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Attenuation of ultrasound near incommensurate phase transitions
PhysicsLettersA 168 (1992) 437—442 North-Holland
PHYSICS LETTERS A
Attenuation of ultrasound near incommensurate phase transitions A.M. Schorgg and F. Schwabl Institut für Theoretische Physik, Physik-Department der Technischen Universitat Munchen, James-Franck-Strasse, W-8046 Garching, Germany Received 30 March 1992; revised manuscript received 7 July 1992; accepted forpublication 8 July 1992 Communicated by L.J. Sham
We present a theory for the attenuation of ultrasound near incommensurate phase transitions. The dynamics in the phase of broken continuous symmetry are severely influenced by phason modes. Contrary to earlier assertions these massless Goldstone modesdo not alterthe hydrodynamic frequency dependence of the coefficient ofsound attenuation. However, coexistence anomalies show up as a cusp singularity of the scaling function. Application of our theory to Rb 2ZnCI4 reveals agreement over several orders ofmagnitude of the scaling variable.
The interesting dynamical properties near incommensurate phase transitions induced by the phason modes have been the subject of thorough investigations in recent years (for a recent review, see ref. [1]). However, the experimental data obtained by ultrasonic measurements have not been explained, up to now, by a quantitative theory beyond the mean-field level. On the contrary, due to the peculiarities of the phason dynamics, wrong predictions have been made concerning the low frequency behaviour of the coefficient of sound attenuation [2], i.e. an co312 instead of the hydrodynamic co2 law. Also in ref. [3] the attenuation seems to have been overestimated, prompting the assumption of a phason mass. While infrared divergent contributions appear at intermediate steps of a perturbative treatment, earlier investigations [41showed already that these cancel in the final result leading to a finite universal amplitude ratio of sound attenuation [5]. Furthermore, Dengler and one of the authors have shown that the Goldstone modes lead to a cusp singularity ofthe scaling function [5]. We confirmed these findings by an explicit solution of the spherical model [4,6]. The present work develops a renormalization group theory that puts the prediction of the Goldstone singularity on a firm basis (the main results of this article have already been presented elsewhere [7]). We determine the scaling law of the coefficient of sound attenuation and as a result we calculate the ensuing scaling functions. The comparison with experiments on Rb2ZnCl4 demonstrates that acoustic attenuation does not require to introduce a phason mass as suggested in the analysis of K2SeO4 [3]. For a large group of incommensurate solids (e.g., the A2BX4-family, with a one-dimensional incommensurate modulation along a high-symmetry direction), critical behaviour belongs to the universality class of the isotropic (n = 2)-component Heisenberg model [81, described by the well known ~ ~ free-energy functional of an n-component order parameter ø~,, (1)
.~p= $d1x{~[roø~+(Vøo)2]+(üo/4!)ø~},
with 2=~(V~
ø~=~ ~X’~J~ (Vø0)
0~)2,
Ø~=(~~)2,
r0cc(T—T~°)
where T~° is the mean field transition temperature and ü0 is positive. 0375-9601/92/S 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
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Our theory is based on an interaction being linear in the sound mode and quadratic in the order parameter. This is the relevant case for longitudinal sound modes propagating along a principal axis perpendicular to the direction of the incommensurate modulation. Thus in the displacement field u(x,
t)= -:j~ ~
Qk.~(t)Ek,Kexp(—ikx)
P the wave vector k and polarization EkK lie parallel to these axes. The energy functional .r= .J~+.~~honon,with the phonon part ~~honon thus given by ~honon=
~
[~/7RCo,cI2lQkK(t)l2+V~CopcEk,c.kYKQkK(t)~(_k,t)]
.~“
is taken to be (2)
.
The first term is the purely elastic energy with the mass density p, the bare sound velocity cOK and k= 1k I and in the second term YK is the phonon coupling strength. Dynamical properties are determined by equations of motion of the Langevin type [9]. Because the phasons are overdamped at small wavenumbers [10], purely relaxational dynamics are appropriate for the order parameter [2,3] +r~(k,t), j=(l,...,n).
(3)
The equation of motion of the acoustic phonon reads fl~,K(t) =
—
6Q~K(t)~flDKk2Q~,K(t)+RK(k, t),
(4)
where DK is the bare damping. The stochastic forces r~(k,t) of (3) and RK(k, 1) of (4) produce a Gaussian white noise obeying the Einstein relations. From this complete model, one can derive the phonon dispersion relation and an expression for the coefficient of critical sound attenuation, (5) has been obtained [11], where c is the interacting sound velocity. Here1~(k,w) is the ~ i-correlation function, determined by the pure order parameter dynamics with a shifted coupling coefficient ü 0-÷ u0 = 1 2~y~. In the low-temperature phase, the expectation value of the order parameter is finite, implying spontaneous order even in the absence of an external field. It is convenient to introduce purely fluctuational fields —
•0(k, w)_~(,,/~-j~mÔ~~~)+a(k
~)
(a=l,
...,
n—i),
(6)
where <~,> = ~ m0ô~,.From now on we use instead of r.~,the order parameter m~,to characterize the separation from the critical temperature T~.The nonlinear interactions can be treated using a field-theoretic method by introducing response fields {~ (k, o) } and ö0(k, w) [12,13]. In this approach, 1~(k,w) can be expressed in terms of vertex functions according to w)= —f~:~(k, a)+fft83(k, co)f~*~(—k, —co)/t~?’?~(—k, —w)
.
(7)
In the vertex functions ~ 2 specifies the number of (x0 ~ + o, ô~)-insertions, M that of (2r~+ c~)-insertions and ~ N count external fields ã~,,o~respectively. We draw attention to an important feature of the theory in the coexistence limit [14,15], where the Goldstone anomalies appear for k-40, w—~~0 (region of spontaneously broken symmetry). Since the fluctuations of 438
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the longitudinal amplitudon mode freeze out, certain nonlinear interactions of the energy functional drop out in this limit. The theory becomes essentially a Gaussian theory and is exactly soluble. Recently, an extended dynamical renormalization scheme, presented elsewhere [14,16], has been developed where the exact treatment of the coexistence limit is equivalent to a one-loop theory with respect to the fields {ir~}, o~,{ft~}, ö~ and which also incorporates the critical point anomalies as well. In one-loop order, the vertex functions (renormalized quantities without index 0) contain fluctuational contributions of order parameter modes, i.e.
r’~’~(k, w)
=
AdIL
[(n—i )I-r(k, w)+IL(k, w)]
(8)
.
Here p is a wave number scale, e = 4— d, and 112 xe/2 Ad= 2°’~’x’ f’(d/2) sin(7te/2) is a geometric factor. For later application the k= 0 expressions are required. In case of the transverse Goldstone modes we obtain
(9) which obviously is divergent at low frequency for 0< g <2. The longitudinal order parameter bubble yields 1+ 2 —e/2 2~. 2 2 IL(O,a))=(m2)8/2[( mT + pm [(l_iw/2)~P2m2)1_e12_l]]. (10)
)
The renormalized vertex functions obey the renormalization group equation which as usual can be solved by the method of characteristics. Thus we obtain the generalized scaling transformation ~ ft(ic/p, w/?~p~, m~,u) =
~
ft(k/u(l), w/.Z(1),u2(1), m2(1), u(l) ) —r(1)
(11)
.
Here / is an arbitrary scale parameter inducing a linear variation of the effective wave number scale p (1) = and ft is the renormalized and dimensionless counterpart of 11 By virtue of the renormalization scheme, the characteristic functions, u (1), m(l), etc. appearing in (11) depend on land the renormalized temperature variable m. The initial conditions m (1) = m, u (1) = u, etc. hold. One can show from the renormalization group flow equations that near the transition point, i.e. for values of m ~ 0.1, these are homogeneous functions, u(l) = U(1~),
m2(l) =M2(l~),
2(1) =21~2A(l 5),
(12a)
s(1) =l~~S(1~) , r(1) = (v/a)E~2—l’~R(l~),
(12b)
where 1 and m enter via the scaled flow parameter 4~ Im _22 and the critical exponents are the Heisenberg exponents. The functions U, M, etc. follow from the flow equations. They can be determined analytically in limiting cases and numerically in the whole parameter range. For instance, fig. 1 shows the flow of the coupling constant, which contains an unstable plateau at the Heisenberg fixed point u~and crosses over to the coexistence fixed point u~= 6e/ (n —1). We impose the matching condition -‘i)
2+(P~/))2=1, (13) (l);2(l)) which prevents the right-hand side of the generalized scaling law (11) from being infrared singular. Inserting 2(l) from eq. (1 2a) and using the definitions of the correlation length ~= jc ‘m “i and the characteristic —
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Fig. 1. Flow of the coupling constant for a two-component order parameter (n = 2) in three dimensions (e= 1), increasing from the plateau at the Heisenberg fixed point u~to the coexistence fixed point u~.Universal crossover behaviour appears through utilization ofthe scaled flow parameter 4.
3
10
10
7 September 1992
10
S
eq. (13) turns into an equation for the scaled flow parameter I. with the result 4= L (k~, frequency Wch =2,s2mz~~~, w/wCh).
Now we are ready to obtain from eq. (11) the scaling law for the renormalized 0 ~-corre1ationfunction ~ft(k/p,o/2p2,m2,u)=_(v/a)~.
(14a)
2+m_a/PP(k~,w/wCh),
with the scaling function P(x, y) given by
2(L(x,y)), U(L(x,y)))]
(14b)
.
.41
Our primary goal is the coefficient ofcritical sound attenuation given by (5), whose frequency and temperature dependence is dominated by the factor ã(k/p, w/2p2, m)— ~
Im{(2n/Ad)17(k/p, w/2p2, m2, u)}.
The remaining part in eq. (5) depending on the non-universal coupling constant y,~yields a numerically small contribution, which nevertheless has to be taken into account in analyzingexperimental data. Using eqs. (1 4a), (1 4b), the scaling law of the coefficient of sound attenuation reads ã(k/~u,w/2p2, m) =
(co/2p2) 2m
~‘~g(k~,
w/a)Ch)
,
(iSa)
where the scaling function is given by g(x, y)=y~Im{P(x, y)}. From our exact treatment of the coexistence limit, it follows that g(0, 0) is finite and the hydrodynamic w2 law is recovered. This result is remarkable because individual vertex functions as shown in eq. (8) contain divergent contributions of the Goldstone modes (see eq. (9)), which, however, cancel each other in the final result for fl. This finding is in accord with low order perturbation theory [4,5] and with the exact solution of the spherical model (n—pcc) [6], all of which disagree with Zeyher’s prediction [2]. The coexistence anomalies that truly can be attributed to the Goldstone modes are much more subtle and lead to a cusp singularity of the scaling function. By setting k~=0 we find for small frequencies 440
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, (l5b) g(0, co/wCh)=aO—al(n— 1) (co/aCh)~2_a2(w/wCh)_e~~2~/’2 with positive constants a. This result is in accord with eq. (3.21) of ref. [5]. As can be seen from fig. 2, the cusp singularity ofg(0, y) produces a significant effect. The cusp singularity of the Goldstone modes also has been obtained in the spherical model limit [6]. Finally, we turn to the critical region (T—~.Ta). The characteristic behaviour in this region is exhibited more clearly by the alternative scaling function G (k~,co/wCh) = (o/w~h)’~°~”g(k~, W/&Ch), which is finite on approaching T~.Hence, the attenuation becomes temperature independent in this limit, whereas the frequency dependence satisfies a universal power law. The scaling law can then be written in the equivalent form â(k/p,w/2p2, m)=(w/2p2)~_~x1’zPG(k~, w/wCh)
(15c)
.
The scaling function G(0, y) displays a characteristic maximum whose height decreases with increasing component number n and finally is absent in the spherical model limit [61. It is shown for n = 2 by the solid line in fig. 3. We now apply our theory to ultrasonic measurements of the c 33-mode in Rb2ZnCl4 [17], which is a well studied member of the A2BX4-family [1]. First, we determine the experimental scaling function G. To this end the frequency dependence according to eq. (1 Sc) and the non-universal contributions contained in (5) have to be removed from the attenuation data. As usual in low frequency ultrasonic experiments, the scaling variable kc~of the sound mode can be set to zero. The experimental temperature values r= (T— T~) / Tare related to the order parameter by m =Am I I P~By an appropriate choice of the non-universal parameters Am, )412, y3 and c,, the data points for frequencies 30, 50 and 70 MHz coalesce fairly well on a single scaling function. Here we should emphasize that the data have not been normalized to their asymptotic critical value. There is some scatter for large scaling arguments, but one has to realize that measurements performed by different groups [18] at the same frequency (30 MHz) show discrepancies of comparable size. In the presentation of the data we use T~= 302.65 K. The solid line in fig. 3 shows the theoretical scaling function. The experimental scaling 3.5
3.0
/
2.5
6
>~ 2.0 0
___________
: 0.:
,102
10_i
20 10°
302
10’
102
306
3 0
2
4
6
8
10
y 10 ~ Fig. 2. Scaling function g(0, y) at small scalingarguments y= w/ Weh for n = 2, exhibiting the Goldstone cusp singularity, the top ofwhich is marked by the arrow.
Fig. 3. Theoretical scaling function G(0, y) for n=2 (solid line) versus the scaling variable y=w/co~,experiments on Rb2ZnCI4 (0) 30MHz, (<)) 50MHz, (~)70MHz. Inset: Complete theoretical attenuation a and the experiments in [dB/cm] forthese frequencies versus the temperature.
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function is in agreement with the theoretical over almost four order of magnitude of the scaling variable. The insert shows the complete coefficient of sound attenuation as a function of temperature for the above frequencies including the high temperature phase. Our analysis confirms the hydrodynamic w2 law for the coefficient of sound attenuation. However, the present experimental data are outside the range where the cusp singularity ofg(0, y) shows up. Verification of the cusp singularity is thus a challenge for future experiments. Recent Brillouin scattering experiments in K 2SeO4 [3] are also in3”2 qualitative agreement with our predictions. Most important, the proper theoretical treatment deviation from hydrodynamics and thus there is no need to introduce a phason gap which is free hint of theata impurity pinning. We mention that our theory also gives the frequency and temperature dewould pendence of the sound velocity. We briefly comment on other systems with 0(2) symmetry, i.e. charge density wave (CDW) systems and 4He. Theoretically 4He is more complicated, since the relaxational dynamics are replaced by the 0(2) symmetric oscillatory dynamics involving superfluid velocity and entropy. There is a notable theory by Ferrell and Bhattacharjee [19], who however have neglected the propagating character of second sound. By some severe approximations the experimental data [20] could be fitted fairly well. The cusp singularity has not been found in this theory. Concerning CDW systems we mention ultrasonic work on TiS 2 etc.that [211. There is again 312 deviation from hydrodynamics [22]. It is likely a reanalysis of thisa thecaloretical prediction w theory will show the absence of such an anomaly. culation in the lightofofan our This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) under contract number Schw 348/4-1.
References [1] H.Z. Cummins, Phys. Rep. 185 (1990) 211, and references therein. [2] R. Zeyher, Ferroelectrics 66 (1986) 217. [3] G. Li, N. Tao, L. Van Hong, H.Z. Cummins, C. Dreyfus, M. Hebbache, R.M. Pick and J. Vaguer, Phys. Rev. B 42 (1990) 4406. [4] R. Dengler, Ph.D. Thesis, Technische Universität Munchen (1987). [S]R. Dengler and F. Schwabl, Z. Phys. B 69 (1987) 327. [6) A.M. Schorgg and F. Schwabl, Goldstone anomalies ofdynamic susceptibilities and sound attenuation in the spherical model limit, submitted to Phys. Rev. B. [7) A.M. Schorgg and F. Schwabl, at Workshop on Structural phase transitions in betaine compounds, especially in BCCD, Wurzburg (February 1991). [8] R.A. Cowley and A.D. Bruce, J. Phys. C 11(1978) 3577. [9] P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49 (1977) 435. [10] R. Zeyher and W. Finger, Phys. Rev. Lett. 49 (1982) 1833. [11) B. Drossel and F. Schwabl, Critical ultrasound attenuation in systems with conservation laws, to be published. [l2JH.K.Janssen,Z.Phys.B23 (1976) 377; R. Bausch, H.K. Janssen and H. Wagner, Z. Phys. B 24 (1976) 113. [13] P.C. Martin, E.D. Siggia and H.A. Rose, Phys. Rev. A 8 (1973) 423. [141 U.C. TSuber and F. Schwabl, Critical dynamics ofthe 0(n)-symmetric relaxational models belowthe transition temperature, Phys. Rev. B (1992), to be published. [lS]I.D. Lawrie, J. Phys. A 14(1981)2489. [16] A.M. Schorgg and F. Schwabl, Theory of ultrasonic attenuation at incommensurate phase transitions, to be published. [17] 5. Hirotsu, K. Toyota and K. Hamano, Ferroelectrics 36 (1981) 319; Z. Hu, C.W. Garland and S. Hirotsu, Phys. Rev. B 42 (1990) 8305. [18] V.V. Lemanov and S.K. Esayan, Ferroelectrics 73 (1987) 125. [19] R.A. Ferrell and J.K. Bhattachai~ee,Phys. Rev. B 23 (1981) 2434. [20] R.D. Williams and I. Rudnick, Phys. Rev. Lett. 25 (1970) 276; K. Tozaki and A. Ikushima, J. Low Temp. Phys. 32 (1978) 379. [21] A. Caillé, Y. Lepine, M.H. Jericho and A.M. Simpson, Phys. Rev. B 28 (1983) 5454. [22]R. Zeyher, Phys. Rev. Lett. 61(1988)1009.
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PHYSICS LETTERS A
Attenuation of ultrasound near incommensurate phase transitions A.M. Schorgg and F. Schwabl Institut für Theoretische Physik, Physik-Department der Technischen Universitat Munchen, James-Franck-Strasse, W-8046 Garching, Germany Received 30 March 1992; revised manuscript received 7 July 1992; accepted forpublication 8 July 1992 Communicated by L.J. Sham
We present a theory for the attenuation of ultrasound near incommensurate phase transitions. The dynamics in the phase of broken continuous symmetry are severely influenced by phason modes. Contrary to earlier assertions these massless Goldstone modesdo not alterthe hydrodynamic frequency dependence of the coefficient ofsound attenuation. However, coexistence anomalies show up as a cusp singularity of the scaling function. Application of our theory to Rb 2ZnCI4 reveals agreement over several orders ofmagnitude of the scaling variable.
The interesting dynamical properties near incommensurate phase transitions induced by the phason modes have been the subject of thorough investigations in recent years (for a recent review, see ref. [1]). However, the experimental data obtained by ultrasonic measurements have not been explained, up to now, by a quantitative theory beyond the mean-field level. On the contrary, due to the peculiarities of the phason dynamics, wrong predictions have been made concerning the low frequency behaviour of the coefficient of sound attenuation [2], i.e. an co312 instead of the hydrodynamic co2 law. Also in ref. [3] the attenuation seems to have been overestimated, prompting the assumption of a phason mass. While infrared divergent contributions appear at intermediate steps of a perturbative treatment, earlier investigations [41showed already that these cancel in the final result leading to a finite universal amplitude ratio of sound attenuation [5]. Furthermore, Dengler and one of the authors have shown that the Goldstone modes lead to a cusp singularity ofthe scaling function [5]. We confirmed these findings by an explicit solution of the spherical model [4,6]. The present work develops a renormalization group theory that puts the prediction of the Goldstone singularity on a firm basis (the main results of this article have already been presented elsewhere [7]). We determine the scaling law of the coefficient of sound attenuation and as a result we calculate the ensuing scaling functions. The comparison with experiments on Rb2ZnCl4 demonstrates that acoustic attenuation does not require to introduce a phason mass as suggested in the analysis of K2SeO4 [3]. For a large group of incommensurate solids (e.g., the A2BX4-family, with a one-dimensional incommensurate modulation along a high-symmetry direction), critical behaviour belongs to the universality class of the isotropic (n = 2)-component Heisenberg model [81, described by the well known ~ ~ free-energy functional of an n-component order parameter ø~,, (1)
.~p= $d1x{~[roø~+(Vøo)2]+(üo/4!)ø~},
with 2=~(V~
ø~=~ ~X’~J~ (Vø0)
0~)2,
Ø~=(~~)2,
r0cc(T—T~°)
where T~° is the mean field transition temperature and ü0 is positive. 0375-9601/92/S 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
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Our theory is based on an interaction being linear in the sound mode and quadratic in the order parameter. This is the relevant case for longitudinal sound modes propagating along a principal axis perpendicular to the direction of the incommensurate modulation. Thus in the displacement field u(x,
t)= -:j~ ~
Qk.~(t)Ek,Kexp(—ikx)
P the wave vector k and polarization EkK lie parallel to these axes. The energy functional .r= .J~+.~~honon,with the phonon part ~~honon thus given by ~honon=
~
[~/7RCo,cI2lQkK(t)l2+V~CopcEk,c.kYKQkK(t)~(_k,t)]
.~“
is taken to be (2)
.
The first term is the purely elastic energy with the mass density p, the bare sound velocity cOK and k= 1k I and in the second term YK is the phonon coupling strength. Dynamical properties are determined by equations of motion of the Langevin type [9]. Because the phasons are overdamped at small wavenumbers [10], purely relaxational dynamics are appropriate for the order parameter [2,3] +r~(k,t), j=(l,...,n).
(3)
The equation of motion of the acoustic phonon reads fl~,K(t) =
—
6Q~K(t)~flDKk2Q~,K(t)+RK(k, t),
(4)
where DK is the bare damping. The stochastic forces r~(k,t) of (3) and RK(k, 1) of (4) produce a Gaussian white noise obeying the Einstein relations. From this complete model, one can derive the phonon dispersion relation and an expression for the coefficient of critical sound attenuation, (5) has been obtained [11], where c is the interacting sound velocity. Here1~(k,w) is the ~ i-correlation function, determined by the pure order parameter dynamics with a shifted coupling coefficient ü 0-÷ u0 = 1 2~y~. In the low-temperature phase, the expectation value of the order parameter is finite, implying spontaneous order even in the absence of an external field. It is convenient to introduce purely fluctuational fields —
•0(k, w)_~(,,/~-j~mÔ~~~)+a(k
~)
(a=l,
...,
n—i),
(6)
where <~,> = ~ m0ô~,.From now on we use instead of r.~,the order parameter m~,to characterize the separation from the critical temperature T~.The nonlinear interactions can be treated using a field-theoretic method by introducing response fields {~ (k, o) } and ö0(k, w) [12,13]. In this approach, 1~(k,w) can be expressed in terms of vertex functions according to w)= —f~:~(k, a)+fft83(k, co)f~*~(—k, —co)/t~?’?~(—k, —w)
.
(7)
In the vertex functions ~ 2 specifies the number of (x0 ~ + o, ô~)-insertions, M that of (2r~+ c~)-insertions and ~ N count external fields ã~,,o~respectively. We draw attention to an important feature of the theory in the coexistence limit [14,15], where the Goldstone anomalies appear for k-40, w—~~0 (region of spontaneously broken symmetry). Since the fluctuations of 438
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the longitudinal amplitudon mode freeze out, certain nonlinear interactions of the energy functional drop out in this limit. The theory becomes essentially a Gaussian theory and is exactly soluble. Recently, an extended dynamical renormalization scheme, presented elsewhere [14,16], has been developed where the exact treatment of the coexistence limit is equivalent to a one-loop theory with respect to the fields {ir~}, o~,{ft~}, ö~ and which also incorporates the critical point anomalies as well. In one-loop order, the vertex functions (renormalized quantities without index 0) contain fluctuational contributions of order parameter modes, i.e.
r’~’~(k, w)
=
AdIL
[(n—i )I-r(k, w)+IL(k, w)]
(8)
.
Here p is a wave number scale, e = 4— d, and 112 xe/2 Ad= 2°’~’x’ f’(d/2) sin(7te/2) is a geometric factor. For later application the k= 0 expressions are required. In case of the transverse Goldstone modes we obtain
(9) which obviously is divergent at low frequency for 0< g <2. The longitudinal order parameter bubble yields 1+ 2 —e/2 2~. 2 2 IL(O,a))=(m2)8/2[( mT + pm [(l_iw/2)~P2m2)1_e12_l]]. (10)
)
The renormalized vertex functions obey the renormalization group equation which as usual can be solved by the method of characteristics. Thus we obtain the generalized scaling transformation ~ ft(ic/p, w/?~p~, m~,u) =
~
ft(k/u(l), w/.Z(1),u2(1), m2(1), u(l) ) —r(1)
(11)
.
Here / is an arbitrary scale parameter inducing a linear variation of the effective wave number scale p (1) = and ft is the renormalized and dimensionless counterpart of 11 By virtue of the renormalization scheme, the characteristic functions, u (1), m(l), etc. appearing in (11) depend on land the renormalized temperature variable m. The initial conditions m (1) = m, u (1) = u, etc. hold. One can show from the renormalization group flow equations that near the transition point, i.e. for values of m ~ 0.1, these are homogeneous functions, u(l) = U(1~),
m2(l) =M2(l~),
2(1) =21~2A(l 5),
(12a)
s(1) =l~~S(1~) , r(1) = (v/a)E~2—l’~R(l~),
(12b)
where 1 and m enter via the scaled flow parameter 4~ Im _22 and the critical exponents are the Heisenberg exponents. The functions U, M, etc. follow from the flow equations. They can be determined analytically in limiting cases and numerically in the whole parameter range. For instance, fig. 1 shows the flow of the coupling constant, which contains an unstable plateau at the Heisenberg fixed point u~and crosses over to the coexistence fixed point u~= 6e/ (n —1). We impose the matching condition -‘i)
2+(P~/))2=1, (13) (l);2(l)) which prevents the right-hand side of the generalized scaling law (11) from being infrared singular. Inserting 2(l) from eq. (1 2a) and using the definitions of the correlation length ~= jc ‘m “i and the characteristic —
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Fig. 1. Flow of the coupling constant for a two-component order parameter (n = 2) in three dimensions (e= 1), increasing from the plateau at the Heisenberg fixed point u~to the coexistence fixed point u~.Universal crossover behaviour appears through utilization ofthe scaled flow parameter 4.
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eq. (13) turns into an equation for the scaled flow parameter I. with the result 4= L (k~, frequency Wch =2,s2mz~~~, w/wCh).
Now we are ready to obtain from eq. (11) the scaling law for the renormalized 0 ~-corre1ationfunction ~ft(k/p,o/2p2,m2,u)=_(v/a)~.
(14a)
2+m_a/PP(k~,w/wCh),
with the scaling function P(x, y) given by
2(L(x,y)), U(L(x,y)))]
(14b)
.
.41
Our primary goal is the coefficient ofcritical sound attenuation given by (5), whose frequency and temperature dependence is dominated by the factor ã(k/p, w/2p2, m)— ~
Im{(2n/Ad)17(k/p, w/2p2, m2, u)}.
The remaining part in eq. (5) depending on the non-universal coupling constant y,~yields a numerically small contribution, which nevertheless has to be taken into account in analyzingexperimental data. Using eqs. (1 4a), (1 4b), the scaling law of the coefficient of sound attenuation reads ã(k/~u,w/2p2, m) =
(co/2p2) 2m
~‘~g(k~,
w/a)Ch)
,
(iSa)
where the scaling function is given by g(x, y)=y~Im{P(x, y)}. From our exact treatment of the coexistence limit, it follows that g(0, 0) is finite and the hydrodynamic w2 law is recovered. This result is remarkable because individual vertex functions as shown in eq. (8) contain divergent contributions of the Goldstone modes (see eq. (9)), which, however, cancel each other in the final result for fl. This finding is in accord with low order perturbation theory [4,5] and with the exact solution of the spherical model (n—pcc) [6], all of which disagree with Zeyher’s prediction [2]. The coexistence anomalies that truly can be attributed to the Goldstone modes are much more subtle and lead to a cusp singularity of the scaling function. By setting k~=0 we find for small frequencies 440
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, (l5b) g(0, co/wCh)=aO—al(n— 1) (co/aCh)~2_a2(w/wCh)_e~~2~/’2 with positive constants a. This result is in accord with eq. (3.21) of ref. [5]. As can be seen from fig. 2, the cusp singularity ofg(0, y) produces a significant effect. The cusp singularity of the Goldstone modes also has been obtained in the spherical model limit [6]. Finally, we turn to the critical region (T—~.Ta). The characteristic behaviour in this region is exhibited more clearly by the alternative scaling function G (k~,co/wCh) = (o/w~h)’~°~”g(k~, W/&Ch), which is finite on approaching T~.Hence, the attenuation becomes temperature independent in this limit, whereas the frequency dependence satisfies a universal power law. The scaling law can then be written in the equivalent form â(k/p,w/2p2, m)=(w/2p2)~_~x1’zPG(k~, w/wCh)
(15c)
.
The scaling function G(0, y) displays a characteristic maximum whose height decreases with increasing component number n and finally is absent in the spherical model limit [61. It is shown for n = 2 by the solid line in fig. 3. We now apply our theory to ultrasonic measurements of the c 33-mode in Rb2ZnCl4 [17], which is a well studied member of the A2BX4-family [1]. First, we determine the experimental scaling function G. To this end the frequency dependence according to eq. (1 Sc) and the non-universal contributions contained in (5) have to be removed from the attenuation data. As usual in low frequency ultrasonic experiments, the scaling variable kc~of the sound mode can be set to zero. The experimental temperature values r= (T— T~) / Tare related to the order parameter by m =Am I I P~By an appropriate choice of the non-universal parameters Am, )412, y3 and c,, the data points for frequencies 30, 50 and 70 MHz coalesce fairly well on a single scaling function. Here we should emphasize that the data have not been normalized to their asymptotic critical value. There is some scatter for large scaling arguments, but one has to realize that measurements performed by different groups [18] at the same frequency (30 MHz) show discrepancies of comparable size. In the presentation of the data we use T~= 302.65 K. The solid line in fig. 3 shows the theoretical scaling function. The experimental scaling 3.5
3.0
/
2.5
6
>~ 2.0 0
___________
: 0.:
,102
10_i
20 10°
302
10’
102
306
3 0
2
4
6
8
10
y 10 ~ Fig. 2. Scaling function g(0, y) at small scalingarguments y= w/ Weh for n = 2, exhibiting the Goldstone cusp singularity, the top ofwhich is marked by the arrow.
Fig. 3. Theoretical scaling function G(0, y) for n=2 (solid line) versus the scaling variable y=w/co~,experiments on Rb2ZnCI4 (0) 30MHz, (<)) 50MHz, (~)70MHz. Inset: Complete theoretical attenuation a and the experiments in [dB/cm] forthese frequencies versus the temperature.
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function is in agreement with the theoretical over almost four order of magnitude of the scaling variable. The insert shows the complete coefficient of sound attenuation as a function of temperature for the above frequencies including the high temperature phase. Our analysis confirms the hydrodynamic w2 law for the coefficient of sound attenuation. However, the present experimental data are outside the range where the cusp singularity ofg(0, y) shows up. Verification of the cusp singularity is thus a challenge for future experiments. Recent Brillouin scattering experiments in K 2SeO4 [3] are also in3”2 qualitative agreement with our predictions. Most important, the proper theoretical treatment deviation from hydrodynamics and thus there is no need to introduce a phason gap which is free hint of theata impurity pinning. We mention that our theory also gives the frequency and temperature dewould pendence of the sound velocity. We briefly comment on other systems with 0(2) symmetry, i.e. charge density wave (CDW) systems and 4He. Theoretically 4He is more complicated, since the relaxational dynamics are replaced by the 0(2) symmetric oscillatory dynamics involving superfluid velocity and entropy. There is a notable theory by Ferrell and Bhattacharjee [19], who however have neglected the propagating character of second sound. By some severe approximations the experimental data [20] could be fitted fairly well. The cusp singularity has not been found in this theory. Concerning CDW systems we mention ultrasonic work on TiS 2 etc.that [211. There is again 312 deviation from hydrodynamics [22]. It is likely a reanalysis of thisa thecaloretical prediction w theory will show the absence of such an anomaly. culation in the lightofofan our This work has been supported by the Deutsche Forschungsgemeinschaft (DFG) under contract number Schw 348/4-1.
References [1] H.Z. Cummins, Phys. Rep. 185 (1990) 211, and references therein. [2] R. Zeyher, Ferroelectrics 66 (1986) 217. [3] G. Li, N. Tao, L. Van Hong, H.Z. Cummins, C. Dreyfus, M. Hebbache, R.M. Pick and J. Vaguer, Phys. Rev. B 42 (1990) 4406. [4] R. Dengler, Ph.D. Thesis, Technische Universität Munchen (1987). [S]R. Dengler and F. Schwabl, Z. Phys. B 69 (1987) 327. [6) A.M. Schorgg and F. Schwabl, Goldstone anomalies ofdynamic susceptibilities and sound attenuation in the spherical model limit, submitted to Phys. Rev. B. [7) A.M. Schorgg and F. Schwabl, at Workshop on Structural phase transitions in betaine compounds, especially in BCCD, Wurzburg (February 1991). [8] R.A. Cowley and A.D. Bruce, J. Phys. C 11(1978) 3577. [9] P.C. Hohenberg and B.I. Halperin, Rev. Mod. Phys. 49 (1977) 435. [10] R. Zeyher and W. Finger, Phys. Rev. Lett. 49 (1982) 1833. [11) B. Drossel and F. Schwabl, Critical ultrasound attenuation in systems with conservation laws, to be published. [l2JH.K.Janssen,Z.Phys.B23 (1976) 377; R. Bausch, H.K. Janssen and H. Wagner, Z. Phys. B 24 (1976) 113. [13] P.C. Martin, E.D. Siggia and H.A. Rose, Phys. Rev. A 8 (1973) 423. [141 U.C. TSuber and F. Schwabl, Critical dynamics ofthe 0(n)-symmetric relaxational models belowthe transition temperature, Phys. Rev. B (1992), to be published. [lS]I.D. Lawrie, J. Phys. A 14(1981)2489. [16] A.M. Schorgg and F. Schwabl, Theory of ultrasonic attenuation at incommensurate phase transitions, to be published. [17] 5. Hirotsu, K. Toyota and K. Hamano, Ferroelectrics 36 (1981) 319; Z. Hu, C.W. Garland and S. Hirotsu, Phys. Rev. B 42 (1990) 8305. [18] V.V. Lemanov and S.K. Esayan, Ferroelectrics 73 (1987) 125. [19] R.A. Ferrell and J.K. Bhattachai~ee,Phys. Rev. B 23 (1981) 2434. [20] R.D. Williams and I. Rudnick, Phys. Rev. Lett. 25 (1970) 276; K. Tozaki and A. Ikushima, J. Low Temp. Phys. 32 (1978) 379. [21] A. Caillé, Y. Lepine, M.H. Jericho and A.M. Simpson, Phys. Rev. B 28 (1983) 5454. [22]R. Zeyher, Phys. Rev. Lett. 61(1988)1009.
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