Review of hydrodynamic tunneling issues in high power particle accelerators

Nuclear Inst, and Methods in Physics Research B 427 (2018) 70–86

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Nuclear Inst. and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb

Review of hydrodynamic tunneling issues in high power particle accelerators

T

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N.A. Tahira, , F. Burkartb, R. Schmidtb, A. Shutovc, A.R. Pirizd a

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

A R T I C LE I N FO

A B S T R A C T

Keywords: High power accelerators Hydrodynamic tunneling High energy density physics Warm dense matter Beam-matter heating

Full impact of one Large Hadron Collider (LHC) 7 TeV proton beam on solid targets made of different materials including copper and carbon, was simulated using an energy deposition code, FLUKA and a two-dimensional hydrodynamic code, BIG2, iteratively. These studies showed that the penetration depth of the entire beam comprised of 2808 proton bunches significantly increases due to a phenomenon named hydrodynamic tunneling of the protons and the shower. For example, the static range of a single 7 TeV proton and its shower is about 1 m in solid copper, but the full LHC beam will penetrate up to about 35 m in the target, if the hydrodynamic effects were included. Due to the potential implications of this result on the machine protection considerations, it was decided to have an experimental verification of the hydrodynamic tunneling effect. For this purpose, experiments were carried out at the CERN HiRadMat (High Radiation to Materials) facility in which extended solid copper cylindrical targets were irradiated with the 440 GeV proton beam generated by the Super Proton Synchrotron (SPS). Simulations of beam-target heating considering the same beam parameters that were used in the experiments, were also performed. These experiments not only confirmed the existence of the hydrodynamic tunneling, but the experimental measurements showed very good agreement with the experimental results as well. This provided confidence in the work on LHC related beam-matter heating simulations. Currently, a design study is being carried out by the international community (with CERN taking the leading role) for a post LHC collider named, the Future Circular Collider (FCC) which will accelerate two counter rotating proton beams up to a particle energy of 50 TeV. Simulations of the full impact of one FCC beam comprised of 10,600 proton bunches with a solid copper target have also been done. These simulations have shown that although the static range of a single 50 TeV proton and its shower in solid copper is around 1.8 m, the entire beam will penetrate up to about 350 m in the target. Feasibility studies of developing a water beam dump for the FCC have also been carried out. A review of this work and its implications on machine protection system are presented in this paper.

1. Introduction Intense particle beams delivered by high power accelerators are an efficient tool to research many areas of basic and applied physics including, particle physics, nuclear physics, High Energy Density (HED) physics, Inertial Confinement Fusion (ICF) and many others. However, the protection of equipment from beam impact is a very important issue when dealing with such powerful beams. The most powerful accelerator is the CERN Large Hadron Collider (LHC), which accelerates two counter rotating proton beams up to a particle energy of 7 TeV. Each of these beams carries 362 MJ energy, sufficient to melt 500 kg copper. An accidental release of even a small fraction of this energy can seriously

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Corresponding author. E-mail address: [email protected] (N.A. Tahir).

https://doi.org/10.1016/j.nimb.2018.04.009 Received 22 January 2018; Received in revised form 9 April 2018; Accepted 9 April 2018 Available online 04 May 2018 0168-583X/ © 2018 Elsevier B.V. All rights reserved.

damage accelerator components. One of the worst case scenarios is if the total beam energy is lost at a single point. The probability for such an accident is very low, nevertheless it is important to understand the consequences, if it ever happens. For this purpose the full impact of one LHC beam on solid copper and solid carbon targets were simulated using an energy deposition code FLUKA [1], together with a two-dimensional hydrodynamic code, BIG2 [2]. Initially, these calculations were performed using some approximations [3–5], but later, the problem was simulated using the two codes iteratively [6]. In this scheme, first the FLUKA code is used to calculate the specific energy deposition assuming solid target density. This data is then used as input to the BIG2 code to calculate the

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thermodynamic and the hydrodynamic response of the target. As the successive proton bunches and their shower deposit energy in the target, the temperature rises rapidly that leads to high pressure along the beam path. This high pressure generates radially outgoing shocks that cause density depletion along the path. As a consequence, the protons that are delivered in subsequent bunches penetrate deeper into the target. The process continues till the last bunch hits the target. The penetration depth of the beam and the shower is substantially increased as compared to the static range of a single proton and its shower. This phenomenon has been named as hydrodynamic tunneling of the beam and it has important implications on the machine protection system design. In our calculations, when the density along the beam path is reduced by 10–15%, the BIG2 code is stopped and the modified density distribution generated by BIG2 is used in the FLUKA code to calculate the modified energy deposition, which is then used in BIG2 as energy input for the next iteration. In this manner the two codes are used iteratively until the last bunch hits the target. The phenomenon of hydrodynamic tunneling was first observed in simulation studies of the interaction of 20 TeV proton beam with a solid carbon beam dump for the Superconducting Super Collider (SSC) [7,8]. This effect was later observed when simulating the LHC proton beam interaction with solid targets. It was estimated that although the static range of a single 7 TeV proton and its shower is about 1 m in solid copper, the full LHC beam which is comprised of 2808 proton bunches may penetrate up to 40 m into the target [3]. More advanced simulations showed that the penetration length was about 35 m [5,6]. This result has important implications on machine protection design for LHC. In particular, the hydrodynamic tunneling effect needs to be considered for the design of a sacrificial beam stopper, if deemed to be necessary. In order to have confidence in these simulations, it was necessary to have experimental verification of the hydrodynamic tunneling. For obvious reasons it was not possible to do such experiments using the LHC beam. It was thus decided to carry out beam-heating experiments at the CERN HiRadMat (High Radiation to Material) facility using the 440 GeV proton beam generated by the Super Proton Synchrotron (SPS). The SPS is used as injector to the LHC and has the same bunch structure as the LHC. These experiments provided first experimental confirmation of the hydrodynamic tunneling [9,10]. The measurements showed very good agreement with the numerical simulations [11,12]. Numerical simulations of the beam-matter heating problem for the Future Circular Collider (FCC) proton beam using the same technique of employing the FLUKA and the BIG2 codes iteratively have also been performed. These studies showed that although the static range of a single FCC proton and its shower is around 1.8 m, the full FCC beam comprising of 10,600 proton bunches will penetrate up to 350 m in solid copper [13,14]. In this paper we present a review of the theoretical and experimental work done to study the phenomenon of hydrodynamic tunneling. We note that most of the work in this field has been done during the past two decades by the authors of the present paper. This problem was also studied at the Fermi Lab in connection to the Superconducting Super Collider (SCC) in the 1980’s. It is worth noting that in all the above studies, a substantial part of the target was converted into High Energy Density (HED) matter. Extensive theoretical work has been reported elsewhere [15–27] to assess the feasibility of intense particle beams to research the important field of HED physics. These studies have shown that high power particle beams are an efficient and versatile tool to generate large samples of HED matter in the laboratory. Moreover, due to the high efficiency and high repetition rate of the particle accelerators, bunched high power particle beams are considered to be a viable driver for any future ICF reactor [28–36]. Production of rare radioactive isotopes is another important branch of physics that has made significant advances due to the availability of high power particle beams [37–39]. In Section 2 we present the nominal parameters of the SPS, the LHC

Table 1 Comparison between SPS, LHC and FCC Beam Parameters.

Proton Energy (TeV) Bunch Intensity Bunch Length (ns) Bunch Separation (ns) Number of Bunches Focal Spot σ (mm) Beam Duration (μ s) Total Beam Energy (GJ) Accelerator Tunnel Circumference (km)

Super Proton Synchrotron

Large Hadron Collider

Future Circular Collider

0.45 1.5× 1011 0.5 25 (50) 288 0.2 7.2

7 1.15× 1011 0.5 25 2808 0.2 89 0.362

1011 0.5 25 (5) 10,600 0.2 265 8.5

28

100

3.8 × 10−3 6.9

50

and the FCC beams. The physics included in the FLUKA and the BIG2 codes is briefly described in Section 3. The historical perspective of the work is discussed in Section 4, whereas a review of the work related to the simulations of impact of the LHC beam on solid targets of different materials is presented in Section 5. A review of the beam-matter experimental work at the HiRadMat facility is given in Section 6 while in Section 7we review the work related to the FCC beam-target impact simulations. Conclusions drawn from this work are noted in Section 8. 2. Nominal beam parameters for the SPS, LHC and FCC In Table 1 we present the nominal beam parameters for SPS, LHC and FCC. The SPS synchrotron is used as LHC injector, but also to accelerate and extract protons and ions (such as lead and other ion species) for fixed target experiments and for producing neutrinos (CNGS). When the SPS operates as LHC injector, up to 288 bunches are accelerated, each bunch with 1.15 × 1011 protons (nominal parameters). The bunch length is 0.5 ns and two neighboring bunches are separated by 25 ns so that the duration of the entire beam is 7.2 μ s. The maximum bunch intensity for fixed target experiments could be up to 1.5 × 1011 protons. There is an option of 50 ns bunch separation as well. The focal size can go down to 0.1 mm σrms , thus providing a very dense beam (energy/size). The beam size can be tuned from 0.1 mm to 2 mm. The beam is accelerated in a tunnel with a circumference of 6.9 km while the total beam energy is about 3.8 MJ. The LHC accelerates two counter rotating proton beams in a 28 km long circular tunnel to a particle energy of 7 TeV. Each beam is comprised of 2808 proton bunches and it carries an energy of 362 MJ. The bunch intensity is 1.15 × 1011 protons, bunch length is 0.5 ns and two neighboring bunches are separated by 25 ns. The duration of the bunch train is 89 μ s. The FCC is designed to accelerate two counter rotation proton beams to maximum particle energy of 50 TeV. The circumference of the accelerator tunnel is around 100 km. Each beam is comprised of 10,600 bunches with each bunch having an intensity of 1011 protons. Bunch length is considered to be 0.5 ns while neighboring bunches are separated by 25 ns. It is to be noted that there is an option of 5 ns bunch separation as well. The total energy stored in each beam is about 8.5 GJ, which is equivalent to the kinetic energy of an Airbus A380 (560 t) flying at a typical speed of 850 km/h. It is worth noting that the FCC design study also includes an option for an electron-positron (e + e−) collider to be installed in the FCC tunnel before the proton–proton (pp), the e + e− collider could be used as Higgs factory. However, in the present paper we only deal with the proton-proton case. In Table 2 we present the physical parameters achieved in solid copper due to energy deposited by one bunch of the SPS, the LHC and the FCC, respectively. Although the SPS can accelerate the protons to a maximum particle energy of 450 GeV, in this study we consider 440 GeV energy because that was the value used in the experiments 71

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calculated by the SRIM code [40] is used as energy input. The elastic plastic effects are treated assuming an ideal plasticity model based on Hooks law complemented with von Mises yield criterion. Different phases of the target material during and after the irradiation are treated using a semi-empirical EOS model [41–43].

Table 2 Physical parameters induced in solid copper by SPS, LHC and FCC beams.

Considered Proton Energy (TeV) Bunch Intensity Specific Energy Deposition by a Single Proton Calculated by FLUKA (GeV/g) Specific Energy Deposition by one Proton Bunch (kJ/g) Temperature (K) Calculated by BIG2 Pressure (GPa) Calculated by BIG2

Super proton synchrotron

Large hadron collider

Future circular collider

0.44

7

40

1.5 × 1011

1.15 × 1011

1011

3.6

134

960

0.09 515

2.4 5019

14.4 27440

1.50

30

95

4. A historical perspective of the work In this section we present a historical overview of the work that introduced the phenomenon of hydrodynamic tunneling of the particle beam and the shower. 4.1. Superconducting Super Collider Numerical simulations of interaction of 20 TeV Superconducting Super Collider (SSC) proton beam with a carbon beam dump were reported in [7,8]. These simulations were performed using a Monte Carlo program MARS [44] together with a two- and three-dimensional hydrodynamic code, MESA [45]. The purpose of these studies was to assess the damage caused to the beam dump in case of the failure of the high frequency deflecting magnet, which can result in dumping of the beam at a single spot. In these calculations, the beam duration was assumed to be 290 μ s and two fluences, namely, 4.5 × 1017 and 1.0 × 1019 protons/s were considered. These correspond to a total number of 1.3 × 1014 and 2.9 × 1015 protons, respectively. The standard deviation of the beam’s transverse Gaussian profile was considered to be 2 mm. The calculations showed significant beam penetration in the beam dump due to hole boring (hydrodynamic tunneling). In case of the lower fluence, the penetration wave proceeded with a speed of 7 cm/μ s, which meant a penetration distance of around 20 m. In case of the higher fluence, the penetration speed was 70 cm/μ s, that implied that the beam would penetrate up to about 200 m in the material. This was not acceptable because the length of the beam dump was only 8 m.

described later. According to the FLUKA calculations, a single proton deposits an energy of 3.6 GeV/g in solid copper. One SPS bunch comprised of 1.5 × 1011 protons will deposit a specific energy of 0.09 kJ/g in the target. According to the semi-empirical Equation-of–State (EOS) model used in these studies [41–43], a temperature of about 500 K and a pressure of 1.5 GPa are induced in the target. The table also shows that one LHC bunch deposits a specific energy of 2.4 kJ/g in solid copper that generates a temperature of about 5000 K and a pressure of 30 GPa. It is to be noted that when the work on FCC beam-matter heating was started, the final FCC beam parameters were not yet fixed. At that time, it was considered that 40 TeV is a reasonable energy. Therefore, the calculations were done using a particle energy of 40 TeV, but the results were later extrapolated to the final design energy of 50 TeV. Table 2 shows that one FCC bunch with 40 TeV particle energy will deposit about 14.5 kJ/g specific energy in a solid copper target that will generate a temperature of about 27,500 K and a pressure of around 1 Mbar.

4.2. Large Hadron Collider This problem was addressed in the case of the LHC 7 TeV proton beam using the energy deposition code FLUKA [1] and a two–dimensional hydrodynamic code, BIG2 [2]. Initially, a simple approximation was considered in which the FLUKA code was employed to calculate the specific energy deposition of a single 7 TeV proton assuming solid copper density. These calculations have shown that the proton and the shower it generates penetrates up to 1 m in solid copper and the maxima of the energy deposition distribution lies at a longitudinal position of 16 cm at the axis. Considering the maximum value of the energy deposition, the BIG2 code was used to calculate the hydrodynamic and the thermodynamic response of the material in the cylinder cross section at L = 16 cm [3,4]. These studies lead to an estimated penetration distance of about 40 m in solid copper by the entire LHC beam. The work was later improved by performing the BIG2 calculations in a length-radius plane of the cylinder assuming an axial symmetry, while the energy deposition distribution provided by the FLUKA code was normalized with the line density along the axis in each simulation cell at every time step. These calculations showed that the LHC beam and the shower will penetrate about 35 m into the solid copper target [5]. In these calculations the standard deviation, σ , of the Gaussian transverse intensity distribution was assumed to be 0.2 mm. Finally, a major improvement was made in these studies by using the FLUKA and the BIG2 codes iteratively. In this scheme, the FLUKA code was employed to calculate the energy deposition assuming a solid target density. This energy deposition data was used in the BIG2 code as energy input and the code was allowed to calculate the thermodynamic and the hydrodynamic response of the target. The code was stopped when the density along the axis was reduced by 10–15% due to the radial shock wave. The modified density distribution provided by the

3. Physics included in the FLUKA and the BIG2 codes FLUKA is a fully integrated particle physics and multi-purpose Monte Carlo simulation package, capable of simulating all components of the particle cascades in matter up to TeV energies. FLUKA has many applications in high energy experimental physics and engineering, shielding, detector and telescope design, cosmic ray studies, dosimetry, medical physics and radio-biology, as well as allows to simulate the interaction of beams with matter over a very wide energy range. The most relevant energy range for these applications being:

• Hadron and ion beams from as low as a few MeV/u 10,000 TeV/u. • Neutrons down to thermal energies. • Electromagnetic radiation from 1 keV up to 10,000 TeV. • Muons up to 10,000 TeV.

up to

More details about the used models and their performances, as well as a vast amount of benchmarking can be found in [1]. It is to be noted that the models used in FLUKA also include nuclear size correction to the stopping power at very high energies. BIG2 is a two-dimensional hydrodynamic computer code based on a Godunov type numerical scheme. It uses a stable and versatile numerical mesh that can handle simple as well as complicated geometries of single and multi-layered targets. A heat conductivity package and an ion beam energy deposition module are also included. In case of the SPS, LHC and FCC proton beams, energy loss data provided by the FLUKA code is taken, whereas for the heavy ions, the energy loss data 72

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Table 3 Overview of LHC calculations. Cu Target FLUKA: Energy deposition at solid density. BIG2: Hydrodynamic calculations along radius at one fixed point on axis [3]

C Target

Hydrodynamic calculations in r-z geometry, energy deposition in BIG2 normalized with axial line density (analytic approximation) [5]

Hydrodynamic calculations in r-z geometry, energy deposition in BIG2 normalized with axial line density (analytic approximation) [46]

Hydrodynamic calculations in r-z geometry running FLUKA and BIG2 codes iteratively [6]

0.2 mm was considered. Feasibility studies of the development of a water beam dump for the FCC have also been carried out. For this purpose the full impact of 50 TeV FCC proton beam on a cylindrical water target has been calculated using the FLUKA and the BIG2 codes iteratively. The σ of the transverse Gaussian intensity distribution in the focal spot is considered to be 0.4 mm. These studies have shown that the length of the water tube should be about 1.3 km in order to stop the entire beam. An overview of these calculations is presented in Section 7. 5. A review of beam-matter heating simulations using the LHC beam In this section, an overview of the simulation studies of the full impact of one LHC beam on cylindrical targets made of solid copper as well as carbon is presented. 5.1. Copper target irradiated by the LHC beam

Fig. 1. Energy deposition per proton in solid copper as a function of depth into the target and radial coordinate.

The FLUKA code [1] was employed to calculate the energy deposition by 7 TeV protons in a solid copper cylindrical target having a radius of 1 cm and a length of 5 m [3]. The energy deposition was obtained using a realistic two-dimensional beam distribution, namely, a Gaussian beam σ (horizontal and vertical σrms = 0.2 mm) that was incident perpendicular to the front face of the cylinder. The calculated local energy deposition (bin size radial: 0.01 cm, longitudinal: 2 cm) per proton caused by showering in the first part of the block is shown in Fig. 1. The energy density deposited in cylinders of radius 0.01, 1, and 100 cm is shown as a function of the depth in the target in Fig. 2. The specific energy deposited by a single LHC bunch along the axis is shown in Fig. 3. The longitudinal peak of the energy deposition is about 1200 GeV/ cm3/proton and occurs at about 16 cm. After 1.5 m the energy deposition is a factor of 1250 lower than in the peak which indicates that a 1.5 m length of copper is already long enough to effectively stop a single LHC bunch. For comparison, the inelastic nuclear scattering length of copper at 7-TeV protons is about 15 cm, so the primary proton beam will be attenuated after this distance by a factor of exp(150/ 15) = 22,000.

BIG2 code was used in the FLUKA code to calculate modified energy deposition distribution, which was again used in the BIG2 code. This process continued until the last bunch hit the target. Using this technique, impact of the LHC beam on a solid carbon cylindrical target was simulated assuming a focal spot σ of 0.4 mm. These calculations showed that although the static range of a single 7 TeV proton and its shower in solid carbon is around 3 m, the full LHC beam will penetrate up to about 25 m in the target [6]. An overview of the history of the LHC beam-target simulation work is given in Table 3. A review of these calculations will be presented in Section 5. 4.3. Beam-matter heating experiments at the CERN HiRadMat facility Due to the potential important implications of the hydrodynamic tunneling on machine protection, it was decided to have experimental confirmation of the simulations. Beam–matter heating experiments were performed at the CERN HiRadMat facility using the SPS 440 GeV proton beam that irradiated extended solid copper targets. Two different focal spot sizes characterized by σ = 0.2 mm and 2 mm, respectively, were considered while the beam comprised of 144 bunches. Numerical simulations of these experiments were also performed using the FLUKA and the BIG2 codes iteratively. The simulations showed excellent agreement with the experimental measurement which has provided confidence in the LHC beam-heating simulations [9–12]. A review of this work will be given in Sec. VI. 4.4. Future Circular Collider Recently, the full impact of the FCC beam on solid copper cylindrical target was simulated employing the FLUKA and the BIG2 codes iteratively [13,14]. In these calculations a particle energy of 40 TeV was considered and they showed that although the static range of a single 40 TeV proton and its shower in solid copper is around 1.6 m, the full FCC beam will penetrate up to about 295 m in the target. An extrapolation of these results to a proton energy of 50 TeV leads to a penetration distance of the order of 350 m. In these studies a value for σ of

Fig. 2. Energy deposition along the beam axis in solid copper as a function of the depth into the target averaged over circles of 0.01, 1 and 100 cm radius. 73

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At 16 cm in the target along the beam

Temperature (K)

100000

t = 500 ns t = 1000 ns t = 1500 ns t = 2000 ns t = 2500 ns

10000

1000 Fig. 3. Specific energy deposition versus depth into the target by a single LHC bunch with 1.15 ×; 1011 protons.

0

0.2 0.4 0.8 0.6 Target Transverse Coordinate (cm)

1

(a)

First estimates for the time-dependent target response were made using the FLUKA results as energy input for BIG2. Fig. 4 shows the specific energy deposition profiles versus the transverse coordinate of the target at different longitudinal positions, at t = 1 ns that corresponds to the impact of the first bunch. The energy deposition profiles at later times were obtained by scaling these curves by the appropriate number of bunches (40 bunches for 1 μ , 80 for 2.0 μ and so on). In view of the three-dimensional nature of the problem and the limitation that BIG2 code is two dimensional, the target response along the radial direction at longitudinal positions, L = 16 cm and L = 36 cm, respectively, were studied [3,4]. Fig. 4(a) shows the target temperature along the transverse coordinate of the target at a longitudinal position of 16 cm at different times. At t = 2500 ns, when 100 bunches have been delivered, the temperature at the center of the beam-heated region is of the order of 10 eV. The pressure and density profiles corresponding to these temperature distributions are shown in Fig. 4(b) and (c), respectively. It is seen from Fig. 4(b) that the pressure at t = 500 ns (within the inner 4 mm of the heated region) is of the order of 35 GPa. This drives a shock wave outwards which removes material from the central part of the heated region leading to a reduction in the target density. This is clearly seen from the density curve at t = 500 ns in Fig. 4(c). As the shock wave spreads out, the density decreases further. It is also seen from Fig. 4(c) that at t = 2500 ns, the density at the center of the beam-heated region has been reduced by a factor of about 10 compared to the initial solid density. Note that by this time only the energy from 100 out of 2808 bunches has been deposited. Under these conditions the subsequent bunches will encounter very less target mass to interact with and will therefore penetrate deeper into the target. The substantial target density reduction observed from the simulations clearly demonstrates that the dynamic density effect must be considered in order to have a reasonable estimate of the penetration depth. Other longitudinal positions were also considered and results were reported in [3,4]. With the help of simple analytic approximations and using the FLUKA and BIG2 simulations, it was estimated that the full LHC beam with 2808 bunches will penetrate between 10 and 40 m in solid copper. Details can be found in [3]. A major progress in this research was made when BIG2 was employed to simulate the beam-matter heating considering a solid copper cylindrical target having a radius = 5 cm and a length = 5 m. The beam was incident perpendicular to one face of the cylinder and the calculations were done in the length-radius plane assuming axial symmetry. The FLUKA energy deposition data shown in Fig. 1 was used and the energy deposition in the target was normalized with the line density along the axis. This allowed modeling of the hydrodynamic tunneling of the protons and the shower in the material [5]. Fig. 5(a) and (b) present the specific energy deposition distributions in the target on length-radius plane at t = 500 ns and 9500 ns,

At 16 cm in the target along the beam

40

Pressure (GPa)

30

t = 500 ns t = 1000 ns t = 1500 ns t = 2000 ns t = 2500 ns

20

10

0 0

0.2 0.4 0.8 0.6 Target Transverse Coordinate (cm)

1

(b) At 16 cm in the target along the beam

10

3

Density (g/cm )

8 6 4

t = 500 ns t = 1000 ns t = 1500 ns t = 2000 ns t = 2500 ns

2 0 0

0.2 0.4 0.8 0.6 Target Transverse Coordinate (cm)

1

(c) Fig. 4. Physical parameters vs transverse coordinate at a longitudinal position of 16 cm at different times, (a) temperature, (b) pressure and (c) density.

respectively as produced by the BIG2 code. These two graphs represent the delivery of 20 and 380 LHC bunches. It is seen in Fig. 5(a) that a maximum specific energy of about 19 kJ/g has been deposited at the axis and the beam penetrates up to about 1.3 m. Fig. 5(b) shows that the beam has penetrated up around 4.5 m due to the hydrodynamic tunneling, while the maximum specific energy deposition is of the order of 25 kJ/g. 74

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Fig. 6. Temperature distribution on depth-radius plane by BIG2 (a) at t = 500 ns [20 bunches delivered] and (b) at t = 9500 ns [380 bunches delivered].

Fig. 5. Specific energy deposition distribution on depth–radius plane by BIG2 (a) at t = 500 ns [20 bunches delivered] and (b) at t = 9500 ns [380 bunches delivered].

to a density of 9.35 g/cm3 in the radial direction due to the shock propagation. Fig. 8(b) shows that after the delivery of 380 out of 2808 proton bunches, a large low-density cavity has been created along and around the axis and the entire length of 5 m has been affected by the hydrodynamic tunneling. The minimum density is about 0.1 g/cm3 which is about 1% of the solid copper density. The material in this region is weekly ionized and is in strongly coupled plasma state. In fact a substantial part of the target is converted into different phases of HED matter. To illustrate this further, the physical state of the target at t = 9500 ns is shown in Fig. 9. It is seen that the strongly coupled plasma state extends up to 3 m into the target, surrounded by hot liquid material that extends up to 5 m (the entire length of the cylinder). Due to several limitations, for example, the required computing time, it is not currently possible to do these calculations considering all 2808 LHC bunches. Nevertheless, one can estimate the penetration length by doing the calculations using a smaller number of bunches until the penetration velocity of the protons and the shower along the axis becomes constant. From this velocity, it is possible to calculate the distance the protons and the shower will penetrate considering the entire beam [5,6,13]. Fig. 10 shows the density profiles along axis at different times. At t = 1 μ s, the density has decreased significantly at the target center to up to L = 2 m. The density depletion surface moves to a position L = 5 m at t = 8.5 μ s. It is seen from the profiles plotted at later times that the density depletion front moves with an average speed

The corresponding temperature distributions are shown in Fig. 6(a) and (b), respectively. Fig. 6(a) shows that at t = 500 ns, a maximum temperature of about 3.5 × 10 4 is generated on the target axis, whereas Fig. 6(b) shows that the maximum temperature is of the order of 4 × 10 4 K. The latter figure also shows deeper penetration of the beam that generates high temperature along the entire length of the target. Moreover, extension of the heated region in the radial direction is also seen which is the result of the energy deposition in the target by the shower outside the beam radius. Fig. 7(a) and (b) present the pressure distributions in the target at t = 500 ns and 9500 ns, respectively. Fig. 7(a) shows that a maximum pressure of about 30 GPa is generated at the axis and a radially outgoing shock is forming. Fig. 7(b) shows that the pressure wave has arrived at the cylinder boundary. Moreover, the pressure has increased along the entire length of the target due to the hydrodynamic tunneling. It is also interesting to note that the energy deposition is a localized phenomenon and it only affects the region that is directly irradiated by the beam and the shower. The pressure increase, on the other hand, does not remain localized, but generates waves that propagate throughout the target (see Fig. 7(b)). The density distributions at t = 500 ns and 9500 ns are shown in Fig. 8(a) and (b), respectively. It is seen in Fig. 8(a) that after the delivery of 20 proton bunches, the density along a part of the axis has been reduced to about 4 g/cm3 while the material has been compressed 75

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Fig. 7. Pressure distribution on depth-radius plane by BIG2 (a) at t = 500 ns [20 bunches delivered] and (b) at t = 9500 ns [380 bunches delivered].

Fig. 8. Density distribution on depth-radius plane by BIG2 (a) at t = 500 ns [20 bunches delivered] and (b) at t = 9500 ns [380 bunches delivered].

of 0.35 m/μ s. From these considerations it has been calculated that the LHC beam (all 2808 bunches and their shower) will penetrate up to 35 m in the target.

5.2. Carbon target irradiated by the LHC beam Simulations of the full impact of one LHC beam with a solid carbon cylindrical target have been reported in [13]. These simulations have been carried out using the FLUKA and the BIG2 codes iteratively. In this scheme, the FLUKA code was used to calculate the energy deposition distribution assuming solid material density. This data was used as energy input to BIG2 which was employed to calculate the thermodynamic and the hydrodynamic response of the target. Since the high pressure produced due to the material heating in the deposition region generated an outgoing radial shock wave, the density along and around the axis continuously depleted. As a consequence, the protons that were delivered in the subsequent bunches penetrated deeper in the target. A continuation of this process lead to a substantial increase in the penetration distance of the beam and the shower (hydrodynamic tunneling). As the density along the axis was reduced by 10–15%, the hydrocode was stopped and the modified density distribution it provided, was used in the FLUKA code to generate new energy deposition data for the next iteration of the hydrodynamic calculations. The procedure allowed to simulate the beam matter heating and the hydrodynamic tunneling of

Fig. 9. Physical state of target material at t = 9500 ns.

the beam with reasonable accuracy. It is to be noted that carbon is an important material because the LHC beam dump is made of solid carbon. The beam is strongly diluted before impacting the beam dump to avoid any material damage. It is thus important to analyze the damage caused by the focused beam in 76

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12 1 : at t = 500 ns 2 : at t = 1500 ns 3 : at t = 2500 ns

3

Density (g/cm )

10

8

1

6

2

3

150

200

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case of malfunctioning of the dilution magnets. Another failure mode is a wrong deflection of the beam by the extraction kicker. A carbon absorber with a length of 6 m, the TCDQ, has been installed to capture the particles. One objective of the simulation results summarized in this section was to address the question, if such an absorber can absorb the entire beam if the extraction kicker deflects the beam by a wrong angle. The geometry for the FLUKA calculations was a cylinder of solid carbon with radius = 5 cm and length = 6 m [13]. The solid density of carbon was considered to be 2.28 g/cm3. The energy deposition was calculated using a two-dimensional Gaussian beam distribution (horizontal and vertical, σrms = 0.5 mm) that was incident perpendicular to the front face of the cylinder. This beam size was selected for the simulations since it corresponds to the size of the beam at beam absorbers downstream of the extraction kicker, which is the most likely point of impact in case of a failure. Fig. 11(a) presents the energy deposition distribution per 7 TeV proton in units GeV/g as calculated by FLUKA assuming solid material density. This data shows that the range of the shower is about 3 m in the target and the peak of the distribution is around 12 GeV/p/g. The FLUKA calculations also suggest that approximately 54% beam energy escapes while 46% is absorbed in the target. Fig. 11(b) shows the energy deposition data obtained with FLUKA, but using the density distribution provided by BIG2 at t = 5 μ s (second iteration). The energy deposition distribution has been substantially modified with a significant broadening of the energy peak that indicates deeper penetration of the protons and the shower into the target. Moreover, the distribution has two peaks and the higher peak lies in the beam direction where the material density is much higher. The energy deposition distribution plotted in Fig. 11(c) has been calculated by FLUKA using the density distribution obtained from BIG2 at t = 10 μ s (fourth iteration). This figure shows a much longer penetration of the particle shower and the contrast between the two peaks is much more pronounced. Detailed hydrodynamic simulations have been reported in [13]. The specific energy deposition profiles along the axis at different times during irradiation are shown in Fig. 12(a). It is seen that between t = 1–3 μ s, the maximum specific energy increases from about 1.5 kJ/g to almost 4 kJ/g. In the interval t = 5– 7.2 μ s, on the other hand, the specific energy deposition increases from 6 kJ/g to about 7 kJ/g. The reason for reduction in the rate of increase of the specific energy during later part of the pulse is the deeper penetration of the beam into the target that results in distribution of the beam energy over an ever increasing mass. The corresponding temperature curves presented in Fig. 12(b), show

Fig. 11. FLUKA calculations of energy deposition of a single 7 TeV LHC proton in a solid carbon cylinder having radius, r = 5 cm, length, L = 6 m, with facial irradiation, beam spot size characterized by standard deviation, σ = 0.5 mm; (a) using solid density of 2.28 g/cm3; (b) using the density distribution provided by the BIG2 at t = 5 μ s (second iteration) and (c) using the density distribution provided by the BIG2 at t = 10 μ s (fourth iteration).

a very interesting behavior. The curve labeled with 4 μ s has a hump shaped part between L = 100 and 200 cm which represents the gaseous phase of carbon. The constant temperature regions extending towards the left and the right of the hump, on the other hand, represent the twophase liquid-gas state of the material. The two-phase region on the right side of the hump continuously moves towards the right due to the higher energy deposition as a result of deeper penetration of the beam. The temperature in the gaseous region increases with time and saturates at around 9000 K. Fig. 13(a) shows that the peak in the pressure profile continuously moves towards the right as a result of the hydrodynamic tunneling effect. The density profiles plotted in Fig. 13(b) show that after 6 μ s, the density depletion front moves along the axis with a constant speed of about 25 cm/μ s. This means that during the remaining 83 μ s when the other bunches hit the target, the penetration depth will be about 21 m. Since in the first 6 μ s the beam has penetrated up to 4 m, the total penetration depth will be around 25 m. This suggests that the 6 m length of the TCDQ beam stopper is not sufficient to stop the beam completely, nevertheless, the beam will be significantly weakened after 77

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Fig. 13. (a) Pressure vs depth into the target at different rimes and (b) Density vs depth into the target at different rimes.

emerging through this stopper and therefore the damage to other equipment will be reduced.

144 bunches with a beam focal spot characterized by σ = 2 mm. Target–2 was irradiated with 108 bunches, whereas Target–3 was irradiated with 144 bunches while in both these cases, the beam had a much smaller focal spot size characterized by σ = 0.2 mm. A summary of the beam parameters used in these three experiments is presented in Table 4. The temporal profile of the beam is presented in Fig. 15 which shows that the protons were delivered in sets of 36 bunches each, while a separation of 250 ns was considered between the neighboring bunch packets. Due to the high level of material activation after irradiation, a cool down time of 8 months was allowed before the Al cover was removed for the visual inspection of the targets in February 2013. Droplets and splashes of molten and evaporated copper have been found on the copper cylinders, the aluminum housing at the position of the gaps between cylinders and in the front aluminum caps [9,10]. In Fig. 16(a), photograph of the Al cover that was placed on top of the target is shown. After the beam impact, molten/evaporated material was projected outwards and was deposited on the top cover. The traces of the projected copper between the 10 cm long cylinders are clearly visible. It is seen that in case of the experiment using 144 bunches and beam focal spot σ = 2.0 mm (bottom picture), the splash of 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 the second experiment 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

6. Review of beam-matter heating experiments at the CERN HiRadMat facility Beam-matter heating experiments were performed at the CERN HiRadMat facility. In these experiments, extended solid copper cylindrical targets were irradiated by 440 GeV SPS beam at the CERN HiRadMat facility [9,10]. The target assembly used in these experiments is shown in Fig. 14. It is seen that it consists of three targets, each comprised of fifteen copper cylinders with a spacing of 1 cm in between that allowed for visual inspection of the targets after irradiation. Each cylinder has a radius, r = 4 cm and length, L = 10 cm. The three target cylinders are enclosed in an aluminum housing that provides rigidity to the setup and prevents contamination of the facility. The front face of the first cylinder and the rear face of the last cylinder in the three target assemblies are covered with cylindrical aluminum caps. Each cap has a radius of 4 cm, a length of 18.5 cm and a 1 cm diameter hole that allows the beam to pass through. 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. For all the three experiments, the proton energy was 440 GeV, bunch intensity was = 1.5 × 1011 protons, bunch length was = 0.5 ns and the bunch separation was = 50 ns. Target–1 was irradiated with 78

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Fig. 16. Top cover of the experimental set-up 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. Table 5 Comparison between measured and expected length of the molten zone, simulations are done using FLUKA [1] (static model without hydrodynamics).

Fig. 14. 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 separation in between.

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Expectation

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running the FLUKA and the BIG2 codes iteratively, using an iteration step of 700 ns [11,12]. This was the time during which the target density was reduced by about 15% at the target center due to the hydrodynamic processes. For the simplicity of the calculations a single solid copper cylindrical target was considered that was 150 cm long and had a radius of 4 cm, which is a good approximation to the target used in the experiments (Fig. 14) as the 1 cm gaps between neighboring cylinders does not affect the energy deposition. Moreover, the results could not be affected as the hydrodynamic processes are much stronger in the radial direction than in the axial direction in this type of problems. The results are summarized in Fig. 17. Fig. 17(a) shows 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 parameters of Experiment–1. It is seen that the flat part 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 escaped from the left face of cylinder number 6 and collided with the liquefied material ejected from the right face of cylinder number 5. As a result of this collision, the material was splashed vertically and was 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. 17(b), are plotted the density and the temperature, along the axis at t = 5800 ns obtained from BIG2, when 108 bunches were delivered for the beam parameters of Experiment–2 (σ = 0.2 mm). It is seen that in this case that 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 escaped from the left face of cylinder number 8 and collided with the melted/gaseous material ejected from the right face of cylinder number 7. The splashed material was thus deposited at the inner surface of the target cover above the gap between cylinder number 7 and 8. The simulations are thus in full agreement with the measurements of Experiment–2. Fig. 17(c) shows same variables as Fig. 17(b), but at t = 7850 ns, when 144 bunches were delivered. The melting region now lies

Table 4 Experimental beam parameters used in the three experiments. Target

Target

Fig. 15. Time structure of the proton beam.

the 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 length of 85 ± 5 cm. Table 5 shows a significant discrepancy between the experimental measurements and the simulations based on a static approximation. In order to have a better understanding of the problem, detailed numerical simulations were also done using the above beam parameters and 79

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Table 6 Comparison between experimental measurements and simulation results using FLUKA and BIG2 iteratively.

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3

with the measurements of Experiment–3. These experiments therefore confirmed the existence of the hydrodynamic tunneling in case of the SPS beam in accordance with the theoretical predictions and the experimental measurements showed good agreement with the simulations results. This provided confidence in the numerical simulations for the more powerful LHC beam reported earlier. A comparisons between the experimental measurements and the simulation results is shown in Table 6. Once the activation was reduced to acceptable levels, the targets were declared safe for further investigations. After cleaning, different parts of the cylinders were dissected in order to discover the finer and more interesting details about the beam penetration depth and damage to the target through visual and microscopic inspection of the interior part [10]. In Fig. 18(a)–(c) the pictures of the left and the right faces of the first three cylinders of the target used in Experiment–3 are shown. It is seen in Fig. 18(a) that the beam generates a small hole on the left face of the first cylinder which is in accordance with the traces of material deposited at the inner surface of the aluminum cap around cylinder

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(c) Fig. 17. Density and temperature vs depth into the target, (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].

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

Fig. 18. Experiment–3, front face (left) and back face (right) of; (a) first cylinder, (b) second cylinder and (c) third cylinder. 80

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Fig. 20. Experiment–3, front face (left) and back face (right) of; (a) seventh cylinder, (b) eighth cylinder and (c) ninth cylinder.

Fig. 19. Experiment–3, front face (left) and back face (right) of; (a) fourth cylinder, (b) fifth cylinder and (c) sixth cylinder.

face, as reported in [9]. On the right face, however, a much wider hole exists and traces of the ejected material around the hole that has been solidified after the cooling are clearly visible. The temperature curve in Fig. 17(b) shows that the material in the left half of the first cylinder (0–5 cm) is liquefied, whereas a two phase liquid-gas state exists in the right half region (5–10 cm). The ejected material from the right face is ejected with high speed and collides with the material ejected from the left face of the second cylinder that is partly deposited on the inner side of the aluminum cover (Fig. 16) and is partly splashed on the opposite faces of the two cylinders. The material deposited on the faces of the cylinders solidifies as seen in the right part of Fig. 18(a) and left part of Fig. 18(b). Similar behavior is seen in Fig. 18(c). In Fig. 19(a)–(c) are presented the pictures of the left and the right faces of the fourth to sixth cylinders of the target used in Experiment–3. Again it is clearly seen that the beam has penetrated through these cylinders generating holes at both faces of the cylinders. The traces of the material that solidifies after forceful ejection are also visible. Fig. 20(a)–(c) show the left and the right faces of cylinders seven to nine of the target used in Experiment–3. It is seen that holes exist on both faces of cylinders seven [Fig. 20(a)] and eight [Fig. 20(b)]. In case of the ninth cylinder [Fig. 20(c)], a hole is only seen on the left face, whereas on the right face no hole is visible. This is in agreement with the previous experimental measurements [9,10] and numerical simulations [11,12]. We have also carried out detailed microscopic analysis of a few selected cylinders in order to view the damaged caused by the beam to the interior of the targets and the results are presented in the following. In Fig. 21(a)–(c), are presented different sections of the first cylinder of

Fig. 21. Pictures produced by microanalysis of different parts of the first cylinder in Target–3 used in Experiment–3; (a) first (left) one third, the beam entrance from left, micro cavities exist till L = 2.3 cm, downstream start of a bell mouthed hole, (b) middle one third, showing solidified liquefied copper after cooling and the continuation of the hole, (c) right one third of the cylinder, showing that the hole continues till the end of the cylinder (Magnification 50×).

Target–3 that was used in Experiment–3. Fig. 21(a) shows the left one third, whereas Fig. 21(b) and (c) represent the middle one third and the right hand side one third length of the cylinder, respectively. It is seen in Fig. 21(a) that micro cavities have been formed till 2.3 cm along the

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Table 7 Comparison between final measurements and hydrodynamic simulations using FLUKA and BIG2 iteratively. Target

Simulated Penetration (cm)

Measured Penetration (cm)

Δ Measure to Simulated

1 2 3

56 80 90

58 79.5 85

2 0.5 5

used in the experiments. 7. A review of beam-matter heating simulations using the FCC beam A review of the simulations of interaction of the FCC protons with solid and liquid targets is presented in the following. 7.1. Solid copper target irradiated by the FCC beam Detailed numerical simulations of the full impact of one FCC beam on a solid copper cylinder were reported in [13,14]. The target length was assumed to be 5 m while the radius was considered to be 2 cm. The beam was incident perpendicular to one face of the cylinder. As mentioned previously, in these simulations the proton energy was considered to be 40 TeV whereas, the nominal FCC proton energy is now fixed at 50 TeV. The FLUKA and the BIG2 codes have been used iteratively to study this problem. It is important to note that a single 40 TeV FCC bunch deposited a specific energy deposition of about 14 kJ/g that generated a pressure of the order of 1 Mbar in the heated zone. The shock generated by this pressure was so strong that initially, an iteration step of 25 ns was required. Later, when the hydrodynamics achieved a steady state, a longer iteration step of 100 ns could be used. One-dimensional profiles of the specific energy deposition calculated with BIG2 are presented in Fig. 23. In Fig. 23(a), the specific energy deposition along the axis (r = 0) at different times during irradiation is plotted. It is seen that due to the continuous delivery of the bunches to the target, the specific energy deposition increases with time, but the rate of increase of the maxima of the curve slows down with time because the energy delivered by the subsequent bunches is deposited in an ever increasing mass. Moreover, deeper penetration of the protons and the shower as a function of time is clearly seen. Fig. 23(b) shows the same parameters as Fig. 23(a), but at a radial position, r = 1 mm. It is seen that even at this distance from the axis, large amount of energy is deposited by the shower that is sufficient to evaporate the material. It is interesting to note that the curves show that the maximum value of the deposited energy continues to increase at a faster rate, as compared to that in Fig. 21(a). This is because the material density at this radial position is significantly higher than that at the axial position. In Fig. 23(c), are plotted the specific energy deposition curves along the cylinder length at a radial position, r = 2 mm, at different times. It is seen that even at this relatively large distance from the beam axis, enough energy is deposited that can severely damage the material. However, due to a higher material density compared to the other two radial positions, the maximum value of the energy deposition still increases at a steady rate. Fig. 24(a)–(c) show the corresponding profiles of the temperature vs distance into the target at different radial positions including, r = 0, 1 and 2 mm, respectively. It is seen in Fig. 24(a) that the temperature along the axis steadily increases with time and the heating front moves deeper into the target due to the hydrodynamic tunneling. It is also interesting to note that the peak of the curve number 5 (t = 1250 ns) has shifted towards right compared to that in curve number 4 (t = 1000 ns). This is because of the strong reduction in the target

Fig. 22. First cylinder, Target–3, Experiment–3; (a) 3D picture of the hole between L = 2.4 – 2.6 cm (magnification 200x), (b) between L = 9.2–10 cm (magnification 50×) , (c) radius of the hole which is 2.3 mm (magnification 50×) at the exit (right face of the cylinder) at L = 10 cm.

length that is followed by a bell mouthed hole that extends till the end of the picture. Fig. 21(b) and 21(c) show that the hole continues along the cylinder axis till the end and solidification of the liquefied copper is also clearly visible. Fig. 22(a)–(c) show three dimensional high resolution pictures of hole drilled by the beam in the first cylinder of Target–3 used in Experiment–3. These pictures have been made by the camera in the microscope. Fig. 22(a) represents the region between L = 2.4–2.6 cm while Fig. 22(b) shows the hole between L = 9.3–10 cm. Fig. 22(c) shows the radius of the hole which is 2.3 mm. These detailed microscopic studies of the targets improved the previous measurements. A comparison between the final measurements and numerical simulations is presented in Table 7. In the above table, it is considered that the melting zone in the target also represents target damage due to beam penetration. Therefore in view of Fig. 17(a)–(c), the beam penetration in the three different cases of simulations is 56, 80 and 90 cm, respectively. These values show very good agreement with the beam penetration distance obtained by the microscopic analysis of the targets in corresponding 82

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1.5e+05

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Fig. 24. Target temperature profiles calculated by BIG2, (a) at t = 250 ns [10 bunches delivered], (b) at t = 500 ns [20 bunches delivered], (c) at t = 1000 ns [40 bunches delivered] and (d) at t = 1250 ns [50 bunches delivered].

density in that region for the curve corresponding to 1250 ns. Fig. 24(b) shows that at a radial position of 1 mm, the maximum temperature increases to about 70000 K (curve 5) after the delivery of 50 FCC bunches. The corresponding curve in Fig. 24(c) shows that the temperature at the radial position of 2 mm becomes 40000 K due to the high level of energy deposition by the shower. It has also been considered important to study the material state at the boundary of the beam-heated and the cold material. For this purpose Fig. 25(a)–(c) show magnified view of the temperature profiles vs distance into the target at different radial positions including, r = 0, 1 and 2 mm, respectively at different times. These figures show that every

temperature profile has a flat part which represents the melting of the target material at that location. It is also seen that this flat region (melting front) continuously shifts towards the right as the boundary of the beam heated region extends in the beam direction due to the hydrodynamic tunneling. Fig. 25(a) shows that at the axis, the melting region lies between L = 300–310 cm at t = 1250 ns [delivery of 50 FCC bunches]. At a radial positions 1 mm and 2 mm, the corresponding melting zones are located between L = 280–290 cm and 260–270 cm, 83

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8000

6000

Temperature (K)

Fig. 26(a) has been plotted at t = 250 ns, which shows that the entire beam heated region has been liquefied while the melting zone surrounding this liquid region is clearly seen. Fig. 26(b) shows that at t = 500 ns, the liquid region has been significantly extended and the melting front has further propagated, both in the radial as well as in the longitudinal direction. The extension of the liquefied zone in the longitudinal direction is due to the deeper penetration of the protons and the shower due to the hydrodynamic tunneling. In the radial direction, as more and more energy is deposited by the shower generated by the incoming proton bunches, the material melts when the melting threshold is achieved. Fig. 26(c) shows that at t = 1000 ns, a significant increase in the volume of the liquefied material has taken place. Moreover, a small pencil shaped region of gaseous/plasma phase has also appeared along the cylinder axis. In Fig. 26(d), the physical state at t = 1250 ns is presented. It is seen that a substantial part of the target has been liquefied and a noticeable gaseous/plasma zone also exists at, and around the axis. The beam heated part of the target is thus converted into different phases of HED matter, including, melting, expanded as well as compressed hot liquid, gas and weakly ionized plasma. This is also a very important field of research. It is therefore clear that in case of an uncontrolled release of the beam energy, only a few bunches of the FCC can cause irreversible damage to the accelerator components and other equipment. This demands great care in the handling of these beams that store a very large amount of energy and a secure and robust machine protection system must be designed. In Fig. 27, the density on axis vs target length at different times during irradiation is presented. Curve “a” represents the time when 2 bunches have been delivered while later curves are plotted using an interval of 150 ns. A density of 6 g/cm3 is considered as the reference point on the density curve and the speed with which this point moves towards the right, is calculated. At t = 800 ns, the depletion front achieves a steady speed as in each of the following 150 ns time intervals it covers equal distances, x1, x2 and x3, respectively. In the simulations, x1 = 16.6 cm, x2 = 16.4 cm and x3 = 16.8 cm which amounts to a total penetration length of 49.8 cm covered in 450 ns. This leads to a steady average speed of 1.1 × 106 m/s. The total beam duration considering all 10,600 bunches is 265 μ s that leads to a penetration distance of about 290 m. This means that in case of wrong deflection of the beam, the beam and the shower will penetrate through about 290 m of solid copper. If one considers 50 TeV proton beam, the penetration distance could be up to 350 m.

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6000 5000 4000 1

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7.2. Water target irradiated by the FCC beam

2000

For the LHC, a carbon beam dump is being used. The beam is diluted before it reaches the beam dump and the bunches are delivered on an extended spiral shaped path so that the energy is distributed over a large volume of the beam dump to avoid material damage. For the FCC beam, such a scheme will require 20 m long spiral path for the bunches, which is very challenging. An alternate beam dump design that uses ordinary water to absorb the beam energy, has been studied with the help of numerical simulations done employing the energy deposition code, FLUKA and a 2D hydrodynamic code, BIG2, iteratively [47]. With this option, it would be possible to avoid a highly complex and expensive beam dilution system. The geometry of this scheme is shown in Fig. 28. It is a Cu pipe with an inner radius of 15 cm, an outer radius of 17 cm and is 14 m long. It is filled with ordinary water and the beam is incident on the left face of the cylinder along its axis. The focal spot is circular with a Gaussian transverse intensity distribution with σ = 0.4 mm. Fig. 29 shows density along the target axis at different times. It is seen that a density depletion front propagates towards the right and the position and the speed of this front is presented in Table 8 at different

1000 0

0

100

200 300 Distance in Target (cm)

400

(c) Fig. 25. Magnified view of temperature profiles across heated and cold material calculated by BIG2, (a) at t = 250 ns [10 bunches delivered], (b) at t = 500 ns [20 bunches delivered], (c) at t = 1000 ns [40 bunches delivered] and (d) at t = 1250 ns [50 bunches delivered].

respectively. This is consistent with the Gaussian intensity distribution in the focal spot. This analysis showed that the target will not only be damaged at the axis, but melting and evaporation of the material could occur in large volume around the axis. To illustrate this point, the physical state of the material at different times during the irradiation is shown in Fig. 26(a)–(d). 84

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Fig. 26. Target physical state calculated by BIG2, (a) at t = 250 ns [10 bunches delivered], (b) at t = 500 ns [20 bunches delivered], (c) at t = 1000 ns [40 bunches delivered] and (d) at t = 1250 ns [50 bunches delivered].

1

10

a

c

d

i: 1250 ns h: 1100 ns g: 950 ns f: 800 ns e: 650 ns d: 500 ns c: 350 ns b: 200 ns a: 50 ns

e f g h i x1 x2

6

x3

x0

0,8

4

c

100

150

200

250

300

350

l: 1300 ns k: 1200 ns j: 1100 ns i: 1000 ns h: 900 ns g: 800 ns f: 700 ns e: 600 ns d: 500 ns c: 400 ns b: 300 ns a : 200 ns

l

i

d

0,2 50

k

e

0,4

0

j

0,6

f g h

2

0

x1 x2 x3 x4 x5 x6 x7 x8 x9

b 3

b

Density (g/cm )

a

3

Density (g/cm )

8

400

0

200

400

600

800

1000

1200

1400

Distance in Target (cm)

Distance in Target (cm)

Fig. 29. ρ vs depth into the target at different times.

Fig. 27. Density on axis vs depth into the target at different times, calculation of the penetration distance of the FCC beam in a solid copper target.

Table 8 Position, x, Distance travelled in 100 ns, Δ x and speed, v of depletion front. Time (ns)

x (cm)

Δ x (cm)

v (m/s)

700 800

713.00 769.70

56.7

5.67 × 106

900

822.80

53.1

5.31 × 106

1000

872.6

49.8

5.31 × 106

1100

920.5

47.9

4.79 × 106

1200

968.7

48.2

4.82 × 106

1300

1016.7

48.0

4.80 × 106

times. It is seen that the speed decreases with time and achieves a constant value of about 4.80 × 106 m/s. Since the duration of the bunch train is 265 μ s, the total penetration distance of the beam and the shower will be about 1.3 km, which should be the length of the beam

Fig. 28. Beam dump geometry. 85

Nuclear Inst, and Methods in Physics Research B 427 (2018) 70–86

N.A. Tahir et al.

dump. It is interesting to note that the static range of a single 50 TeV proton and its shower is about 7 m in water. Such a long target is not practical, therefore it is planned to repeat the calculation with an increase beam size to reduce the length by at least of magnitude.

[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

8. Conclusions Beam-matter heating is an important problem related to powerful accelerators for a number of reasons. First, in the case of an accidental and uncontrolled release of the beam energy, the equipment and some accelerator components may be irradiated by energetic particles. One of the worst case scenario is when the entire beam is lost at a single point. To assess the damage caused by such an accident, interaction of the beam with relevant materials needs to be studied. Second, the safe disposal of the beam needs a stable and robust beam-dump that can absorb the entire beam energy without being damaged. Designing such a system requires detailed theoretical and experimental studies of interaction of the beam with the material to be used in construction of the beam-dump. Moreover, intense and energetic particle beams are used as a modern tool to research numerous areas of basic and applied physics which directly or indirectly involve beam-matter interaction. Detailed simulation studies have been reported in the literature to address the first problem in case of the 20 TeV SSC proton beam [7,8], 7 TeV LHC proton beam[3–6] and the 50 TeV FCC proton beam [13,14,47]. These studies have shown that the penetration distance of the beam and the shower in case of all these accelerators is substantially increased compared to the static range of the particles and their showers due to the hydrodynamic tunneling phenomenon. The existence of the hydrodynamic tunneling was experimentally confirmed at the CERN HiRadMat facility where extended solid copper cylindrical targets were irradiated by the 440 GeV proton beam generated by the SPS [9,10]. It was also shown that the experimental measurements agreed very well with the simulations of these experiments [11,12]. A detailed review of this work is presented in the present paper.

[41] [42] [43] [44] [45] [46] [47]

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