Experimental observation of triple correlations in fluids
We present arguments for the hypothesis that under some conditions, triple correlations of density fluctuations in fluids can be detected experimentally by the method of molecular spectroscopy. These correlations manifest themselves in the form of so-called 1.5-(or sesquialteral) scattering. The lat...
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irk-123456789-1210742017-06-14T03:05:21Z Experimental observation of triple correlations in fluids Sushko, M.Ya. We present arguments for the hypothesis that under some conditions, triple correlations of density fluctuations in fluids can be detected experimentally by the method of molecular spectroscopy. These correlations manifest themselves in the form of so-called 1.5-(or sesquialteral) scattering. The latter is of most significance in the pre-asymptotic vicinity of the critical point and can be registered along certain thermodynamic paths. Its presence in the overall scattering pattern is demonstrated by our processing available experimental data for the depolarization factor. Some consequences of these results are discussed. Наведено аргументи на користь гiпотези, що при певних умовах методом молекулярної спектроскопiї можна зареєструвати потрiйнi кореляцiї флуктуацiй густини в рiдинах. Цi кореляцiї проявляють себе у виглядi так званого 1.5- (тобто пiвторакратного) розсiяння, яке є найбiльш суттєвим в передасимптотичнiй областi критичної точки та може бути зареєстроване вздовж певних термодинамiчних шляхiв. Його присутнiсть у загальнiй картинi розсiяння демонструється результатами обробки вiдомих експериментальних даних для коефiцiєнта деполяризацiї. Обговорено деякi наслiдки цих результатiв. 2013 Article Experimental observation of triple correlations in fluids / M.Ya. Sushko // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С. 13003:1–12. — Бібліогр.: 32 назв. — англ. 1607-324X PACS: 05.40, 05.70.Fh, 05.70.Jk, 64.70.Fx, 78.35.+c DOI:10.5488/CMP.16.13003 arXiv:1303.5224 http://dspace.nbuv.gov.ua/handle/123456789/121074 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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We present arguments for the hypothesis that under some conditions, triple correlations of density fluctuations in fluids can be detected experimentally by the method of molecular spectroscopy. These correlations manifest themselves in the form of so-called 1.5-(or sesquialteral) scattering. The latter is of most significance in the pre-asymptotic vicinity of the critical point and can be registered along certain thermodynamic paths. Its presence in the overall scattering pattern is demonstrated by our processing available experimental data for the depolarization factor. Some consequences of these results are discussed. |
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Sushko, M.Ya. |
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Sushko, M.Ya. Experimental observation of triple correlations in fluids Condensed Matter Physics |
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Sushko, M.Ya. |
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Sushko, M.Ya. |
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Experimental observation of triple correlations in fluids |
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Experimental observation of triple correlations in fluids |
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Experimental observation of triple correlations in fluids |
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Experimental observation of triple correlations in fluids |
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Experimental observation of triple correlations in fluids |
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experimental observation of triple correlations in fluids |
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Інститут фізики конденсованих систем НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/121074 |
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Experimental observation of triple correlations in fluids / M.Ya. Sushko // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С. 13003:1–12. — Бібліогр.: 32 назв. — англ. |
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Condensed Matter Physics |
| work_keys_str_mv |
AT sushkomya experimentalobservationoftriplecorrelationsinfluids |
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2025-07-08T19:09:03Z |
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2025-07-08T19:09:03Z |
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1837106980986028032 |
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Condensed Matter Physics, 2013, Vol. 16, No 1, 13003: 1–12
DOI: 10.5488/CMP.16.13003
http://www.icmp.lviv.ua/journal
Experimental observation of triple correlations
in fluids
M.Ya. Sushko∗
Mechnikov National University, Department of Theoretical Physics,
2 Dvoryanska St., 65026 Odesa, Ukraine
Received July 3, 2012, in final form October 29, 2012
We present arguments for the hypothesis that under some conditions, triple correlations of density fluctuations
in fluids can be detected experimentally by the method of molecular spectroscopy. These correlations manifest
themselves in the form of the so-called 1.5- (i.e., sesquialteral) scattering. The latter is of most significance
in the pre-asymptotic vicinity of the critical point and can be registered along certain thermodynamic paths.
Its presence in the overall scattering pattern is demonstrated by our processing experimental data for the
depolarization factor. Some consequences of these results are discussed.
Key words: density fluctuations, critical opalescence, 1.5-scattering, depolarization factor
PACS: 05.40, 05.70.Fh, 05.70.Jk, 64.70.Fx, 78.35.+c
1. Introduction
The intensity I of light scattered by a one-component fluid drastically increases as the critical point
is approached [1, 2]. The physical nature of this critical opalescence phenomenon is well-known, i.e.,
an increase in the magnitude of permittivity (in fact — density) fluctuations and the development of
long-range correlations between them. The result is that I is contributed to by not only single scattering
effects, but also by those of higher multiplicities. One could expect that I provides certain information
about higher-order correlation functions for the density fluctuations.
However, a common view is that it is only double scattering [3–8] together, probably, with Andreev’s
scattering (due to fluctuations in the distribution function of thermal fluctuations) [9–11] and triple scat-
tering [12] that contribute significantly near the critical point. The pertinent theories are based upon
quasi-Gaussian statistics for the fluctuations. As a consequence, any information on the irreducible parts
of higher-order correlation functions is lost and multiple scattering is viewed as parasitic.
By contrast, we pursue the idea that the so-called 1.5- (sesquialteral) scattering, caused by triple cor-
relations of density fluctuations, contributes significantly to I . In this report, we support this statement
by our results of processing extensive experimental data [13] on the depolarization factor near the criti-
cal point. In our view, these results strongly indicate that the 1.5-scattering is noticeably present, under
certain conditions, in the overall scattering pattern. Moreover, the idea of 1.5-scattering allows us to give
simple explanations for the anomalies in the behavior of I−1 in the gravity field [14, 15] and those of the
Landau-Plazcek ratio near the λ-line [16–18], which were observed long ago, but have been interpreted
controversially.
It is also important to note that, according to our own theory and in view of earlier estimates [4],
from among different three-point configurations of density fluctuations δρ (located at points r, r1, and
r′), the major contribution to the integrated 1.5-scattering intensity I1.5 is made by those in which two
successively scattering fluctuations merge on the macroscopic scale (|r − r1| → 0; technically, this is a
consequence of the replacement of the internal electromagnetic field propagators in the iterative series
for the scattered field with their leading short-range singularities). As a result, I1.5 is determined by the
∗E-mail: mrs@onu.edu.ua
©M.Ya. Sushko, 2013 13003-1
http://dx.doi.org/10.5488/CMP.16.13003
http://www.icmp.lviv.ua/journal
M.Ya. Sushko
Fourier transform of the pair correlation function 〈
[
δρ(r)
]2
δρ(r′)〉. Provided Polyakov’s hypothesis [19]
(see also [20, 21]) of conformal symmetry of critical fluctuations is valid, the latter is expected to vanish
at the critical point (see appendix A). It follows that the recovery of the 1.5-scattering contribution from
I and a scrutinized study of its behavior along appropriate thermodynamic paths ending up at the crit-
ical point provide a unique opportunity for experimental verification of Polyakov’s hypothesis [19] for
systems with scalar order parameters.
2. Theory of 1.5-scattering
2.1. General expression
The theory of 1.5-scattering was proposed in [22] and developed further in [23, 24]; some additional
numerical estimates were made in [25, 26]. We assume that molecular light scattering from condensed
matter is a result of re-emission of light not only by single fluctuations, but also by compact groups of
fluctuations. By compact we understand any group of fluctuations all the distances between which are
much shorter than the wavelength λ of the probing light in the medium. Physically, scattering by such a
group is single. The overall polarized single-scattering spectrum is, therefore, given by the series [23]
I (q,ω) =
∑
n,mÊ1
Inm (q,Ω), (2.1)
where
Inm (q,Ω) ∝
(
−
1
3ε0
)n+m−2
×
1
π
Re
+∞∫
0
dt
∫
V
dr
〈
[δε(r, t)]n [δε(0,0)]m
〉
eiΩt−iq·r (2.2)
is the contribution from a pair of compact groups of n andm permittivity fluctuations [attributed further
to density fluctuations, δε≈
(
∂ε/∂ρ
)
T δρ], ε0 is the equilibrium value of the permittivity,Ω and q are the
changes in the light frequency and wavevector due to scattering, and the scattering volume V is included
into the proportionality coefficient.
It is only the term I11(q,Ω) in equation (2.1) that has been associated so far with the single scattering.
The 1.5-scattering intensity is defined as I1.5(q,Ω) = I12(q,Ω)+ I21(q,Ω).
2.2. Hydrodynamic region, qrc ≪ 1
Far enough from the critical point, where the correlation radius rc ≪ λ and nonlocal correlations
between fluctuations can be ignored, the integrated 1.5-scattering intensity can be expressed in terms of
the third moment of thermodynamic density fluctuations ∆ρ [22]:
I1.5 ∝−
2
3ε0
(
∂ε
∂ρ
)3
Ṽ 〈(∆ρ)3〉 =−
2
3ε0
(
ρ
∂ε
∂ρ
)3 k2
B
T 2
Ṽ
[
2β2
T +
(
∂βT
∂P
)
T,V
]
, (2.3)
where βT is the isothermal compressibility of the fluid and Ṽ is a macroscopic volume over which the
fluctuations δρ are averaged to single out their thermodynamic parts ∆ρ. We suggest that Ṽ is slightly
dependent on temperature far away from the critical point, but Ṽ ∝ r 3
c ∝β3/2
T
in the critical region.
Calculations with the van der Waals and Dieterici equations of states give the estimates
I1.5 ∝−
1
ε0
(
ρ
∂ε
∂ρ
)3 k2
B
T 2
Ṽ
·6Pcωβ
3
T
and
I1.5 ∝−
2
3ε0
(
ρ
∂ε
∂ρ
)3 k2
B
T 2
Ṽ
·
(
β2
T +4Pcωβ
3
T
)
,
respectively, where ω ≡ ρc/ρ− 1, |ω| ≪ 1 is the deviation of ρ from the critical value ρc and Pc is the
critical pressure. It follows that the 1.5-scattering can become of significance in those domains in the
13003-2
Observation of triple correlations in fluids
(τ,ω)-plane where ω , 0, but βT is sufficiently large. Then, I1.5 ∝ ωβ3/2
T
for a non-critical isochore, but
I1.5 ∝β1/2
T
or even I1.5 → 0 for the critical one.
A distinctive feature of the 1.5-scattering contribution is that it is not positive definite: for instance,
I1.5 < 0 in the region where ω> 0 and τ> 0, at least.
2.3. Fluctuation region, qrc ≫ 1
Understanding, in this section, ρ as a scalar order parameter,we see that formulas (2.1) and (2.2) agree
with the hypothesis of algebra of fluctuating quantities [20]. Then, within the first order of ǫ-expansion
and in the long-wave limit q → 0, the critical index of I1.5, defined by I1.5 ∝ |τ|−µ21 , is estimated to be
µ21 ≈ 0.67 for ω = 0 [22]. This value is close to an earlier estimate of 0.7 given in [20]. Correspondingly,
I1.5 ∝β1/2
T
on the critical isochore and in the immediate vicinity of the critical point.
This result can be refined using the algebra of fluctuating quantities (see appendix A). However, it
is more important to emphasize that it disregards the conformal invariance hypothesis [19]. If the latter
is indeed valid, then the orthogonality relation holds for fluctuating quantities with different scaling
dimensions (see [20, 21]), that is, I1.5 → 0 as the critical point is approached.
2.4. Intermediate region, qrc. 1
This region is of special interest to us because it is typical of actual experiment. Taking into account
that correlations between fluctuations δρ remain relatively weak, we argue [23] that the convolution-type
approximation
〈
ρk1
ρk2
ρk3
〉
≈−
2c ′
kB T
p
V
G(k1)G(k2)G(k3)δk3,−k1−k2
can be used for the three-point correlation function of density fluctuations. Here, ρk is the Fourier compo-
nent of δρ(r), G(k) ≡ 〈|ρk|2〉, and c ′ is a k-independent function of temperature and density. Calculations
with the Ornstein-Zernike expression for G(k) then give:
I1.5(q) ∝
c ′
3πε0
(
ρ
∂ε
∂ρ
)3 ρ3k2
BT 2β3
T
qr 4
c
(
1+q2r 2
c
) arctan
( qrc
2
)
. (2.4)
Requiring that in the limit qrc ≪ 1, equation (2.4) transforms into equation (2.3), we recover c ′ through
the third thermodynamic moment of density fluctuations, with an accuracy to a positive proportionality
constant:
c ′ ∝−
[
2β2
T +
(
∂βT
∂P
)
T,V
](
ρβT
)−3
. (2.5)
Extrapolation of formulas (2.4) and (2.5) on the fluctuation region shows that (see appendix B) c ′ → 0
and, therefore, I1.5 → 0 as both τ→ 0 and ω→ 0, which is in accordance with the conformal invariance
hypothesis.
The structure of the 1.5-scattering spectrum in the intermediate region is discussed in [24].
3. Depolarization factor
3.1. Theoretical considerations
Now, we are in a position to scrutinize the effect of 1.5-scattering on the depolarization factor ∆ as
a function of temperature (in fact, βT ) and the geometrical size L ∼ V 1/3 (volume V ) of the scattering
system. Suppose that the following contributions to I are present in the intermediate region qr . 1:
(1) the “standard” intensity I11 ∝ V βT of polarized single scattering due to density fluctuations [1, 2];
(2) the intensity I1.5 of polarized 1.5-scattering [22–24]; (3) the intensity Ia ∝ V of depolarized single
scattering due to anisotropyfluctuations [1] (it is virtually insensitive to the critical point); (4) the intensity
I2p ∝ V 4/3β2
T of polarized double scattering due to density fluctuations [3–8]; (5) the intensity I2d ∝
V 4/3β2
T of depolarized double scattering due to density fluctuations [3–8]; (6) the intensity IA ∝ V β1/2
T
13003-3
M.Ya. Sushko
of depolarized single scattering due to fluctuations in the distribution function of thermal fluctuations
(Andreev’s scattering) [9]. Then, ∆ is given by
∆=
Ia + I2d + IA
I1 + I1.5 + I2p
. (3.1)
In view of the individual temperature dependences of the above contributions and under the condi-
tion I1.5 = 0, ∆ as a function of βT is expected to decrease first, then reach a minimum, and then increase
again. Such a behavior, indeed observed in the experiment, is considered as a manifestation of double
scattering effects. However, as we show later, the presence of the 1.5-scattering contribution does not
alter this qualitative behavior of ∆ as a function of βT .
Thus, expression (3.1) should be transformed in order to obtain an experimentally-measurable func-
tion whose behavior significantly depends on whether the 1.5-scattering contributes to ∆ or not [22].
Rewriting (3.1) as
I2d
I1∆
=
1+ I1.5 I−1
1 + I2pI−1
1
1+ Ia I−1
2d
+ IAI−1
2d
(3.2)
and taking into account the specific features of the intensity contributions, we immediately arrive at the
relation
LβT
∆
∝
1+aβ1/2
T
+bLβT
1+cL−1β−2
T
+dL−1β−3/2
T
(3.3)
valid for the intermediate region qr . 1. Here, the coefficients b, c, and d are practically temperature-
independent and positive constants. The coefficient a ∝ c ′ arctan
( qrc
2
)
/rc is due to the 1.5-scattering
contribution and is not positive definite. If the 1.5-scattering is negligible, then a = 0 and the right-hand
side in formula (3.3) is a monotonous increasing function of βT . With the 1.5-scattering present, this
monotonous behavior is expected to be violated. The effect should be most pronounced in the following
two cases.
(1) The critical point is approached along a noncritical isochore ω > 0. Then, I1.5 ∝ −β3/2
T
and a is
close to a negative constant.
(2) The critical point is approached along the path where τ → 0, ω → 0, and I1.5 > 0. The relative
magnitude I1.5/I1 of the 1.5-scattering should start decreasing somewhere due to the temperature law
I1.5 ∝β1/2
T
coming into play [20, 22] (see section 2.3) or as a consequence of the conformal invariance [19,
20]. As such a path, the liquid branch of the coexistence curve can be quoted.
Thus, by varying the temperature (βT ) and density (ω) of the scattering system, we hope to “stick out”
the 1.5-scattering contribution from among the others. It should manifest itself as a non-monotonous
behavior of the experimentally-measurable quantity LβT /∆−1 with βT . The fact that the scattering con-
tributions involved depend differently on L, provides an additional powerful option for analysis. Some
results obtained by processing the extensive depolarization factor data [13] are presented in figures 1–12.
They generalize our earlier results [25].
3.2. Data processing
3.2.1. Noncritical isochores ρ <ρc, τ> 0
Figure 1 represents the log-linear plots of the quantity L(D∆)−1 ∝ LβT ∆
−1 as a function of the pa-
rameter (k0 is the wavevector in vacuum of the incident light)
D−1 =
k4
0
144π2
(ε0 −1)2 (ε0 +2)2 kBTβT ∝βT
for xenon along the ω = 6.8 × 10−3 isochore and five values of L (in cm). The parameter D (in m) is
a convenient measure of the distance to the critical point [27]. It is evaluated in [13] as a function of
temperature and density by using the Clausius-Mossotti relation for ε0 and the restricted linear model
equation of state [28] for βT . These calculations are claimed to be most reliable for the region not very
close to and not far away from the critical point, i.e., the one of special interest to us.
13003-4
Observation of triple correlations in fluids
Figure 1. L(D∆)−1 versus D−1 along the ω= 6.8×10−3 isochore of xenon for five values of L, based on
experimental data [13]. From left to right, τ decreases from 7.8×10−2 to 1.2×10−5 . Three segments can
be distinguished on each of these log-linear plots.
As τ reduces from 7.8× 10−2 to 1.2× 10−5, three typical temperature intervals are clearly seen in
figure 1. We shall refer to them as segments A (leftmost), B (central) and C (rightmost). It is easy to note
that simple division of L(D∆)−1 by L and changing to (D∆)−1 as a function of D−1 for different values of
L transforms the original plots dissimilarly on these segments: the plots tend to merge on A and C, but
invert the vertical ordering and disperse on B (figure 2). This fact implies that (D∆)−1 is contributed to
by terms with differing functional dependences on L. Our further analysis of them is guided by relation
(3.3).
Suppose that on segments A, i.e., the most distant from the critical point, I1 prevails much over I1.5
and I2p. Then, relation (3.3) takes the form
LβT
∆
∝
1
1+cL−1β−2
T
+dL−1β−3/2
T
.
It follows that the dependence of (D∆) upon D2 should be close to a linear one, with the slope independent
of L and, if the Andreev contribution is noticeable, a slight concavity: (D∆) ∝ const+ cD2 +d(D2)3/4.
Figure 2. (D∆)−1 versus D−1 for the data shown in
figure 1.
Figure 3. (D∆) versus D2 for segments A. From right
to left, τ decreases from 7.8×10−2 to 3.3×10−3 .
13003-5
M.Ya. Sushko
Figure 4. L(D∆)−1 versus D−1/2 for segments A and
B. From left to right, τ decreases from 7.8×10−2 to
1.4×10−4 .
Figure 5. L(D∆)−1 versus D−1/2 for the portions of
segments B where τ decreases from 9.8 × 10−4 to
1.4×10−4 .
Figure 3 does exhibit, at least approximately, such kind of dependence. The study of the latter could, in
principle, provide experimental estimates for the magnitude of IA. However, the discussion of this issue
is beyond the scope of the present report.
On segments B, where we expect I2d to dominate over Ia and IA, but I2p to remain relatively weak as
compared to I1 and I1.5, relation (3.3) takes the form
LβT
∆
∝ 1+aβ1/2
T .
At ω > 0, L(D∆)−1 should decrease with D−1/2 by the linear law L(D∆)−1 ∝−D−1/2, with negative and
equal slopes for different values of L (figures 4 and 5). Correspondingly, the dependences of (D∆)−1 upon
−D−1/2 should be linear on B, with slopes inversely proportional to L, but, as was already mentioned,
merge on A (figure 6).
Finally, segments C are formed mainly by single and true double scatterings. Relation (3.3) takes the
form
LβT
∆
∝ 1+bLβT
Figure 6. (D∆)−1 versus D−1/2 for the data shown in figure 4.
13003-6
Observation of triple correlations in fluids
Figure 7. L(D∆)−1 versus D−1 for the entire ω =
6.8×10−3 data plotted with a linear D−1 scale. Seg-
ments C stand out as τ decreases rightwards from
8.6×10−5 to 1.2×10−5 .
Figure 8. (D∆)−1 versus D−1 for the entire ω= 6.8×
10−3 data plotted with a linear D−1 scale.
and we expect L(D∆)−1 to increase with D−1 by a linear law, with slope proportional to L (figure 7).
The (D∆)−1 versus D−1 plots for different values of L should approach a single straight segment as D−1
increases (figure 8).
3.2.2. Liquid branch of the coexistence curve
The dependence of L(D∆)−1 upon D−1 along the liquid branch of the coexistence curve of xenon is
shown in figure 9. It agrees well with our expectations.
Thus, the above processing of experimental data [13] clearly reveals the presence in the overall scat-
tering pattern of a contributionwhich we associate with the 1.5- (sesquialteral) molecular light scattering.
Figure 9. L(D∆)−1 versus D−1 along the liquid branch of the coexistence curve of xenon; L = 0.547cm.
From left to right, τ changes from −9.2×10−2 to −2.9×10−5 .
13003-7
M.Ya. Sushko
3.3. Numerical estimates
Now, we present quantitative estimates of the magnitude of 1.5-scattering intensity. They were ob-
tained by fitting the L(D∆)−1 versus D−1 data for the entire ω = 6.8× 10−3 isochore and then used to
reproduce the original ∆ versus D data [13].
In view of formulas (3.2) and (3.3), the fitting function was taken in the form
f (x) =
x2
C + x2
(
K + Ax1/2 +B x
)
, x ≡ D−1,
and the following two sets of coefficients were chosen: K1 = 581.066, A1 =−63.6968, B1 = 3.66569, C1 =
0.012 and K2 = 607.508, A2 = −89.0932, B2 = 5.79532, C2 = 0.012. The relative magnitudes r1.5 ≡ I1.5/I1
and r2p ≡ I2p/I1 of the 1.5-scattering and polarized double scattering, as compared to the single scattering,
were estimated as
r1.5 =
Ax1/2
K
, r2p =
B x
K
.
To calculate ∆ with fi , our theoretical estimate [11] I2d ≈ 1
8 I2p was additionally used. Then,
∆=
B x
8K f (x)
.
Figure 10. Fitting the (D∆)−1 versus D−1 data (ω =
6.8×10−3 , L = 0.547cm) with f1 (dashed line) and f2
(solid line).
Figure 11. Fitting the∆ versusD data (ω= 6.8×10−3 ,
L = 0.547cm) with f1 (dashed line) and f2 (solid
line).
The results are demonstrated by figures 10–12. They clearly show that in the intermediate region,
the intensities I1.5 and I2p reach magnitudes comparable with that of I1, but are opposite in sign and
tend to compensate for each other. These facts are surprising. They contradict the common view that
multiple scattering contributions come into play gradually as the critical point is approached. In other
words, they imply an asymptotic nature of the iterative series for the overall scattering intensity near the
critical point. They can also be interpreted in the sense that triple and quadruple correlations in fluids
contribute, at least to light scattering effects, in opposite directions.
The agreement of our fitting results with the ∆-data [13] (figure 11) is also unexpectedly good.
4. Conclusion
The above estimates have prompted us to identify other experiments where the situation is favorable
for 1.5-scattering to come into play. First of all, of interest are the studies on light scattering from critical
13003-8
Observation of triple correlations in fluids
Figure 12. I1.5/I1, I2p/I1, and
(
I1.5 + I2p
)
/I1 along the ω= 6.8×10−3 isochore of xenon, L = 0.547cm, as
estimated with f1 (dashed line) and f2 (solid line).
fluids under the earth’s gravity. Due to the gravity effect, the system is spatially inhomogeneous in the
vertical direction. A negative 1.5-contribution is expected to appear in the light scattered from the fluid
layers located above the level of critical density. For such a layer, the total scattering intensity I should, as
a function of temperature, start decreasing somewhere as the critical point is approached. In other words,
the I−1 versus τ dependence should have a minimum at some τ0. The effect was indeed registered, for
instance, in freon 113 [14, 15]. Our estimations for the minimum location, τ0 ∼ 10−3 for heights up to
20mm (τ> 0), agree well with experiment.
We suggest that a specially-designed processing of the gravity-induced height- and temperature de-
pendences of I , obtained for systems with a scalar order parameter, is a feasible opportunity for singling
out the 1.5-scattering contribution and verifying Polyakov’s conformal invariance hypothesis.
To finish, we mention that the studies of the spectral distribution in critical opalescence spectra are
of great interest as well. In particular, we have proved that in the presence of 1.5- and double scattering
effects, the ratio of the integrated intensities of the Rayleigh and Brillouin components takes the form
Rexp = R
1+a1.5r1.5 +a2pr2p
1+b1.5r1.5 +b2pr2p
, (4.1)
where R = γ−1 is the well-known Landau-Placzek ratio [30] for single scattering (γ≡ cP /cV , cP and cV
being the specific heats at constant pressure and volumes). The coefficients a1.5 and b1.5 are given in [24],
whereas a2p and b2p can be recovered from the results [31]:
a2p = 1−
3
2γ
+
1
2(γ−1)
, b2p = 2−
3
2γ
. (4.2)
Suggesting that r1.5 = 0, it is not difficult to verify, based on experimental data [32] for He4, that the double
scattering alone should cause Rexp to exceed R as the λ-line is approached along a high-pressure isobar.
Such a fact was indeed registered [18], but in most other experiments in this series the tendency was
direct opposite [16–18]. We attribute the reduction in Rexp to the effect of 1.5-scattering.
Our detailed calculations of the above effects will be presented elsewhere.
13003-9
M.Ya. Sushko
Appendix A
Let Ak (r) be a complete set (algebra) of fluctuating scalar quantities with scaling dimensions ∆k : un-
der the scaling transformations r → λr, Ak (λr) → λ−∆k Ak (r). According to the local algebra hypothesis
(see, for instance, [20]), the scalar function δρ(r) and, therefore, the scalar function
[
δρ(r)
]2
can be de-
veloped into the series
δρ(r) =
∞∑
k=1
ak Ak (r),
[
δρ(r)
]2 =
∞∑
m,n=1
bmnBmn (r),
where the coefficients bmn are simply related to the coefficients ak , and Bmn (r) = Am(r)An(r) are scalar
quantities with definite scaling dimensions ∆mn = ∆m +∆n : Bmn (λr) → λ−∆mn Bmn (r). Correspondingly,
the pair correlation function 〈δρ(r′)
[
δρ(r)
]2〉 is given by a linear combination of pair correlators
Kk ,mn(|r′−r|) ≡ 〈Ak (r′)Bmn (r)〉.
For a d -dimensional system with scalar order parameter, only the Fourier transforms of those Kk ,mn
can reveal a singular behavior near the critical pointwhich satisfy the condition∆k+∆mn < d . In the first-
order ǫ-expansion ∆k = 1
2
k(2−ǫ)+ 1
6
k(k −1)ǫ, where ǫ= 4−d [20]. If d = 3, then the relevant correlators
are K1,11, K1,12 = K1,12, and K2,11, each involving two scalar quantities with different scaling dimensions.
Once Polyakov’s conformal symmetry hypothesis holds true and the system is spatially homogeneous
and isotropic, they all vanish at the critical point due to the orthogonality relation (see [20, 21] and the
literature cited therein).
Appendix B
As one of the ways for evaluating c ′ in the fluctuation region, we can use the asymptotic equation of
state [29]
π(τ,ω) = Mτ+
1
2Γ0
1
(1+ω)2
τ |τ|γ−1 −D0ω|ω|δ−1,
which immediately follows from the requirements that it leads to (a) the correct asymptotic behavior
of a limited number of the fluid parameters along the selected thermodynamic paths (the susceptibility
χ = ρ2βT along the critical isochore, the critical isotherm equation, and the derivative of pressure with
respect to temperature at the critical point) and (b) reveal a Van-der-Waals-type loop below the critical
point. In this equation, π= P
/
Pc−1, τ= T
/
Tc−1, ω=V
/
Vc−1, Γ0 and D0 are the critical amplitudes for
χ and the critical isotherm, respectively, and M is a constant. The definition
βT =−
1
V
(
∂V
∂P
)
T,V
=−
1
Pc(1+ω)
(
∂ω
∂π
)
τ
,
relations (
∂βT
∂P
)
T,V
=
(
∂βT
∂P
)
T,N
=
(
∂βT
∂V
)
T,N
(
∂V
∂P
)
T,N
=−(1+ω)βT
(
∂βT
∂ω
)
τ
,
and formula (2.5) then yield
c ′ ∝−PcD0δ(δ−1)ρ−3ω|ω|δ−3.
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13003-11
http://dx.doi.org/10.3367/UFNr.0140.198307b.0393
http://dx.doi.org/10.1070/PU1983v026n07ABEH004448
http://dx.doi.org/10.1016/0009-2614(77)80207-8
http://dx.doi.org/10.1016/0378-4371(80)90187-9
http://dx.doi.org/10.1103/PhysRevLett.30.427
http://dx.doi.org/10.1088/0022-3719/8/2/003
http://dx.doi.org/10.1080/00018737800101444
http://dx.doi.org/10.1134/1.1854804
http://dx.doi.org/10.1063/1.2784150
http://dx.doi.org/10.1016/j.molliq.2011.07.006
http://dx.doi.org/10.1016/0009-2614(79)80159-1
http://dx.doi.org/10.1016/j.molliq.2010.10.014
http://dx.doi.org/10.1103/PhysRevA.8.530
M.Ya. Sushko
Експериментальне спостереження потрiйних кореляцiй
у рiдинах
М.Я. Сушко
Одеський нацiональний унiверситет iменi I.I. Мечникова, вул. Дворянська, 2, 65026 Одеса, Україна
Наведено аргументи на користь гiпотези, що при певних умовах методом молекулярної спектроскопiї
можна зареєструвати потрiйнi кореляцiї флуктуацiй густини в рiдинах. Цi кореляцiї проявляють себе у
виглядi так званого 1.5- (тобто пiвторакратного) розсiяння, яке є найбiльш суттєвим в передасимптотич-
нiй областi критичної точки та може бути зареєстроване вздовж певних термодинамiчних шляхiв. Його
присутнiсть у загальнiй картинi розсiяння демонструється результатами обробки вiдомих експеримен-
тальних даних для коефiцiєнта деполяризацiї. Обговорено деякi наслiдки цих результатiв.
Ключовi слова: флуктуацiї густини, критична опалесценцiя, розсiяння кратностi 1.5, коефiцiєнт
деполяризацiї
13003-12
Introduction
Theory of 1.5-scattering
General expression
Hydrodynamic region, qrc1
Fluctuation region, qrc1
Intermediate region, qrc1
Depolarization factor
Theoretical considerations
Data processing
Noncritical isochores < c, >0
Liquid branch of the coexistence curve
Numerical estimates
Conclusion
|