Simulations of Magnetic Fields and Forces in Highly Adjustable Magnet (HAM) Undulator Concept Using COMSOL

Department of

Physics and Astronomy Uppsala University P.O. Box 516

FREIA Report 2019/01 26 February 2019

Simulations of Magnetic Fields and Forces in Highly Adjustable Magnet

(HAM) Undulator Concept Using COMSOL.

DEPARTMENT OF PHYSICS AND ASTRONOMY UPPSALA UNIVERSITY

Simon Fahlström, Mathias Hamberg

Dept. Physics and Astronomy

Uppsala University, Uppsala, Sweden

Abstract

A design for a new type of undulator insertion device has been proposed. The undulator would consist of a stack of disks that each contain a pair of magnet structures, each disk being a half period of the minimum planar case. The disks could rotate independently about the beam axis, and the distance of the magnets from the center line is adjustable, so the magnetic field is adjustable in magnitude and orientation within the transverse plane along the beam axis. This would allow the undulator to be configured for generating a wide variety of radiation. Plane polarization would be attained with undulator periods as integer multiples of the base period. In the base case the direction of magnetization alternates between each disk, and for the case of multiple periods the magnetization alternates each M:th disk. Helical polarization would be attained with a period greater than two times the base period such that an integer number of periods fit inside the undulator structure. Initial numerical simulations have been made, and are reported in this paper. Building on a previous study [1] further simulations were requested to study the behavior of the magnets closer, and to calculate the forces acting on the magnet structures for use in feasibility assessment. In the previous study simple magnetic structures with uniform direction of magnetization were simulated. In this study a more complex structure was also investigated: a type of partial transverse Halbach configuration. This structure would concentrate the magnetic flux along the beam line and could lead to a more compact design, and limit the magnetic field outside the device.

The simulations were made using COMSOL Multiphysics modelling software.

Contents

1 Introduction 1

1.1 HAM undulator concept . . . . 2

2 Theory 2 2.1 Synchrotron radiation . . . . 3

2.2 Insertion devices and the Free Electron Laser . . . . 4

2.3 Magnetostatics . . . . 5

2.3.1 Force calculation using Maxwell’s Stress Tensor . . . . 6

2.3.2 Halbach arrays . . . . 6

3 Modelling and simulations 7 3.1 Computational model . . . . 8

3.2 Magnet pair . . . . 8

3.2.1 Varying the height of the magnets . . . . 9

3.3 Array of disks . . . 11

3.3.1 Magnetic fields in plane polarized configuration . . . 11

3.3.2 Magnetic field in helical configuration . . . 14

3.3.3 Forces on magnets in array . . . 16

3.3.4 Forces in multiple period planar configuration. . . 19

4 Outlook and conclusion 19

Appendix A Derivation of the Maxwell Stress Tensor 22

1 Introduction

The purpose of this study was to build a simple computational model of the magnetic elements in a new conceptual undulator setup, the Highly Adjustable Magnet (HAM) undulator [1]. Undulators are insertion devices used for generating synchrotron radia- tion. They are made up of arrays of magnets, usually strong permanent magnets, with alternating polarity. An electron beam from a particle accelerator is passed through the undulator, such that the magnetic field is perpendicular to the beam. Through the Lorentz force the path of the electrons is deflected and the accelerated charges produce radiation, called syncrotron radiation. For relativistic particles the intensity of the radi- ation is concentrated in the forward direction. In an undulator the particles in the beam are made to go in a slalom like trajectory by the alternating magnetic field. The emit- ted light overlaps with the beam as it travels through the undulator, and the intensity is concentrated in the forward direction. In figure 1 we see the rows of magnets with alternating polarity, and the path of the beam through the undulator.

If the undulator is long enough, and the beam fulfills tight demands on beam quality the induced radiation constructively interfere for the characteristic wavelength and it’s harmonics, producing coherent laser like light, making a Free Electron Laser (FEL) [2]. The resonant wavelengths depend on the energy of the particles, period length and magnetic field. Existing synchrotron light facilities are creating light ranging from microwaves to the hard X-ray regime.

Figure 1: A schematic of an electron beam travelling through a planar undulator, generating synchrotron radiation. The magnitude of the undulation is exaggerated for illustrative purposes.

In traditional undulators with fixed magnets the emitted light is plane polarized.

Some newer versions, e.g. Apple [3], Delta [4], have rows of magnet that are split lon-

gitudinally and can be translated less than a period which leads to helical fields that

produce circular and elliptically polarized light.

1.1 HAM undulator concept

The HAM undulator would consist of pairs of magnets in disks stacked along the beam axis, where the disks are free to rotate around the same axis. Each disk constitutes one half base period, the base period is defined as the smallest periodicity possible in planar configuration, where the magnetic field is rotated 180 degrees between consecutive disks.

The design would make it possible to configure the undulator for both helical and plane polarization, with adjustable periods. In the helical case the period can be adjusted with the limitation that the beam must exit the undulator with no net turning, an integer number of periods must fit within the undulator. The period for plane polarized light can be an integer factor of the base period, also with the limitation that it provide no net turning.

Alternating the magnetic field between each disk would reproduce a traditional planar undulator. Integer multiples of the base period λ

u

can be achieved by pairing several disks with the same direction of magnetization. If the angular displacement α between each disk is 90

^o

helical polarization with a period double that of the base period is achieved. The helical period can be increased from that point with smaller values of α, given that for the whole undulator there should be no net turning of the beam.

Figure 2 shows some of these basic configurations: the base planar configuration with field alternating between each disk; a planar configuration where the field alternates every third disk; and a helical configuration with 45

^o

rotation between each disk.

Compared to existing undulators, which have the rows of magnets rigidly mounted to strong girders, the HAM undulator would have more moving parts and a less rigid construction. The forces acting on the magnets must therefore be known to judge the feasibility of keeping the mechanical deformation small enough so that the tight tolerances of the magnetic fields are met.

In the previous study the magnetic elements consisted of a single section with uni- form magnetization, in this study we will also investigate a transverse Halbach-like setup with each element consisting of three sections to increase the flux density through the center of the undulator.

The simulations were made using COMSOL Multiphysics modeling software [5]

which was previously [1] bench-marked against the “industry standard” RADIA software [6, 7].

Since the disks can be adjusted independently it is also possible to create regions with different characteristics within the same undulator, make quasi periodic fields, im- plement chicanes, and easily adjust linear and non-linear tapers. All inside the same undulator allowing a great variety of use.

2 Theory

This section is a short review of the theoretical framework behind synchrotron light

sources and undulators. First a look at synchrotron radiation and FEL:s. After that some

theory on magnetostatics which is used for computing the fields inside the undulator.

Figure 2: Examples of basic configurations. a) Side view of the base planar configuration.

b) Side view of planar configuration with three times the base period. c) Front view of helical configuration with 45

^o

angular displacement between disks.

2.1 Synchrotron radiation

Accelerating charges emit radiation. Synchrotron radiation occurs when highly energetic electrons are deflected by a magnetic field perpendicular to their trajectory. Relativistic effects mean the light is emitted in a narrow cone in the direction of travel. The wave- length of this radiation spans from microwaves to hard x-ray [8], and was originally seen only as a hindrance to accelerators as it was the main contribution to the loss of particle energy. It was later found that the generated radiation could be of great use scientifically as it has desirable qualities as a light source for microwaves up to x-rays.

The use of x-rays in scientific research date back to it’s discovery, and various methods of spectroscopy and crystallography has yielded great results in areas from condensed matter physics and engineering to chemistry and biology. Several Nobel prizes have been awarded for x-ray research.

The radiated power, P , from a particle traveling in a curved path is given by the Schwinger formula [9]:

P = 2 3 ω

⁰

e

²

R β

³

 E mc

²



⁴

. (1)

ω

⁰

and R are instantaneous angular velocity and radius of curvature. e is the charge, β

is the velocity as a fraction of the speed of light, c. E is energy of particle, and m is the mass. The radiated power is inversely proportional to mass to the fourth power, making accelerating electrons in a ring very hard due to radiation losses. This same property makes electrons ideally suited for generating radiation.

2.2 Insertion devices and the Free Electron Laser

It was discovered that by using periodic magnetic assemblies with alternating magnetic fields, making the electrons travel in a slalom like path, you could increase the output compared to simple dipole magnets. These are called insertion devices since they were inserted in the straight parts between the bending magnets of synchrotrons. The first kind were the wigglers where the magnetic field strength and period combined resulted in the electrons taking a wide path, emitting synchrotron radiation each turn. The other type is the undulator, where the width of the path is narrower, and the emitted light overlaps with the traveling beam, interacting with it each period [10]. This allows for a narrower wavelength band. If the undulator is long enough, the light will interact repeatedly with the electron beam creating coherent, laser like light. This is called a Free Electron Laser, and was invented by John Madey [2] at Stanford University in the early 1970’s and demonstrated later that decade [11].

To understand the working principle, assume an ideal undulator with period λ

_u

where the magnetic field along the beam-axis (z-axis) is given by

B

_y

= B

0

sin(k

_u

z). (2)

Where k

u

= 2π/λ

u

.

The relativistic electrons arrive from the accelerator in bunches having been accel- erated by a strong alternating field. The requirements of the bunches include high peak current, low relative energy spread and low emittance [12]. To obtain this beam quality linear accelerators are used and the beam is sent through bunch-compressors that shorten the bunches to increase the linear density and with it, the peak current.

When entering the undulator, the electrons will spontaneously emit synchrotron radiation. Radiation with the resonant wavelength λ

_r

will overtake the electron beam by a whole period for each undulator period, and positively interfere and amplify along the undulator. The resonant frequency is given by

λ

r

= λ

u

2γ

0²



1 + K

0²

2 + γ

0²

φ

²



. (3)

where φ is the observation angle relative to the beam axis, K

⁰

= eB

⁰

mck

u

(4)

is the undulator strength parameter, which is dimensionless, and γ

⁰

is the Lorentz factor

of the electrons [13].

Through this mechanism energy is transferred from the electron bunches to light of the resonant frequency and to some degree its harmonics. The light enters an exponential gain regime until saturation is reached. When the beam is saturated the electrons in the bunch will have experienced micro bunching, where the electrons are in bands with a longitudinal period the same as the resonant frequency radiation. Since the light is amplified along the electron beam, the FEL yields a narrow laser like beam [13].

2.3 Magnetostatics

Starting from Maxwell’s equations we can build our computational model, using the constitutive relation between magnetic flux B, magnetic field H and magnetization field M ,

B ≡ µ

0

(H + M). (5)

The model used in this project deals only with permanent magnets and magnetostatics, i.e time independent and without currents, so we have

∇ · B = 0 (6)

and

∇ × H = 0. (7)

It follows from 7 that the magnetic field, H can be expressed as the gradient of a scalar magnetic potential, V

m

,

H = −∇V

m

. (8)

From eq.s 5, 6, 8 the following equation can be derived:

∇

²

V

m

− ∇ · M = 0. (9)

Which in this project is solved using Finite Element Methods, FEM, with COMSOL software [5]. It can also be solved analytically for certain geometries or semi-analytically as is done in the RADIA software [6, 7].

The magnetization field, M is defined as M = dm

dV (10)

where dm is elementary magnetic moment, and dV is the infinitesimal volume element.

The magnetization field is related to the remanent flux B

r

according to M = 1

µ

⁰

B

_r

. (11)

2.3.1 Force calculation using Maxwell’s Stress Tensor

The method used by COMSOL Multiphysics for calculating forces acting on magnets is based on the Maxwell Stress Tensor on a surface, S, in air (treated as vacuum) surround- ing the magnet. [5]. The equation reads as follows:

F = 1 µ

⁰

I

S

 (B · ˆ n )B − 1 2 B

²

ˆ n 

dS. (12)

A derivation can be found in Appendix A.

2.3.2 Halbach arrays

Instead of having simply alternating polarities in planar undulators, i.e. with two polari- ties per period, often a configuration with four or more polarities, called a Halbach array [14] is used. See figure 3. This allows for higher peak strength along the beam, and has the added benefit of minimizing the flux on the outside of the magnets.

The formula for peak strength along the beam axis is B

0

= B

r

×

"

sin(πε/M ) ε/M

#

× {1 − exp [− (2πh/λ

w

)]} × exp [− (πg/λ

w

)] . (13) Where λ

w

is the period of the Halbach array, M is the number of segments per period,h is the height of the magnets and g is the gap. ε is the packing factor, the proportion of each structure that is filled with magnets. ε = 1 if the magnet segments are touching, and less than one if there are gaps [15]. In figure 3 a side view of a Halbach array is shown. The arrows indicate the direction of magnetization for the different elements. In a regular planar undulator the beam would pass between the rows of magnets, from left to right in the figure.

Figure 3: Schematic of magnetic elements in a Halbach array with M = 4 segments per period.

The arrows show the direction of magnetization for the individual segments.

One of the simulated magnetic structures derives from this, using 3/4 of a Halbach array, in the transverse direction, to increase the field in the center from a given height of the magnets.

3 Modelling and simulations

The simulations were made using COMSOL Multiphysics 5.2a on machines with Intel Core i7-4770s CPU and 16 GB memory, running 64-bit Microsoft Windows 7 Enterprise Operating System.

The coordinate system is defined such that the beam travels along the z-axis in the positive direction. The disks are orthogonal to the beam, laying in xy-planes. Positive y direction is defined as the direction of the magnetization of the first disk. The origin is at the center of the first disk. See figure 4 for reference. Each individual disk also has a local "primed" coordinate system with the y’-axis in the direction of it’s own primary magnetization. The value of remanent flux, B

r

, used for all simulations is 1.1 T, an arbitrary value within the range of typical values for rare earth magnets. The dimensions used, unless otherwise specified, are as follows: The whole magnet structure is 90 mm wide (x) by 30 mm high (y) by 20 mm deep (z); Each section within the magnets are 30 mm wide; The transverse gap is 30 mm; The undulator base period is 50 mm, i.e. a longitudinal gap of 5 mm between each magnet pair.

Figure 4: Array of 10 disks in planar configuration. The coordinate directions are shown in the

lower left corner.

3.1 Computational model

The model was built in COMSOL selecting 3D, the physics selected were AC/DC>Magnetic Fields, No Currents (mfnc) and a stationary study from Preset Studies>Stationary.

The magnet regions were built using blocks, with spheres and cylinders used to create fillets along the edges to avoid sharp corners which may introduce singularities when using FEM. The region of air around the magnet was defined as a block enclosing the magnets with dimensions 0.3 m ×0.3 m in the xy-plane, and 0.55 m in the z direction.

In the surrounding air a free tetrahedral mesh is used with size: predefined - fine, along the beam axis (z-axis) an edge with maximum element size 0.002 m is used, and for the magnets a free tetrahedral mesh is used with size: predefined - finer.

3.2 Magnet pair

Two different structures for the magnetic elements were simulated. A simple structure where all magnetization is along the y-axis, and a complex structure, where the outer thirds of the structure have magnetization along x-axis to increase flux density in the origin. This structure took it’s inspiration from Halbach arrays, but where Halbach arrays are oriented longitudinally, this structure is transverse. In figure 5 a schematic of the structures is shown. Figure 6 shows slices of the magnetic field for the two structures.

In the simple case, the flux is spread over the width of the structure, giving a weaker, but more homogeneous field. In the complex structure the flux is concentrated in the middle, reaching a narrow peak of higher flux around the origin.

Figure 5: Schematic of single pairs in xy-plane. The arrows indicate the direction of magnetiza-

tion. (a) A simple magnetic structure with the magnetization in only one direction.(b) Halbach

type structure.

(a) (b)

Figure 6: Simulated fields for single pairs, xy-slice. (a) A simple magnetic structure. (b) Halbach type structure that concentrates the flux along the beam line.

3.2.1 Varying the height of the magnets

Simulations were done over a range of heights, h, for complex structures. The flux density in the origin is shown in figure 7. The total force on one of the magnets in the pair is shown in figure 8, along with a fitted function of the form

F

_y

= C (1 − exp[−λ (h − h

0

)]) .

This form of the fitted function was chosen with the height-dependent factor in equation 13 in mind and is for illustrative purposes. That equation deals with peak field strength at the beam line, and this is not to infer that a linear relationship is generally the case.

Rather to show the trend for larger values of h.

The forces are calculated using the method outlined in section 2.3.

Figure 7: The y-component of magnetic flux density in the center of the gap, B

⁰

, against the height of the magnets.

Figure 8: The y-component of the attractive force acting on one of the magnets in a single pair.

Plotted against the height of the magnets.

As can be seen in figure 8, the data points are fairly scattered, this is due to the discretization of the volume. A finer mesh would give better results. As it stands the results are for a mesh size slightly below where the computer would run out of available memory. In this respect the simulations can be improved either by running on better hardware of increasing the density only in the desired region. These results are still sufficient to place some upper bounds on the expected forces.

3.3 Array of disks

To model a whole undulator section the base pair was copied and translated along the z-axis with a displacement of 25 mm per half period. The magnets had the standard dimensions of 90 mm width (y), 30 mm height (x) and 20 mm depth (z). The transverse gap between the magnets is 30 mm. Each pair had angular displacement, α, with regard to the previous pair. For single period plane polarization α = 180

^o

. In the case of helical polarization 0

^o

< α < 90

^o

. For multiple period planar configuration, the magnetization is unchanged for M pairs, followed by a 180

^o

rotation. 10 pairs were simulated, giving 5 base periods. The standard configuration of complex magnetic structure was used for all simulations. Figure 4 shows a 3D view of the magnets in planar configuration, and figure 9 shows a zoomed in view with the direction of magnetization included for planar and helical (α = 45

^o

) configuration.

(a) (b)

Figure 9: The magnetization direction for (a) single period planar configuration and (b) α = 45

^o

helical configuration.

3.3.1 Magnetic fields in plane polarized configuration

Simulations results for the single period plane polarized configuration were made. Figure 10 shows a 3D view with a slice in the yz-plane showing the magnitude of the B-field, and quiver along z-axis. In figure 11 the y-component of the B-field along the z-axis is plotted.

The field is stronger at the end pairs, while inside the undulator the overlapping fields

from neighboring pairs partially cancel out. This could be solved by either increasing the

gap for the outer pairs, or making the magnets thinner.

Figure 10: Single period planar, slice of B-field and quiver along beam line.

Figure 11: Single period planar, y-component of B-field along z-axis

Results from simulations of double period planar is shown with yz-slice of B-field

magnitude and quiver along z-axis in figure 12. In figure 13 the y-component of the

B-field along the z-axis for M = 1, .., 5. The maximum field strength increases with the

pairing, since the superposition of the field from each pair is constructive. Above M = 3

the increase is only slight, since the field from a single disk tapers off quickly and is small

at distances grater than to the nearest neighbours. Due to the gap between the magnets the field is not even, but varies slightly with period equal to half the base period. This is seen most clearly for M = 4 and M = 5. Whether this has a negative impact on the quality of light produced is a question for further study.

Figure 12: Double period planar, slice of B-field and quiver along beam line.

Figure 13: M period planar, y-component of B-field along z-axis

3.3.2 Magnetic field in helical configuration

The magnetic field was simulated for values of α between 0 and 90 degrees. Figure 14 shows yz-slice of magnetic flux density, and quiver along the z-axis. In figure 15 and 16 the x and y-components and magnitude of the B-field along the z-axis shown for the case of α = 45

^o

and α = 90

^o

respectively. As was the case with multiple period planar, the magnitude shows an unevenness due to the gaps between the disks. It can be seen that the magnitude of the field is lower for the 90 degree configuration, with a peak magnitude of 0.35 T compared to 0.45 T for the 45 degree case. For the case of 90 degrees the magnitude also has a slight decrease in the middle of the undulator.

Figure 14: Helical configuration, 90 deg., slice of B-field and quiver along beam line.

Figure 15: 45 degree helical configuration. x- and y-components and norm of B-field evaluated along z-axis.

Figure 16: 90 degree helical configuration. x- and y-components and norm of B-field evaluated along z-axis.

In a perfectly helical magnetic field the x and y-components would be sinusoidal,

and for long periods the field from the array of disks closely resemble this. But for

shorter periods, especially for the α = 90

^o

the discreet transitions between the disks

are more noticeable. This can also be seen in figure 17 where a quiver plot along the

z-axis is shown head on. As α grows the field is less circular, and for 90 degrees it is

almost square. In figure 18 the polar argument of the magnetic field, when expressed

in cylindrical coordinates is shown, also demonstrating the deviation from ideal helical

behavior, which would be straight lines when plotted against z.

Figure 17: View of magnetic field down beam line. Notice how when α gets bigger, the profile gets less circular.

Figure 18: Polar argument of B-field in cylindrical coordinates along z-axis. A perfect helix would describe straight lines. With greater α the field deviates more and more from the ideal case.

3.3.3 Forces on magnets in array

Simulations were done for values of α between 0 and 180 degrees. The forces were

calculated acting on a magnetic element in the first, third and fifth pair. In figure 19

the magnitude of the force is displayed. The greatest forces are on the outer disk, since

longitudinal forces cancel out for the inner disks. The largest force being 230 N on a

magnet in the outer pair when α = 0

^o

The third and fifth pairs experience similar forces.

Figure 19: The norm of the force on magnets in first, third and fifth pair, over different values of angular displacement, α.

The components of the force in the respective disk fixed systems are shown in figure

20. It can be seen that for the third and fifth disk the forces are similar, and all but the

local y component vanish. In the edge disk, the z component is the greatest around 0

and 180 degrees.

(a)

(b)

(c)

Figure 20: x’, y’ and z components of the force on magnet in the disk-local coordinate system

as α changes. (a) first pair, (b) third pair, (c) fifth pair.

3.3.4 Forces in multiple period planar configuration.

The greatest forces are observed in the transition between direction of magnetization in the case of multiple period planar configuration. For the case of M = 5 the magnitude of the force on each element in a transition disk was 443 N. F

_x

= −2 N, F

_y

= 44 N, F

z

= 440 N. The forces in y-derection on both magnets will cancel out within the disk, but the force in the z-direction will be twice that of one magnet, meaning the disks must be rigidly mounted. In figure 21 you can see that the flux density between the disks with opposite polarity is much higher than elsewhere, leading to large forces.

Figure 21: Side view of the undulator in planar configuration with M = 5. Slice shows magnitude of B-field.

4 Outlook and conclusion

Within one magnet pair, the forces and field strength grow with the height asymptotically, so increasing the height above a certain limit serves little purpose.

In the single period planar configuration the field cancels out somewhat, and the longitudinal forces between disks are only an issue at the edges. For multiple period planar the greatest forces are between the disks where the field alternates, and this is where we observe the highest forces for all the simulations. In the helical case, certain values of α produce higher forces, this needs to be considered when switching between configurations.

So far the forces do not seem to be prohibitively large in simulations where the field

strength is in the order of 0.4 T. Some use cases would require higher fields than that,

which would lead to higher forces. The design needs further development, and mechanical

simulations made before an upper limit on the forces and torques and consequently field

strength can be made.

Due to the gaps between the magnets in each pair, some unevenness is observed in the field along the z-axis. How much of an issue this would cause for generating light is unknown, and further analysis of the magnetic fields should be made, along with calculations of the beam dynamics through the undulator.

References

[1] M Hamberg. “The Highly Adjustable Magnet Undulator”. In: Proc. 37th Inter- national Free Electron Laser Conference (FEL2015). Daejeon, Republic of Korea, August 2015. doi: doi:10.18429/JACoW-FEL2015-TUP057.

[2] John MJ Madey. “Stimulated emission of bremsstrahlung in a periodic magnetic field”. In: Journal of Applied Physics 42.5 (1971), pp. 1906–1913.

[3] Shigemi Sasaki. “Analyses for a planar variably-polarizing undulator”. In: Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrom- eters, Detectors and Associated Equipment 347.1 (1994), pp. 83–86. doi: https:

//doi.org/10.1016/0168-9002(94)91859-7.

[4] Alexander B. Temnykh. “Delta undulator for Cornell energy recovery linac”. In:

Physical Review Special Topics - Accelerators and Beams 11.12 (2008), p. 120702.

doi : https://doi.org/10.1103/PhysRevSTAB.11.120702.

[5] COMSOL Inc. COMSOL Multiphysics Reference Manual, version 5.2a. url: https:

//www.comsol.com.

[6] Pascal Elleaume, Oleg Chubar, and Joel Chavanne. “Computing 3D magnetic fields from insertion devices”. In: Particle Accelerator Conference, 1997. Proceedings of the 1997. Vol. 3. IEEE. 1997, pp. 3509–3511.

[7] P Elleaume et al. “A 3-Magnetostatics Computer Code for Insertion Devices”. In:

J. Synchrotron Rad. (1998), pp. 481–484.

[8] Evgeny L Saldin, EV Schneidmiller, and Mikhail V Yurkov. The physics of free electron lasers. Springer Science & Business Media, 2013, p. 1.

[9] Julian Schwinger. “On the Classical Radiation of Accelerated Electrons”. In: Phys.

Rev. 75 (12 June 1949), pp. 1912–1925. doi: 10.1103/PhysRev.75.1912.

[10] AL Robinson. “X-ray Data Booklet: Section 2.2. History of Synchrotron Radiation”.

In: Center for X-ray Optics and Advanced Light Source, Berkeley, CA (2009).

[11] David AG Deacon et al. “First operation of a free-electron laser”. In: Physical Review Letters 38.16 (1977), p. 892.

[12] J Arthur et al. Linac coherent light source (LCLS) conceptual design report. Tech.

rep. 2002. Chap. 4.2.

[13] Zhirong Huang and Kwang-Je Kim. “Review of x-ray free-electron laser theory”.

In: Physical Review Special Topics-Accelerators and Beams 10.3 (2007), p. 034801.

[14] Klaus Halbach. “Physical and optical properties of rare earth cobalt magnets”. In:

Nuclear Instruments and Methods in Physics Research 187.1 (1981), pp. 109–117.

[15] G. Rakowsky et al. “High-performance pure permanent-magnet undulators”. In:

Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 296.1 (1990), pp. 597–602. doi:

https://doi.org/10.1016/0168-9002(90)91273-E.

[16] Jack Vanderlinde. Classical electromagnetic theory. Vol. 145. Springer Science &

Business Media, 2006, pp. 257–251.

Appendix A Derivation of the Maxwell Stress Tensor

To derive the Maxwell Stress Tensor [16], we begin from Maxwell’s equations in vacuum (free of polarization and magnetization)

∇ · E = ρ ε

0

, (14)

∇ · B = 0, (15)

∇ × E = − ∂B

∂t , (16)

∇ × B = µ

⁰

J + ε

⁰

µ

⁰

∂E

∂t , (17)

and the electromagnetic force on a continuous medium, the Lorentz force density,

f = ρE + J × B. (18)

The total force on an object is then the integral over its volume V , F =

Z

V

f dV = Z

V

dV h

ρE + J × B i

. (19)

Substituting from Maxwells equation’s 14, 17, F =

Z

V

dV h

ε

⁰

(∇ · E)E + 1 µ

0

(∇ × B) × B − ε

⁰

dE dt × B i

. (20)

We use the product rule to get dE

dt × B = d

dt (E × B) − E × dB dt ,

along with Maxwell’s equation 16. And use ∇ · B = 0 to add the term (∇ · B)B for symmetry,

F = Z

V

dV h

ε

⁰

((∇ · E) E + (∇ × E) × E)+ 1

µ

⁰

((∇ · B) B + (∇ × B) × B) B−ε

⁰

d

dt (E × B) i . (21) Recall the Poynting vector S = 1

µ

⁰

E × B, and write F +ε

0

µ

0

Z

V

dS dt dV =

Z

V

dV h

ε

0

((∇ · E) E + (∇ × E) × E)+ 1 µ

0

((∇ · B) B + (∇ × B) × B) i

.

(22)

The integrand on the RHS can be seen as the divergence of a tensor. We focus on the

term containing B, and write using Einstein’s summation notation, where we assume

summation of non-repeating indices, so for a vector F we have F = F

i

e

_i

,

∇ = e

i

∂

∂x

_i

,

∇ · F = e

_i

· e

_j

∂F

j

∂x

_i

,

∇ × F = e

_i

× e

_j

∂F

_j

∂x

i

. The B part of the integrand is then

(∇ · B) B + (∇ × B) × B

=

 e

_i

· e

j

∂B

j

∂x

_i



B

_k

e

_k

+

 e

_i

× e

j

∂B

j

∂x

_i



× B

_k

e

_k

=

 δ

_ij

∂B

j

∂x

_i



B

_k

e

_k

+ (e

_i

· e

_k

B

_k

) ∂B

j

∂x

_i

e

_j

−  ∂B

j

∂x

_i

e

_j

· e

_k

B

_k



e

_i

. (23) i, j, k = 1, 2, 3.

(∇ · B) B + (∇ × B) × B

= ∂B

i

∂x

i

B

k

e

_k

+ δ

ik

B

k

∂B

_j

∂x

i

e

_j

−

 δ

jk

B

k

∂B

_j

∂x

i

 e

_i

= ∂B

i

∂x

i

B

k

e

_k

+ B

i

∂B

j

∂x

i

e

_j

− B

j

∂B

j

∂x

i

e

_i

. (24)

Because we are summing over indices we are free to interchange them, and get (∇ · B) B + (∇ × B) × B =

 B

i

∂B

_j

∂x

j

+ B

j

∂B

_i

∂x

j

− B

j

∂B

_j

∂x

i



e

_i

. (25) Continue with the i:th component

[(∇ · B) B + (∇ × B) × B]

i

= B

_i

∂B

j

∂x

j

+ B

_j

∂B

_i

∂x

j

− B

_j

∂B

j

∂x

i

= ∂

∂x

j

(B

i

B

j

) − 1 2

∂

∂x

i

(B

j

B

j

)

= ∂

∂x

j



B

_i

B

_j

− δ

_ij

1 2 B

_k

B

_k



= ∂

∂x

_j



B

i

B

j

− δ

ij

1 2 B

²



(26) The same procedure can be made for the E part of the integrand, and 22 can be written as

F +ε

0

µ

0

Z

V

dS dt dV =

Z

V

dV ∂

∂x

j

h ε

0



E

_i

E

_j

− δ

_ij

1 2 E

²

 + 1

µ

⁰



B

_i

B

_j

− δ

_ij

1 2 B

²

 i

e

_i

, (27)

where each component of the integrand can be seen as the divergence of a vector making up a row of a tensor, which is the Maxwell Stress Tensor, T

ij

,

T

_ij

= ε

0



E

_i

E

_j

− δ

_ij

1 2 E

²

 + 1

µ

⁰



B

_i

B

_j

− δ

_ij

1 2 B

²



. (28)

Using Gauss’s theorem we can express the integral of a divergence over volume, V as a surface integral over the boundary S as follows

Z

V

∂

∂x

_j

T

ij

dV = I

S

T

ij

n

i

dS, (29)

where n

_i

is the normal vector to the surface S. In the computational model for this project, we are dealing with magnetostatics, so the time derivative on the LHS and the electric field terms are zero, we can then finally write the relevant equation that will be used for calculating the force on a permanent magnet,

F

i

= 1 µ

0

I

S



B

i

B

j

− δ

ij

1 2 B

²



n

i

dS, (30)

or in more familiar vector notation F = 1

µ

0

I

S



(B · ˆ n )B − 1 2 B

²

ˆ n



dS. (31)

The bounding surface is chosen to be in in the air surrounding the magnet (air is treated

as vacuum).