2. Interaction of 50 MeV - 50 TeV proton with solid copper target at CERN hadron accelerator complex
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Thursday, August 25, 2016
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3. Outline
Introduction
Beam parameters to be studied for CERN hadron accelerator complex
Energy deposition of single proton in copper: FLUKA simulations
Comparison with the Bethe equation
Specific energy deposition of multi-protons (bunches)
Summary and future works
Thursday, August 25, 2016
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4. I. Introduction
Studies on beam-matter interaction are essential for machine protection
Numerical simulations have been performed especially for the SPS, the LHC, and FCC (40TeV), but the beam parameters (energy & size) studied are still limited
We intend to make an integral study (database) that covers all typical beam energies and beam sizes in the CERN hadron accelerator complex
Therefore, FLUKA is used to calculate energy depositions of protons in copper (in graphite as well, show in the future)
No thermodynamics and hydrodynamic simulation is included (static approximation)
Thursday, August 25, 2016
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5. II. Beam parameters to be studied for CERN hadron accelerator complex
Related parameters of the hadron accelerator complex at CERN
Accelerator
PSB PS
SPS
LHC
FCC
Injection energy
50 MeV a)
1.4 GeV
26 GeV
450 GeV
3.3 TeV
Extraction energy
7 TeV
50 TeV
# bunches
1 / ring
72 c)
288
2808
10600
Bunch intensity
16.2×1011 b)
1.35×1011
1.15×1011
1.0×1011
Beam power
0.4 kJ per bunch
40 kJ
2.4 MJ
362 MJ
8500 MJ
Norm. emittance
2.5 µm
3 µm
3.5 µm
3.75 µm
2.2 µm
Pulse length
bunch length
~190 ns
1.8 µs
7.2 µs
88.9 µs
327 µs
Circumference
157 m (4 rings)
1/4 PS
628 m
1/11 SPS
6.9 km
7/27 LHC
26.7 km
97.97 km
11/3 LHC
a) Will be upgraded to 160 MeV (H- from LINAC4)
b) Ref.: B. Mikulec, et al., CERN-ATS
c) Debunching and rebunching (splitting) 6 bunches from PSB to make 72 bunches spaced by 25 ns at PS.
Ref.: M. Benedikt, et al., EPAC 2000; R. Garoby, et al., EPAC 2000
Thursday, August 25, 2016
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6. II. Beam parameters to be studied for CERN hadron accelerator complex
For the Gaussian distribution: 𝜎 𝑥,𝑦 = 𝛽 c 𝜀 rms = 𝛽 c 𝜀 n,rms /𝛾 where 𝛽 c is the betatron function of the beam optics, 𝜀 n,rms = 𝛾𝜀 rms is the normalized rms beam emittance.
For 450 GeV SPS beam, 𝜀 n,rms ~3.5μm, assuming 𝛽 c =100m, 𝜎 𝑥,𝑦 =0.85mm; For 7 TeV LHC beam, 𝜀 n,rms ~3.75μm, assuming 𝛽 c =100m, 𝜎 𝑥,𝑦 =0.22mm; For 50 TeV FCC beam, 𝜀 n,rms ~2.2μm, assuming 𝛽 c =200m, 𝜎 𝑥,𝑦 =0.09mm; The typical beam size from the LINAC, PSB and PS is up to a few mm.
Therefore, we first study such a case that the beam size is kept to be 0.2mm for all the energies. In addition, different beam sizes are studied depending on the beam energy, which will be listed later.
Emittance history for LHC
D. Möhl, Transverse dynamics II: Emittances, CAS2005
Thursday, August 25, 2016
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7. III. Energy deposition of single proton in copper: FLUKA simulations
TARGET
VACUUM
BLACK BODY
Here, the energy deposition is calculated only by FLUKA (static approximation), i.e., the hydrodynamic tunneling effect is not considered, or equivalently, the beam energy is deposited instantaneously.
"The FLUKA Code: Developments and Challenges for High Energy and Medical Applications“,
T.T. Böhlen, F. Cerutti, M.P.W. Chin, A. Fassò, A. Ferrari, P.G. Ortega, A. Mairani, P.R. Sala, G. Smirnov and V. Vlachoudis,
Nuclear Data Sheets 120, (2014);
"FLUKA: a multi-particle transport code“, A. Ferrari, P.R. Sala, A. Fasso`, and J. Ranft,
CERN (2005), INFN/TC_05/11, SLAC-R-773.
Thursday, August 25, 2016
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8. III. Energy deposition of single proton in copper: FLUKA simulations
Energy deposition (GeV/cm3/p) of 50MeV (PSB injection) proton in copper, σx,y=0.2mm
Thursday, August 25, 2016
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9. III. Energy deposition of single proton in copper: FLUKA simulations
Energy deposition (GeV/cm3/p) of 50TeV (FCC top energy) proton in copper, σx,y=0.2mm
Thursday, August 25, 2016
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10. III. Energy deposition of single proton in copper: FLUKA simulations
Simulation results for 50 MeV proton beam with different transverse beam sizes
Energy deposition vs. target radius, at L=0.39cm (position of the Bragg peaks).
Energy deposition vs. target axis
Bin size σx,y/4,
primaries 1000*50*20 (event*cycle*spawn)
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11. III. Energy deposition of single proton in copper: FLUKA simulations
Simulation results for 50 TeV proton beam with different transverse beam sizes
Energy deposition vs. target radius, around L=23cm, where the on-axis energy deposition is the maximum.
Energy deposition vs. target axis
Bin size σx,y/4,
primaries 10*100*10 (event*cycle*spawn)
Thursday, August 25, 2016
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12. III. Energy deposition of single proton in copper: FLUKA simulations
Energy depositions (GeV/cm3/p) of protons with various energies in copper, σx,y=0.2mm
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13. III. Energy deposition of single proton in copper: FLUKA simulations
The peak energy depositions (GeV/cm3/p, for σx,y=0.2mm) of protons in copper as a function of incident kinetic energy over 6 orders of magnitude. The corresponding values of 𝛽𝛾 are plotted as well.
Thursday, August 25, 2016
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14. III. Energy deposition of single proton in copper: FLUKA simulations
Energy (GeV)
RMS beam size (mm)
Max. energy deposition (GeV/cm3/p)
Max. specific energy (×10-9J/g/p)
Peak location (cm)
0.05
0.2
75.4
1.35
0.39 (Bragg Peak)
0.4
30.4
0.54
1.0
6.0
0.11
0.16
12.9
0.23
0.1
3.5
0.06
0.3
0.67
0.01
3.0 (Bragg Peak)
1.4
5.7
0.10
1.5
0.03
0.26
0.005 26
6.8
0.12
1.9
1.3
0.42
0.008
4.2
450
59.2
1.06
11.0
35.1
0.63
12.6
0.8
12.3
0.22
15.8
3300
814.7
14.5
472.7
8.4
17.3
252.9
4.5
18.3
7000
2059.1
36.7
1147.2
20.5
18.8
606.5
10.8
19.8
50000
353.4
21.8
185.0
23.3
5043.5
90.1
23.8
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15. IV. Comparison with the Bethe equation
The mean energy loss rate of moderately relativistic (0.1≤𝛽𝛾≤1000) charged heavy particles in intermediate-Z materials is well-described by the Bethe equation:
− 𝑑𝐸 𝑑𝑥 =𝐾 𝑧 2 𝑍 𝐴 1 𝛽 ln 2 𝑚 e 𝑐 2 𝛽 2 𝛾 2 𝑇 max 𝐼 2 − 𝛽 2 − 𝛿 𝛽𝛾 2
where 𝑇 max is the maximum kinetic energy that can be transferred to a free electron in a single collision, 𝛿 𝛽𝛾 is the density effect correction to ionization energy loss:
𝑇 max = 2 𝑚 e 𝑐 2 𝛽 2 𝛾 𝛾 𝑚 e 𝑀 + 𝑚 e 𝑀 2
𝛿 𝛽𝛾 = 2 ln 10 𝑥− 𝐶 if 𝑥≥ 𝑥 1 ; 2 ln 10 𝑥− 𝐶 +𝑎 𝑥 1 −𝑥 𝑘 if 𝑥 0 ≤𝑥< 𝑥 1 ; if 𝑥< 𝑥 0 (nonconductors); 𝛿 𝑥− 𝑥 if 𝑥< 𝑥 0 (conductors);
with 𝑥= log 10 𝑝/𝑀𝑐 = log 10 𝛽𝛾 .
𝑻 𝒎𝒂𝒙 can be simplified to 𝟐 𝒎 𝒆 𝒄 𝟐 𝜷 𝟐 𝜸 𝟐 when 𝟐𝜸 𝒎 𝒆 ≪𝑴, while 𝜹 𝜷𝜸 /𝟐 is small at low energies.
The other variables are defined in the following table.
K.A. Olive et al. (PDG), Chin. Phys. C38, (2014) (
D.E. Groom et al., Atomic Data and Nuclear Data Tables 76, 1-37(2001)
Thursday, August 25, 2016
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16. IV. Comparison with the Bethe equation
Summary of the variables used (the unit of −𝑑𝐸/𝑑𝑥 is hence 𝐌𝐞𝐕 𝐠 −𝟏 𝐜𝐦 𝟐 )
Symbol
Definition
Units or Value 𝑲
4𝜋 𝑁 𝐴 𝑟 𝑒 2 𝑚 𝑒 𝑐 2
MeV mol −1 cm 2 𝒛
Charge number of the projectile particle
1 for proton 𝒁
Atomic number of the target material
29 for copper 𝑨
Atomic mass of the target material
g/mol for copper
𝒎 𝐞 𝒄 𝟐
Electron mass ×𝑐 2
0.511 MeV 𝑰
Mean excitation energy of the target atom
𝐼≅ 10±1 eV 𝑍 for 𝑍≥16, 322 eV for copper 𝑴
Incident particle mass
MeV/ 𝑐 2 for proton 𝒑
Incident particle momentum 𝑝=𝑀𝛽𝛾𝑐
MeV/𝑐
𝑵 𝐀
Avogadro’s number
6.022× mol −1
𝒓 𝐞
Classical electron radius 𝑒 2 /4𝜋 𝜀 0 𝑚 e 𝑐 2
2.818× 10 −15 m 𝝆
Density of the target material
8.96 g/ cm 3 for solid copper 𝒂
Sternheimer’s parameter for certain 𝑍
for copper 𝒌
for copper
𝒙 𝟎
for copper
𝒙 𝟏
for copper 𝑪
for copper
𝜹 𝟎
0.08 for copper
R.M. Sternheimer and R.F. Peierls, PR B, 1971
Thursday, August 25, 2016
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17. IV. Comparison with the Bethe equation
Using the Bethe equation, we can calculate the mean energy loss rate of one proton in copper, in the units of MeV g −1 cm 2 . Meanwhile, from FLUKA, we get the energy deposition in the units of GeV/ cm 3 , in the predefined volume of the copper target. In order to compare the two values, we multiply the mean loss rate by the copper density (g/cm3) to obtain the energy loss per penetration length, in the units of GeV/cm. On the other hand, for the energy deposition values from FLUKA, we perform an integration in the cross section at the entrance of the target to obtain the energy loss per penetration length, in the same units of GeV/cm. Such an energy loss rate is independent on the beam size, as illustrated by the figure below.
For example, the areas under the three curves (the integrals) are the same (𝟎.𝟎𝟔𝟖 𝐆𝐞𝐕/𝐜𝐦), corresponding to three different beam sizes 𝜎 𝑥,𝑦 =0.2 mm, 𝜎 𝑥,𝑦 =0.4 mm and 𝜎 𝑥,𝑦 =1.0 mm, respectively, for a 50 MeV proton in copper.
Thursday, August 25, 2016
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18. IV. Comparison with the Bethe equation
− 𝑑𝐸 𝑑𝑥 =𝐾 𝑧 2 𝑍 𝐴 1 𝛽 ln 2 𝑚 e 𝑐 2 𝛽 2 𝛾 2 𝑇 max 𝐼 2 − 𝛽 2 − 𝛿 𝛽𝛾 2
The results are in good accordance with each other when the proton kinetic energies are 50 MeV, 160 MeV and 1.4 GeV.
The values from Bethe equation are more and more under-estimated for higher energies. For 26 GeV (𝛽𝛾≈30), it becomes 70% of that in FLUKA.
The reason is that at higher energies, radiative effects (bremsstrahlung, pair production and photonuclear interactions) begin to be important, which are not included in the Bethe equation.
Thursday, August 25, 2016
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19. V. Specific energy deposition of multi-protons (bunches)
Using non-constant specific heat capacity, 674J/g is needed to make the copper melted from 300K, and 6250J/g to vaporize it.
The number of protons needed to melt the copper material at the maximum energy deposition point (denoted as 𝑛 p, max ) as well as at the entrance of the target (denoted as 𝑛 p, entrance ) are listed in the following table.
In addition, the corresponding number of proton bunches (denoted as 𝑁 p, max and 𝑁 p, entrance , respectively) are presented as well, based on the nominal bunch intensity listed previously.
For the SPS (450 GeV, 𝜎 𝑥,𝑦 =0.8mm), the safe beam intensity (no melting) is 3.1×1012, which agrees well with the damage test experiments (Kain et al., Chamonix 2009 Workshop on LHC Performance).
Analogously, the threshold is 3.3×1010 for the beam size of 0.2 mm at the LHC top energy of 7 TeV, and 1.9×109 for 0.1 mm at the FCC top energy of 50 TeV, as suggested in the following table.
Thursday, August 25, 2016
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20. V. Specific energy deposition of multi-protons (bunches)
Number of protons and proton bunches needed to melt the copper target at the maximum energy deposition location as well as at the target front surface
Energy (GeV)
RMS beam size (mm)
𝒏 𝐩, 𝐦𝐚𝐱
𝑵 𝐩, 𝐦𝐚𝐱
𝒏 𝐩, 𝐞𝐧𝐭𝐫𝐚𝐧𝐜𝐞
𝑵 𝐩, 𝐞𝐧𝐭𝐫𝐚𝐧𝐜𝐞
0.05
0.2
5.0×1011
0.34
1.3×1012
0.79
0.4
1.2×1012
0.77
5.1×1012
3.15
1.0
6.3×1012
3.88
3.1×1013
18.94
0.16
2.9×1012
1.81
3.0×1012
1.86
1.1×1013
6.66
1.2×1013
7.52
5.6×1013
34.77
7.3×1013
44.81
1.4
6.6×1012
4.09
6.8×1012
4.20
2.5×1013
15.53
1.5×1014
89.61 26
5.6×1012
41.12
6.0×1012
44.38
2.0×1013
147.15
2.2×1013
159.76
9.0×1013
665.68
1.1×1014
822.31
450
0.1
6.4×1011
5.54
1.4×1012
11.89
1.1×1012
9.35
5.0×1012
43.76
0.8
3.1×1012
26.68
6.7×1013
586.09
3300
4.6×1010
0.46
13.73
8.0×1010
0.80
4.1×1012
40.58
1.5×1011
1.49
1.3×1013
134.80
7000
1.8×1010
10.90
3.3×1010
0.29
35.67
6.2×1010
0.54
1.5×1013
131.28
50000
1.9×109
0.02
8.3×1011
8.33
3.6×109
0.04
2.8×1012
27.96
7.5×109
0.07
120.20
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21. V. Specific energy deposition of multi-protons (bunches)
440 GeV (in HiRadMat experiment):
One bunch, bunch intensity 1.5×1011 protons, σx,y= 0.2 mm
Peak specific energy: 93 J/g
440 GeV (in HiRadMat experiment):
144 bunches, bunch intensity 1.5×1011 protons, σx,y= 0.2 mm
Penetration depth of the beam: 69 cm (static approximation)
Note that the melting and boiling energies are 613 J/g and 5236 J/g in the reference.
Agree with the results in Ref.:
R. Schmidt, et al., Physics of Plasma, 2014
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22. V. Specific energy deposition of multi-protons (bunches)
3.3 TeV (FCC injection):
One bunch, bunch intensity 1.0×1011 protons, σx,y= 0.2 mm
Peak specific energy: 844 J/g > 674 J/g
3.3 TeV (FCC injection):
10600 bunches, bunch intensity 1.0×1011 protons, σx,y= 0.2 mm
Penetration depth of the beam: 214 cm (static approximation)
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23. V. Specific energy deposition of multi-protons (bunches)
50 TeV (FCC top energy):
One bunch, bunch intensity 1.0×1011 protons, σx,y= 0.2 mm
Peak specific energy: J/g > 6250 J/g >> 674 J/g
50 TeV (FCC top energy):
10600 bunches, bunch intensity 1.0×1011 protons, σx,y= 0.2 mm
Penetration depth of the beam: 315 cm (static approximation)
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24. VI. Summary and future works
The energy depositions of 50 MeV-50 TeV protons in copper have been simulated and analysed. Typical energies and beam sizes were selected to cover the CERN hadron accelerator complex. They serve as a starting point for further analyses. Another target material of graphite is being studied.
Based on this, estimation can be performed for beam energies / sizes that haven’t been listed. For instance, the higher-energy LHC (HE-LHC) has been proposed to reach a beam energy of 16.5 TeV, of which the maximum energy deposition can be assessed to be around 3000 GeV/ cm 3 /p at 21 cm into the solid copper target for σx,y= 0.2 mm.
The simulation here was performed only using FLUKA, i.e., the static approximation was assumed and the hydrodynamic tunnelling hadn’t been considered. We are (only) able to predict the minimum penetration depth into the target for a certain beam based on such static approximation.
The existence of hydrodynamic tunnelling has been confirmed by the HiRadMat experiment, for high energy beam with a long bunch train. To simulate it, BIG2 has been used, and we will try to use ANSYS Autodyn / LS-Dyna as well.
A case study using Autodyn / LS-Dyna is in progress in collaboration with EN-MME. One copper target & beam parameter of the 2012 HiRadMat experiment is taken as reference.
Thursday, August 25, 2016
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