Modelling Stripper Foil Heating for ESSνSB

M. Martini, CERN, Geneva, Switzerland

March 9, 2015

ESS higher energy complex layout

1

List of Slides

3 Introduction

4 Accumulator injection parameters

5 Foil heat model: Power density modelling

8 Foil heat model: Temperature calculation

11 H− injection scenario (no circulating H+)

13 Foil heating from just stripped H−

15 H+ circulating scenario (with injected H−)

20 Foil heating from stripped H− andcirculating H+

24 Summarizing remarks

25 What next

2

Introduction

• Proposal is to use the 2 GeV, 5 MW proton linac, 2.86 ms long pulses at 14 Hz

of the European Spallation Source [1] being built in Lund to deliver,

alternately with the spallation neutron production a very intense neutrino

beam for the discovery of leptonic CP violation. For this the linac must be

upgraded to supply, in addition to the 2.86 ms proton pulses at 14 Hz, four

0.72 ms H short pulses at 70 Hz for neutrino production.

• Pulse length used for neutrino production must be compressed to a few µs via

an accumulator ring due to the high current needed in the pulsed horn.

• Charge exchange injection of H− pulses will be used, the linac delivering

1.1×1015 H−/pulse (ions are assumed to be fully stripped for energy deposit).

• Focus is on the maximum temperatures that the stripper carbon foil will

undergo during the many accumulator re-fills [3].

• The approach carried out here follows an analytical treatment of foil heating

by means of a rough implementation of pseudo-painting schemes and

circulating proton foil hits, concurrently with ion injection.

3

Accumulator injection parameters

Parameter Assumptions for the analysis

Ion kinetic energy 1-ring intensity Ek=2 GeV∣∣∣N =2.8×1013 H−/pulse

H− pulse duration (one-ring filling) τp = 2.86/4 ms = 0.715 ms

H− pulse repetition rate fc = 70 Hz, τc = 14.29 ms

H−mean pulse current Iav = 62.5 mA

RMS normalized emittances εnx,y = 0.33 µm

Beta functions at injection βx = 9 m, βy = 18 m (at foil location)

Core beam sizes (H− spot at±1σx,y) σx = 1 mm, σy = 1.4 mm (at foil)

Stripper foil dimensions Lx = 17mm, Ly = 25 mm, tc = 3.95µm

Carbon foil thickness density tc = 750 µg/cm2

Figure 1: Ion pulse distribution for neutrinos (5th cycle for neutron spallation)

4

Foil heat model: Power density modelling

• 750µg/cm2 carbon foil: 2 GeV/H−.

SH+=1.76 MeVcm2/g stp power.

Se−= 1.61 MeVcm2/g [4]. Hence

β(H−)=β(e−)→1.09 MeV/e− at foil.

SH−= SH+ +2 Se−=4.98 MeVcm2/g.

• Recent studies [5]-[6] show that

(i) first 2/3 H− foil crossing is

treated as 1 e−& 1 H0, last 1/3

crossing as 2 e−& 1 H+.

(H0 stopping power is assumed to

be half of that for H+).

(ii) knock-on e−correction:≈84%

of the energy deposited in the foil.

SH− = 0.84×(SH+ +2 Se−

)×2/3

P0H− = SH− tcIav/(4σxσy) (1)

• P0H−= 23.2 MW/m2 (Eq. 1)

(1 charge=1.6×10−19C→1eV×1A=1W)

Figure 2: Foil layout (right-handed

system): inward arrow (red) is the beam

power inflow, outward arrows (green) are

the radiative power outflow.

5

Foil heating model: Power density modelling (2)• A beam power density distribution PH− (x, y, n) [W/m2] can be cast as the

product of a power density P 0H− times a super-Gaussian function [8], [9].

This choice is made to estimate at best the foil temperature due to injecting

ion pulse foil hits.

• Flat-topped power density distributions can be modelled using 2-D super

Gaussians (SG) with foil spot center impact position µx,y and standard-like

deviation σx,y(n).

• Un-normalized SG functions of order n > 0 are defined for u=x,y as

f(u, σu, µu, n) = exp

[−

y∑u=x

|u− µu|n

2λnσnu(n)

]with λ(n) =

[π

2

][1− 2n

]j

and j = 0, 1 (2)

• λ(n) aims to keep the derived SG standard deviation σSG,u(n) close to the

right σu of an observed distribution where σ2SG,u(n)=〈u2(n)〉−〈u(n)〉2 and

〈uk(n)〉 =

+∞∫∫−∞

ukf(x, y, σx,y, µx,y, n) dx dy

+∞∫∫−∞

f(x, y, σx,y, µx,y, n) dx dy

(3)

• Normalized SG functions can be written

fN(x, y, σx, σy, µx, µy, n) =f(x, y, σx, σy, µx, µy, n)

2πλ2(n)σx(n)σy(n)(4)

6

Foil heating model: Power density modelling (3)• Normalized SG are used to fit distributions of more rectangular, compact

shape (n≥2) i.e. with non-negative kurtosis) (e.g. particle beam distributions)

and also with small peak and wide tails (n<2 i.e. with negative kurtosis).

• Un-normalized SG are suited to fit power core distributions incident onto a

surface (e.g. the irradiance [W/m2] of laser beams) to quantify the power

deposit effects on materials (e.g. stripper foils).

• SGs are Gaussians for n=2 and hard-edge rectangular distributions if n→∞.

Case j=1: Integrals of f(u,σu,µu,n) (blue, Eq. 2),

fN(u,σu,µu,n) (red,Eq. 4),σSG,u(n) (green,Eq. 3).

1. For j=1 σSG,u(n)near≈ σu=1 since

σSG,u(n)/σu∈(0.86, 1) for

n=2 . . .∞ (Eq. 2) with u=(x, y).

σSG,u(2, 3, 5,∞)/σu =

(1, 0.89, 0.86, π/√

12 = 0.91).

2. For j=0 σSG,u(n)far≈ σu=1 since

σSG,u(n)/σu∈(0.58, 1) ,n=2. . .∞.

σSG,u(2, 3, 5,∞)/σu =

(1, 0.77, 0.65,√

1/3 = 0.58).

SG computations will be done for j=1 to better mirror the rms deviation σx,y

7

Foil heat model: Temperature calculation

• A Mathematica [11] 3-D model of foil heat-conduction equation with beam

power heat source and radiation cooling was set up (ignoring vacuum pipe

effects and heat convection).

• The variations of the carbon foil thermal conductivity k(T ) W/m K and heat

capacity c(T ) J/kg K vs. temperature T K have been considered (4th order

polynomials measurement fit [3]). The heat equation is (units: W/m3)

c(T (~u, t))ρ∂tT (~u, t)−∇ · [k(T (~u, t)~∇T (~u, t)] =(PH− (~u, n, t)− 2σεc(T

4(~u, t)− T 4

0 ))/tc (5)

t is the time, ~u=x,y,z the foil axes, ρ the carbon density kg/m3, tc the foil thickness

m, σ=5.67 10−8 W/m2K4 the Boltzmann constant, εc=0.8 the carbon emissivity.

• PH− is the heat source inflow from beam power density at foil entry W/m2.

• σεc(T4(~u, t)− T 4

0 ) is the foil radiation power outflow (black body radiation)

W/m2 (Stefan-Bolzmann law).

• The linac delivers repeatedly 4 pulses (τp=0.715 ms) during 4 cycles (τc=14.29

ms) followed by a 5th cycle for neutron spallation (forming a 71.4 ms long

cycle, cf. Fig. 1).

8

Foil heat model: Temperature calculation (2)• The beam power function PH− s made of a piecewise super-Gaussian Eq. 2

where k is the number of long cycle used.

PH− (x, y, n, t) ={P 0

H−f(x, y, σx,y, µx,y, n) if (tmod τc<τp)∧ (tmod (k+1)τc<kτc)

0 otherwise(6)

with P 0H− = 0.84SH− tcIav/(4σxσy) and

(SH+ +2 Se

)2/3 is the total ion stopping

power [4]-[5] (i.e. SH−=2.79 MeVcm2/g).

• The initial condition T (~u, 0) = T0 and the 6 foil boundary conditions are:

k(T (x, y, 0, t))∂zT (x, y, 0, t) = σεc(T4(x, y, 0, t)− T 4

0 )

−k(T (x, y, tc, t))∂zT (x, y, tc, t) = σεc(T4(x, y, tc, t)− T 4

0 )

k(T (x, 0, z, t))∂yT (x, 0, z, t) = σεc(T4(x, 0, z, t)− T 4

0 ) (7)

−k(T (x, Ly, z, t))∂yT (x, Ly, z, t) = σεc(T4(x, Ly, z, t)− T 4

0 )

k(T (0, y, z, t))∂xT (0, y, z, t) = σεc(T4(0, y, z, t)− T 4

0 )

−k(T (Lx, y, z, t))∂xT (Lx, y, z, t) = σεc(T4(Lx, y, z, t)− T 4

0 )

9

Foil heat model: Temperature calculation (3)

• The heat equation is solved with

Mathematica by the method of

lines for PDE discretization across

the space variables and an ODE

solver along the time.

• Stiffness effects can be observed

when accuraccy following from

smoothness allow for larger

time-steps wheras stability

requires smaller time-steps (e.g. if

the actual time scale is much larger

than small time scales existing in a

problem to solve) [10], [11].

Figure 3: Carbon thermal conductivity,

heat capacity and foil temperature during the

very first 0.715ms H− injection cycle where the

peak temperature is 1651 K at end of cycle.

• Neglecting heat conduction Eq. 5 reduces to a 1-D model with initial condition

T (x, 0) (the factor 2 accounts for the 2 large radiative cooling foil sides)

c(T (t))ρdT (t)

dt=

tc[PH− − σεc(T

(t)− T )] (8)

10

H− injection scenario (no circulating H+)

σx,y=(1,1.4) mm P0H−

= 23.2 MW/m2

• Un-normalized super-Gaussian

power density profiles PH− (x, y, n)

for n=2, 3, 10 with σx,y = (1, 1.4)

mm and µx,y = (4, 5.6) mm.

• SG of orders n=3 & n=2 will be

chosen to model the succeeding

H−& H+ power densities.

• Peak foil temperatures at the

end of the first 8 injections for 5

super-Gaussian power density

orders n=2,/,3, 4, 5, 100.

• The highest temperature difference

of 86 K occurs at the 2nd injection

(14.29 ms) for the SG order n=100.

Gaussian and flat-topped SG power density distributions

11

H− injection scenario (no circulating H+) (2)σx,y=(1,1.4) mm P0

H−= 23.2 MW/m2

Figure 4: H− beam power density

distribution PH− (x, y, 3, t) (Eq. 6) for

linac spot core size σx,y=(1, 1.4) mm cen-

tred at µx,y=(4, 5.6) mm.

Figure 5: Projection of the beam power

density distribution of Fig. 4 over the

whole injection pulse (τp=0.715 ms).

Snapshots of H− linac pulse power density

12

Foil heating from just stripped H−

σx,y=(1,1.4) mm P0H−

= 23.2 MW/m2

Figure 6: Foil temperature evolution

during the first two long cycles (71.45 ms

each) and along the 17 mm foil width

(similar plot hold for the foil height).

Figure 7: Foil temperature evolution

during the first two long cycles and along

the 3.9 µm foil thickness.

The maximum foil temperature is 2108 K (4th injection, 43.6ms)

13

Foil heating from just stripped H− (2)

σx,y=(1,1.4) mm P0H−

= 23.2 MW/m2

• 1-D simplified model (Eq. 8

without heat conduction): during

the first eight accumulator fillings

the peak foil temperature attains

2483 K.

• 3-D model (Eqs. 6-8 with heat

conduction): during the same eight

accumulator fillings the peak foil

temperature attains 2108 K, i.e.

375 K (≈15 %) lower than that of

the 1-D model.

Figure 8: Peak foil temperature

evolution of the H− linac spot center de-

rived from the 3-D model (green continu-

ous line) and the 1-D model without con-

duction (blue dotted line).

14

MOTIVATION

H+ circulating scenario (with injected H−)

• Pseudo-painting analogy with painting via tracking simulation : During the

painting the H+ beam recirculate through a surface called beam shape and

size (BSS for short) overlaping the foil near the injection.

• The BSS bounds the circulating H+ beamlets resulting from the transverse

painting phase space betatron motion.

• Correlated painting yield rectangular beam profiles (and BSS shape).

• H+transverse plane: end painting.

Courtesy A. Drozhdin, B. Wang [16]

[15] (top&bottom-left, top-right).

• H+vertical phase-plane: turn 6

from painting start (top-left).

• H+transverse plane: turn 33

from painting start (bottom-left).

15

H+ circulating scenario (with injected H−) (2)

• Pseudo-painting and circulating proton scheme : During the H− injection

superimpose a distribution of circulating protons which mimics a correlated

painting process [12], [13] to get the required transverse emittances at

extraction (H+ closed orbit µC,x,y(t) and beam size σC,x,y(t) move as√t).

• Roughly, circulating proton foil hits can be thought of as a crude model needing

no tracking simulations: Adding the H+ power density PH+ to PH− gives the

total power densityPH∓ (x, y, n−, n+, t)=PH− (x, y, n−, t)+PH+ (x, y, n+, t)

PH+ (x, y, n+, t) =

nH+∑i=

PH+× (9)

{f(x, y, σx,y, µ

ix,y(tmodτc), n+) if (tmodτc<τp)∧ (tmod(k+1) τc<kτc)

otherwise

• nH+=5 is the number of foil hits per proton [2], [7], µix,y are the nH+ initial

H+ foil impact centres, n−= 3 and n+= 2 are the PH+ and PH− SG orders with

PH+ = SH+ tcIav/(σxσy) with σx,y

def= nσ σx,y (nσ ≥ ) (10)

• nσ is an evenness index typifying the mean core size cross-section σx,y of foil

hit incident H+ power density. For nσ=2 the H+ power density for each of

the 5 beamlets is P0H+= 3.08 MW/m2 (SH+

knock-on e−−→ 0.84×SH+ )

16

H+ circulating scenario (with injected H−) (3)

Figure 9: Cross-section view of the injec-

tion painting scheme with a 17×25mm×3.9µm

foil : (i) Stripper foil (gray), (ii) H− injected

linac spot (tiny-red rectangle), (iii) H+ beam

snapshots (continuous rectangles), (iv) beam-

shape-and-size areas (dotted recangles).

Figure 10: Pseudo-painting zoom of

bottom-left foil corner: CirculatingH+ power

density foil impacts are mimicked using 5

imbricated beamlet power densities (at

σx,y=(1,1.4)mm) distributed within the paint-

ing domain from top (violet beamlets) to bottom

(red beamlets) and horizontally swept during

the injection within the crossing of the foil and

beam-shape-and-size areas.

17

H+ circulating scenario (with injected H−) (4)σx,y=(1,1.4) mm P0

H−= 23.2 MW/m2 P0

H+= 3.1 MW/m2 (nσ = 2)

Figure 11: Joined H∓ beam

power densities. The 5 intertwined

H+ beamlet power densities are aligned

along the y-axis and moved vs time along

the x-axis.

Figure 12: Contour plot of joined H∓

beam power densities PH∓ (x, y, 3, 2, t) of

Fig. 11 at injection end (t=τp=0.715 ms).

Snapshots of joined H− pulse and circulating H+ power densities

18

Just H+ circulating scenario (no injected H−) (5)

σx,y=(1,1.4) mm P0H−

= 23.2 MW/m2 P0H+= 3.1 MW/m2 (nσ = 2)

Figure 13: H+ circulating beamlet

power density PH+ (x, y, 2, t) (Eq. 10)

(no H− power density PH− (x, y, 3, t).

• Pseudo-painting drawbacks:

• (i) The number of foil its by

proton was not found analytically

and must be computationally

derived from beam painting

simulations.

• (ii) The true transverse particle

distribution and beam

shape-size contour during the

pseudo-painting are not known.

• (iii) Magnet errors, resonance,

coupling, non-linearity and

intensity dependant effects (e.g.

space charge) are not included in

the modelling.

Snapshots of just circulating H+ power density

19

Foil heating from stripped H− and circulating H+

σx,y=(1,1.4) mm P0H−

= 23.2 MW/m2 P0H+= 3.1 MW/m2 (nσ = 2)

Figure 14: Foil temperature evolution

along the first two long cycles over the 17

mm foil width (x-direction). Similar plot

holds for the height.

Figure 15: Temperature on the foil

surface at end of the 1st (0.7 ms, 1850 K)

and 8th (115 ms, 2253 K) injections and at

end of the 10th cycle (143 ms, 656 K).

Maximum foil temperature is 2309 K (3rd injection, 29.3ms)

20

Foil heating from stripped H− and circulating H+(2)

σx,y=(1,1.4) mm P0H−

= 23.2 MW/m2 P0H+= 3.1 MW/m2 (nσ = 2)

• Modulation of the foil temperature

over the first two long cycles (made

of 4 consecutive accumulator filling

cycles of 14.29 ms each plus 1 14.29

ms empty cycle gap) is quite stable.

• During this time the temperature

varies between 656 K (end of 10th

cycle) and 2309 K (end of 3rd

injection).

• The effects of the circulating

proton foil hit increase the foil

temperature by 234 K (11.4 %)

(at end of 2nd injection).

Figure 16: Peak foil temperature plots

along the first two long cycles, consid-

ering the H∓ beam (red) and just the

H− (green, no circulating H+ foil hits).

21

20% larger beam size

Foil heating from stripped H− and circulating H+(3)

σx,y=(1.2,1.7) mm (20% more P0H−

= 16.1 MW/m2 P0H+= 8.5 MW/m2 (nσ = 1)

Figure 17: Foil temperature evolution

along the first three long cycles over the

17 mm foil width (x-direction). Similar

plot holds for the height.

Figure 18: Foil temperatures at

end of the 1st (0.7 ms, 1500 K), and

12th (186 ms, 1948 K) injections and of

15th cycle (214 ms, 749 K).

Maximum foil temperature is 1983 K (4th injection, 43.6ms)

22

Foil heating from stripped H− and circulating H+(4)

σx,y=(1.2,1.7) mm (20% more P0H−

= 16.1 MW/m2 P0H+= 8.5 MW/m2 (nσ = 1)

• Modulation of the foil temperature

over the first two long cycles (made

of 4 consecutive accumulator filling

cycles of 14.29 ms each plus 1 14.29

ms empty cycle gap) is quite stable.

• During this time the temperature

varies between 749 K (at the end

of the 10th cycle) and 1983 K (at

end of 4th injection).

• The effects of the circulating

proton foil hits increase the foil

temperature by 174 K (9.6 %)

(end of 2nd injection).

Figure 19: Peak foil temperatures over

the first three long cycles covering the

H∓ (red) and H− beams (green, no proton

foil hits).

23

Summarizing remarks

σx,y (1,1.4) mm (1.2,1.7) mm ∆T

nσ=2 2309 K 1961 K 48 K

nσ=1 2336 K 1983 K 353 K

∆T 27 K 36 K

• Peak foil temperatures for the 4

cases studied: nominal linac beam

size and 20% larger size both for

the evenness index nσ=1&2 .

• Peak temperature results show that the evenness index doesn’t have a too

strong impact on the foil heating. So nσ=1 could be the best choice as it

yields mean core circulating proton beamlet sizes quasi-equal to that of

injected ion linac spots.

• The pseudo-painting scenario built is presumably based on a realistic enough

approach as it delivers peak temperature differences between H− linac spots

and combined H− with H+ comparable to that of other studies [2]-[3].

• Particle tracking code PyORBIT [18], [19] would certainely be a good choice

to benchmark the foil heat modelling study presented here.

(i) Small changes in the parameters may make big differences in the results

(ii) Peak allowable foil temperature seems to be kept below ≈2300 K [5]

24

What next

• The accumulator lattice design must be flexible enought that sufficiently

large linac spot sizes can be made at ring injection.

• The stopping power figure (and thus beam power density via the linac

current and the ion spot size) is a key issue as it determines the foil heating

so that experiments on the chosen foil material may be needed.

• Evaluate the consequences and possible upgrading of the accumulator in

case of linac beam energy change.

• This study sounds like showing the possibility to do a first setting up of the foil

stripping system (later research would permit an upgrade to laser stripping).

• The lattice design should be made such that optics for both laser and foil

stripping can be used.

25

References

[1] E. Bouquerel et al., ”A new facility concept for the production of very intense

neutrino beams in Europe”, Proceedings of IPAC2014, Dresden, Germany,

2014.

[2] E. Wildner et al., ”The accumulator of the ESSNUSB for neutrino

production”, IPAC’14, Dresden, Germany, June 2014.

[3] C. J. Liaw et al.,”Calculation of the maximum temperature on the carbon

stripping foil of the Spallation Neutron Source”, PAC, New York, 1999, pp.

3300.

[4] J. M. Berger, ”Stopping Power and Range Table for Electrons, Protons and

Helium Ions”, NISTIR 4999,

http://physics.nist.gov/PhysRefData/Star/Text/contents.html.

[5] M. Plum et al., ”SNS stripper foil development program”, NIM A590 (2008)

pp. 43 and private communication, 2014.

[6] H. Ogawa et al.,”Measurements of energy losses of 10-MeV neutral hydrogen

atoms in carbon foils”, Phys. Rev. A 54, 5027 (1996).

[7] H. Schonauer, CERN, private communication, 2014.

[8] F.-J. Decker, ”Beam distribution beyond RMS”, SLAC-PUB-6684, 1994 (A).

26

[9] B. Lu, S. Luo, ”Beam propagation factor of apertured super-Gaussian

beams”, International Journal of Light and Electron Optics, Volume 112, Issue

11, 2001, pp. 503-506.

[10] R.M. Mattheij, S.W. Rienstra, J.H.M. ten Thije Boonkkamp, ”Partial

differential equations, Modeling, analysis, computation”, SIAM, 2005.

[11] Wolfram Mathematica R© Tutorial Collection, ”Advance numerical differential

equation solving in Mathematica”, Wolfram Research, Inc., 2008.

[12] J. Wei et al., ”Low-loss design for the high intensity accumulator ring of the

SNS”, PRSTAB 080101, 2000.

[13] J. Beebe-Wang et al., ”Beam properties in the SNS accumulator ring due to

transverse phase space painting”, EPAC 2000.

[14] J. Beebe-Wang., ”Oscillating injection painting and related technical issues”,

BNL/SNS Technical Note No. 081, BNL 2000.

[15] J. Beebe-Wang., ”Injection painting, foil & target distribution”, SNS ASAC

Review, BNL 2000.

[16] A. Drozhdin, O. Krivosheev., ”The Fermilab proton driver painting injection

simulations”, FERMILAB-FN-0694 2001.

[17] J. Qiu et al., ”Studies of transverse phase space painting for the CSNS RCS

injection”, High Energy Physics and Nuclear Physics, Vol. 31, No. 7, 2007.

[18] J.A. Holmes et al., ”Status of the ORBIT code: recent developments and

27

plans”, EPAC 2008.

[19] S. Webb et al., ”Simulating high-intensity proton beams in nonlinear lattices

with PyORBIT”, IPAC 2012.

28