Emittance

The emittance is the product of the angular divergence and the cross-sectional area of a particle or light beam : ${\displaystyle \epsilon }$

${\displaystyle \epsilon =\mathrm {Div} \cdot A}$

More precisely, the emittance describes the volume that a particle or light beam fills in the phase space . The term is mainly important in electron optics and accelerator physics as a measure of the cross section and concentration of a particle beam in an accelerator .

In large particle accelerators, it is mostly a matter, an as small to achieve beam emittance, so to create a highly focused beam of particles has a small cross-sectional area. On its way through the particle accelerator, the particle beam should also remain focused (ie not “smear” or diverge).

A related term from light optics is brilliance .

introduction

First, consider the particles of a particle beam that was generated by a particle source, assuming that they move almost parallel to the positive z-axis, and (for the sake of simplicity) only in one plane, the xz-plane . Then the phase space is two-dimensional: it consists of the plane that is spanned by the coordinates  x and  p _x , where p _{x is} the x component of the particle momentum (which in this case is much smaller than the z component p _z ) . Instead of the momentum component p _x one can also use the small angle θ that the direction of flight of the particle makes with the z-axis:

${\displaystyle \tan \theta ={\frac {p_{x}}{p_{z}}}\approx \theta }$( Small angle approximation ),

where θ is measured in radians , i.e. in radians (1 rad = 57.3 °).

The emittance is the size of the area of ​​the x-θ plane filled with dots

If we now consider this x-θ-plane at a certain point  z , every particle in this plane is characterized by a point. All particle points are in a certain area of ​​the plane, i.e. H. of the phase space; the size of this area in the x-θ plane is the emittance, measured e.g. B. in mm · mrad.

In practice, the particles of the beam (which essentially flies in the z-direction) will not only have an x-component, but also a y-component; the phase space is then four-dimensional instead of two-dimensional. Then the particle points do not fill an area, but a four-dimensional volume, measured in (mm mrad) ² .

Example : if a parallel light beam is focused in the z-direction by a converging lens on a small focal spot , the width of the beam is reduced, but the angle of the individual beams to the z-axis increases: the product, the emittance, remains constant. This is an example of Liouville's theorem : when using linear optics, the emittance is a conserved quantity , i. that is, it does not change along the beam, in the z-direction.

Applications

The emittance denotes the phase space volume of an ensemble of particles, which z. B. was generated by a particle source or is located in a particle accelerator. The emittance is determined by the properties of the (light or particle) source.

With high jet currents, i.e. H. in an ensemble that consists of many charged particles, the electromagnetic interaction between the particles of the particle beam can no longer be neglected. The space charge effects that occur here lead to an increase in emittance.

Likewise, other nonlinear effects, e.g. B. energy losses due to friction or release of synchrotron radiation , to a change in emittance, d. In other words, the emittance is then no longer a conserved quantity.

The emittance also changes when the particles accelerate in the direction of the beam: it becomes smaller because the longitudinal component of the momentum vector increases with the acceleration, but the transverse components do not change, and thus the divergence of the beam becomes smaller.

The normalized emittance takes this emittance reduction into account and is - if the particle acceleration is the only non-linear effect - a conserved quantity. (Since all charged particles emit synchrotron radiation, the normalized emittance is, however, only a conservation quantity to a good approximation). It is defined as: ${\displaystyle \epsilon _{n}}$

${\displaystyle {\begin{aligned}\epsilon _{n}&={\frac {p_{0}}{m_{0}\cdot c}}\cdot \epsilon \\&=\gamma \cdot \beta \cdot \epsilon \end{aligned}}}$

With

• the rest pulse ${\displaystyle p_{0}}$
• the rest mass ${\displaystyle m_{0}}$
• the speed of light ${\displaystyle c}$
• the Lorentz factor ${\displaystyle \gamma }$
• the particle speed relative to the speed of light. ${\displaystyle \beta =v/c}$

Particularly in the case of beams of lightly charged particles (such as electrons), the synchrotron radiation attenuation creates an emittance equilibrium (equilibrium emittance).

There are different definitions of emittance as the edge of the area is not precisely defined:

• 95% or 90% emittance: 95% (90%) of the particles are within the phase space volume
• RMS emittance: based on the standard deviation
• Convention : is usually added to the unit, it is then z. B. mm mrad. ${\displaystyle \pi }$ ${\displaystyle \pi }$

For free-electron lasers , the “slice emittance” is an important parameter: Instead of a beam, the accelerator contains bunches of particles . These packages can be seen in smaller slices ( slices cut), and each of these discs assign an emittance. This is the slice emittance, which is especially important for lasing the FEL.

Individual evidence

1. ↑ DA Edwards, MJ Syphers: An Introduction to the Physics of High-Energy Accelerators . Wiley, 1993, ISBN 0-471-55163-5 .
2. ^ David M. Pozar: Microwave Engineering . Wiley, 1993, Elsevier 2003, ISBN 0-471-17096-8 , p. 593.
3. ↑ R. Bakker et al .: First beam measurements at the photo injector test facility at DESY Zeuthen . In: K.-J. Kim, SV Milton, E. Gluskin (eds.): Free Electron Lasers 2002. Elsevier, 2003, ISBN 0-444-51417-1 , pp. 210-214 ( article in the CERN publication database (PDF; 246 kB), accessed on May 5, 2013).
4. ↑ K. Abrahamyan, J. Bähr, I. Bohnet , K. Flöttmann, M. Krassilnikov, D. Lipka, V. Miltchev, A. Oppelt, F. Stephan, Iv. Tsakov, Transverse emittance measurements at the photo injector test facility at DESY Zeuthen (PITZ) . In: Proceedings DIPAC 2003, Mainz. ( Article in the CERN publications database (PDF; 784 kB), accessed on June 4, 2013).
5. ↑ Frank Hinterberger: Physics of the particle accelerator and ion optics . Springer, 2008, ISBN 978-3-540-75281-3 , pp. 363-381.
6. ^ SY Lee: Accelerator Physics , World Scientific, Singapore 1999, ISBN 981-02-3710-3 , p. 489
7. ↑ DH Dowell et al .: Slice emittance measurements at the SLAC gun test facility . In: K.-J. Kim, SV Milton, E. Gluskin (eds.): Free Electron Lasers 2002. Elsevier, 2003, ISBN 0-444-51417-1 , pp. 327-330 ( article in the SLAC publication database (PDF; 161 kB), accessed on May 5, 2013).
8. ↑ DH Dowell et al .: Longitudinal emittance measurements at the SLAC gun test facility . In: K.-J. Kim, SV Milton, E. Gluskin (eds.): Free Electron Lasers 2002. Elsevier, 2003, ISBN 0-444-51417-1 , pp. 331-334 ( article in the SLAC publication database (PDF; 72 kB), accessed on May 5, 2013).

Remarks

1. ↑ An ensemble of particles describes a number of particles that belong together in a certain way, whereby one understands by particles not only elementary particles , but also very general and abstract elements that are not characterized except by their location where they are. An example is a particle package consisting of billions of electrons that moves through a particle accelerator - the electrons in the particle package have the same elementary properties, but are located in different places within the package.