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Ferrielectric phase transition in ammonium sulphate: A study of two-sublattice model by EPR of polar VO2+ probes
FERRIELECTRIC PHASE TRANSITION IN AMMONIUM SULPHATE: A STUDY OF TWO-SUBLATTICE MODEL BY EPR OF POLAR V02+ PROBES? M. FLIJIMOTO, T. J. Yu and K. FURUKAWA~ Department
of Physics, University
(Recessed 3 Map 1977; Abstract-EPR
of Guelph, Guelph, Ontario NIG 2W1, Canada
acceptedin r&cd form 10August 19773
spectra from VO” -doped (NH&SO,
crystals were studied in the range between -40 and - 150°C.
VO” impurities substituted fog NHa’ ions are accompanied with lattice defects, forming complexesoriented in crystals. Below the transition temperature ( -50°C) the spectra exhibited splittings into two branches with unequal intensities, as prevjously observed in ferroelectric TGS. In ferrielectric (NH~)~SO~, however, the temperature dependence of the spectra was markedly different from the TGS case. While the internal field determined from the dipolar energies of VO”-probes was consistent with anomalous polarization observed by Unruh(l], the displacement in VO”-directions showed a monotonic increase when the temperature was lowered. We consider that the local order in the molecular arrangement is reflected in the VO’+-displacement. The two-sublatticemodel for the ferrielectric (NH&SO4 is reviewed on the basis of the present EPR results, and the validity of the concept of sublattice polarization is discussed.
1. INTRODUCTION At
temperatures
below
- 50°C
to
ammonium
sulphate
spontaneous polarization with exhibit temperature dependence. On the basis of this finding Unruh [ I, 21 has suggested that the polar phase of (NH&SO4 is ~errie~ec~~jc.The presence of two crystallographically independent NH4’ ions in the unit cell[3] may tempt one to assume that these ions form two dipolar sublattices. However, the concept of sublattice polarization has not been established in the ferrielectric (NH&S04. Sawada et al.[4] studied the properties of mixed crystals [KX(NH4),-J2S04, and found that the spontaneous polarization appeared to become ~o~rnu~ as the potassium concentration approached to l/2. Unruh and Ayere[S] also reported similar results on Rb-substituted crystals. These authors considered that potassium and rubidium impurities occupy ammonium sites of one kind, resulting in normal polarization associated with unsubstituted NHI’ ions of the other type. Their results imply support of the two-sublattice idea; however, it is unknown how NR’ ions compose the elementary dipoles for the observed poiarization. Nevertheless the sublattice model explains anomalous properties of (NH&SO4 to a satisfactory extent as discussed recently by Dvorak and Ishibashi [6]. Apart from Unruh’s finding, Sawada et al. 171proposed a soft-mode theory of the phase transition in (NH&SOc suggesting that a collective librational motion of NR’ ions with the BI, symmetry softens at the center of the Brillouin zone as the transition point T, is approached from above. Although the relation to the sublattice model was not made clear in this theory, we felt it was essential crystals
anomalous
%upported by a grant from the National Research Council of Canada. *Permanent address: Department of Physics, Kagoshima University. Kagoshima, Japan. 345
investigate the proposed mode in detail. We therefore carried out an EPR experiment using paramagnetic NHX’ probes placed at ammonium sites[S]. The j4N hyperfine tensor in NH,* radicals was found to possess a well-defined unique axis, thereby representing the corresponding NH.++orientations. With this assumption we analyzed the NH3+ spectra, and the results indicate that the orientations of two dissimilar NHI’ ions become nearly parallel belay T,. This may be considered as evidence for an aniiparailel dipolar arrangement of these ions in the polar phase, although the EPR spectra by no means indicated such a polarity relation. Nevertheless the change in the NH>+ spectra at T, implies that softening occurs in the A, and B,, librational modes in the NH4+ sublattices. According to Sawada et al.[7], the latter mode should be accompanied with a translational displacement of the same irreducible representation, constituting the origin for thi spontaneous polarization. We thought that NHx+ radicals are polar: at least instantaneously, and their dipole moment would respond to the spontaneous electric field. However the failure to detect such an effect may be attributed to a relatively small magnitude of the hyperfine separation in the NHI’ spectra ( - 15OG). In our previous studies on vanadyl ions (VO”) in ferroelectric TGS (Triglycine Sulphate), in contrast, the effect was clearly resolved in their widely spread spectra f + lSO0G). Use of VO” probes, as applied to (NH&SO+ should be advantageous over NHX’ in order to study the nature of the spontaneous polarization. Pandey and Ven~tesw~u~~O] reported EPR spectra from VO*‘-doped (NH&S04 crystals at room temperature. We have extended their work to, a lower temperature region including the phase change at - 50°C. Below - 50°C we observed that ail the VO’+ spectra reported in Ref. [lo] exhibited extra structures, which were interpreted in the light of the principle of the polar probes proposed for TGS[9]. The results allowed us to define
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347
Ferrielectric phase transition in ammonium sulphate Table l. ‘IV hypefine interaction tensor values for (VO’+), complexes in paraelectric and ferrielectric ammonium sulphate crystals Princip&l Values
Temperature -45%
ISO.
iso.
200.0
(?0.904,
0.403,
0.141)
(?0.904,
0.403,
74.2
(70.322,
0.861, -0.395)
(70.322,
0.861,
0.395)
74.8
c50.2a1,
0.311,
0.908)
(70.281,
0.311,
-0.908)
210.0
(iO.900,
0.414,
0.133)
(t0.900,
0.414,
-0.133)
74.2
(70.375,
0.894, -0.245)
74.8
(10.281,
0.311,
(~0.375, (70.281,
0.894, 0.311,
-0.908)
l
unit:gauss, *
-0.141)
(ko.888,
(rO.360,
0.432, 0.867,
(?0.888, (T0.360,
0.432, 0.867,
-0.160) 0.345)
(To.287,
0.249,
(70.287,
0.249,
-0.925)
0.908)
0.245)
119.7 208.7 73.9
l
**
116.3
-105Oc
lS0.
Direction cosines
0.160) -0.345) 0.925)
A:; error * 0.5 gauss
Ref. axes [lOOI, [OlOl, [OOll
Table 2. “V hyperfine interaction tensor values for a member of (VO”)z spectra?in paraelectric and ferroelectric ammonium sulphate crystals Principfl values
Temperature -45Oc
iso.
Direction Cosines
**
73.4
(r0.971,
-0.229,
(tO.971,
-0.229,
-0.067)
199.5
(ko.198,
0.308)
(t0.198,
(70.133,
0.949)
(io.133,
0.931, -0.286,
-0.308)
73.6 115.5
0.931, -0.286,
0.067)
-0.949)
-105Oc
iso. t
* **
74.6
(r0.883,
-0.261,
0.391)
(i0.883,
-0.261,
-0.391)
200.1
(iO.171,
0.250)
(?0.171,
(50.438,
0.886)
(70.438,
0.953, -0.154,
-0.250)
71.6 115.4
0.953, -0.154,
-0.886)
The figures are only for a member spectrum with the largest intensity. unit:ln gauss; error f 0.5 gauss Ref. axes [lOOI, COlOl, COO11
which may however be inadequate when applied to the present case. As will be discussed later, the VO*’ displacement in (NH&SO4 was not directly proportional to the spontaneous polarization. Because of difficulties mentioned above, the values of hyperfine tensors were determined only for (VO*‘), and a member of (V02’)2 spectra with the largest intensity. Tables 1 and 2 show some results which may suffice to indicate basic features of these VO*‘-complexes. It is noticed from these figures that the molecular axis is evidently represented by the axis for the largest principal value. Nearly constant isotropic values imply that reorientation of virtually rigid VO*’ complexes occurs as the temperature is altered. On the basis of impurity-defect pair models, the impurity sites can be inferred from the VO”-axis directions as compared with the unit cell geometry. A slight distortion in the surroundings is inevitable so that an exact agreement is not expected in such a comparison. However, the method has been proven in many other examples in leading to an approximate geometry closely describing the impurity situation in solids. Figure 3 shows a sketch of the unit cell, in which the models proposed for (VO*‘), and (VO*‘)* complexes are
indicated. Considering a point charge lattice, the models shown here lie on the mirror plane ab, while the observed VO*‘-directions are symmetrically tilted about 8” out of the plane, resulting in four magnetically inequivalent directions for each type of VO*+ complexes. Although not evident from the spectra, the tilting mechanism could be attributed to bondings to the immediate neighborhood. In the above models, a (VO*‘), complex consists of a VO*+ ion and a defect both at (NH4+),, sites, while a (V02’), is related to (NH4+)1and (NH4’),,. (Two crystallographically different NH,’ ions are distinguished by suffices I and II as in Ref.[8].) In these models, the polarity should be reversible in order for the spectrum to split into two branches below 7’,. The reversed polarity must be thermally accomplished by exchanging the impurity and the defect, and the presence of such a mechanism has been assumed. The multiple structure of the (V02’)2 spectra may be ascribed to two dissimilar ammonium sites involved and/or a (NH4+),-(NH4+), pair with a similar orientation in ab-plane; however, no further hint of the complex structures was indicated in the spectra. In fact we weretiterested in finding probes to represent dipolar sublattices, so that their relation to
- 49.6
0
0 ---
-- (NH,&
(NH;),
1 Fig. 3. A sketch of unit ceils of (NH&SO, crystals. Crystallographically different NH,’ ions are distinguished with thin and thick circles. Also proposed models for (V02’), and (VO*‘), complexes are shown.
the macroscopic properties can be investigated. The (VO”), complex may be considered as a probe in the (NH.,‘),,-sublattice, if dictated only by the short-range interaction. The (VO*‘)z complex, on the other hand, is located between the two sublattices. It may therefore be difficult to find a clear-cut interpretation of the results. The phase transition as observed through the VO*’ spectra appeared to be discontinuous at T = T, as shown in Figs. 5-7. Figure 4 shows a closer look at the transition region of (VO”b spectra, in which there is a narrow, but clearly recognizable boundary phase with about 1.5-2°Cwidth. We thought that it might be due to a long thermal relaxation in this region, but this possibility has been ruled out after careful studies.
P
0 I
30G I
I
Fig. 4. A “V hype&e component line of (V02’),complexes with the highest resonance field observed at temperatures in the boundary phase. p and f are the lines in para- and ferrielectric phases. Notice that p :f composition varies with temperature.
where u and u0 are unit vectors along the VO*’ axis at temperatures T and T,, respectively (T < T, ). The tensor E. as in the theory of elastic continuum, may be regarded as representing the local strain at the impurity site. An internal stress and/or electric field associated with the structural change in host crystals must be responsible for such an internal strain. While the stress displaces VO*+-axes, the two-way shifts observed in the polar phase are doubtless due to dipole moments of VO*’ ions in the internal electric field. Unequal intensities of the branch spectra are therefore attributed to the energy difference AU between the dipoles with opposite polarities, p = t pu, i.e. AU =?p,E.
su= u -
uo =
EUU,
(1)
I
f
4.DISPLACEMENTSAND INTENSITIES OF V@+ SPECTRA
In ferrielectric (NH4)*S04, the VO’+ spectra split into two branches with unequal intensities as observed in ferroelectric TGS. The temperature dependence showed, however, a marked difference from the TGS case. We therefore focussed attention on the change in VO*+ directions with temperature to investigate if the sublattice model can be used to interpret the EPR results. The displacement vector 6u may be defined in terms of a strain tensor E in the equation
I
(2)
Assuming that the impurity dipole is reversible, the equilibrium intensity ratio of the spectra r = 1,/l* where 1, > 12, is given by r =
exp (AU//CT).
(3)
Hence the quantity T In r as plotted against T should provide information about AU and the internal field E. In ferroelectric TGS, AU and the corresponding displacement vector Su were found to possess a common temperature factor n, which is best described within
Ferrielectric phase transition in ammonium sulphate
349
experimental errors by Tj =
(T, -
7-y.
(4)
The spontaneous polarization P in TGS is well known as the order parameter for the phase transition showing a temperature dependence as in eqn (4). Therefore the VO’+ displacement 6u in TGS is believed to be a direct consequence of the internal linear field. In ferrielectric (NH&SO+ in contrast, the EPR spectra showed that AU and Su do not share a common temperature profile. Figures 5-7 show plots of T In r vs T determined for the three VO*+ complexes. The curves are not identical, but similar in that there is a maximum followed by a monotonic decline as the temperature is lowered. The polarization for undoped (NH&S04 observed by Unruh[l] resembles these curves, but different in detail. The internal field E, in fact, should depend on the probe site, although believed to be dictated by Lorentz’ field due to long-range electrostatic interactions. Therefore, it may not be surprising that different complexes indicate different T In r-curves. It is also worth mentioning at this point that the polarization
100
TInr
II
1
m
I
I
120
I
I
I
I
140
.
160
I
I
180
I
I
200
I
I-1 220%
Fig. 7. Plots of T In r and A vs temperature for (VOZ+)lcomplexes. Bt (30”,90”,60”).(Angles from a, b and c axes). Scales are arbitrary.
of VO*‘-doped crystals should depend on the impurity concentration and the distribution between the two dissimilar ammonium sites. Because of these unknown factors, we did not attempt to make any further justification of the observed curves. We have, however, every reason to believe that the observed T In r represents the local field E at the impurity sites in the polarized crystals. In Figs. 5-7 also shown are plots of the splitting A. which is defined in Ref. [9] as A= l-C,=-K;, where K, and KZ are the hyperfine splitting constants in the two component spectra for a fixed direction of the static magnetic field. It can be shown [ 111that A = A,& + hz(Su)*
Fig. 5. Plots of T In r and A vs temperature for (Vd’), complexes. Bt[lOO].Scales are arbitrary.
Fig. 6. Plots of T In r and A vs temperature for the strongest component of (Vd’), spectra. B,[OlO].Scales are arbitrary.
JPCS Vol. 39 No. 4-B
where 6u is the magnitude of the difference vector between the displacement vectors au, and 6~2 for these component spectra. The splitting A is therefore a convenient measure of the temperature dependence of the displacement vector if A, and AZ are temperature independent. Although not quite exact, plotting A is convenient for a quick look at the temperature profile of Su, when the temperature is not much different from T,.. In contrast to the curves of T In r, all the A-curves show a monotonic increase as the temperature is lowered. To determine the mode of the displacement more accurately than A-curves imply, we carried out determination of the 5’V hyperfine tensor at various temperatures. In spite of tedious data processing, the accuracy was only fair because of accumulated experimental errors after combining a large number of spectra. Figures 8 and 9 show plots of calculated polar and azimuthal angles, (r/2) - 0 and 4, of the directions of (VO’+), and the strongest component of (VO”), spectra against temperatures. Although the points are scattered, the curves representing them are indeed consistent with Acurves shown in Figs. 5-7 in that they all show monotonic change with temperature. This implies that the displacement Su is dictated by the short-range interaction surrounding impurities, and that the local order
350
M. FUJIMOTO et al. and
The sublattice polarizations may be defined in terms of n1 and ~2, respectively, since the long-range dipolar order will make a separation into two dipolar sublattices unrealistic. The spontaneous polarization in the ferrielectric polar phase may then be written as P=P,-Pz
where (9)
/
I
I
c
I
-150
I
I
I
I
11
I1
-100
and
-5O’C
Fig. 8. Calculated polar and azimuthal angles n/2 - 0 and $ from the (V02’), tensor directions as a function of temperature. Black and open circles are for the two branches of (VO”), spectra. (Instead of polar angles, angles from the mirror plane ab, 0 are plotted in the figure.)
I -150
I
I
I
I
I
I
I
I
-100
I
,114 -5O’C
Fig. 9. Calculated polar and azimuthal angles, 42 - 0 and 4 from the strongest component of (VO*+)s spectra. The other branch spectra were not always identifiable, and only the angles for the main branch spectra are calculated.
increases monotonically after a discontinuous change at T, when the temperature is decreased. In such a case, the local-order parameter ns may be written as 7)$= (T, - T)@r
(7)
where the exponent /% depends on the site. The value of p. was not determinable with accuracy, but is estimated as 0.4 + 0.1. Although in this experimental work not all the possible impurity sites were sampled, there must be two particular sites representing the two sublattices. The local-order parameters at such sites could be written as q1 =
(T< - T)@’
Pz = Aqz + Bz.
Here (A,, B,) and (AZ, &) are sets of constants characterizing two sublattices. These values together with the exponent PI and /3~ may be fit to the observed data empirically. B, and B2 are the sublattice polarizations appearing at the critical points. According to Dvorak and Ishibashi[6] it is not necessary to assume that both sublattices are critical, in order to obtain anomalous polarization as reported by Unruh(l]. Alternatively, if two sublattices are critical at a common transition point T, as above, /3, and @2must be unequal to obtain a ferrielectric state. In this work p, and p2 are still undetermined, and further investigations are needed to clarify the ambiguity. It should be noted that the sublattice polarizations PI and PI are defined in terms of local orders in the molecular arrangement rather than the long-range dipolar order as usually considered in ferroelectric cases. The local-order, when extended to the entire domain, could, in principle, be correlated to the long-range order in dip&r arrangement. The elementary dipole is, however, not usually an easily abstractable concept in the unit cell geometry. Furthermore the sublattice polarization is not a quantity directly measurable by polar probes, because of long-range electrostatic interaction between the sublattices. Hence, it seems logical to consider the shortrange orders as variables for the sublattices. PI nnd Pz defined in eqn (9) are therefore only theoretically conceivable when used to express ordering in dipolar sublattices. Another feature of the curves shown in Fig. 8 is that angular displacements in branch spectra of (VO”), complexes are clearly asymmetrical with respect to the direction at 7’,, whereas in TGS, in contrast, the two-way displacements were symmetrical and described by &I*
= +- EUO.
In ferrielectric (NH&SO+ the asymmetrical shifts can be decomposed into two; i.e. SUf = su, _tEUo
(10)
351
Ferrielectric phase transition in ammonium sulphate
are not. However, NH3+ probes are, perhaps, representing the lattice modes in question more faithfully than VO*+ probes.
where &I, = tcsu+ t su-). In eqn (IO) 6u, evidently represents a contribution independent
of
the
polarity
of
probes,
implying
the
of an internal stress below T,., while * EUOare doubtless due to the linear spontaneous field in the polar phase. .The ferrielectric (NH&SO4 crystals are known to have a twin structure[l2], and Su, may therefore be taken as evidence for spontaneous stress. Further studies are required to obtain the relation with the ferroelasticity of (NH&Sod. When the mode of (VO*‘), displacements at all the four inequivalent sites in the unit cell (Table I) is reviewed at various temperatures, it is immediately evident that the changes with temperature involve softening in lattice modes with A, and B,, symmetries, being consistent with those observed with NH3+ probes. The basic difference is, of course, that VO*‘-dipoles are responding to the spontaneous field, while NH,‘-dipoles presence
REFERENCES
I. Unruh H.-G, Solid St. Comm. ti, 1951(1970). 2. Unruh H.-G and Riidieer U.. J. Phvs. 33. C2-77 (1972). 3. Schlemper E. 0. and Hamilton W. C., J. Chem. &ys. ‘444498 (1966). 4. Sawada A., Ohya S., Ishibashi Y. and Takagi Y., .I. Phys. Sot. Japan 38. 1408 (1975).
5. Unruh H. -G and Ayere O., Ferroelecttics 12, 181(1976). 6. Dvorak V. and Ishibashi Y.. J. Phvs. Sot. Jaoan 41. 548 .
(1976). 7. Sawada A., TakagiY. and Ishibashi Y., 1. Phys. Sot. lapan 34, 748 (1973). 8. Fujimoto M., Dressel L. A. and Yu T-J., J. Phys. Chem. Solids 3B, 97 (1977). 9. Fujimoto M. and Dressel L. A.. Ferroelectrics 8. 61I (1974). 10. Pandey S. D. and Venkateswalu P., 1. Msg. Res. 17, 137
(1975). II. Fujimoto M. and Dressel L. A.. Ferroelectrics 9. 221 (1975). 12. Makita Y., Sawada A. and Takagi Y., J. Phys. Sot. Japan 41, 167(1976).
of Physics, University
(Recessed 3 Map 1977; Abstract-EPR
of Guelph, Guelph, Ontario NIG 2W1, Canada
acceptedin r&cd form 10August 19773
spectra from VO” -doped (NH&SO,
crystals were studied in the range between -40 and - 150°C.
VO” impurities substituted fog NHa’ ions are accompanied with lattice defects, forming complexesoriented in crystals. Below the transition temperature ( -50°C) the spectra exhibited splittings into two branches with unequal intensities, as prevjously observed in ferroelectric TGS. In ferrielectric (NH~)~SO~, however, the temperature dependence of the spectra was markedly different from the TGS case. While the internal field determined from the dipolar energies of VO”-probes was consistent with anomalous polarization observed by Unruh(l], the displacement in VO”-directions showed a monotonic increase when the temperature was lowered. We consider that the local order in the molecular arrangement is reflected in the VO’+-displacement. The two-sublatticemodel for the ferrielectric (NH&SO4 is reviewed on the basis of the present EPR results, and the validity of the concept of sublattice polarization is discussed.
1. INTRODUCTION At
temperatures
below
- 50°C
to
ammonium
sulphate
spontaneous polarization with exhibit temperature dependence. On the basis of this finding Unruh [ I, 21 has suggested that the polar phase of (NH&SO4 is ~errie~ec~~jc.The presence of two crystallographically independent NH4’ ions in the unit cell[3] may tempt one to assume that these ions form two dipolar sublattices. However, the concept of sublattice polarization has not been established in the ferrielectric (NH&S04. Sawada et al.[4] studied the properties of mixed crystals [KX(NH4),-J2S04, and found that the spontaneous polarization appeared to become ~o~rnu~ as the potassium concentration approached to l/2. Unruh and Ayere[S] also reported similar results on Rb-substituted crystals. These authors considered that potassium and rubidium impurities occupy ammonium sites of one kind, resulting in normal polarization associated with unsubstituted NHI’ ions of the other type. Their results imply support of the two-sublattice idea; however, it is unknown how NR’ ions compose the elementary dipoles for the observed poiarization. Nevertheless the sublattice model explains anomalous properties of (NH&SO4 to a satisfactory extent as discussed recently by Dvorak and Ishibashi [6]. Apart from Unruh’s finding, Sawada et al. 171proposed a soft-mode theory of the phase transition in (NH&SOc suggesting that a collective librational motion of NR’ ions with the BI, symmetry softens at the center of the Brillouin zone as the transition point T, is approached from above. Although the relation to the sublattice model was not made clear in this theory, we felt it was essential crystals
anomalous
%upported by a grant from the National Research Council of Canada. *Permanent address: Department of Physics, Kagoshima University. Kagoshima, Japan. 345
investigate the proposed mode in detail. We therefore carried out an EPR experiment using paramagnetic NHX’ probes placed at ammonium sites[S]. The j4N hyperfine tensor in NH,* radicals was found to possess a well-defined unique axis, thereby representing the corresponding NH.++orientations. With this assumption we analyzed the NH3+ spectra, and the results indicate that the orientations of two dissimilar NHI’ ions become nearly parallel belay T,. This may be considered as evidence for an aniiparailel dipolar arrangement of these ions in the polar phase, although the EPR spectra by no means indicated such a polarity relation. Nevertheless the change in the NH>+ spectra at T, implies that softening occurs in the A, and B,, librational modes in the NH4+ sublattices. According to Sawada et al.[7], the latter mode should be accompanied with a translational displacement of the same irreducible representation, constituting the origin for thi spontaneous polarization. We thought that NHx+ radicals are polar: at least instantaneously, and their dipole moment would respond to the spontaneous electric field. However the failure to detect such an effect may be attributed to a relatively small magnitude of the hyperfine separation in the NHI’ spectra ( - 15OG). In our previous studies on vanadyl ions (VO”) in ferroelectric TGS (Triglycine Sulphate), in contrast, the effect was clearly resolved in their widely spread spectra f + lSO0G). Use of VO” probes, as applied to (NH&SO+ should be advantageous over NHX’ in order to study the nature of the spontaneous polarization. Pandey and Ven~tesw~u~~O] reported EPR spectra from VO*‘-doped (NH&S04 crystals at room temperature. We have extended their work to, a lower temperature region including the phase change at - 50°C. Below - 50°C we observed that ail the VO’+ spectra reported in Ref. [lo] exhibited extra structures, which were interpreted in the light of the principle of the polar probes proposed for TGS[9]. The results allowed us to define
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347
Ferrielectric phase transition in ammonium sulphate Table l. ‘IV hypefine interaction tensor values for (VO’+), complexes in paraelectric and ferrielectric ammonium sulphate crystals Princip&l Values
Temperature -45%
ISO.
iso.
200.0
(?0.904,
0.403,
0.141)
(?0.904,
0.403,
74.2
(70.322,
0.861, -0.395)
(70.322,
0.861,
0.395)
74.8
c50.2a1,
0.311,
0.908)
(70.281,
0.311,
-0.908)
210.0
(iO.900,
0.414,
0.133)
(t0.900,
0.414,
-0.133)
74.2
(70.375,
0.894, -0.245)
74.8
(10.281,
0.311,
(~0.375, (70.281,
0.894, 0.311,
-0.908)
l
unit:gauss, *
-0.141)
(ko.888,
(rO.360,
0.432, 0.867,
(?0.888, (T0.360,
0.432, 0.867,
-0.160) 0.345)
(To.287,
0.249,
(70.287,
0.249,
-0.925)
0.908)
0.245)
119.7 208.7 73.9
l
**
116.3
-105Oc
lS0.
Direction cosines
0.160) -0.345) 0.925)
A:; error * 0.5 gauss
Ref. axes [lOOI, [OlOl, [OOll
Table 2. “V hyperfine interaction tensor values for a member of (VO”)z spectra?in paraelectric and ferroelectric ammonium sulphate crystals Principfl values
Temperature -45Oc
iso.
Direction Cosines
**
73.4
(r0.971,
-0.229,
(tO.971,
-0.229,
-0.067)
199.5
(ko.198,
0.308)
(t0.198,
(70.133,
0.949)
(io.133,
0.931, -0.286,
-0.308)
73.6 115.5
0.931, -0.286,
0.067)
-0.949)
-105Oc
iso. t
* **
74.6
(r0.883,
-0.261,
0.391)
(i0.883,
-0.261,
-0.391)
200.1
(iO.171,
0.250)
(?0.171,
(50.438,
0.886)
(70.438,
0.953, -0.154,
-0.250)
71.6 115.4
0.953, -0.154,
-0.886)
The figures are only for a member spectrum with the largest intensity. unit:ln gauss; error f 0.5 gauss Ref. axes [lOOI, COlOl, COO11
which may however be inadequate when applied to the present case. As will be discussed later, the VO*’ displacement in (NH&SO4 was not directly proportional to the spontaneous polarization. Because of difficulties mentioned above, the values of hyperfine tensors were determined only for (VO*‘), and a member of (V02’)2 spectra with the largest intensity. Tables 1 and 2 show some results which may suffice to indicate basic features of these VO*‘-complexes. It is noticed from these figures that the molecular axis is evidently represented by the axis for the largest principal value. Nearly constant isotropic values imply that reorientation of virtually rigid VO*’ complexes occurs as the temperature is altered. On the basis of impurity-defect pair models, the impurity sites can be inferred from the VO”-axis directions as compared with the unit cell geometry. A slight distortion in the surroundings is inevitable so that an exact agreement is not expected in such a comparison. However, the method has been proven in many other examples in leading to an approximate geometry closely describing the impurity situation in solids. Figure 3 shows a sketch of the unit cell, in which the models proposed for (VO*‘), and (VO*‘)* complexes are
indicated. Considering a point charge lattice, the models shown here lie on the mirror plane ab, while the observed VO*‘-directions are symmetrically tilted about 8” out of the plane, resulting in four magnetically inequivalent directions for each type of VO*+ complexes. Although not evident from the spectra, the tilting mechanism could be attributed to bondings to the immediate neighborhood. In the above models, a (VO*‘), complex consists of a VO*+ ion and a defect both at (NH4+),, sites, while a (V02’), is related to (NH4+)1and (NH4’),,. (Two crystallographically different NH,’ ions are distinguished by suffices I and II as in Ref.[8].) In these models, the polarity should be reversible in order for the spectrum to split into two branches below 7’,. The reversed polarity must be thermally accomplished by exchanging the impurity and the defect, and the presence of such a mechanism has been assumed. The multiple structure of the (V02’)2 spectra may be ascribed to two dissimilar ammonium sites involved and/or a (NH4+),-(NH4+), pair with a similar orientation in ab-plane; however, no further hint of the complex structures was indicated in the spectra. In fact we weretiterested in finding probes to represent dipolar sublattices, so that their relation to
- 49.6
0
0 ---
-- (NH,&
(NH;),
1 Fig. 3. A sketch of unit ceils of (NH&SO, crystals. Crystallographically different NH,’ ions are distinguished with thin and thick circles. Also proposed models for (V02’), and (VO*‘), complexes are shown.
the macroscopic properties can be investigated. The (VO”), complex may be considered as a probe in the (NH.,‘),,-sublattice, if dictated only by the short-range interaction. The (VO*‘)z complex, on the other hand, is located between the two sublattices. It may therefore be difficult to find a clear-cut interpretation of the results. The phase transition as observed through the VO*’ spectra appeared to be discontinuous at T = T, as shown in Figs. 5-7. Figure 4 shows a closer look at the transition region of (VO”b spectra, in which there is a narrow, but clearly recognizable boundary phase with about 1.5-2°Cwidth. We thought that it might be due to a long thermal relaxation in this region, but this possibility has been ruled out after careful studies.
P
0 I
30G I
I
Fig. 4. A “V hype&e component line of (V02’),complexes with the highest resonance field observed at temperatures in the boundary phase. p and f are the lines in para- and ferrielectric phases. Notice that p :f composition varies with temperature.
where u and u0 are unit vectors along the VO*’ axis at temperatures T and T,, respectively (T < T, ). The tensor E. as in the theory of elastic continuum, may be regarded as representing the local strain at the impurity site. An internal stress and/or electric field associated with the structural change in host crystals must be responsible for such an internal strain. While the stress displaces VO*+-axes, the two-way shifts observed in the polar phase are doubtless due to dipole moments of VO*’ ions in the internal electric field. Unequal intensities of the branch spectra are therefore attributed to the energy difference AU between the dipoles with opposite polarities, p = t pu, i.e. AU =?p,E.
su= u -
uo =
EUU,
(1)
I
f
4.DISPLACEMENTSAND INTENSITIES OF V@+ SPECTRA
In ferrielectric (NH4)*S04, the VO’+ spectra split into two branches with unequal intensities as observed in ferroelectric TGS. The temperature dependence showed, however, a marked difference from the TGS case. We therefore focussed attention on the change in VO*+ directions with temperature to investigate if the sublattice model can be used to interpret the EPR results. The displacement vector 6u may be defined in terms of a strain tensor E in the equation
I
(2)
Assuming that the impurity dipole is reversible, the equilibrium intensity ratio of the spectra r = 1,/l* where 1, > 12, is given by r =
exp (AU//CT).
(3)
Hence the quantity T In r as plotted against T should provide information about AU and the internal field E. In ferroelectric TGS, AU and the corresponding displacement vector Su were found to possess a common temperature factor n, which is best described within
Ferrielectric phase transition in ammonium sulphate
349
experimental errors by Tj =
(T, -
7-y.
(4)
The spontaneous polarization P in TGS is well known as the order parameter for the phase transition showing a temperature dependence as in eqn (4). Therefore the VO’+ displacement 6u in TGS is believed to be a direct consequence of the internal linear field. In ferrielectric (NH&SO+ in contrast, the EPR spectra showed that AU and Su do not share a common temperature profile. Figures 5-7 show plots of T In r vs T determined for the three VO*+ complexes. The curves are not identical, but similar in that there is a maximum followed by a monotonic decline as the temperature is lowered. The polarization for undoped (NH&S04 observed by Unruh[l] resembles these curves, but different in detail. The internal field E, in fact, should depend on the probe site, although believed to be dictated by Lorentz’ field due to long-range electrostatic interactions. Therefore, it may not be surprising that different complexes indicate different T In r-curves. It is also worth mentioning at this point that the polarization
100
TInr
II
1
m
I
I
120
I
I
I
I
140
.
160
I
I
180
I
I
200
I
I-1 220%
Fig. 7. Plots of T In r and A vs temperature for (VOZ+)lcomplexes. Bt (30”,90”,60”).(Angles from a, b and c axes). Scales are arbitrary.
of VO*‘-doped crystals should depend on the impurity concentration and the distribution between the two dissimilar ammonium sites. Because of these unknown factors, we did not attempt to make any further justification of the observed curves. We have, however, every reason to believe that the observed T In r represents the local field E at the impurity sites in the polarized crystals. In Figs. 5-7 also shown are plots of the splitting A. which is defined in Ref. [9] as A= l-C,=-K;, where K, and KZ are the hyperfine splitting constants in the two component spectra for a fixed direction of the static magnetic field. It can be shown [ 111that A = A,& + hz(Su)*
Fig. 5. Plots of T In r and A vs temperature for (Vd’), complexes. Bt[lOO].Scales are arbitrary.
Fig. 6. Plots of T In r and A vs temperature for the strongest component of (Vd’), spectra. B,[OlO].Scales are arbitrary.
JPCS Vol. 39 No. 4-B
where 6u is the magnitude of the difference vector between the displacement vectors au, and 6~2 for these component spectra. The splitting A is therefore a convenient measure of the temperature dependence of the displacement vector if A, and AZ are temperature independent. Although not quite exact, plotting A is convenient for a quick look at the temperature profile of Su, when the temperature is not much different from T,.. In contrast to the curves of T In r, all the A-curves show a monotonic increase as the temperature is lowered. To determine the mode of the displacement more accurately than A-curves imply, we carried out determination of the 5’V hyperfine tensor at various temperatures. In spite of tedious data processing, the accuracy was only fair because of accumulated experimental errors after combining a large number of spectra. Figures 8 and 9 show plots of calculated polar and azimuthal angles, (r/2) - 0 and 4, of the directions of (VO’+), and the strongest component of (VO”), spectra against temperatures. Although the points are scattered, the curves representing them are indeed consistent with Acurves shown in Figs. 5-7 in that they all show monotonic change with temperature. This implies that the displacement Su is dictated by the short-range interaction surrounding impurities, and that the local order
350
M. FUJIMOTO et al. and
The sublattice polarizations may be defined in terms of n1 and ~2, respectively, since the long-range dipolar order will make a separation into two dipolar sublattices unrealistic. The spontaneous polarization in the ferrielectric polar phase may then be written as P=P,-Pz
where (9)
/
I
I
c
I
-150
I
I
I
I
11
I1
-100
and
-5O’C
Fig. 8. Calculated polar and azimuthal angles n/2 - 0 and $ from the (V02’), tensor directions as a function of temperature. Black and open circles are for the two branches of (VO”), spectra. (Instead of polar angles, angles from the mirror plane ab, 0 are plotted in the figure.)
I -150
I
I
I
I
I
I
I
I
-100
I
,114 -5O’C
Fig. 9. Calculated polar and azimuthal angles, 42 - 0 and 4 from the strongest component of (VO*+)s spectra. The other branch spectra were not always identifiable, and only the angles for the main branch spectra are calculated.
increases monotonically after a discontinuous change at T, when the temperature is decreased. In such a case, the local-order parameter ns may be written as 7)$= (T, - T)@r
(7)
where the exponent /% depends on the site. The value of p. was not determinable with accuracy, but is estimated as 0.4 + 0.1. Although in this experimental work not all the possible impurity sites were sampled, there must be two particular sites representing the two sublattices. The local-order parameters at such sites could be written as q1 =
(T< - T)@’
Pz = Aqz + Bz.
Here (A,, B,) and (AZ, &) are sets of constants characterizing two sublattices. These values together with the exponent PI and /3~ may be fit to the observed data empirically. B, and B2 are the sublattice polarizations appearing at the critical points. According to Dvorak and Ishibashi[6] it is not necessary to assume that both sublattices are critical, in order to obtain anomalous polarization as reported by Unruh(l]. Alternatively, if two sublattices are critical at a common transition point T, as above, /3, and @2must be unequal to obtain a ferrielectric state. In this work p, and p2 are still undetermined, and further investigations are needed to clarify the ambiguity. It should be noted that the sublattice polarizations PI and PI are defined in terms of local orders in the molecular arrangement rather than the long-range dipolar order as usually considered in ferroelectric cases. The local-order, when extended to the entire domain, could, in principle, be correlated to the long-range order in dip&r arrangement. The elementary dipole is, however, not usually an easily abstractable concept in the unit cell geometry. Furthermore the sublattice polarization is not a quantity directly measurable by polar probes, because of long-range electrostatic interaction between the sublattices. Hence, it seems logical to consider the shortrange orders as variables for the sublattices. PI nnd Pz defined in eqn (9) are therefore only theoretically conceivable when used to express ordering in dipolar sublattices. Another feature of the curves shown in Fig. 8 is that angular displacements in branch spectra of (VO”), complexes are clearly asymmetrical with respect to the direction at 7’,, whereas in TGS, in contrast, the two-way displacements were symmetrical and described by &I*
= +- EUO.
In ferrielectric (NH&SO+ the asymmetrical shifts can be decomposed into two; i.e. SUf = su, _tEUo
(10)
351
Ferrielectric phase transition in ammonium sulphate
are not. However, NH3+ probes are, perhaps, representing the lattice modes in question more faithfully than VO*+ probes.
where &I, = tcsu+ t su-). In eqn (IO) 6u, evidently represents a contribution independent
of
the
polarity
of
probes,
implying
the
of an internal stress below T,., while * EUOare doubtless due to the linear spontaneous field in the polar phase. .The ferrielectric (NH&SO4 crystals are known to have a twin structure[l2], and Su, may therefore be taken as evidence for spontaneous stress. Further studies are required to obtain the relation with the ferroelasticity of (NH&Sod. When the mode of (VO*‘), displacements at all the four inequivalent sites in the unit cell (Table I) is reviewed at various temperatures, it is immediately evident that the changes with temperature involve softening in lattice modes with A, and B,, symmetries, being consistent with those observed with NH3+ probes. The basic difference is, of course, that VO*‘-dipoles are responding to the spontaneous field, while NH,‘-dipoles presence
REFERENCES
I. Unruh H.-G, Solid St. Comm. ti, 1951(1970). 2. Unruh H.-G and Riidieer U.. J. Phvs. 33. C2-77 (1972). 3. Schlemper E. 0. and Hamilton W. C., J. Chem. &ys. ‘444498 (1966). 4. Sawada A., Ohya S., Ishibashi Y. and Takagi Y., .I. Phys. Sot. Japan 38. 1408 (1975).
5. Unruh H. -G and Ayere O., Ferroelecttics 12, 181(1976). 6. Dvorak V. and Ishibashi Y.. J. Phvs. Sot. Jaoan 41. 548 .
(1976). 7. Sawada A., TakagiY. and Ishibashi Y., 1. Phys. Sot. lapan 34, 748 (1973). 8. Fujimoto M., Dressel L. A. and Yu T-J., J. Phys. Chem. Solids 3B, 97 (1977). 9. Fujimoto M. and Dressel L. A.. Ferroelectrics 8. 61I (1974). 10. Pandey S. D. and Venkateswalu P., 1. Msg. Res. 17, 137
(1975). II. Fujimoto M. and Dressel L. A.. Ferroelectrics 9. 221 (1975). 12. Makita Y., Sawada A. and Takagi Y., J. Phys. Sot. Japan 41, 167(1976).