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Influence of defects on the phase transition in squaric acid (H2C4O4)
0038-1098/81/200749-05502.00/0
Solid State Communications, Vol. 38. pp. 749-753. Pergamon Press Ltd. 1981. Printed in Great Britain.
INFLUENCE OF DEFECTS ON THE PHASE TRANSITION IN SQUARIC ACID (H,.C~O4) D. Suwelack and M. Mehring Institut fiar Physik, Postfach 500500, 4600 Dortmund 50, Germany (Received 17 September 1980; in revised form 5 December 1980 by B. Miihlschlegel) We have investigated the influence of defects induced by chromium and deuterium doping on the phase transition in solid squaric acid (H2C404) by high resolution ~3C NMR. Deuterium doping alone is observed to increase the phase transition temperature Tc linearly with the 2H concentration, whereas chromium defects, destroy the local order and lead to clusters of the high temperature phase. Correspondingly the critical temperature decreases with increasing chromium doping and the critical temperature region is smeared out.
1. INTRODUCTION THE INFLUENCE of defects, clusters and their interrelation near phase transitions has attracted much interest since the discovery of a "central peak" in the phonon spectrum of strontium titanate (SrTiO3)at the phase transition by Riste et al. [1, 2] by means of neutron scattering. Also in light scattering experiments the central peak phenomena has been observed [3]. There is a wealth of theoretical explanations for this phenomena available [4-6] based either on "intrinsic" (i.e. solitonlike excitations) or on "extrinsic" defects introduced by impurities. We are considering in this publication the "extrinsic" nature of cluster formation near the phase transition only. Halperin and Varma [4] investigated the possibility of a "frozen defect cell" near the phase transition. Such a "frozen defect" cannot adjust itself to the orderparameter. Orderparameter fluctuations are therefore enhanced in the vicinity of such a defect. The critical temperature Tc in crystals with "frozen defects" is therefore expected to decrease with defect concentration [4]. The increase of the central peak of the phonon spectra in SrTiO3 with increasing defect concentration has recently been observed by Hastings et al. [10] in neutron scattering experiments. A decrease of T~ with increasing defect concentration was observed as well. Only resonance methods, however, such as ESR [7], NQR [8, 9] and high resolution NMR [11, 12] are capable of distinguishing microscopic clusters of different order parameter. Blinc and Bjorkstam [8] were able to observe polar clusters in KDA far above the phase transition temperature. No concentration dependence was investigated and the "extrinsic" or "intrinsic" nature of the clusters could not be decided. An explanation allowing for both types of clusters was recently 749
given by Blinc [91. Mailer and Berlinger [7] using ESR were able to observe an "extrinsic" slowing down of order-parameter fluctuations near the defect. Using high resolution 13C NMR spectroscopy [ 13] we have recently given a preliminary account on the influence of defects on the cluster formation near the phase transition in squaric acid [11]. The goal of this publication is to present a more detailed experimental investigation of' the influence of chromium and deuterium doping on the phase transition in this substance. Squaric acid undergoes a structural phase transition at Tc = 373 K as was first observed by Semmingsen and Feder [14] and Samuelsen and Semmingsen [15]. Although a continuous transition to the disordered phase was suggested originally [14, 15], more recent investigations [ 11, 16, 17] have revealed a discontinuous jump of the order parameter at Tc, suggesting a "first order" transition. The "intrinsic" slowing down of order parameter fluctuations when approaching Tc from above has recently been demonstrated by ~3C NMR [19]. 2. EXPERIMENTAL Single crystals of squaric acid were grown by slowly cooling an aqueous solution. Deuterium doping - with different concentrations - was achieved by varying the D20/H20 ratio of the aqueous solution. A three step recristallization procedure was used. The deuterium concentration XD corresponds to the ratio of D/H in the solution. The chromium doping was achieved by dissolving CrCI3 in the aqueous solution. The chromium doping of the crystals manifests itself in a green coloring of the crystals, a shortening of the proton spin-lattice relaxation time and the appearance of ESR-spectra [20]. The chromium concentration Xcr was obtained in the following way:
750
INFLUENCE OF DEFECTS IN (H2C404)
(i) we have choosen six different samples with a nominal concentration Xcr = 0.01 tool. % in the solution and have assumed that their average concentration corresponds to 0.01 mol. %. (ii) We have measured for all our samples the proton spin-lattice relaxation rate which is proportional to the chromium concentration [ 12]. Although the absolute chromium concentration Xc, is not precisely known, the relative chromium concentration with respect to the set (i) could nevertheless be determined• We have applied a high-resolution 13C NMR technique [13] to obtain ~3C spectra of naturally abundant lSC (1.1%). The measurements were performed in two different superconducting magnets with 6,3 T and 4.2 T respectively• The corresponding resonance frequencies were: 13C (67.9 MHz), ~H (270 MHz) and LSc (45.3 MHz), tH (180 MHz) respectively. The temperature regulation was achieved within -+ 0.1 K using preheated nitrogen gas. Temperature inhomogeneity across the sample was less then + 0.1 K [22]• The single crystals of squaric acid were oriented in the magnetic field in such a way as to obtain a maximal line splitting of the four magnetically non-equivalent ~3C nuclei• The line separation has been shown to be temperature dependent and to vary proportional to the order parameter [23]•
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3. RESULTS Typical 13C NMR spectra near the phase transition temperature as shown in Fig. 1 were obtained in all samples. All four lines corresponding to the ordered phase are visible below the transition temperature (bottom), whereas only two lines are observed in the high temperature phase (top). In between a coexistence of both phases is observed and no continuous reduction of the linesplitting (order parameter) takes place• As pointed out before [ 11 ] there is a jump of the order parameter at Tc suggesting a discontinuous phase transition [16, 17]. The number of squaric acid molecules which belong to high- and low-temperature phase clusters in the sample near the phase transition is directly proportional to the integrated intensities Ih, IZ of the corresponding NMR lines• The "concentration" of these molecules can therefore be expressed as Ca = Ia/Io for the high-temperature phase clusters and Cz = IJIo for the low-temperature phase, with Io = Ih + II. Experimental spectra as shown in Fig. 1 were fitted by a computer analysis to a set of spectral lines belonging to either the low- or hightemperature phase respectively• Numerical integration of the fitted spectral set then led to In and It.
Vol. 38, No. 8
t
*
Fig. 1. ~3C spectra of a squaric acid sin~e crystal doped with 0.017 mol. % Cr 3÷ at different temperatures near the phase transition temperature To. Below the transition temperature four lines (bottom) are observed due to the four non-equivalent 13C nuclei in the unit cell. Above the transition two lines (top) reflect the higher symmetry, whereas the coexistence of both phases is visible in between. The variation of Ch and Cz with temperature could be described for all our samples in the whole temperature range where cluster spectra were observable by
Ch/Cl = exp (ICe)
(1)
with =
(T--
r~*)/r$
(2)
and K being a dimensionless constant. The critical temperature T* is defined here as the temperature where Ch = Cl = 1/2• The relative temperature interval + AT~T* in which both phases coexist is related to the constant K by
K - ' = AT~T*.
(3)
Vol. 38, No. 8
INFLUENCE ON DEFECTS IN (H2C404)
751 385,
Cr3"molGr cone I0
_(Q) 37~%
- / /
(b)
/ /
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/ / C 17.10 [] 0 8
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375~-OlI l
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370
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Fig. 2. Cluster concentration Cz of the low temperature phase versus temperature according to equations (4), (5) for different chromium concentrations.
O C-S
i 0
! 2
%
I a
Xo,
I 6 %
Fig. 3. Variation of the critical temperature Tc for (a) chromium (Xc~) as well as (b) deuterium (Xo) doping. Notice the different scale on the Xc~ and XD axis respectively. The dashed line corresponds to equation (11).
Table 1. Evaluation o f the parameter T* and K according to equations (4) and (5)for different samples o f squaric acid doped with chromium and deuterium respectively
is observed. Deuterium doping on the other hand increases the transition temperature. Again a linear dependence of T* on the defect concentration (XD: deuterium) which follows
Xcr (tool.%)
X o (mol.%)
T* (K)
10 3 K - l
T* = Tc(I + 0.295XD);
is obtained. Notice, however, that the temperature dependence on doping is by about a factor 100 larger in the case of chromium as for deuterium. A certain set of partially deuterated samples contained also a small amount of chromium in order to reduce the proton spin-lattice relaxation time. This was done for experimental convenience. Therefore the data of the kind shown in Fig. 3 had to be corrected for the chromium doping. The assumption that chromium and deuterium doping have an additive effect on T* and K-~ leads to the relations:
0.055 0.044 0.017 0.008 0.000
0 0 0 0 0
367.0 370.9 371.3 372.6 373.2
7.93 3.44 1.32 0.90 0.81
0.013 0.006 0.014 0.021 0.009
I 2 3 5 I0
373.7 375.6 375.6 375.8 384.9
1.30 2.22 1.40 3.51 1.32
T* = Tc(I + 0.295XD -- 24.6Xcr);
The temperature variation of Ch and Cl follows from equation (1) and the definition (2) as [11]
Ch = [exp (-- Ke) + I ] - t,
and (4)
and
Ct = [exp (Ke) + I I -~
(5)
Figure 2 shows this variation of cluster concentration with temperature for a typical set of data. Evaluation of the parameters K and T* for different samples is summarized in Table I. A dramatic decrease of the critical temperature T* is observed with increasing chromium concentration Xcr. This is evident from Fig. 3 where T* is plotted vs Xcr and XD respectively. A linear dependence of the form T* = Tc(1 -- 24.6Xcr);
Tc = 373.3 K
Tc = 373.5K
(6)
K -t = K o 1 + K c r I + K D l
(7)
Tc = 373.4 K (8) (9)
where a background value Ko- ~ has been introduced to account for the effects observed in the undoped crystals. As was pointed out before [ 11] even in completely undoped crystals cluster formation could be observed near the phase transition and follows equations (4) and (5). There may be an "intrinsic" cause for the cluster formation in terms of"heterophase fluctuations" [24]. However, also defects like dislocations and point defects can be responsible for the formation of clusters. The data analysis shows, that K~ ~ is temperature independent and small. An average over all samples leads to g D = 0.7 x 10 -3 which is much smaller than the background Ko--1. We have therefore plotted only
752
INFLUENCE OF DEFECTS IN (H2C404) i
(b)
7 I ! I I I
35'3
\ L \
it
\
I it
25O
I // .p
200
i
052
O'Oa %Cr
i50
3÷E
3~
3~3
TIK
Fig. 4. Left: The relative temperature interval of the coexistence region K -t vs. chromium concentration according to the data listed in the table. Right: 13C linewidth 6v of the central line, belonging to hightemperature phase clusters versus temperature (see text). K~ + KcrL vs chromium concentration Xcr in Fig. 4. A dramatic increase of the "smearing-out parameter" K - ~ with increasing chromium doping Xcr is observed. Another feature is evident from Fig. 1, namely the narrowing of the central line with increasing temperature. This feature is more clearly shown in the right part of Fig. 4 where the linewidth 6v of the central line is plotted versus temperature for a doped crystal with 0.017 mol.% chromium. Homogeneous broadening of the line, e.g. by transversal spin-lattice relaxation time 7"2 could be excluded by spin-echo experiments [22]. The observed broadening is therefore due to inhomogeneous local fields, probably caused by the r -3 dependence of the Cr 3+ local field at the nuclear site [25]. (i) Chromium defects: The concentration of chromium defects used in this investigation corresponds to an average distance of 8 0 - 1 5 0 A. Defect-defect interaction can therefore be excluded. The chromium replaces hydrogen bonds. Since chromium does not take part in the H-bond dynamics, it therefore interrupts the H-bond chains which transmit the order through the crystal. The chain interruption can be related to the bond percolation problem of the Ising model. There magnetic atoms are randomly replaced by nonmagnetic atoms with concentration X. Brooks-Harris [26] calculated the variation of the critical temperature Tc with concentration X in the percolation problem of the Ising model. A linear temperature variation of the type T* = T e ( l - - s X )
(10)
was obtained with s = 1.329 for the 2-dimensional quadratic Ising-model and s = 1.060 for the 3-dimensional simple cubic Ising model. Landau [27] performed a Monte Carlo calculation on the same model
Vol. 38, No. 8
and confirmed equation (10). For higher defect concentration near the percolation threshold a deviation from equation (10) is expected [28]. A smearing out of the phase transition with increasing detect concentration has been found in those theoretical calculations as well as in [26, 29, 30]. The theoretical reasoning leading to equation (10) is not directly transferable to the chromium doping of squaric acid, however. The chromium ion seems to "wipe out" several bonds, making the down-shift in the critical temperature much more effective than in the simple percolation problem. By comparing equations (6) and (i0) we find a "wipe out ratio'R = 24.6/1.329 = 18.5 for the two-dimensional lattice. In essence many H-bonds are effected in the vicinity of the chromium ion leading to a reduced "mean field" of H-bonds in the neighourhood of the chromium ion. In the classification of Halperin and Varma [4] the chromium defect in squaric acid corresponds to a "frozen defect cell". The conjecture of Halperin and Varma [4] that "frozen impurities will have a strong tendency to depress Tc below the value for the pure system" is clearly seen in our data. Recent ESR measurements by Stankowski et al. [20] on the chromium defect in squaric acid have shown, that the Cr 3÷ center occupies interstitial sites between the layers in the center of oxygen octahedra. Thus at least eight next neighbour hydrogen are "frozen", perhaps even more are affected. In contrast to ESR experiments [7, 12, 31], where the defect ion itself is investigated, high resolution 13C NMR allows to investigate the molecules in the neighbourhood as well as in the undistorted areas of the sample individually. (ii) Deuterium defects: Deuterium 2H replaces quite naturally the protons in the sample. Because of a drastic reduction of the tunnelling integral for deuterium in the hydrogen bond with respect to protons an increase of the phase transition temperature is expected [32]. The deuterium concentration XD in the crystal may not be, however, identical to the D20/H20 ratio of the crystal growing solution. In KH2POa (KDP), e.g. Loiacono et al. [33] found, that the deuterium concentration in the crystal XD was related to the deuterium concentration in the solution XDS by XD = O.0068XDs exp (0.00382XDs).
(11)
If we assume the same relation to hold in our case, the data shown in Fig. 3 referring to the deuterated samples have to be corrected according to equation (11). The corrected data are shown in Fig. 3(b). It is interesting to find the data for shifted transition temperature near the straight line, connecting simply the two transition temperatures TCH = 373.4 K (XD = O) and TCD = 516 K
Vol. 38, No. 8
INFLUENCE OF DEFECTS IN (H~_C~O~)
(XD = 1). The transition temperature Tc of partially deuterated squaric acid is therefore given by Yc = TCH(I --XD) + TcDXD
7. 8. (12)
with TCH = 373.4 K and TCD = 516K. Deuterium replacing hydrogens in the hydrogen bond creates a defect cell which hops back and forth between positions which break the symmetry in opposite senses. It therefore can be considered to be a "relaxing defect cell" in the sense of Halperin and Varma [4]. The change in temperature for dilute defect concentrations XD can be expressed as [4]
drc
rc = rcH + XD - -
dAD
(13)
(14)
-= TCHA dXD
where the parameter A is positive in the case of a relaxing defect cell and can be related to model parameters of the system. Our values obtained for the deuterium defect correspond to A = 0.38. It would be interesting to verify this relation for the full range of deuteration (0 ~< XD ~< 1). Acknowledgements - Discussions with Professors J. Petersson - on the nature of the phase transitions and R. Blinc - on the duster problem - are gratefully acknowledged. The Deutsche Forschungsgemeinschaft (DFG) has given a financial support. REFERENCES
3. 4. 5. 6.
12. 13. 14.
16.
dTc
2.
11.
15.
with
1.
9. I0.
T. Riste, E.J. Samuelsen, K. Otnes & J. Feder, SolidState Commun_ 9, 1455 (1971). S.M. Shapiro, J.D. Axe, G. Shirane & T. Riste, Phys. Rev. B6, 4332 (1972). N.C. Lagakos & H.Z. Cummins, Phys. Rev. BIO, 1063 (1974); K.B. Lyons & P.A. Fleury,Phys. Rev. BI7, 2403 (1978). B.I. Halperin & C.M. Varma,Phys. Rev. B14, 4030 (1976). K.H. H6ck & H. Thomas, Z. Physik B27,267 (1977). H. Schmidt & F. Schwabl, Phys. Lett. 61A, 476 (1977).
17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.
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K.A. Mialler & W. BerIinger. Z. Physik B31, 151 (1978). IL B[inc & J.L. Bjorkstam, Phys. Rev. Lett. 23, 788 (1969). R. Blinc, J. Magn. Res. 30, 251 (1978). J.B. Hastings, S.M. Shapiro & B.E. Frazer, Phys. Rev. Lett. 40,237 (1978). M. Mehring & D. Suwelack, Phys. Rev. Lett. 42, 317(1979). M. Mehring, High Resolution NMR Spectroscopy in Solids, NMR: Basic Principles and Progress, Vol. 1 I. Springer-Verlag, Berlin (1976). A. Pines, M.G. Gibby & J.S. Waugh, J. Chem. Phys. 56, 1776 (1972);3". Chem. Phys. 59,569 (1973). D. Semmingsen & J. Feder, Solid State Commun. 15, 1369 (1974). E.J. Samuelsen & D. Semmingsen, SolidState Commun. 17,217 (1975). D. Barth, J. Helwig, D.H. Maier, H.E. Maser & J. Petersen, Z. Physik B34,393 (t 979). W. Kuhn, H.D. Maier & J. Petersson, SolM State Commun. 32,249 (1979). J.D. Becker, M. Mehring & D. Suwelack, Solid State Commun. (to be published). D. Suwelack & M. Mehring, SolM State Commun. 33,207 (1980). J. Stankowski (private communication). A. Abragam, Principles of Nuclear Magnetism. Oxford University Press, London (1961 ). D. Suwelack, Ph.D. Thesis, Univ. Dortmund, 4600 Dortmund/W.-Germany (March 1979). J.D. Becket, D. Suwelack & M. Mehring, Solid State Commun. 25, 1145 (1978). H.E. Cook, Phys. Rev. BI5, 1477 (1977). A. Abragam, Principles of Nuclear Magnetism, p. 381. Oxford University Press, London (196t). A. Brooks-Harris, J. Phys. C7, 1671 (1974). D.P. Landau,Physica 86-88B, 731 (1977). M.A. Klenin, Phys. Rev. B19, 3586 (1979). B.Ya. Bala~rov & V.G. Vaks, Soy. Phys. JETP 38,799 (1974). Y. Imry & M. Wortis, Phys. Rev. BI9, 3580 (1979). K. Binder, SolidState Commun. 24,401 (1977). R. Blinc & B. Zeks, Soft Modes in Ferroelectrics and Antiferroelectrics. North-Holland, Amsterdam (1974). G.M. Loiacono, J.F. Balscio & W. Osborne,AppL Phys. Lett. 24,455 (1974).
Solid State Communications, Vol. 38. pp. 749-753. Pergamon Press Ltd. 1981. Printed in Great Britain.
INFLUENCE OF DEFECTS ON THE PHASE TRANSITION IN SQUARIC ACID (H,.C~O4) D. Suwelack and M. Mehring Institut fiar Physik, Postfach 500500, 4600 Dortmund 50, Germany (Received 17 September 1980; in revised form 5 December 1980 by B. Miihlschlegel) We have investigated the influence of defects induced by chromium and deuterium doping on the phase transition in solid squaric acid (H2C404) by high resolution ~3C NMR. Deuterium doping alone is observed to increase the phase transition temperature Tc linearly with the 2H concentration, whereas chromium defects, destroy the local order and lead to clusters of the high temperature phase. Correspondingly the critical temperature decreases with increasing chromium doping and the critical temperature region is smeared out.
1. INTRODUCTION THE INFLUENCE of defects, clusters and their interrelation near phase transitions has attracted much interest since the discovery of a "central peak" in the phonon spectrum of strontium titanate (SrTiO3)at the phase transition by Riste et al. [1, 2] by means of neutron scattering. Also in light scattering experiments the central peak phenomena has been observed [3]. There is a wealth of theoretical explanations for this phenomena available [4-6] based either on "intrinsic" (i.e. solitonlike excitations) or on "extrinsic" defects introduced by impurities. We are considering in this publication the "extrinsic" nature of cluster formation near the phase transition only. Halperin and Varma [4] investigated the possibility of a "frozen defect cell" near the phase transition. Such a "frozen defect" cannot adjust itself to the orderparameter. Orderparameter fluctuations are therefore enhanced in the vicinity of such a defect. The critical temperature Tc in crystals with "frozen defects" is therefore expected to decrease with defect concentration [4]. The increase of the central peak of the phonon spectra in SrTiO3 with increasing defect concentration has recently been observed by Hastings et al. [10] in neutron scattering experiments. A decrease of T~ with increasing defect concentration was observed as well. Only resonance methods, however, such as ESR [7], NQR [8, 9] and high resolution NMR [11, 12] are capable of distinguishing microscopic clusters of different order parameter. Blinc and Bjorkstam [8] were able to observe polar clusters in KDA far above the phase transition temperature. No concentration dependence was investigated and the "extrinsic" or "intrinsic" nature of the clusters could not be decided. An explanation allowing for both types of clusters was recently 749
given by Blinc [91. Mailer and Berlinger [7] using ESR were able to observe an "extrinsic" slowing down of order-parameter fluctuations near the defect. Using high resolution 13C NMR spectroscopy [ 13] we have recently given a preliminary account on the influence of defects on the cluster formation near the phase transition in squaric acid [11]. The goal of this publication is to present a more detailed experimental investigation of' the influence of chromium and deuterium doping on the phase transition in this substance. Squaric acid undergoes a structural phase transition at Tc = 373 K as was first observed by Semmingsen and Feder [14] and Samuelsen and Semmingsen [15]. Although a continuous transition to the disordered phase was suggested originally [14, 15], more recent investigations [ 11, 16, 17] have revealed a discontinuous jump of the order parameter at Tc, suggesting a "first order" transition. The "intrinsic" slowing down of order parameter fluctuations when approaching Tc from above has recently been demonstrated by ~3C NMR [19]. 2. EXPERIMENTAL Single crystals of squaric acid were grown by slowly cooling an aqueous solution. Deuterium doping - with different concentrations - was achieved by varying the D20/H20 ratio of the aqueous solution. A three step recristallization procedure was used. The deuterium concentration XD corresponds to the ratio of D/H in the solution. The chromium doping was achieved by dissolving CrCI3 in the aqueous solution. The chromium doping of the crystals manifests itself in a green coloring of the crystals, a shortening of the proton spin-lattice relaxation time and the appearance of ESR-spectra [20]. The chromium concentration Xcr was obtained in the following way:
750
INFLUENCE OF DEFECTS IN (H2C404)
(i) we have choosen six different samples with a nominal concentration Xcr = 0.01 tool. % in the solution and have assumed that their average concentration corresponds to 0.01 mol. %. (ii) We have measured for all our samples the proton spin-lattice relaxation rate which is proportional to the chromium concentration [ 12]. Although the absolute chromium concentration Xc, is not precisely known, the relative chromium concentration with respect to the set (i) could nevertheless be determined• We have applied a high-resolution 13C NMR technique [13] to obtain ~3C spectra of naturally abundant lSC (1.1%). The measurements were performed in two different superconducting magnets with 6,3 T and 4.2 T respectively• The corresponding resonance frequencies were: 13C (67.9 MHz), ~H (270 MHz) and LSc (45.3 MHz), tH (180 MHz) respectively. The temperature regulation was achieved within -+ 0.1 K using preheated nitrogen gas. Temperature inhomogeneity across the sample was less then + 0.1 K [22]• The single crystals of squaric acid were oriented in the magnetic field in such a way as to obtain a maximal line splitting of the four magnetically non-equivalent ~3C nuclei• The line separation has been shown to be temperature dependent and to vary proportional to the order parameter [23]•
I I ,(~-"
:: T: 3re Z ~
•o
.
•
._,_,,,_,,,./
T: 37J 6 K
..'.." : ,: ,-:
/~,.d./ox:.~
o
~-,..~.._j%j
~ "--....,.,..._~
..:"
2~
?.
:"
• •
..
.• .
.
.
. -
-
.
. .
~:.-
-
.
:
-,,:
:.
..
.'•
.•
T : 3 7 i 2K
•
2
.-
,.. -,.v,v,F"
-
,
.•
:"~
•
"...
..- . \
,
i;
41
v,~W'
T: 3tO 5 K
3. RESULTS Typical 13C NMR spectra near the phase transition temperature as shown in Fig. 1 were obtained in all samples. All four lines corresponding to the ordered phase are visible below the transition temperature (bottom), whereas only two lines are observed in the high temperature phase (top). In between a coexistence of both phases is observed and no continuous reduction of the linesplitting (order parameter) takes place• As pointed out before [ 11 ] there is a jump of the order parameter at Tc suggesting a discontinuous phase transition [16, 17]. The number of squaric acid molecules which belong to high- and low-temperature phase clusters in the sample near the phase transition is directly proportional to the integrated intensities Ih, IZ of the corresponding NMR lines• The "concentration" of these molecules can therefore be expressed as Ca = Ia/Io for the high-temperature phase clusters and Cz = IJIo for the low-temperature phase, with Io = Ih + II. Experimental spectra as shown in Fig. 1 were fitted by a computer analysis to a set of spectral lines belonging to either the low- or hightemperature phase respectively• Numerical integration of the fitted spectral set then led to In and It.
Vol. 38, No. 8
t
*
Fig. 1. ~3C spectra of a squaric acid sin~e crystal doped with 0.017 mol. % Cr 3÷ at different temperatures near the phase transition temperature To. Below the transition temperature four lines (bottom) are observed due to the four non-equivalent 13C nuclei in the unit cell. Above the transition two lines (top) reflect the higher symmetry, whereas the coexistence of both phases is visible in between. The variation of Ch and Cz with temperature could be described for all our samples in the whole temperature range where cluster spectra were observable by
Ch/Cl = exp (ICe)
(1)
with =
(T--
r~*)/r$
(2)
and K being a dimensionless constant. The critical temperature T* is defined here as the temperature where Ch = Cl = 1/2• The relative temperature interval + AT~T* in which both phases coexist is related to the constant K by
K - ' = AT~T*.
(3)
Vol. 38, No. 8
INFLUENCE ON DEFECTS IN (H2C404)
751 385,
Cr3"molGr cone I0
_(Q) 37~%
- / /
(b)
/ /
• 5 5 I0 "a
/ / C 17.10 [] 0 8
/
372
"a
38o }--
I 0 "a
/
,-.
,j
t"--
/
}"
I.-
37'0
Q/" e//
0
375~-OlI l
368
0 02
0 0.,.1.
X:,, 365
370
375 TIK
Fig. 2. Cluster concentration Cz of the low temperature phase versus temperature according to equations (4), (5) for different chromium concentrations.
O C-S
i 0
! 2
%
I a
Xo,
I 6 %
Fig. 3. Variation of the critical temperature Tc for (a) chromium (Xc~) as well as (b) deuterium (Xo) doping. Notice the different scale on the Xc~ and XD axis respectively. The dashed line corresponds to equation (11).
Table 1. Evaluation o f the parameter T* and K according to equations (4) and (5)for different samples o f squaric acid doped with chromium and deuterium respectively
is observed. Deuterium doping on the other hand increases the transition temperature. Again a linear dependence of T* on the defect concentration (XD: deuterium) which follows
Xcr (tool.%)
X o (mol.%)
T* (K)
10 3 K - l
T* = Tc(I + 0.295XD);
is obtained. Notice, however, that the temperature dependence on doping is by about a factor 100 larger in the case of chromium as for deuterium. A certain set of partially deuterated samples contained also a small amount of chromium in order to reduce the proton spin-lattice relaxation time. This was done for experimental convenience. Therefore the data of the kind shown in Fig. 3 had to be corrected for the chromium doping. The assumption that chromium and deuterium doping have an additive effect on T* and K-~ leads to the relations:
0.055 0.044 0.017 0.008 0.000
0 0 0 0 0
367.0 370.9 371.3 372.6 373.2
7.93 3.44 1.32 0.90 0.81
0.013 0.006 0.014 0.021 0.009
I 2 3 5 I0
373.7 375.6 375.6 375.8 384.9
1.30 2.22 1.40 3.51 1.32
T* = Tc(I + 0.295XD -- 24.6Xcr);
The temperature variation of Ch and Cl follows from equation (1) and the definition (2) as [11]
Ch = [exp (-- Ke) + I ] - t,
and (4)
and
Ct = [exp (Ke) + I I -~
(5)
Figure 2 shows this variation of cluster concentration with temperature for a typical set of data. Evaluation of the parameters K and T* for different samples is summarized in Table I. A dramatic decrease of the critical temperature T* is observed with increasing chromium concentration Xcr. This is evident from Fig. 3 where T* is plotted vs Xcr and XD respectively. A linear dependence of the form T* = Tc(1 -- 24.6Xcr);
Tc = 373.3 K
Tc = 373.5K
(6)
K -t = K o 1 + K c r I + K D l
(7)
Tc = 373.4 K (8) (9)
where a background value Ko- ~ has been introduced to account for the effects observed in the undoped crystals. As was pointed out before [ 11] even in completely undoped crystals cluster formation could be observed near the phase transition and follows equations (4) and (5). There may be an "intrinsic" cause for the cluster formation in terms of"heterophase fluctuations" [24]. However, also defects like dislocations and point defects can be responsible for the formation of clusters. The data analysis shows, that K~ ~ is temperature independent and small. An average over all samples leads to g D = 0.7 x 10 -3 which is much smaller than the background Ko--1. We have therefore plotted only
752
INFLUENCE OF DEFECTS IN (H2C404) i
(b)
7 I ! I I I
35'3
\ L \
it
\
I it
25O
I // .p
200
i
052
O'Oa %Cr
i50
3÷E
3~
3~3
TIK
Fig. 4. Left: The relative temperature interval of the coexistence region K -t vs. chromium concentration according to the data listed in the table. Right: 13C linewidth 6v of the central line, belonging to hightemperature phase clusters versus temperature (see text). K~ + KcrL vs chromium concentration Xcr in Fig. 4. A dramatic increase of the "smearing-out parameter" K - ~ with increasing chromium doping Xcr is observed. Another feature is evident from Fig. 1, namely the narrowing of the central line with increasing temperature. This feature is more clearly shown in the right part of Fig. 4 where the linewidth 6v of the central line is plotted versus temperature for a doped crystal with 0.017 mol.% chromium. Homogeneous broadening of the line, e.g. by transversal spin-lattice relaxation time 7"2 could be excluded by spin-echo experiments [22]. The observed broadening is therefore due to inhomogeneous local fields, probably caused by the r -3 dependence of the Cr 3+ local field at the nuclear site [25]. (i) Chromium defects: The concentration of chromium defects used in this investigation corresponds to an average distance of 8 0 - 1 5 0 A. Defect-defect interaction can therefore be excluded. The chromium replaces hydrogen bonds. Since chromium does not take part in the H-bond dynamics, it therefore interrupts the H-bond chains which transmit the order through the crystal. The chain interruption can be related to the bond percolation problem of the Ising model. There magnetic atoms are randomly replaced by nonmagnetic atoms with concentration X. Brooks-Harris [26] calculated the variation of the critical temperature Tc with concentration X in the percolation problem of the Ising model. A linear temperature variation of the type T* = T e ( l - - s X )
(10)
was obtained with s = 1.329 for the 2-dimensional quadratic Ising-model and s = 1.060 for the 3-dimensional simple cubic Ising model. Landau [27] performed a Monte Carlo calculation on the same model
Vol. 38, No. 8
and confirmed equation (10). For higher defect concentration near the percolation threshold a deviation from equation (10) is expected [28]. A smearing out of the phase transition with increasing detect concentration has been found in those theoretical calculations as well as in [26, 29, 30]. The theoretical reasoning leading to equation (10) is not directly transferable to the chromium doping of squaric acid, however. The chromium ion seems to "wipe out" several bonds, making the down-shift in the critical temperature much more effective than in the simple percolation problem. By comparing equations (6) and (i0) we find a "wipe out ratio'R = 24.6/1.329 = 18.5 for the two-dimensional lattice. In essence many H-bonds are effected in the vicinity of the chromium ion leading to a reduced "mean field" of H-bonds in the neighourhood of the chromium ion. In the classification of Halperin and Varma [4] the chromium defect in squaric acid corresponds to a "frozen defect cell". The conjecture of Halperin and Varma [4] that "frozen impurities will have a strong tendency to depress Tc below the value for the pure system" is clearly seen in our data. Recent ESR measurements by Stankowski et al. [20] on the chromium defect in squaric acid have shown, that the Cr 3÷ center occupies interstitial sites between the layers in the center of oxygen octahedra. Thus at least eight next neighbour hydrogen are "frozen", perhaps even more are affected. In contrast to ESR experiments [7, 12, 31], where the defect ion itself is investigated, high resolution 13C NMR allows to investigate the molecules in the neighbourhood as well as in the undistorted areas of the sample individually. (ii) Deuterium defects: Deuterium 2H replaces quite naturally the protons in the sample. Because of a drastic reduction of the tunnelling integral for deuterium in the hydrogen bond with respect to protons an increase of the phase transition temperature is expected [32]. The deuterium concentration XD in the crystal may not be, however, identical to the D20/H20 ratio of the crystal growing solution. In KH2POa (KDP), e.g. Loiacono et al. [33] found, that the deuterium concentration in the crystal XD was related to the deuterium concentration in the solution XDS by XD = O.0068XDs exp (0.00382XDs).
(11)
If we assume the same relation to hold in our case, the data shown in Fig. 3 referring to the deuterated samples have to be corrected according to equation (11). The corrected data are shown in Fig. 3(b). It is interesting to find the data for shifted transition temperature near the straight line, connecting simply the two transition temperatures TCH = 373.4 K (XD = O) and TCD = 516 K
Vol. 38, No. 8
INFLUENCE OF DEFECTS IN (H~_C~O~)
(XD = 1). The transition temperature Tc of partially deuterated squaric acid is therefore given by Yc = TCH(I --XD) + TcDXD
7. 8. (12)
with TCH = 373.4 K and TCD = 516K. Deuterium replacing hydrogens in the hydrogen bond creates a defect cell which hops back and forth between positions which break the symmetry in opposite senses. It therefore can be considered to be a "relaxing defect cell" in the sense of Halperin and Varma [4]. The change in temperature for dilute defect concentrations XD can be expressed as [4]
drc
rc = rcH + XD - -
dAD
(13)
(14)
-= TCHA dXD
where the parameter A is positive in the case of a relaxing defect cell and can be related to model parameters of the system. Our values obtained for the deuterium defect correspond to A = 0.38. It would be interesting to verify this relation for the full range of deuteration (0 ~< XD ~< 1). Acknowledgements - Discussions with Professors J. Petersson - on the nature of the phase transitions and R. Blinc - on the duster problem - are gratefully acknowledged. The Deutsche Forschungsgemeinschaft (DFG) has given a financial support. REFERENCES
3. 4. 5. 6.
12. 13. 14.
16.
dTc
2.
11.
15.
with
1.
9. I0.
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