TRIUMF Document: 10049

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1 TRIUMF Document: DESIGN NOTE TRI-DN Beam Line 2A Optics: Calculations and Measurements Legacy Document Document Type: Design Note Name: Signature: Date: Approved By: N/A Note: Before using a copy (electronic or printed) of this document you must ensure that your copy is identical to the released document, which is stored on TRIUMF s document server Template: Document Rel.1

2 TRIUMF 4004 WESBROOK MALL, VANCOUVER, B.C. V6T 2A3 PAGE 1 of 17 DESIGN NOTE SUBJECT NAME DATE Y.-N. Rao, R. Baartman June 27, 2007 Beamline 2A Optics: Calculations and Measurements FILE NO. TRI-DN Abstract In a recent revisit of 2A front-end optics which was initiated to image the scattering angle in the stripping foil at one of the monitors VM6, we discovered that one of the quads Q4 s excitation had to detune from the theoretical value by 10%, in order to achieve the imaging. This discrepancy invoked us to re-investigate the front-end optics, so as to obtain a better understanding and more accurate description of 2A optics and the extracted beam from the cyclotron. To that end, we performed systematic measurements with beam: we used the cyclotron B r trim coils to move the beam vertically and then measured the shift of beam centroid at the monitors in 2A; also we measured the beam sizes as a function of the stripping foil thickness. From this work, we discovered that the transfer matrix, involved in the optics calculations, between the stripping foil and 2A entrance was incorrect. After correcting this error, we obtained good agreement between the measured and calculated envelopes. We report on the details of this work as well as on a measurement of the beam characteristics as a function of stripper foil thickness.

3 TRI-DN Page 2 of 17 1 Fringe Field Transfer Matrix As we stated in the design note TRI-DN , in terms of the geometry of the extracted reference trajectory of 500 MeV proton in the cyclotron, the whole extraction path of 2A from the stripper through the combination magnet exit can be well approximated with a dipole of an edge angle, followed by a drift. We found this bending angle and radius and the drift length to be 23.38, 6.7 m and 3.0 m respectively. Playing with the edge angle only, we obtained a matrix as following (in cm and mrad): where the edge angle is 59. This matrix closely agrees, in the vertical plane, with the one that is calculated from STRIPUBC; errors are within 1% for all elements: But STRIPUBC calculation involves a fringe field map and this field map is a duplication of beamline 1 extraction port, because there was no fringe field survey for 2A extraction port. It is uncertain that the fringe field is the same for both extraction ports. This suggests that we use this bend-edge-drift approximation in the optics modeling, so that we can treat the edge angle as a fitting parameter. The optics in the vertical plane is very sensitive to the edge angle. 2 Measurements In order to achieve a more accurate description on the edge angle and thereafter the beamline optics, we carried out systematic measurements with beam in the vertical plane because the vertical plane has no energy dispersion involved and also the vertical plane is more sensitive to the edge angle of the fringe field. 2.1 Beam Centroids We used a B r trim coil doublet 48/52 in the cyclotron to move the beam vertically at energy of 500 MeV, and then measured the vertical shift of the centroid of beam down to 2A at monitors VM1,2,3,5 and 6; while the vertical shift of beam in the cyclotron was measured using an internal high energy probe (HE1). To make the measurement clean, 2A was left alone as single user, i.e. 1A foil was taken out, so there was no radial shadowing present; and extra care was taken while moving the beam vertically so that no beam got lost in the tank due to this movement and the same amount of beam current was pulled out to 2A.

4 TRI-DN Page 3 of 17 Fig.1 shows the movement of the vertical profiles measured at the monitors. Fig.2 shows the shift of beam centroid at each monitor as a function of the change in the trim coil 52 excitation (Here we use TC52 s excitation as a reference; in practice, the changes of current in TC48 and TC52 remains at 1/5). Fig.3 shows the HE1 probe measured finger currents vs the excitation of TC52. A change of 60 or 66 Ampere-turn in the trim coil moves the beam centroid from one finger to the next. Each finger is 0.25 inch wide vertically, so we find the coil s driving strength to be ± inch/a.t. in average. For 30, 60 and 90 Ampere-turn changes, the vertical shift of beam centroid at HE1 is respectively 3.02, 6.05 and 9.08 mm. Whereas at 2A stripper location, the shift would be 1.14 times, according to the results of static equilibrium orbit calculation with CYCLOP. 2.2 Beam Sizes We measured beam distributions with profile monitors through the whole beamline and then calculated the rms sizes. This was done with 3 foils of different thicknesses, i.e. 4.43, 2.87 and 1.99 mg/cm 2. As an example, Fig.4 shows the profiles measured at monitor VM6 for these 3 foils. As for the size of the beam on the foil, this was found by differentiating the current as the beam was moved from finger to finger on HE1. See Fig.3; we used F4+F5, plotted in Fig.5 (top), and differentiated (bottom). We did the same exercises with F1+F2, F1+F2+F3 and F3+F4+F5, and obtained distributions that all look similar. In this way, we obtained the 2*rms size of the beam of 0.20 inch(2rms). Since β y 31 m, this implies a 4*rms circulating emittance of 0.8 πmm-mrad. At 2A stripper location, β y 40 m, so the size would be /31 = 0.23 inch. 3 Optics Calculations Beamline 2A consists of 4 dipoles and 16 quadrupoles (of which,however, only 14 are used). All the quadrupoles had been mapped and these maps were used to determine effective lengths and fringe field integrals. The remaining uncertainty is due to hysteresis effects and amounts to no more than about 1%. We first calculated 2A optics with a single particle so as to model the beam centroid movement in the vertical plane. In the calculations, the measured shifts of centroid at the stripper, and at monitors VM1,2,3,5,6, are fitted. We treated, as unknown, the angle of the beam centroid vertically as it leaves the foil, because the trim coil doublet does not only shift the position of the beam, but also changes the angle of the beam, according to the calculation of static equilibrium orbit. Using the data measured at 30 Ampere-turn change, the fit results in a optimum vertical angle of 0.18 mrad, but the fit is not good, as shown in Fig.6. Then, we did the fit allowing the edge exit from the cyclotron magnet to vary as well. The fit is good (See Fig.6), but the edge angle is 68.3, instead of 59. The vertical angle of beam is 0.65 mrad. We then did the same calculations for the 60 and 90 Ampere-turn both cases. As a result, the edge angle is 68.5 for both fits, while the vertical angle of beam is respectively 1.32 and 2.00 mrad. Both fits are as good as for the 30 Ampere-turn case. See Fig.7. In summary, all the 3 fits result in almost an identical edge angle, i.e. 68.5, and also the vertical angle of the beam has almost the same scale factor as the Ampere-turn changes in the trim coils.

5 TRI-DN Page 4 of 17 Figure 1: Measured movement of vertical profile of beam at monitors VM1,2,3,5,6 in 2A, with the change of excitation in Br trim coils 48/52 in cyclotron. Note that the movement on VM6 is within error. The optics was tuned in the experiment for zero movement of beam profile on VM6. Figure 2: Vertical shift of beam centroid at monitors VM1,2,3,5 as a function of the change in B r trim coils 48/52 excitation. The dots represent the measured result where the random error is ±0.2 mm; the dash lines denote a linear fit through the measured points and the zero point as well. All the measured points at each monitor almost stay on a straight line crossing the zero point.

6 TRI-DN Page 5 of 17 Figure 3: HE1 probe measured finger currents vs the excitation of B r trim coils 48/52. A change of 60 or 66 Ampere-turn in the trim coil moves the beam centroid from one finger to the next. Each finger is 0.25 inch wide vertically, so we find the coil s driving strength to be ± inch/a.t. in average.

7 TRI-DN Page 6 of 17 Figure 4: Top: The measured vertical profile of beam on monitor VM6 for 3 stripping foils of different thicknesses. The difference between 4.43 and 2.87 mg/cm 2 foils being relatively larger is because the multiple scattering in the 4.43 mg/cm 2 foil dominates the emittance. Bottom: A comparison between the measured profile and a Gaussian distribution. The core is just a Gaussian and the tails are due to the large angle scattering.

8 TRI-DN Page 7 of 17 Figure 5: Top: HE1 probe measured finger currents sum (F4+F5) vs the vertical position of the beam at 500 MeV; Bottom: vertical distribution of the beam, derived by differentiating the current sum (F4+F5) w.r.t. the vertical position Z. The 2*rms size of the bottom curve is 0.20 inch.

9 TRI-DN Page 8 of fitted vertical shift measured (mm/30a-t) distance (m) fitted vertical shift measured (mm/30a-t) distance (m) Figure 6: Beam centroid shift (in mm) measured vertically up to the monitor VM6 and the trajectory fitted. The vertical lines represent the location of the monitors. The fit with edge angle 59 is not good (top), but the fitted 68.3 angle is satisfactory (bottom).

10 TRI-DN Page 9 of fitted vertical shift measured (mm/30a-t) distance (m) fitted vertical shift measured (mm/60a-t) distance (m) fitted vertical shift measured (mm/90a-t) distance (m) Figure 7: Beam centroid shift (in mm) measured vertically up to the monitor VM6 and the trajectory fitted. The vertical lines represent the location of the monitors. Top: 30 Ampereturn change in TC52. The fit results in an edge angle of 68.3, and the beam vertical angle of 0.65 mrad as it leaves the foil. Middle: 60 Ampere-turn change in TC52. The fit results in an edge angle of 68.5, and the beam vertical angle of 1.32 mrad. Bottom: 90 Ampere-turn change in TC52. The fit results in an edge angle of 68.5, and the beam vertical angle of 2.00 mrad. All the 3 fits are good; they give almost an identical edge angle, i.e. 68.5, and also the vertical angle of the beam has a consistent scale factor with the Ampere-turn changes in TC52.

11 TRI-DN Page 10 of 17 Further, we calculated the optics by doing fits to the beam sizes measured. Since the envelope in the beamline involves the scattering in the foil, we first calculated the scattering angle of the foil. Fig.8 shows the GEANT calculated in-plane scattering angle distribution of 500 MeV H beam penetrating a 5 mg/cm 2 carbon foil 2. Figure 8: Distribution of in-plane scattering angle of 500 MeV proton beam exiting out of a 5 mg/cm 2 carbon foil. red: GEANT calculated result where the rms angle is mrad for angle range of ±2 mrad; green: a Gaussian fit with σ = mrad; blue: another Gaussian fit with σ = mrad. Note that the GEANT distribution (red) has a broader tail than the blue Gaussian. So one can speculate that the rms angle is somewhere between and mrad. What is the range of the scattering angle that we can use to calculate the rms angle? In 2A, the transfer matrix element R 34 arrives at about 2 cm/mrad at maximum, where the beam distribution as measured on the profile monitor is up to ±2 cm wide, namely, we only use the distribution of within ±2 cm to calculate the rms size of the beam. So, the scattering angle is within ±1 mrad. To this end, we made a cut at ±1 mrad in Fig.8, and then calculated the statistics. This results in an angle of 0.35 mrad(2rms). This agrees with the value calculated from Molière s formula. The rms scattering angle is proportional to the foil thickness. We use this to scale the result to foils of other thicknesses. For instance, for 1.99 mg/cm 2, θ scat = mrad(2rms); for 2.87 mg/cm 2, θ scat = mrad(2rms); for 4.43 mg/cm 2, θ scat = mrad(2rms). Using each of these 3 foils, beam profiles in 2A were measured with monitors through 2A. The beam sizes that were calculated from these measured profiles are listed in table 1. We started a fit in the vertical plane only. For the 1.99 mg/cm 2 foil case, we used the measured beam sizes at monitors VM1,2,3,5,6, of values 0.48, 0.68, 0.57, 0.43, 0.24 cm (as listed in Table 1). Also, we used the vertical beam size(2rms) at the stripper of value cm, as described above. We used the scattering angle of mrad. We treated, as unknowns, the circulating emittance and Y of the beam at stripper but before scattering. As for the edge angle of the fringe field, although it has been found out, i.e. 68.5, from

12 TRI-DN Page 11 of 17 Table 1: The 2*rms beam sizes calculated from the profiles measured monitor x and y beam sizes(2rms) (cm) foil: 4.43mg/cm mg/cm mg/cm 2 2AVM1 0.27, , , 0.48 VM2 0.09, , , 0.68 VM3 0.34, , , 0.57 VM5 0.49, , , 0.43 VM6 0.52, , , 0.24 M9 0.56, , , 0.93 M , , , 1.28 M , , , 1.01 M , , , 0.91 M , , , 0.33 M , , , 0.25 M , , , 0.26 M , , , 0.37

13 TRI-DN Page 12 of 17 fits made to the beam centroid, we still treated it as an unknown and allowed it to vary in the fit. The fit is good, as shown in Fig.9; the result is an edge angle of 68.4 which is consistent with the previous result, the circulating emittance of 1.0 πmm-mrad, and Y of 1.2 mrad. Overall, there were 3 fitting parameters: circulating emittance, Y, and edge angle. There were 6 vertical profiles. We also tried a fit with edge angle fixed at 59, but the fit is obviously unsatisfactory. See Fig distance (m) fitted vertical size measured (1.99mg/cm^2) distance (m) fitted vertical size measured (1.99mg/cm^2) Figure 9: Beam sizes (in mm) measured vertically up to monitor VM6 and the envelopes fitted. Top: The fitted edge angle is 68.4, and the circulating emittance ɛ y is 1.0 πmmmrad; Bottom: The edge angle was fixed at 59 in the fit, as a result, the circulating emittance ɛ y is 2.9 πmm-mrad; but this fit is obviously unsatisfactory. Both fits have the scattering angle fixed at mrad. Then, the question is raised: if the edge angle is indeed 68.5 instead of 59, how come, for the past year, the matrix of 59 could always give a good fit to the beam sizes measured? In order to find answer to this question, we did a computer experiment, in which we had already known the solution to be sought. We treated following parameters as knowns: edge angle ea = 68.5, circulating emittance ɛ y =1 πmm-mrad, lattice functions β y =32.5 m, α y =7.0, foil scattering angle(2rms) θ scat =0.329 mrad; and used these to calculate envelopes and generate a list of beam sizes at 13 monitors VM1,VM2,..., M19. And then, we did fit to these sort of measured sizes to calculate the initial condition. Having ea = 68.5 fixed in the fit yields exactly the same initial condition as given above, which is of course. Nevertheless, fixing ea = 59 results in a fit that looks good too, see Fig.10. The reason why we say looks good too is because the objective function χ resulted from this fit is small as well (only 0.30), and it is about the same as the one that is obtained from a fit with edge angle of 68.5 but made for the truly measured beam sizes. In other words, with the truly measured beam sizes, a fit

14 TRI-DN Page 13 of 17 with edge angle of 59 can give the same magnitude of χ as the fit with edge angle of 68.5, provided the scattering angle is treated as a variable parameter in the fit of edge angle 59. That is why it is critical to have either the scattering angle or the edge angle or both fixed in the fit, because they are not orthogonal parameters. But, a large difference between these 2 fits is that the 59 edge angle results in a very small circulating emittance ɛ y, i.e. 0.3 πmm-mrad, and a very large scattering angle, i.e mrad, which is 2 times of the GEANT result. This is exactly the situation that we had in the past year. Our understanding to such a false result is that, because the edge angle, i.e. 59, is smaller, which implies less defocussing vertically, so the vertical size (as well as the emittance) of the circulating beam has to be smaller; but as a compensation, the scattering angle in the foil has to become larger. In other words, mathematically there exist two different solutions to the same problem. That s why it is critical to assure the foil thickness and let the scattering angle fixed in the fit. However, these two different solutions converge to almost a same solution. It appears that, at the end-point of the transfer matrix (just before entry into quad Q1), the σ matrices are approximately the same for both fits, particularly in the vertical plane. This can explain the fact that we still could achieve the desired beam size at the monitor M19 which was actually calculated based on a false initial condition and an incorrect edge angle. The next question is: can we obtain more evidences about the edge angle being 68.5 instead of 59. Let s look back at above-mentioned two parameters that are obtained from the fit, i.e. the circulating emittance of 1.0 πmm-mrad and Y of 1.2 mrad. Since the Y-size is 5.79 mm, we can easily calculate that, at stripper, the lattice functions of the machine are: β y = 32.5 m, α y = 7.0. To compare them with the theoretical values: 40.5 m and 3.5, we plot out two ellipses of an identical emittance of 1 πmm-mrad in the (y, y ) plane, see Fig.11. Both look not very much different in terms of the shape and orientation; the rotation angle between these two ellipses is only 7.2. This would mean that the practical tune ν z of the cyclotron at 500 MeV is slightly larger than the theoretical value(0.2) by 0.05; focusing is somewhat stronger in practice. Qualitatively, this is consistent with the result that HE probe measured regarding the vertical shift of beam centroid under the action of Br trim coils. Further, we used the edge angle of 68.5 and the initial condition obtained from the fit made for the vertical sizes measured up to VM6, to extend the calculations for the vertical envelope through the whole beamline, and compared with the sizes measured at the subsequent monitors from M9 to M19. As well, we did fit in the horizontal plane, allowing the β x, α x and ɛ x to vary. In the vertical plane the calculated beam sizes agree very well with the measured ones. In the horizontal plane, the fit is good as well, see Fig.12. The resulting emittance ɛ x on the foil but before scattering is 0.62 πmm-mrad, smaller than the vertical one. This is qualitatively consistent with COMA simulation result. COMA simulation shows that, as long as the foil is fully dipped into the beam vertically which is true most of the time in the cyclotron operation, then, for the beam hitting on the foil but before suffering scattering, ɛ y is larger than ɛ x. However, the fit with edge angle of 59 results in ɛ x constantly larger than ɛ y, for example, ɛ x was as large as 0.7 πmm-mrad, while ɛ y was only ( )πmm-mrad. This is not consistent with the calculated circulating emittance: since the Y-size is 5.08 mm at HE1, if β y at HE1 is equal to the theoretical value 30.8 m, then ɛ y = /30.8 = 0.84 πmm-mrad; but if β y at HE1 has a similar scale as what we found at 2A stripper, then it is 24.7 m, so the emittance is 1.0 πmm-mrad. Anyhow, one cannot believe that at 500 MeV the cyclotron is largely weakened in the vertical focusing so that the beta function becomes as large as /0.3 = 86 m, a factor of 2.8 bigger than the theoretical machine. Another interesting discovery is related to the fudge factor for the quad Q11. In the fits which involves edge angle of 58, in order to get good agreement between the fitted and

15 TRI-DN Page 14 of fit with edge angle= 68.5 : 2. fit with edge angle= 59 : 1 Figure 10: A computer experiment in which we did two fits with edge angle of 68.5 (top) and 59 (bottom) separately. The fit with edge angle of 68.5 yields exactly the same initial condition as the knowns, i.e. the circulating ɛ y is 1 πmm-mrad and the scattering angle θ scat is mrad. Whereas the fit with edge angle 59 results in very different ɛ y and θ scat, i.e. 0.3 πmm-mrad and 0.71 mrad. But both fits look good.

16 TRI-DN Page 15 of 17 Figure 11: Plot showing two phase ellipses of an identical emittance of 1πmm-mrad in the (y,y ) plane. One is having the theoretical values of β and α of 40.5 m and 3.5; the other is having β and α of 32.5 m and 7.0, obtained from the fit. The shape and orientation of these two ellipses does not look very much different; the rotation angle between both is only 7.2.

17 TRI-DN Page 16 of 17 Figure 12: Measured and calculated beam sizes agree very well in the whole beamline. A total of 6 parameters were fitted to obtain the best fit. measured beam sizes at the last 5 monitors behind Q11, we had to artificially introduce a scaling factor for Q11 and treated it as a fitting parameter. This scaling factor appeared to be 1.15 all the time. There was no interpretation for this in the past, because there was no evidence showing that the power supply for this quad was off by 15%; no evidence either that the effective length of this quad was longer by 15% than other quads of the same type. But, by using the edge angle of 68.5, this scaling factor gets reduced to Either 1.15 or 1.04 does not mean that this quad is wrong by that amount; instead, it suggests that there exist errors in the upstream either in the transfer matrix or in the beam sizes measured, and these errors get propagated to the downstream and peak up at Q11. In fact, we have observed many times in practical tuning of 2A that the size on monitor M19 is very sensitive to Q11 s excitation. We repeated the fits for the 2.87 mg/cm 2 foil and 4.43 mg/cm 2 stripping foils, and again, found the best fit to be with edge angle of Besides, we noticed from table 1 that, in the vertical plane, thicker foil leads to larger beam sizes down to the beamline. This is because the scattering in the foil dominates. The vertical emittance in 2A is approximately 1.6, 1.8, 2.2 πmm-mrad respectively with the 3 foils. In the horizontal plane, the size of the beam down to 2A is fairy sensitive to the circulating beam, e.g. the emittance. Experimentally, we observed that only changing the aperture of ISIS emittance limiting slits resulted in sizable changes of beam profiles horizontally measured at all the monitors in 2A.

18 TRI-DN Page 17 of 17 4 Discussions 4.1 Fringe Field The fringe field map that was used in the STRIPUBC calculations for the transfer matrix is a duplication of the map surveyed for the beamline 1 extraction port, because there was no similar survey for 2A. The 2A stripper at 500 MeV is about 60 apart from that of beamline 1, which is approximately on the symmetry of the cyclotron, but it is not certain that the fringe field is exactly the same for both extraction ports. In reality, the extraction port structure is very complicated and both are different. Moreover, the fringe field for the beamline 1 was actually surveyed from 324 inch, the maximum radius for the cyclotron internal field survey, outward up to the pivot point where the radial position is inch. This implies that even if the fringe field has been dramatically changed somehow, it does not mean that the cyclotron internal field has been changed dramatically too. 4.2 Angle Imaging In the imaging experiment, it was by tuning Q4 that we got to a point where the vertical profile of beam at VM6 arrived at a best overlap before and after shifting the beam with the Br trim coils. But in fact this is not really the angle imaging tune, because the angle of the beam entering the foil is also changed in the vertical plane in parallel with the shift of the beam position on the foil. As a consequence, the vertical beam size as measured at VM6 does not really reflect the scattering angle in the foil. For the moment we seem to have difficulty finding a good way to independently measure the elements of transfer matrix between the stripper and monitor VM6, so to speak, but still we can further check up and find more evidence for the magnet edge angle. For example, we can systematically change the excitation of quad Q4, and then record the 2*rms Y-size of beam at VM6. The Y-size square vs the the Q4 excitation would be a parabola, and the vertex of such a parabola is approximately the point where the angle imaging occurs, provided the foil being used is relatively thick enough. If the magnet edge angle is 68.5, then the Q4 excitation would be in a range of A±1.5 A. But if the edge angle is 59, then Q4 would be 132 A, 14% difference. This should be easy to identify. References [1] R. Baartman, Y.-N. Rao, Corrections to TRIUMF cyclotron stripper to combination magnet transfer matrices and computer program STRIPUBC, TRIUMF design note TRI- DN-05-4, Feb.7, [2] Private communication, Oct.2006.