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