Critical sound attenuation of three-dimensional Ising systems
The field-theoretic description of dynamic critical effects of the disorder on acoustic anomalies near the temperature of second-order phase transition is considered for three-dimensional Ising-like systems. The calculations of sound attenuation in high temperature phase for pure and diluted Ising...
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| Cite this: | Critical sound attenuation of three-dimensional Ising systems / P.V. Prudnikov, V.V. Prudnikov // Condensed Matter Physics. — 2006. — Т. 9, № 2(46). — С. 403–410. — Бібліогр.: 23 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1213302025-06-03T16:27:04Z Critical sound attenuation of three-dimensional Ising systems Затухання критичного звуку тривимірних ізингоподібних систем Prudnikov, P.V. Prudnikov, V.V. The field-theoretic description of dynamic critical effects of the disorder on acoustic anomalies near the temperature of second-order phase transition is considered for three-dimensional Ising-like systems. The calculations of sound attenuation in high temperature phase for pure and diluted Ising-like systems near the critical point are presented. The dynamic scaling function for critical attenuation coefficient is calculated. The effect of the quenched disorder on asymptotic behaviour of the critical ultrasonic anomalies is discussed. This work has been supported by the Russian Foundation for Basic Research through Grants (No. 04–02–17524, No. 04–02–39000 and No. 05–02–16188), and by Grant No. MK–8738.2006.2 of Russian Federation President. 2006 Article Critical sound attenuation of three-dimensional Ising systems / P.V. Prudnikov, V.V. Prudnikov // Condensed Matter Physics. — 2006. — Т. 9, № 2(46). — С. 403–410. — Бібліогр.: 23 назв. — англ. 1607-324X PACS: 64.60.Ak, 64.60.Fr, 64.60.Ht, 43.35.Cg DOI:10.5488/CMP.9.2.403 https://nasplib.isofts.kiev.ua/handle/123456789/121330 en Condensed Matter Physics application/pdf Інститут фізики конденсованих систем НАН України |
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The field-theoretic description of dynamic critical effects of the disorder on acoustic anomalies near the temperature
of second-order phase transition is considered for three-dimensional Ising-like systems. The calculations
of sound attenuation in high temperature phase for pure and diluted Ising-like systems near the critical
point are presented. The dynamic scaling function for critical attenuation coefficient is calculated. The effect
of the quenched disorder on asymptotic behaviour of the critical ultrasonic anomalies is discussed. |
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Article |
| author |
Prudnikov, P.V. Prudnikov, V.V. |
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Prudnikov, P.V. Prudnikov, V.V. Critical sound attenuation of three-dimensional Ising systems Condensed Matter Physics |
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Prudnikov, P.V. Prudnikov, V.V. |
| author_sort |
Prudnikov, P.V. |
| title |
Critical sound attenuation of three-dimensional Ising systems |
| title_short |
Critical sound attenuation of three-dimensional Ising systems |
| title_full |
Critical sound attenuation of three-dimensional Ising systems |
| title_fullStr |
Critical sound attenuation of three-dimensional Ising systems |
| title_full_unstemmed |
Critical sound attenuation of three-dimensional Ising systems |
| title_sort |
critical sound attenuation of three-dimensional ising systems |
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Інститут фізики конденсованих систем НАН України |
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Critical sound attenuation of three-dimensional Ising systems / P.V. Prudnikov, V.V. Prudnikov // Condensed Matter Physics. — 2006. — Т. 9, № 2(46). — С. 403–410. — Бібліогр.: 23 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT prudnikovpv criticalsoundattenuationofthreedimensionalisingsystems AT prudnikovvv criticalsoundattenuationofthreedimensionalisingsystems AT prudnikovpv zatuhannâkritičnogozvukutrivimírnihízingopodíbnihsistem AT prudnikovvv zatuhannâkritičnogozvukutrivimírnihízingopodíbnihsistem |
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2025-11-30T20:50:10Z |
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Condensed Matter Physics 2006, Vol. 9, No 2(46), pp. 403–410
Critical sound attenuation of three-dimensional Ising
systems
P.V.Prudnikov, V.V.Prudnikov
Omsk State University, Omsk, 644077 Russia
Received August 15, 2005, in final form November 14, 2005
The field-theoretic description of dynamic critical effects of the disorder on acoustic anomalies near the tem-
perature of second-order phase transition is considered for three-dimensional Ising-like systems. The calcu-
lations of sound attenuation in high temperature phase for pure and diluted Ising-like systems near the critical
point are presented. The dynamic scaling function for critical attenuation coefficient is calculated. The effect
of the quenched disorder on asymptotic behaviour of the critical ultrasonic anomalies is discussed.
Key words: renormalization group, disordered systems, field-theoretic approach, ultrasonic attenuation
PACS: 64.60.Ak, 64.60.Fr, 64.60.Ht, 43.35.Cg
1. Introduction
The progress achieved in understanding the critical phenomena has largely been due to theo-
retical and experimental works devoted to studying the critical dynamics of condensed media. We
have seen from experiments (figure 1) [1] that for a solid an anomaly peak of ultrasonic attenuation
is observable in the vicinity of critical point. The critical anomalies exhibited by sound attenuation
have long been recognized as an important study of dynamic critical phenomena.
Figure 1. Ultrasound anomaly for experimental attenuation in Rb2ZnCl4 (dots) as function of
temperature in comparison with the theoretical results (solid lines) [7].
The important physics processes are described by multi-spin correlation functions such as sound
propagation and attenuation, fluctuations of energy density. Some experimental methods, such as
EPR, NMR, propagation of ultrasonic waves are capable of studying the behaviour of multi-spin
correlation functions. Ultrasonic methods permit simultaneous measurements of both static and
c© P.V.Prudnikov, V.V.Prudnikov 403
P.V.Prudnikov, V.V.Prudnikov
dynamic properties. Measurements of sound velocities give information on the equilibrium proper-
ties, while measurements of sound attenuation yield the information on the dynamic properties of
material.
The main difficulty in the theoretical discussion of critical propagation of sound waves consists
in the estimation of the four-spin correlation function. The method, which is based on the repre-
sentation of the four-spin correlation function through two-spin correlation functions by means of
its decoupling, leads to overestimated values of critical fluctuations.
There are a lot of theories and phenomenological descriptions [1–8] of ultrasonic anomalies
in solids with good agreement with the experiments [9–11] but real materials and crystals have
many structural defects and the effect of such defects or disorder on the dynamic process of sound
propagation in solid media should be taken into consideration.
Structural disorder, presence of impurities or other defects play an important role in real ma-
terials and physical systems. They may induce new types of phase transitions, universality classes
as well as modify the transport properties of dynamic.
According to the Harris criterion [12], critical behaviour of Ising systems is changed by the
presence of a weak quenched disorder. The problem of the effect of disorder on the critical sound
propagation in Ising-like systems has been discussed in [13] with the use of ε-expansion in the lowest
order of approximation. However, our pilot analysis of this phenomenon showed that in [13] there
were not considered some diagrams which are indispensable for a correct description of disorder
effect. Furthermore, our numerous investigations of pure and disordered systems performed in the
two-loop and higher orders of approximation for a three-dimensional system directly and together
with the use of series summation methods show that the predictions made in the lowest order
of approximation, especially based on the ε-expansion can strongly differ from the real critical
behaviour [14,15]. Therefore, the results from [13] must be reconsidered with the use of a more
accurate field-theoretic approach in the higher orders of approximation.
Figure 2. Three-dimensional disordered Ising system.
In this paper we have realized the correct field-theoretic description of dynamic effects of the
disorder on acoustic anomalies near the temperature of second-order phase transition for a three-
dimensional Ising-like systems (figure 2) in the two-loop approximation.
2. Model and RG analysis
In this paper, we extended the model of phase transition in a disordered system with a coupling
between nonfluctuating variables [16,17] to the case (physically important for structural phase
transitions) of a compressible three-dimensional Ising model with frozen-in lattice defects that is
considered by the renormalization-group methods in the two-loop approximations.
404
Critical sound attenuation of three-dimensional Ising systems
Interaction of the order parameter with elastic deformations plays a significant role in the
critical behaviour of the compressible system. For the first time in [18] it was shown that the
critical behaviour of the system with elastic degrees of freedom is unstable with respect to the
connection of the order parameter with acoustic modes and the first-order phase transition is
realized. However, the conclusions of [18] are only valid at low pressures. It was shown in [19] that
in the range of high pressures, starting from a threshold value of pressure, the deformational effects
induced by the external pressure lead to a change in the type of the phase transition.
The Hamiltonian of a disordered Ising model with allowance for elastic degrees of freedom may
be specified as
H = Hel + Hop + Hint + Himp , (1)
consisting of four contributions.
The elastic part is determined by
Hel =
1
2
∫
ddx
C0
11
∑
α
u2
αα + 2C0
12
∑
αβ
uααuββ + 4C0
44
∑
α<β
u2
αβ
, (2)
where uαβ(x) are components of the strain tensor and Ck
ij are the elastic moduli.
Hop is a magnetic part in the appropriate Ginzburg-Landau form
Hop =
∫
ddx
[
1
2
τS2 +
1
2
(∇S)
2
+
1
4
u0S
4
]
, (3)
where S(x) is the Ising field variable which associated with the spin order parameter, u0 is a
positive constant and τ ∼ (T − Tc) /Tc with the phase transition temperature Tc.
The term Hint describes the spin-elastic interaction
Hint =
∫
ddx
[
g0
∑
α
uααS2
]
, (4)
which is bilinear in the spin order parameter and linear in deformations. The g0 is the bare coupling
constant.
The term Himp of Hamiltonian determines the effect of disorder and it is considered as follows:
Himp =
∫
ddx
[
∆τ(x)S2
]
+
∫
ddx
[
h(x)
∑
α
uαα
]
, (5)
where the random variables ∆τ(x) and h(x) are the local transition temperature fluctuations and
induced random stress, respectively.
The Fourier transformed variables become
uαβ = u
(0)
αβ + V −1/2
∑
q 6=0
uαβ(q) exp (iqx) , (6)
with u(q) = i/2 [qαuβ + qβuα]. We have to isolate in (6) tensor u
(0)
αβ of homogeneous striction [18]
and then make the integration in the partition function with respect to the nondiagonal components
of the uniform part of the deformation tensor u
(0)
αβ that are insignificant for the critical behaviour
of the system in an elastically isotropic medium.
After all transformations [20] the effective Hamiltonian of the system becomes as follows:
405
P.V.Prudnikov, V.V.Prudnikov
H̃ =
1
2
∫
ddq
(
τ + q2
)
Sq S−q +
1
2
∫
ddq ∆τ−q Sq1 Sq−q1
+
1
4
u
∫
ddq Sq1 Sq2 Sq3 S−q1−q2−q3 +
∫
ddq q h−q Qq
−
1
2
w
∫
ddq (Sq S−q) (Sq S−q) − g
∫
ddq q Q−q Sq1 Sq−q1
+ a
∫
ddq q2 Qq Q−q . (7)
For a weak disorder, the distribution of random fields ∆τq and h−q can be considered to be
Gaussian
P [∆τq, h−q] = A
∫
ddq exp
[
−
1
8b1
∆τ2
q −
1
8b2
h2
−q −
1
8b3
∆τqh−q
]
, (8)
where A is the normalization constant and bi are positive constants which are proportional to the
impurity concentration.
The critical dynamics of the system in the relaxation regime can be described by the Langevin
equations [21] for the spin order parameter S (q) and deformation variables Q (q)
Ṡq = −Γ0
∂H̃
∂S−q
+ ξq + Γ0hS , Q̈q = −
∂H̃
∂Q−q
− q2D0Q̇q + ηq + hQ , (9)
where Γ0 and D0 are the initial kinetic coefficients, ξq(x, t) and ηq(x, t) are Gaussian white noises.
The quantities of interest are the response functions G(q, ω) and D(q, ω) of spin and deformation
variables, respectively. They can be obtained by linearization on corresponding fields
D(q, ω) = δ [〈Qq,ω〉] /δhQ = [〈Qq,ωQ−q,−ω〉] , (10)
G(k, ω) = δ [〈Sq,ω〉] /δhS = [〈Sq,ωS−q,−ω〉] , (11)
where 〈· · · 〉 denotes the averaging over Gaussian white noises, [· · · ] denotes the averaging over
random fields ∆τq and h−q.
The response functions may be expressed in terms of self-energy parts
G−1(q, ω) = G−1
0 (q, ω) + Π(q, ω), D−1(q, ω) = D−1
0 (q, ω) + Σ(q, ω), (12)
where the free response functions G0(q, ω) and D0(q, ω) ar as follows:
D0(q, ω) = 1
/(
ω2 − aq2 − iωD0q
2
)
, G0(q, ω) = 1
/(
iω /Γ0 +
(
τ + q2
))
.
Characteristics of critical sound propagation are defined by the response function D(q, ω). Thus,
the coefficient of ultrasonic attenuation is determined through imaginary part of Σ(q, ω)
α(ω, τ) ∼ ωImΣ(ω). (13)
In diagrammatic representation the expression for the self-energy part Σ(q, ω) in two-loop
approximation is as follows:
Σ(q, ω) = 4g2 – 96g2u
+ 16g2v + 16g2v + 16g2v
+ 16g2v + 16g2v + 16g2v ,
406
Critical sound attenuation of three-dimensional Ising systems
where we have used the following diagrammatic rules
G0(q, ω) ,
2
Γ0
G0(q, ω)G0(−q,−ω) ,
v =
[
(∆τ)2
]
.
The Feynman diagrams involve momentum integrations on dimension d (in our case d = 3).
Near the critical point the correlation length ξ increases infinitely. When ξ−1 � Λ, where Λ is
a cutoff in momentum-space integrals (the cutoff Λ serves to specify the basic length scale), the
vertex functions are expected to display an asymptotic scaling behaviour for wave-numbers q � Λ.
Therefore, one is lead to consider the vertex functions in the limit Λ → ∞.
Now our purpose is that all variables become finite. We have applied the matching method
introduced for statics in [23] and generalized for critical dynamics in [4]. At first, we use the
dynamic scaling property of the response function
D(q, ω, τ) = e2lD(qel, (ω/Γ0)e
zl, τe(l/ν)), (14)
and then calculate the right-hand side of this equation for some value l∗ = l, where not all the
arguments vanish simultaneously [23]. The choice of l∗ is determined by
(ωl /2Γ0 )
4/z
+ χ−2
l = 1. (15)
The equation (14) guarantees that at least one of the arguments of the right-hand side of (15)
is finite. The particular form of the matching condition (15) containing the exponents z and ν
permits an explicit solution for l∗
el = τ−ν
[
(
1 − b + bτ−α
)−2
+ (y/2)
4/z
]−1/4
≡ τ−νF (y, τ). (16)
In (16) b is an initial parameter and the abbreviation y = (ω/Γ0) τ−zν is introduced and F (y, τ)
is defined. The values for exponents z,α and ν were taken from [20] for a corresponding fixed point.
The response function D(qel, (ω/Γ0)e
zl, τe(l/ν)) on the right hand side of (14) is represented
by Dyson equation (12) and for self-energy part we obtain
ImΣ(ω)
ω
= el (α+zν)
ν
ImΣ(ωezl)
ωezl
. (17)
It may be argued [4] that condition (15) with a well-known expression for susceptibility [13]
provides an infrared cutoff for all diverging values.
It was shown in the later theoretical works [3,11] that in asymptotic regions the coefficient of
attenuation is described by a simple scaling function of the variable y. The experimental investi-
gations performed on three-dimensional crystals Gd [10] and MnP [11] confirmed the validity of
the concepts of dynamic scaling.
3. Scaling and asymptotic behaviour
Thus after renormalization procedure (17) we can define the scaling relation in the form
ImΣ(ω) /ω = τ−α−zνφ(y), (18)
where φ(y) is a dynamic scaling function
φ(y) =
g∗2Γ0
π
Fα/ν+1/2ν−z
y2
1 −
[
1
2
(
1 +
y2F 2z−2/ν
4
)1/2
+
1
2
]1/2
−
12g∗2u∗Γ2
0
π2
Fα/ν+1/ν−2z
y3
[
1
2
(
1 +
y2F 2z−2/ν
4
)1/2
−
1
2
]1/2
−
yF z−1/ν
4
+
8g∗2v∗
(4π)3
Fα/ν−z
y2
φimp(y), (19)
407
P.V.Prudnikov, V.V.Prudnikov
where v∗, g∗, and u∗ are values of vertices in critical point [20], φimp (y) is the numerically calculated
contribution of diagrams in Σ (ω) with disorder inserting.
The dynamic scaling function φ(y) is plotted against y on a double logarithmic scale for pure
and disordered systems in figure 3. We can determine two asymptotical regimes: “hydrodynamical”
(y ∼ ωξz ∼ (qξ)z � 1) with φ(y → 0) = const and “critical” regime (y ∼ ωξz � 1) with
characteristic power law φpure(y → ∞) ∼ y−0.96, φdis(y → ∞) ∼ y−0.78. Thus we have seen that
the presence of disorder is irrelevant for the scaling behaviour in the hydrodynamic region with
y � 1 but it produces a drastic effect in the critical region with y � 1 (T → Tc).
Figure 3. Scaling functions for the critical sound attenuation φ(y) in double-logarithmic plotting
for pure (1) and disordered (2) Ising systems, φ0 = φ(0).
We hope that these theoretical results will create a demand for the ultrasonic experimental
investigations in the diluted Ising-like systems, for example FepZn1−pF2, MnpZn1−pF2. In figure
it was used that the presence of disorder causes the reduction of the phase transition temperature
Tc in relation to the pure system. The model presentation of calculated attenuation coefficient
for pure system is shown in figure 4 in comparison with experimental data (3) in Gd [8]. The
adjustment on experimental data permited us to determine the value of theoretical parameter
Γ0 and calculate then the attenuation coefficient for disordered model. We must note that the
observable differences with experimental results below Tc is explained by contribution of order
parameter relaxation effects to attenuation which are presence always below Tc. The relaxation
effects are not considered in this paper. Whereas, the contribution of fluctuations to the attenuation
coefficient is a relevant in the whole critical range.
Table 1. Asymptotic behaviour of the attenuation coefficient in the hydrodynamical and the
critical regions.
System Region α(ω, q)
Pure hydrodyn. ω2τ−1,38
critical ω1.04τ−0.16
Disordered hydrodyn. ω2τ−1,44
critical ω1.23τ−0,26
Pure (ε-exp.)[5] hydrodyn. ω2τ−1,34
critical ω0.858
From equation (13) we find that the sound attenuation coefficient obeys the asymptotic scaling
relation
α (ω, τ) ∼ ω2τ−α−νzφ (y) . (20)
408
Critical sound attenuation of three-dimensional Ising systems
Figure 4. Thermal dependencies of the ultrasonic attenuation at the critical point for pure (1)
and disordered (2) systems in comparison with experimental results (3) for Gd [8].
Based on the obtained dependencies (19) we defined the exponents of asymptotic behaviour
of the attenuation coefficient in hydrodynamical and critical regions (table 1). The discrepancies
between our results for the pure system and results in [5] are determinated by the use in [5] the
critical exponents calculated on the basis of the ε-expansion, whereas we use the more accurate
values of the critical exponents and vertices in fixed point for the three-dimesional system directly.
From the table we see that in the critical region the anomalies of the attenuation coefficient
should be observe both in pure and disordered systems. However, for disordered systems the tem-
perature and frequency dependent anomalies of the attenuation coefficient should be expressed
stronger than in pure systems. These conclusions are also demonstrated in figure 4.
The obtained theoretical estimates create a great demand for detecting the anomalies which
are found to be induced by quenched disorder in ultrasonic experimental investigations of critical
dynamics.
4. Acknowledgement
This work has been supported by the Russian Foundation for Basic Research through Grants
(No. 04–02–17524, No. 04–02–39000 and No. 05–02–16188), and by Grant No. MK–8738.2006.2 of
Russian Federation President.
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PACS: 64.60.Ak, 64.60.Fr, 64.60.Ht, 43.35.Cg
410
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