Space-charge Effects for a KV-Beam Jeffrey Eldred

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Presentation transcript:

Space-charge Effects for a KV-Beam Jeffrey Eldred 1 1 Space-charge Effects for a KV-Beam Jeffrey Eldred Classical Mechanics and Electromagnetism June 2018 USPAS at MSU 1 1 1 1 1 1

Space-charge Effects From KV-Distribution 2 2 2 2 2 2 2 2 2 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 2 2 2 2 2 2

Cylindrical Approximation for a Beam Typically a bunch is much longer than it is wide. As the beam accelerates it will compress longitudinally but the electric fields from the beam will also compress longitudinally. Since the longitudinal beam distribution is much more gradual than the transverse beam distribution, we can use a Gaussian cylinder to approximate the local electric field. 3 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 3 3 3 3

Cylindrical Approximation for a Beam Gaussian cylinder with a line charge at origin: Gaussian cylinder with a line charge displaced: Gaussian cylinder with transverse line-charge distribution: 4 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 4 4 4 4

Kapchinskij-Vladimirskij (KV) Distribution Normalized coordinates: Beam envelopes: KV-Distribution: A 4D ellipsoid shell projects onto the x-y plane as 2D uniform ellipse. This statue in Calgary illustrates the profile of a hollow 3D shape. 5 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 5 5 5 5

Field from KV Beam Electric field from transverse distribution: Transverse distribution from KV beam: Electric field from KV beam: Magnetic field from KV beam: Force from KV beam: 6 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 6 6 6 6

Equations of motion for KV Beam Force from KV beam: Transverse focusing: Linear! Space-charge perveance Classical particle radius 7 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 7 7 7 7

Beam Envelope Equation Transverse focusing: Floquet Transformation: Beam Envelope Equation: 8 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 8 8 8 8

Beam Envelope Equation Beam Envelope Equation for a=b: Space-charge parameter 9 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 9 9 9 9

Betatron Oscillation vs. Envelope Oscillation S. Lund 10 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 10 10 10 10

Space-charge Tune-shift for KV Beam Equations of motion with round beam at equilibrium: Change to focusing force, small κ: Space-charge Tune-shift for KV beam: 11 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 11 11 11 11

Other Space-charge Effects 12 12 12 12 12 12 12 12 12 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 12 12 12 12 12 12

Core-Halo Model T. Wangler et al. PRSTAB 1998 13 13 13 13 13 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 13 13 13 13

Space-charge Tune-shift for Gaussian Beam Form Factor F For a KV beam, F = 1. For any other distribution, F is the ratio between the peak charge density of that distribution, to the charge density of a KV beam. Bunching Factor B 14 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 14 14 14 14

Space-charge Tune-spread with Synchrotron Oscillation G. Franchetti et al. PRSTAB 2017 15 Classical Mechanics and Electromagnetism | June 2018 USPAS at MSU 11/18/2018 15 15 15 15