Numerical simulations of energy deposition caused by 50 MeV—50 TeV proton beams in copper and graphite targets

The conceptual design of the Future Circular Collider (FCC) is being carried out actively in an international collaboration hosted by CERN, for the post–Large Hadron Collider (LHC) era. The target center-of-mass energy of proton-proton collisions for the FCC is 100 TeV, nearly an order of magnitude higher than for LHC. The existing CERN accelerators will be used to prepare the beams for FCC. Concerning beam-related machine protection of the whole accelerator chain, it is critical to assess the consequences of beam impact on various accelerator components in the cases of controlled and uncontrolled beam losses. In this paper, we study the energy deposition of protons in solid copper and graphite targets, since the two materials are widely used in magnets, beam screens, collimators, and beam absorbers. Nominal injection and extraction energies in the hadron accelerator complex at CERN were selected in the range of 50 MeV–50 TeV. Three beam sizes were studied for each energy, corresponding to typical values of the betatron function. Specifically for thin targets, comparisons between fluka simulations and analytical Bethe equation calculations were carried out, which showed that the damage potential of a few-millimeter-thick graphite target and submillimeter-thick copper foil can be well estimated directly by the Bethe equation. The paper provides a valuable reference for the quick evaluation of potential damage to accelerator elements over a large range of beam parameters when beam loss occurs.

Figure 1

Schematic view of the CERN accelerator complex.

Figure 2

Two-dimensional energy deposition in the units of $\mathrm{GeV}/{\mathrm{cm}}^3$ per proton in a solid copper target at the energies of (a) 50 MeV and (b) 50 TeV. The rms beam size is $\sigma =0.2\mathrm{mm}$.

Figure 3

Energy deposition per 50 MeV proton in a cylindrical copper target for three different beam sizes (a) in the longitudinal direction at $r=0$ and (b) in the radial direction at $L=0.39\mathrm{cm}$ (position of the Bragg peaks). The three beam sizes are 0.2, 0.4, and 1.0 mm.

Figure 4

Energy deposition per 50 TeV proton in a cylindrical copper target for three different beam sizes (a) in the longitudinal direction at $r=0$ and (b) in the radial direction at $L=21.3\mathrm{cm}$ for the beam size of 0.1 mm, $L=22.3\mathrm{cm}$ for 0.2 mm, and $L=23.8\mathrm{cm}$ for 0.4 mm, where we have the maximum energy deposition.

Figure 5

Energy deposition per incident proton as a function of the depth into the solid copper target at $r=0$. The beam size is constant (0.2 mm) for the energies from 50 MeV to 50 TeV. It is interesting to note that the results shown here are significantly higher than that reported in Ref. [33]. We have clarified that this is mainly due to the coarse bin settings (large scoring steps) in the previous work.

Figure 6

Maximum energy deposition in copper as a function of the incident proton kinetic energy. The corresponding values of $\beta \gamma$ are plotted as well. The beam size is constant (0.2 mm) for the energies from 50 MeV to 50 TeV.

Figure 7

Energy deposition of 50 TeV proton in copper along the target axis with and without linking fluka and DPMJET-III. Without linking, the maximum energy deposition is 6% higher than that for the case of linking.

Figure 8

Maximum energy deposition in copper as a function of the beam size for the energies from 50 MeV to 50 TeV.

Figure 9

Energy deposition or loss rate per proton in a thin copper target as a function of kinetic energy ranging from 50 MeV to 50 TeV. Comparisons are made between fluka simulations (energy deposition per target length) and direct calculations using the Bethe equation (energy loss rate). For the simulation results, three target thicknesses are considered as denoted in the figure.

Figure 10

Specific energies of (a) one bunch with $1.0\times {10}^{11}$ protons and (b) one full FCC beam with 10600 bunches. The top energy of the FCC (50 TeV) is used. The beam size is 0.2 mm.

Figure 11

Two-dimensional energy deposition in units of $\mathrm{GeV}/{\mathrm{cm}}^3$ per proton in a solid graphite target at the energies of (a) 50 MeV and (b) 50 TeV. The rms beam size is $\sigma =0.2\mathrm{mm}$.

Figure 12

Energy deposition per 50 MeV proton in a cylindrical graphite target for three different beam sizes of 0.2, 0.4, and 1.0 mm (a) in the longitudinal direction at $r=0$ and (b) in the radial direction at $L=1.09\mathrm{cm}$ (position of the Bragg peaks).

Figure 13

Energy deposition per 50 TeV proton in a cylindrical graphite target for three different beam sizes (a) in the longitudinal direction at $r=0$ and (b) in the radial direction at $L=135.8\mathrm{cm}$ for the beam size of 0.1 mm, $L=155.3\mathrm{cm}$ for 0.2 mm, and $L=164.3\mathrm{cm}$ for 0.4 mm.

Figure 14

Energy deposition per incident proton as a function of the depth into the solid graphite target at $r=0$. The beam size is 0.2 mm for all the presented energies from 50 MeV to 50 TeV.

Figure 15

Maximum energy deposition in graphite as a function of the incident proton kinetic energy. The corresponding values of $\beta \gamma$ are plotted as well. The beam size is 0.2 mm for all the presented energies from 50 MeV to 50 TeV.

Figure 16

Energy deposition of 50 TeV proton in graphite along the target axis with and without linking fluka and DPMJET-III. Without linking, the maximum energy deposition is 11% higher than that for the case of linking.

Figure 17

Maximum energy deposition in graphite as a function of the beam size for the energies from 50 MeV to 50 TeV.

Figure 18

Energy deposition or loss rate per proton in a thin graphite target as a function of kinetic energy ranging from 50 MeV to 50 TeV. Comparisons are made between fluka simulations (energy deposition per target length) and direct calculations using the Bethe equation (energy loss rate). For the simulation results, three target thicknesses are considered as denoted in the figure.