Comprehensive review of the effects of ionizing radiations on the ZERODUR® glass ceramic

Antoine Carré; Rule Kirchhoff; Tony Hull; Janina Krieg; Daniela Eppers; Marty Valente; Thomas Westerhoff

doi:10.1117/1.JATIS.9.2.024005

3 May 2023 Comprehensive review of the effects of ionizing radiations on the ZERODUR^® glass ceramic

Antoine Carré, Rule Kirchhoff, Tony Hull, Janina Krieg, Daniela Eppers, Marty Valente, Thomas Westerhoff

Journal of Astronomical Telescopes, Instruments, and Systems, Vol. 9, Issue 2, 024005 (May 2023). https://doi.org/10.1117/1.JATIS.9.2.024005

Abstract

The low thermal expansion glass ceramic ZERODUR^® has a long and successful history in spaceborne optical applications. This material is especially used where precise shape invariance is required, i.e., for the mirror substrates dimensional stability when subject to temperature gradients and transients. In space, temperature may not be the exclusive driving force impacting the form stability; influence from ionizing radiations require considerations. The real impact of ionizing radiation on ZERODUR^® has become a matter of reconciling, on the one hand, in situ experience, e.g., that the secondary mirror of low Earth orbit (LEO) Hubble Space Telescope crossing the South Atlantic Anomaly or the overall optics of geosynchronous equatorial orbit (GEO) Chandra X-ray Observatory are not reporting any specific problems related to dimensional stability at the optical form level. On the other hand, finite element simulation based on early lab experiments of ZERODUR^® compaction are suggesting the opposite. This debate was brought to the forefront with the SILEX mission, where radiative ageing models were significantly overestimating the deformation experimentally observed on the lab replicas and were in even stronger disagreement with the observations collected over the mission. It has been speculated that an erroneous form factor in the physical model used to derive the phenomenological compaction law was responsible for these discrepancies. Following this hypothesis, we readdressed the effect of ionizing radiation induced by γ, electron, and proton fluences on ZERODUR^® compaction. For each of these, we present and discuss the irradiation source, the experimental setup, the sample design, and the measurement procedure as well as the observations. Consistent with the feedback gathered over many different space missions, we confirm that the compaction observed is significantly smaller than the estimations available in the prior literature.

1. Introduction

For more than 50 years, the glass-ceramic ZERODUR^®¹ has played an important role in design of stable mirror substrates and structures dedicated to astronomical applications.²^,³ For much of this period, ZERODUR^® has also been successfully used in space. ZERODUR^® is a lithium-aluminum-silica glass-ceramic, initially cast then annealed while remaining in an amorphous state. Next, a precise ceramization process enables the homogeneous distributed crystallization of a solid solution of eucryptite. Locally, this glass matrix has a net positive coefficient of thermal expansion (CTE), while the solid solution of eucryptite has a negative one.⁴ Macroscopically the interplay between these two coexisting phases results in material having a net macroscopic CTE close to $0 ppm . K^{- 1}$ over temperature range that may be centered on $T = 20 ° C$ and spanning tens of degrees.¹ This supports form stability under thermal perturbations, but furthermore provides the critical features of high homogeneity, high isotropy and availability in large monolithic lightweigthed forms made of ZERODUR^® the material of choice for sensitive telescope mirrors, typically in orbiting missions.⁵^,⁶ For these reasons ZERODUR^® continues to be a widely employed material in space where high stability is needed. It is a critical part of more than 40 orbiting missions²^,³ among these two NASA great observatories, Hubble-M2 and all Wolter mirrors on Chandra.²^,³

In spite of these successful applications, we note that the form stability of this material is not exclusively determined by its thermomechanical properties. Ionization due to space radiation should also be evaluated for possible impact on the optical figure.

A comprehensive understanding of the resilience of ZERODUR^® under ionizing radiations is relevant to properly estimate the optical systems errors budget over the duration of the space mission. Mirror substrate materials can respond to local changes in their CTE induced by radiation. A CTE gradient, coupled with the temperature gradient, has the potential to drive local deformation of the optical surface depending of course on the susceptibility of the material properties to ionization.⁷ Another effect of radiation induced change, namely compaction, may alter the surface form; here the deformations are not caused indirectly by local material shrinkage/dilatation under thermal load but are directly caused by the irradiation itself. The material shrinks under irradiation, i.e., its density ( $ρ$ ) increases. The compaction effect $Δ ρ / ρ$ can be quantitively defined as follows:

Eq. (1)

\frac{Δ ρ (D)}{ρ} = - \frac{ρ (D) - ρ_{0}}{ρ_{0}},

with

ρ_{0} = ρ (D = 0)

being the density of the material prior irradiation and

ρ = ρ (D)

the density of the material after any given deposited total ionizing dose (TID)

D

. This effect has been extensively investigated for fused silica (see R.E. Schenker and W.G. Oldham⁸ who gathered the different studies on this material⁹^–¹³).

Quantifying the impact of ionizing radiation upon a specific spaceborne mission require extensive a priori information. Factors include mission duration, orbit, solar activity, as well as optical and telescope design (due to shielding) and orientation. All these parameters could drive to very different irradiation scenarios that obviously cannot be comprehensively addressed in a general discussion. A more general approach has been proposed by development groups¹⁴^–¹⁸ where a phenomenological law binds the deposited ionizing dose to the induced variation of material properties of interest. Coupled with finite element methods (FEMs) analysis these phenomenological laws offer much more flexibility in that the need for experimental testing on real size optics can then be to some extent circumvented via interpretation. One of the first phenomenological studies of ionizing radiations on ZERODUR^® was initiated 40 years ago by Bourrieau and Romero.¹⁵ This pioneering work proposed to estimate the indirect effect of ionizing radiation on the compaction by measuring the change induced in the vertex radius of curvature (VRoC) of flat discs. In this work, the compaction $Δ ρ / ρ (D)$ was phenomenologically described by a power law of type: $Δ ρ / ρ (D) = A \times \sqrt{D}$ . This contribution is laudable, but the lack of computational power in that time constrained the authors to coarsely approximate the irradiated ZERODUR^® discs into a bimetallic system. The continuous gradient of deposited ionizing dose through the thickness of the coupon was approximated by a heavyside function (whose step position corresponds to the relaxed solarized/unsolarized interface within the glass ceramic). In this model, the radial distribution was neglected and the coupons were assumed to be thin plates (radius $> 10 \times$ thickness). The authors derived the compaction law using the Timoschenko bimetallic approach¹⁹ on this ideal 1D approximation. However, this compaction law has been challenged some years after by Higby et al.,¹⁶ who used a more conventional but less sensitive approach where coupons of ZERODUR^® have been irradiated and the shrinkage has been estimated by densimetry. The compaction law proposed by this research group was in disagreement with the one proposed by Bourrieau and Romero.¹⁵ A third contribution weakened further the phenomenological compaction law proposed by Bourrieau and Romero.¹⁵ During the preparatory phase of the ESA’s Artemis mission, a consortium made of the ESA/ZEISS/ONERA/SCHOTT was able to cross-validate neither the experimental results observed on the SILEX mirror replica nor the numerical multilayer modeling based on the Bourrieau compaction law.¹⁵ The computed defocus contributions attributed to compaction were found to be significantly higher than the experimental ones. These inconsistencies drove Davis and Fainberg¹⁴ to reassess the compaction law initially proposed by Bourrieau.¹⁵ In their methodological contribution, these authors suggest that a form factor in the bimetallic equation could be erroneous ( $5 \times$ to $10 \times$ ) and cause a systematic overestimation of the compaction. Interestingly, all the experimental efforts made in order to derive an empirical compaction law²⁰ were made at the beginning of the 1980s during the US/Soviet Union missile race; thus, these research efforts were mostly addressing potential consequences of weapons effects rather than environmental impact on spaceborne telescope or optical assembly. This context explains why the levels of radiation considered in these publications were very high compared to most space natural environments achieved and those measurements involved using a high flux of particles heating coupons (up to 70°C¹⁶). In addition, measurements were made considering that a nuclear blast rich in soft x-rays absorbed in the mirror’s shallow level, often were sufficient to cause surface spalling.

Review of the technical contributions over the past four decades exposes the need to refine the understanding on the compaction law as applied to spaceborne mirrors. This is the aim of the present contribution.²¹ Here we derive compaction laws using both the densimetric approach (the experimental observable is the density) and the interferometric approach (the experimental observable is then the deformation of the optical substrate, i.e., closer to a real case in orbit). These two approaches are very different in their concept and can be viewed as being complementary. Neither is suited for all types of ionizing radiations. Densimetry can be more reliably measured on homogeneously irradiated coupons, ideally large coupons (about 1000 g), while interferometry is more adapted to very heterogeneous dose distributions and can be applied to small coupons. Densimetry is more adapted to where the ionizing particle has a large penetration depth, contrary to situations where interferometry is employed. Densimetry will give direct access to the compaction law whereas interferometry will require more physical insight in order to derive a compaction law. The definition of a compaction law implicitly assumes a definition of the deposited dose into the Device Under Test (DUT), here into the ZERODUR^® coupons. Each of the experimental setups for $γ$ , $e^{-}$ , and $H^{+}$ has been modeled using Geant4²²^,²³ version 10.03p03 using the standard EM package for the description of the physics interaction processes. This simulation toolkit is also embedded in the internet interface SPENVIS supported by the ESA²⁴ for modeling the space environment (e.g., cosmic rays, natural radiation belts, etc.) and its effects.²⁴ These dose(-rate)-depth profiles once simulated via Monte-Carlo methods, give a quantitative estimate of the deposited dose equivalent to those provided by SPENVIS tool²⁴ (for example MULASSIS or GRACE) on ZERODUR^®.

We observe that the ionizing particle energy spectra over low Earth orbit (LEO)/geosynchronous equatorial orbit (GEO) is dominated by low-energy particles having a small penetration depth. Therefore, the resulting TID-depth profile has dose asymptotically decreasing with depth. In the space environments considered, deposited ionizing dose may span several orders of magnitude over a depth of few micrometers. As a practical matter, any design of experiment (DOE) addressing the ageing behavior under space ionizing radiation should explicitly cover this range of magnitude. Later, we discuss the effect of monoenergetic laboratory generated dose-depth profiles, which are several orders of magnitude larger than typical dose-depth profile expected on orbit. As mentioned, the experiments presented here do not define any specific concept of operations (CONOPS), our results serve the purpose of fitting a phenomenological compaction law that can then be used to estimate the resilience of ZERODUR^® over the targeted mission.

In the following sections, we will present the different types of radiation used to characterize the compaction law(s), first introducing the method and then the results collected with $γ$ , electron, and proton particles. Next, we introduce the mathematical and physical approaches used to derive compactions laws related to the different types of particles. The significance of these results is discussed.

2. Experimental Results

All the experiments summarized here are designed to define compaction laws describing the response of ZERODUR^®. Since these results cannot be directly applied to a specific CONOPS at this time, we summarize these differentiators in Table 1 below.

Table 1

The contrast between the space and laboratory irradiations is summarized. Spaceborne conditions are from the AP8 and AE8 Max estimates made using a METEOSAT10 GEO orbit computed with SPENVIS24 using the AE-8 and AP-8 particle distribution models.

Particle	Env.	Angular distribution	Energy distribution	Typical flux
Particle	Env.	Angular distribution	Energy distribution	Energy (MeV)	Flux (cm−2.s−1)
$e^{-}$	Space	Omnidirectional	Continuous	$> 1.0$	$[10^{4}; 10^{7}]$
$e^{-}$	Lab	Point source, normally distributed	Monoenergetic	1.5	$1 \times 10^{11}$
$H^{+}$	Space	Omnidirectional	Continuous	$> 1.0$	$[10; 10^{7}]$
$H^{+}$	Lab	Collimated	Monoenergetic	23.0	$1 \times 10^{10} - 1 \times 10^{11}$

The irradiation scenario timescale between the laboratory and the space conditions obviously is very different, and it is impractical to continuously irradiate coupons over a decade because of cost and irradiation stability considerations. These accelerated particle fluxes potentially could introduce two main experimental biases. First, the energy imparted in the target by accelerated fluence of the ionizing particles is converted into heat, and any overheating of the samples, even on a microscopic scale, may lead to unwanted thermally activated healing and induce additional failures that would not have occurred under much more moderate environmental fluences. Large fluxes may induce instantaneously large defect concentrations that may eventually heal and relax over an extended time scale, implying the effects observed directly after irradiation may be misleading, as they do not represent the steady state. To mitigate the contribution of these two effects on the characterization of the resilience of ZERODUR^®, we performed all irradiation experiments under controlled temperature (samples were maintained at room temperature). Moreover when possible, we characterized the impact of irradiation for up to a 1-year time span. Discrepancies in the damaging/healing kinetic of ZERODUR^® and characterization of steady-state fluence have already been investigated on ZERODUR^® by Doyle et al.²⁵ In this work, the authors observed similar coupons in deformation under two very distinct $β$ -irradiation levels, by means of Sr90-Y90 decay and a Van de Graaf accelerator ( $100 \times$ temporal acceleration).

In all following subsections, the local deposited TID ( $D$ ) within the sample can formally be approximated

Eq. (2)

D (x, y, z, t; E_{particle}) = \frac{d E}{d m} (x, y, z, t; E_{particle}) = \frac{1}{ρ S} \frac{d E}{d z} (x, y, z, t; E_{particle}),

with:

• $x, y$ being the position of the volume element on the plane normal to the direction of the incident ionizing particles;
• $z$ being the depth of the elementary unit of material considered;
• $t$ the duration of the exposure;
• $E_{particle}$ the energy of the incident ionizing particle;
• $d E$ the elementary amount of ionizing energy deposited;
• $d m$ the elementary mass considered, presently ZERODUR^® ;
• $ρ$ the material density, presently ZERODUR^®;
• And $S$ the surface area being irradiated;

The amount of energy deposited is directly proportional to the fluence of ionizing particle fired onto the target. Maintaining the experimental setup constant over the irradiation time, we vary only the fluence of particle being fired onto the target ( $n (t)$ ) yielding:

Eq. (3)

E (x, y, z, t; E_{particle}) = e (x, y, z; E_{particle}) * n (t),

with

e (x, y, z; E_{particle})

being the average ionizing energy imparted in the system per fired incident ionizing particles and consequently

Eq. (4)

D (z, t; E_{particle}) = \frac{1}{ρ S} \frac{d E}{d z} (z, t; E_{particle}) = \frac{1}{ρ} \frac{d e (x, y, z; E_{particle})}{d z} \int_{u = t_{0}}^{u = t} \frac{1}{S} \frac{d n (u)}{d u} d u .

In the above equation, $\frac{1}{S} \frac{d n (t)}{d t}$ represents the particle flux in number of particles being fired per second and per unit of surface, and $\int_{u = t_{0}}^{u = t} \frac{1}{S} \frac{d n (u)}{d t} d u$ corresponds to the particle fluence over the temporal range $[t_{0}; t]$ . In the subsequent sections, namely for $γ$ , $e -$ , and $H +$ irradiations, we systematically kept the experimental setups constant (positioning, particle energy, i.e., the deposited energy distribution profile $e (x, y, z, E_{particle})$ was constant) and we only modulated (decadic) the fluence either by changing the duration of the coupon exposure $t$ or via the particle flux $\frac{d n}{d u}$ ; the shape of the dose profile within the DUT remained unchanged over the given ionizing experiment.

In the following subsections $γ$ , $e -$ , and $H +$ irradiations are introduced according to the experimental chronological order, $α$ , $e -$ , and $H +$ , respectively. The $e -$ and $H +$ irradiation experiments are based on the same measurement principle. However, the observations made during electron irradiation showed that we could loosen the specifications regarding the manufacturing of the $H^{+}$ coupons without impairing significantly the quality of the results. A short overview of the samples design considered in the following subsections is provided on Table 2.

Table 2

Irradiation and measurement procedure used for the investigation of the compaction. Some well characterized commercial charges have been selected to manufacture these samples, namely Charge H12132 Article No. 1010088 having a CTE=−0.004 ppm.K−1 and I10414 Article No. 1066717 with a CTE=0.021 ppm.K−1, both charges belonging to the expansion class 0 and having a standard striae, inclusion, and bubble level.

Particle	ZERODUR® charge	Geometry (mm)	Surface quality	Coating	Measure	Dose range (Gy)
$γ$	H12132	$ϕ 90 \times 62.3 mm$	P3 (P1-P2 for edge polishing)	No	Densimetry	$1 \times 10^{5}$
$e^{-}$	I10414	$ϕ 50 \times 5 mm$	P3 (optical surface) – remaining surface acid etched	Yes (Al 120 nm)	Interferometry	$1 \times 10^{3} - 1 \times 10^{6}$
$H^{+}$	I10414	$ϕ 50 \times 5 mm$	P3	No	Interferometry	$3 \times 10^{4} - 1 \times 10^{7}$

2.1.

Gamma Irradiation

Gamma radiation is routinely considered when dealing with material or component resilience in space. The large penetration depth of gamma particles enables the direct characterization of the compaction by means of densimetry, thus large samples provide better results. First, DOE methods have been used to define the electron and proton irradiation experiments.

2.1.1.

Irradiation configuration

The $γ$ irradiation experiments presented here have been achieved using the radio-isotope of Cobalt, ${}^{60}{Co}$ . The decay of ${}^{60}{Co}$ into ${}^{60}{Ni}$ is associated with emission of two $γ$ photons having energies of 1.17 and 1.33 MeV. These high-energy particles interact with the electronic structure of the mirror material and generate electronic defects (ionization) as well as secondary electrons via three main mechanisms

• photoelectric effect;
• Compton scattering;
• and marginally, pair production.

These irradiation experiments have been conducted at the ${}^{60}{Co}$ facility of the ESTEC. The ESTEC ${}^{60}{Co}$ facility is accredited by RvA (Dutch Accreditation Council) according to ISO/IEC 17025.2005(L517) for methods to accurately determine TID and Dose Rate.

The ZERODUR^® coupons have been embedded in ZERODUR^® material surrounds and placed at a distance of $l = 400 mm$ from the $γ$ source. The dose rate in water is measured at this position by means of a PTW TW30012-10 ionization chamber giving a value of ${\dot{D}}_{water} = 112.5 Gy . h^{- 1}$ (the uncertainty budget is $4.4 %$ with $k = 2$ for absorbed dose rate in water, i.e., $\pm 4.5 Gy . h^{- 1}$ ).

Six coupons have been irradiated with this setup, all with different exposure times. All exposure times are approximate and integer multiples of each other. However, the overall duration of the experiment was limited by the irradiation facility available time, and because of this, it was not possible to repeat some experimental points. As a first approximation, the attenuation of electromagnetic radiation in the material is expected to follow a Lambert’s law²⁶ $\propto e^{- (\frac{μ}{ρ}) ρ x}$ , consequently the dose deposited in the cylindrical coupon exhibits a similar decreasing exponential profile along the coupon rotation axis. This profile is characterized by the total mass attenuation coefficient $\frac{μ}{ρ}$ whose value has been estimated by measuring experimentally the dose in water behind ZERODUR^® coupons exhibiting different thicknesses, thus determining $\frac{μ}{ρ} = 0.0562 {cm}^{2} . g^{- 1}$ for $E = 1.25 MeV$ . With this estimate of the mass attenuation coefficient, we can reasonably approximate the dose profile as linearly decreasing profile over the length of the DUT, $e^{- (\frac{μ}{ρ}) ρ x} ≃ 1 - μ x$ . To mitigate the axial changes introduced by the dose gradient, the coupons have been turned around after one half of the total irradiation time. The cumulated dose in water has been measured directly at the back of the assembly (coupon + build-up material) during the first half of the irradiation (see Fig. 1), and then again after the duration of the second half of the irradiation. This duration has been corrected so that the cumulated dose in water measured matches after the second half that measured during the first half. By doing so, we also compensated for the marginal decrease of activity of the ${}^{60}{Co}$ source over the irradiation time.

Fig. 1

Picture of the DUT (ZERODUR^® cylinder embedded into the ZERODUR^® surround material) is placed in front of the ${}^{60}{Co}$ source. The absorbed dose monitoring is achieved by means of a ionization chamber placed at the back of the DUT.

The temperature of the irradiation facility has been kept constant around $T = 22.4 ° C \pm 0.2 ° C$ over the duration of the irradiation.

2.1.2.

Coupons design

Since the ${}^{60}{Co}$ is implemented as an isotropic emitting point source, the irradiation field in the plane normal to the ${}^{60}{Co}$ source axis is then axisymmetric.²⁷ To take full advantage of this feature we designed cylindrical pieces of ZERODUR^®. Beyond this irradiation constraint, we also had metrology constraints that drove this geometrical approach. The density of the coupons is measured at Physikalisch-Technische Bundesanstalt (PTB) using a specific hydrostatic balance. High measurement precision requires that the coupons fulfill prerequisites: The weight of each coupon must be within the $[1000; 1005] g$ range, since this range corresponds to the tight calibration range of the scale. The symmetry and weight constraints, combined with demand for high fluence (i.e., compactness of the irradiated cross section close to the ${}^{60}{Co}$ source), resulted in design freeze with $Ø = 90$ and thickness $= 62.3 mm$ (the individual thicknesses of the coupons have been controlled and adjusted to meet the weight requirement). The hydrostatic metrology principle involves immersion of the coupon. The measurement can be impacted if wetting of the glass into the immersion oil is imperfect, since then the presence of air cavities would bias the measurement. Tests made on ZERODUR^® coupons having different level of ISO10110 polishing quality P1, P2, and P3, showed that a quality of P2 or higher was required to ensure the required surface wetting and to avoid air cavities. Practically the coupons have been polished to a P3 surface quality level for the front and back faces, P2 for the sidewall, and technically it was only possible to achieve a P1 quality for the chamfers of the coupons. However the smallness of the chamfered area made its contribution negligible ( $\leq 1 %$ of the overall of the ZERODUR^® coupon area).

Irradiation by means of $γ$ particles is associated with the build-up effect, with the maximum dose deposited a few millimeters deep in the material. To minimize the impact of this dose-deposition gradient, we manufactured a surround structure/holder also made of ZERODUR^® that tighlty encases the DUT coupons. This is implemented to promote the Compton scattering equilibrium (see Fig. 2) and is believed to entirely mitigate Compton scattering variance on the coupon to the accuracy of the experiment.

Fig. 2

Sketch representing the surround assembly implemented for $γ$ irradiation of ZERODUR^®. The ZERODUR^® DUT (yellow) is embedded into ZERODUR^® surround material (orange).

The surround consists of a large cylindrical ring, as mentionned made of ZERODUR^®, that tightly fits around the DUT diameter. Additionally, the DUT is surrounded front and back by disks of ZERODUR^®. The contact surfaces between the build-up structure and the DUT have all been polished to avoid scratches (and associated later air cavities).

2.1.3.

Measure

Density measurements of solids are carried out with an automatically operating measuring apparatus at PTB.²⁸ The measurements have been all performed in a thermally stable environment with a temperature $T = 20.02 ° C \pm 0.01 ° C$ . The density is obtained by determining the mass and the volume, whereby the mass is measured by comparison with mass standards and the volume is measured by hydrostatic weighing, see Fig. 3. The mass of the coupon is determined by differential weighing using the substitution method, in which the mass standard and the solid coupon are compared successively on the same weighing pan of a high-resolution mass comparator. The masses to be determined are traceable to the SI base unit kilogram. Hydrostatic weighing is a metrology method that determines the density or volume of solids by measuring the buoyant force acting on a solid in a liquid. This method follows from Archimedes’ principle, which states that a fully immersed solid appears to lose as much weight as the amount of liquid displaced. Thus, by means of the apparent weight loss or buoyancy $ρ_{l} V g$ (liquid density $ρ_{l}$ , volume of the solid $V$ , and local gravitational field of earth $g$ ), the density or volume of a solid coupon can be determined. For this purpose, the solid DUT is placed between two density standards (1 kg spheres of natural silicon) of known density and volume in an organic metrology liquid (here n-pentadecane $C_{15} H_{32}$ , $ρ ≃ 769 kg . m^{- 3}$ at 20°C). The two density standards and the solid coupon are weighed alternately with a mass comparator. The solids are deposited via a suspension gear at different heights, connected to the balance by a wire. An exchange mechanism for hydrostatic weighing allows the solids to be weighed independently by placing on and removing the solids from the suspension gear. The two density standards are used to determine the liquid density. The liquid density at the position of the coupon can be obtained by interpolating the liquid density of the position of the density standards to the position of the coupon. With the calculated liquid density, the density or volume of the coupon can be determined by taking into account the buoyancy force and the weight reading. To achieve small measurement uncertainties, a mass comparator with a measuring range of 1 kg and a resolution of $1 μ g$ is used. For this purpose, the weight readings must be adjusted to the measuring range of the mass comparator. An automatic handler in the balance is used to place additional standards (stainless steel weights) on the weighing pan, the weight of which is added to the weights of the coupon or the density standards. If the coupon and standards are very similar in mass and volume, the highest accuracy is obtained when the apparent weights of the coupon and standard are compared in both liquid and air and only the differences are measured. In this way, the densities (and volumes) of the coupons of ZERODUR^® ( $V = 396.3 {cm}^{3}$ ) studied here can be compared with an uncertainty for the density of $ρ = 0.0022 kg m^{- 3}$ , i.e., 2 ppm ( $k = 2$ ).

Fig. 3

(a) The cloche protecting the mass comparator from thermal noise and convective flow. (b) ZERODUR^® coupon in the weighing basket. (c) Overview of the hydrostatic weighing scale. (d) Closer view if the hydrostatic scale with a ZERODUR^® coupon located in the middle basket -below and above Si density standard used for determining the actual density of the immersion fluid.

To investigate both potential drift over the metrological chain as well as any external effect that could have induced a density change we purposely kept a reference coupon unirradiated throughout the experimental process. This coupon testifies the absence of changes induced by environmental parameters over the duration experiments. This demonstrates a density metrology drift not greater than $0.0004 kg . m^{3}$ . The average density of all six samples prior to irradiation has been determined being $ρ = 2534.0425 kg . m^{- 3}$ .²⁹

2.1.4.

Simulation

Our experimental setup has been simulated using the Geant4 software package.²²^,²³ The ${}^{60}{Co}$ source has been approximated²⁷ as a point source located at 400 mm from the target, while the surrounding space is regarded to be made exclusively of air. To reduce computational time we have made a reasonable approximation of only considering the $γ$ particles whose initial direction crosses the target (the target corresponding to a solid angle $Ω_{Cylinder}$ ). The density of particles fired per unit of solid angle has been set constant for the simulation. Here, the simulated target is a cylinder representing both the assembly coupon and surround, i.e., a $Ø = 130 mm$ , height 75-mm cylinder of ZERODUR^®. The $γ$ particles energy distribution is represented by two Dirac delta functions centered on 1.17 and 1.33 MeV. For this purpose ZERODUR^® has been defined using the individual National Institute of Standards and Technology (NIST) element definition set and weighting by the ZERODUR^® composition,¹ and the density has been set to $ρ = 2.53 g . {cm}^{- 3}$ .¹ The number of particles $N$ fired onto the target over the simulation can be summed into a duration $Δ t$ following:

Eq. (5)

Δ t ≃ \frac{1}{a A} (\frac{4 π}{Ω_{Cylinder}}) (\frac{N}{2}),

where activity

A

corresponds to the activity of the source at the time considered and

a

being the source self absorption coefficient.³⁰ This equation assumes that the overall activity

A

does not decay significantly over the duration of the irradiation experiment

\frac{Δ t}{A} \frac{\partial A}{\partial t} < 1

, i.e., experiments lasted about 1 month, to be compared with a half-life of

{}^{60}{Co}

of 5 years. The local dose rate can then be expressed as being the deposited dose per volume element over the simulation divided by the exposure time. Due to this simplification, the contribution of particles scattered by the surrounding environment is not taken into account (back-scattered particles on irradiation facility, table, mount, etc.) and is believed negligible. The corresponding computed dose-depth profile is represented on Fig. 4. Based on this profile, the average TID-rate in ZERODUR^® has been estimated being about

68 Gy . h^{- 1}

(see Fig. 4 horizontal dashed line).

Fig. 4

Dose-depth (TID) rate calculated using Geant4 on the ZERODUR^® cylinder. The two vertical lines represents the limit of the shielded domain (build-up material). The horizontal line corresponds to the estimated average TID (mean of a function) within the ZERODUR^® DUT.

The computed dose-depth profile of water has also been added to Fig. 4, the simulated value read at the charged particle equilibrium (CPE) in water is $107 Gy . h^{- 1}$ in reasonable agreement with the experimental measure achieve with the PTW TW30012-10 ionization chamber, $112.5 Gy . h^{- 1}$ , i.e., about 5% mismatch between the simulation and the measure.

We also used the mass-attenuation coefficient to provide an estimate of expected dose in ZERODUR^®. To this end, we need to rescale the dose in water as read with the ionization chamber. This has been accomplished by referencing the tabulated experimental mass energy absorption coefficient of a well-calibrated material like Borosilicate glass (similar to ZERODUR^® in term of density, ${SiO}_{2}$ and alkali contents) divided by the corresponding experimental coefficient of water at an energy of 1.25 MeV available in the NIST database,³¹ $\frac{\frac{μ_{en}}{ρ} |_{glass}}{\frac{μ_{en}}{ρ} |_{water}} = \frac{2.650 * 1 \times 10^{- 2}}{2.9655 * 1 \times 10^{- 2}} = 0.8937$ . Thus, we can propose an estimate of the dose rate in glass under conditions of secondary CPE by about $98 Gy . \min^{- 1}$ , also in qualitative agreement within 14% to the calculated CPE TID dose rate in ZERODUR^® presented on Fig. 4.

Based on those value, we estimated the overall uncertainty budget (measure + simulation) for the TID dose in ZERODUR^® about 20%.

2.1.5.

Results

The graphical representation of these density variations²⁹ as a function of both the particle fluence and the estimated average deposited TID is provided on Fig. 5. The variation of density for 10 and 100 Gy exhibit large uncertainties. Above 100 Gy, the linear aspect of the compaction data in Fig. 5 strongly suggests that the phenomenological law binding the ionizing dose to the compaction can be optimally described by a power law.

Fig. 5

Compaction of ZERODUR^® as a function of the $γ$ fluence (bottom abscissa) and corresponding estimated TID (top abscissa) achieved by means of ${}^{60}{Co}$ . The fluence $ϕ$ of primary $γ$ particles has been calculated $Φ = A a \frac{Ω}{4 π} \frac{1}{π r^{2}} Δ t$ , with $Ω ≃ π (\frac{r}{l})^{2}$ being the solid angle of the coupon (radius $r = 45 mm$ ) seen from the source (distance $l = 400 mm$ ), $A$ the source activity ( $A = 58.5 TBq$ at the time of the irradiation and $a$ the source self absorption coefficient of $a = 0.8352$ .³⁰

2.2.

Electron Irradiation

For required precision, densimetry requires large coupons ( $m ≃ 1000 g$ , i.e., typically ${(m / ρ_{ZERODUR})}^{1 / 3} ≃ 75 mm$ large coupons). While not possible to homogeneously irradiate such large coupon with electrons without massively activating the coupons (activation energy threshold $\geq 10 MeV$ ), nevertheless, the effect compaction can still be indirectly determined by measuring the bending of superficially irradiated discs, as originally proposed by Bourrieau et al.¹⁵ This approach can easily be achieved using $≃ 1 MeV$ incident kinetic energy electrons followed by ultraprecise measurement of radiation induced coupon curvature. While the curvature comprises mostly optical power, this directly relates to the expectation of optical surface error over a spaceborne telescope’s mission.