Moving Solid Metallic Targets for Pion Production in the Muon Collider /
Neutrino Factory Project
P.A. Thieberger and H.G. Kirk
Brookhaven National Laboratory
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
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.
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
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
Target radius (mm)
Heat Transfer in Invar, Inconel
Heat Transfer in Vascomax
Heat Transfer in Iron
Sound Travel Time
50 ns Beam Pulse
5 ns Beam Pulse
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.)
10 15 20
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.
Table 2. Mechanical an thermal characteristics of the materials
Table 3. Typical Chemical Compositions (%).
Vasco Max C-350
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.
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.
10-6 / 0K
8.15 0.63 144 88.9 0.23 0.515 10.5 276 ~138
Maximum initial stress as % of yield stress for the 1 MW option.
Target Radius (mm)
% of Yield Stress
Maximum initial stress as % of yield stress for the 4 MW option.
Target Radius (mm)
% of Yield Stress
Maximum initial stress as % of fatigue limit for the 1 MW option.
Target Radius (mm)
% of Fatigue Limit
Maximum initial stress as % of fatigue limit for the 4 MW option.
Target Radius (mm)
% of Fatigue Limit
Available energy to yield energy ratio for the 1 MW option
Target Radius (mm)
Available energy / Yield energy
Available energy to yield energy ratio for the 4 MW option
Target Radius (mm)
Available energy / Yield Energy
Available energy to fatigue energy ratio for the 1 MW option
Target Radius (mm)
Avail. energy / Fatigue energy
Available energy to fatigue energy ratio for the 4 MW option
Target Radius (mm)
Avail. energy / Fatigue Energy
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
Table 5 4 MW-option results for a 7.5 mm radius target
Iron Inconel 718 Vascomax C-350 Super Invar
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
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
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
Fig 12a Fig. 12b
Comparison of a doubly Gaussian beam profile (a) with an ideal flat profile (b) containing the same number
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.
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.
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.
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
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
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.
1 MW OPTION
4 MW OPTION
15 pulses per second
66.7 kJ per pulse
17.3 TP per pulse
Fig. 16 Proton beam pulse sequences for the proposed 1 MW and 4 MW options.
We first consider the case of Super Invar, using the example shown in Fig.2 for a 3mm rms radius beam impinging
on a 7.5 mm radius target. Using the data for the 1MW and the 4 MW options shown in Fig.16 and the specific heat
of Super Invar listed in Table 2, we obtain the core temperature increments along the 40 cm target shown in Fig.17
30 pulses per second
133 kJ per pulse
34.7 TP per pulse
Temperature increase for multiple pulses on a 1.5
cm diameter Super Invar Target
∆ ∆T for the 1MW option (°C)
∆ ∆T for the 4MW option (°C)
every 10 cm
every 15 cm
every 20 cm
every 30 cm
every 40 cm
The left temperature scale is for the 1 MW option and the right one for the 4 MW option. The lowest curve labeled
"every 40 cm" corresponds to a single beam pulse per target section, avoiding any overlap. We see that for the 4
MW option the maximum temperature increment in this case is 86 0C. If the hottest spot in the target is cooled
down to, say, 30 0C before returning to the reaction chamber then the maximum temperature reached could be 116
0C. In reality this temperature will be somewhat lower if the rotation of the chain or cable is such that the hottest
spot from one cycle doesn't coincide with the maximum energy deposition point in the next cycle. In any case,
considering the shape of the linear expansion coefficient curve (Fig. 11), one must conclude that overlapping of
beam pulses must be avoided in this case. Then the minimum target velocity becomes 40 cm / 33.3 ms = 12.0 m/s
which is one of the entries in Table 6 below.
For the 1MW option we can allow overlap. We see from fig. 17 that an acceptable ~87 0C degree increment is
reached for the "every 15 cm" case. Accordingly the velocity entered in Table 6 for this case is 15 cm / 20 ms = 7.5
Next we follow the same approach for Vascomax C-350. In this case we are less constrained in the allowable
temperature increments since we don't rely on an abnormally low linear expansion coefficient of Super Invar which
only holds over a very limited temperature range (see Fig. 11). Still we can't allow the material to become too hot
because of the degradation with temperature of the mechanical strength. In Fig. 18 we show how the tensile strength
varies with temperature. We presume that the fatigue endurance limit will vary in a similar fashion.
Vascomax-350 Yield Strength as
Function of Temperature
0 100200 300 400500600
0.2% Yield Strength (MPa)
For this discussion we arbitrarily assume that no more than 20% of the room-temperature value of the tensile stress
must be lost and therefore, according to the data shown in Fig. 18, a maximum of 300 0C temperature increment is
The results of the maximum temperature increase calculations for Vascomax C-350 are shown in Fig. 19
Temperature increase for multiple pulses on a
1.5 cm diameter Vascomax-350 Target
0 1020 3040
∆ ∆T for the 1MW option (°C)
∆ ∆T for the 4MW option (°C)
every 0 .0 cm (1 M W o nly)
every 1 .7 cm (1 M W o nly)
every 1 0 cm
every 2 0 cm
every 3 0 cm
every 4 0 cm
Here the uppermost two curves correspond to the 1 MW option only, taking advantage of the intermittent sequence
of six pulses during 100 ms followed by a 300 ms pause. For the curve labeled "every 0.0 cm" the target is
stationary for the 6-pulse sequence (see Fig. 16), and then advances by one target length (40 cm) during the 300 ms
interval. This mode would be particularly advantageous for a target made of 40 cm long links such as the one
sketched in Fig. 15. The curve labeled "every 1.7 cm" moves continuously just fast enough (1.7 cm/ 20 ms = 0.85
m/s) to get a fresh portion of the target into position at the beginning of the next 6-pulse burst. We see that in both
cases we stay within the ~300 0C limit established above. For the 4 MW option we can go up to one pulse every 10
cm (10cm / 33.3ms = 3 m/s) to stay below the 300 0C increment.
We now summarize these examples in Table 6:
Table 6. Examples of target velocities and lengths according to assumptions explained in the text.
Super Invar - 1 MW Option 7.5 175
Vascomax C-350 - 1 MW Option 0.85 16.8
Super Invar - 4 MW Option 12.0 274
Vascomax C-350 - 4 MW Option 3.0 34
The minimum target lengths were estimated from the velocities by assuming the chain or cable travels through the
cooling bath for the characteristic 1/e cooling time (Table 1) which is 22 s for Invar and 8 s for Vascomax C-350. It
was further assumed that there is a minimum 10 m of target outside the bath. The need for the much longer Invar
target chains arises from the higher velocities required to keep the temperature in a narrow interval, combined with
the lower heat conductivity requiring longer cooling. These length estimates can vary rather considerably with
better heat transport calculations based on actual chain geometries. Also the shorter Vascomax C-350 lengths may
not be adequate if one wants to extend the life of the chains by spreading radiation damage over large total volumes.
Eddy Currents and Magnetic Forces
Eddy currents will be induced in conductors moving through magnetic field gradients. These currents will cause
some heating and will also give rise to forces opposing the motion. Additional large forces will be present when a
ferromagnetic material such as Vascomax C-350 enters and exits a 20T solenoidal magnetic field. We will estimate
these effects for the worst case example listed in Table 6, namely a 1.5 cm diameter cylindrical Vascomax C-350
target moving at 12 m/s. We will limit this discussion to on-axis motion to obtain orders of magnitude of the forces
involved, recognizing that more complex situations, including forces perpendicular to the target, will occur for
inclined targets and for portions of the target moving over pulleys.
For the present calculations we use the axial magnetic field profile depicted in Fig. 20 calculated for a 20T
solenoidal field generated by an array of superconducting coils, followed by a region where the field, shaped by
additional coils, gradually tapers off. For the present example we (somewhat arbitrarily) consider a ~5m long target
cylinder starting 1 m before the solenoid exit plane.
Axial Magnetic Field
-1000 100200300 400
Magnetic Field (T)
Fig. 20 Axial magnetic field calculated for the 20T solenoid arrangement described in the Feasibility Study1).
To calculate de Eddy currents and the forces they generate we use equations developed for these quantities during a
previous study17) of the behavior of a mercury jet entering a magnetic field:
j = -σr/2c Vz δBz/δz
where j is the current density at a radius r, σ is the conductivity, c the velocity of light and Vz the velocity of the
fz = -σr2/4c2 Vz (δBz/δz)2
where fz is the axial force density due to the interaction of the Eddy current with the field.
We evaluate these quantities as well as the derived power density σj2 and temperature increase using the values for
Vascomax C-350 in Gaussian units shown in Table 7:
Table 7. Values in Gaussian units used in the
calculations for a Vascomax C-350 target
Electrical conductivity σ 1.023E+16 1/s
Specific Heat 4.50E+6 erg/(gK)
Saturated Magnetization 1.86E+4 Oersteds
Target velocity Vz 1200 cm/s
Light velocity c 3E+10 cm/s
The results are shown in figs. 21, 22 and 23
Breaking Forces due to Eddy Currents
-1000 100200 300 400
Force per unit length
Fig. 21 Breaking forces per unit target length due to Eddy currents in a 1.5 cm diameter Vascomax C-target
moving axially at 12 m/s through the field shown in fig. 20.
We can see that the forces due to eddy currents are small. The total force obtained by integrating the values shown
in fig. 21 is 1.99 Newtons or 0.2 kg force. The total power dissipation due to these currents is 6 W and the
maximum temperature increase is only ~ 6 × 10-3 oC.
In contrast, the magnetic forces on Vascomax C-350 which is a ferromagnetic material are rather large:
Magnetic Forces on Vascomax
-1000 100 200300 400
Force per unit length
Fig. 22 Magnetic forces on a 1.5 cm diameter Vascomax C-350 target placed in the field shown in fig. 20.
We can see that, as expected, these forces tend to pull the material into the high field region of the solenoid. For
this 1.5 cm diameter Vascomax C-350 target, the force per unit length reaches a maximum of 12.5 Newton/cm or
~1.3 kg force per cm. To maintain the chain or cable under tension everywhere we must apply external tension. An
example of this is shown in fig.23 where an external ~3.9 MPa tension was applied to maintain a minimum of 1
MPa at the maximum field point. The actual value of the required external tension will depend on the desired
trajectory of the target and on the maximum allowable deviation from a straight line. In the case of an inclined
target such deviations will be due to transverse components of the magnetic forces and to a smaller extent to gravity.
Chain Tension to maintain a minimum of 1 MPa
-1000 100 200 300400
Fig. 23 Tension as a function of position for a 1.5 cm diameter Vascomax C-350 target placed in the field
shown in fig.20. In this example, an external tension is applied to achieve a minimum tension of 1 MPa.
Tensions of ~4 MPa such as shown in fig. 23 are negligible when compared to the yield strength (~2200 MPa), but
they may complicate the engineering and required lubrication of the bearings joining the links of such a chain. The
force corresponding to this tension is ~707 Newton or ~71 kg force. This is one more reason, in the case Vascomax
C-350 is used, why a cable15) may be preferable to a chain.
After the strong vibrations following each beam pulse subside there will still be a non-uniform temperature
distribution present in the target, causing residual stresses which will later disappear on a longer time scale (see Fig.
1). In the cases where multiple successive pulses impinge on the same portion of the chain or cable, pulses
subsequent to the first one will encounter already stressed material, and the transient vibrational stresses will be
superimposed on residual stresses. The worst such case contemplated above was the one depicted as the dotted
curve in fig. 19 for Vascomax C-350. Here six pulses spaced by 20ms (see fig. 16) hit a 7.5 mm radius stationary
target leading to a maximum on-axis temperature increment of ~300 0C. Taking into account the radial dependence
of the energy deposition (fig. 2) at Z= 9 cm, and the specific heat of the material, we calculate de temperature
distribution after 5 pulses, and from these values we obtain the residual stresses that will be encountered at the
arrival of the 6th pulse. The results of this calculation are shown in Fig. 24
Worst-case residual stresses for a Gaussian beam
The same approach was followed as we previously used for carbon rods 4) using thermal stress equations 18) for
cylindrical geometry with arbitrary radial temperature profiles. In this figure compression is indicated by positive
stress values and tension by negative values. We see that even in this worst case, the residual stresses are not very
significant when compared to the yield stress or to the fatigue endurance limit.
For the "flat" beam case (Fig. 12b) we get similar residual tangential stress values as can be seen from fig. 21 below:
Worst-case residual stresses for a "flat" beam
Here we assumed that the same total beam energy density at z=9 cm is uniformly distributed within a 7 mm radius
circle leaving a 0.5 mm wide un-irradiated annular sheet. Since the residual stresses aren't any worse, one could
take full advantage of the reduced transient stresses discussed before.
Questions about Invar
The small linear expansion coefficients of these materials seems to continue puzzling the specialists, even though
progress has been made over the years. One question that came up regarding our possible application is the time
response of the material given the short pulse lengths involved. In other words does the small expansion result from
a larger expansion followed by a later contraction or vice versa. Or are such processes (if they exist) much faster
than our time scale. After consulting a specialist in this field19) we obtained the following answer:
"The Invar effect is a strong magneto-volume coupling effect, and therefore, the time scales in which the effect
actuates correspond to lattice fluctuation times in the order of 10(-13 s). 10 ns is too long compared to this time
scale, so that the overall thermal expansion would still be valid."
This answer suggested the next question. Namely, if the Invar effect is a "magneto-volume coupling effect", will the
response be modified by the presence of a strong external magnetic field such as we will have in our system.
Fortunately that question had been answered experimentally by H. Zähres. M.Acet, W. Stamm and E.F.
Wassermann20) for fields up to 6 Tesla . We reproduce part of their results in fig. 26:
Fig. 26 Invar linear expansion coefficient as a function of temperature and magnetic field 19).
We see that there is a pronounced magnetic effect on the linear expansion coefficient at cryogenic temperatures up
to about 30 K, but that effect disappears at higher temperatures. These experiments were done with Invar. We
presume that Super Invar will show a similar behavior. In Fig. 27 we compare these data (left side of the figure)
with an approximate representation of the known expansion coefficient values around room temperature, both for
Invar and for Super Invar.
Fig. 27 Low temperature data20) from Fig. 26 combined with known expansion coefficients at higher
We conclude that neither time response limitations nor magnetic effects would affect the usefulness of super invar
for our application..
Discussion and Conclusions
Special solid metallic targets with very large tensile strength or with very low coefficients of expansion are viable
candidates for the 1 MW option, and very probably also for the 4 MW option. It was also shown that more
conventional solids such as iron can not be expected to work at all. This may help to explain the rupture of some
metallic targets at CERN21) which probably led to the prevailing pessimism regarding solid targets for this
The present stress and temperature estimates are very conservative since no credit was taken for the beneficial
effects of inclined beam incidence on the target, nor for the advantageous frequency dependence of fatigue
tolerance, nor for the possibility of using non-Gaussian beam profiles. As was discussed before4) special forms of
graphite my also work. It seems therefore that the complications and uncertainties associated with mercury jets3)
can be avoided.
The advantage of the graphite option is that a stationary target is simple and reliable since moving parts are avoided.
Disadvantages are the 50% yield loss compared to heavy targets, the fact that a cooled jacket will line the chamber
taking up some of the volume otherwise available to the spiraling particles, and that the highest radiation dose will
be constantly delivered to the same volume element of the target probably leading to premature failure due to
The advantages of the metallic targets discussed here are that most of the yield available for mercury targets will be
realized14), that the radiation damage will be much more spread out and that target cooling can take place outside the
reaction chamber, far away from the very high radiation environment. The disadvantage is the complication and
potential reliability problems with moving mechanical parts.
The thermal shock advantage of Super Invar over Vascomax C-350 can only be realized with thorough cooling
between beam exposures. This together with the lower heat conductivity of Super Invar leads to much longer total
target chains. If Vascomax C-350 survives the repeated thermal shock (and we have seen that there is an excellent
chance that it will) it may become the preferred option. Inconel 718, while similar to Vascomax C-350 in its fatigue
endurance, shares the lower heat conductivity of Super Invar. For Vascomax C-350 the chain length may have to be
increased beyond minimum values required for cooling in order to spread radiation damage over a larger volume to
extend the lifetime of the chain. The Vascomax C-350 chain may thus become long enough to match the minimum
length requires for cooling a Inconel 718 chain of cable. In this case Vascomax C-350 and Inconel 718 become
equivalent assuming their radiation-damage rates are similar. Inconel 718 may then be preferable to avoid possible
problems associated with large magnetic forces. The main comparative advantages and disadvantages of the three
solid metallic targets are summarized in Table 8:
Table 8. Comparison of advantages and disadvantages the three alloys considered in this paper.
MATERIAL ADVANTAGES DISADVANTAGES
Largest margin for thermal shock tolerance.
Absence of large magnetic forces.
Narrow temperature range and low heat
conductivity leading to the need for long chains
The largest uncertainty regarding deleterious
effects of radiation damage.
Largest tensile strength.
Good heat conductivity allowing the use of
relatively short chains or cables.
The material is ferromagnetic and will be
subject to large magnetic forces
Inconel 718 Good fatigue endurance limit similar to
Absence of large magnetic forces.
Poor heat conductivity (similar to Super Invar)
requiring long chains of cables, but not as long
as for Super Invar since the temperature range
is not as small.
These three alloys appear to be viable candidates for targets in a future Muon Collider or Neutrino Factory.
Radiation damage studies10) as well as further experiments, more detailed simulations, and engineering
considerations will help in the selection of one of these materials.
We derive here the expressions used in equations 3, 4 and 5 to calculate Pmax, the maximum pressure following
beam heating , Eσ , the energy densities required to reach the yield stress or the fatigue endurance limit, and Em the
energy density available in the initially compressed volume element, respectively.
Calling εmax the maximum total energy density (per unit mass) deposited by the beam in a volume element V0 of
the target of density ρ , and cv the specific heat at constant volume of the target material, the temperature increase
∆Tmax of that volume element will be:
∆Tmax = εmax / cv
If we allowed V0 to expand freely it would occupy a volume V1 = V0 + ∆V where
∆V = 3 α × ∆Tmax × V0 (7)
α being the linear expansion coefficient.
Before V1 expands, the compression is therefore ∆V/ V0 and the pressure is:
Pmax = B × ∆V/ V0 (8)
where B is the bulk modulus. From 6), 7) and 8, we get the equation we used before for the maximum pressure:
Pmax =3 × εmax × Β × α / cv 3)
We now turn to the mechanical energy density, Eσ , contained in a volume element stressed to a stress σ. For that
purpose we imagine a rod of initial length x0 and cross-section s, stretched to a length x by applying a force F(x) to
F(x) = Y × s × (x - x0) / x0 9)
Where Y is the Young modulus. The work W done to reach x starting form x0 will be:
W = ⌡Xo F(x) dx = 1/2 Y × s × (x - x0)2 / x0 10)
If (x - x0)/x0 is the strain corresponding to the stress σ then:
(x - x0)/x0 = σ / Y
and the work Wσ to reach this strain is from (10) and (11):
Wσ = 1/2 × Y × s × (x0 × σ / Y)2 / x0
But x0 × s is the volume. Therefore the energy per unit mass is:
Wσ / ( x0 × s × ρ ) = Eσ = σ2 / (2 Y ρ)
Which is the expression for the energy per unit mass used before for σ = σ0.2 (the yield stress) or σ = σ-l (the
fatigue endurance limit), to compare with the maximum mechanical energy density available.
We finally will derive the expression used to calculate that maximum mechanical energy density Em available in the
initially inertially confined compressed volume. This mechanical energy will be only a fraction of the total
maximum energy εmax deposited by the beam, most of which goes directly into heating the material.
As the first step we calculate the temperature decrease ∆T when we allow a volume element of initial density
ρ0, temperature T0 and pressure P0 to expand adiabatically. We do this in two imaginary steps:
First we remove some heat ∆Q1 at constant volume obtaining a temperature T1 and pressure P1
T1 = T0 - ∆T1
∆T1 = ∆Q1/cv
P1 = P0 - ∆P1
Now we add some heat ∆Q2 at constant pressure:
∆T2 = ∆Q2/cp T2 = T0 - ∆T1 + ∆T2
The volume increases by:
∆v = 3 α × v × ∆T2 14)
Where v is the specific volume (1/ρ) and α the linear expansion coefficient
Setting ∆Q1 = ∆Q2 to get an overall adiabatic expansion we get:
cv ∆T1 = cp ∆T2
∆T1/∆T2 = cp/cv
(∆T1 - ∆T2)/ ∆T2 = (cp - cv)/cv
∆T2 = (∆T1 - ∆T2) cv / (cp - cv) = 1/(3 α) ∆v/v from 14) above we get
∆T = ∆T2 - ∆T1 = - (cp - cv)/cv 1/(3 α) ∆v/v or dT = - (cp - cv)/cv 1/(3 α) dv/v
for an infinitesimal adiabatic expansion.
Fig. 28 Diagram used to calculate the mechanical energy released during an adiabatic expansion following
inertially confined heating of the target (see text).
We now use the diagram of fig. 28 to plan the calculation of the mechanical energy released during the adiabatic
expansion (B → C) following the inertially confined heating of the target by the beam (A → B). The final point C
could in principle also be reached by heating at constant pressure (A → C). We will calculate the total energies per
unit mass EAB and EAC required to go from A to B and from A to C respectively. The difference EAB - EAC will be
the energy released from B to C. Since this is an adiabatic transition, there is no heat exchange and this energy EBC
is therefore entirely mechanical.
EBC = ⌠Tb cv dt - ⌠Tc cp dt = ⌠Tc (cv - cp ) dt - ⌠Tb cv dt 16)
⌡Ta ⌡Ta ⌡Ta ⌡Tc
We know that 17) :
cp - cv = v T B α2 17)
and therefore the first term becomes:
v B α2 (Tc
To calculate the second term we use (15) and the fact that:
∆v/v = (vc - va ) / va = 3 α (Tc -Ta )
TB - TC = - (cp - cv)/cv 1/(3 α) 3 α (Tc -Ta )
Since TB -TC << TB -TA we can use 17) to substitute the value cp - cv at TC:
TB - TC = - (v Tc B α2 )/cv (Tc -Ta )
Assuming cv ≈ constant over the small temperature interval TB - TC we get the second term of 16):
v TC B α2 (Tc -Ta ) 22)
Combining 18 and 22:
EBC = v B α2 (Tc
Approximately Tc ≈ Ta + EAB /cp and we get:
EBC ≈ EAB v B α2 ( 2 Ta + 3/2 EAB/cp ) /cp
Which is equivalent to (5) used in the text.
We would like to acknowledge stimulating discussions with R.J. Weggel, N. Simos R. B. Palmer. And K.
McDonald. R.J. Weggel suggested the investigation of Vascomax C-350 as one of the strongest materials available.
2 - Ta
2 ) / 2 18)
/2 - Ta
/2 + Tc
- Tc Ta ) 23)
31 Download full-text
1. N. Holtkamp and D. Finley: editors, A Feasibility Study of a Neutrino Factory Based on a Muon Storage
2. AGS experiment E951
3. P. Thieberger, MUC note# 212: http://www-mucool.fnal.gov/mcnotes/muc0212.pdf
4. P.A. Thieberger and H.G. Kirk, MUC note (to be published)
5. Simos windows modeling.
6. Cylindrical targets are assumed here for simplicity even though the final design may be different if stress-
focusing effects are significant.
7. Handbook of Physical Quantities, I.S. Grigoriev and E.Z. Meilikhov, Editors, CRC Press, Inc. 1997, page
8. N.V. Mokhov and A. Van Ginneken, Pion Production and Targetry at mu+mu- Colliders, Fermilab-Conf-
98/041 (Jan. 1998).
9. N.V. Kadobnova and A.M. Bratkovskii, Handbook of Physical Quantities, I.S. Grigoriev and E.Z.
Meilikhov, Editors, CRC Press, Inc. 1997, page 83.
10. Radiation Damage experiments to be performed with 200 MeV protons.
11. N. Tsoupas in Booster Applicatins Facility Report,Phase II - BNL-52291
12. N. Simos and R.J. Weggel, private communication
13. B.J. King, N.V. Mokhov, N. Simos and R.V. Weggel, April 9,2001,
14. N.V.Mokhov, Private communication to N. Simos
15. R.B. Palmer, private communication.
16. Pelletron chain charging system. National Electrostatics Corporation, Middleton, Wisconsin.
17. P.Thieberger, MUC note #0182: http://www-mucool.fnal.gov/mcnotes/muc0182.pdf
18. Elements of Thermal Stress Analysis, David Burgreen, C.P. Press 1971, pages 246 and 251.
19. M.Acet, Tieftemperaturphysik, Universität Duisburg, Germany, private communication.
20. H. Zähres. M.Acet, W. Stamm and E.F. Wassermann, Journal of Magnetic Materials 72 (1988) 80-84
21. H. Haseroth, private communication.