MAGNETIC DESIGN OF AN IN VACUUM

UNDULATOR SPECIFICATIONS AND MAGNETIC

OPTIMIZATION

Z e u s M ar t í

M ique l T r ave r i a

Xav i e r Q ue r a l t

J o s ep C amp many

S P A N I S H M I N I S T R Y O F C I E N C E P R O J E C T F P A 2 0 0 3 - 0 6 7 0 3

2

3

abstract ........................................................................................4

Undulator type ...........................................................................5

Market survey.............................................................................6

Undulator length and minimum gap ....................................11

Period length and range of gap variation.............................13

Central period optimization...................................................17

End poles design......................................................................22

Mechanical tolerances .............................................................25

List of specifications ...............................................................27

4

Abstract

This report describes the magnetic design of an in-vacuum undulator, as well as the derivation of specifications useful for its mechanical design.

In November 2003, the Spanish Ministry of Science and Technology funded the project entitled “Design of an In-vacuum undulator for the synchrotron light production in the energy range of 1 to 10 keV”, FPA2003-06703, within the National Plan for Scientific Research, Development and Technological Innovation 2000-2003. The project should be accomplished in 2006 and the main target is to elaborate a detailed design of an In-vacuum undulator, useful enough for the construction of such a device by the Spanish industry.

In the first year we have finished a conceptual magnetic design, as well as an exhaustive market survey on magnetic materials supply. We present the results of this work in this paper. Moreover, we also had finished a conceptual mechanical design, which is presented in the paper entitled “Conceptual design of an In-vacuum undulator. Mechanical and vacuum aspects”, by ELYTT.

The project is managed through the Laboratori de Llum del Sincrotró (LLS), a consortium between the Universitat Autònoma de Barcelona and the Generalitat de Catalunya. This LLS institution has a memorandum of understanding with CELLS, the institution responsible for construction, equipping and exploitation of the future synchrotron light facility to be built near Barcelona. Because of this agreement, CELLS is indirectly involved in this work, and supports and benefits from the acquired know-how, in order to apply it in the new synchrotron facility.

The calculations and conceptual designs included in this paper have been mainly done by Zeus Martí, with the advice and support of Miquel Traveria, Xavier Queralt and Josep Campmany. The market survey was done by the local company Atipic.

5

Undulator type

The first election is related to the undulator type, pure permanent magnet (PPM) or hybrid.

The main advantage of hybrid technology is that the maximum field that can be reached inside the gap is higher than that obtained using PPM technology.1

The main disadvantage of hybrids is that it is difficult to implement passive end-poles correction in order to minimize the variation of field integrals when the gap is changed. However, this disadvantage can be sorted out implementing active feedback and using local corrector magnets placed just before and after the insertion device.

In fact, the current trend is to use hybrid technology for in-vacuum undulators, mainly because the target is to produce light energies as high as possible, thus, magnetic field inside the gap as high as possible. So, we have decided to go for a hybrid undulator.

The second election is related to magnetic material used for permanent magnets and the type of iron used for poles.

The permanent magnet materials required to build IDs need to have both a large spontaneous magnetization and a large coercitivity. Two possible materials are widely used: NdFeB and Sm2Co17 alloys. In case of NdFeB, the remanent field can achieve high values (1.1 to 1.4 T) whilst the intrinsic coercitivity cJH0µ can achieve values between 1 and 3.5 T In case of Sm2Co17 high

coercitivity materials show a remanent field of ca. 1.03 T and a coercitivity cJH0µ of ca. 2 T.2

In next chapter we present a market survey in order to find the availability and cost of the most adequated permanent magnets.

With respect to the ferromagnetic material used for poles, in a first step we focused the market survey into the high saturation field materials, i. e., Fe-Co family materials (PERMENDUR, 49% Fe, 30% Co, 2% Va), with a Bs = 2,35 T. In a second step, mainly because economical reasons, the survey was extended to lower saturation field materials. The alternative in this case will be ARMCO iron (Bs = 2,15 T).

1 Joel Chavanne, Pascal Elleaume, Technology of insertion devices, «Undulators, wigglers and their applications», H. Onuki and P. Elleaume Ed., Taylor & Francis, London 2003, p. 174. 2 J. Chavanne, B. Plan, c. Penel, P. Vanvaerenbergh, Magnetic design considerations for in-vacuum undulators at ESRF, Proceedings of EPAC 2002, Paris, france.

6

Market sur vey

In a first step, a list of specifications (Table 1) were sent to different suppliers asking for a proposal on the best magnetic material available. All suppliers coincided in propose NdFeB alloy. They are making a lot of progress in their products, mainly in the increase of remanent field and energy product, as well as in the increase of the intrinsic coercitivity, in the range of operational temperatures. All the offers claim to reach without losses a working temperature of 150ºC.

Table 1. Specifications for the permanent magnets Min Max Working temperature (ºC) 20 125 Br (T) 1,05 >1,1 HcJ (kA/m) 1900 >2175 Radiation resistence 3GeV Irreversible losses 2,75% Remanent field value dispersion between blocks

±1.5%

Remanent field axis dispersion between blocks

<1,5º

Block coating No electrolytic or polymeric The losses due to radiation is the only specification that more of the suppliers have been able

to guarantee. Usually it is said that SmCo is more resistive to radiation than NdFeB, but recent publications remark that NdFeB with high coercitivity gives also good results, possibly because the origin of materials radiation damage has a thermal origin.3

With respect to the ferromagnetic material used for poles, the specifications issued to the companies are listed in Table 2.

Table 2. Specifications for the pole pieces iron Min Max Working temperature (ºC) 20 125 Bs (T) 2,3 2,4 Coating thickness 10 – 15 µm Maximum anisotropy <1%

A. Permanent magnets

17 companies were contacted in order to get information, but only 4 offered materials within

our specifications. In Table 3 we show the characteristics of each offer. These companies are the usual suppliers for ID laboratories and industries.

3 R. D. Brown and J. R. Cost, Radiation-induced changes in magnetic properties of NdFeB permanent magnets, IEEE Transactions on magnetics, 25 (1989) 3117-3120. James Spencer and SLAC MMG, and James Volk, Permanent magnets for radiation damage studies, PAC 2003 proceedings.

7

In order to compare the properties of the materials manufactured by each company, some

Figures have been set up. In Figure 1 we plot the magnetic properties (remanence field and intrinsic coercitivity of magnets). The area of the circle is proportional to price.

Table 3. Technical specifications of magnets for each supplier (NdFeB).

Mechanical tolerances

(mm) Comapny Cost(€/block)

T max. (ºC)

Br (T)

Hcj (kA/m)

Tolerance in

romanence

(%)

Tolerance in axis of

magnetization

(º)

Coating thickness

(µm)

Coating type

Direction of magnetization

Other directions

VAC 12,42 190 1,12-1,18 >2230 ±2 <1.5º 10-15 IVD Al ±0.1 ±0.1

DEXTERMAG 5,30 150 1,14-1,17 >1990 ±1,5 <1.5º 10-15 IVD Al ±0.1 ±0.1

ARNOLD 2,10 150 1,05-1,08 >1990 ±3.0 <2º 10-15 IVD Al ±0.1 ±0.1

SUMITOMO

(Neomax 32 EH) 21,95 180 1,11-

1,19 >2175 ±1,5 <1º 5-7 TiN ±0.05

±0,1mm @45mm

±0.05

±0,1mm @45mm

Magnetic properties

5.30 €

2.10 €

12.42 €

21.95 €

1.04

1.06

1.08

1.1

1.12

1.14

1.16

1.18

1900 2000 2100 2200 2300

Minimum coercitivity (kA/m)

Typi

cal r

eman

ent f

ield

(T)

c

VACDEXTERMAGARNOLDSUMITOMO

Figure 1. Magnetic properties of the NdFeB materials for each supplier.

8

In figure 2 we present the tolerances associated to remanent field and magnetization direction of the blocks.

In Figure 3 we present the tolerances in remanent field associated to the tolerances in the

mechanization of blocks.

In summary, the material offered by SUMITOMO is clearly the best in properties, but it is

expensive. In case the coercitivity was not a critical factor, then DEXTERMAG is the best offer because of the excellent relation quality/price.

Tolerances

2.10 €

21.95 €

12.42 €5.30 €

0

0.02

0.04

0.06

0.08

0.1

0.12

0.0% 1.0% 2.0% 3.0% 4.0%

Tolerance in remanent field

Mec

hani

cal t

oler

ance

s(m

m)

c

VAC

DEXTERMAG

ARNOLD

SUMITOMO

Figure 3. Magnetic and mechanical tolerances for each supplier

Tolerances of magnetic properties

12.42 €

2.10 €

21.95 €

5.30 €

0

0.5

1

1.5

2

2.5

0.0% 1.0% 2.0% 3.0% 4.0%

Remanent field tolerance

Tole

ranc

e in

dire

ctio

n of

mag

netiz

atio

n(º)

VACDEXTERMAGARNOLDSUMITOMO

9

B. Pole pieces. Two options are analyzed here: in the first case the pole pieces are mechanized by suppliers (Table 4). In the second case, bulk material is supplied and pieces should be mechanized in house (Table 5).

In case of mechanized polar pieces, we compare in Figure 4 the offers of different companies,

taking into account the saturation field and the mechanical tolerances.

Table 5. bulk material offers

Company

Bs (T)

Bulk piece appearence

Dimensions of supplied

bulk piece

# Polar blocks / Bulk piece

Price(€/Bulk

piece)

Price

(€/Kg)

CARPENTER (Permendur)

2, 40

Rectangular

48 x 18 x 2250 mm

~ 320 Blocks /Bulk piece

~ 2800

€/Bulk piece

167,13 €/Kg

IMPHY

(Permendur)

2,35

Cylindrical

∅19mm x long. 700 mm

~ 30 Bloks /

Bulk piece

~ 455 €/Bulk

piece

283,29 €/KG

AK Steel (ARMCO)

2,15

Cylindrical

∅45mm x long.2000 mm

~ 610 Blocks /

Bulk pieces

~ 125

€/Bulk piece

4,98

€/Kg

Table 4. Suppliers providing mechanized FeCoV pole pieces.

Company

Price per pole

(including

thermal

treatment)

Material type Bs

(T)

Coeating

thickness

(µm)

Coating type

Mechanical

tolerances

(mm)

VAC 90,00 € VACOFLUX 2,35 10-15 Sn, Cu,Ag-Sn,

Ag-Cu ±0.05

IMPHY 33,82 € (1) AFK 502 2,35 NO NO

[4.16]+/-0.09

[15]+/-0.13

[45]+/-0.20

ARNOLD 47,82 € (1) HIPERCO

Alloy 50 2,35 NO NO ±0.2

AMES 46,00 € FeCoV 2,2

(2) 3-5 TiN ±0.01

SUMITOMO 28,10 € Permendur 2,30 5 TiN ±0.05

(1) These quotations include 2,82 € that is the cost for TiN coating at the local company TTC. (2) The saturation field can be increased improving the densification process. We consider here the worst case.

10

In case of

bulk material, we compare in Figure 6 the offers of different companies, taking into account the saturation field and the price of each piece.

As a conclusion, for Permendur, SUMITOMO offers the best quality/price rated offer.

Polar pieces

90 €

34 €

48 €

46 €

28 €

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

2.15 2.2 2.25 2.3 2.35 2.4

Saturation field (T)

Mec

hani

cal t

oler

ance

(mm

) c

VACIMPHY+TTCARNOLD+TTCAMESSUMITOMO

Figure 5. Comparation of mechanical and magnetic properties of pole materials for each offer.

Bulk material

2,800 €

4,853 €

125 €- €2 €4 €6 €8 €

10 €12 €14 €16 €18 €

2.1 2.15 2.2 2.25 2.3 2.35 2.4 2.45

Saturation field (T)

Pric

e fo

r pie

ce(€

) c CARPENTER

IMPHYAK Steel

Figure 6. Price and saturation field for different bulk materials. Note that the AK Steel refers to ARMCO iron.

11

Undulator length and minimum gap

The aim of this chapter is to determine the magnetic length of the undulator and the minimum gap.

The first number is closely related to the machine optics functions. According to our current lattice design, we show the machine layout in the straight sections available for IDs in Figure 7, and we present the beta-functions for ALBA_03 at that place in Figure 8.

Figure 7. Machine layout at the straight section as well as vacuum chamber layout. The free space available for IDs is 3.2 m long.

Figure 8. Optical beta-functions in the straight section.

Distance from cell origin

βx , βy

12

The value of the vertical beta-function in the center of the straights is 1.02 m. Using the classical expressions derived by Brown, Winick and Eisenberger,4 we can derive the minium achievable gap and the maximum admitted length for the undulator. We note that these expressions are only valid when the physical aperture is given by the undulator gap (which will not be our case).

02 yL β=

yLg ε20>

The ALBA_03 lattice has the following values:

• 0yβ = 1.019 m

• yε = 3.7·1011 m·rad

Thus, the maximum achievable length is 2 m in order to have an optimized physical aperture in the vertical direction, and the minimum gap is 0.2 mm in order to have enough quantum lifetime.

However, a minimum gap of 0.2 mm is not realistic, because the forces between magnetic arrangements will be so high that mechanical deformations will be impossible to minimize in the space available inside the vacuum chamber. We present in Figure 9 the magnetic force and maximum magnetic field generated by a hybrid undulator with a period of 24 mm versus the gap value.

0

10000

20000

30000

40000

50000

60000

70000

0 2 4 6 8 10 12

Gap aperture (mm)

Tota

l for

ce o

n th

e be

am (N

)

0

0.5

1

1.5

2

2.5

3

Max

imum

mag

netic

fiel

d (T

)

Figure 9. Force and maximum magnetic field between magnetic arrangements versus gap aperture, for a hybrid undulator with period of 24 mm, calculated with Radia.

There are other limiting factors such as the magnets demagnetization. Also, we have a real

electron beam, with a finite size and a dynamic aperture and also an imperfect orbit. In addition, in in-vacuum undulator, the minimum gap is a key factor to be accounted for the vacuum quality

In order to facilitate the mechanical design, we limit the force on the beam to 20.000 N (2

tones). Because of this constraint, along with the above considerations, we set the minimum achievable gap at 5 mm.

4 George Brown, Herman Winick, Peter Eisenberger, The optimization of undulators for synchrotron radiation, Nuclear Instruments and Methods, 204 (1983) 543-547.

13

Per iod length and range of gap var iat ion

The aim of this chapter is to determine two of the basic specifications needed for the mechanical conceptual design of the undulator: the period length and the range of variation of the gap.

To this end, some simulations have to be done in order to know the performance of the undulator from the point of view of optics. The objective is to obtain high on-axis brightness (Ph/s/mm2/mrad2/0.1%BW) in a continous range from 1 to 10 keV, even more if possible. This will determine the range of gap variation as well as the length of the undulator period.

To calculate the Brilliance of a photon beam produced by an undulator, we use the typical expression given by Kim:5

( ) 2'

2'

2'

2'

22222

2

2

2

212

2

21

2217

02

214

214

21

110·431.1yrxryrxr

nn KnKJ

KnKJ

KnKNIBr

σσσσσσσσπωω

++++

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛+

∆=

+−

[1]

Where:

• K is the deflection parameter characterizing the undulator (see eq. [4]). • N is the number of periods.

• ωω∆

is the relative bandwith used in calculations, 0.1% in our case.

• n is the order of the harmonic • I is the intensity of the electron beam current (in mA). • e is the electron charge • J are the Bessel functions • )4/( πλλσ Ur N= and )/(' Ur Nλλσ = are the effect of size and angular divergence

of photons, where Uλ is the length of the period of the undulator andλ is the wavelength of the observed light. In this approximation, we choose λ to be the wavelength of the interference peaks, i. e. we only account for the peak photon flux.

• xσ and yσ are the horizontal and vertical electron beam sizes in the center of the undulator, provided it is placed in the center of the straight section.

• 'xσ and 'yσ are the horizontal and vertical electron beam divergences in the center of the undulator, provided it is placed in the center of the straight section.

5 Kwang-Je Kim, Characteristics of synchrotron radiation, AIP Conference Proceedings, 184 “Physics of Particle Accelerators”, M. Month & M. Dienes Ed., AIP 1989, p. 565-632.

14

The expression [1] takes into account the angular divergence of the electron beam, but it does not take into account the Brilliance dilution because of the electron beam energy dispersion Eσ . To this end, it should be corrected by the following factor:6

( )( )2

1,1 2 2

E

E

D nNnN

σπ σ

=+

[2]

Using the corrected expression, we can calculate the curve of Brilliance versus K. In order to obtain the Brilliance tuning curves (Brightness versus λ ) we have to use the relation between K and λ :

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+=

21

000511.0/2

2

2

KEn

Uλλ [3]

Where E is the energy of the electron beam, expressed in GeV.

In order to scan the values of K, physically the undulator gap should be varied. Thus, there are clear physical limitations to the values that K can achieve: the maximum achievable K corresponds to the minimum gap, and the minimum achievable K corresponds to the maximum gap available.

The relationship between the gap and the K values is not easy to calculate. For a given length of the undulator period Uλ , K depends on the value of the maximum magnetic field in the gap,

0B :

uBK λ09344.0= [4]

However, the value of 0B for a hybrid undulator cannot be analytically related to the gap, mainly because pole saturation depends itself on gap. So, it should be figured out numerically. There are some approximations, valid within certain K ranges,7 but in case of In-vacuum undulators, characterized by low values of the period, the empirical formulae are not useful. So, the best way to proceed is to calculate 0B using a finite element model.

In Figure 10 we show the results of such a calculation for different undulator period lengths, assuming a cell with dimensions given in Table 6.

Table 6. Parameters used to simulate the maximum field within the gap

PPM height 14 mm Pole height 10 mm PPM width 45 mm Pole width 35 mm Pole length uλ / 3

6 Josep Nicolas, Light from undulators, private communication 7 Richard P. Walker, Insertion Devices: undulators and wigglers, Synchrotron Radiation and free electron lasers, CERN 98-04, CERN accelerator School 1996, S. Turner Ed., p. 152.

15

0

0.2

0.4

0.6

0.8

1

1.2

15 17 19 21 23 25 27 29

Period length (mm)

Max

imum

fiel

d (T

) c

5 mm 10 mm 15 mm 20 mm 25 mm

Figure 10. Maximum field for different gap apertures and period lengths (non-optimized undulators)

Using these results, we calculate the tuning curves for different cases,

1.E+18

1.E+19

1.E+20

1000 10000 100000

Energy (eV)

On

axis

bril

lianc

e at

cen

tre

Ph/s

/mra

d^2/

mm

^2/0

.1%

BW

30 mm 25 mm 20 mm

Figure 11. Tuning curves for different undulator period lengths. Note that for short periods (red curve, 20 mm) there is not a continous curve from 1 keV to 10 keV. Note also that for long periods (brown curve, 30 mm) the obtained brightness at 10 keV is significatively lower than that obtained with short periods. The total magnetic length in the three cases is ca. 2 m.

It is clear that to achieve high brilliances at high energies, we have to go down to short

undulator period lengths. However, we impose the requirement that a continous tuning curve can be obtained during the operation of the undulator. Thus, smallest periods should be avoided.

The smallest period giving a continous tuning curve is 24 mm, as shown in Figure 12. This is the selected undulator period length.

Gap =

Period lenght =

16

1.E+16

1.E+17

1.E+18

1.E+19

1.E+20

100 1000 10000 100000

Energy (eV)

On

axis

bril

lianc

e at

cen

tre

Ph/s

/mra

d2̂/

mm

^2/0

.1%

BW

Figure 11. Tuning curve for the selected period and gap variation, calculated using maximum field given by Radia (non optimized central cell, without passive and-poles correction).

With regards to the maximum gap, we choose a value of 25 mm, which is enough to ensure a wide scan between 1 keV and 20 keV.

17

Centra l period opt imization

A. Parameter definition and simulation details

Once the period length uλ is selected, 5 parameters define the central cell:

• L: PPM length in the z direction.

• H: PPM height in the y direction.

• Hp: pole height in the y direction.

• G: PPM width in the x direction.

• Gp: pole width in the x direction.

Pole overhanging is not allowed. Figure 13 shows these parameters and the coordinate system:

To find the most appropriated values for these parameters, an optimization has to be done. The optimization procedure is described elsewhere,8 and uses the simplex algorithm. We use opera-3d to simulate the magnetic field within the gap. Figure 14 shows a pole and a ppm piece as they are modeled in the magnetic field simulator. We make use of the multiples symmetries to reduce the time of the numeric computation. First the up-down symmetry, which allows to simulate only the upper array of magnets, and y = 0 is set as a boundary of the model with Bx = Bz = 0. Left-right symmetry is also used, allowing to simulate only the right side. To this end, x = 0 is set as a boundary of the model with Bx = 0. The periodicity can be simulated by introducing two symmetry 8 Z.Martí, J.Campmany, M.Traveria “New improvements in 2d magnet pole design”, RSI, in press.

L Gp

G

Hp H

x

y

z x

y

z

Figure 13. Different views of the blocs arrangement in the central region. The main parameters defining the blocs are shown. Pole is in dark blue and ppm in light blue.

18

planes perpendicular to the beam axis (z axis): the center of the pole and the center of the ppm. The center of the pole (z = 0) is chosen as boundary of the model with Bz = 0. The center of the ppm ( 4/uz λ= ) is chosen as boundary of the model with Bx = By = 0. See Figure 14.

In these simulations homogeneous non linear materials are used. For the pole we use Permendur (the Opera’s HB magnetization curve is used), and for the ppm we use NdFeB alloy (the Opera’s HB magnetization curve is used).

Figure 15. Half a pole bloc and half a ppm bloc, as they are simulated in Opera-3d.

x y z

Figure 14. A pole bloc and a ppm bloc, the non transparent part, with a non dashed line, is the simulated part.

19

B. The error function

In order to optimize the central cell dimensions, we define an error function. This function is a figure of merit of the quality of the field. It is composed of many functions. All of them are described in this section.

B.1 The effective field

To enlarge as much as possible the spectrum range of the undulator, one needs a peak field as big as possible at minimum gap. However, as the trajectory of the electrons acts as a low-pass frequency filter (double integral), the important magnitude is not the peak field but the effective field (Beff),9 that takes into account this effect.

∑≥

+ ⎟⎠⎞

⎜⎝⎛

+=

1

212

12i

ieff i

BB [5]

B2i+1 are the harmonic components of the magnetic field, because we consider the magnetic field along the central path as a periodic function. In the real case it is not true for two reasons:

1. The undulator is finite; the periodicity is broken at the undulator ends.

2. The blocs present defects and errors in magnetization and size. These defects break the periodicity.

(1) is the subject of the next chapter and (2) is not in the scope of this report, but it is maintained under control using good-quality magnetic elements with the specifications shown in chapter 3, Table 3. The contribution of the magnetic field to de figure of merit is given by:

effBF −=1

This means that higher the field, lower the value of the function to be minimized.

B.2 Field quality

One important requirement is that within a region transversal to the beam, the magnetic field be homogeneous enough. Electrons traveling in that region should produce light spectra closer to ∆λ/λ = 10-3, this quantity given by the optical specifications. To know what this means in terms of the magnetic field, we use the expression [3]:

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+=

21

000511.0/2

2

2

KEn

Uλλ [3]

Then, we obtain:

λλ∆+

=∆

2

2

0

0 21

K

K

BB

[6]

9 The ALS Insertion Device Group U10.0 Undulator Conceptual Design Report, Lawrence Berkeley laboratory internal report, PUB-5390 UC-410.

20

We have, in the worst case: at minimum gap (maximum K value), the relative error in the magnetic field should be below 10-3 approximately (K ≈ 1). At maximum gap (minimum K value) the coefficient (1+K2/2)/K2 is quite big, it is about 14 in our case, then the magnetic the magnetic error should be below the 10-2 limit. We do not know much about the field quality dependence when varying the gap, but it is well known that the quality increases with decreasing the gap. In addition, the final quality depends also in the blocks magnetization homogeneity; this will decrease the quality of the simulated field. To play safe, we impose a field quality of 10-3 at maximum gap; which is the required quality at minimum gap. In this way, whatever the quality dependence with the gap, the quality of the simulation will fulfill the specifications for all gaps. Figure 16 shows schematically this strategy.

This implies that each time the error function is evaluated we have to make two simulations, one simulation at minimum gap, from which the peak magnetic field is evaluated, and another at the maximum gap where the magnetic field quality is evaluated.

We have set the magnetic field quality border, but we haven’t set yet in which region this has to be fulfilled. The quality region is an area transversal to the beam motion. This region is related with the beam size. The vertical size is negligible (σy = 6 µm), but the horizontal size (σx = 138 µm) have to be considered. As region of field quality we choose ∆x = ± 2 mm, this sufficiently contains the beam.

Now we have all the elements to define mathematically the contribution of the field quality to the error functiont:

BB

eF ∆∆−

=δ20

2

Here δ is the magnetic field quality at y = 0, z = 0, x = 2 mm at maximum gap, in our case 25 mm, and ∆B = 10-3. The numerical factor 20 in the exponent is set to have a very sharp edge in the parameters space, in order to avoid solutions with δ > ∆B.

1E-4

1E-3

0,01

∆B/B

desired design requirements

minimum gap maximum gap

quality specifications

gap

Figure 16. During the optimization, field error at maximum gap is below 10-3. In this way the final design accomplish by far the specifications at all gaps.

21

B.3 Geometry constrains

This design is for an in vacuum undulator, the volume of the device is an important issue. Thus a constraint in the blocks height and width should be included. The limitations are detailed below:

1. H, Hp < 15 mm.

2. G,Gp < 50 mm.

To produce solutions always accomplishing these limitations, the term to be introduced in the error function is:

XXG

XXG

YYH

YYH pp

eeeeF ∆

∆−

∆∆−

∆

∆−

∆∆−

+++=20202020

3

Here ∆Y = 15 mm and ∆X = 50 mm. This term acts as a wall that maintains the parameters within the constraints.

To summarize, the expression of the error function is:

321 FFFF ++=

Once the object function has been mathematically defined, we start the optimization procedure.

C. Results.

Using this methodology and Opera 3D simulation code, we obtain the following optimized block dimensions:

hp l h G Gpole

10.799430 8.916256 14.399174 43.537815 31.617540

Using these dimensions of the central cell, the maximum field inside the gap is B0 = 1.1293 T. This is the optimized value, significatively higher that that shown in Figure 10 for the non-optimized case, where B0 = 1.0170 T.

The Effective field as defined in [5] is Beff = 1.026 T

22

End poles des ign

Once the parameters of the central block are chosen, the non periodic part of the structure has to be considered. This is a crucial part of the design since determines the field integrals. In absence of magnetic and mechanical errors, a periodic structure does not add any field integral, but a finite structure it does. Here we try to implement the best possible passive integrals correction for a hybrid undulator. In the ideal case, for all gaps the first and second integrals will be zero and the averaged trajectory is zero at the central part.

To decrease as much as possible the field integrals for all gaps, an optimization has to be done of the end geometry. To run an optimization, some parameters defining the geometry of the ends are needed, and a model should be proposed. For each proposed model we run an optimization. The procedure is described elsewhere,10 and uses the simplex algorithm.

In this case the parameters depend on the particular geometrical model we choose. Here we present the methodology for a particular model which has given the best results of the tested models.

A. Parameter definition and simulation details

We consider a model that adds one additional ppm blocs and two pole pieces at each branch end. Figure 16 shows schematically the geometry used. This model is based in other well known geometries.11

10 Z.Martí, J.Campmany, M.Traveria “New improvements in 2d magnet pole design”, RSI, in press. 11 J. Chavanne, P. Ellaume, P. Van Vaerenbergh segmented quality undulators PAC 1995, Dallas, Texas, USA.

Figure 16: Scheme of the end design. Pole material is represented by blocks in grey, and ppm are represented by white blocks.

L2

Lx L1

H1

y

z

23

In this case the magnetic field and its integrals are calculated with Radia. In the simulation, eight periods are considered plus some additional blocs at the ends. Figure 17 shows the Radia model with the used end-pole design.

In these simulations homogeneous non linear materials are used. For the pole we use Permendur (the Radia’s HB magnetization curve is used), and for the ppm we use Nd Fe B alloy (the Radia’s HB magnetization curve is used).

B. The error function

By construction, if the second integral I2 = 0, then the first integral I1 = 0. So, the target is to minimize I2. The second field integral gives the electrons displacement, )(2 sI xδ= :

( ) ( )∫ ∫∞−

′

∞−

′′=s s

ye

x sdsdsBcm

esγ

δ

Here s and s’ represent the longitudinal coordinate (in the direction of nominal electron trajectory). δx gives an idea of the goodness of the field, but to avoid the oscillations and make the optimization algorithm run faster and safer, we consider the averaged trajectory T(z). This magnitude has also been used elsewhere.12

( ) ( )∫=+

−

2/

2/

1 u

u

z

zx

u

dsszTλ

λδ

λ

Figure 18 shows schematically the comparison between T(z) and δx(s).

12 Ingvar Blomqvist Magnetic Design of a 20 mm Hybrid Undulator for CLS. Proceeeding PAC 2003.

Figure 17. Radia modelization of the undulator used to optimize the pole ends. It has 16 periods.

24

The optimization should decrease the offset of the trajectory at the central part. Thus, we define the function to be minimized as:

( )∫=−

u

u

dzzTFu

endλ

λλ

4

481

C. Results.

Using this methodology and Radia simulation code, we obtain the following optimized end pole dimensions (in mm):

Lx L1 L2 H1

1.69 5.30 5.43 13.02

With this pole-ends for the passive correction, the variation of the field integrals when the gap is scanned are shown in Figure 19.

-400

-200

0

200

400

600

800

1000

1200

0 5 10 15 20 25

Gap aperture (mm)

Seco

ng in

tegr

al (G

·cm

2)

-1

1

3

5

7

9

11

13

15

Firs

t int

egra

l (G

·cm

)

Second integral First integral

Figure 19. First and second integral values versus gap aperture, using the optimized end-poles for passive correction.

Figure 18. Comparison between T(z) (dashed line) and δx(z)(continuous line). Note that this is not a scaled drawing. The oscillations in δx(z) have an amplitude of some microns, whilst the offset of T(z) is in the range of nanometers.

z

x

25

Mechanical tolerances

1. Energetic tolerance: we accept energy deviations of emmited light within the undulator line

bandwidth.

σ2σ2

The total line bandwidth is a contribution from 3 factors: diffraction, energy spread and divergence. We define the undulator line bandwidth as:

222φσσσσ ++= ED

Where:

We take ∆E/E = 10-3, n = 9, N = 80, K = 0.144, γ = 5.9·103, βx = 5.3, these values corresponding to the most stringent lattice considered for ALBA up to now. Within these values, we consider for calculations:

Worst case: εx = 1,8·10-9 Best case: εx = 5,2·10-9

Contribution to bandwidth Worst case Best case

Diffraction σD = 1.39·10-3 σD = 1.39·10-3

Energy spread σE = 2·10-3 σE = 2·10-3

Beam divergence σφ = 6·10-3 σφ = 2·10-2

Then, the maximum tolerable ∆E/E = 2σ = 1.3·10-2 (worst case) or 3.3·10-2 (best case).

Diffraction

nNEE

D2

=∆

=σ

Energy Spread

e

eE E

EEE ∆

=∆

= 4σ

Beam divergence

X

X

KEE

βεγσ φ 2

21

2

1 +≅

∆

26

Energy dependence from mechanical errors Energy of nth peak:

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

21

95.02

2

KnEE M

n

λ

Where n = 9, λ = 2.5 cm, EM = 3.0 GeV, and K = 0.93·λ·Β1 To compute the errors, we take the approximate expression giving the maximum field B1 inside the gap for a hybrid undulator,13 is:

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛+−

=

2

·1λλgcgb

eaB Where a = 3.33, b = 5.47, c = 1.88 for SmCo5 with Br = 0.9 T a = 3.44, b = 5.08, c = 1.54 for NdFeB with Br = 1.1 T and g = 5 mm is the gap, Or, for a PPM structure,

λπ

λπ

ππ gh

r eeM

MBB−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

2

1 1/

)/sin(2

Where M = 4, number of poles per period h = 1.5 cm, magnetic block height Br = 0.9 T for SmCo5 blocks and Br = 1.1 T for NdFeB blocks and g = 5 mm is the gap Thus, deriving and isolating ∆g:

EE

BBg ∆+

=∆πλ

λλ

21

22

21

22

93.093.0·5.01

for a PPM structure

EE

gcbBBg ∆

+−

+=∆

λ

λλ

λ

293.093.0·5.01

21

22

21

22

for hybrid structure

Filling the values

Worst case Best case PPM 83 µm 214 µm

Hybrid 44 µm 110 µm

13 Richard P. Walker, Insertion Devices: undulators and wigglers, Synchrotron Radiation and free electron lasers, CERN 98-04, CERN accelerator School 1996, S. Turner Ed., p. 152.

27

List of specif icat ions

As a conclusion, for mechanical design, the specifications to be followed are:

1. Magnetic structure: hybrid

2. Period: 24 mm

3. Undulator length: 2 m

4. Minimum gap: 5 mm

5. Maximum gap: 25 mm

6. Maximum temperature on magnets: 125°C

7. Working temperature: 20°C

8. Maximum magnetic field within the gap: 1.13 T

9. Material used for ppm: NdFeB

10. Material used for poles: Permendur

11. Vacuum conditions: 10-8 Pa

12. Field quality: ∆E/E < 10-3 within 2 mm horizontal

13. Dispersion of magnetization of blocks (module): < ±1.5%

14. Dispersion in the direction of magnetization: < ±1.5°

15. Total mechanical allowance in the vertical direction: 40 µm

16. Allowance for rolling of inner beam: ±0.025 µm