- Email: [email protected]
Magnetic design of a 14 mm period prototype superconducting undulator
Nuclear Instruments and Methods in Physics Research A 846 (2017) 13–17
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Magnetic design of a 14 mm period prototype superconducting undulator a,⁎
a,b,c
Mona Gehlot , G. Mishra
b
MARK
c
, Frederic Trillaud , Geetanjali Sharma
a
Insertion Device Development Laboratory, School of Physics, Devi Ahilya University, Indore 452001, MP, b Institute of Engineering, UNAM, Mexico c Soleil, Paris, France
A R T I C L E I N F O
A BS T RAC T
Keywords: Superconducting undulator Undulator Free electron laser Radia
In this paper we report the design of a 14 mm period prototype superconducting undulator that is under fabrication at Insertion Device Development Laboratory (IDDL) at Devi Ahilya Vishwavidyalaya, Indore, India. The field computations are made in RADIA and results are presented in an analytical form for computation of the on axis field and the field on the surface of the coil. On the basis of the findings, a best fit is presented for the model to calculate the field dependence on the gap and the current density. The fit is compared with MoserRossmanith formula proposed earlier to predict the magnetic flux density of a superconducting undulator. The field mapping is used to calculate the field integrals and its dependence on gap and current densities as well.
1. Introduction In recent years there are interests and advantages of using super conducting undulator (SCU) for synchrotron radiation and free electron laser (FEL) applications over pure permanent magnet (PPM) or hybrid undulator (HU) due to a variety of reasons. Higher magnetic fields allow reducing the undulator lengths which are often required for table top FEL facility. Short period undulators are feasible, but not in the case with PPM or HU structure due to finite size of the magnets and reduced magnetic field strengths. The superconducting undulator is lesser sensitive to radiation damage thus allowing a longer life of operation at smaller gaps. The SCU leads to a simpler K – control through the current flowing in the coils as compared to the massive and difficultly adjustable gap in PPMs and HUs. A superconducting magnet [1–4] is built using coils wound with superconducting commercial wires. They are cooled down to cryogenic temperatures between 1.8 K and 6 K typically. At this temperature range, they can produce stronger magnetic fields than ordinary iron-core electromagnets due to the ability to carry larger current densities without electrical losses. In superconducting undulators, the magnetic field is created by a pair of identical electromagnets wound on ferromagnetic cores. The two poles are separated by a gap. Each electromagnet is a series of racetrack coils with alternating current pattern to create an undulatory magnetic field on axis. Several superconducting undulators were built around the globe and operated in liquid helium at 4.2 K [5–16]. Over the years, the interests and efforts have grown on using superconducting technology on developing advanced SC undulator schemes such as a transverse gradient superconducting undulator [17] and elliptically polarized ⁎
undulator as well [18,19]. Wallen et al. [20,21] using RADIA model reported calculation of magnetic flux density and field integrals of an SCU suitable for installation at the ESRF storage ring. In this paper we follow the software package RADIA to model a 14 mm period proto-SCU at IDDL, DAVV, Indore, India. In Section 2, the magnetic design layout is detailed with the end field scheme. In Section 3, the field computations are made and results are presented in an analytical form for computation of on axis field and the field on the surface of the coil. An empirical fit is obtained and compared with Moser-Rossmanith formula for the magnetic flux density at different current densities in Section 3. The field mapping is used to calculate the field integrals versus gap and current densities as well. 2. Mechanical design & RADIA modeling It is proposed to build a 10 periods, 14 mm long each period protoSCU at DAVV for field integral measurement studies. The superconducting magnet will be composed of racetrack coils connected in series and wound on two ferromagnetic poles made of carbon steel. To produce the undulatory field on axis, from one coil to the next, the current direction is required to be inverted. It will be done by alternating the winding for each adjacent coil. Superconducting commercial NbTi wire with a cross section of 1 mmx0.5 mm including its insulation are used in the calculation. The SCU will consist of 26 poles and 25 coils which are numbered from 1 to 51. Fig. 1 shows 3D view of the mechanical design, without coil packs of the superconducting undulator. The regular pole is 2 mm
Corresponding author. E-mail address: [email protected] (M. Gehlot).
http://dx.doi.org/10.1016/j.nima.2016.11.070 Received 11 September 2016; Received in revised form 29 November 2016; Accepted 30 November 2016 Available online 07 December 2016 0168-9002/ © 2016 Elsevier B.V. All rights reserved.
Nuclear Instruments and Methods in Physics Research A 846 (2017) 13–17
M. Gehlot et al.
Fig. 1. SCU structure without coil packs.
in length (longitudinal direction), 40 mm in width (horizontal direction) and 8 mm in height (vertical direction). The regular coil length with five turns is 5 mm (5 turnsx1 mm) and the coil height with 16 layers is 8 mm (16 layersx0.5 mm). The coil width is same as the pole width. The undulator begins with a pole and runs with pole-coil–pole arrangement numbered from 1 to 51 and ends with a pole in an asymmetric field configuration. The poles-coils numbered from 5 to 47 are regular in size. The end field configuration in our scheme is 1:3/4:1/4. The end poles–coils are numbered as 1-2-3-4 at the left end and numbered as 48-49-50-51 at the right end (Fig. 1). The end pole at 1 is 1.6 mm in length and pole 3 is 1.96 mm in length. The coils numbered as 3 and 4 are 5 mm in longitudinal length. The pole 1 and 3 is 2 mm (¼) and 6 mm (¾) in height respectively. The coil 2 will be 2 mm in height (0.5 mmx4) and coil No. 4 will be 6 mm in height (0.5 mmx12). The right end of the SCU has similar end-design. The total length of the magnetic structure (22 regular poles=44 mm, 21 coils=105 mm, end design=2×13.56 mm (2x(1.6 mm+5 mm+1.96 mm+5 mm))) reaches a total length of 176.12 mm. Fig. 2 presents a longitudinal view of the complete superconducting undulator assembly. The structure will be held by a magnet support stand in an adjustable gap ranging from 5 to 8 mm. The groove dimension is 5 mmx8 mm. The core grooves and the pole widths will be machined within a tolerance of 10 µm. The flatness of the grooves will be kept within 10 µm as well. The dimensions of the poles and coils are used in RADIA to estimate the performance of the proto-SCU. The magnetic flux density at a gap of 3–11 mm is plotted in Fig. 3 for a current density of 800 A/ mm2. The analysis predicts a field of > 1 T at 5 mm gap. The integrals i.e.
I1 =
∫0
z
By(ξ )dξ,
I2 =
∫0
Fig. 3. Magnetic flux density versus longitudinal position.
−e 0.298 −1 −1 = T m , with E is in GeV γmc E
(2)
The first field integral and the second field integral versus gaps and current densities are evaluated and plotted in Figs. 4–7. In Fig. 8 we calculate the magnetic flux density at 5 mm gap at different current densities. In Fig. 9, on-axis magnetic flux density versus gap is plotted for current densities from 700 to 1800 A/mm2. In Fig. 10, the on-axis magnetic flux density versus current density is plotted for different gaps. In Figs. 11 and 12 the calculations are made at the surface of the coil.
3. Results & discussion
z
I1(ξ )dξ
(1)
The design details of a proto-SCU structure have been discussed. The code RADIA has been used extensively for the computation of the
are called first and second field integrals, respectively and are the quantities of interests for design of a good quality undulator. These integrals are proportional to the angular and change of position of the electron beam at the undulator exit. The above equations when multiplied by -e/(γmc) gives the angular and trajectory offset. Setting -e/mc=565 T−1 m−1and γ=1957E (GeV), we get,
gap 5mm 8mm 11mm
first integral(Tmm)
2
0
-2
-4
-200
-100
0
100
200
300
z(mm) Fig. 4. First field integral for different gaps at current density of 800 A/mm2.
Fig. 2. SCU structure with coil packs.
14
Nuclear Instruments and Methods in Physics Research A 846 (2017) 13–17
M. Gehlot et al.
2.4
gap
Gap=5mm
5mm 8mm 11mm
j (A/mm )
1.6
700 800 900 1000 1200 1400
-50
0.8
B (T)
second integral(Tmm)
0
-100
0.0
-0.8 -150
-200
-100
0
100
200
-1.6
300
z(mm) Fig. 5. Second field integral for different gaps at current density of 800 A/mm2.
-2.4 -180-150-120 -90 -60 -30 0 30 60 90 120 150 180 210 240 270 300 330
z(mm)
2
je(A/mm )
4
2
6.0
on axis
5.5
j (A/mm ) 700 800 900 1000 1200 1400
5.0
0
4.5 4.0
(T)
-2
B
First integral(Tmm)
Fig. 8. Magnetic flux density versus longitudinal position at several current densities.
700 800 900 1000 1200 1400
-4
3.5 3.0 2.5 2.0 1.5
-6 -180-150-120 -90 -60 -30 0
1.0
30 60 90 120 150 180 210 240 270 300 330
0.5
z(mm)
0.0
Fig. 6. First field integral with different current density at 5 mm gap.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
gap(mm) 2
je(A/mm ) 700 800 900 1000 1200 1400
2
second integral(Tmm )
0
Fig. 9. Magnetic flux density versus gap at different current density. 2.5
on axis
gap 5mm 8mm 11mm
2.0
1.5
B
(T)
-100 1.0
0.5
-200 -200
-100
0
100
200
300 0.0
z(mm) Fig. 7. Second field integral for different current density at 5 mm gap.
-200
0
200
400
600
800
1000
1200
1400
1600
1800
2000
j (A/mm )
magnetic flux density of the SCU device. The calculations are done with λu = 14 mm and pole length of 2 mm. The results for magnetic field density in Fig. 9 and Fig. 10 are presented by an empirical fit formula with a, b, c coefficients similar to a hybrid undulator structure [22]. The results fit an analytical formula
Fig. 10. Magnetic flux density versus current density at different gaps.
Baxis(T ) = a(Je )exp[−b(Je )g + c(Je )g2] a(Je ) = 2.018 + 0.0031Je 15
(3)
Nuclear Instruments and Methods in Physics Research A 846 (2017) 13–17
M. Gehlot et al. 6.0
on surface of the coil
j (A/mm )
5.5
4.5
5 4
(T)
5.0
4.0
B
B (T)
6
700 800 900 1000 1200 1400
3.5
2
3.0
1
2.5 2.0 2
4
6
8
10
12
j (A/mm )
0
gap(mm)
Fig. 11. Magnetic flux density versus gap at the surface of the coil. 4.4
on surface of the coil
4.0
gap 5mm 8mm 11mm
3.6 3.2
(T)
2.8
B
3
0 1800 1600 1400 1200 1000 900 800 700 13 12 11 10
8
9
7
6
5
4
3
gap(mm) 1.4
2.4 2.0
1.2
1.6
1.0
700A/mm 800A/mm 900A/mm
1.2 0.8
Δ B(T)
0.8 0.4 0.0 -0.4 -200
0
200
400
600
800
0.4 0.2
1000 1200 1400 1600 1800 2000
j (A/mm )
0.0
Fig. 12. Magnetic flux density versus current density at the surface of the coil.
-0.2
b(Je ) = 0.24731 + 0.10436exp( − 0.0012Je )
-0.4 3
c(Je ) = 0.00142 + 0.00682exp( − 0.0012Je )
4
5
6
7
8
9
10
11
12
gap(mm)
In the formula g is in mm and current densities are in A/mm2. Moser et.al. [5] reported an empirical fit through SRW developed by ESRF to calculate the magnetic flux density from an analytical formula for the SC structure. The model compares the results with the field obtained from that of the pure permanent magnet (PPM) undulator. The present empirical formula is compared with Moser formula and the results are presented in Fig. 13a for the range of the current densities and undulator gaps of our interest. It is observed that the Moser result over estimates the flux density at smaller gaps in comparison with the present calculation. However at wider gaps the present formula (Eq. (3)) calculates higher magnetic flux density than that of the Moser formula as seen in Fig. 13b where we define ΔB(T ) = BMoser − BMona where BMona is calculated from Eq. (3). The difference is about 1400 Gauss for a gap of 3 mm and the difference drops to 300 Gauss for a gap around 8 mm. The magnetic flux density at the surface of the coil is another important design issue. The results in Fig. 11 and Fig. 12 can be seen through an empirical fit as,
Bcoil (T ) = A1(Je ) + A2 (Je )exp[ − B(Je )g]
0.6
Fig. 13. a. Comparison of peak field on axis as a function of gap and current density, solid for Moser formula & blue for present result. b. Difference between Moser formula and present result. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
(4)
A1(Je ) = 1.02017 + 0.00155Je
Fig. 14. First and second field integral versus current densities.
A2 (Je ) = 0.93484 + 0.00155Je design tool of the magnetic structure. For an unbalanced PPM magnet structure the gap dependence of the first field integral exhibits a bump like pattern with a sharp decline at the low gap side and a slow decay
B(Je ) = 0.4807 + 0.12691exp( − 0.0015Je ) The evaluation of the field integrals versus gap is an important 16
Nuclear Instruments and Methods in Physics Research A 846 (2017) 13–17
M. Gehlot et al.
1.42 T. The superconducting magnetic structure will be operated at ≈1000 A/mm2. The straightness of the electron trajectories, phase error and shimming [24–26] is an issue and option.
Table 1 Variation of magnetic flux density, first and second field integral with gap. Gap
3 mm
5 mm
8 mm
11 mm
Magnetic flux density –on axis (T) @ 1000 A/mm2 Magnetic flux density –surface of the coil (T) @ 1000 A/mm2 First field integral (T mm) @ 1000 A/ mm2 Second field integral (T mm2) @ 1000 A/mm2 First field integral (T mm) @ 850 A/ mm2 Second field integral (T mm2) @850 A/ mm2
2.25
1.42
0.709
0.36
3.25
2.78
2.61
2.58
Acknowledgment This work is supported by SERB-DST grant -EMR/2014/00120. Dr. Mona Gehlot acknowledges the financial support from UGC No. F.151/2014-15/PDFWM-2014-15-GE-MAD-26801(SA-II). Delhi, and Govt. of India to carry out the simulation study presented in the manuscript. Frederic Trillaud thanks the DGAPA of the National Autonomous University of Mexico for financial support provided through grant TA100617.
−04
0.30
0.014
0.004
1.17×10
95.54
91.13
86.04
81.14
0.577
0.49
0.41
0.035
164.87
140.9
118.67
104.50
References towards zero for large gaps where magnetic field also approaches zero [23]. The SCU field integrals are computed for different current densities in Fig. 14 at different gaps. It reflects a similar unbalanced magnetic structure behavior. The left axis of the plot corresponds to the value of the first field integral. The right side axis of the plot corresponds to second field integral values. The field integral values are taken at a longitudinal point close to the structure (z=195 mm). The first field integral shows dip around 850 A/mm2. Increasing the current density beyond this point the magnetic flux density increases and decreasing the current density from this value the value of the magnetic flux density decreases. The sudden dip 850 A/mm2 corresponds to imperfect magnetic flux density with unequal slopes at both the sides and to a mismatch end design. By increasing the gap between the coils the magnetic flux density decreases, the slope flattens causing the dip to flatten. The results show a minimum first field integral of 0.01414 T mm (5 mm gap) and 0.00417 T mm (8 mm gap) at ≈1000 A/mm2. This corresponds to 1.68 µ radian and 0.5 µ radian at 5 mm and 8 mm gap respectively. The second field integral shows a minimum value of 10.8 µm and 10.25 μm at 5 mm and 8 mm gap respectively. Both the calculations are done at beam energy of 2.5 GeV. To conclude, Radia modeling of a 14 mm, ten periods SCU is carried out in the present paper. An analytical fit is predicted from the calculated field dependence on the gap and the current density and a comparison of the fit is made with the results by Moser et.al. The dependence of the field integrals versus current density and gap has been determined. A summary of the results is presented in Table 1. The device at IDDL, DAVV is proposed to work at 5 mm gap and can be varied up to 8 mm for a magnetic flux density in the range 0.709–
[1] R.B.Palmer, in: Proceedings of the IEEE Particle Accelerator Conference, SanFrancisco, CA, May 6–9, 1991. [2] A.V. Grudiev, et al., Nucl. Instrum. Methods Phys. Res. A 359 (1995) 101. [3] V.M. Borovikov, et al., Nucl. Instrum. Methods Phys. Res. A 405 (1998) 208. [4] V.M. Borovikov, et al., J. Synchrotron Radiat. 5 (1998) 440. [5] Moser, Rossmanith, Nucl. Instrum. Methods Phys. Res. A 490 (2002) 403. [6] A.Grau, etal., in:Proceedings of IPACTHPC163, Spain, 2011. [7] A.Grau, etal., in: Proceedings of IPACKyoto, Japan, WEPD019, 2010. [8] E.Mashkinaetal., in: Proceedings of EPAC08,Genoa,Italy WEPC 121 [9] E. Mashkina, et al., IEEE Trans. Appl. Supercond. 18 (2) (2008). [10] B.Kostka, R.Rossmanith, in: Proceedings of APAC, Gyeongju, Korea, 2004. [11] S.H.Kim, et al., in: Proceedings of IEEE Particle Accelerator Conference, 2003, pp. 1020–1022. [12] E. Trakhtenberg, et al., Diam. Light Source Proc. 1 (2010) 1. [13] Y. Ivanyushenkov, et al., IEEE Trans. Appl. Supercond. 22 (3) (2012). [14] Y. Ivanyuschenkov, et al., Phys. Rev. Spl. Top. –Accerlerators Beams 18 (2015) 040703. [15] Y.Ivanyuschenkov, et al., in: Proceedings of IPAC Dresden, WEPRO048, Germany, 2014. [16] D. Dietderich, et al., IEEE Trans. Appl. Supercond. 17 (2) (2007). [17] V. Afonso Rodriguez, et al., IEEE Trans. Appl. Supercond. 23 (3) (2013). [18] S.D.Chen, J.C.Jan, C.S.hwang, K.S.Liang, in: Proceedings of the 9th European Conference on Applied superconductivity, EUCAS-09, IOP Publishing [19] S.D. Chen, et al., J. Phys. Ser. 234 (2010) 032006. [20] E. Wallen, J. Chavanne, P. Elleaume, Nucl. Inst. Methods Phys. Res. A 541 (2005) 630. [21] Erik Wallen, Nucl. Instrum. Methods Phys. Res. A 495 (2002) 58. [22] J.Qi-ka, Z.Shancai, L.Shengkuan, H.Duohui, , Proceedings of the FEL Conference, TUPOS38, 2004, p. 494 [23] M.Tischer, J.Pflueger, TESLA-FEL 2000–2012,December, 2000. [24] O.Bilani, et al., in: Proceedings of IPAC, Dresden, TUPRO084, 2014. [25] S.Chunjarean, C.S.Hwang, H.Wiedemann, in: Proceedings of the 2nd International Particle Accelerator conference IPAC-2011, San Sebastian, Spain, sept 4–9, 2011. [26] D. Arbelaez, et al., IEEE Trans. Appl. Supercond. 23 (3) (2013).
17
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima
Magnetic design of a 14 mm period prototype superconducting undulator a,⁎
a,b,c
Mona Gehlot , G. Mishra
b
MARK
c
, Frederic Trillaud , Geetanjali Sharma
a
Insertion Device Development Laboratory, School of Physics, Devi Ahilya University, Indore 452001, MP, b Institute of Engineering, UNAM, Mexico c Soleil, Paris, France
A R T I C L E I N F O
A BS T RAC T
Keywords: Superconducting undulator Undulator Free electron laser Radia
In this paper we report the design of a 14 mm period prototype superconducting undulator that is under fabrication at Insertion Device Development Laboratory (IDDL) at Devi Ahilya Vishwavidyalaya, Indore, India. The field computations are made in RADIA and results are presented in an analytical form for computation of the on axis field and the field on the surface of the coil. On the basis of the findings, a best fit is presented for the model to calculate the field dependence on the gap and the current density. The fit is compared with MoserRossmanith formula proposed earlier to predict the magnetic flux density of a superconducting undulator. The field mapping is used to calculate the field integrals and its dependence on gap and current densities as well.
1. Introduction In recent years there are interests and advantages of using super conducting undulator (SCU) for synchrotron radiation and free electron laser (FEL) applications over pure permanent magnet (PPM) or hybrid undulator (HU) due to a variety of reasons. Higher magnetic fields allow reducing the undulator lengths which are often required for table top FEL facility. Short period undulators are feasible, but not in the case with PPM or HU structure due to finite size of the magnets and reduced magnetic field strengths. The superconducting undulator is lesser sensitive to radiation damage thus allowing a longer life of operation at smaller gaps. The SCU leads to a simpler K – control through the current flowing in the coils as compared to the massive and difficultly adjustable gap in PPMs and HUs. A superconducting magnet [1–4] is built using coils wound with superconducting commercial wires. They are cooled down to cryogenic temperatures between 1.8 K and 6 K typically. At this temperature range, they can produce stronger magnetic fields than ordinary iron-core electromagnets due to the ability to carry larger current densities without electrical losses. In superconducting undulators, the magnetic field is created by a pair of identical electromagnets wound on ferromagnetic cores. The two poles are separated by a gap. Each electromagnet is a series of racetrack coils with alternating current pattern to create an undulatory magnetic field on axis. Several superconducting undulators were built around the globe and operated in liquid helium at 4.2 K [5–16]. Over the years, the interests and efforts have grown on using superconducting technology on developing advanced SC undulator schemes such as a transverse gradient superconducting undulator [17] and elliptically polarized ⁎
undulator as well [18,19]. Wallen et al. [20,21] using RADIA model reported calculation of magnetic flux density and field integrals of an SCU suitable for installation at the ESRF storage ring. In this paper we follow the software package RADIA to model a 14 mm period proto-SCU at IDDL, DAVV, Indore, India. In Section 2, the magnetic design layout is detailed with the end field scheme. In Section 3, the field computations are made and results are presented in an analytical form for computation of on axis field and the field on the surface of the coil. An empirical fit is obtained and compared with Moser-Rossmanith formula for the magnetic flux density at different current densities in Section 3. The field mapping is used to calculate the field integrals versus gap and current densities as well. 2. Mechanical design & RADIA modeling It is proposed to build a 10 periods, 14 mm long each period protoSCU at DAVV for field integral measurement studies. The superconducting magnet will be composed of racetrack coils connected in series and wound on two ferromagnetic poles made of carbon steel. To produce the undulatory field on axis, from one coil to the next, the current direction is required to be inverted. It will be done by alternating the winding for each adjacent coil. Superconducting commercial NbTi wire with a cross section of 1 mmx0.5 mm including its insulation are used in the calculation. The SCU will consist of 26 poles and 25 coils which are numbered from 1 to 51. Fig. 1 shows 3D view of the mechanical design, without coil packs of the superconducting undulator. The regular pole is 2 mm
Corresponding author. E-mail address: [email protected] (M. Gehlot).
http://dx.doi.org/10.1016/j.nima.2016.11.070 Received 11 September 2016; Received in revised form 29 November 2016; Accepted 30 November 2016 Available online 07 December 2016 0168-9002/ © 2016 Elsevier B.V. All rights reserved.
Nuclear Instruments and Methods in Physics Research A 846 (2017) 13–17
M. Gehlot et al.
Fig. 1. SCU structure without coil packs.
in length (longitudinal direction), 40 mm in width (horizontal direction) and 8 mm in height (vertical direction). The regular coil length with five turns is 5 mm (5 turnsx1 mm) and the coil height with 16 layers is 8 mm (16 layersx0.5 mm). The coil width is same as the pole width. The undulator begins with a pole and runs with pole-coil–pole arrangement numbered from 1 to 51 and ends with a pole in an asymmetric field configuration. The poles-coils numbered from 5 to 47 are regular in size. The end field configuration in our scheme is 1:3/4:1/4. The end poles–coils are numbered as 1-2-3-4 at the left end and numbered as 48-49-50-51 at the right end (Fig. 1). The end pole at 1 is 1.6 mm in length and pole 3 is 1.96 mm in length. The coils numbered as 3 and 4 are 5 mm in longitudinal length. The pole 1 and 3 is 2 mm (¼) and 6 mm (¾) in height respectively. The coil 2 will be 2 mm in height (0.5 mmx4) and coil No. 4 will be 6 mm in height (0.5 mmx12). The right end of the SCU has similar end-design. The total length of the magnetic structure (22 regular poles=44 mm, 21 coils=105 mm, end design=2×13.56 mm (2x(1.6 mm+5 mm+1.96 mm+5 mm))) reaches a total length of 176.12 mm. Fig. 2 presents a longitudinal view of the complete superconducting undulator assembly. The structure will be held by a magnet support stand in an adjustable gap ranging from 5 to 8 mm. The groove dimension is 5 mmx8 mm. The core grooves and the pole widths will be machined within a tolerance of 10 µm. The flatness of the grooves will be kept within 10 µm as well. The dimensions of the poles and coils are used in RADIA to estimate the performance of the proto-SCU. The magnetic flux density at a gap of 3–11 mm is plotted in Fig. 3 for a current density of 800 A/ mm2. The analysis predicts a field of > 1 T at 5 mm gap. The integrals i.e.
I1 =
∫0
z
By(ξ )dξ,
I2 =
∫0
Fig. 3. Magnetic flux density versus longitudinal position.
−e 0.298 −1 −1 = T m , with E is in GeV γmc E
(2)
The first field integral and the second field integral versus gaps and current densities are evaluated and plotted in Figs. 4–7. In Fig. 8 we calculate the magnetic flux density at 5 mm gap at different current densities. In Fig. 9, on-axis magnetic flux density versus gap is plotted for current densities from 700 to 1800 A/mm2. In Fig. 10, the on-axis magnetic flux density versus current density is plotted for different gaps. In Figs. 11 and 12 the calculations are made at the surface of the coil.
3. Results & discussion
z
I1(ξ )dξ
(1)
The design details of a proto-SCU structure have been discussed. The code RADIA has been used extensively for the computation of the
are called first and second field integrals, respectively and are the quantities of interests for design of a good quality undulator. These integrals are proportional to the angular and change of position of the electron beam at the undulator exit. The above equations when multiplied by -e/(γmc) gives the angular and trajectory offset. Setting -e/mc=565 T−1 m−1and γ=1957E (GeV), we get,
gap 5mm 8mm 11mm
first integral(Tmm)
2
0
-2
-4
-200
-100
0
100
200
300
z(mm) Fig. 4. First field integral for different gaps at current density of 800 A/mm2.
Fig. 2. SCU structure with coil packs.
14
Nuclear Instruments and Methods in Physics Research A 846 (2017) 13–17
M. Gehlot et al.
2.4
gap
Gap=5mm
5mm 8mm 11mm
j (A/mm )
1.6
700 800 900 1000 1200 1400
-50
0.8
B (T)
second integral(Tmm)
0
-100
0.0
-0.8 -150
-200
-100
0
100
200
-1.6
300
z(mm) Fig. 5. Second field integral for different gaps at current density of 800 A/mm2.
-2.4 -180-150-120 -90 -60 -30 0 30 60 90 120 150 180 210 240 270 300 330
z(mm)
2
je(A/mm )
4
2
6.0
on axis
5.5
j (A/mm ) 700 800 900 1000 1200 1400
5.0
0
4.5 4.0
(T)
-2
B
First integral(Tmm)
Fig. 8. Magnetic flux density versus longitudinal position at several current densities.
700 800 900 1000 1200 1400
-4
3.5 3.0 2.5 2.0 1.5
-6 -180-150-120 -90 -60 -30 0
1.0
30 60 90 120 150 180 210 240 270 300 330
0.5
z(mm)
0.0
Fig. 6. First field integral with different current density at 5 mm gap.
0
1
2
3
4
5
6
7
8
9
10
11
12
13
gap(mm) 2
je(A/mm ) 700 800 900 1000 1200 1400
2
second integral(Tmm )
0
Fig. 9. Magnetic flux density versus gap at different current density. 2.5
on axis
gap 5mm 8mm 11mm
2.0
1.5
B
(T)
-100 1.0
0.5
-200 -200
-100
0
100
200
300 0.0
z(mm) Fig. 7. Second field integral for different current density at 5 mm gap.
-200
0
200
400
600
800
1000
1200
1400
1600
1800
2000
j (A/mm )
magnetic flux density of the SCU device. The calculations are done with λu = 14 mm and pole length of 2 mm. The results for magnetic field density in Fig. 9 and Fig. 10 are presented by an empirical fit formula with a, b, c coefficients similar to a hybrid undulator structure [22]. The results fit an analytical formula
Fig. 10. Magnetic flux density versus current density at different gaps.
Baxis(T ) = a(Je )exp[−b(Je )g + c(Je )g2] a(Je ) = 2.018 + 0.0031Je 15
(3)
Nuclear Instruments and Methods in Physics Research A 846 (2017) 13–17
M. Gehlot et al. 6.0
on surface of the coil
j (A/mm )
5.5
4.5
5 4
(T)
5.0
4.0
B
B (T)
6
700 800 900 1000 1200 1400
3.5
2
3.0
1
2.5 2.0 2
4
6
8
10
12
j (A/mm )
0
gap(mm)
Fig. 11. Magnetic flux density versus gap at the surface of the coil. 4.4
on surface of the coil
4.0
gap 5mm 8mm 11mm
3.6 3.2
(T)
2.8
B
3
0 1800 1600 1400 1200 1000 900 800 700 13 12 11 10
8
9
7
6
5
4
3
gap(mm) 1.4
2.4 2.0
1.2
1.6
1.0
700A/mm 800A/mm 900A/mm
1.2 0.8
Δ B(T)
0.8 0.4 0.0 -0.4 -200
0
200
400
600
800
0.4 0.2
1000 1200 1400 1600 1800 2000
j (A/mm )
0.0
Fig. 12. Magnetic flux density versus current density at the surface of the coil.
-0.2
b(Je ) = 0.24731 + 0.10436exp( − 0.0012Je )
-0.4 3
c(Je ) = 0.00142 + 0.00682exp( − 0.0012Je )
4
5
6
7
8
9
10
11
12
gap(mm)
In the formula g is in mm and current densities are in A/mm2. Moser et.al. [5] reported an empirical fit through SRW developed by ESRF to calculate the magnetic flux density from an analytical formula for the SC structure. The model compares the results with the field obtained from that of the pure permanent magnet (PPM) undulator. The present empirical formula is compared with Moser formula and the results are presented in Fig. 13a for the range of the current densities and undulator gaps of our interest. It is observed that the Moser result over estimates the flux density at smaller gaps in comparison with the present calculation. However at wider gaps the present formula (Eq. (3)) calculates higher magnetic flux density than that of the Moser formula as seen in Fig. 13b where we define ΔB(T ) = BMoser − BMona where BMona is calculated from Eq. (3). The difference is about 1400 Gauss for a gap of 3 mm and the difference drops to 300 Gauss for a gap around 8 mm. The magnetic flux density at the surface of the coil is another important design issue. The results in Fig. 11 and Fig. 12 can be seen through an empirical fit as,
Bcoil (T ) = A1(Je ) + A2 (Je )exp[ − B(Je )g]
0.6
Fig. 13. a. Comparison of peak field on axis as a function of gap and current density, solid for Moser formula & blue for present result. b. Difference between Moser formula and present result. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
(4)
A1(Je ) = 1.02017 + 0.00155Je
Fig. 14. First and second field integral versus current densities.
A2 (Je ) = 0.93484 + 0.00155Je design tool of the magnetic structure. For an unbalanced PPM magnet structure the gap dependence of the first field integral exhibits a bump like pattern with a sharp decline at the low gap side and a slow decay
B(Je ) = 0.4807 + 0.12691exp( − 0.0015Je ) The evaluation of the field integrals versus gap is an important 16
Nuclear Instruments and Methods in Physics Research A 846 (2017) 13–17
M. Gehlot et al.
1.42 T. The superconducting magnetic structure will be operated at ≈1000 A/mm2. The straightness of the electron trajectories, phase error and shimming [24–26] is an issue and option.
Table 1 Variation of magnetic flux density, first and second field integral with gap. Gap
3 mm
5 mm
8 mm
11 mm
Magnetic flux density –on axis (T) @ 1000 A/mm2 Magnetic flux density –surface of the coil (T) @ 1000 A/mm2 First field integral (T mm) @ 1000 A/ mm2 Second field integral (T mm2) @ 1000 A/mm2 First field integral (T mm) @ 850 A/ mm2 Second field integral (T mm2) @850 A/ mm2
2.25
1.42
0.709
0.36
3.25
2.78
2.61
2.58
Acknowledgment This work is supported by SERB-DST grant -EMR/2014/00120. Dr. Mona Gehlot acknowledges the financial support from UGC No. F.151/2014-15/PDFWM-2014-15-GE-MAD-26801(SA-II). Delhi, and Govt. of India to carry out the simulation study presented in the manuscript. Frederic Trillaud thanks the DGAPA of the National Autonomous University of Mexico for financial support provided through grant TA100617.
−04
0.30
0.014
0.004
1.17×10
95.54
91.13
86.04
81.14
0.577
0.49
0.41
0.035
164.87
140.9
118.67
104.50
References towards zero for large gaps where magnetic field also approaches zero [23]. The SCU field integrals are computed for different current densities in Fig. 14 at different gaps. It reflects a similar unbalanced magnetic structure behavior. The left axis of the plot corresponds to the value of the first field integral. The right side axis of the plot corresponds to second field integral values. The field integral values are taken at a longitudinal point close to the structure (z=195 mm). The first field integral shows dip around 850 A/mm2. Increasing the current density beyond this point the magnetic flux density increases and decreasing the current density from this value the value of the magnetic flux density decreases. The sudden dip 850 A/mm2 corresponds to imperfect magnetic flux density with unequal slopes at both the sides and to a mismatch end design. By increasing the gap between the coils the magnetic flux density decreases, the slope flattens causing the dip to flatten. The results show a minimum first field integral of 0.01414 T mm (5 mm gap) and 0.00417 T mm (8 mm gap) at ≈1000 A/mm2. This corresponds to 1.68 µ radian and 0.5 µ radian at 5 mm and 8 mm gap respectively. The second field integral shows a minimum value of 10.8 µm and 10.25 μm at 5 mm and 8 mm gap respectively. Both the calculations are done at beam energy of 2.5 GeV. To conclude, Radia modeling of a 14 mm, ten periods SCU is carried out in the present paper. An analytical fit is predicted from the calculated field dependence on the gap and the current density and a comparison of the fit is made with the results by Moser et.al. The dependence of the field integrals versus current density and gap has been determined. A summary of the results is presented in Table 1. The device at IDDL, DAVV is proposed to work at 5 mm gap and can be varied up to 8 mm for a magnetic flux density in the range 0.709–
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