Moving solid metallic targets for pion production in the Muon Collider/Neutrino Factory project

Abstract

The production of large fluxes of pions and muons using high-energy, high-intensity proton pulses impinging on solid or liquid targets presents unique problems which have not yet been entirely solved. We investigate the possibilities of using solid targets by choosing a metal of either extremely low thermal expansion coefficient or exceptionally high mechanical strength. Candidates are respectively Super-Invar and Vascomax 350 or Inconel 718. Moving targets in the form of chains or cables would be required for cooling purposes. These materials seem easily capable of surviving the beam pulses required for the largest beam power contemplated. Questions regarding radiation damage effects are being investigated.

Full text

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MUC-NOTE-TARGET-234

Moving Solid Metallic Targets for Pion Production in the Muon Collider /
Neutrino Factory Project
P.A. Thieberger and H.G. Kirk
Brookhaven National Laboratory

Introduction

The production of large fluxes of pions and muons using high energy, high intensity proton pulses impinging on
solid or liquid targets1) presents unique problems which have not yet been entirely solved. The large required power
and power density deposited in the material as well as the short pulse duration produce large, almost instantaneous
local heating, and the resulting sudden thermal expansion can result in damage-causing stresses in solids and in the
violent disruption of liquid jets.

So far three types of target materials have been proposed and investigated to some extent, both theoretically and
experimentally. They are steel, graphite and mercury with the largest recent emphasis on the last two. While none
of these alternatives can be ruled out at this point, we will briefly mention below some of the problems observed2)
and envisaged with mercury3) and graphite4), which provided the incentive for the present, new and more detailed
look at high strength steel and Invar alloys.

During a recent series of experiments2) with 24 GeV proton pulses hitting both stationary and moving mercury
targets, we observed the dispersal of the liquid targets with beam pulses up to 4 TP (less than the 16 TP and 34 TP
pulses required for the 1 MW and 4 MW options to be discussed). While the dispersal velocity was seen to be
modest enough to mitigate concerns for damage to the interior of the target vessel, new concerns arose 3) that the
residual mercury drops within the target chamber will not sufficiently clear the vessel. The mercury jet will be
flowing at about 2 liters per second and much of this mercury, following the interaction with the beam, will collide
with the walls and remain in the target chamber before settling into the pool of mercury which will serve as the
beam dump. This residual material could interfere with the produced pions spiraling within the solenoidal capture
system and with subsequent portions of the mercury jet.

The recently measured2), extremely small, beam-induced strains in a carbon-carbon composite indicate that such a
material may perhaps survive the thermal shock induced by a much more intense beam. Since these targets are
envisaged as being stationary one must consider the problem of removing the power deposited by the beam without
interfering too much with the particles being extracted. Recent estimates4) indicate that the average temperatures
reached by these targets will exceed the range where the linear expansion coefficient is sufficiently small. However
other special high-strength graphites4), somewhat similar to the ATJ carbon used in the experiments2) may actually
work. Insufficient data is available at present for a full evaluation.

The conditions created by the short beam pulses (rms width ~50 ns during recent tests2) and <5 ns for the final
system1)) are very unusual. Intense almost instantaneous beam heating causes a fraction of the target volume to
suddenly be in a highly compressed, inertially confined state. Subsequently this volume expands initiating strong
vibrations in the material. The amplitude of these oscillations is such that large negative pressures (tension) or shear
stresses can be generated exceeding the strength of the material and thus causing mechanical damage. To evaluate
this situation in detail for a given geometrical configuration and a given material, extensive computer simulations
are required such as the ones recently performed5) for the windows in experiment E9512).

Here we attempt to develop simple, more general criteria, suitable for a preliminary screening of possible materials
by evaluating their relative merits. Computer modeling will still be required once a candidate material is selected
and specific target geometries are considered. The two criteria we will discuss are:

1) Assume that negative pressures (tension) will arise in the oscillations which are similar in magnitude to the
initial compression.

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2) Assume that the compressive energy initially available can appear later in a similar volume (same energy
density) as tensile energy available to do damage.

Both criteria may be violated if significant vibration focusing effects take place such as were observed in
calculations5) for circular windows rigidly constrained at their periphery. We will assume here that such highly
symmetric geometries with natural foci can be avoided by adopting appropriate target geometries, thus avoiding
such situations. In fact the natural tendency in most cases should be for the energy initially concentrated in a fraction
of the target volume to rapidly spread over the entire volume thus reducing subsequent peak values. Before
describing and further justifying these criteria we first evaluate the characteristic vibration periods and heat transfer
times in the target materials.


Characteristic Times

The times over which energy is deposited by the beam are of the order of ~50 ns rms for the recent E951
experiment 2) and ~5 ns for the proposed facility 1) The nature of the phenomena and of the approximations we can
make depend on how these times compare with the thermal time constant of the target and the vibrational period or
sound transit time across the diameter of the target6).

The results for sound transit times shown for iron in Fig. 1 correspond to a radial sound velocity of 5100 m/s, which
is within ± 2% of values for different types of steel.7).


Fig. 1 The sound travel times are the same within ± ±2% for the materials considered here and the value used
for the velocity is 5100 m/s. The order-of-magnitude heat transfer times are 1/e decay times for the
exponential decay of a temperature difference across a characteristic distance equal to the radius of the
target.


We see from fig. 1 that the sound transit times and therefore the oscillation periods are much longer than the 5ns
beam pulse length planned for the final system. For example for a 5 mm radius target the ratio of these times is 392.
Characteristic times for targets heated by beam pulses
1.E-09
1.E-08
1.E-07
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
12345678910
Target radius (mm)
Time (seconds)
Heat Transfer in Invar, Inconel
Heat Transfer in Vascomax
Heat Transfer in Iron
Sound Travel Time
50 ns Beam Pulse
5 ns Beam Pulse

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Or, looking at this in another way, for a velocity of 5100 m/s the distance traveled by the disturbance in 5 ns is only
25.5 µm compared to a beam size of several mm. Therefore, to a very good approximation, nothing moves
appreciably during the pulse and the thermal expansion is inertially confined, leading to a large initial compression.

Large temperature differences will arise inside the target following a beam pulse. The time it takes for such
temperature differences to decay will depend on the gradients, the specific heat cp and the thermal conductivity κ of
the material. Here we only make rough order-of-magnitude estimates to determine whether or not heat conduction
will play a significant role in the time interval after the pulse during which large vibrational stresses may occur. For
this purpose we replace our target by an infinite sheet of thickness r (equal to the radius), into one side of which
flows a constant heat flux φ, the other side being attached to a heat sink maintained at temperature T0 which will
receive a flux φ' = φ .The temperature T of the input side will then be such that:

φ = κ × (Τ − T0) / r 1)

If we now suddenly turn off the input heat flux, φ' will initially stay at its previous value but will start to decrease as
the object cools down. The initial rate of this cooling, dTav/dt, will depend on the heat capacity per unit area cp
× ρ × r of the sheet, where ρ is the density :

dTav/dt = φ / (cp × ρ × r ) = κ × (Τ − T0) / (cp × ρ × r 2 )

The rate of decay of a temperature difference across any object of characteristic dimension r is proportional to that
temperature difference, with a constant of proportionality roughly equal to ~ κ / (cp × ρ × r 2 ). The inverse of this
quantity , cp × ρ × r 2 / κ , i.e. the 1/e decay time, is plotted as the upper three curves of Fig.1.

We see that not only will single oscillations be adiabatic, but heat conduction will not be a significant factor even
after hundreds of oscillations.


Criteria for Selecting a Material

The most straightforward comparison for evaluating a material for this application is between the value of the initial
compression and the yield stress or the fatigue limit. A symmetric compression is of course not the primary cause of
damage, but the idea is that, as the vibrations evolve, the Fourier components of the initial pressure pulse may later
recombine generating tensions of the same order of magnitude. We will later address the limitations of this
criterion. In any case it should provide a rough idea of the suitability, and a reasonable comparison between the
relative merits of various materials.

To determine the initial compression we must first find values of the energy density deposited by the beam. This
was done by using the MARS code8) for a number of different iron target radii, and by assuming a beam σ 2.5
times smaller in each case. An example of such a calculation is shown in Fig. 2 for a target radius of 7.5 mm, a
beam rms radius of 3 mm and a proton pulse of 16 TP (1TP = 1012 protons.)


2)

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0
5
10 15 20
Z(cm)
25
30
35
0
r(mm)
2
4
6
0
5
10
15
20
25
J/g

Fig. 2 Three-dimensional view of energy deposition MARS data for a 3 mm rms radius 16TP beam on a 7.5
mm radius iron target.




The result of these MARS calculations for iron are summarized in Table 1

Table 1. Maximum energy density deposited by a 16TP, 24 GeV beam in an iron target.

Beam width [rms mm] .5 1 1.5
Target radius [mm] 1.25 2.5 3.75
Maximum energy density [J/g] 305 105 55.6



These calculations are time consuming, and the values obtained for iron are representative enough for the alloys
considered here since the densities are similar and the atomic number of the main constituents cover a relatively
narrow range (see Table 3). Once the maximum value εmax of the energy density (per unit mass) is found from these
calculations for each case we calculate the corresponding maximum compression Pmax for each material.

Pmax =3 × εmax × Β × α / cv 3)

Where B is the bulk modulus, α the linear expansion coefficient and cv the specific heat at constant volume. Values
for these parameters, and for others used later are listed in Table 2 for the materials of interest. The chemical
compositions of these alloys are detailed in Table 3.



2
5
36.0
2.5
6.25
26.5
3
7.5
22.1
3.5
8.75
16.5

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Table 2. Mechanical an thermal characteristics of the materials
Densit
y Exp.
Coeff.
Linear
Young
Modulus
Bulk
Modu-
lus
Poisson
Ratio
Specific
Heat @
constant
pressure
cp
J/(g 0K)
0.478
0.435
0.450
Thermal
Conducti-
vity
Yield
Stren-
gth
Fatigue
Endu-
rance
Limit
σ-l
M Pa
~85
586
758
Symbol
Unit
Iron
Inconel 718
VascoMax
C-350
Super Invar


Table 3. Typical Chemical Compositions (%).
Element
Atomic Number
Inconel 718
Vasco Max C-350
Super Invar


We consider the second criterion mentioned above because the effects of oscillations along the three axes are not
really independent, and because "mode mixing" is possible, i.e. initial oscillations along the y-axis, for instance, can
later contribute to the amplitude along the y-axis. Such effects are included in detailed computer simulations for
each particular geometry. To arrive at a general criterion, we first calculate the energy Eσ per unit volume required
to stress the material to reach the yield stress σ0.2 or the fatigue endurance limit σ-l :

Eσ = σ2/ (2 Y) (4)

Where σ is either σ0.2 or σ-l according to which comparison we want to make. We then compare this energy
density to the mechanical energy Em initially available in the compressed, inertially confined, volume. This is only a
fraction of the total energy Etot deposited by the beam, the rest being converted immediately to heat:

Em = Etot × Β × α2 × ( 2 T0 + 3/2 Etot /cp ) / (cp × ρ )

where T0 is the absolute temperature before beam heating, cp is the specific heat and ρ is the density.

Derivations of equations 3), 4) and 5) are given in the appendix.

One final comment regarding the above criteria is in order. Our application of these criteria will probably be
conservative or pessimistic because values for yield stresses and fatigue endurance limits we will use have all been
experimentally determined at frequencies which are orders of magnitude smaller than the hundreds of kHz
characteristic of the transverse oscillations of interest here. There is evidence9) that fatigue endurance limits increase
with frequency already at the much lower frequencies used for these tests. The same is probably true for yield
stresses which are normally determined under essentially static conditions.


Stress Estimates
The results of the calculations outlined in the previous section are shown in Figs. 3 through 10. Figs. 3 through 6
correspond to the first criterion in terms of maximum compressive stresses as % of yield stress and % of fatigue
endurance limit, both for the 1 MW and the 4 MW options. Figs. 7 through 10 correspond to the second criterion
in terms of available energy to yield- and fatigue endurance limit ratios, also for the 1 MW and the 4 MW options.
ρ
α
Y B
µ

λ
σ0.2
M Pa
170
1034
2242
g/cm3
7.87
8.19
8.08
10-6 / 0K
12.5
13.1
15.0
G Pa
205
200
200
G Pa
171
158
167
W/(m 0K)
80
11.2
25.2
0.30
0.29
0.30
8.15 0.63 144 88.9 0.23 0.515 10.5 276 ~138
C
6

.02
.05
Al
13
0.5
0.1
.07
Si
14

.05
.09
S
16

.005
.01
Ti
22
1
1.4

Cr
24
19

.03
Mn
25

.05
.4
Fe
26
19
63
62
Co
27

12
5.4
Ni
28
52.5
18.5
31.8
Cu
29


.08
Nb
41
5


Mo
42
3
4.8

(5)

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Maximum initial stress as % of yield stress for the 1 MW option.
1
10
100
1000
10000
123456789
Target Radius (mm)
% of Yield Stress
Iron
Inconel 718
Vascomax 350
Super Invar

Fig. 3

Maximum initial stress as % of yield stress for the 4 MW option.
1
10
100
1000
10000
123456789
Target Radius (mm)
% of Yield Stress
Iron
Inconel 718
Vascomax 350
Super Invar

Fig. 4

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Maximum initial stress as % of fatigue limit for the 1 MW option.
1
10
100
1000
10000
123456789
Target Radius (mm)
% of Fatigue Limit
Iron
Inconel 718
Vascomax 350
Super Invar

Fig. 5

Maximum initial stress as % of fatigue limit for the 4 MW option.
1
10
100
1000
10000
100000
123456789
Target Radius (mm)
% of Fatigue Limit
Iron
Inconel 718
Vascomax 350
Super Invar

Fig. 6

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Available energy to yield energy ratio for the 1 MW option
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
123456789
Target Radius (mm)
Available energy / Yield energy
Iron
Inconel 718
Vascomax 350
Super Invar

Fig. 7

Available energy to yield energy ratio for the 4 MW option
1.E-03
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
123456789
Target Radius (mm)
Available energy / Yield Energy
Iron
Inconel 718
Vascomax 350
Super Invar

Fig 8

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Available energy to fatigue energy ratio for the 1 MW option
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
123456789
Target Radius (mm)
Avail. energy / Fatigue energy
Iron
Inconel 718
Vascomax 350
Super Invar

Fig. 9

Available energy to fatigue energy ratio for the 4 MW option
1.E-02
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
123456789
Target Radius (mm)
Avail. energy / Fatigue Energy
Iron
Inconel 718
Vascomax 350
Super Invar

Fig. 10

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The results for a 7.5 mm radius target and a 3 mm rms radius beam are summarized in Tables 4 and 5 for the 1MW
and the 4 MW options respectively.

Table 4 1 MW-option results for a 7.5 mm radius target

Iron Inconel 718 Vascomax
C- 350
Maximum Stress/
Yield Stress
Maximum Stress/
Fatigue Limit
Available Energy/
Yield Energy
Available Energy/
Fatigue Energy


Table 5 4 MW-option results for a 7.5 mm radius target

Iron Inconel 718 Vascomax C-350 Super Invar
Maximum Stress/
Yield Stress
Maximum Stress/
Fatigue Limit
Available Energy/
Yield Energy
Available Energy/
Fatigue Energy



One can see that iron is inadequate even for the 1 MW option, but the other three materials are probably viable, even
for the 4 MW option. Only according to the most stringent criterion, (available energy/fatigue energy) could there
be a problem in the 4 MW case with Inconel 718 and Vascomax C-350 for which these ratios are > 1. However, as
was mentioned before, the fatigue endurance limits at high frequencies are expected to be considerably higher than
the tabulated values used here. If these high-frequency values were known the computed energy ratios may well
end up being <1. Another possibility for improving the situation even more is to use proton beams of non-Gaussian
cross section. That possibility is discussed in the next section. The approach of simply adopting larger targets and
correspondingly larger beams is limited by the increasing outgoing particle loss due to absorption.

Another observation is that Vascomax C-350 is superior to Inconel 718 if their respective yield stresses are
considered, but they become almost equivalent if the fatigue endurance limit is take as the relevant factor.

Finally it is obvious from the Figs. 3 through 10 and from Tables 4 and 5 that Super Invar is vastly superior to the
other materials regarding thermal shock, as long as the operating temperature doesn't exceed ~150 0C where the
linear expansion coefficient starts to increase rapidly (see Fig.11). In Figs. 3 through 10 the curves corresponding to
Super Invar do not extend to small radii because the higher energy density would heat the material too much. The
assumption is that each portion of the target is thoroughly cooled to room temperature or below before returning to
the reaction chamber. Important concerns regarding Super Invar are that rather careful heat treatments are required
to achieve the very low values of the expansion coefficient, and it remains to be seen to which extent this property
will be affected by repeated temperature cycling and by radiation damage. Experiments are planned10) to explore
these questions.


Super Invar
1.7 0.27 0.16 0.025
3.3 0.48 0.47 0.050
10.9 0.28 0.09 0.004
43.6 0.87 0.81 0.014
3.4 0.54 0.32 0.05
6.7 0.95 0.94 0.10
24.2 0.61 0.21 0.008
96.8 1.93 1.79 0.03

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Fig. 11 Linear expansion as a function of temperature for Invar and Supper Invar alloys.



Non Gaussian Beams

One way to reduce maximum stresses is to reduce the peak energy density at the center of the target by spreading
out the beam more uniformly over the target cross section. Ideally one would strive for a totally uniform beam,
which, when compared to a Gaussian profile, would reduce the central energy density by a factor ~3 for a Gaussian
beam with rms radius = 0.4 × target radius. The comparison of the intensity profile of such a "flat" beam with a
Gaussian beam of equal total intensity is shown in Figs. 12a and 12b

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-3
-2
-1
0
1
2
3
-3
-1
1
0.00
0.05
0.10
0.15
0.20

-3
-2
-1
0
1
2
3
-3
-1
1
0.00
0.05
0.10
0.15
0.20


Fig 12a Fig. 12b

Comparison of a doubly Gaussian beam profile (a) with an ideal flat profile (b) containing the same number
of particles.

A perfectly flat beam such as shown in Fig. 12b can of course not be realized, but using octupole lenses one can
generate profiles which are fairly close to this goal. Fig 13 shows one of the projections of such a distribution which
was calculated11) for larger "uniform" beams required for the irradiation of biological materials.

Fig. 13 Octupole-lens generated beam profile compared to a Gaussian profile with the octupole turned off.

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While a factor of 3 in peak energy density reduction is out of reach, it seems reasonable to expect a factor ~2, which
could be very significant (see Table 5)

One objection12) to such non-Gaussian beams is that larger temperature gradients can induce unacceptably large
thermal stresses. In our rather unique situation these stresses are however much smaller than the large dynamic
stresses that can be induced following the inertially confined initial compression. This is discussed further below, in
the section on residual stresses.


Stress Focusing

The stresses estimated in the previous sections can be exceeded in systems with highly symmetric geometries such
as was observed5) in simulations for circular windows rigidly constrained at the periphery and impacted by a circular
beam centered on the window. In such a case reflected waves can interfere constructively generating large localized
stresses before the oscillations die down. Similar calculations need to be performed for rod-shaped targets. If the
problem appears then the solution will probably involve the selection of a cross-sectional shape without natural foci,
i.e. circles and ellipses would need to be avoided. For example, rectangular cross sections such as suggested for the
"band saw" target will probably work.

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Possible Implementations

One possible implementation of a moving metallic target is the "Band Saw" system which has been previously
described13) . One potential disadvantage of that design is that the rigid circular target must enter and exit the
extremely high radiation area at points where the radiation is still very high, and this fact leads to shielding
difficulties. Also the geometry of the coils needs to be modified in a rather critical area to accommodate the "band
saw".

Here we suggest the possibility of rather compact metallic chains such as sketched in Fig. 14 or the use of a metallic
cable as originally suggested by Palmer 15) .



Fig. 14 Schematic examples of metallic chain links showing rather compact designs with large metal to gap
volume rtatios.

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Fig.15 Schematic example of a chain with long links that would allow the beam to be coaxial with the target.


Such chains or cables could be as long as necessary to allow sufficient time for thorough cooling, and they could
enter and exit the high radiation area at points sufficiently far removed from the reaction chamber so as to minimize
the impact on shielding and on coil locations. In the case of chains, lubrication of the joints could be achieved with
graphite powder or using graphite bushings. The reliability of such chains could be extremely high. We note that,
in a different application, much lighter and weaker chains using plastic joints operate reliably for years at
comparable velocities as part of the charging systems in many electrostatic accelerators16).

To change such chains or cables, the old target can be used to pull the new one through, except in the unlikely event
of an unexpected rupture. For such an eventuality it may be possible to leave in place auxiliary pull wires or cables
which would not be exposed to the direct beam, and would therefore not be radiation damaged.

Finally, it should be mentioned that for most of these chain or cable configurations (except the one shown in Fig.
15), the beam would probably be inclined at a shallow angle with respect to the axis of the target. Nevertheless, all
the stress estimates above, and also the cooling requirements estimated in the next section correspond to centered
beams parallel to the target. The peak energy densities deposited by inclined beams are considerably smaller than
assumed here. For such geometries we are therefore overestimating stresses and temperatures increments by up to a
factor ~2, which is a nice safety factor.


Target Velocity and Length Considerations

The 1 MW and the 4 MW scenarios1) considered as options for this project would have different pulse sequences
and different beam intensities. These factors will obviously impact the choice of material and also the overall length
and velocity of the target to avoid excessive heating and to allow sufficient time for cooling. The pulse sequences
and intensities for both cases are represented schematically in Fig 16.