Dipole current, bending angle and beam energy in bunch compressors at TTF/VUV-FEL P. Castro August 26, 2004 Abstract We apply a rectangular (hard edge) magnetic model to calculate the trajectory in the bunch compressor 2 and 3 in TTF/VUV-FEL [1]. The parameters of the model are based on magnetic measurements. In this paper, we analyse these magnetic measurements and present equations that describe the relationship between beam energy, beam position and dipole current in both bunch compressors. 1 Introduction: dipole magnets as spectrometers The trajectory of a charged particle in a magnetic field (perpendicular to the trajectory) describes an arc of radius r given by 1 r = e B (1) p where B is the field component perpendicular to the trajectory, e is the charge and p is the momentum of the particle. Thus, the trajectory of the particle traveling through a dipole magnet changes its direction by an angle of α = ds B ds r = e p where the integral goes along the trajectory. A measurement of the particle momentum can be obtained from the measurement of trajectory angle change α and the measurement of the magnetic field integral along the trajectory. In terms of typical units used in particle accelerators we can write B[T] ds[m] p[mev/c] 299.8 (3) α[rad] In the case of ultra-relativistic particles, one uses the approximation E = pc (where c is the light velocity) to replace p by E (2) E[MeV] 299.8 1 B[T] ds[m] α[rad] (4)
A hard edge model of the magnetic field avoids the numerical integral and simplifies the calculation of the particle trajectory. This model is explained in section 2 and is used in section 3. 2 Magnetic measurements of dipoles Bunch compressor 2 (BC2) and 3 (BC3) consist of dipole magnets of the so-called type TDD 1 produced by DANFYSIK [2]. The longitudinal position z, name in the linac and serial number of each magnet is shown in Table 1. The magnetic measurements presented here have been carried out at DESY by Y. Holler et al. The measurement data can be found in the EDMS at DESY [3]. name z[m] serial number D1BC2 21.14 97190 D2BC2 22.14 97189 D3BC2 23.60 97192 D4BC2 24.60 97191 D1BC3 67.58 02098 D4BC3 70.46 02100 D5BC3 71.46 02096 D10BC3 77.22 02094 D11BC3 78.22 02097 D14BC3 81.10 02095 Table 1: List of dipole magnets installed in BC2 and BC3 including its name in the linac, the longitudinal position of its center and its serial number. Measurements of the longitudinal magnetic field profile of dipole TDD-02099 (which is kept as spare magnet) are shown in Fig. 1 for several currents. In order to simplify the calculations, we use the so-called hard edge approximation. That is, we approximate the magnetic field profile to a rectangular field profile, whose strength is the same as the strength of the magnetic field in the center of the dipole B o. The length of the rectangular field profile (so-called effective length l eff ) is calculated so that the field integral remains the same, so B(z) dz l eff = (5) B o Longitudinal profile measurements have been carried out also for dipole TDD-02098 with currents of 50, 80 and 100 A and for ten other dipoles with currents of 50 and 100 A. The effective length obtained from all these measurements are plotted in Fig. 2 as function of the magnet current. Based on these results, we choose l eff = 0.501 m for all currents, which introduces a maximum error of 0.2% for 10 A and 120 A. 1 T stands for TTF or TESLA, D for dipole and the second D for DANFYSIK. 2
measured magnetic field [T] 0.5 120 A 100 A 0.4 0.3 0.2 50 A 0.1 10 A 0-600 -400-200 0 200 400 600 longitudinal position z [mm] Figure 1: Measured magnetic field profile of dipole TDD-02099 along the longitudinal axis z. z = 0 refers to the center of the magnet. Measurements of the field strength (at the center of the magnet) as function of the magnet current have been performed in regular steps of 5 A from 0 to 100 A and down to 0 A for BC2 dipoles in 1998. Additional measurements have been done for the spare dipole TDD-02099 for currents from 0 to 120 A and down to 0. In order to visualize the hysteresis and saturation effects and to see the differences between magnets, we select the measurements done at D1BC2 and take the line between the magnetic strength measured at 0 A and measured at 100 A as a reference line. The difference of the field strength measurements with respect to this reference line is shown in Fig. 3 as function of the magnet current. We observe that saturation effects start to play an important role at currents above 50 A, therefore, the field strength deviates from a linear current dependence. Hysteresis effects cause a maximum field difference of about 4 mt for the same current and the same magnet. We observe also that the measurements done at dipole D1BC2 (red line) differ from the other three dipoles up to about 2 mt, that is, 1% of the field strength at 50 A. In order to obtain reproducible magnetic fields, the dipole magnets in BC2 and BC3 are cycled first to zero, then to 100 A and finally set to its target value. Immediately after, the trim coils are set to the values presented in [4] to compensate the field strength differences between dipoles. The current of the trim coils is arranged so that 1. the field in the magnet decreases, and therefore stays on the same branch of the hysteresis curve (the upper one in Fig. 3) 3
effective length [mm] 502.5 502 501.5 501 500.5 500 0 20 40 60 80 100 120 magnet current [A] Figure 2: Effective length (Eq. (5)) obtained from longitudinal profile measurements for all 11 dipole magnets of TDD type. 2. the field of all magnets are equal to the field of the weakest magnet (D1BC2 in BC2 and D4BC3 in BC3). Therefore, we select only the measurements done for dipole D1BC2 which present the smallest field strength for any given current. Moreover, we select a range of currents between 0 and 90 A for interpolating the measurements points with a 3rd order polynomial B = a 0 + a 1 I + a 2 I 2 + a 3 I 3 (6) Saturation effects are dominant for a current larger than 90 A and forces to use a polynomial fit of higher order which becomes impracticable. Additionally, the working range for BC2 is between 40 and 80 A (for beam energies between 100 and 140 MeV and for bending angles between 15 and 21 o ) and the working range for BC3 is between 10 and 80 A (for beam energies between 200 and 500 MeV and for bending angles between 1.7 and 5.4 o ). As result of the fit, we obtain the coefficients a 0 = 3.17 10 3 T a 1 = 4.2545 10 3 T/A a 2 = 2.1 10 7 T/A 2 a 3 = 1.21 10 8 T/A 3 (7) 4
field difference [T] 0.01 0.008 0.006 0.004 0.002 0-0.002-0.004-0.006 D1BC2 (TDD-97190) D2BC2 (TDD-97189) D3BC2 (TDD-97192) D4BC2 (TDD-97191) reserve (TDD-02099) 0 20 40 60 80 100 120 magnet current [A] Figure 3: Difference between field strength (measured at the center of the magnet) and a reference line between the measured value at 0 A and at 100 A for dipole TDD-97190 as function of the magnet current. There are two lines for each magnet: the lower line with the measurements done from 0 A to 100 A (or 120 A) and the upper line with the measurements taken in steps from 100 A (or 120 A) to 0 A. In the case that one needs to calculate the magnet current for a given magnetic field strength, it is useful to use inverse function of Eq. (6) in the form of the polynomial with the coefficients I = a 0 + a 1 B + a 2 B 2 + a 3 B 3 (8) a 0 = 0.84 A a 1 = 239.03 A/T a 2 = 29.3 A/T 2 a 3 = 102 A/T 3 (9) Both polynomials presented above describe precisely the measured curve of field strength versus magnet current for dipole D1BC2 (if cycled correctly) and for all dipoles in BC2 if cycled and trimmed. However, if BC3 dipoles are cycled and trimmed with the currents indicated in [4], their field strength is equal to that of D2BC3, which is about 0.32% larger than the field strength of D1BC2 for 50 A in both dipoles. 5
3 Beam energy measurement in BC2 and BC3 A schematic layout of both BC2 an BC3 is shown in Fig. 4 D2BC2 x D3BC2 D1BC2 d D4BC2 D1BC3 d D4BC3 D5BC3 f D10BC3 D11BC3 d D14BC3 Figure 4: Schematic layout (not to scale) of BC2 (top) and BC3 (bottom) viewed from the top. Dipoles are represented as rectangles of length l eff. The beam enters the first dipole D1BC2 (D1BC3) from the left and exits at the dipole D4BC2 (D14BC3). A measurement of the beam position in both bunch compressors is possible after the 2nd dipole with a beam position monitor (BPM) and a screen. Eventually, the beam position can be measured in BC2 by positioning a collimator in the path of the beam. Assuming that the beam has zero offset and angle at the entrance of the bunch compressor, the measured beam position leads to a measurement of the bending angle α of a single dipole, which in turn results in a measurement of the beam energy (as seen in section 1). In the following, we assume that the dipoles have a rectangular magnetic field profile (hard edge approximation) in the longitudinal direction. The trajectory of the beam in BC2 and BC3 is derived in a previous paper [5]. The horizontal displacement introduced by (half of) the bunch compressor is x = 2 l eff 1 cos α sin α + d tan α (10) where d is the length of the field free space between the first and second dipoles. The sum l eff + d is the distance between the centers of the dipoles, which is 1000 mm for BC2 and 2880 mm for BC3. Eq. (10) establishes the relationship between the dipole bending angle and the beam position offset from the linac axis. 6
α r Assuming a model of rectangular magnetic longitudinal profile with a length l eff, the beam trajectory describes an arc inside the dipoles (see adjacent figure) with radius r and length larc given by larc = r α = l eff α sin α (11) where α is the direction change in the beam trajectory after a single dipole. The field integral along the trajectory inside the dipole is B ds = B larc (12) beam larc and the beam momentum can be calculated using Eq. (3) leff p = e B l arc α = e B l eff sin α Alternatively, the bending angle can be determined from the measurement of the so-called time-of-flight of the beam. The path length difference of the beam between its trajectory through the bunch compressor and the straight line along the linac axis is derived in [5]. The path difference in BC2 is given by and in BC3 is l = 4 l eff ( ) ( ) α 1 sin α 1 + 2 d cos α 1 ( ) ( ) α 1 l = 6 l eff sin α 1 + (2 d + f) cos α 1 (15) where f = 5.258 m is the length of the field free space between dipole D5BC3 and D10BC3. The path length difference introduces a time-of-flight delay of the beam arriving at the downstream RF structures which can be measured by a phase monitor. In terms of a 1.3 GHz RF with a wavelength of λ = c/f RF = 230.61 mm, the difference in path length corresponds to a phase shift of and in time units φ[ o ] = 360o λ l 1.56 l[mm] φ[ o ] = 360 o f RF t 0.468 t[ps] 4 Beam energy from beam position measurements A measurement of the beam position in the bunch compressor is done relative to the center of its vacuum chamber, which has a offset of 321.5 mm in BC2 and of 180 mm in BC3 from the linac axis. Thus, centering the beam in the vacuum chamber of BC2 corresponds to x = 312.5 mm, which is obtained with a bending angle of α = 18.036 o from Eq. (10), which introduces a time-of-flight delay of φ = 213.6 o. For this angle, the relationship (13) (14) 7
between magnet current and particle momentum can be obtained using Eqs. (13) and (6) with coefficients of Eq. (7) BC2: p[mev/c] 1.54 + 2.064 I[A] 1.02 10 4 (I[A]) 2 5.87 10 6 (I[A]) 3 (16) To set up the magnet current of BC2 for a given value of beam energy and bending angle, it is useful to use the Eq. (8) with coefficients of Eq. (9) to write Eq. (13) in the form I[A] 0.84 + 0.4927 p[mev/c] 1.24 10 4 (p[mev/c]) 2 + 8.97 10 7 (p[mev/c]) 3 (17) For example, the nominal energy for the commissioning of the TTF/VUV-FEL is 130 MeV in BC2, then we need to set the current to 62.82 A (after ramping up to 100 A) to obtain a bending angle of 18.036 o and, therefore, to center the beam in the bunch compressor. The width of the vacuum chamber allows for angles between 15 o and 21 o approximately. Similarly in BC3, centering the beam ( x = 180 mm) is achieved with a bending angle of 3.577 o (Eq. (10)), which introduces a time-of-flight delay of l = 35.6 o. Thus, centering the beam with the help of the screen or/and beam position monitor installed between these dipoles, one can estimate the beam momentum from the magnet current BC3: p[mev/c] 7.64 + 10.24 I[A] 5.08 10 4 (I[A]) 2 2.9 10 5 (I[A]) 3 (18) To set up the magnet current of BC3 for a given value of beam energy and bending angle, it is useful to use the Eq. (8) with coefficients of Eq. (9) to write Eq. (13) in the form I[A] 0.84+9.93 10 2 p[mev/c] 5.05 10 6 (p[mev/c]) 2 +7.3 10 9 (p[mev/c]) 3 (19) For example, the nominal energy for the commissioning of the TTF/VUV-FEL is 380 MeV in BC3, then we need to set the current to 36.38 A (after ramping up to 100 A) to obtain a bending angle of 3.577 o, that is, to center the beam in the bunch compressor. The width of the vacuum chamber allows for angles between 1.7 o and 5.4 o approximately. 5 Validity of the hard edge model A particle tracking program has been used to investigate the validity of the hard edge model in predicting the position of the beam in the BC2. The field profile measurements done at dipole TDD-02099 with 50 A have been used in the tracking through the first two dipoles. For a particle with p = 104.92 MeV/c the bending angle is exactly 18.0 o (in the tracking), which gives a horizontal displacement of 320.75 mm. For the same momentum and field strength (B = 0.21574 T) Eq. (13) results in a bending angle of 17.989 o and Eq. (10) yields 320.63 mm which represents a difference below 0.04%. 8
6 Summary The relationship between beam momentum (or energy), magnetic field strength and bending angle in BC2 and BC3 is given in Eq. (13). The relation between field strength and magnet current is described in terms of fitting polynomials given in Eqs. (6) and (8). The relation between horizontal beam displacement and bending angle is calculated with Eq. (10) valid for BC2 with d = 499 mm and for BC3 with d = 2379 mm. These relationships are sketched in Fig. 5. p (or E) Eq.(13) B Eq.(6) Eq.(8) I α Eqs.(14),(15) Eq.(10) x Figure 5: Diagram of the relation between quantities involved in the measurement of the beam energy in bunch compressors. A full line symbolizes an analytic equation and a dashed arrow represents a polynomial fit. Taking these expressions we have derived the relation between beam momentum and magnet current for the case of having the beam centered in the vacuum chamber. 7 Conclusions The bunch compressor dipoles can be used as spectrometers with the measurement of the horizontal beam displacement from the linac axis or with the measurement of the bunch arrival time (time-of-flight). The expressions used to calculate the horizontal beam offset and time-of-flight delay are derived in a previous paper [5] using a hard edge model for dipole fields. The use of a hard edge model to calculate the beam trajectory in the bunch compressors is proved to be appropriate. The magnetic properties of the dipoles TDD (used in the bunch compressors) have been analysed to extract a hard edge model. The relation between the magnet current and the magnetic field are described in terms of polynomials of 3rd order fitted to the data, taking into account the hysteresis of the magnet. Magnetic hysteresis may introduce a magnetic field difference of up to 4 mt (about 2% of the field strength for 50 A), therefore, a cycling of the magnets is advised. Magnetic measurements reveal differences in the field strength larger than 1% at 50 A. In order to compensate this difference, we need to use the trim coils with currents calculated in [4]. 9 l
Using all these expressions, we have obtained the equations which provide the beam momentum from the current applied to the magnet and vice versa, in the case of a well centered beam. The analysis of systematic errors derived from errors in the beam position measurement, beam misalignment, etc. have still to be done. 8 Acknowledgments We would like to thank J.P. Carneiro, B. Faatz and Y. Holler for their careful reading of the manuscript and many useful comments. References [1] The TESLA Test Facility FEL team; SASE FEL at the TESLA Facility, Phase 2, TESLA-FEL-2002-01, 2002. [2] DANFYSIK A/S, Mollehaven 31, DK-4040 Jyllinge, Denmark. [3] Engineering Data Management System at DESY; http://edms.desy.de [4] P. Castro; About the use of trim coils in bunch compressors at TTF/VUV-FEL, DESY Technical Note 04-01, August 2004. [5] P. Castro; Beam trajectory calculations in bunch compressors of TTF2, DESY Technical Note 03-01, April 2003. 10