Technical Considerations

Apart from the interaction of the primary and secondary particles with the monitor, several important design issues should also be addressed in this study. These include foil heat load, thickness choice and deformation due to electrostatic forces.

Foil Heating

The beam stopped in the foil exchanges its kinetic energy with electrons of the medium causing heating of the target. As discussed in Section 5.2.5, the only possible process of cooling down of electrically and thermally isolated components in vacuum environment is thermal radiation. In equilibrium, the absorbed heat is equal to the emitted heat and the approximate temperature can be expressed by the balance condition of black body radiation [264]:

Qabsorbed =Qemitted=A·σ·β·(T4−T04), (6.2)

whereAis the heated area of the foil,σis Stefan-Boltzmann constant,βis the emissivity

coefficient, andT0 is the ambient radiation temperature. The emissivity coefficient of

The absorbed heat can be calculated as the energy of a single particle times the number of extracted particles per second. From equation 6.2, the foil temperature rise can be estimated to less than 0.2 degree for a 1 mm diameter beam incident at the foil surface

at 45◦ and is negligible for larger beams.

Foil Thickness

The influence of annihilation could be minimised by the use of an ultra-thin foil with a thickness smaller than the range of projectiles in matter. As discussed in Section 6.3.1,

a sub-µm thick film is required in order to reduce the number of particles stopped in

the monitor. The foil needs to be at least 50 mm in diameter to ensure an area large enough for imaging 20 mm diameter beams and mechanical support at the edges.

A solution developed at CERN was used for the SLIM beam monitor [119]. The thin foil produced for the SLIM detector consists of a support of 100–200 nm thick

aluminium oxide (Al2O3), coated on each side with 10–50 nm thick layer of aluminium,

and has a diameter of 65–70 mm. The pure aluminium evaporation process leads to discharging to electrical ground any electrostatic build-up within the oxide [119].

The use of a 30 µg/cm2 (about 150 nm) thick carbon foil coated by a ∼10 µg/cm2

lithium fluoride (LiF) layer was reported [266]. The latter was added to improve the secondary electron emission of pure carbon, yet no quantitative results were provided.

At Grand Acc´el´erateur National d’Ions Lourds (GANIL), 0.5–0.9 µm mylar foils

with ∼100 nm Al evaporated on one face for electrical conductivity were used [267].

The number of secondary electrons is enhanced by a factor of about 5 by a ∼50 nm

thin CsI coating on the Al.

Foil Deflection due to Electrostatic Forces

The foil and mesh assembly can be simplified and seen as two parallel plates at a

distance dand a constant voltage. The pressure between them due to an electrostatic

potential differenceU can therefore be written as:

p= 0

2 ·

U2

d2. (6.3)

According to [268], the deflection of a solid circular plate of radiusa, clamped atr=a,

due to a uniform pressure p can be calculated as:

w(r) = p· a

2_−r22

6.3 Design Considerations

whereD is the bending stiffness of a plate and is defined as:

D= E·h

3

12·(1−ν2₎, (6.5)

with the modulus of elasticity E, also known as Young’s modulus, Poisson’s ratio

ν and plate thickness h [268]. Young’s modulus is a measure of the stiffness of an

elastic material, whereas Poisson’s ratio is the ratio of transverse contraction strain to longitudinal extension strain in the direction of the applied force. Table 6.1 lists values of both parameters for selected materials [265]. It is not possible to provide a single value of Young’s modulus for carbon, because it depends on a particular structure and composition under investigation [269]. It is even more complicated for graphene for which the modulus of elasticity depends also on the definition of the material thickness [270–274]. Currently, graphene films of 50 mm diameter are not available, but it is not

unlikely that ongoing studies will lead to an extremely stiff solution with E >1 TPa

and thickness of a single or only few atomic layers [275].

Material Modulus of elasticity [GPa] Poisson’s ratio

Aluminium 69 0.33 Aluminium oxide 300 – 400 0.21 Beryllium 290 0.03 Carbon <100 –>500 ∼0.3 Copper 117 0.36 Gold 74 0.42 Nickel 214 0.31 Silver 72 0.37 Stainless steel 180–200 0.30

Table 6.1: Moduli of elasticity and Poisson’s ratios for selected materials. See text for details.

It should be noted that equation 6.4 is only a first-order approximation of a real deformation, because the pressure, as a function of distance, changes with the bending of the plate. Furthermore, this linear approximation is valid only for small deflections, typically less than one-half of the plate thickness. For larger deformations, stretching and bending of the plate are coupled and in-plane stress resultants are not constant but location-dependent. In such a case, the analysis should be carried out numerically [268]. However, the linear approximation can give an idea of the significance of the foil deflection.

Carbon and Al2O3 films offer the highest stiffness and can be considered the most

to mesh d= 5 mm, equation 6.4 leads to at least 10 mm deformation for a 3µm thick foil and voltage of –5 kV. In reality, it may mean that the foil breaks.

A way around the problem is to place the foil symmetrically between two grids so that the electrostatic forces are present in both directions and cancel out. However, such a solution would be sensitive to inhomogeneities and any asymmetry would expose the foil to considerable forces. Also, the grids would still be bent towards the foil, thus their thickness cannot be too small. From this point of view, it is therefore preferred to use a thick foil or even a plate.