Field Precision title

Emittance of a circular beam

Emittance characterizes the degree of disorder in a charged particle beam. The transverse emittance parametrizes the thermal spread of transverse momentum. Random motion results in an effective pressure that determines focusing forces necessary to confine a beam as well as the minimum spot size of a focused beam. My text Charged Particle Beams (available for download at http://www.fieldp.com/cpb.html) gives a comprehensive review of definitions and applications of emittance.

High-energy accelerators employ quadrupole lenses and bending magnets. The convention is to designate the local beam axis as z, with transverse motion in x (bending plane) and y (vertical plane). In such a system, particle motions in x and y are decoupled. In this case, we can define separate emittance values εx and εy. To make an ideal emittance calculation, we make a plot of x (displacement from the main axis) and x' = dx/dz (angle with respect to the main axis) for the particle distribution at a given location in z. The quantity εx is the trace-space area of the minimal ellipse that encloses the distribution divided by π. An ideal distribution at a beam waist is one that uniformly fills ellipses with dimensions x0-x0' and y0-y0'. In this case, the emittances are

Generally, the non-uniform trace-space distributions of real beams add ambiguity to the definition of the bounding ellipse. A resolution is to employ the RMS (root-mean-squared) expressions of Lapostolle (P. Lapostolle, IEEE Trans. Nucl. Sci. NS-18, 1101 (1971).):

The overline symbols on the right-hand side denote averages over the particle distribution. The equations hold for tilted distributions (diverging or converging beams) as well as upright distributions. The factor of 4.0 is included to ensure that the expressions give the same value as Eq. 1 for a uniform distribution inside an upright ellipse.

Equations 2 and 3 are included in most books on beam physics and listed on several Internet sites. Unfortunately, they do not apply to circular beams where motions in x and y are coupled. Beams in solenoid magnet lenses have an ordered azimuthal velocity. In this case, the results of Eqs. 2 and 3 are meaningless. There are many important applications for circular electron beams guided by solenoid lenses, from the low-current beams of electron microscopes to the intense beams of pulsed radiographic accelerators. It is essential to have a consistent figure for radial emittance to use in the cylindrical paraxial ray equation.

For radial RMS emittance calculations in the Trak code, Carl Ekdahl of Los Alamos National Laboratory suggested I use the expression derived in E. Lee and R. Cooper, Particle Accelerators 7, 83 (1976). Although this result has great practical importance, I could not find it on the Internet or in any accelerator texts (including my own). Therefore, I thought it would be valuable to record it here as a potential search target:

I have written the equation in a useful form for particle codes like Trak. Such codes trace a large number of model particles to represent a beam. The codes generates values of position (x,y,z) and momentum (px,py,pz). The Lee-Cooper expression has three useful features:

  1. The value of RMS radial emittance remains invariant for beams moving through solenoid lenses.
  2. The result does not depend on whether the azimuthal distribution of particles is uniform. This is important in 2D codes like Trak where a circular beam is represent by a collection of model particles at a single azimuth. For calculations of the beam-generated electric and magnetic field, the model particles represent an annulus of charge and current.

I added the radial emittance calculation to Trak and GenDist. The tutorial Emittance Calculations for Circular Beams describes numerical tests I carried out to confirm validity and consistency.

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