Yves Garrabos
email: garrabos@icmcb-bordeaux.cnrs.fr

Carole Lecoutre-Chabot

Samuel Marre

Romain Guillaument

Daniel Beysens
email: dbeysens@cea.fr

Inseob Hahn
email: inseob.hahn@jpl.nasa.gov

C Lecoutre

Crossover equation of state

Keywords: Critical crossover function, Parametric equation of state, Turbidity, Xenon

come

Introduction

In their experimental paper devoted to a first unambiguous evaluation of the corrections-toscaling in the susceptibility of xenon using turbidity measurements along the critical isochore, Güttinger and Cannell (GC) [START_REF] Güttinger | Corrections to scaling in the susceptibility of xenon[END_REF] (reference also labeled as I herein) concluded that any theoretical progress, which would effectively reduce the number of adjustable parameters, would be extremely valuable. Moreover, it seems that the GC turbidity data are the last published data studying a critical one-component fluid with the required experimental precision on Earth's gravity. It is thus the main objective of the present paper to show that the present theoretical status of the critical point phenomena can predict the GC turbidity data near the xenon liquid-gas critical point without adjustable parameter. The theoretical estimation can also be extended to any other one-component fluid for which the experimental location of its liquid-gas critical point is known in the p, V, T phase surface.

Indeed, it is now well-established (see, for example Ref. [START_REF] Anisimov | 11 Critical region[END_REF]). that the asymptotic singularity of the field and density properties of the one-component fluids can be described by the classical-to-critical crossover functions provided by the φ 2 2 field theory framework (see, for example Ref. [START_REF] Zinn-Justin | Quantum Field Theory and Critical Phenomena[END_REF]). Asymptotically close to the liquid-gas critical point, each onecomponent fluid exhibits the universal features predicted for the complete {d = 3, n = 1} universality class of the three-dimensional (3D), symmetrical, Ising like systems, where d is the space dimension and n is the dimension of the order parameter. Moreover, recent theoretical improvements [START_REF] Bagnuls | Classical-to-critical crossovers from field theory[END_REF][START_REF] Garrabos | Mean crossover functions for uniaxial 3D Ising-like systems[END_REF] have extended the applicability range of the crossover functions far from the critical point. Nevertheless, their fundamental interest remains their conformity with the accurate Ising-like universal values of the critical (leading and lowest confluent) exponents and (leading and first confluent) amplitude combinations [START_REF] Guida | Critical exponents of the N-vector model[END_REF], only valid very close to the φ 2 2 singular point. Consequently, only the crossover description within the so-called Ising-like preasymptotic domain (PAD) [START_REF] Bagnuls | Classical-to-critical crossovers from field theory[END_REF] can be precisely characterized by a limited number (3) of free, system-dependent, nondimensional parameters. In addition, this description in terms of field and density variables needs to define the energy and length scale units to make any physical properties in a dimensionless form, requiring thus the introduction of the measurable physical critical parameters such as the critical temperature T c , the critical pressure p c , the critical density ρ c , etc. The subscript c refers to a critical quantity. In that dimensionless scheme where the uniqueness of the length unit is essential [START_REF] Privman | Universal critical point amplitude relations[END_REF], the Ising-like singular behaviors within the PAD are then characterized by two leading amplitudes and one-confluent amplitude of the two-term Wegner like expansions [START_REF] Wegner | Corrections to scaling laws[END_REF]. This three-parameter characterization results from universal scaling laws and universal amplitude combinations with three independent universal exponents (two leading exponents and the lowest exponent of the confluent singularities) and, correlatively, the same number (three) of related fluid-dependent amplitudes.

In general, a renormalizable theory cannot predict the location of the critical point or estimate the true extension of the fluid PAD. Therefore, experiments remain the only way to validate the Ising-like behavior and to estimate the limited number of readily independent parameters that (i) localize the critical point and (ii) characterize the crossover phenomena. This challenging situation introduces a crucial difficulty in analyzing the experimental data obtained at "finite" distance to the critical point, i.e., in a field (and/or a density) range that can overpass the unknown extension of the PAD. Indeed, beyond the PAD, many other correction terms can also contribute to a Wegner-like expansion at any order beyond the first order term of the confluent corrections to scaling, leading to a non-conclusive Ising-like fluid characterization.

However, the thermodynamic information contained in the experimental location of the isolated liquid-vapor critical point on the experimental phase surface of equation of state p (T, V, N ) = ∂ A ∂ V T,N , permits to overcome the above challenge, as proposed in Refs. [START_REF] Garrabos | Facteurs d'échelle phénoménologiques pour la transition critique liquide-gaz des fluides purs[END_REF][START_REF] Garrabos | see also: Phenomenological Scale Factors for the Liquid-Vapor Critical Transition of Pure Fluids[END_REF][START_REF] Garrabos | Scaling behaviour of the fluid subclass near the liquid-gas critical point[END_REF]. The total Helmholtz free energy A is here expressed in terms of its three natural variables, namely, the temperature T , the total volume V , and the total number of particles N . After the normalization of the fluid description per constitutive particle, it is then thermodynamically equivalent to define a minimal number (four) of finite critical coordinates of the critical point location in the particle phase surface of equation of state p T, v p = ∂a p ∂v p T and to introduce a limited number (five) of asymptotic system-dependent parameters in a renormalizable theory [START_REF] Bagnuls | Nonasymptotic critical behaviour from field theory for Ising like systems in the homogeneous phase: theoretical framework[END_REF][START_REF] Bagnuls | Experimental data analysis on xenon above the critical temperature from nonlinear renormalization group[END_REF]. a p = A(T,V,N ) N is the Helmholtz free energy of the fluid particle, v p = V N is the volume occupied by the fluid particle and the subscript p then refers to a particle quantity. Then, the term-to-term mapping of the physical and theoretical singular behaviors within the PAD completes the critical thermodynamic description by introducing three well-defined "master" (i.e., constant) numbers for all the one-component fluids, which provides the master crossover functions [START_REF] Garrabos | Master crossover functions for onecomponent fluids[END_REF], especially for the crossover behaviors of the isothermal compressibility κ T ( τ * , ρ = 0) and the coexisting liquid-gas density difference ρ LV (| τ * | , ρ = 0), along the critical isochore. Here

κ T = -1 V ∂ V ∂ p T,N , τ * = T -T c
T c , ρ = ρ-ρ c ρ c , and ρ LV = ρ L -ρ V 2ρ c , where ρ L and ρ V are the coexisting liquid and vapor densities. We are thus able to estimate the PAD extension of any critical fluid, only using its four well-defined critical coordinates. In addition, while the correlation length ξ remains a non-explicit physical parameter in a thermodynamic description, we emphasize that this master description can also provide access to the crossover function ξ ( τ * , ρ = 0). In the above procedure, the xenon behaves as a standard (non-quantum) Ising-like fluid [START_REF] Bagnuls | Experimental data analysis on xenon above the critical temperature from nonlinear renormalization group[END_REF][START_REF] Garrabos | Characteristic parameters of xenon near its liquid-gas critical point[END_REF]. Accordingly, the comparison between the predicted master crossover behaviors (without adjustable parameters) and the experimental measurements of the turbidity of xenon are of basic interest.

Another well-developped approach in representing thermodynamic fluid-properties by a crossover formulation is provided by the crossover parametric model (CPM) [START_REF] Agayan | Crossover parametric equation of state for Ising-like systems[END_REF]. The latter model is issued from a transformed Landau expansion for the Helmholtz energy density, so-called the crossover Landau model (CLM), originally formulated by Chen et al. [START_REF] Chen | Crossover from singular critical to regular classical thermodynamic behavior of fluids[END_REF][START_REF] Chen | Global thermodynamic behavior of fluids in the critical region[END_REF]. In such a phenomenological crossover approach, Chen et al. have implemented a matchpoint method proposed by Nicoll et al. [START_REF] Nicoll | Crossover functions by renormalization-group matching: O( 2 ) results[END_REF][START_REF] Nicoll | Crossover functions by renormalization-group matching: three-loop results[END_REF][START_REF] Nicoll | Background fluctuations and Wegner corrections[END_REF]. The resulting crossover formulation also incorporates the accurate universal features of the Ising-like systems close to their critical point. It is then of importance for the renormalization-group methods to provide accurate matching for the Ising-like description within the preasymptotic domain, as well using the above master crossover functions as using the symmetrical CPM functional form. According to the Ising-like similarity, we can also define the crossover master model (CMM) of the one-component-fluid subclass, where the free Ising-like parameters in the CPM model are now calculated from the four well-defined critical coordinates of each pure fluid. Therefore, the comparison with the measured singular behaviors of xenon is important to validate the results predicted from symmetrical CMM. This basic comparison can also provide access to the thermodynamic description of the complete phase surface surrounding the vicinity of the liquid-gas critical point of any one-component fluid. If the correlation length is estimated without any supplementary adjustable parameter, we will then open a new route for the complete crossover description of the singular behaviors of the one-component fluid subclass, which will be expected valid for any point in the vicinity of the liquid-vapor critical point of the fluid phase surface.

We shall proceed as follows. In Sect. 2, we provide the main characteristic parameters of the GC experiment and we discuss the conditions to perform theoretical analysis of the turbidity in conformity with the two-scale-factor universality. In Sect. 3, we give the master theoretical estimation of the Ising-like singular behaviors 1 of ξ and κ T only using the four critical coordinates that localize the gas-liquid critical point of xenon. We define precisely the temperature range of the PAD extension where the three-parameter characterization of the fluid is made in accordance with the universal features of the {d = 3, n = 1} universality class. After a brief presentation in Sect. 4 of the fluid turbidity calculated from the Ornstein-Zernike (OZ) theory [START_REF] Ornstein | Accidental deviations of density and opalescence at the critical point of single substance[END_REF][START_REF] Ornstein | The linear dimension of density variations[END_REF], we apply in Sect. 5 the Puglielli and Ford functional form [START_REF] Puglielli | Turbidity measurements in SF6 near its critical point[END_REF] to estimate the xenon turbidity. From a detailed analysis of the asymptotic turbidity behavior when τ * → 0, we show that, at large temperature distance from T c , the contribution due to the confluent corrections to scaling provides the agreement with the experimental measurements within the claimed experimental uncertainty. We compare also ξ and κ T to the previous literature results along the critical isochore. In Sect. 6, we extend the comparison to the results predicted by the crossover master model of the equation of state (EOS), especially for the top-shape of the density coexistence curve, by introducing the master-consistent estimation of the adjustable CPM critical parameters. We conclude in Sect. 7.

Güttinger and Cannell Experiment

The following analysis is focused on ξ and κ T obtained analyzing the optical (diffusion and transmission) responses of a xenon critical cell illuminated by a laser beam. Indeed, the GC experiment provides very accurate measurements of the turbidity, i.e., the light attenuation intensity by transmission through a xenon layer of 19.5 mm thickness, and the light scattering intensity at θ = 13 • scattering angle. The reduced temperature range was 9.607 × 10 -5 ≤ τ * ≤ 1.001×10 -1 (i.e., 28 mK ≤ T -T c ≤ 29 K). The turbidity differences τ (T )-τ (T or ) from a selected turbidity value τ (T or ) at a reference temperature T or are given as a function of τ * in Table I of Ref. [START_REF] Güttinger | Corrections to scaling in the susceptibility of xenon[END_REF]. This table also reports the data of the relative susceptibility per mass unit ( χ ρ χ ρ (T or ) ) obtained from light scattering intensity measurements. To define χ ρ , we consider a system maintained at constant total volume V , characterized by the Helmholtz free energy per volume unit a T, n p = A(T,V,N ) V . In such a thermodynamic description, the chemical potential per particle μ p = ∂a ∂n p T and the chemical potential per mass unit

μ ρ = ∂a ∂ρ T = μ p
m p are the conjugated variables to the particle number density n p and the mass density ρ = m p n p , respectively. m p is the particle mass. The subscript ρ now refers to a specific quantity. Therefore, the inverse susceptibility per particle χ p -1 =

∂μ p ∂n p T = ∂ 2 a ∂n 2 p T
is associated to the particle number density while the inverse susceptibility per mass 1 Universal central values of the critical exponents used for ξ and κ T in this work are ν = 0.6303875 and γ = 1.2395935 (see Ref. [START_REF] Garrabos | Mean crossover functions for uniaxial 3D Ising-like systems[END_REF]). Accordingly, = 0.50189 is the lowest universal value of the confluent exponent.

unit χ ρ

-1 = ∂μ ρ ∂ρ T = ∂ 2 a ∂ρ 2 T
is associated to the mass density, with χ ρ = m p2 χ p , χ p = n p 2 κ T and χ ρ = ρ 2 κ T . Here, χ p (or χ ρ ) are expected symmetric around the critical density to be compared with the magnetic-like susceptibility function of the O(1) vector of the φ 2 2 field theory. In I, the χ ρ data at ρ = ρ c were obtained by the authors using the Ornstein-Zernike (OZ) theory [START_REF] Ornstein | Accidental deviations of density and opalescence at the critical point of single substance[END_REF][START_REF] Ornstein | The linear dimension of density variations[END_REF] to approximate the light scattering intensity due to the local fluid density fluctuations. It was simultaneously accounted for the singular behavior of the correlation length. Consequently, τ ( τ * ), ξ ( τ * ) and κ T ( τ

* ) = χ ρ ( τ * ) (ρ c ) 2
along the xenon critical isochore were stated to conform to the OZ theory in spite of its questionable validity when τ * → 0. Moreover, it is also noticeable that the universal features of the Ising-like systems are not completely accounted for in their ξ ( τ * ) and κ T ( τ * ) fitting functions, especially at the first order of the confluent corrections-to-scaling.

Alternatively, in the present approach, all these Ising-like universal features can be precisely controlled without any adjustable parameter through the non-dimensional master crossover functions ξ ( τ * ) and κ T ( τ * ) given in Ref. [START_REF] Garrabos | Master crossover functions for onecomponent fluids[END_REF] (see also Eqs. ( 1) and (2) below). Therefore, ξ ( τ * ) and κ T ( τ * ), (or χ ρ ( τ * ) and χ p ( τ * )), being known a priori, our following calculation of τ ( τ * ) is only depending on the theoretical functional form that connects ξ and κ T (or χ ρ , χ p ) to the light scattering intensity I (θ ) as a function of the scattering angle θ.

In a general fluid theory assuming spatial isotropy, the magnitude and the spatial character of the local density fluctuations are described in terms of the pair correlation function g (r ) or, equivalently, in terms of the static structure factor S (q), where q is the wavenumber amplitude of the momentum transfer wave vector. S (q) is proportional to the intensity I (q) of the scattered electromagnetic radiation by the transparent fluid f of refractive index n f , for the Bragg condition q = 4π n f λ sin θ 2 at (generally small) angle θ (see, for example Ref. [START_REF] Fabelinskii | I. Molecular Scattering of Light[END_REF]). The q-scattering experiments provide thus a distinction between light scattering experiments (typical wavelength range λ ∼ 500-700 nm) and X-ray or neutron scattering experiments (typical wavelength range λ ∼ 0.4-1 nm). Accordingly, S (q) is a measurable quantity for any local fluid theory able to calculate the thermodynamic equilibrium from the representation of the fluid microscopic structure by the radial distribution of the particles. We recall that S (0) is a purely thermodynamic quantity proportional to the scattered intensity I (0) at zeroangle (q = 0). In a one-component fluid description per particle, S (0) = n p k B T χ p , since the zeroth-moment of the pair correlation function is related to χ p , due to the fluctuationdissipation theorem.

In a critical fluid theory, the normalization of S(q) k B T appears essential in order to correctly account for the physical dimension of the selected symmetrized susceptibility 2 in pure fluids (see Ref. [START_REF] Levelt-Sengers | Scaling predictions for thermodynamic anomalies near the gas-liquid critical point[END_REF]). As previously mentioned, when the susceptibility χ p (or χ ρ ) appears symmetric around the zero value of the related order parameter [START_REF] Garrabos | Master crossover behavior of parachor correlations for one-component fluids[END_REF], the singular behaviors of a one-component fluid very close to the liquid-gas critical point must behave similarly to the ones of the O(1) vector model of the symmetric φ 2 2 field theory and the simple cubic scalar (N = 1) lattice model of 3D-Ising systems. In such an ideal Ising-like asymptotic description which complies with the two-scale factor universality when T → T c , the expected behaviors of the static structure factor [START_REF] Green | On the theory of the critical point of a simple fluid[END_REF][START_REF] Fisher | Correlation functions and the critical region of simple fluids[END_REF][START_REF] Bray | Predicted critical correlation function for three-dimensional phase transitions[END_REF][START_REF] Bray | Dispersion-theory approach to the correlation function for critical scattering[END_REF][START_REF] Ferrell | Determination of the anomalous dimension Green-Fisher critical exponent η by light scattering in a fluid[END_REF][START_REF] Martin-Mayor | Critical structure factor in Ising systems[END_REF], and the turbidity [START_REF] Ferrell | Determination of the anomalous dimension Green-Fisher critical exponent η by light scattering in a fluid[END_REF][START_REF] Martin-Mayor | Critical structure factor in Ising systems[END_REF], must involve only three asymptotic amplitude-exponent pairs (neglecting thus the contribution of the confluent sin-gularities in the Wegner-like expansions). These three asymptotic amplitude-exponent pairs are: i) + 0 , γ pair of the singular pure power law κ T = + 0 ( τ * ) -γ describing the asymptotic temperature dependence of the physical isothermal compressibility, ii) ξ + 0 , ν pair of the singular pure power law ξ = ξ + 0 ( τ * ) -ν describing the asymptotic temperature dependence of the physical correlation length, and iii) C + ∞ , η universal pair describing the Ising-like asymptotic dependence

G (x) = C + ∞ x 2-η of the universal scaling function for x → ∞. G (x) = S[q, τ * ,ξ ( τ * )] S[0, τ * ,ξ ( τ * )]
only depends on the single variable x = qξ (x is non-dependent of the normalization scheme of the physical quantities), while η is the so-called Green-Fisher critical exponent [START_REF] Green | On the theory of the critical point of a simple fluid[END_REF][START_REF] Fisher | Correlation functions and the critical region of simple fluids[END_REF].

In that asymptotic Ising limit, + 0 and ξ + 0 are used as the fluid-dependent parameters. The two-scale factor universality is accounted for by the universal scaling law γ = ν (2 -η) and the universal amplitude combination

Q 3 = Dρ,0 ξ + 0 2-η ρ 2 c + 0 ≡ C + ∞ = χ(q)ξ 2-η χ(0) = G (x) x 2-η
[ [START_REF] Fisher | Correlation functions and the critical region of simple fluids[END_REF][START_REF] Bray | Predicted critical correlation function for three-dimensional phase transitions[END_REF][START_REF] Bray | Dispersion-theory approach to the correlation function for critical scattering[END_REF][START_REF] Martin-Mayor | Critical structure factor in Ising systems[END_REF], where: i) γ and ν are the two leading critical exponents, ii) + 0 and ξ + 0 are the two entry physical amplitudes, iii) Dρ,0 , η is the resulting (physical) amplitude-(universal) exponent pair in the thermodynamic description per mass unit.

The critical point condition ξ ∼ ∞ at τ * = 0 and ρ * = 0 is here labelled by the hat decoration. Therefore, in momentum space at the critical point, the fluid-dependent amplitude Dρ,0 , of χ ρ (q, ξ = ∞) = Dρ,0 q 2-η can be estimated unambiguously from γ , ν, + 0 and ξ + 0 . Moreover, as ξ and κ T are hereafter predicted without any adjustable parameter, we are also able to verify that the temperature range τ * ≥ 9.6 × 10 -5 (T -T c ≥ 28 mK) covered by GC experiments is well beyond the upper PAD limit L Xe PAD 9 × 10 -5 (T -T c ≤ 26 mK, see below). It is also easy to show that the light scattering experiments at θ = 13 • , i.e., q = 2.23 × 10 6 m -1 , with λ 0 = 632.8 nm and critical refractive index of xenon n Xe,c = 1.1375 [START_REF] Chapman | Temperature and density dependence of the molar polarizability of xenon[END_REF][START_REF] Garside | Refractive index and Lorentz-Lorenz function of xenon liquid and vapour[END_REF], are covering the range 2.0 × 10 -3 ≤ qξ ≤ 0.14 (see below). The condition qξ < 1 being satisfied, the truncated Taylor serie expansion of the universal scaling function is expected similar to the OZ approximant [G (x)] -1 = 1 + x 2 . In such a situation, GC measurements can then be analyzed introducing the complete Wegner expansion given by a classical-to-critical crossover theory. As a final consequence, GC turbidity measurements can demonstrate unambiguously that the deviation from the Ising-like turbidity is entirely accounted for by the contribution of the confluent corrections to scaling.

Ising-Like Singular Behaviors of ξ and κ T for Xenon Case

The estimation of ξ and κ T following the method proposed in Ref. [START_REF] Garrabos | Master crossover functions for onecomponent fluids[END_REF] only needs to know 4 critical parameters of xenon, namely, the critical temperature T c , the critical pressure p c , the critical mass density ρ c , and the common critical slope

γ c = ∂ p ∂ T ρ=ρ c = dp sat dT T →T ± c
of the critical isochore (T → T + c ) and the saturation pressure p sat (T ) curve (T → T - c ). Their values, given in Table 1 (see Refs. [START_REF] Garrabos | Characteristic parameters of xenon near its liquid-gas critical point[END_REF][START_REF] Gillis | Thermoacoustic boundary layers near the liquid-vapor critical point[END_REF]), localize the vapor-liquid critical point of xenon on the p, v p , T phase surface.

v p = m p ρ . M mol = N A m p is the molar mass of the fluid. N A is the Avogadro's number with R = N A k B . R is the ideal gas constant and k B is
ρ c 1113 ± 3 kg m -3 " γ c 0.1197 ± 0.0006 MPa K -1 " (β c ) -1 = k B T c 4.0002 × 10 -21 J " α c = k B T c p c 1 d 8.81498 × 10 -10 m " Y c = γ c T c p c -1 4 .93846 " Z c = p c m p ρ c k B T c 0.28602 " n Xe,c
1.1375 [START_REF] Chapman | Temperature and density dependence of the molar polarizability of xenon[END_REF][START_REF] Garside | Refractive index and Lorentz-Lorenz function of xenon liquid and vapour[END_REF] L L c 10.5271 cm 3 mole -1 "

A 0 k B (88.49 ± 0.30) J K -1 m -4 " T or -T c 0.6677 K [1] χ ρ (T or ) 2.42 × 10 -7 kg 2 J -1 m -3 " k 0 1.12944 × 10 7 m -1 " τ (T or ) 4.1 m -1 " τ 0 7.975 × 10 -4 m -1 "
the Boltzmann's constant. From these critical coordinates we can calculate 4 unequivocal values of the:

(i) energy unit (β c ) -1 = k B T c ; (ii) length unit α c = k B T c p c 1 d with d = 3, where (α c ) d = k B T c
p c is the characteristic critical volume of the molecular interaction cell [START_REF] Garrabos | Facteurs d'échelle phénoménologiques pour la transition critique liquide-gaz des fluides purs[END_REF][START_REF] Garrabos | see also: Phenomenological Scale Factors for the Liquid-Vapor Critical Transition of Pure Fluids[END_REF][START_REF] Garrabos | Scaling behaviour of the fluid subclass near the liquid-gas critical point[END_REF]

; (iii) first scale factor Y c = γ c T c p c -1 of the dimensionless thermal field τ * = T -T c T c ; (iv) second scale factor Z c = p c m p ρ c k B T c of the dimensionless ordering field μ * p = μ p -μ p,c β c conjugated to the dimensionless order parameter density n * p = n p -n p,c (α c ) d . (Z c ) -1 = n p,c (α c ) d is
the characteristic critical number of particles filling the molecular interaction cell [START_REF] Garrabos | Facteurs d'échelle phénoménologiques pour la transition critique liquide-gaz des fluides purs[END_REF][START_REF] Garrabos | see also: Phenomenological Scale Factors for the Liquid-Vapor Critical Transition of Pure Fluids[END_REF][START_REF] Garrabos | Scaling behaviour of the fluid subclass near the liquid-gas critical point[END_REF].

We recall that μ

* p = Z c μ ρ and n * p = ρ Z c , where μ ρ = μ ρ -μ ρ,c ρ c p c and ρ = ρ-ρ c
ρ c are the practical conjugated ordering field and order parameter density, respectively (for a fluid of mass unit). Accounting now for the extensive nature of the fluid susceptibility as already made in Refs. [START_REF] Garrabos | Master crossover functions for onecomponent fluids[END_REF][START_REF] Garrabos | Characteristic parameters of xenon near its liquid-gas critical point[END_REF], the dimensionless susceptibilities

χ = ∂ ρ ∂ μ ρ τ * = χ ρ p c (ρ c ) 2 ≡ χ ρ (for a fluid of mass unit), and χ * = ∂ n * p ∂ μ * p τ * = χ p (α c ) d β c ≡ χ *
p (for a fluid particle), with χ ρ = (Z c ) 2 χ * p , are related to the dimensionless order parameter densities ρ and n * p , respectively. We underline here the importance of the length and energy normalizations for any thermodynamic description, which can then introduce the non-dimensional scale factor Z c (in analytic powered forms) when another length unit or energy unit is used in addition to α c and (β c ) -1 . That is precisely the case for a quantity decorated by a tilde, in spite of a similar non-dimensional potential density, since

a ρ = a p c = aβ c (α c ) d = a * .
Neglecting quantum effects [START_REF] Garrabos | Universality and quantum effects in one-component critical fluid[END_REF] in xenon case,

ξ * cal ( τ * ) = 1 α c × ξ ( τ * ) and κ * T,cal ( τ * ) = p c × κ T,cal ( τ * ) are thus computed from the following equations ξ * cal τ * = * T * = 1 Z {1 f } ξ [ th (t)] -1
(1)

Z c × κ * T,cal τ * = X * T * = 1 Z {1 f } χ [χ th (t)] -1
(2)

with t = {1 f } T * = {1 f } Y c τ * = ϑ τ * , i.e., T * = Y c τ * and ϑ = {1 f } Y c .
The superscript {1 f } recalls for a constant quantity which characterizes the one-component fluid subclass. In Eqs. ( 1) and ( 2), the master crossover functions * (T * ) and X * (T * ) [START_REF] Garrabos | Master crossover functions for onecomponent fluids[END_REF] correspond to the modifications of the theoretical crossover functions [ th (t)] -1 and [χ th (t)] -1 defined in Refs. [START_REF] Bagnuls | Classical-to-critical crossovers from field theory[END_REF][START_REF] Garrabos | Mean crossover functions for uniaxial 3D Ising-like systems[END_REF]. These modifications only introduce three scale factors {1 f } = 4.288 10 -3 , {1 f } = 1.74 10 -4 and L f = 25.6988, to characterize the onecomponent fluid subclass [START_REF] Garrabos | Master crossover functions for onecomponent fluids[END_REF]. Accordingly, the master values of the prefactors appearing in Eqs. ( 1) and ( 2) are [START_REF] Rabinovich | Experimental determination of the compressibility of krypton and xenon in the 300-720 K range and up to 400 bars[END_REF], while the ones of the theoretical and master leading amplitudes of the related crossover functions

Z {1 f } ξ ≡ L f = 25.6988 and Z {1 f } χ = L f d {1 f } 2 = 1950.
are Z + ξ -1 = 0.471474, Z + ξ = 0.5729, Z + χ -1
= 0.269571 and Z + χ = 0.11975, with

Z + ξ Z + ξ = L f {1 f } ν -1 and Z + χ Z + χ = L f d {1 f } 2 {1 f } γ -1
(the subscript + refers to the homogeneous domain T > T c ).

Equations 1 and 2 can be written in their usual forms ξ = ξ + 0 ( τ * ) -ν L ( τ * ) and κ T = + 0 ( τ * ) -γ K ( τ * ) where the asymptotic Ising power laws are corrected by the confluent crossover functions L ( τ * ) and K ( τ * ). The subscript refers to the lowest critical exponent = 0.50189 [START_REF] Bagnuls | Classical-to-critical crossovers from field theory[END_REF][START_REF] Garrabos | Mean crossover functions for uniaxial 3D Ising-like systems[END_REF]. Indeed, L ( τ

* ) = 1 + ∞ i=1 a i,+ ξ ( τ * ) i and K ( τ * ) = 1 + ∞ i=1 a i,+ χ ( τ * ) i
are the resummations of the complete Wegner-like expansions, so-called the -contribution in the following (see details in Refs. [START_REF] Bagnuls | Classical-to-critical crossovers from field theory[END_REF][START_REF] Garrabos | Mean crossover functions for uniaxial 3D Ising-like systems[END_REF]). The confluent singularities of the classical-to-critical crossover are only due to , with the universal ratio

a 1,+ ξ a 1,+ χ = Z 1,+ ξ Z 1,+ χ = Z 1,+ ξ Z 1,+ χ = 0.68.
This universal ratio means that only a 1,+ χ (or a 1,+ ξ ) is the first-order amplitude characteristic of the fluid, which acts as a single crossover parameter. ξ and κ T singular behaviors for xenon are then predicted from Eq. ( 1) and Eq. ( 2) to be in accordance with the universal features of the three-dimensional Ising-like universality class at the first-order of the confluent corrections (see Table 2).

Accordingly,

ξ = α c ξ + ( τ * ) -ν with ξ + 0 = α c ξ + = α c (Y c ) -ν Z + ξ = 0.184531 nm, (i.e., ξ + = 0.209338) and p c κ T = + ( τ * ) -γ with + = (Z c ) -1 (Y c ) -γ Z + χ = 0.
0578238, (i.e., + 0 = 9.901 MPa -1 ) (using standard notations and values of Ref. [START_REF] Garrabos | Mean crossover functions for uniaxial 3D Ising-like systems[END_REF]). The non-dimensional leading amplitude Dρ = ẐG Z c satisfies the universal amplitude combination [START_REF] Bray | Predicted critical correlation function for three-dimensional phase transitions[END_REF][START_REF] Bray | Dispersion-theory approach to the correlation function for critical scattering[END_REF]. Similarly, the non-dimensional leading

Q 3 ≡ C + ∞ = Dρ (ξ + ) 2-η + 0.92
amplitude B = (Z c ) -1 2 (Y c ) β Z M = 1.46762 involved in the power law ρ LV = B ( τ * ) β satisfies the universal amplitude combination R + C R + ξ -d = + B 2 ξ + -d = 2.
92922. There-

R + ξ = A + ξ + 1 d 0.2696 ± 0.0007 0.2659 -1.37 R + C R + ξ -d = + B 2 ξ + -d (2.92922) ( 3.0851) 5.32 ξ + ξ - (1.96) ( 1.96) ( 0) + -
4.79 ± 0.010 4.94 3.13 Residuals R % = 100

A + α A - α = A + A - 0.
× C P M M R -1 (column 4)
fore, considering the amplitude set ξ + , Dρ , + and B, any dimensionless amplitude value (ξ + and Dρ here above) can be estimated when two of them are known and selected as independent ( + and B in Ref. [START_REF] Garrabos | Characteristic parameters of xenon near its liquid-gas critical point[END_REF]), in agreement with the two-scale factor universality.

Relaying the GC conclusive sentence in I, the main interest of Eqs. ( 1) and (2) remains the estimation of the confluent corrections-to-scaling at finite distance to T c , only using a single crossover parameter associated to . In addition, the -contribution can be computed for the complete classical-to-critical crossover domain without any adjustable parameter, selecting for example a 1,+ χ as the independent first-order amplitude, which acts as a fluid crossover parameter. Since the critical density of the xenon cell was well-controled at the level of the illuminated volume under the field acceleration due to Earth's gravity, Eqs. ( 1) and ( 2) can be directly used to predict the GC results along the critical isochore and then to quantify the -contribution to ξ , κ T , and τ (see below Sect. 4), in the covered experimental range.

in the range τ * ≤ 5 × 10 -2 , corresponds to γ e computed from the GC fit κ T = 0.0577 τ * -1.241 1 + 1.29 τ * 0.496 -1.55 τ * 0.992 + 1.9 τ * 1.488 reported in I. The PAD curves show the exponent decrease only due to the first-order term of the confluent singularities, strictly valid within the preasymptotic domain such as τ * ≤

L Xe PAD = L {1 f } PAD Y c (Xe)
9×10 -5 (see the related arrow in lower axis and Ref. [START_REF] Garrabos | Master crossover functions for onecomponent fluids[END_REF] for the estimation

of L {1 f } PAD 4.43 × 10 -4
). The grey area, bracketed on the T -T c (K) axis, corresponds to the GC temperature range. The expected ξ -variation in I covers the range 63 nm ≥ ξ ≥ 0.9 nm while the γ e -and ν e -variations remain such as γ < γ e < γ +γ MF 2 1.12 and ν < ν e < ν+ν MF 2 0.565. That includes the extended asymptotic domain τ * ≤ L Xe EAD 2 × 10 -2 (see corresponding arrow in lower axis), which was already the object of a detailed analysis in Refs. [START_REF] Garrabos | Characteristic parameters of xenon near its liquid-gas critical point[END_REF][START_REF] Garrabos | Master singular behavior from correlation length measurements for seven one-component fluids near their gas-liquid critical point[END_REF]. We have also reported in Fig. 1a the γ e -values published from fitting κ * T data obtained before GC work using an effective power law κ * T = + e ( τ * ) -γ e over a finite, restricted temperature range 0 < τ * min ≤ τ * ≤ τ * max (see Appendix for the data sources). In Fig. 1a, each γ e value is shown as an horizontal i-segment of extension τ * min -τ * max (with label i related to line #i of Table 6). The position of the (full) circle in the segment corresponds to the geometrical mean value τ * exp = τ * min τ * max 1 2 . In the Ising-like side of the crossover where γ ≤ γ e < γ1 2 (left part of Fig. 1a), the circle follows the MR curve in a very satisfactory manner when τ * exp changes, at least for the extended asymptotic domain τ * ≤ L Xe EAD 2 × 10 -2 where the Ising-like nature of the crossover is prominent.

The left half part of Fig. 1a illustrates the experimental status at the mid-80's when was performed the first crossover analysis [START_REF] Bagnuls | Experimental data analysis on xenon above the critical temperature from nonlinear renormalization group[END_REF] of xenon singular properties with only 3 adjustable physical parameters, using the crossover functions [START_REF] Bagnuls | Experimental data analysis on xenon above the critical temperature from nonlinear renormalization group[END_REF] estimated from the MR scheme (see also below line 5 of Table 4). In such initial situation refering to a single characteristic scale factor ϑ associated to the critical isochoric line of xenon, the crossover behavior of any effective exponent appeared certainly governed by the Ising-like nature of the singularity due to the lowest confluent correction-to-scaling (anticipating then posterior analyses, see lines 6 to 8 of Table 4, which have lead to similar conclusions independently of the theoretical scheme at the origin of the crossover functions). Today, the master crossover prediction of the Ising-like critical scaling for γ e ( τ * ) is made without adjustable parameter and complies with the universal features of the MR crossover functions of Ref. [START_REF] Bagnuls | Classical-to-critical crossovers from field theory[END_REF].

However, the dotted lines labeled CO of Fig. 1 interpolating the γ +γ MF 2 (or ν+ν MF 2 ) and γ MF (or ν MF ) values, are also indicative of the abrupt experimental crossover, generally observed in pure fluids far from the critical point. The related double arrow labeled τ * CO in Fig. 1b, differenciates the high temperature range (right half part of Fig. 1) where a mean-field approximation of a fluid theory is probably questionable. Indeed, the vertical line labelled r LJ e indicates the temperature distance where the calculated correlation length behaves similar to the equilibrium position r LJ e = 0.4 nm ≈ α c 2 between a pair of xenon atoms interacting from a 12-6 Lennard-Jones form of the pairwise additive potential (with r LJ e = 2 1 6 σ LJ and σ LJ = 0.3607 nm [START_REF] Hirschfelder | Molecular Theory of Gases and Liquids[END_REF]). The related temperature T 550 K corresponds to a reduced temperature T * = k B T LJ ≈ 1.9 (i.e., τ * ≈ 0.9). However, for T * ≥ 2, it is well-known that treating the whole Lennard-Jones potential as a small perturbation of a hard sphere appears as a more realistic approximation than the mean-field approximation of the attractive molecular interaction, leading to expect the invalid physical meanning of Eqs. ( 1) and (2) for τ * ∼ 0.3 -1 [see for example the similar (master) Fig. 2 in Ref. [START_REF] Garrabos | Master singular behavior from correlation length measurements for seven one-component fluids near their gas-liquid critical point[END_REF]). Therefore, it was clearly shown from Fig. 1

Fluid Turbidity from OZ Theory

The scattered light intensity is caused by the Rayleigh scattering from large density fluctuations of size ξ . From OZ theory, it can be written as follows

I OZ (q) = A τ κ T sin 2 1 + (qξ ) 2 (3)
where q = 4π n λ 0 sin θ 2 represents the amplitude of the transfer wave vector between incident and scattering wave vectors. λ 0 is the light wavelength in vacuum, n is the refractive index of the fluid, and θ is the scattering angle. is the angle between the polarization wave vector of the incident light and the scattering wave vector.

A τ = A 0 k B T ∼ energy (length) 4 ,
where k B is the Boltzmann constant and where A 0 ∼ length -4 , can be written as

A 0 = π 2 λ 4 0 ρ ∂ n 2 ∂ρ 2 T = π 2 λ 4 0 n 2 -1 n 2 + 2 3 2 (4)
A 0 accounts for geometrical factors and light-fluid scattering cross section calculated from the Lorentz-Lorenz approximation of the effective local field, i.e., introducing the related

critical value L L c = M mol ρ c n 2 c -1 n 2 c +2
for the density expansion of the Lorentz-Lorenz function. The turbidity calculated by Puglielli and Ford [START_REF] Puglielli | Turbidity measurements in SF6 near its critical point[END_REF] corresponds to the integral of Eq. ( 3) over all the scattering angles and writes:

τ PF = π A 0 k B T c 1 + τ * κ T F(a), (5)
where the universal function F (a) is given by the following equation

F (a) = 2a 2 + 2a + 1 a 3 ln (1 + 2a) -2 1 + a a 2 (6)
with a = 2 (k 0 ξ ) 2 and k 0 = 2π n λ 0 (k 0 is the amplitude of the incident light wave vector of order of 10 -7 m -1 for λ 0 ∼ 630 nm). The function F (a) reaches a constant value F (a) ∝ 8 . Therefore, the asymptotic singular turbidity predicted by Eq. ( 5) exhibits an incorrect logarithmic singularity approaching T c (see Ref. [START_REF] Ferrell | Determination of the anomalous dimension Green-Fisher critical exponent η by light scattering in a fluid[END_REF] and the dicussion below).

For the xenon case, all the quantities involved in the above calculation of the fluid turbidity are given in Table 1, where the only differences with the GC's analysis originate from the critical coordinates of xenon. The evaluation of the error-bar attached to the quantities given in Table 1 can be found in I.

GC Turbidity Measurements

The GC data points τ expt of xenon turbidity are plotted (full blue stars) in Fig. 2a as a function of τ * in log-log scale. Our theoretical estimation of τ PF using Eqs. ( 5) and ( 6), with ξ of Eq. ( 1) and κ T of Eq. ( 2), corresponds to the continuous black curve in Fig. 2a. The GC fitting analysis is also given as the dotted red curve in Fig. 2a. The corresponding residuals (expressed in %) are given in the log-linear plot of Fig. 2b. Our estimation of τ PF without adjustable parameter is in excellent agreement with the experimental measurements and fitting analysis of GC.

When T → T c , the very small increase of the residuals is mainly due to the small difference in the respective values of the leading critical exponents ν and γ . In addition, the effect of the uncertainty ( 0.5 mK) in the experimental determination of T c,GC approaching the critical temperature, is illustrated by the full blue squares in Fig. 2b, which correspond to the related behaviors of the residuals due to a T c,GC -change of +0.5 mK. We have also reported the new residuals (full red triangles) due to a change of -0.00025 cm -1 (i.e., -0.6 %) in the initial calibration value (τ (T or ) = 0.041 cm -1 , see Table 1 andI).

In the temperature range T -T c 7 K of Fig. 2b, increasing of the residuals is due to the significant decreasing of the xenon turbidity when T -T c increases. Indeed, accurate measurements of xenon turbidity in this temperature range require a xenon sample cell of optical path larger than a few centimeters.

Nevertheless, we believe that the OZ approach of xenon turbidity with Eq. ( 5), incorporating the ξ and κ T values obtained from theoretical Eqs. ( 1) and (2) without adjustable parameter, is adequate in the T -T c 10 K range covered by the GC measurements. The following section shows that the -contribution to the turbidity behavior is the main contribution in the GC temperature range.

Asymptotic Singular Behavior of Turbidity

The estimation of τ can now be revisited with the objective to illustrate the role of the Ising-like parameters characterizing the asymptotic singular behavior within the PAD. Indeed, for the Ising-like asymptotic limit τ * → 0, the contributions of the confluent singularities associated to the lowest exponent can be neglected, leading to L ( τ * ) ≈ 1, K ( τ * ) ≈ 1, and T = T c (1 + τ * ) ≈ T c in the previous equations. Any theoretical asymptotic approach (labelled X) estimating the fluid turbidity is then restricted to the knowledge of ξ = ξ + 0 ( τ * ) -ν and κ T = + 0 ( τ * ) -γ and can only add the universal contribution of the pair C + ∞ , η in order to satisfy the two-scale-factor universality (see above). Therefore, Ising-like turbidity must take the general asymptotic form [START_REF] Ferrell | Determination of the anomalous dimension Green-Fisher critical exponent η by light scattering in a fluid[END_REF][START_REF] Martin-Mayor | Critical structure factor in Ising systems[END_REF]

τ as,X = τ 0 ( τ * ) -γ y 2 ν H as,X (η, y ν ) (7)
Equation ( 7) introduces the universal quantity y ν = k 0 ξ + 0 ( τ * ) -ν where the subscript ν recalls for a Ising universal quantity only characterized by ξ + 0 , ν . The temperature independent quantity τ 0 = π A 0 k B T c + 0 ∼ length -1 is a fluid dependent quantity, only proportional to + 0 . In Eq. ( 7), the universal scaling function H as,X (η, y ν ) accounts for the explicit contribution of η = 0 and depends on the used approximant form for the universal scaling function G (x) ∼ C + ∞ x η-2 when x 1 [START_REF] Ferrell | Determination of the anomalous dimension Green-Fisher critical exponent η by light scattering in a fluid[END_REF][START_REF] Martin-Mayor | Critical structure factor in Ising systems[END_REF]. Now, the explicit derivation of the where H as,PF (y ν ) = ln (2y ν ) 2 -1 = L -1, with L = ln (2y ν ) 2 . Now, Eq. ( 9) only needs ξ and κ T pure power laws, limiting the validity range of the calculated turbidity that behaves asymptotically with the incorrect logarithmic singularity H as,PF (y) ∼ ln y 2 ν approaching T c . In spite of this incorrect asymptotic behavior, the practical condition γ = 2ν corresponds to apply η = 0 in the complete temperature range, leading to the explicit power law term ( τ * ) -γ +2ν = ( τ * ) ην in Eq. [START_REF] Garrabos | Facteurs d'échelle phénoménologiques pour la transition critique liquide-gaz des fluides purs[END_REF].

It is then also useful to calculate the turbidity resulting from the exact OZ condition γ = 2ν, i.e., the turbidity of the OZ-like fluid satisfying η = 0 in the complete temperature range. In such a case, adding (1 + τ * ) due to T , Eq. ( 10) provides the following reference turbidity

τ ref,η=0 = τ 0 (1 + τ * ) k 0 ξ + 0 2 H PF (y ν ) ( 10
)
where H PF (y ν ) is still calculated using ξ = ξ + 0 ( τ * ) -ν , while κ T follows the incorrect pure power law κ T,OZ = + 0 ( τ * ) -2ν resulting from κ T ∝ ξ 2 . In exact OZ theory, as + 0 is a function of ξ + 0 , the fluid turbidity is only characterized by the single amplitude-exponent pair ξ + 0 , ν . τ ref,η=0 always presents the incorrect logarithmic singularity τ as,η=0 =

τ 0 k 0 ξ + 0 2 (L -1) ∼ ln y 2
ν , similar to the one of τ as,PF . The ratio

τ as,η=0
τ ref,η=0 = L-1 H PF (y ν ) → 1 eliminates the logarithmic singularity when τ * → 0. This asymptotic result corresponds to the blue tireted curve labeled OZ as in the diagram τ τ ref,η=0 ; τ * of Fig. 3. The horizontal axis τ τ ref,η=0 = 1 corresponds to η = 0. For a true Ising-like fluid, the respective contributions due to η = 0 and are then easy to illustrate in Fig. 3, at each temperature distance to T c .

For the Ising-like limiting range τ * ≤ 10 -6

L Xe PAD where the scaling law ην = -γ + 2ν can be used to estimate η, τ as,PF

τ ref,η=0 = ( τ * ) -γ +2ν L-1
H PF (y ν ) → ( τ * ) -γ +2ν (see the black tireted curve PF as in Fig. 3). In this asymptotic range, the relative difference between the OZ as and PF as curves reveals that the amplitude of the contribution due to η = 0 is larger than 25 % and increases when τ * → 0. Indeed, if only the pure power laws of ξ and κ T are used for the complete τ * range, Eq. ( 8) takes the simplified form τ PF,ν,γ = τ 0( 1+ τ * )

k 0 ξ + 0 2
( τ * ) -γ +2ν H PF (y ν ) (labeled by the subscript PF, ν, γ ). Accordingly, τ PF,ν,γ τ ref,η=0 = ( τ * ) ην whatever τ * , as illustrated by the black curve labeled PF 2 in Fig. 3. A pure fluid turbidity without contribution of the confluent singularities associated to is only characterized by both pairs ξ + 0 , ν and + 0 , γ . The subscript 2 recalls for a fluid characterized by only two leading amplitudes (ξ + 0 and + 0 ) over the complete temperature range. Consequently, as τ * decreases, the increasing difference between 1-axis and PF 2 curve is a measure of η = 0.

However, as mentioned previously, the above asymptotic singular behavior where the turbidity is calculated from the PF functional form, involves a fluid characterization without conformity with the two-scale factor universality. To illustrate the amplitude effect of the asymptotic singular behavior of the true Ising-like turbidity, we use as a typical example the Martin-Mayor et al's turbidity results of Ref. [START_REF] Martin-Mayor | Critical structure factor in Ising systems[END_REF] obtained by Monte Carlo simulation of a simple cubic Ising lattice. Such results have validated the analytic derivation of the saturated turbidity at T c , ρ c initially proposed by Ferrell [START_REF] Ferrell | Determination of the anomalous dimension Green-Fisher critical exponent η by light scattering in a fluid[END_REF]. Here using the functional form H as,MM (η,

y ν ) = 2C + ∞ (4y 2 ν ) η 2 η 2 +2η+8 η(η+2)(η+4) -1 η + K C + ∞
given by Eq. ( 34) in Ref.

[33], the asymptotic behavior of the corresponding ratio

τ as,MM τ ref,η=0 = ( τ * ) -γ +2ν H as,MM (η,y ν ) H PF (y ν )
is represented by the red tireted curve labelled MM as in Fig. 3 (using K = 0.16). When τ * → 0, the increasing difference from MM as and PF as curves reveals the increasing C + ∞ , η -contribution to the turbidity of the Ising-like fluid which complies with the twoscale factor universality.

Finally, it remains to illustrate the -contribution to the fluid turbidity when the temperature distance to the critical temperature increases well beyond the PAD, i.e., τ * > 10 -5 . Using Eqs. ( 1) and ( 2

), τ PF τ ref,η=0 = ( τ * ) ην K ( τ * ) [L( τ * )] 2 H PF (y)
H PF (y ν ) , leading to the red full curve labelled PF 3 in Fig. 3. Now, the subscript 3 recalls for a fluid characterized by two leading amplitudes (ξ + 0 and + 0 ) and one first-order amplitude (a 1,+ χ ) acting as a single crossover parameter over the complete temperature range. As τ * increases, the increase of the -contribution is thus measured by the increasing difference from PF 3 and PF 2 curves in Fig. 3. The agreement with τ exp τ ref,η=0 obtained from GC turbidity data (red squares in Fig. 3) is noticeable, demonstrating the interest of the present work made without any adjustable parameter as a correct answer to the GC conclusive sentence in I.

An additional remark can be formulated from a hypothetical turbidity measurement, 0.3 % precision, performed at 1 mK temperature distance above T c , i.e., τ * ∼ 3.45 × 10 -6 in xenon case (see red vertical line in Fig. 3). At such a finite temperature distance well-inside the PAD extension, the -contribution due to the confluent corrections to scaling is lower than the experimental precision and can be certainly neglected. From Puglielli and Ford result, τ as,PF τ ref,η=0 = 0.766 ± 0.019, i.e., ∼ ±2.5 % error-bar (with central value ν = 0.63 and accounting for the η-theoretical error-bar, η = 0.0336 +0.00294 -0.00250 ). From Martin-Mayor result, τ as,MM τ ref,η=0 = 0.774 ± 0.025 (increasing to ±3.5 % error-bar after adding ±1 % error-bar from the theoretical estimation of C + ∞ ). Therefore, when the two asymptotic power laws ξ = ξ + 0 ( τ * ) -ν and κ T = + 0 ( τ * ) -γ are known, the agreement with the experimental value of the turbidity at T -T c = 1 mK is a significant probe of the related value η = 2 -γ ν , whatever the theoretical functional form of the turbidity. However, as the residual 100 × τ as,MM τ as,PF -1 is only +1 %, the distinction between the contributions due to the respective ratios

+ 0 k 0 ξ + 0 2
(involved through the Puglielli and Ford Eq. ( 9)) and 7)) remains not significant at such a small temperature distance. We note that the difficulty can be partly solved when an additional singular property is measured, especially the top-shape ρ LV = B ( τ * ) β of the coexistence curve, which gives access to the ratio

+ 0 k 0 ξ +
+ 0 ξ + 0 3 through the non-dimensional amplitude combination ξ + -d + B 2 .
In Sect. 6, we show that the two leading amplitudes + and B are thus well-adapted for the thermodynamic characterization of the fluid, while the role of turbidity for the correlation characterization of the fluid is discussed in Ref. [START_REF] Garrabos | Test of Ising-like equationof-state models using turbidity measurements in near critical fluids[END_REF], analyzing recent measurements of SF 6 turbidity in the close vicinity of its liquid-gas critical point.

Correlation Length

We have reported in the lines 1 to 8 of Table 3 the published amplitude/exponent values of the leading power law ξ = ξ + 0 ( τ * ) -ν , which have been used in the indirect fitting of the results provided by (static and dynamic) Rayleigh-Brillouin methods [START_REF] Smith | Correlation range and compressibility of xenon near the critical point[END_REF][START_REF] Cannell | Brillouin spectrum of xenon near its critical point[END_REF][START_REF] Zollweg | The spectrum and intensity of light scattered from the bulk phases and from the liquid-vapor interface of xenon near its critical point[END_REF][START_REF] Swinney | The Rayleigh linewidth in xenon near the critical point[END_REF][START_REF] Swinney | Dynamics of fluids near the critical point: decay rate of order-parameter fluctuations[END_REF][START_REF] Güttinger | Correlation range and Rayleigh linewidth of xenon near the critical point[END_REF]. Line 9 results from the GC fitting analysis of their turbidity data with fixed values of the exponents in the following two-term Wegner expansion

ξ = ξ + 0 τ * -ν 1 + a 1,+ ξ τ * (11)
In addition, we have also given in lines 10 and 11 the ξ + 0 , ν values for two fixed pure power laws. Indeed, such pure power laws are fixed in the fitting analyses of the shear viscosity [START_REF] Berg | Frequency-dependent viscosity of xenon near the critical point[END_REF], the thermoacoustic boundary layers [START_REF] Gillis | Thermoacoustic boundary layers near the liquid-vapor critical point[END_REF], and the bulk viscosity [START_REF] Gillis | Bulk viscosity of stirred xenon near the critical point[END_REF] of critical xenon. i.e., experiments which cover the extended temperature range 5 × 10 -7 ≤ τ * ≤ 10 -1 . These applications then extend significantly the GC temperature range, especially by two supplementary decades approaching the critical temperature. Finally, line 12 gives the related values for the two-term form of Eq. ( 1), whose validity is restricted to the PAD extension.

The log-lin plots of Fig. 4a,b show the corresponding residuals Rξ (%) = 100× ξ ξ cal -1 (expressed in %) from reference to ξ cal calculated using Eq. ( 1). The curves labeled 1 to 8 of part (a) illustrate the significant dispersion (larger than 10 %) of the results (lines 1 to 8 of Table 3) provided by using (static and dynamic) Rayleigh-Brillouin methods (including the first fitting analysis [START_REF] Güttinger | Correlation range and Rayleigh linewidth of xenon near the critical point[END_REF] of the correlation length measurements made by using the precise differential technique and apparatus). The residuals with the data measurements performed by Smith et al. [START_REF] Smith | Correlation range and compressibility of xenon near the critical point[END_REF] are represented by the full black circles. This Fig. 4a gives clear evidence that the highly correlated values of ν and ξ + 0 are not obtained with the required precision from all these optical measurements performed at finite distance to T c . In (magnified) part (b) of Fig. 4, the lines (labeled 9 to 11) correspond to the residuals (then lowered at the 1 -2 %-level) using Eq. ( 11) with the Ising-like parameters of lines 9 to 11 of Table 3). In spite of the questionable justification of the universal features related to the lowest order of the Ising-like Wegner expansions at large distance from T c , we note a satisfactory agreement (within the experimental uncertainty of 8 %) in the reduced temperature range 7 × 10 -5 ≤ τ * ≤ 2 × 10 -2 L Xe EAD (Fig. 5). 1) restricted to the two-term contribution of Eq. [START_REF] Garrabos | Scaling behaviour of the fluid subclass near the liquid-gas critical point[END_REF] above crossover function of Eq. ( 2) issued from the MR scheme. More generally, it was already noted the inherent Ising-like similarity between the MR scheme and the approach based on a phenomenological crossover transformation for a classical Landau expansion of the singular contribution to a free energy density [START_REF] Chen | Crossover from singular critical to regular classical thermodynamic behavior of fluids[END_REF][START_REF] Chen | Global thermodynamic behavior of fluids in the critical region[END_REF]. Both approaches account for similar asymptotic universal features at the first-order of the confluent singularities, only using three adjustable physical parameters. On such a theoretical basis, we are in position to write the equations that close the unambiguous determination of the CPM free parameters, using the scale factors of the master crossover functions as entry data. It is then expected that the resulting crossover master model (CMM) works without any Ising-like adjustable parameter. Therefore, our following section is focused on the phenomenological key points of the crossover transformation of the equation of state in order to propose a new tool able to estimate the fluid properties in the complete phase surface surrounding the vicinity of its liquid-gas critical point, only knowing the four critical parameters previously introduced in Sect. 3.

Master form of the Crossover Parametric Model

Main Characteristics of CPM

The CPM developped by Agayan et al [START_REF] Agayan | Crossover parametric equation of state for Ising-like systems[END_REF] results from a generic approach based on a phenomenological crossover transformation for a classical Landau expansion of the singular contribution to a free energy density. This approach was at the origin of the so-called crossover Landau model (CLM) [START_REF] Chen | Crossover from singular critical to regular classical thermodynamic behavior of fluids[END_REF][START_REF] Chen | Global thermodynamic behavior of fluids in the critical region[END_REF], where the van der Waals equation was used to develop the parametric form of the equation of state. Due to the initial implementing by Chen et al. [START_REF] Chen | Crossover from singular critical to regular classical thermodynamic behavior of fluids[END_REF][START_REF] Chen | Global thermodynamic behavior of fluids in the critical region[END_REF] of a match-point method proposed by Nicoll et al. [START_REF] Nicoll | Crossover functions by renormalization-group matching: O( 2 ) results[END_REF][START_REF] Nicoll | Crossover functions by renormalization-group matching: three-loop results[END_REF][START_REF] Nicoll | Background fluctuations and Wegner corrections[END_REF], the CPM appears to be conform with the so-called renormalization-group matching technique, while reproducing now the known theoretical values for the universal amplitudes combinations of leading terms of the scaling behaviors and first-order terms of the confluent singularities in Wegner-like expansions.

In a parametric equation-of-state, any point of the phase surface close to the critical point is characterized by the radial variable r , which measures the distance to the critical point and the angular variable θ, which represents the density distance to the critical density on a contour of constant r . The temperature distance and the density distance of the critical point are thus related to r and θ by parametrics forms. In addition, these parametric forms contain a crossover function able to represent the phenomenological crossover transformation for a classical Landau expansion of the singular contribution of a free energy. Starting then far from the critical point where it is observed the classical-so called mean field-behavior of the singular energy, crossover occurs when approching the critical point, to finally observe the critical-so called Ising-like-behavior of the singular energy very close to the critical point. In the CPM model, this phenomenological crossover behavior is defined by a function Y , formulated to satisfy the following equation

1 -(1 -ū) Y = ū 1 + 2 κ 2 1 2 Y ν s ( 13
)
where κ is the inverse of the dimensionless correlation length ξ * and ν is the corresponding universal critical exponent. s (in Agayan et al's notations) is identical to the lowest value of the critical exponent for the confluent singularities of the Wegner expansion (see for example Eqs. ( 1) and ( 2)).

With a parametrization of κ as a function of r as

κ 2 (r ) = c t rY 2ν-1 s
the crossover function Y , like κ, are only linear functions of r and independent of θ. c t is a non-dimensional (fluid-dependent) scale-factor for the reduced temperature distance to T c along the critical isochore. We note also that Y only depends on two crossover variables, ū, and

(c t ) 1 2
, which determine the crossover shape and the crossover temperature scale, respectively (see also below).

The non dimensional ordering field h 1 , the non dimensional non-ordering field h 2 , and the critical part of the non dimensional thermodynamic potential can thus be described by parametric representations in terms of the variables r and θ and by appropriate scaling forms of Y , as follows

h 1 = r 3 2 Y 2βδ-3 2 s l (θ ) ( 14
)
h 2 = rk (θ ) (15) = r 2 Y -α s w(θ) + 1 2 B cr r 2 1 -b 2 θ 2 2 ( 16
)
where β, δ, and α are the critical exponents for the top shape of the coexistence curve, the critical isotherm, and the heat capacity at constant volume, respectively. Among the ν, η, γ , β, δ, and α exponents involved in this phenomenological description, only two remain independent (using for example the additional hyperscaling laws

dν = 2 -α, 2-η d = δ-1 δ+1
, and the scaling laws γ = ν (2 -η), γ = β (δ -1), when ν and γ are selected as independent). The above functional forms of Eqs. ( 14) to ( 16), also introduce the following analytic functions of θ

l(θ) = l0 θ 1 -θ 2 (17) k(θ) = 1 -b 2 θ 2 (18) w(θ) = m0 l0 w 0 + w 1 θ 2 + w 2 θ 4 + w 3 θ 6 + w 4 θ 8 (19)
and the fluctuation induced constant

B cr = -2 m0 l0 w 0 < 0 ( 2 0 )
where appear six universal parameters (b 2 and w i , with i = {0, 4}) and two rescaled parameters ( m0 = m 0 g β-1 2 and l0 = l 0 g βδ-3 2 ) accounting for the characterization of the physical system. m 0 and l 0 are the two system-dependent parameters that determine the asymptotic critical amplitudes to be conform with the two-scale-factor universality (see below). g = ( ū ) 2 c t is the crossover parameter that combine the two previous crossover parameters ū and

(c t ) 1 2
characterizing the crossover function Y . Here it is essential to note that the parameter b 2 and the five coefficients w i of the parametrized CPM, were chosen such that the asymptotic amplitude ratios between the asymptotic thermodynamic amplitudes A + 0 , A - 0 , + 0 , - 0 , B 0 , and D 0 (in Agayan et al's dimensionless notations), agree with the theoretical predictions of the universal features of a Ising-like system [START_REF] Bagnuls | Non-asymptotic critical behavior from field theory at d=3. II. The ordered phase case[END_REF][START_REF] Fisher | The shape of the van der Waals loop and universal critical amplitude ratios[END_REF]. Finally, among the six universal asymptotic parameters b 2 and w i , only two are independent.

To describe the one-component fluid crossover using the intensive variables p and T associated to the molar density n mol = N N A 1 V , the classical thermodynamic form of the CPM assumes that the scaling fields h 1 and h 2 can be expressed in the linear combinations of the dimensionless Gibbs energy density and temperature distances to their respective critical value, i.e., h 1 = g * and h 2 = T * + b 2 g * , where b 2 is the mixing parameter that measures the asymmetry in the slope of the coexistence curve. Here

T * = 1 -T c T = τ * 1+ τ * and g * = g * -g * c .
The definition of g * , which acts as a conjugated ordering field, needs to recall that the density g (T, p) = G(T, p,N ) V of the Gibbs free energy (distinct from the crossover parameter g) can be expressed as g (T, p) = n p μ p = n mol μ mol (T, p), introducing then the molar chemical potential μ mol (T, p) = N A μ p (T, p). Obviously, μ mol = μ molμ mol,c = N A μ p , where μ p = μ p -μ p,c . Moreover, in the related dimensionless form of the CPM, the density of any thermodynamic potential (i.e., an energy per unit of volume) is expressed in units of p c ≡ (β c ) -1

(α c ) d , while the density of the entropy is expressed in units of (β c ) -1 and α c as energy and length units). Once h 1 , h 2 and = p * -1 are defined in terms of the parametric variables r and θ , thermodynamic quantities can then be derived using well known thermodynamic relations.

However, in order to maintain Ising-like similarity with the symmetrical φ 2 2 field theory, we only consider here the CPM symmetrical form by fixing b 2 = 0 leading to

n mol = ϕ 1 ( 21
)
The xenon properties of present interest (molar isothermal susceptibility and liquid-gas coexisting molar densities, see below), can be calculated from their dimensionless forms

( χ mol ) -1 = ∂h 1 ∂ϕ 1 h 2 = ∂ g * ∂ n mol T = n 2 mol,c p c ∂ μ mol ∂ n mol T = n 2 mol,c p c (χ mol ) -1 , (22)
Equation ( 22) leads to

χ mol = χ ρ = ∂ ρ ∂ μ T
and then to χ mol = (Z c ) 2 χ * p . In addition, ).

χ mol = ∂ϕ 1 ∂h 1 h 2 = ∂ 2 ∂h 2 1 h 2 = ∂ 2 (p * -1) ∂( g * ) 2 h 2 = p c (nmol,c) 2 ∂ 2 p ∂μ 2 mol T with ∂ p ∂μ mol T,V =V mol = ∂ N ∂ V T,
Using our notations for the dimensionless amplitudes and Eqs. ( 22) and ( 12), it is thus obtained [START_REF] Agayan | Crossover parametric equation of state for Ising-like systems[END_REF]

+ = 3.38317m 0 (l 0 ) -1 (23)
for the leading amplitude of Eq. ( 12), and

a 1,+ χ = g + χ g -s (1 -u) ( 24
)
with g + χ = 0.590 for the first order amplitude of the related confluent singularities of Eq. ( 12). We introduce now the asymptotic two-term power law describing the symmetrized top-shape of the liquid-vapor coexistence curve

ρ LV = ρ L -ρ V 2ρ c = B τ * β 1 + a 1,- M τ * (25)
From Eqs. ( 21) and ( 25), it is thus obtained [START_REF] Agayan | Crossover parametric equation of state for Ising-like systems[END_REF] B = 3.28613m 0 [START_REF] Levelt-Sengers | Scaling predictions for thermodynamic anomalies near the gas-liquid critical point[END_REF] for the leading amplitude of Eq. ( 25), and

a 1,- M = g - M g -s (1 -u) ( 27
)
with g - M = 0.529 for the first order amplitude of the related confluent singularities of Eq. ( 25). In a similar manner, the crossover behavior of the dimensionless correlation length is specified through

(ξ * ) 2 χ mol = Y -ην s ( ū ) -2ην a (θ, Y 1 ) (28)
where

Y 1 (r ) = 1 s r Y dY dr and a (θ, Y 1 ) = a 0 + a 1 θ 2 Y 1 + a * 0 + a * 1 θ 2 (1 -Y 1 )
. The parameters a 0 , a 1 , a * 0 , and a * 1 are the constants that can be determined to reproduce theoretical correlation-length amplitude ratios. They can be expressed with l 0 and m 0 , as a 0 = 0.0643 (l 0 ) 3 , and a * 1 = 0.0028 (l 0 )

1 3 (m 0 ) -5 3 , a 1 = 0.0178 (l 0 ) 1 3 (m 0 ) -5 3 , a * 0 = 0.0579 (l 0 ) 1 3 (m 0 ) - 5
1 3 (m 0 ) -5 3 , leading to ξ + = 0.466287 (m 0 l 0 ) -1 3 ( 29
)
for the leading term amplitude of Eq. ( 11), and to

a 1,+ ξ = g + ξ g -s (1 -u) ( 30
)
with g + ξ = 0.260 for the first term amplitude of the related confluent singularities of Eq. [START_REF] Garrabos | Scaling behaviour of the fluid subclass near the liquid-gas critical point[END_REF].

Finally, to carry out the CPM calculation in general, for given reduced temperature and density, the parameters r and θ are first solved numerically. Then the scaling fields, the thermodynamic potential and its derivatives can be calculated in terms of r and θ after numerically solving the crossover equation for Y . The generic approach based on a phenomenological crossover transformation for a classical Landau expansion of the singular contribution to a free energy density involves three free parameters, i.e., two asymptotic scaling parameters (noted c t and c ρ , in the more general CLM approach) and one crossover parameter (noted g). Normally, from a field theory framework introducing two asymptotic Ising-like scaling factors, any dimensionless critical crossover description of a 3D Ising-like system with finite cutoff contains at least two other free crossover parameters [START_REF] Bagnuls | Classical-to-critical crossovers from field theory[END_REF], i.e., a reduced wave number related to the cut-off wavelength 0 , and a reduced scaling factor u related to the coupling constant u 4 . For the theoretical infinite-cutoff approximation in the field theory scheme, → ∞, u → 0 and u remains finite. Due to the basic form of the singular part of the free energy density in the CLM description, g is then necessarily related to the product u . However, due to the analytical Landau scheme expressed at the symmetrical fourth-order (with only two independent critical coupling quantities), all the dimensionless quantities are canonical constants (i.e. independent of the selected van der Waals fluid and the dimension of the system). An implicit connection between and u (or c t ) exists, yielding for example to the explicit relation g = (u ) 2 c t . Moreover, a single scale unit is needed to reduce dimensional length appearing in correlation and thermodynamic functions, independently of any hypothesis on the microscopic structure of the interacting fluid particles. It also imposes = constant for CLM description of the Ising-like systems, leading to a possible arbitrarily choice for the numerical value of the constant provided that u remains finite. For example, in the case of the 3D-Ising models where = 1, a detailed application [START_REF] Kim | Crossover critical behavior in the three-dimensional Ising model[END_REF] of an asymptotic version of CLM has studied the crossover behavior governed by the u dependence on the interaction range. This study shows that a more complex, non-monotonic crossover behavior is possible for a very short range of interactions (u > 1) in complex fluids. In case of the application of the parametric form of the equation of state where g

1 2 = u (c t ) 1 2
to a one-component fluid such as xenon, the only way to monitor the CLM asymptotic critical crossover, is to change u, or, equivalently, c t , via a single independent coupling quantity (see Ref. [START_REF] Agayan | Crossover parametric equation of state for Ising-like systems[END_REF] for details). In spite of the possible non-universal nature of the resulting complete crossover behavior, such a fluid application offers the opportunity to match the Ising-like singular behaviors predicted by the master crossover functions with the ones of the now called crossover master model (CMM), as described in the following section. This CMM model requiring no adjustable parameters is then for simple fluid systems where u < 1 is strongly correlated with .

Crossover Master Model (CMM)

The above unknown scaling parameters l 0 and m 0 are explicitely involved in the determination of the leading amplitudes of the singular properties, while, similarly, the known scale factors Y c and Z c appear explicitely in the determination of these same leading amplitudes in the master form of the MR scheme. Therefore, to provide the unequivocal link between the Y c ;Z c pair and the l 0 ;m 0 pair, we consider the isothermal compressibility and the coexistence curve whose essential role was already mentionned for the singular thermodynamic characterization of the fluid. The respective + and B forms in the master description (see Table I in Ref. [START_REF] Garrabos | Master crossover functions for onecomponent fluids[END_REF]) and in the CPM description (see Table III in Ref. [START_REF] Agayan | Crossover parametric equation of state for Ising-like systems[END_REF]), lead to the matching equations:

+ = Z + χ * qe (Z c ) -1 (Y c ) -γ = 3.38317m 0 l -1 0 (31)
and

B = Z M * qe -1 (Z c ) -1 2 (Y c ) β = 3.28613m 0 (32)
The CPM scaling parameters l 0 and m 0 can thus be calculated without adjustable parameter using the two following relations,

l 0 = 3.38317 Z + χ Z M 3.28613 * qe -2 (Z c ) 1 2 (Y c ) β+γ ( 33
)
m 0 = Z M 3.28613 * qe -1 (Z c ) -1 2 (Y c ) β (34)
Note that Eqs. ( 33) and ( 34) also account for quantum effect correction through * qe [START_REF] Garrabos | Universality and quantum effects in one-component critical fluid[END_REF] (with * qe = 1 in the non-quantum xenon case). Moreover, the universality of the first-order correction-to-scaling scheme can also be closed by identifying, for example, the CPM amplitude [START_REF] Garrabos | Master crossover functions for onecomponent fluids[END_REF]). Using g + χ = 0.590, s = 0.52, and g = (u ) 2 c t (see Table III in [START_REF] Agayan | Crossover parametric equation of state for Ising-like systems[END_REF]), we obtain

+ 1 = g + χ g -s (1 -u) with our selected first-order amplitude a 1,+ χ = Z 1+ (Y c ) (see Table I in Ref.
g + χ u (c t ) 1 2 -2 s (1 -u) = Z 1+ (Y c ) (35)
Equation [START_REF] Garside | Refractive index and Lorentz-Lorenz function of xenon liquid and vapour[END_REF] demonstrates that the fluid-dependent crossover parameter g

1 2 = u (c t ) 1 2
introduced in the phenomenological crossover transformation of the Landau expansion is well related to our asymptotic scale parameter Y c , as already noted [START_REF] Garrabos | Master singular behavior from correlation length measurements for seven one-component fluids near their gas-liquid critical point[END_REF] in the case of several non-quantum pure fluids by analyzing the crossover Landau model [START_REF] Chen | Crossover from singular critical to regular classical thermodynamic behavior of fluids[END_REF][START_REF] Chen | Global thermodynamic behavior of fluids in the critical region[END_REF]. Equations ( 33), [START_REF] Chapman | Temperature and density dependence of the molar polarizability of xenon[END_REF], and (35) are the basic equations which define the master Ising-like nature of the CMM. = π initially adopted by the authors [START_REF] Agayan | Crossover parametric equation of state for Ising-like systems[END_REF], the xenon values of the CPM free parameters l 0 , m 0 and u reported in column 2 of Table 5 are obtained from a joint fitting of xenon isothermal compressibility (from Eq. ( 22) and experimental data of Ref. [START_REF] Güttinger | Corrections to scaling in the susceptibility of xenon[END_REF]) and xenon coexistence curve (from Eq. ( 21) and experimental data of Refs. [START_REF] Närger | Coexistence-curve diameter and critical density of xenon[END_REF][START_REF] Balzarini | Universal ratio of correction-to-scaling amplitudes for Xe[END_REF]). For the CMM case, the calculated values of l 0 , m 0 , u and g using Eqs. ( 33), ( 34), [START_REF] Garside | Refractive index and Lorentz-Lorenz function of xenon liquid and vapour[END_REF] and similar arbitrarily relation

(c t ) 1 2
= π, are given in column 3. We note the close agreement between both parameter sets. This satisfactory agreement is confirmed in Figs. 6 and7 showing the results of the comparison between the experimental data and the two different calculations with free and calculated parameter sets. Finally, the correlation length (from Eq. ( 23)) and the turbidity (from Eq. ( 5)) can also be calculated and the corresponding results are illustrated in Fig. 8 where the agreement remains always noticeable.

Only using the four critical coordinates of the liquid-gas critical point of xenon and the master forms of the Ising-like crossover functions calculated from the massive renormalization scheme for the critical isochore, we have estimated the singular behavior of xenon turbidity in agreement with GC experimental data in the temperature range 28 mK ≤ T -T c 9 K (10 -4 τ * 3 × 10 -2 ), contributing then to the expected theoretical progress in the reduction to zero free parameter in the description of the Ising-like confluent singularities far from the critical temperature. The critical xenon, the N = 1 simple cubic lattice model and the O(1) symmetric φ 2 2 field theory belong to the same universality class. In addition, we have proposed a crossover master model by calculating the adjustable Ising-like crossover parameters of the crossover parametric model to match the master singular behaviors of the one-components fluids, which opens the route to estimate the singular properties for the near-critical fluids without adjustable parameter.

Isothermal Compressibility Data Source

Before the GC work, the isothermal compressibility data for xenon were generally fitted using the following effective power law The γ e -results reported in Table 6 are ordered (see #i in column 1) from γ to γ MF , while column 2 indicates the measurement method (FI: Fraunhofer interferometry; LS: light scattering; pV T : [ p (V )] T measurements). Data reported in lines #1a and #1b are from the interferometric measurements of Hocken and Moldover [START_REF] Hocken | Ising critical exponents in real fluids: an experiment[END_REF] in the reduced temperature range 10 -6 τ * 10 -5 , using a critical xenon cell subjected to the gravitational field which generates Fraunhofer interferograms [START_REF] Hocken | Ising critical exponents in real fluids: an experiment[END_REF] related to the isothermal compressibility. The initial fitting with free values of the exponent and amplitude has given γ = 1.23 and + = 0.062±0.006 [START_REF] Hocken | Ising critical exponents in real fluids: an experiment[END_REF] (line #1b), while a subsequent analysis made for fixed Ising value γ = 1.24 has given + = 0.058 ± 0.002 [START_REF] Sengers | Two-scale-factor universality near the critical point of fluids[END_REF] (line #1a). Line #2 is from χ ρ = ∂ ρ ∂ μ ρ T values [START_REF] Levelt Sengers | Scaled equation of state parameters for gases in critical region[END_REF] obtained after numerical integration of [ p (ρ)] T data of Habgood and Schneider [START_REF] Habgood | PVT measurements in the critical region of xenon[END_REF][START_REF] Habgood | Thermodynamic properties of xenon in the critical region[END_REF] to obtain chemical potential difference μ ρ and to validate antisymmetry of the μ ρ ( ρ) T supercritical isotherms (and then symmetry of χ ρ ( ρ) T ), as functions of ρ = ρ -ρ c (see also Footnote 5). Data reported in line #3 are from dynamic light scattering data and complex analysis of the xenon Brillouin spectra by Cannell and Benedek [START_REF] Cannell | Brillouin spectrum of xenon near its critical point[END_REF] covering the temperature range 0.2 K ≤ T -T c ≤ 20 K. Data reported in line #4 are from static light scattering data of Smith et al. [START_REF] Smith | Correlation range and compressibility of xenon near the critical point[END_REF] covering the temperature range 0.045 K ≤ T -T c ≤ 5.1 K. After calibration of the isothermal compressibility at T = T c + 1 K [START_REF] Garrabos | Contribution à l'étude des propriétés d'état des fluides purs dans leur région critique[END_REF], the effective amplitude for Cannell and Benedek's data was lowered by 1.57 %, while the one for Smith et al's data was increased by 11.2 %, in agreement with around 14 % initial deviation between these two data set. In spite of the importance ( (10 -20) %) of these amplitude corrections, the fitting results confirm that the γ e value is well in the range γ e 1.20 -1.21 for the restricted temperature range of these experiments. Such results were confirmed by the data in line #5 obtained by the effective power law fitting of the GC's measurements of high relative precision in the similar temperature range. As a matter of fact, GC have claimed that the correction to scaling terms are important by demonstrating that the susceptibility deviates systematically from Eq. 36 with γ e = 1.206. More precisely, Fig. 2 of I shows that γ e 1.206 is the slope of the tangent line to the rough experimental behavior of κ * T ( τ * ) at τ * γ e =1.206

κ * T = p c κ T = + e τ * -
4.35 × 10 -3 , i.e., the temperature distance which corresponds to the minimum of the deviation curve in this Fig. 2 (see also Fig. 5 in Ref. [START_REF] Garrabos | Characteristic parameters of xenon near its liquid-gas critical point[END_REF]). The finite value τ * γ e =1.206 is clearly outside the PAD extension (see the related circle in the segment 5 of Fig. 1a). Now we note that the line #6 in Table 6 is also from directly fitting [START_REF] Levelt Sengers | Critical phenomena in classical fluids, chap. 4[END_REF] 3 [ p (V )] T measurements of Habgood and Schneider [START_REF] Habgood | PVT measurements in the critical region of xenon[END_REF][START_REF] Habgood | Thermodynamic properties of xenon in the critical region[END_REF] and Michels et al. [START_REF] Michels | Isotherms of xenon at temperatures between 0C and 150C and at densities up to 515 Amagats (pressures up to 2800 atmospheres)[END_REF], using the wellknown restricted linear model of the scaled "universal" equation of state, where k (Xe) = 1.309, a (Xe) = 16.1 and γ e,eos = 1.190 [START_REF] Levelt Sengers | Critical phenomena in classical fluids, chap. 4[END_REF], leading to + e = k a . In addition, the lines #7 to #9 are from similar pV T analyses [START_REF] Garrabos | Contribution à l'étude des propriétés d'état des fluides purs dans leur région critique[END_REF] using the [ p (V )] T data measurements of Michels et al. [START_REF] Michels | Isotherms of xenon at temperatures between 0C and 150C and at densities up to 515 Amagats (pressures up to 2800 atmospheres)[END_REF], Beattie et al. [START_REF] Beattie | The compressibility of gaseous xenon. I. An equation of state for xenon and the weight of a liter of xenon[END_REF][START_REF] Beattie | The compressibility of gaseous xenon. II. The virial coefficients and potential parameters of xenon[END_REF], and Rabinovich et al. [START_REF] Rabinovich | Experimental determination of the compressibility of krypton and xenon in the 300-720 K range and up to 400 bars[END_REF][START_REF] Rabinovich | Equation of state of a monoatomic liquid at high densities[END_REF][START_REF] Abovskii | Cell model for monoatomic gases and liquids[END_REF]. Increasing thus the covered temperature distance to T c leads to observe the continuous decrease of γ e in the range 2 × 10 -2 τ * 1 (see the corresponding segments 6 to 9 in Fig. 1a). Practically, the τ * range where γ e γ MF = 1 starts in the well-defined temperature range 400 < T (K ) < 500 [START_REF] Garrabos | Contribution à l'étude des propriétés d'état des fluides purs dans leur région critique[END_REF]. Therefore, the physical mean field limit τ * 1 is reached after crossing down the effective mean value γ 1 2 = γ +γ MF 2 1.12 around τ * 0.14 [see Fig. 3 See Table 4.3.4, p. 144 in reference [START_REF] Levelt Sengers | Critical phenomena in classical fluids, chap. 4[END_REF]. 1a] and by following an abrupt γ e decrease to γ e 1, over a well-descrimined sharp domain 0.3 ≤ τ * CO ≤ 0.5 (see the double arrow in Fig. 1b). In Fig. 1a, this mean-field-like side of the physical crossover is illustrated by the interpolating line labelled CO from γ 1 2 in the MR curve and γ e γ MF = 1 at τ * exp 0.5 in the segment 9. However, the predicted mean-field-like side from the theoretical crossover function (right part of the figure) is not compatible with these experimental results, which means that the physical crossover (curve CO) is not accounted for by the MR approach (curve MR) of the classical-to-critical crossover (see also Ref. [START_REF] Bagnuls | Nonasymptotic critical behaviour from field theory for Ising like systems in the homogeneous phase: theoretical framework[END_REF][START_REF] Bagnuls | Experimental data analysis on xenon above the critical temperature from nonlinear renormalization group[END_REF]).

3 for a 1 (

 31 i.e., for ξ comparable or smaller than the range of the microscopic interaction forces), leading to τ PF ∝ T κ T far away from the critical temperature. When a 1, F (a) ∝ 2[ln(2a)-1] a

0 2 -

 2 η (involved through the Martin-Mayor et al. Eq. (

h 2 =∂ ∂h 2 h 1 =

 21 k B (α c ) d . Therefore, g * = n mol,c μ mol p c and g * = n mol,c p c μ mol (with n mol,c = n p,c N A ). The scaling field g * can then be related to the previously introduced dimensionless ordering field μ * p (or μ) by the equation g * = 1 Z c μ * p (or g * = μ). In such a thermodynamic representation, the thermodynamic potential corresponds to the density J V of the Grand potential J = -pV and can be written as = pV V 1 p c = j p c = p * in dimensionless units. The singular thermodynamic potential is = c = p * -1. The corresponding conjugated parameters to h 1 and h 2 are ϕ 1 = -∂ ∂h 1 n molb 2 s and ϕ 2 = s, respectively. We recall here that the order parameter n mol = n mol n mol,c -1 is related to the previous order parameters by n mol = ρ = ρ ρ c -1 and n mol = 1 Z c n * p (with ρ = n mol M mol = n p m p ). In addition, s = ss c is the difference between the dimensionless entropy density s = S V T c p c and its (arbitrary) value s c = S c V T c p c at the critical point (with s ≡ s * and s ≡ s * , where the dimensionless variables with asterisk use

  μ=μ mol = n mol (T, p). Using the previous relations between densities expressed in units of mole, mass, and particle, the corresponding dimensional susceptibilities are related by χ mol = M 2 mol χ ρ = N 2 A χ p (with μ mol = ∂a ∂n mol T and (χ mol ) -1 = ∂μ mol ∂n mol T = ∂ 2 a ∂n 2 mol T

6. 2

 2 Ising-Like Parameters for the Crossover Formulation of the Parametric EOS

6. 4

 4 Comparison of the Calculated Isothermal Compressibility, Coexisting Densities, Correlation Length and Turbidity of Xenon Using CPM and CMM Using the arbitrary relation (c t ) 1 2

Table 1

 1 Critical coordinates and physical quantities for xenon (with M mol (Xe) = 131.29 g mol -1 and N A = 6.02214129 × 10 23 mol -1 ), and characteristic parameters for the light transmission experiments of GC (see also Refs. given in the last column)

		Xe	Ref.
	m p	2.1805 × 10 -25 kg
	T c	289.733 ± 0.015 K	[15,36]
	p c	5.84007 ± 0.00050 MPa	[15]

Table 2

 2 Estimated universal features in the master form (column 2) of the MR scheme and in the CPM

	(column 3)
			MR [4]	C P M [ 16]	R %
	Exponents
	γ		1.2395935	1.239	-0.048
	β		0.3257845	0.3255	-0.087
	α		0.1088375	0.110	1.07
	ν		0.6303875	0.630	-0.061
	η		0.033604	0.0333	-0.90
			0.50189	0.52	3.6
	Combinations
	R + C =	+ B 2 A +	0.0574 ± 0.0020	0.0580	1.05

  that the applicability range of the master crossover functions corresponds to ξ 2.5 nm (leading to ξ α c > 3 -5, i.e., ξ

	and τ * 0.02	L Xe EAD , typically, in the xenon case).	r LJ e	> 6 -10

Table 3

 3

		ν	ξ + 0 (nm)	a 1,+ ξ	τ *	Ref.
	1	0 .60	0.302	n.u.		[43]
	2	0 .58 ± 0.05	0.3 ± 0.01	n.u.		[42]
	3	0 .57 ± 0.03	0.307	n.u.		[44]
	4	0 .64 ± 0.02	0.136 ± 0.006	n.u.		[45]
	5	0 .63	0.2	n . u .		[ 42,46]
	6	0 .63	0.167	n.u.		[46]
	7	0 .58	0.232	n.u.		[46]
	8	0 .62	0.193	n.u.		[47]
	9	0 .63	0.184 ± 0.009	0.55 τ * 0.5	[1]
	10	0.63	0.184 ± 0.009	n.u.		[1,48]
	11	0.63	0.1866 ± 0.0010	n.u.		[1,36,49]
	12	0.6303875	0.184531	0.55 τ * 0.50189	This work
	12: Corresponding asymptotic result from Eq. (

1 to 11: Literature exponent-amplitude values for fitting Eq. (

11

) of xenon correlation length when τ → 0 along the critical isochore (n.u.: not used)

  γ e (36) with adjustable effective (i.e., non-Ising) exponent γ e and adjustable effective amplitude + e . The corresponding γ e and + e values are reported in Table 6, in addition with the related finite extension τ * min -τ * max and mean position τ * exp = τ * min τ *

		1
	max	2 of the experimental
	temperature range.

Table 6 γ

 6 e (column 3) and e (column 4) values published before the date of I, from κ * T = + e τ * -γ efitting in a restricted temperature range τ * min ≤ τ * ≤ τ * max along the critical isochore of xenon for T > T c (() corresponds to a fixed value in the fit) 10 -4 1.76 × 10 -2 1.653 × 10 -3 [42] × 10 -5 1.0 × 10 -1 4.35 × 10 -3 I and [15] 6 pV T γ e,eos = 1.190 0.0793 5.0 × 10 -4 3.0 × 10 -2 3.873 × 10 -3 [63-65,67] 7 pV T 1.1665 0.089 6.2 × 10 -3 8.1 × 10 -2 2.24 × 10 -2 [11,67-72] × 10 -1 1.70 × 10 -1 [66-72] ): ordered number of the segment in Fig. 1a (see text). Measurement method (column 2): pV T : [ p (V )] T analysis; LS: light scattering; FI: Fraunhofer interferometry. τ * min -, τ * max -and τ * exp -values (columns 5 to 7, respectively); References (column 8)

	#	γ e	+ e	τ * min	τ * max	τ * exp	Ref.
	1a FI	γ Ising = 1.24	0.058 ± 0.002	10 -6	10 -5	3 × 10 -6	[50,51]
	1b FI	1.23	0.062 ± 0.006	10 -6	10 -5	3 × 10 -6	[50]
	2 pV T 1.211 (±0.025)	0.0743 (±0.015) 6.0 × 10 -4	6.0 × 10 -3 1.9 × 10 -3	[62-64]
	3 LS	1.21	0.076845	6.9 × 10 -4	6.9 × 10 -2 6.9 × 10 -3	[43]
	4 LS 1.553 × 5 LS 1.21 ± 0.03 0.06742 1.206 0.07551466 9.6 8 pV T γ 1 2 = 1.12 0.10 10 -1 2.9 9 pV T (γ MF = 1) 0.11 0.38 0.72	5.2 × 10 -1	[66,68-72]
	# (column 1

A theoretical description of I q, ξ τ * , n * p needs to generally assume symmetrized singular behaviors of the isothermal susceptibility and the correlation length around the critical isochore n p = n p,c .

Acknowledgments

We acknowledge discussions and helpful comments from M. A. Anisimov. YG acknowledges F. Palencia for its contribution in the data analyses using Mathematica. The research at ICMCB and PMMH was supported by CNRS and CNES, and at the Jet Propulsion Laboratory, California Institute of Techology, by NASA.