1. Space-charge Effects for a KV-Beam Jeffrey Eldred1 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

2. Space-charge Effects From KV-Distribution 2 2 2 2 2 2 2 2 2
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3. 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
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4. 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
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5. 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
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6. 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
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7. Equations of motion for KV Beam
Force from KV beam:
Transverse focusing:
Linear!
Space-charge perveance
Classical particle radius 7
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8. Beam Envelope Equation
Transverse focusing:
Floquet Transformation:
Beam Envelope Equation: 8
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9. Beam Envelope Equation
Beam Envelope Equation for a=b:
Space-charge parameter 9
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10. Betatron Oscillation vs. Envelope Oscillation
S. Lund 10
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11. 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
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12. Other Space-charge Effects 12 12 12 12 12 12 12 12 12
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13. Core-Halo Model T. Wangler et al. PRSTAB 1998 13 13 13 13 13
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14. 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
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15. Space-charge Tune-spread with Synchrotron Oscillation
G. Franchetti et al. PRSTAB 2017 15
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