_{1}

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Laser Wakefield plasma acceleration of electrons to energies above 10 GeV, may be possible in the new high power Laser beam facilities. The design of an Electron Spectrometer with an electro-magnet with adjustable magnetic field is proposed for the characterization of electron energy spectrum with a precision better than 10% for the entire energy range from 0.5 GeV to 38 GeV. The expected precision in the measurement of the electron energy is calculated as a function of the magnetic field, of the electron energy and of the magnet length. To outline the advantages offered by a pulsed electromagnet with high magnetic fields, the mass and the electric power lost in the coils of a 4 m long electromagnet with continuous current and Iron yoke are calculated.

The Laser Wakefield Acceleration (LWFA) of electron beams [^{18} W/cm^{2}, with an under-dense plasma confined in a plasma channel with a diameter less than 1 mm. In the preformed discharged-based plasma channel [^{−15} seconds per pulse and 200 Joules per pulse), with an F/40 parabolic mirror, in the center of the entrance edge of a capillary discharge guide up to 80 cm long, to accelerate electrons by LWFA [^{16} and 10^{19 }cm^{-3}. The first conceptual design of the Electron Specrometer (ES) proposed to be used for measurement of electron energies at ELI-NP was presented in [

The Electron Spectrometer (

X and Y axes are the horizontal and vertical axes normal to the axis of Laser beam. The magnetic field in the gap is almost uniform and aligned with the X axis. In _{0}) of the bunch of electrons makes an angle θ with the Z axis. The initial momenta directions of electrons are in the cone with angle 2α.

The diameter of the electron bunch along its trajectory increases with the distance traveled by electrons, due to the Coulomb interaction and to the electron beam intrinsic angular divergence (2α) expected to be less than 0.01 radians. In addition, the electron momentum can acquire a perpendicular component, if there is a small misalignment of the laser beam and capillary axes. In the horizontal XZ plane (of

long. The blue curved lines in

The charge of a few picoseconds long electron bunch is measured with the Integrating Current Transformer (ICT). The 4 meter long magnet starts at about 40 cm from the exit of the plasma discharge guide (PDG). The electrons with energy bigger than 200 MeV, exit from the 1 Tesla magnetic field inside the gap and interact with the horizontal and vertical Lanex plates SP1(a, b), SP2(a, b). The electron beam energy is calculated from the position of the beam spots on the two horizontal (or two vertical) SPs. The light emitted from the beam spots is collected and transported with lenses to the CCD cameras outside the IC. The CCD cameras are placed in Faraday cages that attenuate the electromagnetic pulse (EMP) emitted upon Laser interaction with plasma in PDG. The EMP has a broad range of frequencies (from tens of MHz up to tens of GHz). A dipole magnet with permanent magnets has a better field stability and zero power lost, compared with an electromagnet. The magnet requires an auxiliary chamber (AC, in

Considering the biggest value expected for the total angle _{max} is the peak magnetic field required for a 5% uncertainty in the energy measurements of electrons, with expected energy up to 35 GeV (

More precise calculations of required B_{max} are presented in sections 2.1. For these calculations, the angle θ and the offset dx are zero. The results are obtained from numerical calculations in C codes, of three electron trajectories (

Compared with an electromagnet with continuous current (with a 20 cm gap and 4 m long poles), a pulsed electromagnet can be more compact because it provides a bigger field, only for a few miliseconds, a time window 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

G (cm) | L (cm) | B_{max} (T) | G (cm) | L (cm) | B_{max} (T) |
---|---|---|---|---|---|

4 | 64 | 43 | 7 | 116 | 12 |

5 | 80 | 25 | 8 | 134 | 9 |

6 | 98 | 18 | 9 | 152 | 7 |

In _{1}, Z_{sp1}), (Y_{2}, Z_{sp1}) and (Y_{c}, Z_{sp1}) where Z_{sp1} is the Z-coordinate of the SP1 plate. The Y-coordinates depend linearly on the magnetic field. The linear fitting functions Y_{1}(B) = a_{0} + b_{1}∙B (for α = 0.0025 rad), Y_{2}(B) = −a_{0} + b_{2}∙B, (for α = −0.0025 rad) and Y_{c}(B) = b_{c}∙B, (for α = 0), were calculated from the y-coordinates (Y_{1}, Y_{2}, Y_{c}) 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 a_{0} was calculated from the intersection with SP1 plane of the electron trajectory starting with α = 0.0025. For the model function Y_{1}(B), Y_{2}(B), Y_{c}(B) the only fitting parameters b_{1}, b_{2} and b_{c} were calculated for the 40 energies mentioned above. All three fiting parameters were invers proportional dependent on energy: b_{1}(E) = a_{1}/E, b_{2}(E) = a_{2}/E and b_{c}(E) = a_{c}/E. In the limit of the fitting errors, a_{1} , a_{2} and a_{3} were almost equal. The fitting parameters a_{0}, a_{1}, a_{2}, a_{3 }in the model functions Y_{1}(E,B) = a_{0} + a_{1}·B/E, Y_{2}(E,B) = −a_{0} + a_{2}·B/E and Y_{c}(E,B) =a_{c}∙B/E (where 1T ≤ B ≤ 20T, 1 GeV ≤ E ≤ 40 GeV) are presented in _{1} < 0 and ΔE_{2} > 0 were calculated from the equations Y_{1}(E + ΔE_{1},B) = Y_{c}(E,B) and Y_{2}(E+ΔE_{2},B) = Y_{c}(E,B). For each energy E and peak field B, the relative errors can be calculated from ΔE_{1}/E = a_{1}·B/(a_{3}∙B − a_{0}∙E) − 1 and ΔE_{2}/E = a_{2}·B/(a_{3}∙B + a_{0}∙E) − 1.

In

L_{mag} (m) | a_{0} | a_{1} | a_{2} | a_{3} |
---|---|---|---|---|

0.5 | 0.00275 | −99.585 | −99.775 | −99.68 |

1 | 0.0040 | −216.44 | −216.64 | −216.54 |

1.5 | 0.00525 | −336.14 | −336.24 | −336.19 |

2 | 0.0065 | −445.41 | −445.4 | −445.41 |

2.5 | 0.0075 | −545.18 | −545.23 | −545.18 |

3.0 | 0.0090 | −640.52 | −640.27 | −640.41 |

3.5 | 0.01025 | −734.46 | −734.1 | −734.31 |

4.0 | 0.0115 | −828.25 | −827.77 | −828.05 |

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 (

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 ^{®}, to calculate the magnetic field (B_{0}) in the centre of the gap as a function of the product of the number of turns per solenoid (N_{w}) and the current in wire (I_{w}). The calculations were done for seven wire gauges and for different number of turns N_{w} in solenoid. In _{W}) and the wire current (I_{W}).

The calculations are done for a 40 cm high Iron poles and Cu wire with diameter 9.266 mm. Considering the product N_{w}∙I_{w} computed in

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.

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

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 C_{1}, C_{2}, C_{3} and C_{4} 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 [

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 Z_{2} − Z_{1} 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 (

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.

Septimiu Balascuta, (2016) A Pulsed Electromagnet for Laser Wakefield Electron Acceleration Experiments. Journal of Electromagnetic Analysis and Applications,08,33-41. doi: 10.4236/jemaa.2016.83004