xz plane.
(a) Transverse trajectory and (b) longitudinal trajectory as a function of time calculated by the relativistic theory at initial velocity ratios of (solid line), 5:5 (dashed line), and 3:7 (dasheddotted line): and . The inset plots show the corresponding velocities and trajectory in the xz plane.
(a) Transverse trajectory and (b) longitudinal trajectory as a function of time in the presence of magnetic field calculated by the relativistic (solid line) and semiclassical (dashed line) theories: and . The inset plots show the corresponding velocities and trajectories in the xz plane.
(a) Transverse trajectory and (b) longitudinal trajectory as a function of time in the presence of magnetic field calculated by the relativistic (solid line) and semiclassical (dashed line) theories: and . The inset plots show the corresponding velocities and trajectories in the xz plane.
(a) Critical magnetic field for cyclotron motion as a function of the incident energy calculated by the relativistic (R) and semiclassical (S) theories at three sets of conditions: (i) , (solid line for R, hollow triangle for S); (ii) , (dashed line for R, hollow circle for S); and (iii) , (dasheddotted line for R, cross for S). (b) Peaktopeak amplitude of x(t) and (c) oscillation period as a function of energy at with four sets of conditions: (i) , (solid line for R, hollow circle for S); (ii) , (dashed line for R, hollow triangle for S); (iii) , (dasheddotted line for R, cross for S); and (iv) , (dotted line for R, hollow square for S).
(a) Critical magnetic field for cyclotron motion as a function of the incident energy calculated by the relativistic (R) and semiclassical (S) theories at three sets of conditions: (i) , (solid line for R, hollow triangle for S); (ii) , (dashed line for R, hollow circle for S); and (iii) , (dasheddotted line for R, cross for S). (b) Peaktopeak amplitude of x(t) and (c) oscillation period as a function of energy at with four sets of conditions: (i) , (solid line for R, hollow circle for S); (ii) , (dashed line for R, hollow triangle for S); (iii) , (dasheddotted line for R, cross for S); and (iv) , (dotted line for R, hollow square for S).
(a) Radiation intensity spectrum of the semiclassical trajectory at , , , , azimuth angle , polar angle , and oscillation number . (b) Radiation spectrum of the trajectory of Esarey et al. [z(t) with the second harmonic term] using the Gaussian quadrature method (triangle) and Jacobi–Anger expansion (solid line) with the same conditions as those in Fig. 6(a).
(a) Radiation intensity spectrum of the semiclassical trajectory at , , , , azimuth angle , polar angle , and oscillation number . (b) Radiation spectrum of the trajectory of Esarey et al. [z(t) with the second harmonic term] using the Gaussian quadrature method (triangle) and Jacobi–Anger expansion (solid line) with the same conditions as those in Fig. 6(a).
Radiation intensity spectrum of the relativistic trajectory using (a) the Jacobi–Anger expansion and (b) the Gaussian quadrature method with the same conditions as those in Fig. 6(a).
Radiation intensity spectrum of the relativistic trajectory using (a) the Jacobi–Anger expansion and (b) the Gaussian quadrature method with the same conditions as those in Fig. 6(a).
Radiation intensity spectrum of (a) the relativistic trajectory and (b) the trajectory of Esarey et al. at . Other conditions are the same as those in Fig. 6(a). The inset of (a) shows the enlarged diagram. The inset of (b) shows the semiclassical result.
Radiation intensity spectrum of (a) the relativistic trajectory and (b) the trajectory of Esarey et al. at . Other conditions are the same as those in Fig. 6(a). The inset of (a) shows the enlarged diagram. The inset of (b) shows the semiclassical result.
Radiation intensity spectrum of the relativistic trajectory (a) at and (b) at . Other conditions are the same as those in Fig. 6(a).
Radiation intensity spectrum of the relativistic trajectory (a) at and (b) at . Other conditions are the same as those in Fig. 6(a).
Radiation intensity spectrum for the low initial velocity ratio of the relativistic trajectory (denoted by solid line) and the trajectory of Esarey et al. (hollow triangle) at (a) , (b) , and (c) . Other conditions are the same as those in Fig. 6(a). The inset plots show the radiation spectra of semiclassical trajectory.
Radiation intensity spectrum for the low initial velocity ratio of the relativistic trajectory (denoted by solid line) and the trajectory of Esarey et al. (hollow triangle) at (a) , (b) , and (c) . Other conditions are the same as those in Fig. 6(a). The inset plots show the radiation spectra of semiclassical trajectory.
Effect of the magnetic field on the radiation intensity spectrum of the relativistic trajectory at (a) and (b) . The spectra with and without the magnetic field are denoted by solid triangle and hollow square, respectively; , , , , and . The inset plots show the results of semiclassical trajectory (with : solid line; without : dasheddotted line) and the enlarged diagram.
Effect of the magnetic field on the radiation intensity spectrum of the relativistic trajectory at (a) and (b) . The spectra with and without the magnetic field are denoted by solid triangle and hollow square, respectively; , , , , and . The inset plots show the results of semiclassical trajectory (with : solid line; without : dasheddotted line) and the enlarged diagram.
Radiation intensity spectrum (solid line) with (60 T) of the relativistic trajectory at for (a) and (b) . Other conditions are the same as those in Fig. 11(a). The inset plots show the enlarged diagrams. The results calculated using the twice integration accuracy better than that of solid line are denoted by solid square. A good agreement between them is demonstrated.
Radiation intensity spectrum (solid line) with (60 T) of the relativistic trajectory at for (a) and (b) . Other conditions are the same as those in Fig. 11(a). The inset plots show the enlarged diagrams. The results calculated using the twice integration accuracy better than that of solid line are denoted by solid square. A good agreement between them is demonstrated.
Radiation intensity spectrum (solid line) with (60 T) of the relativistic trajectory at for (a) and (b) . Other conditions are the same as those in Fig. 11(a). The inset plots show the enlarged diagrams. The results calculated using the twice integration accuracy better than that of solid line are denoted by solid square. A good agreement between them is demonstrated.
Radiation intensity spectrum (solid line) with (60 T) of the relativistic trajectory at for (a) and (b) . Other conditions are the same as those in Fig. 11(a). The inset plots show the enlarged diagrams. The results calculated using the twice integration accuracy better than that of solid line are denoted by solid square. A good agreement between them is demonstrated.
Radiation intensity spectrum (solid line) with (60 T) of the relativistic trajectory at for (a) and (b) . Other conditions are the same as those in Fig. 11(a). The inset plots show the enlarged diagrams. The results calculated using the twice integration accuracy better than that of solid line are denoted by solid square. A good agreement between them is demonstrated.
Radiation intensity spectrum (solid line) with (60 T) of the relativistic trajectory at for (a) and (b) . Other conditions are the same as those in Fig. 11(a). The inset plots show the enlarged diagrams. The results calculated using the twice integration accuracy better than that of solid line are denoted by solid square. A good agreement between them is demonstrated.
Radiation intensity spectrum of the relativistic trajectory for low initial velocity ratio at (a), (b) , and (c) . Other conditions are the same as those in Fig. 11(a). The inset plots show the results of semiclassical trajectory.
Radiation intensity spectrum of the relativistic trajectory for low initial velocity ratio at (a), (b) , and (c) . Other conditions are the same as those in Fig. 11(a). The inset plots show the results of semiclassical trajectory.
Tables
Fourier components of relativistic trajectories at various initial velocity ratios . ; ; ; ; and . Dominant components are denoted by bold letters.
Fourier components of relativistic trajectories at various initial velocity ratios . ; ; ; ; and . Dominant components are denoted by bold letters.
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