Radiation of breathing vortex electron packets in magnetic field

This paper demonstrates that although vortex electrons in magnetic fields exhibit oscillating wave packets that theoretically emit radiation, the resulting energy and orbital angular momentum losses are negligible, confirming that linear accelerators are effective tools for preserving the vorticity of relativistic vortex electrons.

The Gist

The Big Picture: Twisted Electrons and the "Breathing" Problem

Imagine an electron not just as a tiny point of charge, but as a swirling tornado or a corkscrew. In physics, we call these "vortex electrons" because they carry a special kind of spin called Orbital Angular Momentum (OAM). Think of this OAM as the electron's "twistiness." Scientists want to use these twisted electrons for advanced imaging and research, but they need to speed them up to very high energies first.

To speed them up, you usually put them in a linear accelerator (a straight tube with magnets). The problem the authors investigated is this: Does the electron lose its "twist" while it's being sped up?

The Setup: A Bouncing Ball in a Magnetic Field

When a normal electron enters a magnetic field, it usually settles into a calm, steady orbit (like a planet in a stable orbit). But a "vortex" electron is different. Because it starts as a swirling cloud, when it hits the magnetic field, it doesn't settle down immediately.

Instead, the electron's shape starts to breathe.

• The Analogy: Imagine a balloon that is being squeezed and released rhythmically. It expands and contracts over and over again.
• The Physics: The electron's "cloud" expands and shrinks (oscillates) as it moves through the magnetic field. This is called a "breathing" motion.

The Fear: Does the "Breath" Create a Leak?

In the world of classical physics (the rules that govern everyday objects), if you have a charged object that is shaking, vibrating, or breathing, it is supposed to radiate energy. It's like a speaker vibrating and creating sound waves.

The authors asked a critical question:

• If this "breathing" electron radiates energy, does it also radiate away its twist (its OAM)?
• If the electron loses its twist by emitting light (photons), then we can't use these particles for our high-tech applications because they will arrive at the destination "un-twisted."

The Investigation: Solving the Equations

The researchers used a "semi-classical" approach. They treated the electron's wave function (its quantum shape) like a real, physical cloud of electric charge. They calculated:

1. How much energy this breathing cloud emits.
2. How much "twist" (angular momentum) is carried away by that emitted energy.

They looked at two scenarios:

1. Electron Microscopes: Short distances, lower speeds.
2. Linear Accelerators (Linacs): Very long distances (up to 1 kilometer), near the speed of light.

The Results: The "Twist" is Safe!

The findings were surprisingly good news for scientists who want to use these particles.

1. The Energy Loss is Tiny
Even though the electron is "breathing," the amount of energy it leaks out is incredibly small.

• The Analogy: It's like a leaky faucet in a massive swimming pool. Even if the faucet drips for a long time, the pool doesn't lose a noticeable amount of water.
• The Math: For a typical setup, the energy lost is so small that the electron is unlikely to even emit a single photon (a particle of light) during its journey.

2. The "Twist" (OAM) is Safe
This is the most important part. The researchers calculated how much "twist" is lost.

• The Result: For almost all realistic scenarios (where the electron cloud isn't absurdly huge), the electron loses almost zero of its orbital angular momentum.
• The Analogy: Imagine a figure skater spinning with their arms out. Even if they wiggle a little bit, they don't suddenly stop spinning. The "twist" stays with them.
• The Exception: The only time the twist is lost significantly is if the electron cloud is initially massive (much larger than the magnetic field's natural scale). But in real machines, electron clouds are usually small enough that this doesn't happen.

The Conclusion: Linear Accelerators are Safe

The paper concludes that linear accelerators are a safe and reliable tool for speeding up vortex electrons.

• The Takeaway: You can take a "twisted" electron, shoot it down a long, straight magnetic track, and it will arrive at the other end still "twisted." It won't lose its special properties to radiation.
• Why it matters: This confirms that we can build machines to create high-energy vortex electrons for use in materials science and particle physics without worrying that the acceleration process will destroy the very thing that makes them special.

In short: The electron breathes, but it doesn't cough up its soul. Its "twist" remains intact.

Technical Summary

1. Problem Statement

Vortex electrons (twisted electrons) possess a definite projection of Orbital Angular Momentum (OAM), making them valuable for applications in material science, chemistry, and particle physics. A major challenge in utilizing these particles is accelerating them to relativistic energies (GeV range) required for high-energy physics, as current vortex electron energies are limited to ~300 keV.

The primary concern is whether vortex electrons can preserve their OAM (vorticity) while traveling through the electromagnetic fields of a linear accelerator (linac). Specifically, the authors investigate:

• Does the oscillating charge distribution of a vortex electron in a magnetic field emit radiation?
• Does this radiation carry away OAM, leading to a significant loss of the electron's vorticity?
• Is the "breathing" (oscillating) nature of the electron wave packet in a magnetic field a source of significant energy and angular momentum loss?

2. Methodology

The authors employ a semiclassical approach to model the radiation emitted by a vortex electron propagating through a longitudinal magnetic field (typical of solenoids in linacs or electron microscopes).

• Quantum State Modeling:

  • Instead of using stationary Landau states, the electron entering a magnetic field from free space is described by a Non-Stationary Laguerre-Gaussian (NSLG) state.
  • This state is characterized by "breathing" dynamics: the root-mean-square (r.m.s.) radius of the electron wave packet oscillates at the cyclotron frequency ($\omega_c$), similar to betatron oscillations.
  • The wave function $\Psi$ is used to derive the probability density and current.

• Source Terms for Maxwell's Equations:

  • The probability density and current are multiplied by the electron charge ($e$) to form effective charge ($\rho$) and current ($\mathbf{j}$) densities.
  • These densities serve as source terms in Maxwell's equations to calculate the emitted electromagnetic fields.

• Radiation Calculation:

  • The authors solve for the scalar ($\phi$) and vector ($\mathbf{A}$) potentials using the retarded time formalism.
  • They utilize a far-field approximation but crucially retain the next-order term in the expansion of the denominator ($1/|\mathbf{R}-\mathbf{r}|$). This is essential because the leading-order term contributes to radiated power, but the next-order term is required to calculate the angular momentum of the radiation.
  • The Poynting vector ($\mathbf{S}$) is decomposed into far-field ($S_{far} \propto R^{-2}$), interference ($S_{int} \propto R^{-3}$), and near-field ($S_{near} \propto R^{-4}$) components.

• Key Quantities Calculated:

  • Radiated Power ($P$): Derived from the $S_{far}$ term.
  • OAM Loss Rate ($dL/dt$): Derived primarily from the $S_{int}$ (interference) term, as the far-field contribution averages to zero over a cyclotron period.

3. Key Contributions and Theoretical Findings

• Mechanism of Radiation: The paper establishes that the "breathing" motion of the NSLG state (oscillations of the wave packet width) creates a non-uniformly moving charge cloud. Classically, this oscillating distribution must radiate.
• Scaling Laws:
  • Radiated Power: Scales as $\langle P \rangle \propto H^6$ (where $H$ is the magnetic field strength) and is proportional to $s^2(2n + |l| + 1)^2 (\sigma_{st}^4 - \sigma_L^4)$.
  • OAM Loss: Scales as $\langle dL/dt \rangle \propto H^5$.
  • Dependence on Initial Conditions: Radiation is zero for stationary Landau states ($s=0$). It is non-zero only when the initial wave packet width ($\sigma_0$) differs from the equilibrium Landau width ($\sigma_L$), causing the packet to expand or contract.
• Angular Momentum Transfer: The authors demonstrate that the OAM loss is determined by the interference term ($1/R^3$) rather than the far-field term. This highlights a subtle quantum-classical correspondence where the angular momentum flux is a higher-order effect compared to energy flux.

4. Results and Numerical Analysis

The authors applied their model to two scenarios: Transmission Electron Microscopes (TEM) and Linear Accelerators (Linacs).

• TEM Scenario (Short Distance):

  • For a 20 cm path with a 1 T magnetic field, the total radiated energy is extremely small (max $\sim 10^{-5}$ eV), far below the cyclotron energy quantum ($\hbar\omega_c \approx 10^{-4}$ eV).
  • The probability of emitting even a single photon is negligible.

• Linac Scenario (Long Distance):

  • Extrapolating to a 1 km linac section, the total radiated energy increases to $\sim 3.5 \times 10^{-2}$ eV (approx. 350 photons at cyclotron frequency).
  • OAM Loss: Despite the increased energy radiation, the loss of OAM remains negligible for realistic wave packet sizes ($\sigma_0 \gtrsim \sigma_L$).
  • For typical parameters, the total OAM shed over 1 km is $\Delta L_z \approx 10^{-6} \hbar$, which is insignificant compared to the initial OAM ($|l| \geq 1$).
  • Limit of Validity: The semiclassical approximation breaks down only for extremely large initial wave packet widths ($\sigma_0 \gg \sigma_L$, e.g., sub-millimeter scales), where the OAM loss could exceed $1\hbar$. However, such large packets are not typical for standard accelerator beams.

• Fringe Fields: The paper also analyzes the fringe fields at the solenoid boundaries. While these can generate transient bursts of radiation, the effect is distinct from the steady-state radiation and does not significantly alter the conclusion regarding OAM preservation in the main accelerator body.

5. Significance and Conclusion

• Validation of Linacs for Vortex Electrons: The most critical conclusion is that linear accelerators are a robust platform for accelerating vortex electrons to relativistic energies. The radiation-induced loss of Orbital Angular Momentum is negligible for realistic beam parameters.
• Preservation of Vorticity: The study confirms that vortex electrons can maintain their topological charge (vorticity) during propagation through longitudinal magnetic fields, enabling their use in future high-energy experiments.
• Theoretical Insight: The work bridges the gap between classical electrodynamics (oscillating charge clouds) and quantum mechanics (NSLG states), providing a rigorous semiclassical framework for calculating radiation from structured electron beams.
• Future Directions: The authors note that while their semiclassical model is robust, a full Quantum Electrodynamics (QED) treatment is needed to account for spontaneous quantum transitions between energy levels, though they hypothesize the "classical" breathing mechanism will dominate for sufficiently non-stationary states.

In summary, the paper resolves a critical feasibility question: Vortex electrons will not lose their "twist" due to radiation losses in linear accelerators, making them viable candidates for next-generation high-energy physics applications.