Quantum excitation (accelerator physics)

Quantum excitation is the effect in circular accelerators or storage rings whereby the discreteness of photon emission causes the charged particles (typically electrons) to undergo a random walk or diffusion process.

## Mechanism

An electron moving through a magnetic field emits radiation called synchrotron radiation. The expected amount of radiation can be calculated using the classical power. Considering quantum mechanics, however, this radiation is emitted in discrete packets of photons. For this description, the distribution of number of emitted photons and also the energy spectrum for the electron should be determined instead.

In particular, the normalized power spectrum emitted by a charged particle moving in a bending magnet is given by

S(ξ)=938πξ∫ξ∞K5/3(ξ¯)dξ¯

This result has been derived originally by Dmitri Ivanenko and Arseny Sokolov and independently by Julian Schwinger in 1949 [1]

By dividing each power of this power spectrum by the energy, we obtain the photon flux

F(ξ)=1ξS(ξ)=938π∫ξ∞K5/3(ξ¯)dξ¯

The photon flux from this normalized power spectrum (of all energies) is then

N˙norm=938π∫ξ=0∞∫ξ¯=ξ∞K5/3(ξ¯)dξ¯dξ=Γ(11/6)Γ(1/6)938π=1538

Note: The fact that the above photon flux integral is finite implies a discrete photon emission, it is a Poisson process. The emission rate is

rγ=536ree0β4γℏ|ρ| [photon/sec] from [4.4][2] [5.9][2] [5.12][2]

For a travelled distance Δs at a speed close to c (β≈1), the average number of emitted photons by the particle can be expressed as

⟨nγ⟩=536ree0γℏ|ρ|Δsc=536αγ|ρ|Δs where α is the fine-structure constant

The probability that k photons are emitted over Δs is

Pr(nγ=k)=⟨nγ⟩kk!e−⟨nγ⟩

The point where the photon number curve and the power spectrum curve are crossing is called the critical energy

uc=3cℏγ32ρ

where γ=E/e0,E is the total energy of the charged particle, ρ is the radius of curvature, re the classical electron radius, e0=mec2 the particle rest mass energy, ℏ the reduced Planck constant and c the speed of light.

The mean of the quantum energy is given by ⟨u⟩=8153uc and impact mainly the radiation damping. However, the particle motion perturbation (diffusion) is mainly related by the variance of the quantum energy ⟨u2⟩ and leads to an equilibrium emittance. The diffusion coefficient at a given position s is given by

d(s)=55483α(ℏmec)2γ5|ρ(s)|3 [3]

## Further reading

For an early analysis of the effect of quantum excitation on electron beam dynamics in storage rings, see the article by Matt Sands.[2]

## References

1. ↑ Schwinger, Julian (1949) (in en). Qn the Classical Radiation of Accelerated Electrons. https://journals.aps.org/pr/pdf/10.1103/PhysRev.75.1912.
2. ↑ 2.0 2.1 2.2 2.3 Sands, Matthew (1970). The Physics of Electron Storage Rings: An Introduction by Matt Sands. http://slac.stanford.edu/pubs/slacreports/reports02/slac-r-121.pdf.
3. ↑ Carmignani, Nicola; Nash, Boaz (2014) (in en). Quantum Diffusion Element in AT. https://github.com/atcollab/atdoc/blob/master/QuantumDiffusionElement/qdifElemAT.pdf.

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