1
Study of supercritical krypton by small-angle neutron
scattering: Analysis of the cross-over region
M. Bonetti
Service de Physique de l’Etat Condensé
CEA de Saclay
91191 Gif-sur-Yvette cedex, France
P. Calmettes
Laboratoire Léon Brillouin
CEA de Saclay
91191 Gif-sur-Yvette cedex, France
C. Bervillier
Service de Physique Théorique
CEA de Saclay
91191 Gif-sur-Yvette cedex, France
Abstract: Small-angle neutron scattering spectra of supercritical krypton have been measured
in the vicinity of the critical point along the critical isochore and at off-critical densities.
The reduced temperature, τ~ = (T – Tc) / Tc, and density, ρ~ = (ρ – ρc) / ρc, were in the ranges
10−3 < τ~ < 9.5×10−2 and – 0.263 < ρ~ < + 0.246, respectively. T and ρ are the temperature and
the density, and Tc and ρc their critical values, respectively. A fit of a parametric
representation of the structure factor to the spectra measured along the critical isochore allows
the amplitudes of the correlation length and of the isothermal compressibility to be
determined accurately. Within the studied range of wave-number transfers, q, the amplitude of
a q3 term corresponding to the r−6 dependence of both the two-body van der Waals and the
three-body Axilrod-Teller potentials can be determined. However the amplitude of the density
dependent three-body Axilrod-Teller interaction cannot be inferred from the spectra measured
at off-critical densities.
2
I. INTRODUCTION
Understanding the behavior of supercritical fluids requires an accurate description of
the asymptotic critical behavior as well as of the regular one far from the critical point. To
obtain an equation of state that describes the cross-over region between these two regimes is a
challenging task.1,2 Small-angle neutron scattering (SANS) measurements are well suited to
determine the critical correlations length in a very large thermodynamic domain surrounding
the critical point. SANS experiments performed along the critical isochore and at off-critical
densities give information about the form of the corrections to the critical scaling expressions
for the isothermal compressibility and the correlation length3 and allows the validity of the
theoretical scaling function for the pair correlations to be tested accurately.3-5
SANS measurements performed on supercritical D2O pointed out the importance of
contributions from the short-range intermolecular correlations of position and orientation.6
Moreover, because of a pronounced asymmetry associated to a large value of the slope of the
rectilinear diameter of the coexistence curve of D2O and to the mixing of the thermodynamics
variables, odd terms in the corrections to scaling had to be taken into account in the analysis
of the scattering data.3
We performed similar measurements on supercritical krypton (Kr). In this case the
interpretation of the experimental data is greatly simplified compared to that of D2O because
the short-range correlations are isotropic. In addition the slope of the rectilinear diameter of
the coexistence curve of Kr is smaller by a factor of about 3 compared to that of D2O. As a
result, the behavior of Kr is expected to be more symmetric with respect to the critical
isochore. The structure factor of Kr contains a well-documented term proportional to q3, q
being the wave-number transfer.7-14 This term results from the r−6 dependence of both the
two-body van der Waals and the three-body Axilrod-Teller (A-T) potentials.
One of the main results of the present study is that only an even correction to the pure
scaling expression for the isothermal compressibility is necessary to describe the experimental
data whereas the behavior of the correlation length is slightly asymmetric with respect to the
critical isochore. An asymmetric correction to the scaling form of the pair-correlation function
is also required.
The paper is organized as follows. Section II describes the experimental setup used to
perform the SANS measurements on supercritical Kr. An accurate and uniform thermalization
is essential for obtaining a homogeneous density of the sample. Section III briefly reviews the
3
formalism that has been widely developed for a supercritical molecular fluid. For a detailed
derivation of this formalism the reader is referred to the paper of Bonetti et al.3 Section IV is
concerned with the results of the fit of the theoretical expressions to the scattering spectra that
were collected along the critical isochore and those obtained from a fit to the whole set of
spectra measured at the critical and off-critical densities.
II. EXPERIMENT
A. Sample cell for small-angle neutron scattering
99.998% pure
36
Kr was purchased from Air Liquide (France). The Kr sample was
contained in a cylindrical cell, 20 mm in diameter, drilled in a Cu-Be parallelepiped
(34 × 43 × 30 mm3). The two apertures are closed by 10 mm thick sapphire windows and the
fluid path length is 10 mm. The neutron beam is parallel to the bore axis. The small
dimensions of the cell and the close values of the thermal diffusivities of Cu-Be alloy and of
sapphire result in a fast thermalization of the fluid and a minimization of the temperature
gradients in the sample.15,16
The sample cell is placed in the electrolytic copper secondary stage of a continuous
flow cryostat.17,18 The first stage cooled by gaseous nitrogen sucked up from a storage dewar.
A constant nitrogen flow is achieved by means of a mass flow controller (Brooks Instrument,
the Netherlands) and adjusted according to the required temperature which is regulated by
means of a 100 Ω heater imbedded in the body of the stage. In order to generate a temperature
difference between the two stages and to allow for a precise temperature regulation of each
stage, an insulating glass fiber shim is sandwiched between them. Heat transfer between the
stages is achieved by means of three copper pads. The temperature of the secondary stage is
controlled by a 100 Ω Manganin heater wounded around the copper block. The temperature of
each stage is measured with a ±150 mK absolute accuracy by means of 100 Ω platinum
resistances. The temperature stability of the second stage is better than 1 mK at 210 K. The
whole thermostat is placed in a cylindrical vacuum chamber with two opposite quartz
windows, 40 mm in diameter. They allow the homogeneity of the sample and its phase
separation to be observed and the SANS measurements to be performed. For a neutron beam
diameter of 7 mm the maximum forward scattering angle is φmax = 15°.
4
The cell is filled through a small shut-off valve connected by a capillary tube to a
secondary standard pressure gauge with an absolute accuracy of 0.02 bar (PAP31/200 gauge,
HBM, Germany). Before filling, the cell was heated at 45°C and the capillary tubes and the
pressure transducer were thoroughly flushed with Kr. Finally the cell was vacuum pumped for
several days. Condensation of Kr in a vessel at liquid-nitrogen temperature allowed the
density of the sample to be adjusted by means of a micro metering valve.8
B Critical coordinates of krypton
The sample density was adjusted to the critical value by looking at the level at which
the meniscus appears in the cell.19,20 At the critical density the meniscus must be in the middle
of the cell. This method allows the values of the critical coordinates to be determined. The
experimental values of the critical temperature and pressure were: Tc,exp = (209.34 ± 0.15) K
and Pc,exp = (55.05 ± 0.02) bar, respectively. The uncertainty on Tc,exp corresponds
to
the
absolute
accuracy
of
the
temperature
probe.
From
the
value
of
(∂P/∂T)ρ = (1.562 ± 0.002) bar K−1 measured between 209.32 K and 210.15 K, the critical
density is inferred to be ρc,exp = (0.911 ± 0.012) g cm−3 using the equation of state of Lemmon
et al.21 Our experimental values of the critical coordinates, Tc, Pc, and ρc, of Kr compare well
with those already published: 209.29 K, 54.93 bar, and 0.908 g cm−3 in Ref. 22 and
209.35 K, 55.01 bar, and 0.919 g cm−3 in Ref. 23. No discontinuity in (∂P/∂T)ρ was observed
at the critical point. This means that the sample was at critical density.24,25
C SANS measurements and thermodynamic coordinates
The SANS measurements were performed with the PACE spectrometer in the
Laboratoire Léon Brillouin (CEA, Saclay). This spectrometer is equipped with 30 concentric
annular detectors that are centered with respect to the neutron beam by means of a fourquadrant detector in a central position. The neutron wave length was λ = 0.289 nm. Such a
short value was obtained by tilting the neutron selector with respect to the beam. Scattering
spectra were recorded for wave-number transfers, q, ranging from 0.48 to 3.7 nm–1.
q = (4π / λ) sin(φ /2), where φ is the scattering angle.
Because of the low value of the coherent scattering cross-section of Kr, each spectrum
was recorded longer than 6 hours in order to obtain a statistical accuracy better than 1.5%.
5
Each raw spectrum was divided by the corresponding transmission and the contribution of the
empty cell was subtracted from the sample spectra. To account for the lack of uniformity of
the detector efficiency, the resulting spectra were normalized to the spectrum of a light water
sample with a 1 mm path length.26
Table I gives the thermodynamic coordinates at which the SANS measurements were
performed. Off-critical density measurements performed along a near-critical isotherm were
discarded because density stratification occurred in the cell even at well off-critical densities.
Fig. 1-a shows the spectra measured along the critical isochore above Tc,exp. The recorded
spectra at off-critical densities measured at (Tc,exp + 9.81) K and (Tc,exp + 19.81) K are shown
in Fig. 1-b and 1-c, respectively.
III. SCATTERING THEORY FOR AN ATOMIC FLUID
The present analysis of the scattering data is based on the theory developed for a
molecular liquid as thoroughly discussed by Bonetti et al.3 The expression for the intensity
scattered from a unit volume of a monatomic fluid simply is
2
2
I (q ) = n [bcoh
S (q) + binc
],
(1)
where n is the number density. bcoh and b inc are the coherent and incoherent scattering lengths
of the atoms, respectively. S(q) = [1 − n c(q)]−1 is the structure factor of the fluid and c(q) the
Fourier transform of the direct correlation function. In mean field theory it is expanded as
follows:11
c(q) = c0 + c2 q2 + c3 q3 + c4 q4 +….
(2)
In this expansion only the first three coefficients are known. c0 = n−1{1 − [S(0)]−1} with
S(0) = n kB T χT, where kB is Boltzmann’s constant and χT the isothermal compressibility.
c2 = −ξ2 [n S(0)]−1 where ξ is the correlation length of density fluctuations. Finally
c3 = (A2 − n A3) T
−1
,
(3)
6
where the coefficients A2 and A3 are proportional to the intensity of the two-body van der
Waals and the three-body Axilrod-Teller potentials, respectively. These two potentials have
an asymptotic r−6 dependence, where r is the distance between two atoms.7,9-11
Near the critical point Eq. (1) for the scattered intensity can be generalized in the
following form:
−1



3
4
2  1
2 
I (q ) = n bcoh 
− n c3 q − n c 4 q + ... + binc
,
 S c (q )



(4)
where
~
S c (q) = n k B T χ T G (qξ ,θ )
(5)
~
is the critical part of the structure factor. G (qξ ,θ ) is the corresponding pair-correlation
function. θ is a parameter measuring the distance from the critical isochore.27 Contrary to the
expression used in Ref. 3 for I(q), Eq. (4) and (5) ensure that ξ 2 is the second moment of the
pair-correlation function.
In the (R,θ) coordinates system27 used for the parametric description of the
thermodynamic space around the critical point, the general expressions for χT and ξ are
χ T = χ T ,s ( R,θ ) [1 + a χ R g χ (θ ) + ...]
(6)
ξ = ξ s ( R,θ ) [1 + aξ R gξ (θ ) + ...] ,
(7)
and
where χ T ,s ( R,θ ) and ξ s ( R,θ ) are the pure critical scaling approximations for the isothermal
compressibility28 and correlation length,3 respectively. aχ and aξ are the amplitudes of the first
corrections to scaling. Their ratio is universal: aχ / aξ = 0.65.29,30 The exponent ∆ = 0.504 is
also universal.31 The general expressions for the functions gχ(θ) and gξ(θ) reads3
g χ (θ ) = 1 + a χθ θ + a χθ 2 θ 2 + ...
(8)
7
and
g ξ (θ ) = 1 + aξθ θ + aξθ 2 θ 2 + ...
(9)
Along the critical isochore, θ = 0, so that gχ(θ ) = gξ(θ ) = 1. The linear θ dependence
accounts for the mixing of the thermodynamic variables in the relevant scaling fields.32,33 The
values of the four parameters
χθ
,
χθ 2
, aξθ , and aξθ 2 are system dependent. They can be
inferred from scattering measurements performed at off-critical densities.
To account for the asymmetry that may result from the analytic correction
corresponding to a finite slope, D, of the rectilinear diameter of the coexistence curve, the
reduced density, n~’ , involved in the scaling equation of state was assumed to be3
n~’ = n~ − D τ~ ,
(10)
where n~ = (n – nc) / nc and τ~ = (T − Tc) / Tc. nc and Tc are the critical values of the number
density and of the temperature, respectively.
~
In Eq. (5) the critical pair-correlation function G (qξ ,θ ) is approximated as follows:

( qξ ) 2 
~
2
G (qξ ,θ ) = Gs (qξ ) + (eθ θ + eθ 2 θ )
,
1 + ( qξ ) 4 

(11)
where Gs(qξ) is the usual scaling form that may be written34
3
Gs (qξ ) = ∑ c j [1 + a 2j (qξ ) 2 ]
−b j
.
(12)
j =1
with c1 = c2 = 1 and c3 = −1; a1 = 1.040056, a2 = 1.058947, and a3 = 1.053932; b1 = (1 − η /2),
where η = 0.034 is the Fisher exponent,31 b2 = 1.554213, and b3 = 1.627419. Eq. (12)
represents very accurately the result of Bray et al.35 which gives the mathematical expression
of the pair correlation function along the critical isochore (θ = 0).
8
Quite generally the form of the correlation function is different for thermodynamic
states not pertaining to this particular path.4 This accounts for the second term in the righthand side of Eq. (11). The form of this correction is chosen so as to vanish along the critical
isochore and in such a way that the usual asymptotic scaling behavior of the correlation
function is recovered for both qξ → 0 and qξ → ∞. The parameters eθ and eθ 2 are arbitrary.
IV. RESULTS
Eqs. (4-12) were fitted36 to all the neutron scattering spectra measured at the
thermodynamic coordinates given in Table I. The critical exponents were fixed to their
theoretical 3-D Ising values31 as well as the ratio aχ / aξ.29,30 The slope of the rectilinear
diameter in Eq. (10) was computed from the P, V, and T data of Ref. 21 and found to be
D ≅ − 0.78. Such a value leads to a negligible asymmetry of the fluid properties in the
investigated thermodynamic domain. Corrections to scaling of order higher than R∆ in Eqs.
(6) and (7), and the parameter c4 in Eq. (4) were found to be unnecessary to describe the data
and will be neglected hereafter.
A. Critical density
SANS measurements performed along the critical isochore allow the critical
−γ
amplitudes χT,0 and ξ0 of the isothermal compressibility, χ T ,s = χ T ,0 τ~ , and of the
−ν
correlation length, ξ s = ξ 0 τ~ , to be directly determined. The amplitudes, aχ and aξ , of the
first correction to scaling in Eqs. (6) and (7) and the coefficients c3 in Eq. (4) can also be
estimated.
Figure 2-a shows the results of the fit of Eqs. (4-9) and (11-12) to all the scattering
spectra measured along the critical isochore in the temperature range 0.001 < τ~ < 0.095. For
0.43 < qξ < 50 the relative deviations of the fit from the measured spectra are randomly
distributed and smaller than ±2.5%. This compares with the 1%-2% uncertainty of Bray’s
approximation35 as assessed from the comparison with the experimental data of Damay et al.5
and the results of the Monte Carlo simulation of Martin-Mayor et al.4
9
The values of the adjustable parameters were found to be χT,0 = (1.08 ± 0.02)×10−3
bar−1, ξ0 = (0.171 ± 0.001) nm, aχ = (0.96 ± 0.34), and c3T = (0.76 ± 0.16) nm6 K.
The absolute value of χT,0 was obtained by scaling the measurements at T = 229.15 K
with respect to the value calculated from the P, V and T data of Ref. 21. The value
of the critical amplitude of the dimensionless isothermal compressibility is found to be
Γ = χT,0 Pc = (0.060 ± 0.001), which is close to a rough theoretical prediction Γ = 0.058.37
This is the first experimental determination of the critical amplitude of the isothermal
compressibility of Kr.
The value of the amplitude of the correlation length compares rather well to that which
can be estimated from the two-scale factor universality hypothesis which gives38,39
 B 2 Pc 

ξ 0 = R 

k
T
Γ
B
c


−1 / 3
,
(13)
where R = (0.692 ± 0.0043) is a universal constant39 and B is the critical amplitude of the
coexistence curve. Using the value B = 1.42 given by Sengers et al.37 and the present
experimental values of Γ, Pc, and Tc one obtains ξ0 = 0.173 nm.
The value of (0.76 ± 0.16) nm6 K obtained for the coefficient c3T = (A2 − nc A3)
compares pretty well with the theoretical expectation of to Reatto and Tau.11 According to
these authors A2 = 0.7465 nm6 K and A3 = 0.011 nm9 K for Kr, hence c3T = 0.7393 nm6 K.
The importance of the q3 term in Eq. (4) is shown in Fig. 3 where the direct correlation
function, c(q), is plotted as a function of q for three different temperatures and compared to
what would be obtained if c3T = 0.
B. Critical and off-critical densities
Eqs. (4-9) and (11-12) were also fitted to the whole set of scattering spectra measured
at both the critical and the off-critical densities given in Table I. Additional parameters are
now involved in the fit. These are
χθ
,
χθ 2
, aξθ , and aξθ 2 in the functions gχ(θ ) and gξ(θ )
~
defined by Eqs. (8) and (9), eθ and eθ 2 in the scaling function G (qξ ,θ ) , Eq. (11), and A2 and
A3 in Eq. (3) instead of c3T = (A2 − nc A3). χT,0, ξ0 and aχ were set to the values previously
10
obtained at critical density. This allows the contribution of each correction to scaling to be
precisely analyzed. Only the terms
be
necessary
parameters
χθ
to
correctly
χθ 2
and aξθ in the functions gχ(θ ) and gξ(θ ) were found to
describe
all
the
scattering
spectra.
The
redundant
, aξθ 2 and eθ 2 have been neglected. The main results of the fit are given in the
second and third columns of Table II.
When the coefficients A2 and A3 in Eq. (3) are regarded as free parameters, they cannot
be significantly determined. The large uncertainty on A3 results from the fact that the
amplitude of the Axilrod-Teller potential is small compared to that of the two-body term.
Indeed, nc A3 is about ten times smaller than A2 at the critical density. Furthermore the density
range covered by the present measurements is too small to allow the amplitude of the A-T
potential to vary appreciably. This has already been mentioned in the studies of gaseous
krypton by Formisano et al.12 The third column of Table II shows the result of the fit when it
is assumed that c3T = (A2 − n A3) does not vary with the density, n. The well defined value
c3T = (0.77 ± 0.03) nm6 K is very close to the value of the quantity (A2 − nc A3) which can be
computed with the values of A2 and A3 given in the second column.
The maximum relative deviations between the fitted function and the experimental
spectra are lower than ±5%. The largest deviations comes from off-critical data. This may be
due to Eq. (11) which is to crude an approximation to describe the correlation function for
densities different from the critical one.
In Figs. 2-b and 2-c, the scattering spectra measured at off-critical densities along the
two isotherms T = (Tc,exp + 9.81) K and T = (Tc,exp + 19.81) K are compared to the result of the
fit with the values of the adjustable parameters given in the third column of Table II. Table III
gives the corresponding values of the parametric coordinates R and θ, and those of the
correlation length ξ.
Fig. 4 shows the correlation length, ξ, as a function of the density for temperatures
between (Tc,exp + 0.26 K) and (Tc,exp + 19.81 K). ξ is computed from Eqs. (7) and (9) using the
parameter values given in the third column of Table II. With respect to the critical isochore, a
slight asymmetric behavior of ξ can be noticed at the two highest temperatures (Fig. 4-b).
To summarize, Eqs.(4-12) with D = 0 in Eq. (10) fit well to the SANS spectra
measured at the critical density in a large range of qξ values, i.e. between 0.44 and 49. The
deviations are randomly distributed and smaller than ±2.5%. The critical amplitudes of the
dimensionless isothermal compressibility is found to be Γ = (0.060 ± 0.001) and that of the
11
correlation length ξ0 = (0.171 ± 0.001) nm, in excellent agreement with the predictions of the
two-scale factor universality hypothesis.
In the studied q range, the measurements performed along the critical isochore points
out the importance of long-range tail of the interaction potentials (Fig. 3). The contribution of
the density dependent Axilrod-Teller potential cannot be inferred from off-critical density
measurements neither for the gaseous state nor the liquid one. Up to now the amplitude of the
three-body interaction potential have mainly been measured in the gaseous phase13 and more
recently in the liquid one14 far from the critical point where the critical fluctuations are weak.
Using appropriate expressions for the isothermal compressibility and the correlation length,
the present SANS study of near critical Kr shows that the r−6 tails of the three-body and
especially of the two-body interaction potentials give an important contribution to the paircorrelation function. Such a contribution cannot be a priori neglected in the interpretation of
the scattering spectra from any fluid in a near critical state. However this is generally done.
For instance we have neglected this contribution in our previous analysis of the SANS
scattering spectra of D2O.6,3 As a matter of fact these spectra can also be described using the
approach given above.
Assuming that the slope of the diameter of the coexistence curve is D = −1.663 as in
Ref. 3, the quality of the fit remains unchanged. The values of all the adjustable parameters
remain close to those previously obtained but are much better defined. We find that the
amplitude of the correlation length is ξ0 = (0.137 ± 0.002) nm instead of ξ0 = (0.136 ± 0.006),3
and that c3T = (0.84 ± 0.02) nm6 K. A q2 term in the right hand side of Eq. (4) is still
necessary to describe the data. It comes from the orientation averaged cross form-factor,
f2(q) = (1 − A12 q2 − A24 q4 − ...), of a pair of angularly correlated molecules.3,6 We find that
A1 = (0.153 ± 0.005) nm instead of ζ1 = (0.13 ± 0.08) nm and that A2 is no more necessary to
describe the data.3 These results demonstrate that the contribution of the expected r−6 tails of
the interaction potential can be detected and quantified close to the critical point of both
atomic and molecular fluids.
ACKNOWLEDGEMENT
We would like to thank P. Pari for useful advices for the realization of the low-temperature
thermostat and V. Padilla and C. Gasquet for technical assistance.
12
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37. J. V. Sengers and J. M. H. Levelt Sengers, Ann. Rev. Phys. Chem. 37, 189 (1986).
38. D. Stauffer, M. Ferer, and M. Wortis, Phys. Rev. Lett. 29, 345 (1972).
39. C. Bagnuls, C. Bervillier, D. I. Meiron, and B. G. Nickel, Phys. Rev. B 35, 3585 (1987).
14
TABLE CAPTIONS
Table I. Thermodynamic coordinates at which the SANS spectra of supercritical krypton were
measured. T is the temperature, P the pressure, ρ the density, and n the number density. The
values of P and of (∂P/∂T)ρ labeled by a star were measured during the adjustment of the
density to its critical value (see text). The experimental values of the critical coordinates are:
Tc,exp = (209.34 ± 0.15) K, Pc,exp = (55.05 ± 0.02) bar, and ρc,exp = (0.911 ± 0.012) g cm−3.
Table II. Values of the adjustable parameters from the fits of Eqs. (4-12) to the SANS spectra
obtained for the thermodynamic coordinates given in Table I. The uncertainties correspond to
one standard deviation. The first column gives the values of the parameters for the
measurements carried out along the critical isochore for temperatures in the range
0. 21 K < (T − Tc,exp) < 19.81 K. The second and third columns give the values of these
parameters for the spectra measured both at the critical and the off-critical densities listed in
Table I. The values in square brackets were fixed in the fits. The slope, D, of the rectilinear
diameter and the parameters
χθ
, aξθ 2 , eθ 2 , and c4 were set to zero.
Table III. Values of the parametric thermodynamic coordinates R and θ, and of the correlation
length ξ calculated using the values of the fitted parameters given in the third column of
Table II. The last column gives the range of qξ values in which the fits were performed. q is
the wave-number transfer.
15
FIGURE CAPTIONS
Fig. 1. Normalized neutron scattering spectra n−1I(q) of krypton as a function q. n is the
number density and q the wave-number transfer. (a) Spectra at the critical density. (b) and (c)
Spectra at off-critical densities measured at (Tc,exp + 9.81) K and (Tc,exp + 19.81) K,
respectively. The critical temperature is Tc,exp = (209.34 ± 0.15) K. The increase of the
scattered intensity at low values of q is due to the divergence of the critical fluctuations. The
symbols correspond to the experimental data and the full lines are guides for the eye.
Fig. 2. Normalized neutron scattering spectra n−1I(q) as a function of qξ, where q is the wavenumber transfer and ξ the correlation length. (a) Spectra at the critical density. (b) and (c)
Spectra at off-critical densities measured at 9.81 K and 19.81 K above the critical temperature
respectively. The full lines are the results of a fit of Eqs. (4-12) with the experimental data
(symbols). The values of the parameters given in the third column of Table II.
Fig. 3. Direct correlation function c(q) versus the wave-number transfer q. Full line: c(q)
computed from the fit of Eqs. (4-12) with the spectra measured at critical density. Dashed
line: Without the term n c3 q3 in Eq. (4) but with the same values of the adjustable parameters
(Table II, 1st column). Symbols: Experimental data.
Fig. 4. Correlation length ξ as a function of the number density (n − nc). (a) Full lines:
Correlation length computed using the parameter values of the third column of Table II. From
top to bottom, the temperatures are 0.21, 0.36, 1.06, 2.31, 4.81, 9.81 and 19.81 K above the
critical temperature, respectively. Open circles: Data at critical density. (b) Magnification for
the two highest temperatures. Upper line: (Tc,exp + 9.81 K) and lower line : (Tc,exp + 19.81 K).
Symbols: Data at off-critical densities (Table I).
16
T
P
ρ
n
(∂P/∂T)ρ
(T – Tc,exp)
(ρ – ρc,exp)
± 0.15
± 0.02
± 0.012
± 0.09
± 0.002
± 0.006
± 0.012
(bar K )
(K)
(g cm−3)
−3
−3
−1
(K)
(bar)
(g cm )
229.15
-
0.911
6.546
-
19.81
0
219.15
-
0.911
6.546
-
9.81
0
214.15
-
0.911
6.546
-
4.81
0
211.65
-
0.911
6.546
-
2.31
0
210.40
56.67*
0.911
6.546
1.5623*
1.06
0
209.70
55.59*
0.911
6.546
1.5623*
0.36
0
209.55
55.36*
0.911
6.546
1.5623*
0.21
0
229.15
78.40
0.692
4.973
1.1869
19.81
-0.240
219.15
67.41
0.692
4.973
1.1869
9.81
-0.240
229.15
81.97
0.777
5.584
1.3642
19.81
-0.147
219.15
69.27
0.777
5.584
1.3642
9.81
-0.147
229.15
87.02
0.962
6.913
1.6173
19.81
0.056
219.15
71.80
0.962
6.913
1.6173
9.81
0.056
229.15
90.08
1.054
6.913
1.7593
19.81
0.157
219.15
73.60
1.054
6.913
1.7593
9.81
0.157
229.15
94.97
1.115
8.012
1.9354
19.81
0.224
219.15
76.94
1.115
8.012
1.9354
9.81
0.224
Table I
nm
17
ρ = ρc,exp
ρ = ρc,exp
ρ = ρc,exp
and
and
ρ ≠ ρc,exp
ρ ≠ ρc,exp
χT,0 (bar−1)
(1.08 ± 0.02)×10−3
[1.08×10−3]
[1.08×10−3]
ξ0 (nm)
0.171 ± 0.001
[0.171]
[0.171]
aχ
0.96 ± 0.34
[0.96]
[0.96]
a χθ 2
[0]
1.31 ± 0.41
1.25 ± 0.39
aξθ
[0]
-2.02 ± 1.36
-0.83 ± 0.69
eθ
[0]
-0.33 ± 0.19
-0.19 ± 0.15
c3T (nm6 K)
0.76 ± 0.16
−
0.77 ± 0.03
A2 (nm6 K)
−
1.75 ± 1.04
−
A3 (nm9 K)
−
0.15 ± 0.16
−
Deviations (%)
± 2.5
±5
±5
Table II
18
(T – Tc,exp)
(ρ – ρ c,exp)
± 0.006
± 0.012
(K)
(g cm−3)
19.81
0
0.0946
0
0.90
0.434, 3.327
9.81
0
0.0469
0
1.34
0.644, 4.935
4.81
0
0.0230
0
2.02
0.973, 7.461
2.31
0
0.0110
0
3.12
1.505, 11.535
1.06
0
0.0051
0
5.00
2.410, 18.481
0.36
0
0.0017
0
9.71
4.679, 35.872
0.21
0
0.0010
0
13.55
8.166, 50.093
19.81
−0.240
0.1224
−0.4765
0.83
0.695, 3.048
9.81
−0.240
0.0697
−0.5725
1.12
0.807, 4.126
19.81
−0.147
0.1045
−0.3070
0.88
0.638, 3.261
9.81
−0.147
0.0547
−0.3790
1.26
0.909, 4.649
19.81
0.056
0.0960
0.1201
0.88
0.423, 3.244
9.81
0.056
0.0479
0.1506
1.30
0.625, 4.791
19.81
0.157
0.1059
0.3262
0.80
0.580, 2.964
9.81
0.157
0.0559
0.4018
1.15
0.827, 4.230
19.81
0.224
0.1185
0.4487
0.74
0.531, 2.716
9.81
0.224
0.0663
0.5419
1.01
0.732, 3.740
Table III
R
θ
ξ
(qξ)min, (qξ)max
(nm)