Interaction of 50 MeV - 50 TeV proton with solid copper target at CERN hadron accelerator complex Yuancun NIE Thursday, August 25, 2016 Yuancun NIE

Outline Introduction Beam parameters to be studied for CERN hadron accelerator complex Energy deposition of single proton in copper: FLUKA simulations Comparison with the Bethe equation Specific energy deposition of multi-protons (bunches) Summary and future works Thursday, August 25, 2016 Yuancun NIE

I. Introduction Studies on beam-matter interaction are essential for machine protection Numerical simulations have been performed especially for the SPS, the LHC, and FCC (40TeV), but the beam parameters (energy & size) studied are still limited We intend to make an integral study (database) that covers all typical beam energies and beam sizes in the CERN hadron accelerator complex Therefore, FLUKA is used to calculate energy depositions of protons in copper (in graphite as well, show in the future) No thermodynamics and hydrodynamic simulation is included (static approximation) Thursday, August 25, 2016 Yuancun NIE

II. Beam parameters to be studied for CERN hadron accelerator complex Related parameters of the hadron accelerator complex at CERN Accelerator PSB PS SPS LHC FCC Injection energy 50 MeV a) 1.4 GeV 26 GeV 450 GeV 3.3 TeV Extraction energy 7 TeV 50 TeV # bunches 1 / ring 72 c) 288 2808 10600 Bunch intensity 16.2×1011 b) 1.35×1011 1.15×1011 1.0×1011 Beam power 0.4 kJ per bunch 40 kJ 2.4 MJ 362 MJ 8500 MJ Norm. emittance 2.5 µm 3 µm 3.5 µm 3.75 µm 2.2 µm Pulse length bunch length ~190 ns 1.8 µs 7.2 µs 88.9 µs 327 µs Circumference 157 m (4 rings) 1/4 PS 628 m 1/11 SPS 6.9 km 7/27 LHC 26.7 km 97.97 km 11/3 LHC a) Will be upgraded to 160 MeV (H- from LINAC4) b) Ref.: B. Mikulec, et al., CERN-ATS-2009-070 c) Debunching and rebunching (splitting) 6 bunches from PSB to make 72 bunches spaced by 25 ns at PS. Ref.: M. Benedikt, et al., EPAC 2000; R. Garoby, et al., EPAC 2000 Thursday, August 25, 2016 Yuancun NIE

II. Beam parameters to be studied for CERN hadron accelerator complex For the Gaussian distribution: 𝜎 𝑥,𝑦 = 𝛽 c 𝜀 rms = 𝛽 c 𝜀 n,rms /𝛾 where 𝛽 c is the betatron function of the beam optics, 𝜀 n,rms = 𝛾𝜀 rms is the normalized rms beam emittance. For 450 GeV SPS beam, 𝜀 n,rms ~3.5μm, assuming 𝛽 c =100m, 𝜎 𝑥,𝑦 =0.85mm; For 7 TeV LHC beam, 𝜀 n,rms ~3.75μm, assuming 𝛽 c =100m, 𝜎 𝑥,𝑦 =0.22mm; For 50 TeV FCC beam, 𝜀 n,rms ~2.2μm, assuming 𝛽 c =200m, 𝜎 𝑥,𝑦 =0.09mm; The typical beam size from the LINAC, PSB and PS is up to a few mm. Therefore, we first study such a case that the beam size is kept to be 0.2mm for all the energies. In addition, different beam sizes are studied depending on the beam energy, which will be listed later. Emittance history for LHC D. Möhl, Transverse dynamics II: Emittances, CAS2005 Thursday, August 25, 2016 Yuancun NIE

III. Energy deposition of single proton in copper: FLUKA simulations TARGET VACUUM BLACK BODY Here, the energy deposition is calculated only by FLUKA (static approximation), i.e., the hydrodynamic tunneling effect is not considered, or equivalently, the beam energy is deposited instantaneously. "The FLUKA Code: Developments and Challenges for High Energy and Medical Applications“, T.T. Böhlen, F. Cerutti, M.P.W. Chin, A. Fassò, A. Ferrari, P.G. Ortega, A. Mairani, P.R. Sala, G. Smirnov and V. Vlachoudis, Nuclear Data Sheets 120, 211-214 (2014); "FLUKA: a multi-particle transport code“, A. Ferrari, P.R. Sala, A. Fasso`, and J. Ranft, CERN-2005-10 (2005), INFN/TC_05/11, SLAC-R-773. Thursday, August 25, 2016 Yuancun NIE

III. Energy deposition of single proton in copper: FLUKA simulations Energy deposition (GeV/cm3/p) of 50MeV (PSB injection) proton in copper, σx,y=0.2mm Thursday, August 25, 2016 Yuancun NIE

III. Energy deposition of single proton in copper: FLUKA simulations Energy deposition (GeV/cm3/p) of 50TeV (FCC top energy) proton in copper, σx,y=0.2mm Thursday, August 25, 2016 Yuancun NIE

III. Energy deposition of single proton in copper: FLUKA simulations Simulation results for 50 MeV proton beam with different transverse beam sizes Energy deposition vs. target radius, at L=0.39cm (position of the Bragg peaks). Energy deposition vs. target axis Bin size σx,y/4, primaries 1000*50*20 (event*cycle*spawn) Thursday, August 25, 2016 Yuancun NIE

III. Energy deposition of single proton in copper: FLUKA simulations Simulation results for 50 TeV proton beam with different transverse beam sizes Energy deposition vs. target radius, around L=23cm, where the on-axis energy deposition is the maximum. Energy deposition vs. target axis Bin size σx,y/4, primaries 10*100*10 (event*cycle*spawn) Thursday, August 25, 2016 Yuancun NIE

III. Energy deposition of single proton in copper: FLUKA simulations Energy depositions (GeV/cm3/p) of protons with various energies in copper, σx,y=0.2mm Thursday, August 25, 2016 Yuancun NIE

III. Energy deposition of single proton in copper: FLUKA simulations The peak energy depositions (GeV/cm3/p, for σx,y=0.2mm) of protons in copper as a function of incident kinetic energy over 6 orders of magnitude. The corresponding values of 𝛽𝛾 are plotted as well. Thursday, August 25, 2016 Yuancun NIE

III. Energy deposition of single proton in copper: FLUKA simulations Energy (GeV) RMS beam size (mm) Max. energy deposition (GeV/cm3/p) Max. specific energy (×10-9J/g/p) Peak location (cm) 0.05 0.2 75.4 1.35 0.39 (Bragg Peak) 0.4 30.4 0.54 1.0 6.0 0.11 0.16 12.9 0.23 0.1 3.5 0.06 0.3 0.67 0.01 3.0 (Bragg Peak) 1.4 5.7 0.10 1.5 0.03 0.26 0.005 26 6.8 0.12 1.9 1.3 0.42 0.008 4.2 450 59.2 1.06 11.0 35.1 0.63 12.6 0.8 12.3 0.22 15.8 3300 814.7 14.5 472.7 8.4 17.3 252.9 4.5 18.3 7000 2059.1 36.7 1147.2 20.5 18.8 606.5 10.8 19.8 50000 19792.7 353.4 21.8 10357.5 185.0 23.3 5043.5 90.1 23.8 Thursday, August 25, 2016 Yuancun NIE

IV. Comparison with the Bethe equation The mean energy loss rate of moderately relativistic (0.1≤𝛽𝛾≤1000) charged heavy particles in intermediate-Z materials is well-described by the Bethe equation: − 𝑑𝐸 𝑑𝑥 =𝐾 𝑧 2 𝑍 𝐴 1 𝛽 2 1 2 ln 2 𝑚 e 𝑐 2 𝛽 2 𝛾 2 𝑇 max 𝐼 2 − 𝛽 2 − 𝛿 𝛽𝛾 2 where 𝑇 max is the maximum kinetic energy that can be transferred to a free electron in a single collision, 𝛿 𝛽𝛾 is the density effect correction to ionization energy loss: 𝑇 max = 2 𝑚 e 𝑐 2 𝛽 2 𝛾 2 1+2𝛾 𝑚 e 𝑀 + 𝑚 e 𝑀 2 𝛿 𝛽𝛾 = 2 ln 10 𝑥− 𝐶 if 𝑥≥ 𝑥 1 ; 2 ln 10 𝑥− 𝐶 +𝑎 𝑥 1 −𝑥 𝑘 if 𝑥 0 ≤𝑥< 𝑥 1 ; 0 if 𝑥< 𝑥 0 (nonconductors); 𝛿 0 10 2 𝑥− 𝑥 0 if 𝑥< 𝑥 0 (conductors); with 𝑥= log 10 𝑝/𝑀𝑐 = log 10 𝛽𝛾 . 𝑻 𝒎𝒂𝒙 can be simplified to 𝟐 𝒎 𝒆 𝒄 𝟐 𝜷 𝟐 𝜸 𝟐 when 𝟐𝜸 𝒎 𝒆 ≪𝑴, while 𝜹 𝜷𝜸 /𝟐 is small at low energies. The other variables are defined in the following table. K.A. Olive et al. (PDG), Chin. Phys. C38, 090001 (2014) (http://pdg.lbl.gov) D.E. Groom et al., Atomic Data and Nuclear Data Tables 76, 1-37(2001) Thursday, August 25, 2016 Yuancun NIE

IV. Comparison with the Bethe equation Summary of the variables used (the unit of −𝑑𝐸/𝑑𝑥 is hence 𝐌𝐞𝐕 𝐠 −𝟏 𝐜𝐦 𝟐 ) Symbol Definition Units or Value 𝑲 4𝜋 𝑁 𝐴 𝑟 𝑒 2 𝑚 𝑒 𝑐 2 0.307075 MeV mol −1 cm 2 𝒛 Charge number of the projectile particle 1 for proton 𝒁 Atomic number of the target material 29 for copper 𝑨 Atomic mass of the target material 63.546 g/mol for copper 𝒎 𝐞 𝒄 𝟐 Electron mass ×𝑐 2 0.511 MeV 𝑰 Mean excitation energy of the target atom 𝐼≅ 10±1 eV 𝑍 for 𝑍≥16, 322 eV for copper 𝑴 Incident particle mass 938.26 MeV/ 𝑐 2 for proton 𝒑 Incident particle momentum 𝑝=𝑀𝛽𝛾𝑐 MeV/𝑐 𝑵 𝐀 Avogadro’s number 6.022× 10 23 mol −1 𝒓 𝐞 Classical electron radius 𝑒 2 /4𝜋 𝜀 0 𝑚 e 𝑐 2 2.818× 10 −15 m 𝝆 Density of the target material 8.96 g/ cm 3 for solid copper 𝒂 Sternheimer’s parameter for certain 𝑍 0.14339 for copper 𝒌 2.9044 for copper 𝒙 𝟎 -0.0254 for copper 𝒙 𝟏 3.2792 for copper 𝑪 4.4190 for copper 𝜹 𝟎 0.08 for copper R.M. Sternheimer and R.F. Peierls, PR B, 1971 Thursday, August 25, 2016 Yuancun NIE

IV. Comparison with the Bethe equation Using the Bethe equation, we can calculate the mean energy loss rate of one proton in copper, in the units of MeV g −1 cm 2 . Meanwhile, from FLUKA, we get the energy deposition in the units of GeV/ cm 3 , in the predefined volume of the copper target. In order to compare the two values, we multiply the mean loss rate by the copper density (g/cm3) to obtain the energy loss per penetration length, in the units of GeV/cm. On the other hand, for the energy deposition values from FLUKA, we perform an integration in the cross section at the entrance of the target to obtain the energy loss per penetration length, in the same units of GeV/cm. Such an energy loss rate is independent on the beam size, as illustrated by the figure below. For example, the areas under the three curves (the integrals) are the same (𝟎.𝟎𝟔𝟖 𝐆𝐞𝐕/𝐜𝐦), corresponding to three different beam sizes 𝜎 𝑥,𝑦 =0.2 mm, 𝜎 𝑥,𝑦 =0.4 mm and 𝜎 𝑥,𝑦 =1.0 mm, respectively, for a 50 MeV proton in copper. Thursday, August 25, 2016 Yuancun NIE

IV. Comparison with the Bethe equation − 𝑑𝐸 𝑑𝑥 =𝐾 𝑧 2 𝑍 𝐴 1 𝛽 2 1 2 ln 2 𝑚 e 𝑐 2 𝛽 2 𝛾 2 𝑇 max 𝐼 2 − 𝛽 2 − 𝛿 𝛽𝛾 2 The results are in good accordance with each other when the proton kinetic energies are 50 MeV, 160 MeV and 1.4 GeV. The values from Bethe equation are more and more under-estimated for higher energies. For 26 GeV (𝛽𝛾≈30), it becomes 70% of that in FLUKA. The reason is that at higher energies, radiative effects (bremsstrahlung, pair production and photonuclear interactions) begin to be important, which are not included in the Bethe equation. Thursday, August 25, 2016 Yuancun NIE

V. Specific energy deposition of multi-protons (bunches) Using non-constant specific heat capacity, 674J/g is needed to make the copper melted from 300K, and 6250J/g to vaporize it. The number of protons needed to melt the copper material at the maximum energy deposition point (denoted as 𝑛 p, max ) as well as at the entrance of the target (denoted as 𝑛 p, entrance ) are listed in the following table. In addition, the corresponding number of proton bunches (denoted as 𝑁 p, max and 𝑁 p, entrance , respectively) are presented as well, based on the nominal bunch intensity listed previously. For the SPS (450 GeV, 𝜎 𝑥,𝑦 =0.8mm), the safe beam intensity (no melting) is 3.1×1012, which agrees well with the damage test experiments (Kain et al., Chamonix 2009 Workshop on LHC Performance). Analogously, the threshold is 3.3×1010 for the beam size of 0.2 mm at the LHC top energy of 7 TeV, and 1.9×109 for 0.1 mm at the FCC top energy of 50 TeV, as suggested in the following table. Thursday, August 25, 2016 Yuancun NIE

V. Specific energy deposition of multi-protons (bunches) Number of protons and proton bunches needed to melt the copper target at the maximum energy deposition location as well as at the target front surface Energy (GeV) RMS beam size (mm) 𝒏 𝐩, 𝐦𝐚𝐱 𝑵 𝐩, 𝐦𝐚𝐱 𝒏 𝐩, 𝐞𝐧𝐭𝐫𝐚𝐧𝐜𝐞 𝑵 𝐩, 𝐞𝐧𝐭𝐫𝐚𝐧𝐜𝐞 0.05 0.2 5.0×1011 0.34 1.3×1012 0.79 0.4 1.2×1012 0.77 5.1×1012 3.15 1.0 6.3×1012 3.88 3.1×1013 18.94 0.16 2.9×1012 1.81 3.0×1012 1.86 1.1×1013 6.66 1.2×1013 7.52 5.6×1013 34.77 7.3×1013 44.81 1.4 6.6×1012 4.09 6.8×1012 4.20 2.5×1013 15.53 1.5×1014 89.61 26 5.6×1012 41.12 6.0×1012 44.38 2.0×1013 147.15 2.2×1013 159.76 9.0×1013 665.68 1.1×1014 822.31 450 0.1 6.4×1011 5.54 1.4×1012 11.89 1.1×1012 9.35 5.0×1012 43.76 0.8 3.1×1012 26.68 6.7×1013 586.09 3300 4.6×1010 0.46 13.73 8.0×1010 0.80 4.1×1012 40.58 1.5×1011 1.49 1.3×1013 134.80 7000 1.8×1010 10.90 3.3×1010 0.29 35.67 6.2×1010 0.54 1.5×1013 131.28 50000 1.9×109 0.02 8.3×1011 8.33 3.6×109 0.04 2.8×1012 27.96 7.5×109 0.07 120.20 Thursday, August 25, 2016 Yuancun NIE

V. Specific energy deposition of multi-protons (bunches) 440 GeV (in HiRadMat experiment): One bunch, bunch intensity 1.5×1011 protons, σx,y= 0.2 mm Peak specific energy: 93 J/g 440 GeV (in HiRadMat experiment): 144 bunches, bunch intensity 1.5×1011 protons, σx,y= 0.2 mm Penetration depth of the beam: 69 cm (static approximation) Note that the melting and boiling energies are 613 J/g and 5236 J/g in the reference. Agree with the results in Ref.: R. Schmidt, et al., Physics of Plasma, 2014 Thursday, August 25, 2016 Yuancun NIE

V. Specific energy deposition of multi-protons (bunches) 3.3 TeV (FCC injection): One bunch, bunch intensity 1.0×1011 protons, σx,y= 0.2 mm Peak specific energy: 844 J/g > 674 J/g 3.3 TeV (FCC injection): 10600 bunches, bunch intensity 1.0×1011 protons, σx,y= 0.2 mm Penetration depth of the beam: 214 cm (static approximation) Thursday, August 25, 2016 Yuancun NIE

V. Specific energy deposition of multi-protons (bunches) 50 TeV (FCC top energy): One bunch, bunch intensity 1.0×1011 protons, σx,y= 0.2 mm Peak specific energy: 18496 J/g > 6250 J/g >> 674 J/g 50 TeV (FCC top energy): 10600 bunches, bunch intensity 1.0×1011 protons, σx,y= 0.2 mm Penetration depth of the beam: 315 cm (static approximation) Thursday, August 25, 2016 Yuancun NIE

VI. Summary and future works The energy depositions of 50 MeV-50 TeV protons in copper have been simulated and analysed. Typical energies and beam sizes were selected to cover the CERN hadron accelerator complex. They serve as a starting point for further analyses. Another target material of graphite is being studied. Based on this, estimation can be performed for beam energies / sizes that haven’t been listed. For instance, the higher-energy LHC (HE-LHC) has been proposed to reach a beam energy of 16.5 TeV, of which the maximum energy deposition can be assessed to be around 3000 GeV/ cm 3 /p at 21 cm into the solid copper target for σx,y= 0.2 mm. The simulation here was performed only using FLUKA, i.e., the static approximation was assumed and the hydrodynamic tunnelling hadn’t been considered. We are (only) able to predict the minimum penetration depth into the target for a certain beam based on such static approximation. The existence of hydrodynamic tunnelling has been confirmed by the HiRadMat experiment, for high energy beam with a long bunch train. To simulate it, BIG2 has been used, and we will try to use ANSYS Autodyn / LS-Dyna as well. A case study using Autodyn / LS-Dyna is in progress in collaboration with EN-MME. One copper target & beam parameter of the 2012 HiRadMat experiment is taken as reference. Thursday, August 25, 2016 Yuancun NIE