1. Synchrotron Radiation Lecture 3 Undulator Magnet Designs
Jim Clarke
ASTeC
Daresbury Laboratory

2. Generating Periodic Magnetic Fields
So far we have discussed at length what the properties of SR are, when it is generated, and how it can be tailored to suit our needs (wavelength, polarisation, flux, etc)
But, how do we know what magnetic fields are actually achievable?
In this part we will look at how periodic fields are generated and what the limitations are
Later we will look at the present state of the art and some future possibilities

3. What are the possibilities?
To generate magnetic field we can use:
Electromagnets
Normal conducting or
Superconducting
Permanent Magnets
Both types can also include iron if required

4. What is a Permanent Magnet ?
Definition:
A magnet is said to be Permanent (or Hard) if it will independently support a useful flux in the air gap of a device
A material is magnetically Soft if it can only support such a flux with the help of an external circuit (eg iron is soft)
A PM can be considered as a passive device analogous to a spring (which stores mechanical energy)
An electron in a microscopic orbit has a magnetic dipole moment – can be modelled as a current flowing in a loop
In a Permanent Magnet these ‘molecular’ currents can be identified with atoms with unfilled inner shells
eg 3d metals (Fe, Co, Ni) or 4f rare earths (Ce to Yb)

5. Permanent Magnets
Permanent Magnet materials are manufactured so that their magnetic properties are enhanced along a preferred axis
To do this, advantage is taken of crystal lattices
The direction of alignment is called the easy axis
When a magnetic field, B, is applied to a magnetic material each dipole moment tries to align itself with the field direction
When B is strong enough (at saturation) all of the moments are aligned, overcoming other atomic forces which resist this
A Permanent Magnet must be able to maintain this alignment after B is removed

6. An Ideal Permanent Magnet
The characteristics of a Permanent Magnet are determined by its behaviour under an external magnetization force H
Magnetization
2. H reducing, moments stay aligned
1. H large, material saturated, all moments aligned
Intrinsic coercivity
Magnetizing Force
3. H large and negative, material flips – now aligned with opposite direction
Magnetization is magnetic dipole moment per unit volume, B = m0M

7. The Ideal BH Curve Magnetic Flux Density B = m0(H+M) Gradient = m0
Remanent Field
Coercivity
In 2nd quadrant the ideal PM is linear
Magnetizing Force

8. The BH Product
Permanent Magnets are operated in the 2nd quadrant – no external fields are present, moments are aligned along the easy axis
The product BH represents the energy density of the material
Examining the peak BH value in the 2nd quadrant is a good way of comparing the strength of different materials

9. Current Sheet Equivalent Materials (CSEM)
An ideal Permanent Magnet is uniformly magnetized (homogeneous)
The equivalent current model is a sheet of current flowing on the surface with no internal (volume) currents
The relative permeability is ~1 so we can consider the bulk material to be vacuum
This CSEM model implies that the contributions from different magnets can be added linearly (just like adding contributions from currents)
Analytical calculations then become fairly simple because we can calculate the field at a point from every block independently and just add all the individual contributions up

10. Current Sheet Equivalent Materials (CSEM)
Lines of flux for an ideal Permanent Magnet
Lines of flux for a CSEM model

11. At higher temperatures the materials become more non-linear
Temperature Effects
At higher temperatures the materials become more non-linear
So long as the working point stays in the linear region this is a reversible effect
Note that the remanent field drops with increasing temperature – the reverse is also true, cold magnets have a higher remanent field
Day to day temperature variations are important and must be controlled (minimised) Many undulators are operated in air conditioned environments to keep their output more stable

12. Available Materials
Two types of permanent magnet are generally used – Samarium Cobalt (SmCo) and Neodymium Iron Boron (NdFeB)
SmCo
NdFeB
Remanent Field
0.85 to 1.05 T
1.1 to 1.4 T
Coercivity
600 to 800 kA/m
750 to 1000 kA/m
Relative Permeability
1.01 parallel, 1.04 perp
1.05, 1.15
Temperature Coefficient
-0.04 %/C
-0.11 %/C
Max Energy Density
150 to 200 kJ/m3
200 to 350 kJ/m3
Max operating temperature
~300C
~100C
Comment
Brittle, easily damaged, better intrinsic radiation resistance, expensive
Less brittle but still liable to chip, easier to machine, expensive

13. M-H and B-H curves are shown (2nd quadrant)
An Example Material
Vacodym 633 HR
M-H and B-H curves are shown (2nd quadrant)
The material is linear at 20C but non-linear above about 60 C
(courtesy of Vacuumschmelze)

14. Pure Permanent Magnet Undulators
A magnet which contains no iron or current carrying coils is said to be a Pure Permanent Magnet (PPM)
Because of CSEM we can use the principle of superposition
An ideal undulator would have a sinusoidal magnetic field along the direction of the electron beam
To generate a sinusoidal field an ideal PPM would have two sets (arrays) of Permanent Magnet with their easy axis rotating smoothly through 360 per period along the direction of the electron beam
In practice this ideal situation is approximated by splitting the system into M rectangular magnet blocks per period with the easy axis at the relevant set angle

15. Example PPM arrangement, M = 4
Side View
Top Array e-
Bottom Array

16. Lines of Magnetic Flux e-

17. Magnetic Field
The field strength between the two arrays assuming infinite width in the x – direction (2D approximation) is
Where and is a packing factor to allow for small air gaps between blocks
The vertical field on axis (y = 0) is a number of cosine harmonics
As this reduces to a single cosine (ideal case)
The longitudinal (and horizontal) field on axis is zero

18. Note that the fields increase away from the axis
A Practical PPM
The most popular choice is M = 4
This is a good compromise between on axis field strength and quality vs engineering complexity
Higher harmonics then account for < 1% of the field on axis
Away from the axis it is definitely not cosine-like
For an example PPM with 50mm period, block height of 25mm, magnet gap of 20 mm and remanent field of 1.1 T
Note that the fields increase away from the axis

19. Peak Vertical Field vs M
Selecting M = 4 means that you will achieve about 90% of the theoretical limit

20. Simplifying the Magnetic Field
If we assume that only the first harmonic makes a significant contribution (n = 1) – a good approximation in general
Then the equation simplifies greatly on axis to
Important:
Note that so long as all the spatial dimensions scale together the fields on axis do not change
This is not true for electromagnets – there the current densities have to increase to maintain the same field levels

21. Effect of Different Block Heights
A typical block height selection is half the period length
Selecting 0.5 means that you will achieve about 95% of the theoretical limit

22. Peak Field Achievable
The maximum peak field achievable (in theory) is 2Br
In practice with M = 4 and h = lu/2 the peak on axis field is
So even with an ambitious gap to period ratio of 0.1 the peak value is only 1.26Br
Achieving fields of ~1.5T requires very high Br material, small gaps and long periods!
But, higher fields are possible if we include iron in the system
Mixing Permanent Magnets and iron poles is called a hybrid magnet

23. Tuning the Undulator
To vary the output wavelength from the undulator – to map out the tuning curves – we need to alter the field level on the axis
We can now see that the only practical way to do this for a permanent magnet device is to change the magnet gap, g

24. Hybrid Insertion Devices – Inclusion of Iron
Simple hybrid example
Top Array e-
Bottom Array

25. e- Lines of Magnetic Flux
Including a non-linear material like iron means that simple analytical formulae can no longer be derived – linear superposition no longer works!
Accurate predictions for particular designs can only be made using special magnetostatic software in either 2D (fast) or 3D (slow) e-

26. Field Levels for Hybrid and PPM Insertion Devices
Assuming Br = 1.1T and gap of 20 mm
When g/lu is small the impact of the iron is very significant

27. Helical (or Elliptical) Undulators for Variable Polarisation
We need to include a finite horizontal field of the same period so the electron takes an elliptical path when it is viewed head on
We want two orthogonal fields of equal period but of different amplitude and phase
Three independent variables are required for the arbitrary selection of any polarisation state
3 independent variables

28. Helical Undulators
Pure helical fields are suited to a circular magnet geometry (magnets surrounding the vacuum chamber) but these are not generally practical for a light source
A planar geometry is better suited to light sources (all of the magnets in the plane above and below the axis)
This allows the machine to have a narrow vertical gap and a wide horizontal gap – as required for injection or magnet measurements
But, it is not so easy to generate H and V fields
It is not so easy to understand the fields either!
Two degrees of freedom are needed to control the H & V fields independently, ideally three so you can control the phase as well
Hence two (or three) independent motion systems are needed

29. This undulator consists of four standard PPM arrays
The APPLE-2 Design
This undulator consists of four standard PPM arrays
Diagonally opposite arrays move longitudinally together
All the arrays also move vertically like a conventional undulator
The electron beam travels along the central axis of the magnet
Gap is altered also to tune magnet as usual
These two arrays move together
Phase shift, D

30. APPLE-2 Example Trajectories
Trajectories viewed head on – as the observer sees them
3 GeV electron beam
Undulator period is 50 mm
Magnet gap is 20 mm
Fields in circular mode

31. APPLE-2 Examples
SRS HU56 being measured
Diamond HU64

32. Engineering Issues for all PM Undulators & Wigglers
Engineering demands are very high:
Very strong forces present during assembly and when complete
Must have high periodicity
Arrays must be parallel to mm precision and must stay parallel at all gaps
General design themes:
Blocks are held in individual holders – glued or clamped
Fastened to a backing beam
C shaped support frame
Very long magnets (>5m) are usually split into shorter modules (2 – 3m)

33. In-Vacuum Undulators
The minimum magnet gap limits the performance of an undulator
The magnet gap is determined by the needs of the electron beam
In practice this is set by the vacuum chamber
For example:
If an electron beam needs 10mm of vertical space
And the vacuum chamber walls are 2mm thick
With an allowance for mechanical tolerances of 1mm
The minimum magnet gap will be 15mm
So 5 mm is effectively wasted
One option is to put the undulator inside the vacuum chamber, in this example the magnet gap reduces by ~30%

34. In-vacuum and out of vacuum concepts
In-Vacuum Undulators
Magnet blocks are not very good for use within a vacuum system
They must be coated to prevent outgassing (TiN or Ni)
They also should be baked for UHV – this affects the magnet performance and can cause irreversible losses
Generally only bake at ~130 C
The surface resistance of the blocks is high – need a sheet of copper on top of each array to provide low resistance path for the electron’s image current
Magnet measurements are very difficult after full assembly
In-vacuum and out of vacuum concepts

35. In-vacuum undulator – Before vacuum chamber is fitted
In-Vacuum Examples
Diamond U23
In-vacuum undulator – Before vacuum chamber is fitted

36. ALS in-vacuum undulator
In-Vacuum Examples
ALS in-vacuum undulator

37. Cryogenic Undulators
These are a relatively new idea that take advantage of the variation of remanent field with temperature
If the undulator can operate at ~150K then there will be a significant field increase with NdFeB – a slightly awkward temperature to maintain
The intrinsic coercivity increases also which helps with radiation resistance and allows selection of stronger grades
Basically an in-vacuum device with cryogenic cooling attached
H Kitamura, Spring-8

38. Unbaked devices have been used successfully on ESRF and DLS
Cryogenic Undulators
Other materials are more attractive, especially PrFeB, as it can operate at 77K – an easy temperature to maintain
One issue is also wanting to bake the magnets to ~450K to improve the vacuum performance – limits the grades available
Unbaked devices have been used successfully on ESRF and DLS
Kitegi et al, IPAC 2012

39. Electromagnetic Devices
Given that virtually all magnets in particle accelerators are electromagnets (dipoles, quadrupoles, sextupoles, …) it seems surprising that relatively few electromagnetic undulators and wigglers are built
One niche area where electromagnets are used is when there is a requirement for the fields to be changed quickly
This is generally motivated by the need to rapidly change polarisation states – it is much easier to switch the direction of a current quickly than to physically move magnet arrays
Superconductors, of course, have always been used in very high field applications

40. Electromagnetic Devices
Let’s consider a basic conceptual layout of an electromagnetic insertion device
The magnetic field is varied by changing the current in the coils
There is no need to move the arrays
Electromagnetic insertion devices generally have a much lower capital cost but a much higher operating cost
All the coils are connected in series
One individual coil is highlighted in red

41. Simple Electromagnetic Analysis
Consider the design concept as a series of dipoles of alternating polarity
The (approximate) field produced by a dipole with gap, g, driven by NI Ampere-turns is
So the undulator K parameter will be
Note that to reach K ~ 1 we will require NI ~1000 A-t
If the gap is fixed and we want to reduce lu then NI will have to increase to maintain K
But, as the period reduces the space for the coil shrinks as well so the current density increases very rapidly
At some period the resistive losses will be so high that cooling will not be practical

42. Electromagnet Examples
An elliptical wiggler is installed on Elettra
The period is 212 mm and the peak fields are 0.5 T vertically and 0.1 T horizontally
The horizontal field can switch at up to 100Hz
The rapid change in polarisation between left and right circular can be utilised by the experiment to increase the signal to noise level

43. Superconducting Magnets
For fields greater than ~3T superconductors are the only real option
For intermediate fields (~1 to ~3 T) they can have much shorter periods than Permanent Magnets or Normal Conducting Electromagnets
The materials used are only superconducting below ~10K
Hence the magnets always sit inside a cryostat
In the past they would generally have a closed loop liquid Helium refrigerator permanently connected to them
The development of closed cycle cryocoolers has enabled a more ‘stand alone’ approach to be used in some applications – liquid helium is not necessarily required and not always used

44. Superconducting Wavelength Shifter Examples
The highest field has been achieved by the SPring-8 10T wavelength shifter (from BINP)
It uses NbTi and Nb3Sn
The stored energy is 400kJ
The magnet gap is 42 mm

45. Superconducting Multipole Wigglers
Superconducting MPWs are popular in the intermediate energy light sources (~3GeV) because the high field enhances the flux in the hard X-ray region
A key advantage over permanent magnet hybrid wigglers is not just the higher field but also the reduction in period
This lowers the K value and so the radiation is emitted over a narrower horizontal width (±K/g) – making it easier to manage the cooling arrangement within the facility
Note that the majority of the radiation generated by a MPW will not be used by the beamline and so must be absorbed by cooled surfaces

46. Superconducting Multipole Wiggler Examples
Diamond
3.5 T
60 mm period, 45 poles
4.2 T
48 mm period, 45 poles

47. Superconducting Undulators
The motivation of using superconductivity is to generate higher fields on axis than are presently available from the best permanent magnet systems
They have to have a significantly better performance to make them worthwhile
The key region of interest is in short period systems, typically ~15mm
The field quality has to be similar to existing undulators

48. The Advantage of SC Undulators
ASTeC SCU:
period = 15 mm
N = 133 (2m long)
Magnet pole gap = 7.4 mm
Bo = 1.28 T
K = 1.8
SCU/U21
Flux
Brightness
25 keV
6.2
7.6
40 keV
15.4
21.5
R Walker, Diamond

49. The challenge is nearly all engineering
Undulator Design
The standard solution is very simple – currents flowing perpendicular to the beam axis
The challenge is nearly all engineering
Iron yoke
Return paths for currents

50. Superconducting Undulators
A similar scheme has been adopted by several groups
S H Kim, APS

51. Challenges
To achieve the field levels requires the magnets to have a cold beam aperture – no room for an insulating vacuum – any heat transfer from the beam to the wires could trigger a quench
The iron poles need to be accurately machined to minimise field errors
The on axis field is also dependent upon the accurate positioning of the wires
Correction of the field errors is quite tricky – several options have been proposed but they all add an extra level of complication – not as simple as permanent magnets
Ideally the magnet should not need error correction but is this practical? What about transition between iron non-saturation and saturation?

52. Summary
A PM can independently support a flux in an air gap – no coils are needed
To generate a sinusoidal field we use two arrays of PM blocks – one above and one below the electron beam
The field limit for a PPM is ~1.5T but if we include iron (hybrid magnet) then >2 T is easily achievable
The APPLE-2 is the most popular undulator design for generating variable polarisation states
In-vacuum solutions allow for smaller magnet gaps but more complex engineering
Cryogenic undulators at ~150K to 77K generate higher field levels and these are starting to become more popular now
Normal conducting electromagnets cannot compete with permanent magnets, especially at short periods but they do allow fast switching of fields
Wavelength shifters (~5 to 10T) are used to generate the highest energy photons
There are several MPW examples (3.5 to 7T), used to enhance the flux at high photon energies
Superconducting undulators have to be pushed close to their limits to gain a significant advantage over permanent magnet alternatives – on paper SC technology always wins but not true in practice yet!