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* E and Ot max are given in Table 1.10; ^ in Table 8.1. Mohs' hardness scale is maximum at 10 for diamond.

* E and Ot max are given in Table 1.10; ^ in Table 8.1. Mohs' hardness scale is maximum at 10 for diamond.

this may entail a lung desease similar to silicosis. For a surface polish with low micro-roughness, beryllium is electroless nickel plated by a Kanigenprocess, a ther-mochemical process which uses metallic electrochemical potentials with a liquid catalyst; this avoids the difficulty of edge-sharpening encountered in the electrolytic process. For field stabilization at 15 Hz and thermal infrared studies, the ribbed and nickel-plated substrates of the four Cassegrain mirrors of the 8-m Vlt units are beryllium blanks of 1.1 m diameter from Brush-Wellman in USA; more details are given by Stanghellini et al. [84]. The 18 segments of the primary mirror of the Jwst project are being built in beryllium with this technology.

• Stainless steel: Stainless steel is an extremely low cost material compared to the above substrates. Its elastic deformability aTmax/E is the highest of mirror materials (cf. Sect. 8.4.2 and Table 8.2). Its thermal diffusivity makes it weakly sensitive to thermal shocks. This latter advantage was pointed out by Couder [17], and also Mak-sutov [96] whose best and final work is generally regarded to be the Pulkovo 70-cm aperture double-meniscus telescope with a 80-cm diameter stainless steel mirror. The best alloy for a linear stress-strain relationship - Hooke's law - is the marten-sitic steel Fe87Cr13 (in % weight, also known as AISI420) with less than 0.15% carbon. Current applications are turbine blades and springs. Recent developments of variable curvature mirrors, operating since 2006 at the focal plane of the delay lines of the Vlt Interferometer (zoom range [f/^-f/2.5], aspect ratio t/d = 1/60), show that theFe87Cr13 alloy is a stable material. As for Ti90 Al4 V6 alloy, the casting of Fe87Cr13 delivers a fine structural material that can be smoothly polished to high reflectivity without the need of a Kanigen process. Its large elastic deformability ratio (cf. Sect. 2.6.1) would allow the complete aspherization of large mirrors by active optics. For instance, Lemaitre, Wilson et al. [76] proposed an f/1.75 stainless steel primary mirror of diameter 1.8 m and thickness 30 mm (aspect-ratio t/d = 1 /60) for the Vlt auxiliary interferometric telescopes to be at once actively aspherized and actively supported. The cost of the casting and machining of the Fe87 Cr13 meniscus blank - and also in a vase-shaped option - from Ferry-Capitain in France was an order of magnitude lower than that of a Zerodur blank.

• Liquid materials: In contrast to solid materials, it has been known for centuries that liquids at rest naturally provide diffraction limited plane surfaces (in fact of radius that of the Earth). Of course, such surfaces are totally free from ripple surface errors since no figuring is needed. High density liquids, such as glycerine or mercury, are of classical use in the laboratory as high-quality reference surfaces -currently at X/100 - for optical testing such as large Fabry-Perot plates by Marioge [47]; furthermore these are low-cost references. At the surface boundary, the capillarity is an extremely local effect which only requires increasing the liquid surface area by a few percent. In 1909, Wood [95] introduced and built a 50-cm diameter liquid telescope, and performed astronomical observations with it. The mercury mirror was rotated and stabilized to a constant angular velocity for which the hydrody-namic equilibrium equation under gravity shows that the mirror surface is a perfect parabola. The main difficulty is to obtain a high accuracy in the constancy of the angular velocity because ripples arise from accelerations. Solving this requires the use of an extremely low-friction rotating pad and minimum quantity of mercury. Pneumatic pads rotated by belts of ancient magnetic tapes more recently allow zenith observations with 2.5 and 2.7-m liquid mirror telescopes. The last development of liquid mirrors is the 6 m Telescope of the University of Vancouver (cf. Sect. 7.7).

### 8.4 Stiffness and Elastic Deformability Criteria

The duality between (1) low flexural deformation of a mirror to gravity or to wind buffeting and (2) high elastic deformability for stress aspherization of a mirror, though partly incompatible, must be interpreted as two different active optics purposes. In the first one various stiffness criteria of a mirror substrate may be determined from the quantities (E, ¡), the Young modulus and the density. In the second one, because of the ability of a mirror to be actively bent - and possibly completely aspherized - by external forces, an elastic deformability criterion may be determined from the quantities (oTmax, E), the tensile maximum stress and the Young modulus.

### 8.4.1 Mirror Materials and Stiffness Criteria

In order to determine merit factors relative to the stiffness - "raideur" in French -, let us consider a circular plate of diameter d and thickness t in a material of density I and Young modulus E. First, one may determine two merit factor types:

- a merit factor for equal volume plates CSV), where all dimensions are identical,

- a merit factor for equal mass plates (SM), where only diameters are the same.

### The volume and mass of these plates are

Whatever the equilibrium configuration of these plates and provided those configurations are identical for plates in various materials, it is well known from the thin plate theory that the flexure zF due to an external force and the flexure zow due to the own weight under gravity (cf. Sect. 1.13.12) are of the form where dimensionless flexures ZF(p, v) and Zow(p, v) are only dependent on the reduced radial variable p = 2r/d and Poisson's ratio v of the material. Generally Pois-son's ratio contribution in the amplitude of the flexure is quite small so we shall neglect the effect of this dimensionless quantity in a first approximation. Hence, with this restriction, the flexure functions ZF and Zow are not dependent on the material.

• Merit factor for equal volume stiffness: Introducing the volume of the plate into (8.3), we obtain after substitution

1 FdV2 V d5

where the first right-hand fraction is only material dependent. The reciprocal fraction allows defining merit-factors for equal volume stiffness as

where the first one, E, is a stiffness classically introduced in the definition of the rigidity since determining the slope of Hooke's law.

• Merit factor for equal mass stiffness: Introducing the mass of the plate into (8.3), we obtain after substitution v3/2 Fd1/2 V2 d5

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