Statistical Description of Charged Particle Beams and Emittance Measurements Jürgen Struckmeier HICforFAIR Workshop “Aktuelle Probleme der Beschleuniger- und Plasmaphysik” des Instituts für Angewandte Physik der Johann Wolfgang Goethe-Universität Frankfurt am Main Riezlern, 7-13 March 2010

Outline 1.Emittance in geometrical optics 2.Beam moments 3.Invariant associated with Hill’s equation 4.Moment equations 5.Analysis of emittance measurements 6.Conclusions

1. Emittance in geometrical optics “Plane wave” from a distant light source Light from a spatially extended light source (e.g. sun) Sharp focal point: this corresponds to a zero beam emittance Astigmatism: this corresponds to a beam with finite emittance  The emittance measures the “disorder” of the beam. An analytical description will be given in this talk.

2. Beam moments In charged particle beam dynamics, we are commonly not interested in the phase-space location of individual particles  statistical mechanics approach is appropriate. For a given distribution of N beam particles with coordinates x i, we define the moments of this distribution as The similar definition applies for all other coordinates.

2. Beam moments Of particular importance are the second beam moments: and the ratios of the fourth moments to the square of the second beam moments:

2. Beam moments We ask: how are these moments related to physical beam properties? To answer this question, we consider a uniform density of points in the (x,x′)-plane and calculate the second central moments.  we make the transition to a continuous description! Assuming a centered distribution (i.e. zero first moments), the second moment in x is given by: For a uniform density  within an elliptical boundary, this means

2. Beam moments To ease the calculation, we can convert the ellipse into a circle by means of a normal form and a scaling transformation.  The shape of the distribution is maintained.  The second moment is thus proportional to the square of the distribution’s maximum, here the beam envelope. The factor ¼ is a characteristic of the uniform density!

2. Beam moments The 4th moment of a uniform distribution in a phase plane is obtained as The dimensionless ratio of the 4th moment to the square of the corresponding 2nd moment follows as The ratio of 2 is a characteristic of the uniform density!  The ratio characterizes the type of beam distribution (hollow, uniform, gaussian, …)

2. Beam moments Table 1: Characteristic ratios of beam moments

2. Beam moments We have seen that the beam moments have a physical meaning at a fixed position s along the beam axis. Now the dynamical properties of the beam moments will be derived. To this end, we set up their equations of motion from the single particle equations of motions. We first derive the invariant of Hill’s equation.

3. Invariant of Hill’s equation Given a pair of distinct particles whose motion follows from Hill’s equation Then is a non-trivial invariant, i.e. We can prove this easily by direct computation: Remark: the particle’s energy is not invariant.

3. Invariant of Hill’s equation Interpretation: both particles experience the same focusing function k 2 (s)  in a two-dimensional rectangular system, this corresponds to a circular symmetric force, i.e., a central force field The invariant thus corresponds to the conservation of angular momentum in central force fields.  for a system of n particles the sum D 1 is also a constant

4. Moment equations We now set up the equations of motion for the second moments: Inserting the second equation into the first yields the second order equation for the second central moment in x: This form is not useful as it contains two kinds of second moments.

4. Moment equations Yet — in the linear approximation — we can express the unwanted moment in terms of the invariant of Hill’s equation D 1 ≡  x 2,rms : We thus obtain an equation that only contains constants and functions of the second central moment in x: This equation can be simplified making use of the identity

4. Moment equations Defining the abbreviation we obtain the final form for the equation of motion of the variance of the set of beam particles in x: This is the rms envelope equation. It applies to arbitrary phase-space distributions of the beam particles if the particle equations of motion are linear. If the particle motion is non-linear, then  x,rms is an unknown function of s. We can nevertheless use this equation as an approximation as long as the rms emittance is an adiabatic invariant (e.g.: transient effects).

The quantity W and denotes the field energy per unit length of all particles of the actual distribution. W u the describes the field energy for a uniform charge distribution of same rms size. We see that the rms emittance is directly related to physical quantities, hence it is physical on its part. One source of rms emittance change is due to a reversible change of the beam’s field energy. With the scaled beam current K 4. Moment equations this effect is described (for unbunched beams) by the equation

4. Moment equations Mismatched beam in a quadrupole channel, s 0 = 60°

4. Moment equations Matched beam in a quadrupole channel, s 0 = 60°

4. Moment equations We have seen that the beam moments have both a static and a dynamic meaning. It will now be shown that we can directly measure the moments.

5. Analysis of emittance measurements Principle and setup of a “slit and collector” emittance measurement device

5. Analysis of emittance measurements Simple graphical 3D representation of the raw data of a “slit-and- collector” emittance measurement

5. Analysis of emittance measurements Calculation of beam moments from the raw data, given by the current matrix with i nm the collector current as a function of n and m, and  x: step size of the slit position. This defines the spatial resolution of the device.  x′: angle between neighboring collector stripes, which defines the angular resolution.

5. Analysis of emittance measurements The first beam moments are now The second moment follow as

5. Analysis of emittance measurements Finally, the 4th beam moments are given by Higher order moments can be calculated correspondingly. The “rms emittance” follows from the “current matrix” i nm as

6. Conclusions Measuring the properties of charged particle beam dynamics, a statistical approach is appropriate in terms of beam moments. The naïve approach to derive emittance “areas” from the raw data fails as a “boundary” is not well defined and hence so inaccurate that the results are mostly useless (see PhD thesis A. Schönlein). To demonstrate this, we consider the phase-space evolution of a set of points representing solutions of the mathematical pendulum, with each point representing the evolution of a specific initial condition. What would happen if we derived “areas” with a finite resolution measuring device?

6. Conclusions Time evolution of a uniform (top) and a Gaussian (bottom) phase- space distribution for the mathematical pendulum.

6. Conclusions Properties of the rms emittance calculation: no assumption is made with respect to the phase-space distribution of the beam particles all information from the measurement is taken into account there is usually no need to define a cut-off current i min, so that i nm =0 for i nm,raw <i min the calculated moments (and hence the rms emittance) can directly be compared to the moments of multi-particle simulations and to the moments of continuous model distributions. The method can be generalized: if we find a method to measure simultaneously the current at n(  x), m(  x′), j(  y), k(  y′) yielding a current tensor i nmjk, then the entire 4×4 transverse beam matrix could be calculated.