High energy density physics effects predicted in simulations of the CERN HiRadMat beam–target interaction experiments

Accepted Manuscript

High Energy Density Physics Effects Predicted in Simulations of the CERN HiRadMAR Beam–Target Interaction Experiments N.A. Tahir, F. Burkart, R. Schmidt, A. Shutov, D. Wollmann, A.R. Piriz PII: DOI: Reference:

S1574-1818(16)30064-7 10.1016/j.hedp.2016.09.002 HEDP 574

To appear in:

High Energy Density Physics

Received date: Revised date: Accepted date:

23 June 2016 5 September 2016 25 September 2016

Please cite this article as: N.A. Tahir, F. Burkart, R. Schmidt, A. Shutov, D. Wollmann, A.R. Piriz, High Energy Density Physics Effects Predicted in Simulations of the CERN HiRadMAR Beam–Target Interaction Experiments, High Energy Density Physics (2016), doi: 10.1016/j.hedp.2016.09.002

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ACCEPTED MANUSCRIPT High Energy Density Physics Effects Predicted in Simulations of the CERN HiRadMAR Beam–Target Interaction Experiments N.A. Tahir,1 F. Burkart,2 R. Schmidt,2 A. Shutov,3 D. Wollmann,2 and A. R. Piriz4 1

GSI Helmholtzzentrum f¨ ur Schwerionenforschung, Planckstraße 1, 64291 Darmstadt, Germany 2 CERN, 1211 Geneva 23, Switzerland 3 Institute of Problems of Chemical Physics, Russian Academy of Sciences, Institutskii pr. 18, 142432 Chernogolovka, Russia 4 E.T.S.I.Industriales, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain

PACS numbers: 51.50.+v 29.27.-a 51.60.+a

1. INTRODUCTION

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Abstract–Experiments have been done at the CERN HiRadMat (High Radiation to Materials) facility in which large cylindrical copper targets were irradiated with 440 GeV proton beam generated by the Super Proton Synchrotron (SPS). The primary purpose of these experiments was to confirm the existence of hydrodynamic tunneling of ultra–relativistic protons and their hadronic shower in solid materials, that was predicted by previous numerical simulations. The experimental measurements have shown very good agreement with the simulation results. This provides confidence in our simulations of the interaction of the 7 TeV LHC (Large Hadron Collider) protons and the 50 TeV Future Circular Collider (FCC) protons with solid materials, respectively. This work is important from the machine protection point of view. The numerical simulations have also shown that in the HiRadMat experiments, a significant part of the target material is be converted into different phases of High Energy Density (HED) matter, including two–phase solid–liquid mixture, expanded as well as compressed hot liquid phases, two– phase liquid–gas mixture and gaseous state. The HiRadMat facility is therefore a unique ion beam facility worldwide that is currently available for studying the thermophysical properties of HED matter. In the present paper we discuss the numerical simulation results and present a comparison with the experimental measurements.

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The 440 GeV proton beam generated by the Super Proton Synchrotron (SPS) at CERN, is used at the HiRadMat (High Radiation to Materials) facility, for beam– matter interaction experiments using fixed targets. At present, HiRadMat is a unique ion beam facility worldwide that is available for such interesting scientific investigations. In July 2012, a number of experiments were performed at this facility in which large copper cylinders were irradiated with protons to study the problem of beam–target heating and the results of these studies are documented in [1–3]. The main purpose of these experiments was to confirm the existence of ”hydrodynamic tunneling” of ultra–relativistic protons and their hadronic shower in solid targets, as previous numerical simulations predicted a significant range lengthening of the projectile particles due to this phenomenon [4]. This effect is very pronounced in the case of 7 TeV Large Hadron Collider (LHC) protons [4–7]. For example, according to our simulations, the static range of a single 7 TeV proton and the shower it produces in solid copper is about 1 m, whereas, the full LHC beam will penetrate around 35 m in the target due to the hydrodynamic tunneling. This phenomenon, therefore, has important implications on the machine protection design. It is to be noted that experimental verification of the existence of the hydrodynamic tunneling in case of the LHC protons is not possible, which in fact, is a necessary requirement

for the validation of the simulations. The success of the HiRadMat experiments and the excellent agreement between the experimental measurements and the simulation results confirmed the existence of hydrodynamic tunneling and established the accuracy of the corresponding simulations [2]. This provides confidence in the validity of the LHC beam related simulations [4] as one may extrapolate the SPS results to the LHC case. Previously, we published simulation results of selected HiRadMat experiments [2] that were available at that time. We have now completed simulations of the remaining experiment and the results are presented in this paper, emphasizing the HED physics aspect of these experiments. Extensive theoretical work has previously been done to explore the potential of intense heavy ion beams to study HED physics [8–17]. However, the experiments proposed in these papers can not be realized soon because the Facility for Antiprotons and Ion Research (FAIR) at Darmstadt [18], is not yet ready. Ion beams have also been proposed as driver for inertial fusion [19–22]. Another important application of ion beams is the production of radioactive beams [23–26]. In Sec. 2, we present the experimental setup and the beam and the target parameters used in the simulations. The simulation results are presented in Sec. 3 while a brief comparison between the experimental measurements and the simulations is given in Sec. 4. Conclusions drawn from this work are noted in Sec. 5.

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The target assembly used in the experiments before it is installed in the HiRadMat facility is shown in Fig. 1. It consists of three targets, each comprised of fifteen copper cylinders with a spacing of 1 cm in between, to allow for visual inspection of the targets after irradiation. Each cylinder has a radius, r = 4 cm and length, L = 10 cm. The three target assemblies are enclosed in an aluminum housing with a top cover that provides stability to the setup and prevents contamination of the facility as well. The target assembly is mounted onto a movable table which can be moved to four different positions, namely to Target 1, Target 2, Target 3 and off-beam position, thereby leading to transverse irradiation of the left face of the first cylinder of the different targets used in the experiment. In these experiments the proton energy was 440 GeV, the bunch intensity was 1.5 × 1011 protons, the bunch length was 0.5 ns and the bunch separation was 50 ns. Target 1 was irradiated with 144 bunches with a beam focal spot characterized by σ = 2 mm (Experiment–1). Target 2 was irradiated with 108 bunches, whereas Target 3 was irradiated with 144 bunches. In both cases, the beam had a much smaller focal spot of σ = 0.2 mm. The latter two experiments were named Experiment–2 and Experiment–3, respectively. A summary of the beam parameters used in these three experiments is presented in Table I.

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2. TARGET AND BEAM PARAMETERS

FIG. 1: (color on-line) Three target assemblies used in the experiments, each comprised of 15 solid Cu cylinders, every cylinder has radius, r = 4 cm, length, L = 10 cm and 1 cm gap in between.

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TABLE I: Experimental Beam Parameters Used in the Three Experiments

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The temporal profile of the beam is presented in Fig. 2. In fact the protons were delivered in sets of 36 bunches each, while a separation of 250 ns was considered between the neighboring bunch packets. 3. NUMERICAL SIMULATION RESULTS

In the simulations we use the same beam parameters as in the experiments, but a slightly modified target design is considered to simplify the calculations. Instead of considering 15 cylinders in a target, we use a single copper cylinder that has a total length of 150 cm (equivalent

FIG. 2: (color on-line) Time structure of the proton beam.

length of copper) and a radius of 4 cm. This is a good approximation as the hydrodynamic effects in this type of problems are much stronger in the radial direction than in the axial direction. The simulations are done employing a fully integrated Monte Carlo energy deposition code, FLUKA[27, 28] and a two–dimensional hydrodynamic code, BIG2 [29], iteratively. First, the FLUKA code is used to calculate the proton energy loss in the target, assuming solid material density. This data is then used as energy input to

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In order to emphasize the importance of hydrodynamic effects in this type of problems, we also did beam– target simulations excluding hydrodynamics using only the FLUKA code. The results are presented in the following subsection.

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3.1 Static Simulations Using FLUKA

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In Fig. 3 we plot the specific energy deposition along the cylinder employing the FLUKA code, using the beam parameters of Experiment–1 and Experiment–3, respectively. It is seen that in case of Experiment–1, the target material is liquefied between 6 and 47 cm along the axis in the beam direction. This means that the target material remains solid up to 6 cm while the beam penetrates up to 47 cm. In case of Experiment–3, the target is liquefied up to 67 cm which represents the beam penetration distance. Moreover, the material between 2.8 and 32 cm is even evaporated.

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FIG. 3: Specific energy deposition along the target axis calculated by FLUKA [27, 28] (static approximation), case 1: 144 proton bunches, bunch intensity = 1.5 × 1011 protons, σ = 2 mm, Case 3: 144 proton bunches, σ = 0.2 mm, melting energy of copper = 613 J/g, boiling energy of copper = 5236 J/g.

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the BIG2 code that calculates the thermodynamic and the hydrodynamic response of the target. The proton bunches deposit energy in the target that leads to strong heating, which produces huge pressures in the absorption region. This high pressure generates a strong, outmoving radial shock wave that leads to density depletion at, and around the axis. As a consequence, the protons that are delivered in subsequent bunches, penetrate deeper into the target. Continuation of this process of so called hydrodynamic tunneling leads to a substantial prolongation of the projectile range. We note that good accuracy of the calculations requires that the energy deposition in the hydrodynamic calculations should not change too much between two successive iterations. This demands that the density should not be very different between two successive FLUKA calculations. Experience has shown that a 15 % density variation is reasonable for this purpose. Therefore when the density is reduced by about 15 % in the heated zone, the BIG2 code is stopped and the modified density distribution is used to calculate new energy loss distribution using FLUKA that is again used in BIG2 calculations in the next iteration. This procedure is continued till the last bunch hits the target. It is interesting to note that in the simulations using the beam parameters of Experiment–1, an iteration step of 2700 ns is sufficient, whereas in the simulations with beam parameters of Experiment–3, a much shorter iteration step of 700 ns must be used. This is because in the former case, the specific energy deposition along the axis is lower due to a larger beam focal spot that makes hydrodynamic processes slower compared to the latter case.

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3.2 Hydrodynamic Simulations Using FLUKA and BIG2 Iteratively

In Experiment–2 and Experiment–3 the beam had the same focal spot size, bunch length, bunch intensity and bunch separation. The only difference was the total number of bunches, 104 and 144 respectively. In Experiment– 1 we used 133 bunches, but the beam focal spot was 10 times larger that in the above two cases. Therefore to study the effect of the beam focal spot size on the hydrodynamic tunneling, a direct comparison between the results of Experiment–1 and Experiment–3 is sufficient. In Fig. 4(a), we present the two–dimensional specific energy deposition distribution in the target at t = 7850 ns, which is the end of the bunch train (144 bunches delivered), in case of Experiment–1 (σ=2.0 mm). Since energy deposition is a localized phenomenon, we only show the inner 1 cm radius of the target. It is seen that the maximum value of specific energy deposition is 1.8 kJ/g at the axis. In Fig. 4(b) we show the same parameter as in Fig. 4(a), but for Experiment–3 (σ=0.2 mm). In this case the maximum specific energy deposition is around 6 kJ/g at the axis, but the heated zone is significantly longer and thinner than the other case. This is because a smaller focal spot leads to a more focused energy deposition along the axis that makes hydrodynamic tunneling more effective. This leads to a longer beam penetration that increases the dimensions of the heated region in the longitudinal direction, as is seen in Fig. 4(b). The radius of the heated zone is smaller for obvious reasons. The temperature distribution in the target provided by the BIG2 code at t = 7850 ns corresponding to

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FIG. 4: (color on-line) Simulated target physical conditions provided by BIG2 after delivery of 144 proton bunches, (a) specific energy in Experiment–1 (σ=2.0 mm), (b) specific energy in Experiment–3 (σ=0.2 mm), (c) temperature in Experiment–1 (σ=2.0 mm) and (d) temperature in Experiment–3 (σ=0.2 mm).

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Experiment–1 and Experiment–3 is presented in Fig. 4(c) and 4(d), respectively. Fig. 4(c) shows that a maximum temperature of around 3500 K is generated at the target axis which means that the target central part will be seriously damaged. Fig. 4(d) shows that the maximum temperature is about 7600 K, but the heated zone is more localized in this case. In fact, the temperature distributions follow the same pattern as the corresponding energy depositions presented above.

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The high temperature generates high pressures in the heated material which leads to hydrodynamic phenomena that affect the entire target. It is therefore important to study the evolution of the pressure during, and after the irradiation. It Fig. 5(a), we present the pressure distribution provided by the BIG2 code at t = 7850 ns, corresponding to the beam and target parameters used in Experiment–1. It is to be noted that pressure evolution is not a localized phenomenon but it influences the target as a whole, it is therefore important to study the pressure in the entire target. It is seen in Fig. 5(a) that a maximum pressure of 1.22 GPa still exists at the tar-

get axis. It is worth noting that initially, the bunches deposit energy in solid material and the pressure continuously increases due to the rising temperature. However, when the material is liquefied and the density decreases at the axis and around the axis due to the outgoing radial shock wave, the pressure starts to decrease despite energy deposition by the protons delivered in the following bunches. In these simulations, a maximum pressure of about 2.6 GPa is achieved at t = 1000 ns, when 20 proton bunches have deposited their energy, but then it starts to decrease and at the end of the beam it becomes 1.22 GPa. It is also seen in Fig. 5(a) that the radial shock wave has already been reflected from the target boundary. Moreover, the pressure wave has already arrived at the opposite fact of the cylinder. This means the target is not directly heated beyond the penetration length of the beam and the shower, nevertheless the material structure could be changed due to the pressure wave. Fig. 5(b) shows the pressure distribution calculated by BIG2 using the parameters corresponding to Experiment–3. In these simulations, a maximum pres-

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FIG. 5: (color on-line) Simulated target physical conditions provided by BIG2 after delivery of 144 proton bunches, (a) pressure in Experiment–1 (σ=2.0 mm), (b) pressure in Experiment–3 (σ=0.2 mm), (c) density in Experiment–1 (σ=2.0 mm) and (d) density in Experiment–3 (σ=0.2 mm).

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sure of 3.2 GPa is generated on the axis at t = 600 ns, after the delivery of 12 bunches that is reduced to about 1.4 GPa at the end of the bunch train. The hydrodynamic effects generated by the high pressure lead to significant density variations in the target. In Fig. 5(c), we present the density distribution provided by the BIG2 simulations using the beam parameters of Experiment–1. Again we show only inner 1 cm radius of the cylinder where the density changes are significant. It is seen that a low density region with minimum density of 6.74 g/cm3 at the axis exists which extends to about 60 cm in the longitudinal direction and to 0.5 cm in the radial direction. This suggests moderate hydrodynamic tunneling of the protons and the shower. In Fig. 5(d) we show the density target distribution provided by BIG2 simulations at t = 7850 ns using the beam parameters corresponding to Experiment–3. It is seen that in this case, the affected density region extends to about 85 cm in the longitudinal direction and 0.25 cm in the radial direction. The minimum density at the axis is 0.85 g/cm3 . This of course, indicates significant hy-

drodynamic tunneling of the projectile particles. In Fig. 6(a) we present the phase change of the target during and after the beam impact provided by BIG2 simulations using beam parameters of Experiment–1. Semi– empirical EOS model provided in [30, 31] has been used for this purpose. It is seen that a significant part of the target material in the beam heated region has been liquefied that is surrounded by a melting zone with a two–phase solid and liquid mixture. Fig. 6(b) shows the same parameter as in Fig. 6(a), but corresponding to the beam parameters used in Experiment–3. In this case a small gaseous region surrounded by a two–phase liquid–gas zone exists around the axis. This is followed by a liquid region which is enclosed in melting layer. It is also interesting to note that in Fig. 6(b), the liquid region extends in the longitudinal direction and shrinks in the radial direction, as compared to Fig. 6(a). It is thus seen that in these experiments, the beam is powerful enough to generate interesting phases of HED matter including, two–phase solid–liquid phase, compressed as well as expanded hot liquid, two–phase

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FIG. 6: (color on-line) (a) Simulated material physical state provided by BIG2 using beam parameters corresponding to Experiment–1 (σ=2.0 mm), (b) simulated material physical state provided by BIG2 using beam parameters corresponding to Experiment–3 (σ=0.2 mm)

the peak. In Fig. 7(b), the simulated temperature profiles corresponding to the energy profiles shown in Fig. 7(a) are shown. Curve 1 shows that in Experiment–3, after the delivery of all the 144 bunches, a maximum temperature of about 7000 K is achieved along the axis up to about 40 cm, which represents two–phase liquid–gas region. The temperature then steadily decreases and the temperature curve becomes flat, which represents melting of the material. The melting region is located between axial position, L = 85–90 cm. Curve 2 shows that the maximum simulated temperature along the axis, achieved using the beam parameters of Experiment–1, is around 3500 K. The melting zone, in this case, lies between axial position, L = 45–55 cm. Curve 3 shows that the simulated temperature in Experiment–3 at a radial position of 5 mm, is of the order of 950 K that means that the material at this position remains in solid phase. Curve 4, on the other hand, shows that in Experiment–1, the peak temperature at radial position 5 mm, is above 1400 K that leads to melting of the material in this region, as shown by the flat part of the curve. Fig. 7(c) presents density profiles corresponding to the cases plotted in Fig. 7(a). It is seen from profile 1 that in case of Experiment–3, the minimum density at the axis is about 0.8 g/cm3 , whereas curve 2 shows that the minimum density in case of Experiment–1 at the axis is around 6.5 g/cm3 . The corresponding density profiles at radial position of 5 mm are shown in curve 3 and curve 4, respectively. The beam is incident perpendicular to the face of the first cylinder coaxially and the projectile particles, together with their hadronic shower, penetrate deep into the target. A visual inspection of the Al target cover revealed some interesting features of the experiment. A photograph of the inner surface of the cover is presented

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liquid–gas state and gaseous phase. One can therefore emphasize that the HiRadMat facility is suitable for EOS studies of HED matter. It is also worth noting that at present, this is the only proton/ion beam facility worldwide that provides enough beam power to do this type of research. To have more quantitative information about the physical conditions achieved in the experiments, it is useful to study one–dimensional profiles of the physical parameters. In Fig. 7(a), we plot the simulated specific energy deposition along the target length at t = 7850 ns which is the end of the bunch train. Curve 1 represents specific energy deposited along the axis provided by BIG2 simulations using the parameters corresponding to Experiment–3, that shows a maximum value of about 6.4 kJ/g. curve 2 shows the same parameter as curve 1, but using the beam parameters of Experiment– 1. In this case the maximum value of the energy deposition is 1.8 kJ/g. It is to be noted that in Experiment–3, the beam is strongly focused with a σ = 0.2 mm while in Experiment–1, the beam is 10 times less focused with σ = 2.0 mm. This results in a much higher beam energy density in the former case as compared to the latter that leads to such big difference in the energy deposition behavior. In curve 3, the specific energy deposition vs cylinder length at a radial position, r = 5 mm is plotted, which is much larger than the effective beam radius, for the beam with σ = 0.2 mm (Experiment–1). Therefore the energy deposition at this point is only due to the secondary particles and is of the order of 0.3 kJ/g. Curve 4 represents the same parameter as curve 3, but using the beam parameters of Experiment–1. Due to the less focusing of the beam, the considered radial position is closer to the effective beam radius as compared to Experiment–3 that leads to a higher energy deposition of about 0.55 kJ/g at

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FIG. 8: (color on-line) Top cover of the experimental setup after the irradiation. Traces of projected copper between the 10 cm long cylinders of the targets indicate the length of the melting/evaporation zone. For Target–1 (bottom) the molten/evaporation zone ends in the 6 th cylinder, i.e. the copper was molten over a length of 55 ± 5 cm. For Target–2 (mid) the molten zone goes up to cylinder 8, i.e. 75 ± 5 cm. For Target–3 (top) the molten zone goes up to cylinder 9, i.e. 85 ± 5 cm.

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molten copper occurs up to the gap between the fifth and the sixth cylinder. That means that the material was molten/evaporated over a length of 55 ± 5 cm. In Experiment–2 with 108 bunches and beam focal spot, σ = 0.2 mm (middle picture), the molten/evaporation zone goes up to the eighth cylinder that means a damage length of 75 ± 5 cm. In Experiment with 144 bunches and beam focal spot, σ = 0.2 mm (top picture), the molten/evaporation zone is extended to the ninth cylinder, that means a damage length of 85 ± 5 cm.

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FIG. 7: (color on-line) (a) Specific energy deposition profiles along the target length for different cases, (b) temperature profiles along target length for different cases and (c) density profiles along target length for different cases.

in Fig. 8 that shows traces of copper deposited above the gaps between the cylinders. It is seen that in case of Experiment–1 that used 144 bunches and beam focal spot σ = 2.0 mm (bottom picture), the splash of

TABLE II: Comparison between first measurements and static simulations (no hydrodynamics) using FLUKA [27, 28], Sstatic is simulated penetration length, Mf irst is first measured penetration length. Target Sstatic (cm) 1 47 2 64 3 67

Mf irst ∆ Mf irst (cm) to Sstatic (cm) 55 ± 5 8±5 75 ± 5 11 ± 5 85 ± 5 18 ± 5

In Table II we present a comparison between the penetration length of the protons and the shower obtained in these first measurements, Mf irst and the expected penetration length, Sstatic based on a static model (no hydrodynamic) using the energy deposition code FLUKA [27, 28]. It is seen that a large discrepancy exists between the experimental measurements and the simulation results. In order to have a better understanding of the experimental measurements, we carried out full hydrodynamic calculations using the FLUKA and the BIG2 codes iteratively, as explained above. In Fig. 9(a), we plot the density and temperature vs axis at t = 7850 ns (time when 144 bunches have been delivered) obtained from the BIG2 code using the beam

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of the temperature curve that represents melting region lies within L = 45 and 55 cm which is equivalent to the right half of the fifth cylinder and the left half of the sixth cylinder. The liquefied material escapes from the left face of cylinder number 6 and collides with the liquefied material ejected from the right face of cylinder number 5. As a result of this collision, the material is splashed vertically and is deposited at the inner surface of the target cover above the gap between cylinder number 5 and 6. The simulations are therefore in full agreement with the experimental observations. In FIG. 9(b), are plotted the density the temperature, along the axis at t = 5800 ns obtained from BIG2, when 108 bunches have been delivered for using the beam parameters of Experiment–2 (σ=0.2 mm). It is seen that in this case, the flat part of the temperature curve that represents melting region, lies within L = 75 and 80 cm that is equivalent to the right half of the eighth cylinder. The temperature curve also shows that the material along the axis up to 75 cm is liquefied or even evaporated, depending on the value of the temperature. The liquefied material escapes from the left face of cylinder number 8 and collides with the melted/gaseous material ejected from the right face of cylinder number 7. The splashed material is thus deposited at the inner surface of the target cover above the gap between cylinder number 7 and 8. The simulations are also in full agreement with the measurements of Experiment–2. It is also interesting to note that there is a very weak trace of matter in the bottom half of the middle strip (Target–2 in FIG. 8) above the gap between cylinders 8 and 9. This in fact is a tiny amount of matter that originates from the gap between cylinder 8 and 9 of Target–3 and is deposited at this place. Otherwise, the trace would be much thicker and would be along the full length of the gap. Moreover, visual inspection of the target has shown that the opposite faces of cylinders 8 and 9 in Target–2 are not damaged. Fig. 9(c) shows same variables as Fig. 9(b), but at t = 7850 ns when 144 bunches have been delivered. The melting region now lies between L = 85 and 90 cm, which is the right half of cylinder 9 while the left half part (L = 80 – 85 cm) has been liquefied. The simulations thus predict material deposition at the inner surface of the target cover above the region between cylinder 8 and 9, which is in full agreement with the measurements of Experiment–3. These experiments therefore confirm the existence of the hydrodynamic tunneling in case of the SPS beam in accordance with the theoretical predictions. The results thus give confidence in the numerical simulations for the more powerful LHC beam reported in [4]. A comparison between these simulations and the experimental measurements is presented in Table III. Detailed microscopic analysis done after dissection of a few cylinders revealed the precise penetration length of the projectile particles and the damage limit [3]. We note

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FIG. 9: (color on-line) Density and temperature vs axis, (a) Experiment–1 [σ=2.0 mm, 144 bunches], (b) Experiment–2 [σ=0.2 mm, 108 bunches] and (c) Experiment–3 [σ=0.2 mm, 144 bunches],

parameters of Experiment–1. It is seen that the flat part

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Target Shydro (cm) 1 56 2 75 3 85

Mf irst ∆ Mf irst (cm) to Shydro (cm) 55 ± 5 1±5 75 ± 5 0±5 85 ± 5 0±5

Target Shydro Mf inal ∆ Mf inal (cm) (cm) to Shydro (cm) 1 56 58 2 2 80 79.5 0.5 3 90 85 5

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TABLE IV: Comparison between final measurements and full hydrodynamic simulations using FLUKA [27, 28] and BIG2 [29] iteratively, Shydro is simulated penetration length, Mf inal is final measured penetration length.

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that from the machine protection point of view, melting is also considered as material damage. Using this criterion as damage limit in the simulations, we present a comparison between the measured and simulated results in Table IV. The two show very good agreement that provides confidence in the accuracy of the simulations.

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CONCLUSIONS

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Experiments were carried out at the CERN HiRadMat facility on beam–matter heating using the 440 GeV proton beam generated by the SPS, irradiating large solid copper cylindrical target. The beam was delivered in form of a high intensity bunch train while each bunch comprised of 1.5 × 1011 protons. The bunch length was 0.5 ns and two neighboring bunches were separated by 50 ns. The intensity distribution in the transverse direction was Gaussian so the focal spot size could be characterized with standard deviation, σ. Three experiments were done using different beam parameters. In Experiment–1, the target was irradiated with 144 bunches having a relatively large beam focal spot, with σ = 2.0 mm. In Experiment–2, we used 108 bunches but the beam was 10 times better focused, with σ = 0.2 mm. Experiment–3 used the same beam parameters as Experiment–2, except the bunch number, which was 144 in this case. These experiments confirmed the existence of significant hydrodynamic tunneling in case of the SPS protons which prolongs the penetration length

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of the projectile particles and their hadronic shower. It was also confirmed that with larger number of bunches and with better focusing of the beam, this effect becomes more and more pronounced. Numerical simulations were also performed which used exactly the same beam parameters as in the experiments, while a slightly modified target design was considered for the simplicity of calculations. These simulations were done using the energy deposition code FLUKA and a two–dimensional hydrodynamic code, BIG2, iteratively. The simulation results showed very good agreement with the experimental measurements in all the three cases. This provides confidence in similar simulation studies that have been done for the Large Hadron Collider and the Future Circular Collider. The numerical simulations also show that a significant part of the target was converted into different phases of HED matter including, two-phase solid–liquid mixture, compressed as well as expanded hot liquid state, two– phase liquid–phase region and gaseous state. The HiRadMat test facility, therefore, is very much suited to study EOS problems related to HED physics. In fact, this is currently the only facility available worldwide to do this type of research.

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TABLE III: Comparison between first measurements and full hydrodynamic simulations using FLUKA [27, 28] and BIG2 [29] iteratively, Shydro is simulated penetration length, Mf irst is first measured penetration length.

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