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Emittance and Emittance Measurements S. Bernal

Emittance and Emittance Measurements S. Bernal. USPAS 08 U. of Maryland, College Park. Introduction: Phase and Trace Spaces, Liouville’s theorem, etc. IREAP. Vlasov Approx. G -Space. m -Space. 6N dim. 6 dim. p X →. Trace Space. Matrix notation:. or. Liouville’s Theorem :

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Emittance and Emittance Measurements S. Bernal

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  1. Emittance and Emittance MeasurementsS. Bernal USPAS 08 U. of Maryland, College Park

  2. Introduction: Phase and Trace Spaces, Liouville’s theorem, etc. IREAP Vlasov Approx. G-Space m-Space 6N dim. 6 dim. pX→ Trace Space Matrix notation: or Liouville’s Theorem: area conserved in either G or m spaces for Hamiltonian systems Units of area in Phase Space (m-space) are those of Angular Momentum. Units of area in Trace Space are those of length: “metre” (angle measure is dimensionless).

  3. IREAP Transverse coordinate for paraxial particle motion in linear lattice obeys : Further, a, b, g: Courant-Snyder or Twiss Parameters. Review of Courant-Snyder Theory b(s)is a lattice function that depends on k0 (s), but A and φare different for different particles. The ellipse equation defines self-similar ellipses of area pAX2at a given plane s. Thelargest ellipse defines the beam emittance through

  4. IREAP a0: zero-current 2rms beam radius Space charge and emittance (M. Reiser, Chap. 5) If linear space charge is included, then k0(s)→k(s)and the b function is no longer just a lattice function because it now depends on current also. b becomes a lattice+beam function. Further, adding space charge does not require changing the emittance definition…Emittance and current are independent parameters in the envelope equation: The emittance above is the effective or 4rms emittance (also called edge emittance) which is equal to the total (100%) emittance for a K-V distribution. For a Gaussian distribution, the total and RMS emittances are equal when the Gaussian is truncated at 1-s.

  5. IREAP Linear system: Define: RMS Emittance and Envelope Equation(Pat G. O’Shea)

  6. IREAP Without acceleration, rms emittance of a mono-energetic beam is conserved under linear forces (both external and internal): X’ X’ x x X’ X’ x x Emittance Growth (M. Reiser, Chap. 6) The normalized emittance is an invariant: If initial transverse beam distribution is not uniform, rms emittance grows in a length of the order of lp/4. If initial beam envelope is not rms-matched, rms emittance grows in a length of the order of lb.

  7. IREAP Laminar and Turbulent Flows

  8. IREAP Transverse Emittance Measurement Techniques

  9. IREAP By focusing a beam with a lens and measuring the width at a fixed distance z=L downstream from the lens, one can determine the emittance. Without space charge we have: slope radius R0 R(L) We find: Z Z=L Z=0 Basic Optics I

  10. IREAP Beam free expansion with space charge: emittance can be determined as a fit parameter in envelope calculation. 1 cm Z = 1.28 cm Additional beam optics checks possible with solenoid and quadrupole focusing. Z = 3.83 cm Z = 6.38 cm Z = 8.88 cm Z = 11.28 cm Basic Optics II

  11. IREAP Pepper-Pot Technique

  12. IREAP (b) (a) 0.166 mm/pixel (c) Pepper-Pot Technique 10keV, 100 mA, 700 experiment X = 67 μm, Y = 55 μm

  13. IREAP Pepper-Pot Technique Calculating emittance and plotting phase space:

  14. IREAP So beam ellipse can be written as The condition bg-a2=1iswritten Beam (s)and Transfer (R) Matrices Define the beam matrix for, say, x-plane: Consider 2 planes z0 and z1, z1>z0. Particle coordinates transform as: R is the Transfer matrix. Then, the beam matrix is transformed according to:

  15. IREAP d Solenoid IC2 Q2 IC1 Quadrupole Scan

  16. IREAP Quadrupole Scan: results for 6 mA beam

  17. References • Martin Reiser, Theory and Design of Charged-Particle Beams, 2nd Ed, Secs. 3.8.2, 5.3.4 (see Table 5.1) and Chapter 6. 2 D. A. Edwards and M. J. Syphers, An Introduction to the Physics of High Energy Accelerators, Sec. 3.2.4 (see Table 3.1). • Claude Lejeune and Jean Aubert, Emittance and Brightness: Definitions and Measurements, Advances in Electronics and Electron Physics, Supplement 13A, Academic Press (1980). • Andrew M. Sessler, Am. J. Phys. 60 (8), 760 (1992); J. D. Lawson, “The Emittance Concept”, in High Brightness Beams for Accelerator Applications, AIP Conf. Proc. No. 253; M.J. Rhee, Phys. Fluids 29 (1986). • Michiko G. Minty and Frank Zimmermann, Beam Techniques, Beam Control and Manipulation, USPAS, 1999. 6. S. G. Anderson, J. B. Rosenzweig, G. P. LeSage, and J. K. Crane, “Space-charge effects in high brightness electron beam emittance measurements”, PRST-AB, 5, 014201 (2002). 7. Manuals to computer codes like TRACE3D, TRANSPORT, etc. .

  18. IREAP Backup Slides

  19. IREAP Detectors

  20. IREAP Emittance (rms. Norm.) in DC Injection Exps. 24 mA, 10 keV, χ=0.9

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