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Focusing Properties of a Solenoid Magnet

Focusing Properties of a Solenoid Magnet. Simon Jolly UKNFIC Meeting, 12/05/05. Cylindrical Polar Coordinates. Dimensions given in ( r,  ,z ) rather than ( x,y,z). Therefore vector in Cartesian coordinates given by:. (1). C.P. Unit Vectors. In general, unit vectors given by :. So :.

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Focusing Properties of a Solenoid Magnet

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  1. Focusing Properties of a Solenoid Magnet Simon Jolly UKNFIC Meeting, 12/05/05

  2. Cylindrical Polar Coordinates Dimensions given in (r,,z) rather than (x,y,z). Therefore vector in Cartesian coordinates given by: (1) Simon Jolly UKNFIC Meeting, 12/05/05

  3. C.P. Unit Vectors In general, unit vectors given by: So: (2) Simon Jolly UKNFIC Meeting, 12/05/05

  4. C.P. Velocity (3) Simon Jolly UKNFIC Meeting, 12/05/05

  5. C.P. Acceleration (4) Simon Jolly UKNFIC Meeting, 12/05/05

  6. Particle Motion in B-field Particle acceleration, a, in B-field, B, given by: (charge q, mass m, velocity v) In cylindrical polar coordinates: (5) (6) Simon Jolly UKNFIC Meeting, 12/05/05

  7. Solenoid B-field Solenoid field is axially symmetric (no -dependence), so: Define on-axis field: (7) Components of Solenoid field: (8) Simon Jolly UKNFIC Meeting, 12/05/05

  8. Equations of Motion in Solenoid Combine eqns. 6 & 8 and split particle motion into r,  and z components: (9) (focusing) (10) (rotation) (11) (acceleration) Simon Jolly UKNFIC Meeting, 12/05/05

  9. Equations of Motion (2) Since: (12) combining eqns. 8, 10 & 12 gives: (13) Simon Jolly UKNFIC Meeting, 12/05/05

  10. Equations of Motion (3) Now, and So eqn. 13 becomes: (14) Simon Jolly UKNFIC Meeting, 12/05/05

  11. Equations of Motion (4) Integrating eqn. 14 with respect to time: where c is a constant of integration (15) For an on-axis beam, c=0, so eqn. 15 becomes: (16) Simon Jolly UKNFIC Meeting, 12/05/05

  12. Equations of Motion (5) Integrating eqn. 16 with respect to time: (17) Since , (18) Simon Jolly UKNFIC Meeting, 12/05/05

  13. On-Axis Beam Rotation The longitudinal kinetic energy (19) Therefore eqn. 18 becomes: (20) This means that the outgoing beam is rotated with respect to the incoming beam, and this rotation is proportional to the integrated field, Bzdz, and the particle kinetic energy T. Simon Jolly UKNFIC Meeting, 12/05/05

  14. Transverse Beam Motion Now insert eqn. 16 into 9: (9) (21) Simon Jolly UKNFIC Meeting, 12/05/05

  15. Transverse Beam Motion (2) What we actually want is (focusing per unit length): (22) (23) Simon Jolly UKNFIC Meeting, 12/05/05

  16. Solenoid Focusing Strength Substituting eqn. 19 for , and therefore setting , modifies eqn. 23 accordingly, giving the radial ray equation: (24) As such, aside from the instrinsic particle properties of charge, kinetic energy and mass (which the solenoid does not modify), the focusing strength of the solenoid lens is purely a function of the longitudinal B-field, Bz, and the radius r. Simon Jolly UKNFIC Meeting, 12/05/05

  17. Solenoid Focal Length The focal length, f, of the solenoid (using the thin lens approximation) is given by: (25) Since the focal length is proportional to 1/q2, the solenoid lens is only useful at low particle momenta. Simon Jolly UKNFIC Meeting, 12/05/05

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