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Thermal rigid-rotor equilibrium in a uniform magnetic field

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Thermal rigid-rotor equilibrium in a uniform magnetic field

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  1. Adiabatic Thermal Equilibrium of an Axisymmetric Charged-Particle BeamC. Chen, K. Samokhvalova, and J. ZhouPlasma Science and Fusion CenterMassachusetts Institute of TechnologySymposium on Recent Advances in Plasma Physics --- In Celebration of Ronald C. Davidson's 40 Years of Plasma Physics Research and Graduate EducationJune 12, 2007

  2. Thermal rigid-rotor equilibrium in a uniform magnetic field r nb(r) z Davidson and Krall, 1971 Trivelpiece, 1976

  3. Applications of high-brightnesscharged-particle beams • Large Hadron Collider (LHC) • Spallation Neutron Source (SNS) • High Energy Density Physics (HEDP) • International Linear Collider (ILC) • Photoinjectors • High Power Microwave Sources

  4. Periodic focusing channels S/2 N S/2 S S N S N N S Periodic Quadrupole Field Periodic Solenoidal Field

  5. Kapchinskij-Vladimirskij (K-V) equilibrium in an alternating-gradient (AG) magnetic quadrupole focusing channel I.M. Kapchinskij, and V. V. Vladimirskij, Proc. Int. Conf. High Energy Accel. (CERN, Geneva, 1959), p. 274 Delta function distribution in transverse ‘energy’ Generalized KV distribution in an axially varying, linear focusing channel F.J. Sacherer (Ph.D thesis, UC Berkeley, 1968) Rigidly rotating equilibrium in a periodic solenoidal magnetic focusing field C. Chen, R. Pakter and R. C. Davidson, Phys. Rev. Lett. 79, 225 (1997) KV-like distribution Issues Non-physical Beam halos Chaotic-particle motion Periodically focused beam equilibria z 2 2 x y = + T 2 2 a b

  6. Chaotic phase space in a KV beam Qian, Davidson and Chen (1994) Pakter, Chen and Davidson (1999) Zhou, Chen, Qian (2003)

  7. Thermal equilibrium in a periodic solenoidal magnetic field r nb(r) z • Experiment: Recent experiment at UMER demonstrated that the beam focused by a solenoid has a bell-shaped profile • S. Bernal, B. Quinn, M. Reiser, and P.G. O’Shea, PRST-AB 5, 064202 (2002). New: Thermal equilibrium in a periodic solenoidal magnetic field

  8. Kinetic and warm-fluid theories = - = P x P y P const ( ) ( ) = q 1 1 1 1 y x T z r 2 z const ^ brms ( ) 1 E = + + + + f @ 2 2 2 2 self x y P P const ^ 1 1 x 1 y 1 1 2 é ù K ( ) ( ) ˆ self 2 self 2 2 f = f - w z w z r ( ) ê ú ¢ ( ) ^ ^ 1 1 r z 2 2 r z ( ) ë û = + W brms ˆ ˆ V r V e r z e brms ( ) ^ q z r b r z ( ) 2 d w z K 1 brms ( ) ( ) ( ) + k - = z w z w z ( ) ( ) z 2 2 3 dz 2 r z w z brms ( ) [ ] ( ) = - b - w f x , y , P , P , z C exp E P ( ) ì ü e q é ù 2 f 4 b 1 1 x 1 y 1 2 self C r K q r , z ( ) = - + - th , rms n r , z exp í ý ê ú ( ) ( ) ( ) e g b r 2 z 4 2 2 r 2 z 2 k T z b w ë û C , , î þ ^ brms th , rms brms b B Kinetic theory (Zhou, et al., 2007) Warm-Fluid Theory (Samokhvalova, et al., 2007) Equation of state (adiabatic) Constants of motion Angular Momentum Generalized Energy Transverse velocity Beam density Distribution function - constants

  9. Self-Consistent Density Distribution w 2 1 r ( ) ( ) W = - W + z z b b 0 ( ) b cb 2 r 2 z brms ( ) e + w 4 2 2 r 4 V 2 d 2 r z K ( ) ( ) + k - = th , rms b b 0 b z r z brms ( ) ( ) z brms dz 2 2 r z r 3 z brms brms ( ) ( ) 2 k T z r z 2 1 2 ( ) 2 2 N q é ù W 2 = z r r e = = ( ) º 2 const ^ b B brms K k º cb z brms ê ú g th , rms g z 3 2 2 m V 2 b mV 2 c ë û b b b b b Density Poisson’s equation 2 self Ñ f = - p q n 4 b Beam rotation Envelope equation rms beam radius focusing parameter perveance thermal rms emittance

  10. Beam envelope and beam density warm beam cold beam

  11. UMER edge imaging experiment* • 5 keV electron beam focused by a short solenoid. • Bell-shaped beam density profiles • Not KV-like distributions *S. Bernal, B. Quinn, M. Reiser, and P.G. O’Shea, PRST-AB, 5, 064202 (2002)

  12. Density profile comparison for 5 keV, 6.5 mA electron beam Experiment z=6.4cm z=11.2cm z=17.2cm

  13. Conclusion • Established kinetic and warm-fluid equilibrium theories for charged-particle beams in periodic solenoidal focusing channels, extending Davidson’s seminal work on the rigid-rotor thermal equilibrium. • Adiabatic. • Applicable for both high and low intensities. • Found excellent agreement between our theory and the UMER experiment. • Future plans: • Study chaotic particle motion and halo formation in thermal-equilibrium beams in periodic solenoidal focusing channels. • Establish thermal equilibrium theory of charged-particle beams in periodic quadrupole focusing channels. • Develop bunched beam equilibrium theory in rf accelerators. • Pursue high-brightness beam applications.

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