Transverse Magnetic Field Measurements in the CLARA Gun Solenoid

Transcription

1 Transverse Magnetic Field Measurements in the CLARA Gun Solenoid Duncan Scott, Boris Militsyn Ben Shepherd, Alex Bainbridge, Kiril Marinov Chris Edmonds, Andy Wolski STFC Daresbury & Liverpool University and The Cockcroft Institute

2 Transverse Magnetic Field Measurements in the CLARA Gun Solenoid Duncan Scott, Boris Militsyn Ben Shepherd, Alex Bainbridge, Kiril Marinov Chris Edmonds, Andy Wolski STFC Daresbury & Liverpool University and The Cockcroft Institute

3 Transverse Magnetic Field Measurements in the CLARA Gun Solenoid Duncan Scott, Boris Militsyn Ben Shepherd, Alex Bainbridge, Kiril Marinov Chris Edmonds, Andy Wolski STFC Daresbury & Liverpool University and The Cockcroft Institute

4 Transverse Magnetic Field Measurements in the CLARA Gun Solenoid Duncan Scott, Boris Militsyn Ben Shepherd, Alex Bainbridge, Kiril Marinov Chris Edmonds, Andy Wolski STFC Daresbury & Liverpool University and The Cockcroft Institute

5 Contents Clara Gun Solenoids Field Measurements Beam emittance / distribution effects Future Work

6 Solenoids Design 2 coaxial, water-cooled solenoids Fields cancel at cathode The bucking solenoid is stationary The main solenoid can be moved towards and away from the cathode plane.

7 Magnetic Characterisation Main, axial fields perform as expected What about the transverse fields? (Solenoid Transverse Fields Probably not studied very much?)

8 Magnetic Characterisation Main, axial fields perform as expected What about the transverse fields? (Solenoid Transverse Fields Probably not studied very much?)

9 Why do this? Effect Of Transverse Fields? Want a round, symmetric beam distribution and emittance Perturbations Effect: coupling, Eigen-mode emittances Asymmetric beam distributions Halo (?) + other effects Theoretical interest

10 3-Axis Hall Probe Measurements Naive Approach: simply measure with 3-axis hall probe Keep track of the co-ordinate systems Hall probe (x,y,z) Lab frame (x,y,z) Electron beam direction Sample field with Hall probe on (x,y,z) mover Lab Frame +Y (vertically up) Hall Probe (x,y,z) Magnet Measurement Bench solenoid +X (toward measurement bench) +Z (along solenoid) Electron beam direction

11 3-Axis Hall Probe Measurements We are trying to measure very small fields relative to the Axial field Axial Field ~400 mt Transverse Field ~1 mt This leads to two issues: 1. Must take into account Planar Hall Effects 2. We must also cope with misalignments of the Hall Probe relative to the solenoid

12 3-Axis Hall Probe Measurements We are trying to measure very small fields relative to the Axial field Axial Field ~400 mt Transverse Field ~1 mt This leads to two issues: 1. Must take into account Planar Hall Effects 2. We must also cope with misalignments of the Hall Probe relative to the solenoid

13 Transverse Hall Effect Planar Hall Effect?

14 Planar Hall Effect In the presence of a strong longitudinal component the measured voltage for the perpendicular flux density component is distorted Solenoid Transverse Field (small) Solenoid Axial Field (Large)

15 Planar/Transverse Hall Effect Errors Take measurements at 4 different Hall-probe orientations Keeping track of the coordinate systems Take average of these measurements ch1 and ch0 are the Nominal, transverse, Hall-Probe measurement axes

16 Planar/Transverse Hall Effect Errors Take measurements at 4 different Hall-probe orientations Keeping track of the coordinate systems Take average of these measurements ch1 and ch0 are the Nominal, transverse, Hall-Probe measurement axes However, we have now introduced more misalignment errors

17 Misalignments Each measurement is not exactly a 90 degree rotation The Hall probe is not exactly aligned with the Lab Frame How to combine measurements? 0 Degrees 90 Degrees 180 Degrees Lab Frame +Y (vertically up) 270 Degrees +X (toward measurement bench) +Z (along solenoid) Example misalignments between measurements

18 Misalignments Each measurement is not exactly a 90 degree rotation The Hall probe is not exactly aligned with the Lab Frame How to combine measurements? Each Measurement Set can be rotated to an objectively defined system defined by The Solenoid Axial Field Lab Frame +Y (vertically up) 0 Degrees 90 Degrees 180 Degrees 270 Degrees +X (toward measurement bench) +Z (along solenoid) Solenoid Axial Field (Large) Assumed to be constant

19 Misalignments We find the optimal rotation matrix R a, θ for each Measurement Set m n=1 0,0,1 R a, θ B n B n is the measurement at each point, m measurements per set Maximise This a is the rotation axis θ rotation angle R a, θ = a x 2 + cos θ a y 2 + a z 2 a x a y (1 cos θ) a z sin θ a y sin θ + a x a z (1 cos θ) a x a y 1 cos θ + a z sin θ cos θ + a y 2 1 cos θ a y a z 1 cos θ a x sin θ a x a z 1 cos θ a y sin θ sin θ a x + a y a z 1 cos θ cos θ + a z 2 (1 cos θ)

20 Misalignments We find the optimal rotation matrix R a, θ for each Measurement Set m n=1 0,0,1 R a, θ B n B n is the measurement at each point, m measurements per set Maximise This a is the rotation axis θ rotation angle R a, θ = a x 2 + cos θ a y 2 + a z 2 a x a y (1 cos θ) a z sin θ a y sin θ + a x a z (1 cos θ) a x a y 1 cos θ + a z sin θ cos θ + a y 2 1 cos θ a y a z 1 cos θ a x sin θ a x a z 1 cos θ a y sin θ sin θ a x + a y a z 1 cos θ cos θ + a z 2 (1 cos θ) The direction of maximal Bz becomes the nominal Z axis and is assumed to be the same for each measurement set In practise 2 rotations are required: 1. Maximise the global longitudinal field component. 2. Rotate around B z,max to align the transverse components with the nominal X and Y directions.

21 Results (finally!) Transverse fields at 3 different longitudinal positions in the solenoid z = 0 is the nominal centre Plots show expected general behaviour with non-uniform radial component

22 Transverse Multipoles The fields can be decomposed into multipoles It is better to do this in Polar coordinates B θ + ib r = C n r n 1 e inθ n=1 n = 1 is a dipole n = 2 is a quadruple etc. C n are the magnitude of the normal and skew components

23 Transverse Multipoles The fields can be decomposed into multipoles It is better to do this in Polar coordinates For discrete measurements the C n are found with a Discrete Fourier Transform in the polar basis B θ + ib r = C n r n 1 e inθ n=1 n = 1 is a dipole n = 2 is a quadruple etc. C n are the magnitude of the normal and skew components

24 Transverse Multipoles The fields can be decomposed into multipoles It is better to do this in Polar coordinates For discrete measurements the C n are found with a Discrete Fourier Transform in the polar basis Make M measurements at constant radius r, varying the polar angle from 0 to 2π p, where p = M 0, 1, 2, 3 M 1 Interpolate transverse field maps with a spline: continuous derivatives and an approximation to the Maxwell Equations The field at each θ p can be represented by the complex number B p = B θ + i B r B θ + ib r = C n r n 1 e inθ n=1 n = 1 is a dipole n = 2 is a quadruple etc. C n are the magnitude of the normal and skew components C n = 1 M 1 Mr n 1 p=0 B pe 2πin p M

25 Transverse Multipoles The fields can be decomposed into multipoles It is better to do this in Polar coordinates For discrete measurements the C n are found with a Discrete Fourier Transform in the polar basis Make M measurements at constant radius r, varying the polar angle from 0 to 2π p, where p = M 0, 1, 2, 3 M 1 Interpolate transverse field maps with a spline: continuous derivatives and an approximation to the Maxwell Equations The field at each θ p can be represented by the complex number B p = B θ + i B r B θ + ib r = C n r n 1 e inθ n=1 n = 1 is a dipole n = 2 is a quadruple etc. C n are the magnitude of the normal and skew components C n = 1 Mr n 1 M 1 p=0 B pe 2πin p M The accuracy of C n, scales with the accuracy of B p, B p and inversely with r Therefore choose the greatest possible r C n ~ B p r n 1

26 Quadrupole Components Graph Shows convergence as M is increased Unit Value Z position Mm -10, 0, 10 M - 40, 40, 40 Re C 2 mt/m -2.3, -1.7, -1.2 Im C 2 mt/m 4.4, 1.6, -1.6 Not symmetric or similar at different z

27 Effects On Beam Small asymmetry in emittance calculated at 1 st screen Probably not practical to measure, misalignments, laser source asymmetry, etc dominate

28 Effects On Beam Small asymmetry in emittance calculated at 1 st screen Probably not practical to measure, misalignments, laser source asymmetry, etc dominate Also, this may not be an accurate representation of coupling due to skew quadrupole terms in solenoid. Experimentally measured skew quadrupole term varies from -1.6 mt/m to 4.4 mt/m over 20 mm measurement range. This is being investigated more

29 Future Plans Now we have experimental procedure Take more measurements to see how the transverse components vary along the length of the entire solenoid. Magnetic simulations to find source of non-uniform radial components twists in coils, deformation of solenoid Simulate motion of particles with more realistic longitudinal variation of skew quadrupole components. Specification of invariant quantity that can be related to key performance indicators (e.g. power output of a FEL) The challenge in terms of presenting the results is an invariant measure of the emittance. Small errors on coupling terms within the covariance matrix lead to large asymmetries in the calculated Eigen-mode emittances. Some ideas are being investigated

30 Future Plans Now we have experimental procedure Take more measurements to see how the transverse components vary along the length of the entire solenoid. Magnetic simulations to find source of non-uniform radial components twists in coils, deformation of solenoid Simulate motion of particles with more realistic longitudinal variation of skew quadrupole components. Specification of invariant quantity that can be related to key performance indicators (e.g. power output of a FEL) The challenge in terms of presenting the results is finding an invariant measure of the emittance. Small errors on coupling terms within the covariance matrix lead to large asymmetries in the calculated Eigen-mode emittances. Some ideas are being investigated

31 Thanks! End