Innovating in gerontological coordination. The autonomy of older adults at the center of care work ? 1) Presentation : This presentation consists in a synthesis of an in-progress paper, which includes some of the results of my PhD thesis. I followed an innovation process applied to the assistance and care of the older adults during four consecutive years in a french department. It concerns the deployment of an information system for gerontological coordination, conceived to promote more efficient exchanges between professionals. To this end, an online discussion forum allows professionals from different sectors (home hospitalization, nursing and assistance services, etc.) to network and keep track of their exchanges.

Each professional involved in a given situation connects to an interface, chooses the level of emergency of the message to be transmitted (green = not very urgent; red = very urgent) and the degree of confidentiality (all professionals / health professionals only / social workers only).

2) Problematisation: The French gerontological sector has been mainly described so far in social sciences as organised around the biomedical management of ageing bodies (Bungener, 2004). I thus question if the digital innovation process, has allowed an institutional innovation, leading to transformations in the way of considering older patients (Bonny & Giuliani, 2012). In particular, I analyse the attention paid by professionals to the decision-making power of older people in online exchanges. Knowing that older adults and their relatives do not have access to these exchanges. Greater attention to the autonomy of older adults would then be a sign of institutional innovation. To this end, I propose a definition of autonomy in gerontological intervention at home, in the form of a typology of "forms" of autonomy, concerned by this innovation process.

3) Methodology: I deployed a methodological triangulation, crossing participant observations in the home ; a quantitative analysis of online exchanges ; semi-structured interviews with care professionals, older adults and relatives (N=41) for 9 targeted situations, to account for the impact of the innovation process on interactions at home ; semi-structured interviews with the IS project managers. In addition, qualitative analyses of the discussion forums of the patients met (2 or 3 patients / year) were carried out. In this presentation, I will only focus on the qualitative approaches. 4) A focus on body-at-risk : Concerning the situations I followed in 2016, the exchanges and representations of the professionals interviewed mainly concerned the organisational aspect of care, such as the announcement of hospital admissions and discharges. In 2018, the exchanges also concern the monitoring of vital signs and symptoms that the carers formalise online for the General Practioner.

Coordination via the information system is essentially focused on the biomedical management of "bodies-at-risk" (Lòpez and Domènech, 2009), without taking into account the wishes of the older adults. Only functional autonomy, referring to the execution of daily life activities, is supported, by organising the material aspect of assistance and limiting the impact of pathologies. 4) Negotiations on Suzanne's forms of autonomy: I met Suzanne in 2017. She was 91 years old. She has been a widow for about ten years and lives alone. She suffers from cognitive disorders due to Alzheimer's disease, but remains relatively autonomous in terms of functional abilities. She categorically refuses to enter a nursing home. At the time of our meeting, a short-term lethal diagnosis had been made, due to severe cardiovascular issues. The case manager calls the other professionals by telephone, so that they can coordinate on a regular basis via the Information System. The aim is to support her wish to die at home. The General Practioner writes in the discussion forum, in response to the nurse who suggests setting up a preventive registration in a nusing home :

He thus becomes the advocate of her moral autonomy, i.e. the fact of making life choices according to internalised social values, which are the basis of the person's identity (In this case : ageing and dying at home). At the same time, the need to support the securing of home care is asserted. There are also longterm negotiations of her organisational autonomy, i.e. the power she retains over the arrangements of her home. In particular, the case manager writes a note asking whether Susan has agreed to sleep in the healthcare bed. At one point she agreed, but two days later she refused it. In the end, it was her reflexive autonomy that was the subject of long-term multi-partner negotiations, between the different actors involved. Reflexive autonomy refers to a person's ability to assess the correctness of the rules they set for themselves and to conduct a coherent argument. This ability is severely compromised by Susan's Alzheimer's disease.

A negotiation of her decisional autonomy, referring in my definition essentialy to the sovereignty of the will (I want to take a nap / I don't want you in my house), concerning help with the toilet by the home care worker, is reported online: "in most cases she refuses help with the toilet but if you ask her nicely she can accept". By doing this, the professionals involved in the network can recognise the value of the care work carried out by home care workers, which is generally invisible and considered insignificant (Laugier, 2020). I also found this type of formalisation in other discussion forums. 5) Conclusion:At the end of my investigations, I do not observe any notable institutional innovation.

Online exchanges remain mainly focused on supporting functional autonomy, through biomedical management of body-at-risk. The support of other forms of autonomy (decisional, moral, organisational, reflexive) via the Information System is not related to the evolution of the innovation process, but to the power relations between actors around each situation. Nevertheless, if this remains modest at this stage, I observe that the Information System has made the invisible work of care visible.

In my opinion, from the perspective of reflection on care-led innovations in gerontology, it is important to propose a complex definition of autonomy support, by humans networking with non-humans, according to Fabrice Gzil whose work inspired my typology. This definition must take into account the different aspects of the decision-making power of older people, what supports them and what limits them, and not be limited to the compensation of disabilities.

Thank you for your attention


J-M Lagniel

Ganil

EXCITATION OF THE 𝝈𝝈 𝒍𝒍_𝒍𝒍 = 90° RESONANCE BY THE CAVITY RF ACCELERATING FIELDS

Keywords: Non-linear Single Particle Dynamics -Resonances, Tracking, Higher Order, Dynamic Aperture, Code Dev

In rf linacs the longitudinal focalization is done by sinusoidal forces and at high accelerating fields the zero-current longitudinal phase advance per longitudinal focusing period 𝜎𝜎 0𝑙𝑙_𝑙𝑙 can be high. The nonlinear components of the sinusoidal rf field (sextupolar, octupolar and higher order components) can then excite parametric resonances, including the 4th-order resonance (𝜎𝜎 𝑙𝑙_𝑙𝑙 = 90°) when 𝜎𝜎 0𝑙𝑙_𝑙𝑙 is higher than 90°, inducing strong longitudinal emittance growths and acceptance reductions. As pointed out in previous papers, the longitudinal beam dynamics is therefore complex, even when the space-charge forces are ignored. The parametric resonance excitation by the rf field is analyzed before discussing the additional effect of the spacecharge forces, in particular to explain why the zero-current longitudinal phase advance per transverse focusing period 𝜎𝜎 0𝑙𝑙_𝑡𝑡 is not a relevant parameter. Examples are given in the SPIRAL2 linac case.

PARAMETRIC RESONANCES, BASIC DEFINITIONS

As done in [START_REF] Lagniel | Zero-Current Longitudinal Beam Dynamics[END_REF][START_REF] Lagniel | Longitudinal beam dynamics at high accelerating fields, what changes?[END_REF], the choice done here is to study the parametric resonances in the longitudinal plane independently of the study of the radial-longitudinal coupling resonances in order to avoid unnecessary complications and confusions. As usual (e.g. [START_REF] Bruck | Accélérateurs circulaire de Particules[END_REF] chapter X, section IV), the resonance zoology is defined considering the linear equation of motion of individual particles perturbed by linear (n = 1) or nonlinear (n > 1) periodic perturbing forces 𝑑𝑑 2 𝛿𝛿𝛿𝛿 𝑑𝑑𝑧𝑧 2 + 𝜎𝜎 𝑙𝑙 2 𝛿𝛿𝛿𝛿 + 𝑎𝑎 𝑛𝑛,𝑘𝑘 [𝛿𝛿𝛿𝛿 0 sin(𝜎𝜎 𝑙𝑙 𝑧𝑧)] 𝑛𝑛 [𝛿𝛿𝛿𝛿 0 cos(𝜎𝜎 𝑙𝑙 𝑧𝑧)] 𝑛𝑛 cos�𝑘𝑘𝜎𝜎 𝑝𝑝 𝑧𝑧� = 0

with 𝜎𝜎 𝑙𝑙 the phase advance per unit of length of the unperturbed particle motion, 𝜎𝜎 𝑝𝑝 and 𝑘𝑘 the phase advance per unit of length and the harmonic number of the perturbing force respectively. Considering that a resonance condition is fulfilled when a component (n, k) of the perturbing force has a "frequency" equal to the unperturbed particle oscillation "frequency" σ 𝑙𝑙 [START_REF] Bruck | Accélérateurs circulaire de Particules[END_REF], a resonance occurs when (𝑖𝑖 + 1) 𝜎𝜎 𝑙𝑙 = 𝑘𝑘 𝜎𝜎 𝑝𝑝 with i even integer from 0 to n for n even and i odd integer from 1 to n for n odd.

Using 𝜎𝜎 𝑙𝑙_𝑙𝑙 = 𝜎𝜎 𝑙𝑙 𝐿𝐿 𝑙𝑙 the longitudinal phase advance per period of the longitudinal focusing system (gaps or cavi-ties), 𝜎𝜎 𝑙𝑙_𝑝𝑝 = 𝜎𝜎 𝑙𝑙 𝐿𝐿 𝑝𝑝 the longitudinal phase advance per period of the perturbation and 𝜎𝜎 𝑝𝑝 = 2𝜋𝜋 𝐿𝐿 𝑝𝑝 ⁄ with 𝐿𝐿 𝑝𝑝 the period of the perturbation, the resonance condition is

𝜎𝜎 𝑙𝑙_𝑝𝑝 = �𝐿𝐿 𝑝𝑝 𝐿𝐿 𝑙𝑙 ⁄ � 𝜎𝜎 𝑙𝑙_𝑙𝑙 = 2𝜋𝜋𝑘𝑘 𝑖𝑖 + 1 (2)
(𝑖𝑖 + 1) is the order of the resonance and k is the harmonic of the perturbation.

For example, particles with 𝜎𝜎 𝑙𝑙_𝑙𝑙 = 90° can be excited by the 𝑎𝑎 𝑛𝑛=3,𝑘𝑘=1 component of the perturbing force when 𝐿𝐿 𝑝𝑝 = 𝐿𝐿 𝑙𝑙 (fourth order resonance), or can be excited by the 𝑎𝑎 𝑛𝑛=1,𝑘𝑘=1 , 𝑎𝑎 𝑛𝑛=3,𝑘𝑘=1 and 𝑎𝑎 𝑛𝑛=3,𝑘𝑘=2 components of the perturbing force when 𝐿𝐿 𝑝𝑝 = 2𝐿𝐿 𝑙𝑙 (half integer and fourth order resonances).

EXITATION BY THE CAVITY RF FIELD

Following [START_REF] Lagniel | Zero-Current Longitudinal Beam Dynamics[END_REF][START_REF] Lagniel | Longitudinal beam dynamics at high accelerating fields, what changes?[END_REF], the non-linear components of the rf accelerating field must be considered as main perturbing forces in the parametric resonance studies. This can be understood starting from the equation of motion of the phase oscillations around the synchronous particle obtained in smooth approximation ( [START_REF] Bruck | Accélérateurs circulaire de Particules[END_REF], chapter XVI section II-D). The effect of the damping term 𝐾𝐾 𝑑𝑑𝑝𝑝 which has a great importance on the longitudinal dynamics at high acceleration rates [START_REF] Lagniel | Zero-Current Longitudinal Beam Dynamics[END_REF][START_REF] Lagniel | Longitudinal beam dynamics at high accelerating fields, what changes?[END_REF] is not discussed here to focus the analysis on the rf field nonlinear focusing force which can be expressed by its Taylor series

[cos(𝛷𝛷 𝑠𝑠 + 𝛿𝛿ϕ) -cos 𝛷𝛷 𝑠𝑠 ] = -[sin 𝛷𝛷 𝑠𝑠 ] 𝛿𝛿ϕ "quadrupole" (linear focusing) -[cos 𝛷𝛷 𝑠𝑠 /2!] 𝛿𝛿ϕ 2 "sextupole" -[sin 𝛷𝛷 𝑠𝑠 /3!] 𝛿𝛿ϕ 3 "octupole" + [cos 𝛷𝛷 𝑠𝑠 /4!] 𝛿𝛿ϕ 4 "decapole" + … (4)
Equation ( 3) is obtained using the smooth approximation, then averaging the effect of the rf field over the longitudinal period 𝐿𝐿 𝑙𝑙 . The effect of the gap or cavity periodicity must be analyzed including the harmonics of the rf field 𝐸𝐸 𝑧𝑧 (𝑧𝑧) = 𝐸𝐸 0 + ∑ 𝐸𝐸 𝑘𝑘 𝑐𝑐𝑐𝑐𝑐𝑐(𝑘𝑘 2𝜋𝜋𝑧𝑧 𝐿𝐿 𝑙𝑙 ⁄ ) in the longitudinal focusing force of Eq. ( 3), then to each term of the Taylor series of Eq. ( 4). The longitudinal focusing of the rf field can then be seen as a nonlinear focusing Eq. ( 4) and with 𝑘𝑘 = 0 in Eq. ( 1 terms which induce resonances when the resonance conditions are fulfilled. This was demonstrated in [1, 2] using theoretical field distributions, Fig. 1 shows that, as expected, this is also true in the case of the SPIRAL2 superconducting linac cavity field [START_REF] Orduz | Deuteron beam power ramp-up at SPI-RAL2[END_REF]. The different plots in Fig. 1 are Poincaré maps done with the SPIRAL2 low beta cavities (𝐿𝐿 𝑙𝑙 = 1.19 m), working at different fields and tuned in buncher mode (no damping) for deuterons at the RFQ output energy (0.732 MeV/A).

With kE = E/Emax and Emax = 6.5 MV/m the nominal field, for 𝜎𝜎 0𝑙𝑙_𝑙𝑙 = 53.9° at kE=0.06 Fig. 1 shows that, as expected, there is 8 resonance islands at 𝜎𝜎 𝑙𝑙_𝑙𝑙 = 45° and a beginning of acceptance reduction due to the high-order resonance overlap in the separatrix area. As expected, the 6 th order parametric resonance 𝜎𝜎 𝑙𝑙_𝑙𝑙 = 60° is present at 𝜎𝜎 0𝑙𝑙_𝑙𝑙 = 70.3° (kE = 0.10), with a significant acceptance reduction due to higher-order resonance overlap in the separatrix area. As expected, the fourth-order resonance is present at 𝜎𝜎 0𝑙𝑙_𝑙𝑙 = 93.6° (kE = 0.17) and 𝜎𝜎 0𝑙𝑙_𝑙𝑙 = 102.6° (kE = 0.20), with the 4 resonance islands present in the stable region around the synchronous particle in the first case and in the chaotic area in the second case. The drastic longitudinal acceptance reduction due to even more high-order resonance overlap in the separatrix region above 𝜎𝜎 0𝑙𝑙_𝑙𝑙 = 90° is also clearly visible in both cases.

As foreseen by Eq. ( 4) with 𝑐𝑐𝑐𝑐𝑐𝑐 𝛷𝛷 𝑠𝑠 = 0, there is only even order resonances present in Fig. 1. Not shown, but in the 𝜎𝜎 0𝑙𝑙_𝑙𝑙 > 90° cases the chaotic region extends at high energies, particles injected out of the reduced acceptance will then form "high-energy tails" (up to 𝛿𝛿𝛿𝛿 > 1.2 MeV/A above the RFQ output energy of 0.732 MeV/A).

One can finally point out the strong constraint imposed by the 𝜎𝜎 0𝑙𝑙_𝑙𝑙 < 90° rule that can impose the use of cavity fields much lower than the nominal ones in SC linacs, e.g. a limitation at less than 10% of the nominal field for the nominal tuning in the SPIRAL2 linac low energy part.

All of this pushes to better study the effect of the damping reducing the effect of the resonances at high accelerating rates [START_REF] Lagniel | Zero-Current Longitudinal Beam Dynamics[END_REF][START_REF] Lagniel | Longitudinal beam dynamics at high accelerating fields, what changes?[END_REF]. Considering that at the linac design stage, the longitudinal acceptance could be sufficiently preserved to leave the possibility to use higher accelerating fields.

EXITATION BY THE SPACE-CHARGE FIELD

With no loss of generality, the model for the space charge forces assuming the bunch to be a three-dimensional ellipsoid of uniform density introduced by Lapostolle [START_REF] Lapostolle | Effets de la charge d'espace dans un accélérateur linéaire à protons[END_REF] can be used. Following Gluckstern [START_REF] Gluckstern | Linear Accelerators (section C.1.3c)[END_REF], using this model the longitudinal space charge force dependency (via the Poisson equation) on the bunch dimensions is usually (0.8 < 𝜉𝜉 < 5 range in [START_REF] Gluckstern | Linear Accelerators (section C.1.3c)[END_REF]) given by

𝜇𝜇 𝑙𝑙 ∝ 1 � 𝑎𝑎 𝑥𝑥 𝑎𝑎 𝑦𝑦 𝑏𝑏 2 (5)
with 𝑎𝑎 𝑥𝑥 and 𝑎𝑎 𝑦𝑦 the bunch radial sizes and 𝑏𝑏 the bunch longitudinal size.

For matched beams, the longitudinal envelope oscillation has the period 𝐿𝐿 𝑙𝑙 of the longitudinal focusing system (gaps or cavities); all the harmonics at the longitudinal focusing period 𝐿𝐿 𝑙𝑙 are then always present in the 𝑏𝑏 2 (𝑧𝑧) contribution to the space charge "perturbing force" Eq. ( 5).

In linacs the horizontal and vertical emittances are usually equals and the phase advances in the horizontal and vertical planes are traditionally chosen equals. In the case of matched beams in a solenoid or quadrupole doublet focusing (SPIRAL2 case), all the harmonics at the 𝐿𝐿 𝑡𝑡 = 𝐿𝐿 𝑙𝑙 transverse period are also always present in the 𝑎𝑎 𝑥𝑥 𝑎𝑎 𝑦𝑦 contribution to the space charge "perturbing force".

In the case of matched beams in an alternating focusing with quadrupoles (FODO or FOFODODO) and choosing z = 0 such that the lattice has an even symmetry, the bunch radial dimensions can be expressed by their Fourier series

𝑎𝑎 𝑥𝑥 (𝑧𝑧) = 𝑎𝑎 0 + � 𝑎𝑎 𝑘𝑘 cos �𝑘𝑘 2𝜋𝜋𝑧𝑧 𝐿𝐿 𝑡𝑡 � ∞ 𝑘𝑘=1 𝑎𝑎 𝑦𝑦 (𝑧𝑧) = 𝑎𝑎 𝑥𝑥 �𝑧𝑧 + 𝐿𝐿 𝑡𝑡 2 � = 𝑎𝑎 0 + �(-1) 𝑘𝑘 𝑎𝑎 𝑘𝑘 cos �𝑘𝑘 2𝜋𝜋𝑧𝑧 𝐿𝐿 𝑡𝑡 � ∞ 𝑘𝑘=1 (6)
The product 𝑎𝑎 𝑥𝑥 𝑎𝑎 𝑦𝑦 inducing the variations of the longitudinal space charge force with respect to the radial bunch size variations Eq. ( 5) is then of the form

𝑎𝑎 𝑥𝑥 𝑎𝑎 𝑦𝑦 = �𝑎𝑎 0 2 + �(-1) 𝑘𝑘 ∞ 𝑘𝑘=1 𝑎𝑎 𝑘𝑘 2 2 � (7) + � � 𝐴𝐴 1 (𝑘𝑘, 𝑙𝑙) ∞ 𝑙𝑙=1 cos �2𝑘𝑘 2𝜋𝜋𝑧𝑧 𝐿𝐿 𝑡𝑡 � ∞ 𝑘𝑘=1 + � � 𝐴𝐴 2 (𝑘𝑘, 𝑙𝑙) ∞ 𝑙𝑙=1 ∞ 𝑘𝑘=1 cos �2(𝑘𝑘 + 𝑙𝑙) 2𝜋𝜋𝑧𝑧 𝐿𝐿 𝑡𝑡 �
There is therefore no "space-charge excitation" in the longitudinal plane at a period equal to the radial focusing period, the first component (𝑘𝑘 = 1) is at 𝐿𝐿 𝑡𝑡 2 = 𝐿𝐿 𝑙𝑙 ⁄ in the FODO case and at 𝐿𝐿 𝑡𝑡 2 = 2 𝐿𝐿 𝑙𝑙 ⁄ in the FOFODODO case. Figure 2 also shows that the oscillatory part of the radial envelope contribution to the longitudinal space-charge force (blue curve, � 𝑎𝑎 𝑥𝑥 𝑎𝑎 𝑦𝑦 term in Eq. ( 5)) is very low, then has a small contribution compared to the one of the longitudinal envelope oscillations (𝑏𝑏 2 term in Eq. ( 5)). One can then also conclude that in the case of the retained assumptions, the radial envelope oscillations have week effects on the excitation of parametric resonances in the longitudinal plane in both FODO and FOFODODO cases.

SUMMARY

Many publications systematically discuss the excitation of the fourth-order parametric resonance 𝜎𝜎 𝑙𝑙 = 90° by the space-charge forces. After [START_REF] Lagniel | Zero-Current Longitudinal Beam Dynamics[END_REF][START_REF] Lagniel | Longitudinal beam dynamics at high accelerating fields, what changes?[END_REF], it is again pointed out here that the first source of excitation of this resonance (as well as the other parametric resonances in the longitudinal plane) is the rf field of the cavities.

Concerning the space-charge effects, we have shown that in the linac case (equal tune and emittance in the horizontal and vertical planes) there is no excitation in the longitudinal plane at a period equal to the radial focusing period in the FODO and FOFODODO cases, and that the radial envelope oscillations have week effects in both cases.

The periodicity of the main longitudinal motion perturbations (rf field and space-charge) is the longitudinal lattice. As pointed out several times, it is a mistake to consider the 𝜎𝜎 𝑙𝑙_𝑡𝑡 longitudinal phase advances per transverse focusing period speaking of parametric resonances in the longitudinal plane.

For linac tunings at high acceleration rates, the strong effect of the longitudinal phase oscillation damping [START_REF] Lagniel | Zero-Current Longitudinal Beam Dynamics[END_REF][START_REF] Lagniel | Longitudinal beam dynamics at high accelerating fields, what changes?[END_REF] would need to be taken into account.

  cos(𝛷𝛷 𝑠𝑠 + 𝛿𝛿ϕ) -cos 𝛷𝛷 𝑠𝑠 ] = 0[START_REF] Bruck | Accélérateurs circulaire de Particules[END_REF]

Figure 1 :

 1 Figure 1: Longitudinal phase-space portrait for SPIRAL2 low beta cavities tuned in buncher mode (𝛷𝛷 𝑠𝑠 = -90°) at the RFQ output energy (0.732 MeV/A). Top left : 𝜎𝜎 0𝑙𝑙_𝑙𝑙 = 53.9° (kE = 0.06), top right 𝜎𝜎 0𝑙𝑙_𝑙𝑙 = 70.3° (kE = 0.10), bottom left 𝜎𝜎 0𝑙𝑙_𝑙𝑙 = 93.6° (kE = 0.17), bottom right 𝜎𝜎 0𝑙𝑙_𝑙𝑙 = 102.6° (kE = 0.20).

Figure 2 :

 2 Figure 2: Evolution of the Courant-Snyder beta functions (red and green) and of the contribution of the radial envelope oscillations in the longitudinal space-charge perturbing force (blue) for matched beams in FODO (left) and FOFODODO (right) lattices with thin lenses tuned such that 𝜎𝜎 𝑡𝑡_𝑡𝑡 ~ 63°.

Content from this work may be used under the terms of the CC BY 4.0 licence (© 2022). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI

ACKNOWLEDGEMENTS

Many thanks to the GANIL colleagues for their support and to M. Eshraqi (ESS) for his encouragements and judicious suggestions for the improvement of this paper.


A Leduc

T Thuillier

L Maunoury

O Bajeat

ANGULAR DISTRIBUTION MEASUREMENT OF ATOMS EVAPORATED FROM A RESISTIVE OVEN APPLIED TO ION BEAM PRODUCTION

Keywords:

A low temperature oven has been developed to produce calcium beam with Electron Cyclotron Resonance Ion Source (ECRIS). The atom flux from the oven has been studied experimentally as a function of the temperature and the angle of emission by means of a quartz microbalance. The absolute flux measurement permitted to derive Antoine's coefficient for the calcium sample used : 𝐴 = 8.98 ± 0.07 and 𝐵 = 7787 ± 110 in standard unit. The angular FWHM of the atom flux distribution is found to be 53.7 ± 7.3 °at 848K, temperature at which the gas behaviour is non collisional. The atom flux hysteresis observed experimentally in several laboratories is explained as follows: at first calcium heating, the evaporation comes from the sample only resulting in a small evaporation rate. once a full calcium layer has formed on the crucible refractory wall, the calcium evaporation surface includes the crucible's enhancing dramatically the evaporation rate for a given temperature. A Monte-Carlo code, developed to reproduce and investigate the oven behaviour as a function of temperature is presented. A discussion on the gas regime in the oven is proposed as a function of its temperature. A fair agreement between experiment and simulation is found.

INTRODUCTION

This work is dedicated to the study of a low temperature metallic oven dedicated to calcium beam production at the SPIRAL2 facility at GANIL, France [START_REF] Barué | Metallic beam developments for the SPIRAL 2 project[END_REF][START_REF] Thuillier | Status of the SPIRAL2 injector commissioning[END_REF]. The motivation of the study is to better understand the physics and chemistry of such an oven with the ultimate goal to improve the global conversion efficiency of rare and expensive isotope atom (like 48 Ca) to an ion beam in Electron Cyclotron Resonance Ion Source (ECRIS). The long term goal being to build an end-to-end simulation able to optimize and predict the atom to ion conversion yield of oven to produce beams in ECRIS. Here, the differential atom flux from the oven is measured and compared to simulation. Indeed, the angle of atom emission is an important geometrical factor of the whole atom to ion conversion in an ECRIS. In a first part, the calcium oven used is presented in detail. In a second part, the experimental setup used to study the differential atom flux from the oven is described. Next, the Monte Carlo model used to simulate the oven behaviour is presented and the simulation results discussed. In the last section, simulation and experimental results are compared and discussed. The oven design is derived from an existing technology developed at Lawrence Berkeley National Laboratory [START_REF] Benitez | Production of high intensity 48 Ca for the 88-Inch Cyclotron and other updates[END_REF]. A cutaway view of the oven is presented on Fig. 1 with information on its mechanical composition. The oven crucible is made of molybdenum to prevent chemical reaction with the metallic sample. The crucible cavity has a symmetry of revolution and is composed of two parts (see view B of Fig. 1): (a) a 5mm diameter and 11 mm long cylindrical container ending on the last millimeter by a 30°cone (shape imposed by the drilling tool geometry), (b) an 1 mm diameter and 2 mm long extraction channel (also referenced later as a nozzle). The oven is heatable up to 875K when a Joule power of 200W is applied. A thermal simulation done with Ansys software has shown that the crucible temperature is very homogeneous with a maximum temperature gradient of the order of a few degrees Kelvin only. When the electric oven power supply is switched off, the maximum oven temperature at the full microwave power is 500 K, well below the operating range of calcium evaporation in the ECRIS (being 650 to 700 K). Therefore, the design is safe for any experimental operation since, whatever the ion source tuning is, the oven temperature is controlled by adding extra power to the oven cartridge resistor.

CALCIUM OVEN

EXPERIMENTAL MEASUREMENTS

Setup

The oven metallic atom emission has been measured in a dedicated vacuum chamber (see Fig. 2) with a residual pressure 𝑃 0 = 10 -7 mbar. The oven temperature is monitored by the thermocouple included in the oven heater cartridge. The atom flux is measured with a quartz AUDA6 Neyco micro balance. The quartz is inserted into a mechanical support resulting in an active measurement disc diameter of 8.1 mm. The quartz temperature is fixed thanks to a water cooling system. The vertical support is mechanically connected to a rotatable vacuum flange by means of a tube bringing water cooling to the balance. the tube is bent to form two 90°bends so that during a rotation, the quartz describes a circle with a 60 mm radius. The crystal frequency of vibration is 3 MHz when a static electric field is applied. When evaporated metallic atoms are deposited on the quartz surface, its mass increases and thus changes its mechanical resonance frequency. The quartz frequency measurement is done with a dedicated Inficon controller which displays the instantaneous frequency and calculates the mass per cm 2 accumulated during a programmable integration time. During the experiments, the mass flux is calculated after an integration time from 60 to 180 s. In our experimental conditions, The mass limit accuracy of the system has been estimated to be ≈ 0.5𝑛𝑔/𝑐𝑚 2 /𝑠. The oven axis is horizontal and set perpendicular to the balance axis of rotation. The oven position can be translated along the direction of its axis of revolution. The two experimental configurations reported are displayed on Fig. 3.In the configuration (1), the ovenexitis placed on the balance axis of rotation. Duringthe quartzrotation, the distancebetweenthe ovenand the quartzis constant and equalsto 60 mm. The angle betweenthe quartzsurface and the ovensurfaceis noted 𝜃.In the configuration (2), the quartzis set 10 mm awayfrom the ovenexit.The solid angle coveredbythe quartzis then ≈0.459 sr,with amaximum angle of detection 𝜃 1 =22,0°.

Measurements

The oven run presented used a calcium 40 sample weighting 0.0909 g with a surface 𝑠 𝐶 𝑎 = 0.8±0.2 𝑐𝑚 2 . The overall crucible internal surface is 𝑠 𝑐 = 2.8 𝑐𝑚 2 . The differential metallic mass flow

𝑓 (𝜃) = 1 𝑟 2 𝑑 𝑚 𝑑𝜔 (1)
emitted by the oven was measured as a function of 𝜃 at T=848, 873 and 898 K (see protocol [START_REF] Barué | Metallic beam developments for the SPIRAL 2 project[END_REF] in Fig. 3). Here 𝑟 is the radial distance between the oven and the balance and 𝑑𝜔 stands for the differential solid angle taken at 𝜃. Results are displayed in fig. 4. The errorbar on 𝜃 is due to the extended surface of detection (𝜎 𝜃 = 0.9°), the error on angle measurement (𝜎 𝜃 ≈ 2°) and the error on alignment (𝜎 𝜃 ≈ 1.°), leading to a global error 𝜎 𝜃 ≈ 2.4°. The mass flux measurement precision is limited by the resolution of the balance for the chosen time of integration of 180 s. The mass flux error is thus estimated to be 𝜎 𝑚 ≈ 0.5𝑛𝑔/𝑐𝑚 2 /𝑠. The full width half maximum (FWHM) of the distributions are reported in Table 5, along with the reconstructed total mass flux 𝑀 integrated over 2𝜋 sr (and opportunely averaged on the overabundant range of measurement from -𝜋/2 to 𝜋/2) :

𝑀 = 2𝜋𝑟 2 1 2 ∫ 𝜋/2 0 𝑓 (𝜃)𝑠𝑖𝑛𝜃𝑑𝜃 + 1 2 ∫ -𝜋/2 0 𝑓 (𝜃)𝑠𝑖𝑛𝜃𝑑𝜃 (2)
The angular FWHM values are consistent for all data (FWHM 53-63°). Further analysis of the angular mass flow distribution is proposed later in the text helped with a Monte Carlo code. Next, the calcium evaporation rate was measured as a function of the oven temperature in the experimental condition (2) (see Fig. 3). The total calcium flux is reconstructed as a function of the temperature using the experimental measurement of 𝑓 (𝜃), by applying the following

OVEN THERMODYNAMICS AND ANALYSIS

The aperture hole surface of the crucible (0.78 𝑚𝑚 2 ) is small compared to its internal total surface (𝑠 𝑐 = 2.2𝑐𝑚 2 ). See fig. 3, view B to visualise the difference. The cavity is thus sufficiently closed to consider it as a Knudsen cell [START_REF] Knudsen | Die Molekularströmung der Gase durch Offnungen und die Effusion[END_REF][START_REF] Knudsen | Die Gesetze der Molekularströmung und der inneren Reibungsströmung der Gase durch Röhren[END_REF]. Consequently, when the metal sample is heated, the local pressure in the crucible raises rapidly to reach the saturating vapor pressure, well above the residual pressure of the vacuum chamber ( 𝑃 0 = 10 -5 Pa). The calcium saturation vapor pressures 𝑃(Pa) follow the Antoine's law as a function of the temperature :

𝑙𝑜𝑔 10 𝑃 = 𝐴 - 𝐵 𝐶 + 𝑇 (4)
where 𝑇 (𝐾) is the metal temperature and 𝐴, 𝐵, 𝐶 are thermodynamics parameters, unique for each chemical element.

The evaporated mass rate 𝑀 emitted from the solid metal in the crucible can be expressed using the Hertz-Knudsen equation [START_REF] Hertz | Ueber den Druck des gesättigten Quecksilberdämpfes[END_REF][START_REF] De) Martin Knudsen | Experimentelle Bestimmung des Druckes gesättigter Quecksilberdämpfe bei O° und Höheren Temperaturen[END_REF] :

𝑀 = 𝑃 𝑚 2𝜋𝑘𝑇 𝑆 ( 5
)
where 𝑚 is the atom mass, 𝑃 is the metal saturating vapor pressure and 𝑆 is the surface of evaporation. Another important oven parameter is the sticking time of atoms on the hot crucible surfaces defined by the Frenkel equation [START_REF] Frenkel | The Theory of Adsorption and Related Phenomena[END_REF]:

𝜏 = 𝜏 0 𝑒 𝐻 𝑘𝑇 (6)
where 𝜏 0 ≈ 10 -13 𝑠 is the atom vibration period on the surface [START_REF] Glebovsky | Recrystallization in Materials Processing[END_REF], 𝐻 is the enthalpy of the adsorbed atom, 𝑘 the Boltzmann constant and 𝑇 the surface temperature. For the calcium case, two sticking times must be considered. the first is 𝜏 𝐶 𝑎-𝐶 𝑎 , the sticking time of calcium atom on the metallic Ca surface sample. the other is 𝜏 𝐶 𝑎-𝑀 𝑜 , the sticking time of Ca on Mo. The adsorbed Ca enthalpy on Ca is well documented : 𝐻 𝐶 𝑎 = 1.6𝑒𝑉, giving a sticking time of 1 -1000𝑚𝑠 for 𝑇 600 -900𝐾 . On the other hand, the adsorbed Ca on Mo enthalpy value 𝐻 𝐶 𝑎-𝑀 𝑜 was not found in the literature. Nevertheless, 𝐻 𝐶 𝑎-𝑀 𝑜 is expected to be much higher than 𝐻 𝐶 𝑎 . A rough estimate of 𝐻 𝐶 𝑎-𝑀 𝑜 value can be considered from 𝐻 𝑆𝑐-𝑀 𝑜 = 5.5𝑒𝑉 available in [START_REF] Eichler | Adsorption of Volatile Metals on Metal Surfaces and its Application in Nuclear Chemistry, Calculation of Adsorption Enthalpies for Hypothetical Superheavy Elements with Z around 11[END_REF]. In the present experimental conditions, 𝜏 𝐶 𝑎-𝐶 𝑎 ≪ 𝜏 𝐶 𝑎-𝑀 𝑜 on all the temperature range covered. Hence, when the calcium evaporation starts, a layer of calcium should first be formed on the Mo cruciblewalland staystuckon Mo. Once the layeris complete, nextCa adsorptionon the surfaceis done on Ca, whichstronglyreduces the stickingtime and transformsthe cruciblesurfaceto afresh extrasource of calcium. Twoexperimentalconfirmationsof this phenomenon have been observed:(i) at firststart,the calcium flux is verysmall foragiventemperature and, after asufficientlylong time of operation (of the order of an hour), the flux amplifies to reachamuchhigher stablevalue.(ii) Immediatelyafter venting the ovenused and inspecting its crucible,the presence of aCa layeris visible on the cruciblesurfacewhichvery rapidlygetoxidisedto formawhite CaO powder.This hypothesis is furtherinvestigatedhelped with the experimental data from experiment(1) (see Fig. 3).The theoretical mass flowexpectedfrom Eq.4 and Eq.5 using Ca data from [START_REF] Dushman | Scientific foundations of vacuum technique[END_REF] is calculated forthe temperatures 848, 873 and 898 K, consideringthe twoevaporatingsurfaces𝑆=𝑠and 𝑆=𝑠 𝑐 +𝑠 and reportedin Tab. 3along with the experimentalmass flowfrom Tab. 1. Clearly,one can see that the sole metal sample evaporatingsurface(column with 𝑆=𝑠)is insufficient to reproduce the total mass rate from the oven.But when the cruciblesurfaceis added (column with 𝑆=𝑠 𝑐 +𝑠), the matchingbetweentheoryand experimentbecomes very close. The measurement confirmsthe hypothesisof the transient formationof aCa layeron the cruciblesurfacewhich, once completed enhances the evaporationproportionallyto the crucible surface. Result of experiment in condition (2) (see Fig. 3) is next compared with the prediction of eq.5 in the fig. 5. The sensitivity to temperature comes here through the pressure and the Antoine's eq.4. The experimental data is the blue errorbar, the mass flow prediction according to the eq.4 using the known tabulated coefficients for calcium, (see Tab.2), is plotted in solid red line. One can see a fairly good fit with experimental data up to 600°C (873K). Above this value, experimental flux is lower than the semi-empirical prediction. Because the total flux has been reconstructed with a sufficiently high precision, alternative Antoine's coefficients are proposed for calcium to fit the data resulting in the orange solid line. The fit value found for the whole temperature range are 𝐴 = 8.978 ± 0.07, 𝐵 = 7787.5 ± 110 and 𝐶 = 0 in standard units.

MONTE CARLO SIMULATION

The atom angular distribution from the oven has been investigated by means of a Monte Carlo simulation and compared with the experimental data. The exact 3 dimensions crucible geometry is considered. The oven temperature 𝑇 is a free parameter and is assumed to be uniform over all the crucible. The metallic sample geometry is not modelled and the initial atom emission position is done randomly along a line following the bottom part of the crucible part (a). The atom emission from the wall follows the Lambert's cosine law:

𝑃(𝜃) = 𝑐𝑜𝑠𝜃 (7)
where 𝑃(𝜃) is the probability of emission at an angle 𝜃 with respect to the local normal to the wall. A special care must be taken to inverse this probability distribution function appropriately to use it safely in the Monte-Carlo code [START_REF] Greenwood | The correct and incorrect generation of a cosine distribution ofscattered particles for Monte-Carlo modelling of vacuumsystems[END_REF].

Because the atom emission from the wall is done in a cavity with convex walls, a test is added in the code to check if a fresh reemission occurs toward the cavity and not to the wall. The atom velocity is taken as the mean thermal velocity:

𝑣 = 3𝑘𝑇 𝑚 ( 8
)
where 𝑚 is the considered atom mass. No accomodation from the wall is considered as particles are assumed to be isothermal. Each new adsorption at the wall is counted and re-emission is immediate using the cosine law. The pressure in the part (a) of the crucible is considered constant at the saturating vapor pressure 𝑃 of eq.4. The pressure in the extraction channel (b) is assumed to decrease linearly from 𝑃 down to the vacuum chamber pressure 𝑃 0 = 10 -5 Pa. The atom collisions are modelled in the crucible using the atom mean free path 𝜆:

𝜆 = 𝑘𝑇 √ 2𝜋𝑙 2 𝑃 ( 9
)
where 𝑙 is the atom diameter (𝑙 ≈ 360𝑝𝑚 for calcium). At each time step, the local pressure is considered and a random number is generated to check whether the atom collides or not. In case of collision, a new atom velocity direction is randomly generated on a uniform sphere. The metallic gas regime in the oven can be assessed with the Knudsen number:

𝐾 = 𝜆 𝑑 ( 10
)
where 𝑑 is a characteristic length of the system studied.

When 𝐾 > 0.5, the gas is collisionless and atoms exiting the oven are directly those emitted from the walls. An intermediate regime occurs when 0.01 < 𝐾 < 0.5 where the volume collisions start to play a role and finally, when 𝐾 < 0.01, the gas becomes fully collisional and the effect from the wall is secondary. In our case, the crucible has 2 characteristic lengths and hence two Knudsen numbers: 𝐾 𝑖 = 𝜆 𝑑 1 with 𝑑 1 =5 mm for the main cylinder and 𝐾 𝑖𝑖 = 𝜆 𝑑 2 with 𝑑 2 =1 mm for the crucible exiting channel. In this latter case, we considered the mean pressure in the exit channel Nevertheless,forT=900K, 𝐾 𝑖 =0.63is at the threshold and collision effect, thought not dominant should startto play arole. At1000K, the calcium gasis collisional in the bulk area (a) while it remains abovethe transition in the exiting channel.afullydevelopedcollision regime is reachedat 1100K with 𝐾 𝑖𝑖 =0.12.The differential angular distribution of exitingatom per unit of solid angle is displayedin Fig. 6 as afunction of the temperature. The number of particles generatedto produce the curvesis 1million. One can clearly see the transition from the non collisional regime at 600K (see blackcurve)to the intermediateregime at 1000K where the bulk crucibleis collisional while the exitchannelis not (see orangecurve)and finallyabove1100K where the whole ovenvolumeis collisional and the extractionemittance is finallydefined bythe sole exitingchannelgeometry(see blue and purple curves).

COMPARISON OF SIMULATION AND EXPERIMENT

The oven Monte Carlo code is used to simulate the theoretical atom flux expected on the micro scale as a function of the angle 𝜃 (see fig. 3). An example of the comparison between simulation and measurement is proposed on Fig. 8. The simulation reproduces well the general shape of the differential flux as a function of the angle 𝜃. A discrepancy is nevertheless visible for angle larger than 45°where the measurements are higher than the simulation. the difference is nevertheless included in the error bar. The differential flux FWHM is calculated for the simulation and compared with the experimental measurements. The results of the simulation are consistent with the measurements. the experimental FWHM is always higher than the simulation. This systematic difference is likely due to a non simulated physical effect. One could suggest for instance that the actual oven extraction geometry is not exactly as simulated, or that the atom flux continues to collide at the exit of the oven on the first mm. As a conclusion, The Monte Carlo code developed provides satisfactory results compard to experimental measurements and will next be used to study the metallic atom capture in ECR ion source plasma.

Figure 1 :

 1 Figure 1: (A) Calcium oven cut away view . (B) Detailed view of the crucible in light purple.

Figure 2 :

 2 Figure 2: Cutaway view of the vacuum chamber, the rotatable quartz measurement system (blue) and the oven tested (red).

Figure 3 :

 3 Figure 3: Sketch showing the positions of the quartz balance during the experiments. (1) The quartz describes a circle of radius 60 mm around the oven exit. The quartz and the oven axis form an angle with -𝜋/2 ≤ 𝜃 ≤ 𝜋/2. (2) The quartz is set 10 mm away from of the oven hole.

Figure 4 : 1 0

 41 Figure 4: Experimental flux emitted from the oven as a function of the angle 𝜃 for calcium. The black, red and blue plots are respectively measured at the temperatures 850, 875, 900°K.

Figure 5 :

 5 Figure 5: Total calcium flux from the oven as a function of the temperature. Blue symbols: experimental data. Red: theoretical data with 𝑆 = 𝑠 + 𝑠 𝑐 and Ca data from [4]. Orange: least mean square fit of experimental data using 𝑆 = 𝑠 + 𝑠 𝑐 and Antoine's law (see. Eq. 4) coefficients 𝐴 and 𝐵 as fit parameters

Figure 7

 7 Figure7shows the evolution of the mean numbers of bounce and collision versus the temperature. The temperature at which the volume collision becomes dominant is

Figure 7 :

 7 Figure 7: (Black) Mean number of atom adsoption on the crucible wall before atom extraction. (Dashed Black) Number of volume collision in the oven before extraction.

Figure 8 :

 8 Figure 8: Comparison between the experimental differential mass flux measured in the micro-scale and the Monte-Carlo simulation at T=898K.

Table 5 :

 5 Comparison between FWHM (in degree) of Angular Differential Mass Flux Measurements and Simulation for Calcium T (K) exp. (°) simulation (°) 848 53.7 ±7 48.6±0.6 873 55.5 ±7 51.4±0.6 898 62.3 ±7 54.45±0.6 24th Int. Workshop on ECR Ion Sources ECRIS2020, East Lansing, MI, USA JACoW Publishing ISBN: 978-3-95450-226-4 ISSN: 2222-5692 doi:10.18429/JACoW-ECRIS2020-TUXZO03 TUXZO03 Content from this work may be used under the terms of the CC BY 3.0 licence (© 2019). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI

Table 1 :

 1 Mass Flow and FWHM Measurements from Protoc ol(2) (see Fig.3) element T (K) 𝑀 (𝑛𝑔/𝑠) FWHM (deg.)

	Ca	848	148.9±67	53.7 ±7.3
	Ca	873	352.1±67	55.5 ±7.2
	Ca	898	649.3±67	62.3 ±4.2

Table 2 :

 2 Calcium Antoine's Equation Coefficient Accordi ngto [4] and Calculated in this Study, Expressed in Stan dardunits. Ca this work 8.98 ± 0.07 7.79 ± 0.11 × 10 3 0

		A	B	C
	Ca from [4]	10.34	8.94 × 10 3	0

Table 3 :

 3 Theoretical Integrated Mass Flow from the Oven in(ng/s) Derived from Eq.4 and 5, Compared with the Experimental Flux for T=848, 873 and 898 K

	T	P	𝑀 (𝑠) 𝑀 (𝑠 + 𝑠 𝑐 )	𝑀 Exp.
	(°K)	(Pa)	ng/s	ng/s	ng/s
	848 0.627 47.7	179	148.9 ± 67
	873	1.26	94.5	354	352.1 ± 67
	898	2.42	179	671	649.3 ± 67

  Normalizeddifferential distributionof atoms at the exitof the ovenper unit of solid angle as afunction of 𝜃 fordifferent temperatures:600K (black),900K (red), 1000K (orange),1100K (blue), 1200K (purple).to calculate 𝜆.The evolutionsof 𝜆,𝐾 𝑖 and 𝐾 𝑖𝑖 are proposed as afunction of 𝑇and 𝑃in the table 4. Forthe measurements performedon calcium up to 900K, one can deduce that the calcium gasbehaviouris mainlynon-collisional.

	24th Int. Workshop on ECR Ion Sources	ECRIS2020, East Lansing, MI, USA	JACoW Publishing
	ISBN: 978-3-95450-226-4	ISSN: 2222-5692	doi:10.18429/JACoW-ECRIS2020-TUXZO03
	Figure6:
				TUXZO03
	Fundamental Processes

1 2 (𝑃 + 𝑃 0 )

Table 4 :

 4 Evolution of the Atom Mean Free Path 𝜆and the two Knudsen number 𝐾 𝑖 𝐾 𝑖 fo th Calciu Oven as a Function of the Temperature 𝑇 and the Saturating

	Vapor Pressure 𝑃
	T	P	𝜆	𝐾 𝑖	𝐾 𝑖𝑖
	(K)	(Pa)	(m)
	600 2.75 × 10 -5	194.4	38895 285348
	700 3.70 × 10 -3	1.68	337	3365
	800	0.146	0.0488	9.76	97
	850	0.66	0.011	2.2846	22.8
	900	2.55	0.0031	0.63	6.3
	1000	25.19	3.55 × 10 -4 0.071	0.71
	1100	163.2	6.01 × 10 -5 0.012	0.12

Content from this work may be used under the terms of the CC BY 3.0 licence (© 2019). Any distribution of this work must maintain attribution to the author(s), title of the work, publisher, and DOI


Geoffray Battiston

Dominique Beauvois

Gilles Duc

Emmanuel Godoy

Study of the physiological parameters of a regulated oxygen mask (updated version)

Keywords:

This paper is an extension to a previous work concerning the study of a regulated oxygen mask which distributes oxygen in response to an inhalation demand and shows a vibratory behaviour after a certain constant flow demand value. It was shown that it was a chattering effect due to the appearance of an unstability of the model equilibrium trajectory. After a brief summary of the previous results, a focus is made on the physiological parameters of the regulated mask which limit any modification of the mask.

Introduction

The regulated oxygen mask is a purely mechanical device which distributes oxygen in response to an inhalation demand. During an inhalation, the mask volume is depleted, its pressure decreases and an oxygen distribution circuit is activated to compensate this demand. Over a certain demand flow value, the system starts chattering.

There is a classical methodology for studying pneumatic chattering systems. It consists of performing a linear stability analysis to infer the stability of the system equilibrium trajectories and a nonlinear stability analysis (with the use of normal form reductions, center manifold reductions and Lyapunov exponents) to study the trajectories of the system before a collision happens. This event is called a grazing bifurcation. Examples for classical pneumatic components like pressure-relief valves can be found in [START_REF] Macleod | Safety valve dynamic instability: an analysis of chatter[END_REF] [START_REF] Maeda | Studies on the dynamic characteristic of a poppet valve: 1st report, theoretical analysis[END_REF][3] [START_REF] Licsko | Nonlinear Analysis of a Single Stage Pressure Relief Valve[END_REF].

In [START_REF] Battiston | Stability analysis of a regulated oxygen mask[END_REF], only the linear stability analysis was performed. First, a physical model for the regulated oxygen mask was proposed. Then, the equilibria of this nonlinear model were found. Since the mask vibratory behaviour appears after a certain demand flow value, it was chosen to analyse the evolution of the eigenvalues of the linearized varying model to check if some of them were positive after this threshold. It was indeed the case. Simulation showed a valve diverging and striking its seat, which is the chattering effect. The application of the Routh criterion allowed to find inequalities on physical parameters to ensure the stability of the regulated oxygen mask. Among various strategies, it appeared that the increase of the value of a spring stiffness was sufficient to satisfy the Routh criterion. This increase was tested experimentally and it appeared that the trajectory of the regulator was sufficiently stabilized to avoid the chattering. Consequences of this increase like the decrease in the mask pressure, the increase of the amplitude of transient oscillations were observed and partly explained.

One conclusion of the previous work showed the improvement of the mask had to be made with attention to the respect of the physiological confort of the inhalating person. In this paper, we start by quoting the useful results from [START_REF] Battiston | Stability analysis of a regulated oxygen mask[END_REF] in section 2, and study this confort matter in section 3, by analysing the important physiological parameters of the mask. The principles of an optimisation programming to improve the stability of the mask and its physiological performances are quickly presented in section 4.

Conclusions from previous work

This section quickly summarizes the results from [START_REF] Battiston | Stability analysis of a regulated oxygen mask[END_REF]. It first focuses on the physical model of the mask in subsection 2.1. Then, it deals with the equilibrium trajectory of the mask in subsection 2.2 and the Routh conditions to have it stable in subsection 2.3.

Physical model of the mask

Ṗ m = rT V m (W i + W p ) ( 1
)
where T is the temperature of the oxygen, which is considered constant in every subsequent volume, and r is the oxygen gas constant (for simplicity, it is supposed that there is only oxygen in the mask volume, which is not true in practice because of the exhalation of carbon dioxide from the equipped person).

The comparator is a rigid disk of section S m , mass m, tied to a k-stiffness spring and a viscous damper f . It can close a tube of internal section S s from which oxygen can evacuate. The distance between the orifice of this tube and the comparator is x, and equals zero when the tube is closed. ẋ is the comparator speed. Exterior pressure P e and pressure P m apply on the surface S m . On surface S s are applied pressure P p and pressure P m . It can be shown (see [START_REF] Battiston | Stability analysis of a regulated oxygen mask[END_REF] for the details) that the equation of the comparator can be written as:

mẍ + f ẋ + kx = S m (P e -P m -dP 0 ) + S s (P p -P s ) (2)
where dP 0 is the comparator opening threshold. Its role will be studied later.

The evolution of the pilot chamber pressure P p depends on the mass conservation equation:

Ṗp = rT V p (W c -W s ) (3)
where V p is the pilot chamber volume. W s is the mass flow exiting this chamber through the comparator orifice:

W s = k s xP p (4)
where k s is the tube flow constant for a critical compressible flow. W c is the mass flow going through an isothermal Hagen-Poiseuille restriction of coefficient k c :

W c = k c (P 2 s -P 2 p ) (5)
The regulating actuator takes a measurement of P p and sends a mass flow W i proportionnal to it:

W i =
k i (P s -P t -P p ) , P p ≤ P s -P t 0 , P p > P s -P t

where P t is the threshold for P s -P p to start delivering a mass flow.

Equilibrium trajectory of the mask

We use equations (1-6) to write the state equations:

Ẋ =     ẋ -k m x -f m ẋ -Sm m P m + Ss m P p + H -k i rT Vm P p + rT Vm W p + rT Vm k i (P s -P t ) rT Vp (-k c P 2 p -k s xP p ) + rT Vp k c P 2 s     (7)
in which the state vector is defined by:

X =     x ẋ P m P p     (8)
and H is given by:

S m (P e -dP 0 )/m -S s P s /m (9)
Equilibrium state variables are written with a bar:

Pm = S m (P e -dP 0 ) + S s ( Pp -P s ) -kx /S m (10) Pp = W p /k i + P s -P t (11) x = k c /k s (P 2 s / Pp -Pp ) (12)
It was shown in [START_REF] Battiston | Stability analysis of a regulated oxygen mask[END_REF] that two eigenvalues of the 4-dimension evolution matrix of the model linearized around equilibrium state variables were having a positive real part value, which was coincident with the diverging position of the comparator. This one was then colliding with its seat. The time of the first strike was the same as the time of the beginning of chattering. An example is shown in Fig. 234.

Results from the appplication of the Routh criterion

After linearizing the state equations around the equilibrium trajectory and considering the characteristic polynomial of the obtained evolution matrix, it is possible to apply the Routh criterion to find inequalities on the physical parameters to make the mask stable during an inhalation. The first inequality was:

kf m 2 - rT k s S s V p m Pp > 0 (13)
The second inequality was obtained after some complementary assumptions which will not be described here:

rT k i S m V p m 2 f m + 10rT k c P s 3V p 2 < R (14)
with:

R = rT P s (4k c + k s S p ) V p m 2 + rT P s (2k c + k s S p ) V p m . k m + rT f V p m k c 2P s . f m + rT V p k c 2P s (15)
A sufficient condition to ensure these inequalities was to increase the spring stiffness k. This suggestion was tested experimentally, and made the mask stable in simulations and less vibrating in experiments, at the cost of degraded physiological performances [START_REF] Battiston | Stability analysis of a regulated oxygen mask[END_REF].

Study of the mask physiological parameters

Paper [START_REF] Battiston | Stability analysis of a regulated oxygen mask[END_REF] was more focused on the linear stability analysis of the regulated oxygen mask. Presently, we will study the mask physiological parameters, a short name to call physical quantities of the mask which have a physiological influence on a breathing person. There are 4 in total: the opening time t op (section 3.1), the closing time t cl (section 3.2), the equilibrium mask pressure values Pm (section 3.3), the mask pressure overshoot P * m in response to a step-demand (section 3.4). Some of them cannot be derived analitically, so we will sometimes have resort to numerical approximations.

Opening time

The opening time t op is defined as the time between the beginning of an inhalation and the beginning of the oxygen distribution. It must not be too large for the inhalating person to receive oxygen as soon as possible. This value is difficult to obtain analytically. It is estimated through some assumptions which will be presented later. At t = 0, P m = P e , and for the very beginning of an inhalation, the demand is approximated with a ramp function:

W p = -wt (16)
with w the positive slope coefficient of the ramp. The comparator is closed as long as P m > P e -dP 0 . The only working equation is [START_REF] Macleod | Safety valve dynamic instability: an analysis of chatter[END_REF]. Its solution is:

P m (t) = P e - wrT 2V m t 2 (17)
The comparator opens at time t co :

t co = 2V m dP 0 wrT (18)
The next time that is needed is the time t ac between the opening of the comparator and the one of the actuator activation. It is needed to solve (2) and ( 3) at the same time, which is difficult and may lead to a transcandental equation in t ac . Simulations of P m , x and P p are shown in Fig. 567. It can be observed in Fig. 8 that for the actual mask parameter settings, the dynamics of the comparator can be neglected. Then, equation ( 2) can be simplified: Since the actuator is still not sending oxygen, equation ( 17) is used in (19), yielding to:

x = S m k (P e -dP 0 -P m ) + S s k (P p -P s ) (19) Ṗp = rT V p (k c (P 2 s -P 2 p ) -k s xP p ) (20
Ṗp = rT V p k c (P 2 s -P 2 p ) -k s P p S m k ( wrT V m t 2 -dP 0 ) + S s k (P p -P s ) (21)
We change the time origin by writting t = t + t co . The mask pressure can be rewritten:

P m = P e - wrT 2V m (t + t co ) 2 (22)
This is reinserted in (21):

Ṗp = rT V p k c (P 2 s -P 2 p ) -k s P p S m k ( wrT V m (t + t co ) 2 -dP 0 ) + S s k (P p -P s ) (23)
Which we write for simplicity: 2 and where in e the factor P p has been approximated with P s . This approximation is linked to the fact that in practice P s -P p < 0.1P s .

Ṗp -bP 2 p + cP p + d -e(t + t co ) 2 (24) with b = rT Vp (k c + ksSs k ), c = rT ks kVp (S m dP 0 + S s P s ), d = rT Vp k c P 2 s , e = -rT Vp ksPsSm k wrT 2Vm (t +t co )
Solving this Riccati equation is very difficult. This difficulty can be overcome with numerical approximations. The pilot pressure P p will be visible on Fig. 9. P p seems to be quadratic from t co . We want P p to approximate the following time equation:

P p = P s -at 2 (25)
This approximation is inserted in the differential equation ( 24).

-2at -bP

2 s + cP s + d -et 2 co -2et co t + (2bP s a -ca -e)t 2 (26)
where t 4 has been neglected. After a short calculus, it can be shown that the constant term -bP 2 s + cP s + d -et 2 co is equal to 0. a is a constant, so we need to get rid of t from a = (2et co + et )/((2bP s -c)t + 2). In the first case (low t ):

a 1 = et co (27)
In the second one (high t ):

a 2 = e 2bP s -c (28)
These two approximations of P p are plotted in Fig. 9. It appears the first approximation is better that the second one. From there, we seek for the time t ac at which the actuator opens, that is when P p = P s -P t : P s -P t = P s -a 1 t 2 ac (29) which gives:

t ac = P t a 1 (30)
ie:

t ac = kP t V p √ 2V m k s S m P s rT √ rT wdP 0 (31)
It appears that t ac doesn't take into account S s and k c . In fact, it can be checked that their influence is indeed not very remarkable during the fall of P p , see Fig. unchanged. In order to be able to use t ac for an eventual optimisation of the mask parameters, we use a rule of thumb by rewritting it:

t ac = α kP t V p √ 2V m k s S m P s rT √ rT wdP 0 ( 32
)
where α is a correcting factor to make correspond the real t ac and the approximated one in the case of the original parameter settings. This incidently supposes that the variations of the real t ac are followed by its approximating formula.

Deduction of the opening time:

The opening time can finally be written:

t op = t co + t ac (33)
In order to decrease t co , one can decrease V m and dP 0 . To decrease t ac , one can decrease k and P t for instance, and increase dP 0 and S m .

Closing time

The closing time t cl is the duration between the closing of the comparator and the end of oxygen distribution. If it is too high, oxygen still flows after the end of an inhalation and the beginning of the exhalation, which is disturbing for one who breathes in the mask. If the comparator closes its seat, the working equations for the mask are:

Ṗp = rT V p W c = rT k c V p (P 2 s -P 2 p ) ( 34
)
The analytical solution of this differential equation exists. It can be found using the separation of variables:

2P s rT k c V p dt = 2P s dP p (P s -P p )(P s + P p ) = dP p P s -P p + dP p P s + P p ( 35
)
This equation is then integrated between t = 0 and t, P p0 and P p :

P p = P s exp 2P s rT kc Vp t - Ps-Pp0 Ps+Pp0 exp 2P s rT kc Vp t + Ps-Pp0 Ps+Pp0 (36)
for a starting pilot pressure P p (t = 0) = P p0 . The mask stop sending oxygen at the moment the actuator is desactivated, that is to say for P p = P s -P t . We write t cl = {t, P p (t) = P s -P t }, the closing time. Solving (36) for t cl yields:

t cl = V p 2P s rT k c ln 2P s -P t P s + P p0 P s -P p0 P t (37)
The closing time increases with V p and P t and decreases with k c . However, one must not conclude making k c very large is a good idea: if the restriction is small, the opening of the comparator may not be sufficient enough to empty the pilot volume and then send oxygen: someone inhalating could suffocate. A compromise shall be found.

Mask pressure value

For the mask to be easy to breathe in, the equilibrium mask pressure Pm , needs to be as high as possible (high pressure gas flow more easily in lower pressure areas such as the lungs during inhalation). Using (10), ( 11) and (12), the expression of Pm is:

Pm = (P e -dP 0 ) + S s S m W p k i -P t - kk c k s S m P 2 s Wp ki + P s -P t -( W p k i + P s -P t ) (38)
It obviously depends of the demand flow W p . The derivative along W p is:

∂ Pm ∂W p = S s S m k i + kk c k s S m P 2 s k i Wp ki + P s -P t 2 + 1 k i (39)
It is always positive. Nevertheless, one must remember that the convention for W p is a flow entering the mask. Then, it is negative for an inhalation, which also means that Pm decreases with the increase of the demand. If Pm must not decrease too much with W p , one can decrease much as possible this derivative, that is to say increase k i , S m , or decrease S s , k or k c .

Mask pressure overshoot

The last physiological parameter is the mask pressure overshoot P * m . In this case, the demand is not set as a ramp, but a step W p,0 . In Fig. 11, we plot W i and -W p to see at which moment they cross each other, and observe this time t * coincides with the one at which P m reaches its overshoot. In fact, P m reaches its overshoot value P * m when Ṗ m = 0, that is to say W i = -W p,0 (if the demand was not a ramp, we woud not know exactly at which flow value the derivative of P m would be zero). According to (11), we can approximate:

P * p = P s -P t + W p,0 k i (40)
At this point, the mask dynamics is (stars refer to the value at t * ):

Ṗp * = rT V p k c (P 2 s -P * 2 p ) -k s x * P * p (41) mẍ * + f ẋ * + kx * = S m (P e -dP 0 -P * m ) + S s -P t + W p,0 k i (42)
That is:

P * m = P e -dP 0 + S s S m -P t + W p,0 k i - 1 S m (mẍ * + f ẋ * + kx * ) (43)
In order to get * m , the values of x * , ẋ * and ẍ * are needed. It seems difficult to get a sufficiently satisfactory analytical approximation for these derivatives. Nevertheless, the same approximation as (19) can be made. We finally get an approximation for the overshot P * m :

P * m = P e -dP 0 + S s S m (-P t + W p,0 k i ) - k k s S m P * p (k c (P 2 s -P 2 p ) - V p Ṗ * p rT ) (44)
We suppose |rT k c (P Fig. 12: Comparison between the first and second term of the last parenthesis of (44).

The last remaining problem is to get the value of Ṗp * . The same approximation as in (25) for P p is made:

P p = P s -at 2 (45)
where t + t 1 = t and t 1 is the time at which the comparator opens. We give it another name than t co to avoid confusion: these two times are not the same because the demand is a step and not a ramp anymore. This time, the coefficient a is directly set with the activation time t ac :

P p (t = t ac ) = s -P t = P s -at 2 ac (46) ie: a = P t t 2 ac ( 47
)
The time at which P p = P * p is defined by (approx. 40):

P s -at * 2 = P s -P t + W p,0 k i (48) so: t * = 1 a (P t - W p,0 k i ) (49)
This gives the last wanted term as a better approximation:

Ṗp * = -2 P t t 2 ac t * (50)
Equation ( 44) can finally be fully written:

P * m = P e -dP 0 + S s S m (-P t + W p,0 k i ) -2 k k s S m P * p V p rT P t t ac 1 - W p,0 k i P t (51)
Obviously, the time t ac is not the same for a step demand flow as the one for a ramp demand flow. Nevertheless, it is possible to get an approximation of it in the same way as for the previous case. One would have to use (24) and replace t co and e with their new values.

In order to decrease the overshoot, ie increase P * m (because P m decreases with a inhalation and the overshoot is bottom-oriented), one must decrease dP 0 , S s , P t , k or increase S m , k i , k s . These tendencies were confirmed with multiple simulations for different parameter settings.

Improvement of the mask behaviour

As a summary of this work, in order to improve the mask behaviour, it is needed to satisfy conditions (13) and (14) while decreasing the opening time (33), the closing time (37), the mask pressure overshoot (51) and increasing the equilibrium mask pressure (38). As it was already stated in a different way, modifying one parameter may not be sufficient and it is needed to solve an optimisation problem.

This matter was not yet tackled with. Nevertheless, it is possible to precise how it is going to be dealt with. Obviously, one point will concern the choose of the cost function which will at least take into the physiological parameters. Another element must be taken into consideration even if it is not obvious: the cost function must also integrate the size of the region of stability around the position of the minimum. As a matter of fact, the physical of the mask components are not mastered with full accuracy. If the stability-physiologically optimized region is narrow, we may end up with leaving it easily due to experimental settings inaccuracy. There also are uncertainties in the model (one about S s was explained in [START_REF] Battiston | Stability analysis of a regulated oxygen mask[END_REF]). We may find a less physiologically performant region but with more room for parameters settings. At last, one point concerns the geometry limitations for the mask which must be added as additional constraints to the stability conditions.

Conclusion

The stability analysis of the studied regulated oxygen mask performed in [START_REF] Battiston | Stability analysis of a regulated oxygen mask[END_REF] gave stability regions which needed to be refined to take into account physiological performance. The physiological parameters of the mask were given thanks to multiple assumptions in this paper. Future works will be devoted to an optimisation programming in order to find parameters configurations for which both performance and stability are reached.

Fig. 1 :

 1 Fig. 1: Definition sketch of the mathematical model (taken from [5]). Mass flows are drawn in red, the future state variables are written in green. The dashed line represents a pressure measurement.

Fig. 1

 1 Fig.1 represents a simplified diagram of the regulated oxygen mask to construct its mathematical model. The oxygen suppy is made at the opening with pressure P s . The flow demand from the inhalating person is W p . The flow distributed by

Fig. 2 :Fig. 3 :

 23 Fig. 2: Wp as a ramp demand.

Fig. 4 :

 4 Fig. 4: Mask pressure Pm vs time. Near 17s, after a flat transition, it starts chattering.

Fig. 5 :

 5 Fig. 5: Pm at the beginning of a ramp demand. At tco, Pm = Pe -dP0.

Fig. 8 :

 8 Fig. 8: Comparison between kx, f v = f ẋ and ma = mẍ at the beginning of a ramp demand.

Fig. 9 :

 9 Fig. 9: Comparison of Pp and its two approximations 1 and 2 with respective coefficients a1 and a2.

5 Fig. 10 :

 510 Fig. 10: Fall of Pp for different kc.

0 0 . 2 TimeFig. 11 :

 0211 Fig. 11: Plot of normalized Pm and Wi in response to a step demand -Wp. t * is more on the left that it appears.

  ) Pp at the beginning of a ramp demand. At tco, the pressure starts decreasing and at tac it reaches Ps -Pt which is the time at which the actuator is activated.

						kx
						fv
						ma
	Equivalent forces
	0
	0 t co 0 t co P s -P t P s Fig. 7: 0 t co P p	t ac t ac t ac	0.1 0.1 0.1	0.15 Time (s) 0.15 Time (s) 0.15 Time (s)	0.2 0.2 0.2	0.25 0.25 0.25	0.3 0.3 0.3

0 x Fig. 6: x at the beginning of a ramp demand. At tco, the comparator opens.


Chemotherapy-induced cachexia (CIC) is a severe muscle wasting syndrome and aggravates cancer-cachexia in patients [START_REF] Fearon | Definition and classification of cancer cachexia: an international consensus[END_REF] . Currently, there are no official guidelines for the management of muscle wasting. Cannabidiol (CBD) is a bioactive phytocannabinoid produced from Cannabis Sativa. Recent studies demonstrated the beneficial effect of CBD on maintaining muscle function and metabolism in obese models and mdx dystrophic mice [START_REF] Iannotti | Effects of non-euphoric plant cannabinoids on muscle quality and performance of dystrophic mdx mice[END_REF][START_REF] Bielawiec | Attenuation of Oxidative Stress and Inflammatory Response by Chronic Cannabidiol Administration Is Associated with Improved n-6/n-3 PUFA Ratio in the White and Red Skeletal Muscle in a Rat Model of High-Fat Diet-Induced Obesity[END_REF][START_REF] Bielawiec | Chronic Cannabidiol Administration Attenuates Skeletal Muscle De Novo Ceramide Synthesis Pathway and Related Metabolic Effects in a Rat Model of High-Fat Diet-Induced Obesity[END_REF] .

The aim of this study was to evaluate the effect of CBD on muscle atrophy in response to cisplatin, in a model of myotubes in culture.

After differentiation, C2C12 cells (a model of myotubes) were incubated in the presence of 50µM cisplatin for 24-48h with or without increasing dose of CBD (1-5µM). Cell viability was measured using a CellTiter-Glo® 3D Cell Viability Assay. Myotube diameter was analyzed by microscopy. Protein synthesis and proteolysis were measured by Western Blot using the SUnSET technique (puromycin incorporation into nascent proteins), and polyubiquitination of total proteins, respectively. Activation of Akt and S6K1 pathways and cleavedcaspase 3 levels were quantified by Western blot. Results were analyzed by one-way ANOVA and expressed as mean ± sem.

olivier.le-bacquer@inrae.fr These results demonstrate that, in a model of myotubes in culture, cannabidiol prevents cisplatin-induced cell death and myotube atrophy by maintaining protein homeostasis. This suggests cannabidiol could be useful in the treatment of cancer cachexia.

CONCLUSIONS

This study was supported by the INRAE AlimH department.