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

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

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

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)

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

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

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

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

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

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

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

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

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)

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

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

Lines of Magnetic Flux e-

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

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

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

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

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

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

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

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

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-

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

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

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

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

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

APPLE-2 Examples SRS HU56 being measured Diamond HU64

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)

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%

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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?

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!