ow much bigger than the travel time of the electron beam. The peak field of a 1.5 meters long has to be 7 Tesla, in order to obtain a precision better than 10% in the energy measurements of the electron beam, for electron energies from 0.01 GeV to 38 GeV. The dependence of the precision on the magnetic field intensity is calculated in Figure 3(a), for a 2 m long electromagnet and 10 GeV, 20 GeV, 30 GeV, 40 GeV electrons. In Figure 3(b), the peak magnetic field required a 5% precision in electron energy is calculated versus the energy of the electron beam. A 4 meter long magnet has to reach a 3.5 Tesla field in order to assure a 5% precision in the energy measurements of 40 GeV electron beams with angular divergence less than 0.005 radians. The calculations were done for a magnetic field with a Gaussian profile along Y and Z axes, inside and outside the gap. According to both panels in Figure 3, if the peak field is 1 Tesla, the relative uncertainty is bigger than 10%, for the 38 GeV electrons. The dimensions of the electromagnet are constrained in principal by the space available in the Interaction Chamber, the mass load capacity of the cranes and the electric power available at the experimental facility.
Figure 3. The computed relative error for the electron energy measurements is calculated versus the magnetic field, for four energies of the electron beam with angular divergence ±0.0025 rad (in panel (a)). The length of the electron beam spot on the vertical plate (SP2) is calculated versus the electron kinetic energy for four angular divergences 2α of the beam (in panel (b)).
Table 1. The gap width (G), the length of the two poles (L), and the peak magnetic field Bmax (of the electromagnet) required for a 5% uncertainty of electrons with a 35 GeV energy. The poles are 30 cm high.
In Figure 3(a), the relative error ΔE/E was computed versus the magnetic field, from the y-coordinates of the electron spots for the three trajectories starting from the exit of capillary cell with the angle of initial momentum direction α = −0.0025, 0 and 0.0025 radians relative to the Z axis. The three trajectories intersect the SP1 plate in points with coordinates (Y1, Zsp1), (Y2, Zsp1) and (Yc, Zsp1) where Zsp1 is the Z-coordinate of the SP1 plate. The Y-coordinates depend linearly on the magnetic field. The linear fitting functions Y1(B) = a0 + b1∙B (for α = 0.0025 rad), Y2(B) = −a0 + b2∙B, (for α = −0.0025 rad) and Yc(B) = bc∙B, (for α = 0), were calculated from the y-coordinates (Y1, Y2, Yc) computed for 20 magnetic fields [1T, 2T, ... 20T] and a fixed energy E(MeV). The calculatins were done for energies between 1 GeV and 40 GeV, with a 1 GeV step. The parameter a0 was calculated from the intersection with SP1 plane of the electron trajectory starting with α = 0.0025. For the model function Y1(B), Y2(B), Yc(B) the only fitting parameters b1, b2 and bc were calculated for the 40 energies mentioned above. All three fiting parameters were invers proportional dependent on energy: b1(E) = a1/E, b2(E) = a2/E and bc(E) = ac/E. In the limit of the fitting errors, a1 , a2 and a3 were almost equal. The fitting parameters a0, a1, a2, a3 in the model functions Y1(E,B) = a0 + a1·B/E, Y2(E,B) = −a0 + a2·B/E and Yc(E,B) =ac∙B/E (where 1T ≤ B ≤ 20T, 1 GeV ≤ E ≤ 40 GeV) are presented in Table 2. For each electron energy, the absolute errors ΔE1 < 0 and ΔE2 > 0 were calculated from the equations Y1(E + ΔE1,B) = Yc(E,B) and Y2(E+ΔE2,B) = Yc(E,B). For each energy E and peak field B, the relative errors can be calculated from ΔE1/E = a1·B/(a3∙B − a0∙E) − 1 and ΔE2/E = a2·B/(a3∙B + a0∙E) − 1.
In Figure 4(a), the peak magnetic field required for a 5% precision in the energy measurements is computed versus the energy of the electron beam, for a 0.005 rad divergent beam and for six magnet lengths. For the same energy of the electron beam, the required peak magnetic field is smaller for the 4 m magnet than for the 2 m magnet.
Figure 4. (a) The magnetic field in the center of a dipole magnet, required for a 5% precision in electron energy measurement is computed versus the electron energy E (GeV), for six lengths of the magnet; (b) The relative error is calculated for 35 GeV electrons, for three lengths of the magnet, versus the magnetic field in the center.
Table 2. The fitting parameters a1, a2 and ac in the model functions Y1(E,B) = a0 + a1·B/E, Y2(E,B) = −a0 + a2∙B/E, Yc(E,B) = ac∙B/E where Y1,Y2,Yc are the y-coordinates of the electron spot in SP1 plate, for three electron trajectories (with α = −0.0025, 0 and 0.0025 radians). The dipole magnet pole is 0.4 m high and its length is Lmag.
The minimum energy of the electrons that can reach the vertical plate, increases with the length of the magnet. For a 4 meter long magnet, the precision in the energy measurements on the vertical plate, is 5% if the peak field is 3.5 Tesla in the center of the magnet. For 40 GeV electrons and a 2 m long magnet, the computed relative uncertainty is less than 10% if the magnetic field is 6 Tesla. The peak magnetic field required for a 5% measurement, decreases with the divergence of the electron beam (Figure 4(b)). Therefore, by reducing the divergence of the electron beam before it enters in the magnet, the values of the peak magnetic field and magnet length for the required 5%, can be both decreased. In this way, the mass and cost of the electromagnet are reduced. This solution is explored in section 3, where a quadrupole electromagnet is located in front of the magnet to focus the electron beam. The same pulsed electric current flow in the coils of the quadrupole and dipole electromagnets.
2.2. The Mass of the Electromagnet with Iron Yoke
The geometry of the electromagnet with an Iron yoke that can provide a return path for the fringe magnetic field up to 2 Tesla magnets is presented in Figure 5. For this geometry of the Iron yoke and Cu coils, the maximum field is about 2.2 Tesla, for a continuous current mode of operation. To increase the magnetic field intensity in the gap, the two Iron cores are joined with a C-yoke. At ELI-NP, the electromagnet will be located inside the Interaction Chamber where the space is limited. The C-type Iron Yoke is preferred, because it has a smaller volume than other geometries of the Yoke. A 2D model of such electromagnet was build in COMSOL®, to calculate the magnetic field (B0) in the centre of the gap as a function of the product of the number of turns per solenoid (Nw) and the current in wire (Iw). The calculations were done for seven wire gauges and for different number of turns Nw in solenoid. In Figure 6(a) and Figure 6(b) are the dimensions of the cross section in the XY plane of the coils. At the optimum ratio b/a = 2, the mass of the Iron Yoke that assures a given peak magnetic field in the gap, is a minimum. The space between the coils and the top of the Iron Yoke is 5 cm. In Figure 6, the magnetic field averaged over the area −10 cm < X < 10 cm and −10 cm < Y < 10 cm (in the XY plane) is plotted versus the product of the number of wires in a solenoid (NW) and the wire current (IW).
Figure 5. The geometry of the two dimensional COMSOL model of the Iron Yoke (green) and the coils (red) is seen in the XY plane. All the dimensions are in centimetres.
Figure 6. Left panel: The average magnetic field in the gap between two Iron poles is calculated versus the product of the number of turns per solenoid (Nw) and the current in wire (Iw), for 7 wire gauges, 20 cm wide gap and 40 cm high poles with C-yoke. Right panel: The field in the centre of the magnet B0(T) is calculated versus the total current in the solenoid for five gap width and 40 cm high Iron poles.
The calculations are done for a 40 cm high Iron poles and Cu wire with diameter 9.266 mm. Considering the product Nw∙Iw computed in Figure 6(a) for a 1 Tesla electromagnet, the mass of the Cu coils was calculated for four lengths of the electromagnet (1 m, 2 m, 3m and 4 m) versus the gap width in Figure 7(a). The mass of the Iron Yoke is presented in Figure 7(b), for the same range of gap widths.
Figure 7. The mass of Cu coils for a 1 m, 2 m, 3 m and 4 m long electromagnet with a 1 Tesla peak field and Iron Yoke with 40 cm high poles and Cu wire with diameter 9.266 mm, is calculated versus the gap width (left panel). The mass of Iron Yoke and poles is calculated versus the gap width, for 1 m , 2 m, 3 m and 4 m long Iron poles (40 cm high) and B0 = 1 Tesla in the centre of the gap and 9.266 mm diameter wires (right panel).
The mass of a 1 Tesla electromagnet has a small dependence on the wire gauge. For example, for a 4 meter long electromagnet, the mass is 48.1 tons for −1 AWG and 50.3 tons for a 3 AWG. The mass of an 1 meter long electromagnet is 9.14 tons for −1 AWG and 9.61 tons for 3 AWG. In order to decrease the mass of the electromagnet, a pulsed electromagnet with Iron cores and special Iron yoke, is proposed in Section 3. A pulsed electromagnet less than 1.5 meters can fit completely inside the Interaction Chamber. In this case, there is no need to extend the vaccuum volume, by attaching an AC to the IC, and the cost of the IC decreases significantly.
3. The Pulsed Electromagnet
A pulsed electric current in the coils of the electromagnet can be generated by the periodic discharge of a bank of capacitors. To decrease the power lost, the time window for the peak current has to be decreased to a minimum. The rising time for the electric current cannot be decreased below a minimum value that depends on the maximum current that has to be reached and to the available electrical power. The geometry of the pulsed electromagnet projected the top in the horizontal plane XZ and in the vertical plane YZ, can be seen in Figure 8(a) and Figure 8(b) respectively. Figure 8(c) presents the geometry of the quadrupole electromagnet.
The axis of the cylindrical plasma cell is aligned with the common axes of the dipole and quadrupole electromagnets. A water cooling system can be used to absorb the electric power lost in the coils, if the rising and falling times are both less than 10 seconds and the peak current stays constant for 2 milliseconds. This length of the time window when the peak current is constant is required for the synchronization between the arrival time of the Laser beam on the target capillary cell and the turn-on time of the CCD cameras for recording the images of the electron spot on the Scintillating Plates. The four lateral solenoids C1, C2, C3 and C4 increase the horizontal magnetic field in the center of the solenoid. The quadrupole magnet focuses the electron beam and decreases the dimensions of the electron spot on SP plates  . The blue and red arrows in Figure 8 indicate the direction of the magnetization in Iron and the direction of electric currents respectively. The dotted rectangles indicate the rectangular opening holes in the front and back Iron walls, required for the passage of Laser beam. The bottom scintillating plates (SP2) are fixed on the vacuum Aluminum flange. The length of the magnet “L” is bigger than the length of the Iron poles “d” because the Iron yoke is closed in the horizontal plane of the Laser beam.
Due to the Coulomb interaction between the electrons in a bunch and to the initial divergence of the electron beam, the diameter of the electron bunch increases with the distance travelled by the electron beam. Precise measurements of the electron energy are possible if the two diameters of the electron spot along X and Z directions are small. The diameter of the electron spot along Z axis is proportional with the difference Z2 − Z1 between the Z-coordinates of the intersection points of the electron trajectories on horizontal SP1 plates. For comparison, the calculations were done for a 1 T, 4 m long DC electromagnet (Figure 9(a)) and for a 5T, 0.5 m long pulsed electromagnet (Figure 9(b)). In both cases, the entrance face of the electromagnet is at 40 cm distance
(a) (b) (c)
Figure 8. In panel (a) is presented the top view of the pulsed electromagnet with copper coils (red) and Iron yoke (green). In (b), the magnet is seen in the vertical plane through magnet center. In (c), The geometry of the qadrupole electromagnet, is seen from the exit of plasma cell in panel (c).
Figure 9. The length of the electron beam spot on the horizontal plate (SP1) is calculated for a 1 Tesla, 4 m long magnet (panel (a)) and for a 10 Tesla, 0.5 m long pulsed electromagnet (panel (b)).
from the exit of the capillary cell.
The geometry of the coils and Iron yoke of the pulsed electromagnet was designed to minimize the fringe magnetic field in the volume of the target (PDG). The Cu coils wound around the Iron poles, can carry pulsed electric currents with flat-top profile and a peak current intensity that is constant only in a few milliseconds time window. The magnetic field profile along the Z axis is expected to focus the beam of electrons, such that the dimensions of the electron spot are a minimum.
The calculated mass of a 4 meter long electromagnet (with a 1.2 T field in the centre) exceeds 8 tons if the magnet is over 1.5 meters long. One proposed solution is a pulsed electromagnet made from quadrupole and a dipole electromagnets aligned along a common axis parallel with the axis of plasma guide. The quadrupole magnet has to be located in front of the dipole magnet. The mechanical support for each electromagnet has to be strong enough to overcome the repulsive forces between them. The 0.5 meters long pulsed electromagnet presented above is much less heavier than a 4 meter long electromagnet. This simplifies the transport and installation of the electromagnet. The direction of the electric currents in the lateral coils is chosen such that the magnetic field lines are closed through the Iron, and decrease the fringe magnetic field profile. A zero fringe field over the volume of the capillary gas cell is required in the LWFA experiments, where the initial direction of the electron beam has to be parallel with the axis of the capillary cell.
The precision in measurement of the position of the electron spot depends on the dimensions of the spot. On the horizontal SP plates, the length of the spot increases with the length of the electromagnet. The field of the quadrupole magnets decreases the length of the electron spots on the vertical SP plates, in the vertical direction. The light emitted from the spot has to be high enough to assure a good ratio signal/noise for the measurements of the spot position. The brightness of the electron beam spot decreases with the increase in the spot diameter. Because the experiment requires a precision of less than 5%, the length of the pulsed electromagnet is less than 1 meter.
The research work leading to these results was supported by the “Extreme Light Infrastructure Nuclear Physics (ELI-NP) Phase I”, a project co-financed by the Romanian Government and European Union through the European Regional Development Fund.
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