Investigations of the effects of 7 TeV proton beams on LHC collimator materials and other materials to be used in the LHC

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Cecilia Hodges
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1 Russian Research Center Kurchatov Institute Investigations of the effects of 7 ev proton beams on LHC collimator materials and other materials to be used in the LHC A.I.Ryazanov

2 Aims of Investigations: Calculations of energy deposition, determination of electronic loss and electronic excitation produced by a proton beam of 7 ev in a collimator and other materials. Development of theoretical models for the calculation of the effective temperature rise in the collimator materials under irradiation from a 7 ev proton beam. Calculations of radiation damage production in collimator materials irradiated by a 7 ev proton beam taking into account deposited energy, electronic excitation and elastic collisions in these materials. Modeling of microscopic and macroscopic damage formation in different collimator materials and other materials produced by shock wave propagation initiated by a 7 ev proton beam.

3 Development of theoretical models for the calculation of the effective temperature rise in the collimator materials under irradiation from a 7 ev proton beam. Copper Graphite Materials: Physical Processes: Deposited energy by 7 ev proton beam from FLUKA code Electronic excitation of electronic subsystem of materials Electronic thermal conductivity of materials Electron-phonon coupling in materials Phonon thermal conductivity of ionic subsystem

4 Energy deposition per 7 ev proton in copper as a function of the depth into target and the radial coordinate.

5 Energy deposition per 7 ev proton in graphite as a function of the depth into target and the radial coordinate.

6 «hermal Spike» Model Electronic emperature: e e (r,t) r - characteristic distance for deposited energy (r >1 cm) calculated by FLUKA, R (D e _ ep ) _ e (r,t_ ep ) D Z D. mm D min.16 mm 7 ev, p R r Ionic emperature: r i i (r,t _ ep ) i (r,t > _ pp ) i (r,t) 7 ev, p r

7 Characteristic times in «hermal spike» model: _ e ~ 1-16 s - characteristic time of the electron - electron interaction; _ e-ph ~ 1-13 s - characteristic time of the electron - phonon interaction; _ ph-ph ~ s - characteristic time of phonon - phonon interaction;

8 Main equations: Cylindrical geometry [ ] ), ( 1 t r A r rk r r t C i e e e e e + [ ], 1 C i i e i i i r rk r r t + ph e C e _ i is the lattice thermal conductivity; is the thermal conductivity of electrons, _ i is the thermal capacity of ionic subsystem, is the thermal capacity of electronic subsystem, A(r,t) is the effective energy source in electronic subsystem

9 Initial and Boundary Conditions in hermal Spike: d C t r E t r e e dep ), ( ) ( ), ( K t r i 3 ), ( K matr r i r e 3 > < 1 ;... ; ;... ) ( exp ), ( t t t t C t t r r E t r A t dep 1. r i r e r r K r t r t i e 3 ), ( ), (., t ns

10 he values used in the numerical calculations for Cu , m mol Ni m mol Ne < < > K K m Wt K K m Wt K e i e e i e e 1 )]... /( ;...[ )]... /( ;.[ > < K K mol J R K K mol J C e e e e )]... /( );.[ ( )]... /( ;.[ > < < K K mol J R K K mol J C i i i i 8 )];... /( )..[ 3 1(.5 8 )];...1 /(..[ i i K K m Wt

11 Crystal lattice temperature profile near 7 ev proton beam with the beam size D b. mm in Cu collimator material as a function of the radial coordinate at different times on the depth-length into the target L 6 cm. Picture?3 1 3 t1-1 s t1-8 s t1-7 s i, K r, cm

12 Crystal lattice temperature profile near 7 ev proton beam with the beam size D b.16 mm in Cu collimator material as a function of the radial coordinate at different times on the depth-length into the target L 6 cm. 1 3 t1-1 s t1-8 s t1-7 s i, K r, cm

13 Data used in the numerical calculations for Graphite Electronic specific heat: C e 3/ N e k B 1 J cm -3 K -1. Electronic thermal conductivity: K e J cm -1 s -1 K -1. K i, Cal Sec -1 cm -1 K -1.9 y 6.44x % porosity porosit , K emperature dependence of the thermal conductivity of ionic subsystem of high density of graphite with % porosity.

14 K i, Cal Sec -1 cm -1 K y x porosi % porosity emperature dependence of the thermal conductivity of ionic subsystem of graphite with % internal porosity., K

15 hermal Diffusivity--Graphite y 1E-15x 6-6E-1x 5 + 9E-9x 4-9E-6x x x hermal diffusivity, cm^ Sec^ emperature emperature dependence of the thermal diffusivity of ionic subsystem of graphite.

16 Deposited energy in electronic subsystem of graphite produced by one bunch of proton beam with 111protons and the energy of each proton 7 ev as a function of the radial coordinate on the depth into target Z 6 cm. 3 5 E, GeV/cm ,,,4,6,8 1, R, cm

17 Crystal lattice temperature dependence in Graphite in 7 ev proton beam area (R cm) for the beam size Db.16 mm as a function of the time on the depth-length into the target 6 cm i, K t, sec

18 Crystal lattice temperature dependence in Graphite in 7 ev proton beam area (R cm) for the beam size Db.16 mm as a function of the time (up to 1 ns) on the depth-length into the target 6 cm.,k x1-1 1x1-9 1x1-8 1x1-7 t, sec

19 Crystal lattice temperature distribution profile near 7 ev proton beam (one bunch) with the minimum beam size Db.16 mm in Graphite collimator material as a function of the radial coordinate at the time 1ns on the depth-length into the target 6 cm i, K r, cm

20 Modeling of microscopic and macroscopic damage formation in different collimator materials produced by shock wave propagation initiated by a 7 ev proton beam. (L B D SW B ), D SW 1 5 cm/s, B 5 ns, L B.5x1-3 cm. SW is unknown? L B SW 1) Overlapping of shock waves produced by two neighboring bunches: L B SW

21 L B SW ) No overlapping of shock waves produced by two neighboring bunches: L B >> SW

22 heoretical model of shock-wave propagation in materials based on homas-fermi-dirac (FD) microscopic model. Main assumptions of FD model: Jelly model of metal degenerated electron gas with the temperature e : e < F, F is the Fermi temperature, F n /3 e, n e is the electron density. cold ions with the temperature of ions i : e >> i As V F >> C O, (V F 1 8 cm/ sec, C cm/sec ) Where C is the sound velocity. Electronic distribution in the potential is equilibrium and can be described by FD model.

23 n e Main System of Equations: r V t r r + ( V, ) V e M n r + div( nv) t e (1 ) 1/ n e + A + + A µ 4 e( n n) 3, e 1 A Here V, n, M are the velocity, density and mass of ions in material, _ is the electrostatic potential, e is the electron charge, n e is the density of electron gas, A is the term taking into account the exchange correction constant to electron density in FD model, µ is the Fermi energy, µ /3 (3 ), is the Plank constant. ne hm Ry µ

24 At, n e 3 n ne ( + ), 1+ A A n i n + n i, ( n/n << 1) (5) n e n + n e, ( n e /n << 1) (6) From the equations (1)- (4) we get n V r t e M n + n divv r i t 3 n e A i e ( + ) µ (5) (6) (7)

25 Putting (7) into (6) and taking into account (5) we will get the wave equation t C Here C is the sound velocity obtained using FD model C C ( + A), C µ 3M Stationary one dimensional Shock Wave with the velocity D: D Y x x Dt, (8) (9) _ sw t t From the equations (1),() we have e W + M D X

26 In the case e << 1 µ D we have following relations e max C 1+ a 3 µ 1 Equation for the electrical potential a 1 9 ( 8 ( 3 1 4n ( + ) ( ) 1 e A e MC : MD + + A) 1 A) (1) (11) From the equation we can find the width of shock wave U SW ~ C U 1/ ( + A) 1/ ñm, 8 7 ( D C), 1 1 cm µ 6 Here U is the velocity of atoms behind the shock wave n e e (1) 3/ U ( A) max µ + 6 Emax ~ ~ 3(1 1 e C a 1 1/ 7 ) V / cm (13)

27 Dependence of maximum electrical potential in the shock wave front as a function of shock wave velocity D. e µ max , 1,1 1, 1,3 1,4 D/C Characteristic time of electrical field (t r ) is equal M tr 1 1 e max 14 1 sec (14)

28 Using the relation 1 ~ << e e n n n the system of equations (1)-(4) can reduce to ) ( 1 1/3 x nv t n x n x n n n n C x V V t V Where ( ) 3 1 ; A n n e + he solution of equation in the moving coordinate system: z x Ct is equal t V z b V h V t z V sec ), ( C D ñåê ñm V 4 ~ 1., / ~ 1 4 (15) (16)

29 he distribution of velocities of atoms in shock wave front for two velocities of shock waves D 1 (D 1 1.3C ) and D (D 1.45 C ) (D >D 1 ) in Cu. V, cm/cek 1,x ,x1 3 6,x1 3 3,x1 3, -1,x1-6 -5,x1-7, 5,x1-7 1,x1-6 p, cm

30 Space profiles of no overlapping shock waves produced by several bunches (1,, 3) of 7 ev proton beam in Cu with the width of shock wave SW cm and the distance between two bunches L B.5x1-4 cm (L B D SW B ) at shock wave velocity D SW 1 4 cm/s and B 5 ns. L B 1 3 SW ( L B >> SW )

31 he electrical force F acting on a lattice ion (atom) due to the electrical field is equal F ez E max (17) During the life-time of electrical field ( t r ) the lattice ion with the effective charge will receive the momentum p F t r ez E max t r (18) and potential energy E P E P ~ Ze max ~ 1 1eV Criterion for point defect production in shock wave front E k MV (x)/ > E d E d is the threshold displaced energy: E d E P + E F, E F ~ -5 ev MD / Ze _ max > E F (19) ()

32 Propagation of shock wave in cold (ideal) crystal lattice: D Energy barriers for atomic displacements E E d E d ~ 15-3 ev

33 Propagation of shock wave in heated (non ideal) crystal lattice: D Energy barriers for atomic displacements: E E d E d ~ 4-1 ev X

34 Effective emperature and Defect Cluster Production during Shock Wave Propagation 1. emperature rise during shock wave propagation. Main Equations for Description of Shock Wave: Current of Energy: r rv j V + H Here is the density of material; V is the flow velocity of material; H E +PV is the thermal function of unit volume E is the internal energy; _ is the internal pressure; V is the volume of material.

35 hermal Current: k(_) is the thermal conductivity; _ is the temperature of material; Energy of the unit volume: k j ) ( r + E V Q V Equation of the conservation of energy: x k x H V V x E V t ) (

36 Equation (4) in the new coordinate system moving with the shock wave velocity D (YX-Dt): y k y PV w Dw D y PV E w w y ) ( Here wdy/dt is the flow velocity of material in the new coordinate system (VW+D); Main processes at different distances from center of shock wave _) At the flow velocity of material in laboratory coordinate system is equal zero (w-d) and equation (5) has the following form ( ) y k y DE y ) ( ( ) ( ) x k x C t E t ) ( In laboratory coordinate system the equation (6) coincides with the equation of thermal conductivity

37 B) At y the shock wave is described by the following relations: [ ] [ ] [ ] [ ] w P w P w w w H w H Index describes the physical values before the front of shock wave and index «1» behind the front of shock wave. aking into account the relations (8) from the equation (5) we get at y: ) ( y k y he solution of the CdVB equation for shock waves in the frame of homas-fermi-dirac model has the following form, 9 4 sec ) ( D y b W h W y w DC S +4W /9;

38 Let us estimate the width V of distribution W(p). It is equal: ñm ñm V C V b V V ~ 1 ; 1 ~ 1 ~ ~ Assuming that the thermal contribution of pressure and internal energy more then the "cold" contribution the pressure P is written in the form 3 V E P Where E is the internal energy of volume V. From the equation (3) we can get the following equation [ ] y k y D w w y w D E y ) ( 5 3

39 he distribution of atomic velocities in shock wave for two velocities of of shock waves D 1 (D 1 1.3C) and D (D 1.45C) (D >D 1 ) for Cu.

40 able 1 K(), W cm -1 K -1 M (V), K g cm 3 Cu 3K- M M -K K() x x1-1 3 K() x1-7 M >K K().1 Al 3K- M M - V K().4 K() x1-4 M 933 V 74.7 > V K()1.5 Ni 1K- M K() x x x1-1 4 M > M K().5 Fe 1K- M K() x x1-1 3 M > M K().33

41 emperature profile in the front of shock wave propagating in Cu with the different velocities: D 1 1 km/sec, D 1 km/sec and D 3 15 km/sec.

42 emperature profile in the front of shock wave propagating in Al with the different velocities: D 1 1 km/sec and D 1 km/sec.

43 emperature profile in the front of shock wave propagating in Fe with the different velocities: D 1 1 km/sec and D 1 km/sec.

44 emperature profile in the front of shock wave propagating in Ni with the different velocities: D 1 1 km/sec and D 1 km/sec.

45 At t < _ * _/D the thermal wave will overcome the shock wave: _ > L (x,t) D L B D _ ~ ( ) 1/ L B D SW _ x (x,t) D D L B x L _ at : t < _ * _/D At t > _ * _/D the shock wave will overcome the thermal wave: L > _ Cu: _ ~ 1cm /s, D ~ cm/s, _ * s

46 Numerical modeling of microstructure change in collimator materials using molecular dynamic simulations during shock wave propagation in copper. Molecular dynamic method of simulation of shock wave propagation in Cu. he sample is constructed from 8 atoms of copper arranged in FCC lattice having the form of rectangular parallelepiped 1a x 1a x a, (where a is the lattice constant). he periodic boundary conditions are applied along X and Y directions corresponding to short sides of crystal. On the surfaces Z and Z max a the mirror boundary conditions with rigid walls are used. It allows to investigate the reflection of shock wave from the surface. In the present numerical calculations the well known Verlet algorithm is used for the integration of motion equations for moving atoms. he interactions between moving atoms in Cu are described by the so-called Embedded Atom Potential. All necessary numerical data for the calculations are taken from the internet:

47 he changes of initial glass-like microstructure obtained by fast cooling of copper crystal lattice from 3K up to 3K after the penetrating of shock wave having the average ion velocity behind shock wave V cm/s.

48 he dependence of number of displaced atoms as a function of average ion velocity behind shock wave in the initial glass-like microstructure obtained by fast cooling of copper crystal lattice from 3K up to 3K

49 he changes of heated crystal-like microstructure at the temperature ini 8K after the penetraiting of shock wave having the average ion velocity behind shock wave Vm/s. he circles show the displaced atoms.

50 he effect of previous shear deformation on the changes of heated crystal-like microstructure at the temperature ini 6K after the penetrating of shock wave having the average ion velocity behind shock wave Vm/s in Cu. he circles show the displaced atoms. x1e-7 cm, Yx1e-7 cm, Xx1e-7 cm, Z

51 Microstructure of displaced atoms produced by the shock wave initiated by internal electrical field 7 V/A at the temperature 3 K and at the simulation time t.1 ps.

52 he results of numerical simulations for the spatial distribution of displaced atoms produced in proton beam area by the shock wave initiated by an internal electrical field 7 V/A at the temperature 3K at the three different simulation times: t1.3 ps, t.6 ps and t3.1 ps t.3 ps t.6 ps t.1 ps 5 556

53 Formation of channel produced by the shock wave initiated by an internal electrical field Em 1V/A in proton beam area of Cu crystal lattice at the temperature 3 K at the simulation time t.3 ps.

54 he results of numerical simulations for the spatial distribution of displaced atoms produced in proton beam area by the shock wave initiated by an internal electrical field 1 V/A at the temperature 3K at the three different simulation times: t1.3 ps, t.6 ps and t3.1 ps t.3 ps t.6 ps t.1 ps 5 6 7

55 Summary Propagation of shock wave can result in an additional strong temperature rise on the front of shock wave during the propagation of it. he maximum temperature on the front of shock wave in materials:cu, Al, Ni, Fe is increased with the increasing of shock wave velocity. he molecular dynamic simulations of microstructure change in Cu during shock wave propagation show that shock waves can produce stable point defects (displaced atoms). he numerical simulations of shock wave propagation show that comparing with the propagation of shock waves in an ideal crystal lattice (low temperatures) in thermal heated crystal lattice the additional displaced atoms (point defects: vacancies and interstitials) are generated. he numerical simulations of microstructure change demonstrate that atoms in the amorphous non-ideal crystal lattice can remove on some distances during shock wave propagation ( shock wave induced diffusivity ). he concentration of produced displaced atoms during shock wave propagation is increased with shock wave velocity.

56 Distribution profile of stopped electrons in copper per one 7 ev proton as a function of the depth into target and the radial coordinate

57 Distribution profile of stopped electrons in graphite per one 7 ev proton as a function of the depth into target and the radial coordinate

58 Distribution profiles of stopped hydrogen and helium atoms in copper per one 7 ev proton as a function of the depth into target and the radial coordinate.

59 Distribution profiles of stopped hydrogen and helium atoms in graphite per one 7 ev proton as a function of the depth into target and the radial coordinate.

60 Neutron energy spectrum per one 7 ev proton in copper on the several penetration depths of proton.

61 Neutron energy spectrum per one 7 ev proton in graphite on the several penetration depths of proton.

Kinetics. Rate of change in response to thermodynamic forces

Kinetics. Rate of change in response to thermodynamic forces Kinetics Rate of change in response to thermodynamic forces Deviation from local equilibrium continuous change T heat flow temperature changes µ atom flow composition changes Deviation from global equilibrium