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Evidence of a spiral structure of the ferroelectric domain walls in NaNO2 from the quadrupole broadening of the NMR Satellites of 23Na
JOURNAL OF MAGNETIC
RESONANCE
93,36 l-368 ( 199 1)
Evidence of a Spiral Structure of the Ferroelectric Domain Walls in NaN02 from the Quadrupole Broadening of the NMR Satellites of 23Na D.G.HUGHESANDLAKSHMANPANDEY* Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2JI
Received November 6, 1990 An explanation is proposed for the asymmetry of the inhomogeneous quadrupole broadening of the NMR satellites of “Na in single crystals of the ferroelectric material NaNO,, reported earlier (J. Mu@. Reson. 75,272, 1987). It is found that the quadrupole broadening is caused almost entirely by a spread of about 0.1 o in the orientation of the local principal axis of the efg tensor. The distribution is very asymmetric when projected onto the ab plane, but much more symmetric when projected onto the bc plane. This points to a spiraling of the electric polarization in the vicinity of the domain walls in the material. If this is regarded as a precursor of the behavior of the pohuization within the domain walls, the domain wall structure in NaNOr must be similar to the spiral structure of Bloch walls in magnetism. 0 1991Academic press. IX.
It has been known since the work of Betsuyaku (I) that the satellite resonancesof the quadrupole-split NMR spectrum of 23Na in single crystals of the ferroelectric material sodium nitrite, NaN02, exhibit substantial inhomogeneous quadrupole broadening (2). A recent study in our laboratory (3) has shown that this inhomogeneous broadening is highly asymmetric at some orientations of the crystal relative to the external magnetic field. Moreover, the observed asymmetry is inconsistent with the point group symmetry of the 23Na sites, determined using X rays (4). The magnitude and asymmetry of the inhomogeneous broadening were found to be quite similar in two crystals of NaN02, one cleaved and the other sawed,that were obtained from different sources.This led us to suggest(3) that the magnitude and asymmetry of the broadening may be intrinsic properties of the material as normally grown and handled. However, we were unable to explain the observedbehavior at the time, and no explanation seemsto have appearedsince in the literature. Our purpose in this paper is to propose an explanation for these old data, on the basis of the ferroelectric domain structure of the material. THEORETICAL
CONSIDERATIONS
The 23Na nuclei are situated at identical sites in the orthorhombic unit cell of the C,,, (Zm 2m) ferroelectric phase of NaN02 (4)) shown in Fig. 1. The x, y, and z axes * Present address: Department of Postgraduate Studies and Research in Physics, Rani Durgavati Vishwavidyalaya (formerly University of Jabalpur), Jabalpur 482001, India. 361
0022-2364191 $3.00 Copyright Q 1991 by Academic Press,Inc. All rights of reproduction in any form reserved.
362
HUGHES AND PANDEY
l
-Na
0-O
O-N
C
Y FIG. 1. Orthorhombic unit call of NaN02 and the coordinate system used to specify the direction of the external magnetic field.
of the principal coordinate system of the efg tensor of the nuclei (5) lie along the c, a, and b axes, respectively, of the unit cell, as shown in the figure. The frequencies, v,, of the $ ++ f and - i++ - $ satellites, respectively, of a system of I = 3 nuclei such as 23Na, located at identical sites in a single crystal, are given in first order by (3) v+ = v. T (e2qQ/h)(y/4(y1)(3
cos20 - 1 + n sin26 cos 24).
[II
Here, v. is the center line frequency, e2qQ/h is the quadrupole coupling constant (qcc), n is the asymmetry parameter, and y is the magnetogyric ratio, Also, 19and 4 are the polar and azimuthal angles of the external magnetic field measured relative to the principal coordinate system of the efg tensor, as shown in Fig, 1. Inhomogeneous quadrupole broadening is caused by crystal imperfections which give rise to local deviations in 6, #J,or e2qQ/h, from the average values characteristic of the crystal as a whole. Since n is approximately 0.1 for 23Na in NaN02 (5, 6)) we ignore the quadrupole broadening due to variations in 4. If the broadenings associated with variations in B and e2qQ/h are assumed to be independent of each other, it can be shown by differentiating Eq. [l] that the second moment of each satellite, measured in units of (magnetic field)2 relative to its “center of gravity,” about which the first moment is zero, is given by M2 = MZd(fl, 4) + ( 3n/2r)2(
602)(e2qQ/h)2sin22t9
+ (r/2r)*(&(e2qQ/h)‘)(3
cos26 - 1 + n sin28 cos 24)2.
[2]
Here, M2, is the dipolar second moment of each satellite ( 1 ), ( M2) II2 is the rms deviation of L?from the mean, and (6(e2qQ/h)2)‘/2 is the rms deviation of the qcc from the mean. For 8 values well away from zero and ?r radians, ( M2) ‘I* is essentially a measure of the spread in the orientations of the local principal axis projected onto the plane which contains the magnetic field and the principal axis. Therefore (M2) ‘1’ is expected to be a function of 4. On the other hand, (6(e2qQ/h)2)“2
DOMAIN
363
WALLS IN SODIUM NITRITE
should be a unique property of the particular crystal being studied, provided the broadenings associatedwith variations in 0 and e2qQ/ h are independent of each other. In an earlier paper ( 7), we proposed that the third moment of a resonanceplotted as a function of frequency be defined as M3
=
J
(w
-
[31
~0)‘fiwid~/Sl(o)du.
Using this definition, the third moment of the satellites is given by M3 = M3d-t(0, 4,) AZ[(3a/2y)3(683)(e2qQ/h)3sin328 cos*O - 1 + rj sin*8 cos 24)3]. [4] - (~/2r)3(s(e2qQlh)3)(3 Here, the t signs again refer to the ? t-) 4 and - f * - ! satellites, respectively, M3d+(~,~)isthedipolarthirdmoment(7),while(6~3)1~3and(6(e2qQ/h)3)‘~3are the root mean cubed deviations of 0 and the qcc from the mean. RESULTS AND DISCUSSION
Analysis of the asymmetry and linewidth of a satellite. Figure 2a shows the highfield ( -i c* - f ) satellite (3) of the 23Naquadrupole-split spectrum of a single crystal
a
MAGN. FLD. INCR. + I
I
I
I
0
2
4
6
I
I
I
I
,
8 10 12 14 16
1
I
I
I
0
2
4
6
(G) I
I
4
1
I
8 10 12 14 16
(G)
MAGN. FLD. INCR. I
I
I
I
0
2
4
6
I
I
I
I
4
8 10 12 14 16 (G)
I
I
1
I
0
2
4
6
I
I
1
I
,
8 10 12 14 16 W
FIG. 2. Absorption derivative lineshapes of the high-field ( - f ++ - i ) satellite of “Na in a single crystal of NaNO* at the orientations (a) @= 45”, 4 = 90” and 19= 135’, I#I= 90’ and (b) 6 = 45”, #J= 4.5” and l9 = 1350, f$ = 4.5”.
364
HUGHES AND PANDEY
of NaNOz (previously identified as Crystal B and described elsewhere (6))) recorded at the crystal orientations 0 = 45”, 4 = 90” and 8 = 135”, 0 = 90” at room temperature. Figure 2b shows the same satellite recorded at 6 = 45”, 4 = 4.5” and 0 = 135”, 4 = 4.5”. (It was originally believed that $Jwas zero for these resonances, but it was subsequently found that that was not quite the case.) The resonances were recorded as the absorption derivative, great care being taken to minimize the contribution of the dispersion mode, as described earlier (3). The second and third moments of each resonance were measured relative to the center of gravity. Since the “tails” of the resonances make a disproportionate contribution to these moments, they were approximately by the smooth curves shown in the figure. To reduce the random error and to minimize any effects of saturation, the resonances were also recorded with the magnetic field swept in the opposite direction. The mean values of the second and third moments, recorded with the two sweep directions, are listed in Table 1, together with values of the dipolar second and third moments, MZd and M3+, calculated using Eqs. [8] and [9] ofRef. (3). It can be seen from Table 1 that the measured second moments are an order of magnitude larger than the dipolar values, showing that the lineshapes are dominated by inhomogeneous quadrupole broadening. Moreover, the dipolar contribution to the third moment is negligible, so that the asymmetry of the resonances can be entirely attributed to the inhomogeneous quadrupole broadening. According to Eq. [ 21, the second moment of a satellite should be the same at the orientations 8,+ and (?r - 0)) 4. It can be seen from Table 1 that the second moments are indeed equal at the orientations 0 = 45”, $J = 90” and 0 = 135‘, 4 = 90”, as are those at tl = 45”, 4 = 4.5” and 6 = 135”, 4 = 4.5”. By taking e’qQfh, 7, and (6(e2qQ/h)2)1’2 to be 1096 kHz, 0.109 (5, 6), and 2.26 kHz (3), respectively, and substituting the experimental values of the second moment at the orientations 0 = 45”, 4=90”and1?= 135”,~=90”inEq.[2],itisfoundthat(682)1~2is(0.106-t0.004)“. This is a measure of the width of the distribution of the orientations of the local principal axis when projected onto the ab or 4 = 90” plane. Similarly, by substituting the experimental values of the second moment at the orientations 8 = 45”, @J= 4.5” and 0 = 135”, 9 = 4.5”, it is found that (66 2) ‘I2 = (0.131 f 0.002)“. We see that TABLE 1 Experimental Values of the Second and Third Moments of the High-Field Satellite of “Na in a Single Crystal of NaNO2 8 (degrees)
(deices)
45 135 45 135
90 90 4.5 4.5
M2
(G2)
2.2 2.0 3.2 3.3
+- 0.1 + 0.1 + 0.1 * 0.1
Mz.,
G2)
0.203 0.203 0.410 0.410
M3
(G’)
1.6 +- 0.2 -1.6 + 0.2 -0.5 + 0.1 -0.1 2 0.1
M3+
2.8 x 2.8 X 1.0 x 1.0 x
(@I 1O-4 1O-4 10-3 10-3
Note. Also shown are theoretical values of the dipolar second and third moments, M2d and I%&-, of the satellite; 8 and # are the polar and azimuthal angles of the external magnetic field, measured relative to the crystal coordinate system.
DOMAIN
WALLS IN SODIUM NITRITE
365
the width of the distribution of the orientations of the local principal axis when projected onto the C$= 4.5” plane, which is essentially the bc plane, is somewhat larger than the width projected onto the ab plane. The third moment does not display the same symmetry as the second moment, since the second term on the right-hand side of Eq. [4] is of opposite sign at the orientations 8, C#J and (?r - S), 4, whereasthe third term is the sameat both orientations. Since the experimental values are equal in magnitude but opposite in sign at the orientations 0 = 45”, 4 = 90” and 0 = 135”, 6 = 90”, it follows that (fi(e2qQ/h)3) is negligible, as found earlier by measuring the 6 dependence of the third moment when 0 = 90” ( 3). By substituting the experimental third-moment values in Eq. [ 41, 3 ‘I3 is found to be (0.092 + 0.003) ‘. The fact that (68 “) ‘I3 is almost as large as $~*j1,2 reflects the considerable asymmetry of the distribution of the orientations of the local principal axis when projected onto the ab plane. It can be seen from the lineshapesin Fig. 2b and the third-moment values in Table 1 that the satellite is significantly more asymmetric at the orientation 0 = 45”, C#I = 4.5“ than at 6 = 135”, Cp= 4.5”. This seemsto imply an asymmetric distribution of qcc values. By substituting in Eq. [ 41, it is found that (6( e2qQ/h)3) ‘I3 is (-5.3 -t 0.5) kHz and (60 3) ‘I3 = (0 .048 +- 0.005) o in a plane inclined at 4.5” to the bc plane. The former result is inconsistent with the lineshapesshown in Fig. 2a, although that could be explained as a failure to detect a small asymmetry in the presence of a larger one. On the other hand, such a value for (A( e*qQ/ h)3) 1’3implies a contribution of roughly 1 G3 to the third moment of the satellite at orientations where 8 = 90”. There was no evidence of such a contribution in the measurementsdescribed earlier (3)) and we have no explanation for the difference between the asymmetry at ~9= 45’) I#J= 4.5” and that at 19= 135”, 4 = 4.5”. Nevertheless, the fact that (c%?~)‘/~is so much smaller than (60 * ) ‘I* when C$= 4.5” shows that the distribution of the orientations of the local principal axis is much more symmetric when projected onto the bc plane than when projected onto the ab plane. The quadrupole interactions experiencedby nuclei which are situated close to strong defects may be so large that they do not contribute to the observed satellite resonances (2). It is therefore not immediately obvious that the resonancesshown in Figs. 2a and 2b are obtained from the same group of 23Nanuclei, becausethey were obtained using very different values of 4. To check this, the absorption derivative resonancesin Fig. 2 were integrated and the areas under the absorption curves determined. The values are listed in Table 2, together with the peak-to-peak modulation amplitudes used to record the resonances. The area under an absorption curve is proportional to the modulation amplitude used to record the absorption derivative and the number of nuclei that contribute to the resonance (provided other effects such as saturation are unchanged). The mean area under the absorption divided by the peak-to-peak modulation amplitude is 774 arbitrary units/G for the resonances corresponding to 4 = 90” and 778 arbitrary units/G for C$= 4.5”. This strongly suggeststhat the same nuclei contribute to the resonancesshown in Fig. 2a and Fig. 2b, so that our calculated valuesof(68*)‘/*and(66’ 3) Ii3 , for projections onto the ab and bc planes,are properties o:Fthe same group of nuclei. Interpretation of the asymmetry and linewidth in terms of the ferroelectric domain swucture. The inhomogeneous quadrupole broadening that we have observed cannot
366
HUGHES AND PANDEY TABLE 2 Areas under the Absorption Curves of the Satellite Resonance at the Various Crystal Orientations
e
4
(degrees)
(degrees)
Integrated area (arbitrary units)
Peak-to-peak modulation amplitude (G)
45 135 45 135
90 90 4.5 4.5
1512 1476 2103 2190
1.93 1.93 2.76 2.76
Note. Also shown are the peak-to-peak modulation amplitudes used to record the resonances.
be due to a large number of independent defects, since these would give rise to a symmetric distribution of the orientations of the local symmetry axis, when projected onto the ab plane. We believe that the observed asymmetric broadening is associated with the presence of ferroelectric domains (8, 9) normally found in the material. The domain walls in NaN02 are parallel to the bc plane with adjacent domains having antiparallel polarizations (8, 9). The polar axis is parallel to the principal or b axis shown in Fig. 1. It is believed that the polarization is largely due to the dipole moments of the NO2 groups (IO). For purposes of discussion, we assume that the local polarization is parallel to the symmetry axis of the NO2 groups and that this, in turn, is parallel to the orientation of the local principal axis as determined by the quadrupole interaction of the z3Na nuclei, at least in regions well away from a domain wall. It is reasonable to assume that the direction of the polarization changes gradually as the domain wall is approached, rather than abruptly just at the wall. We believe that the small changes in the orientation of the principal axis, which cause the observed inhomogeneous broadening of the satellite resonance, are precursors of the much larger changes in orientation which occur within the domain walls themselves. Figures 3a and 3b show, schematically, two possible configurations of the local electric polarization when projected onto the ab plane. Also shown, schematically, are the regions over which the quadrupole perturbations are sufficiently small for the 23Nanuclei to contribute to the observed satellite resonance. The arrangement in Fig. 3a, repeated throughout the crystal, cannot account for our data, since it implies a symmetric distribution when projected onto the ab plane. However, the arrangement shown in Fig. 3b is consistent with the observed asymmetry, as long as it is repeated through most of the crystal. We note that a single “flaw” of the type shown in Fig. 3a, situated in the middle of the crystal, would lead to a symmetric distribution of orientations of the local polarization, since each half of the crystal would make a contribution of equal magnitude but opposite sign to (~58~)‘13. We also note that the arrangement shown in Fig. 3b would seem to be energetically more favored, since the a component of the polarization is everywhere parallel, rather than antiparallel in adjacent domains as in Fig. 3a. The domain structure cannot simply be the two-dimensional arrangement shown in Fig. 3b, since it implies that (~58~)‘I* would be zero for the bc projection. However,
DOMAIN
WALLS IN SODIUM NITRITE
367
a
/
(I axis
l
FIG. 3. Schematic representation of two possible configurations of the local electric polarization in the nb plane of a multidomain crystal of NaNOz . Also shown, schematically, are the domain walls and, by asterisks, the regions in which the Z3Nanuclei contribute to the observed satellite resonances.
the resonancesin Fig. 2b show that there is a large, but fairly symmetric, distribution of the orientations of the local polarization when projected onto the bc plane. This suggeststhat the local polarization begins to spiral in the vicinity of a domain wall. If this is regarded as a precursor, one must conclude that the polarization also spirals within each domain wall. The structure of the domain walls in NaNOz is therefore similar to that of Bloch walls between magnetic domains ( 11). Our data provide no information about the sense of the spiral in adjacent domains. However, it seems likely that the polarization spirals in the same sensethroughout the crystal. We have seen that the value of ( 60*) ’/* is somewhat larger for the bc projection than for the ab projection. However, when it is remembered that the ab distribution is essentially single sided, so that the center of gravity of the resonancesin Fig. 2a does not coincide with the center of the unperturbed satellite, it is clear that the intrinsic widths of the ab and bc distributions are of very similar magnitude. It therefore seems likely that the “tip angles” of the polarization within the domain walls are almost equal when projected onto the ab and bc planes. This result is at variance with that of Kinase et al. ( 12)) who found theoretically that the interaction energy between the electric dipoles in adjacent atomic layers is much stronger in the a direction than in the c direction. It may be that long-range electric interactions, which were ignored by Kinase et al., are important in determining the domain wall structure in NaNO* .
368
HUGHES
+
Domain Wall
AND
PANDEY
Domain Wall
Domain Wall
a axis
FIG. 4. Schematic three-dimensional representation of the Cartesian components of the spiraling local electric polarization in a multidomain crystal of NaNOz
Figure 4 shows, schematically, the three Cartesian components of the polarization at various locations, corresponding to the proposed domain wall structure. ACKNOWLEDGMENT
The authors are grateful to the Natural Sciences and Engineering Research Council of Canada for financial support during this work. REFERENCES 1. H. BETSUYAKU, J. Phys. Sue. Jpn. 27, 1485 (1969). 2. M. H. COHEN AND F. REIF, in “Solid State Physics” (F. Seitz and D. Turnbull, Eds.), Vol. 5, p. 3 11, Academic Press, New York, 1957. 3. D. G. HUGHES AND L. PANDEY, J. Magn. Reson. 75,272 (1987). 4. G. E. ZIEGLER, Whys. Rev. 38, 1040 ( 1931). 5. A. WEISS,Z. Naturforch. A 15, 536 (1960). 6. D. G. HUGHES AND P. A. SPENCER,J. Phys. C 15,7417 (1982). 7. D. G. HUGHES, J. Chem. Phys. 74,3234 ( 1981). 8. S. NOMURA, Y. ASAO, AND S. SAWADA, J. Phys. Sot. Jpn. 16,917 ( 1961). 9. S. SUZUKI AND M. TAKAGI, J. Phys. Sot. Jpn. 21,554 (1966). 10. K. HAMANO, J. Phys. Sot. Jpn. 35, 157 (1973). Il. A. H. MORRISH, in “The Physical Principles of Magnetism,” p. 367, Wiley, New York, 1965. 12. W. KINASE, W. MAKINO, AND K. TAKAHASHI, Ferroelectrics 64, 173 ( 1985).
RESONANCE
93,36 l-368 ( 199 1)
Evidence of a Spiral Structure of the Ferroelectric Domain Walls in NaN02 from the Quadrupole Broadening of the NMR Satellites of 23Na D.G.HUGHESANDLAKSHMANPANDEY* Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2JI
Received November 6, 1990 An explanation is proposed for the asymmetry of the inhomogeneous quadrupole broadening of the NMR satellites of “Na in single crystals of the ferroelectric material NaNO,, reported earlier (J. Mu@. Reson. 75,272, 1987). It is found that the quadrupole broadening is caused almost entirely by a spread of about 0.1 o in the orientation of the local principal axis of the efg tensor. The distribution is very asymmetric when projected onto the ab plane, but much more symmetric when projected onto the bc plane. This points to a spiraling of the electric polarization in the vicinity of the domain walls in the material. If this is regarded as a precursor of the behavior of the pohuization within the domain walls, the domain wall structure in NaNOr must be similar to the spiral structure of Bloch walls in magnetism. 0 1991Academic press. IX.
It has been known since the work of Betsuyaku (I) that the satellite resonancesof the quadrupole-split NMR spectrum of 23Na in single crystals of the ferroelectric material sodium nitrite, NaN02, exhibit substantial inhomogeneous quadrupole broadening (2). A recent study in our laboratory (3) has shown that this inhomogeneous broadening is highly asymmetric at some orientations of the crystal relative to the external magnetic field. Moreover, the observed asymmetry is inconsistent with the point group symmetry of the 23Na sites, determined using X rays (4). The magnitude and asymmetry of the inhomogeneous broadening were found to be quite similar in two crystals of NaN02, one cleaved and the other sawed,that were obtained from different sources.This led us to suggest(3) that the magnitude and asymmetry of the broadening may be intrinsic properties of the material as normally grown and handled. However, we were unable to explain the observedbehavior at the time, and no explanation seemsto have appearedsince in the literature. Our purpose in this paper is to propose an explanation for these old data, on the basis of the ferroelectric domain structure of the material. THEORETICAL
CONSIDERATIONS
The 23Na nuclei are situated at identical sites in the orthorhombic unit cell of the C,,, (Zm 2m) ferroelectric phase of NaN02 (4)) shown in Fig. 1. The x, y, and z axes * Present address: Department of Postgraduate Studies and Research in Physics, Rani Durgavati Vishwavidyalaya (formerly University of Jabalpur), Jabalpur 482001, India. 361
0022-2364191 $3.00 Copyright Q 1991 by Academic Press,Inc. All rights of reproduction in any form reserved.
362
HUGHES AND PANDEY
l
-Na
0-O
O-N
C
Y FIG. 1. Orthorhombic unit call of NaN02 and the coordinate system used to specify the direction of the external magnetic field.
of the principal coordinate system of the efg tensor of the nuclei (5) lie along the c, a, and b axes, respectively, of the unit cell, as shown in the figure. The frequencies, v,, of the $ ++ f and - i++ - $ satellites, respectively, of a system of I = 3 nuclei such as 23Na, located at identical sites in a single crystal, are given in first order by (3) v+ = v. T (e2qQ/h)(y/4(y1)(3
cos20 - 1 + n sin26 cos 24).
[II
Here, v. is the center line frequency, e2qQ/h is the quadrupole coupling constant (qcc), n is the asymmetry parameter, and y is the magnetogyric ratio, Also, 19and 4 are the polar and azimuthal angles of the external magnetic field measured relative to the principal coordinate system of the efg tensor, as shown in Fig, 1. Inhomogeneous quadrupole broadening is caused by crystal imperfections which give rise to local deviations in 6, #J,or e2qQ/h, from the average values characteristic of the crystal as a whole. Since n is approximately 0.1 for 23Na in NaN02 (5, 6)) we ignore the quadrupole broadening due to variations in 4. If the broadenings associated with variations in B and e2qQ/h are assumed to be independent of each other, it can be shown by differentiating Eq. [l] that the second moment of each satellite, measured in units of (magnetic field)2 relative to its “center of gravity,” about which the first moment is zero, is given by M2 = MZd(fl, 4) + ( 3n/2r)2(
602)(e2qQ/h)2sin22t9
+ (r/2r)*(&(e2qQ/h)‘)(3
cos26 - 1 + n sin28 cos 24)2.
[2]
Here, M2, is the dipolar second moment of each satellite ( 1 ), ( M2) II2 is the rms deviation of L?from the mean, and (6(e2qQ/h)2)‘/2 is the rms deviation of the qcc from the mean. For 8 values well away from zero and ?r radians, ( M2) ‘I* is essentially a measure of the spread in the orientations of the local principal axis projected onto the plane which contains the magnetic field and the principal axis. Therefore (M2) ‘1’ is expected to be a function of 4. On the other hand, (6(e2qQ/h)2)“2
DOMAIN
363
WALLS IN SODIUM NITRITE
should be a unique property of the particular crystal being studied, provided the broadenings associatedwith variations in 0 and e2qQ/ h are independent of each other. In an earlier paper ( 7), we proposed that the third moment of a resonanceplotted as a function of frequency be defined as M3
=
J
(w
-
[31
~0)‘fiwid~/Sl(o)du.
Using this definition, the third moment of the satellites is given by M3 = M3d-t(0, 4,) AZ[(3a/2y)3(683)(e2qQ/h)3sin328 cos*O - 1 + rj sin*8 cos 24)3]. [4] - (~/2r)3(s(e2qQlh)3)(3 Here, the t signs again refer to the ? t-) 4 and - f * - ! satellites, respectively, M3d+(~,~)isthedipolarthirdmoment(7),while(6~3)1~3and(6(e2qQ/h)3)‘~3are the root mean cubed deviations of 0 and the qcc from the mean. RESULTS AND DISCUSSION
Analysis of the asymmetry and linewidth of a satellite. Figure 2a shows the highfield ( -i c* - f ) satellite (3) of the 23Naquadrupole-split spectrum of a single crystal
a
MAGN. FLD. INCR. + I
I
I
I
0
2
4
6
I
I
I
I
,
8 10 12 14 16
1
I
I
I
0
2
4
6
(G) I
I
4
1
I
8 10 12 14 16
(G)
MAGN. FLD. INCR. I
I
I
I
0
2
4
6
I
I
I
I
4
8 10 12 14 16 (G)
I
I
1
I
0
2
4
6
I
I
1
I
,
8 10 12 14 16 W
FIG. 2. Absorption derivative lineshapes of the high-field ( - f ++ - i ) satellite of “Na in a single crystal of NaNO* at the orientations (a) @= 45”, 4 = 90” and 19= 135’, I#I= 90’ and (b) 6 = 45”, #J= 4.5” and l9 = 1350, f$ = 4.5”.
364
HUGHES AND PANDEY
of NaNOz (previously identified as Crystal B and described elsewhere (6))) recorded at the crystal orientations 0 = 45”, 4 = 90” and 8 = 135”, 0 = 90” at room temperature. Figure 2b shows the same satellite recorded at 6 = 45”, 4 = 4.5” and 0 = 135”, 4 = 4.5”. (It was originally believed that $Jwas zero for these resonances, but it was subsequently found that that was not quite the case.) The resonances were recorded as the absorption derivative, great care being taken to minimize the contribution of the dispersion mode, as described earlier (3). The second and third moments of each resonance were measured relative to the center of gravity. Since the “tails” of the resonances make a disproportionate contribution to these moments, they were approximately by the smooth curves shown in the figure. To reduce the random error and to minimize any effects of saturation, the resonances were also recorded with the magnetic field swept in the opposite direction. The mean values of the second and third moments, recorded with the two sweep directions, are listed in Table 1, together with values of the dipolar second and third moments, MZd and M3+, calculated using Eqs. [8] and [9] ofRef. (3). It can be seen from Table 1 that the measured second moments are an order of magnitude larger than the dipolar values, showing that the lineshapes are dominated by inhomogeneous quadrupole broadening. Moreover, the dipolar contribution to the third moment is negligible, so that the asymmetry of the resonances can be entirely attributed to the inhomogeneous quadrupole broadening. According to Eq. [ 21, the second moment of a satellite should be the same at the orientations 8,+ and (?r - 0)) 4. It can be seen from Table 1 that the second moments are indeed equal at the orientations 0 = 45”, $J = 90” and 0 = 135‘, 4 = 90”, as are those at tl = 45”, 4 = 4.5” and 6 = 135”, 4 = 4.5”. By taking e’qQfh, 7, and (6(e2qQ/h)2)1’2 to be 1096 kHz, 0.109 (5, 6), and 2.26 kHz (3), respectively, and substituting the experimental values of the second moment at the orientations 0 = 45”, 4=90”and1?= 135”,~=90”inEq.[2],itisfoundthat(682)1~2is(0.106-t0.004)“. This is a measure of the width of the distribution of the orientations of the local principal axis when projected onto the ab or 4 = 90” plane. Similarly, by substituting the experimental values of the second moment at the orientations 8 = 45”, @J= 4.5” and 0 = 135”, 9 = 4.5”, it is found that (66 2) ‘I2 = (0.131 f 0.002)“. We see that TABLE 1 Experimental Values of the Second and Third Moments of the High-Field Satellite of “Na in a Single Crystal of NaNO2 8 (degrees)
(deices)
45 135 45 135
90 90 4.5 4.5
M2
(G2)
2.2 2.0 3.2 3.3
+- 0.1 + 0.1 + 0.1 * 0.1
Mz.,
G2)
0.203 0.203 0.410 0.410
M3
(G’)
1.6 +- 0.2 -1.6 + 0.2 -0.5 + 0.1 -0.1 2 0.1
M3+
2.8 x 2.8 X 1.0 x 1.0 x
(@I 1O-4 1O-4 10-3 10-3
Note. Also shown are theoretical values of the dipolar second and third moments, M2d and I%&-, of the satellite; 8 and # are the polar and azimuthal angles of the external magnetic field, measured relative to the crystal coordinate system.
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the width of the distribution of the orientations of the local principal axis when projected onto the C$= 4.5” plane, which is essentially the bc plane, is somewhat larger than the width projected onto the ab plane. The third moment does not display the same symmetry as the second moment, since the second term on the right-hand side of Eq. [4] is of opposite sign at the orientations 8, C#J and (?r - S), 4, whereasthe third term is the sameat both orientations. Since the experimental values are equal in magnitude but opposite in sign at the orientations 0 = 45”, 4 = 90” and 0 = 135”, 6 = 90”, it follows that (fi(e2qQ/h)3) is negligible, as found earlier by measuring the 6 dependence of the third moment when 0 = 90” ( 3). By substituting the experimental third-moment values in Eq. [ 41, 3 ‘I3 is found to be (0.092 + 0.003) ‘. The fact that (68 “) ‘I3 is almost as large as $~*j1,2 reflects the considerable asymmetry of the distribution of the orientations of the local principal axis when projected onto the ab plane. It can be seen from the lineshapesin Fig. 2b and the third-moment values in Table 1 that the satellite is significantly more asymmetric at the orientation 0 = 45”, C#I = 4.5“ than at 6 = 135”, Cp= 4.5”. This seemsto imply an asymmetric distribution of qcc values. By substituting in Eq. [ 41, it is found that (6( e2qQ/h)3) ‘I3 is (-5.3 -t 0.5) kHz and (60 3) ‘I3 = (0 .048 +- 0.005) o in a plane inclined at 4.5” to the bc plane. The former result is inconsistent with the lineshapesshown in Fig. 2a, although that could be explained as a failure to detect a small asymmetry in the presence of a larger one. On the other hand, such a value for (A( e*qQ/ h)3) 1’3implies a contribution of roughly 1 G3 to the third moment of the satellite at orientations where 8 = 90”. There was no evidence of such a contribution in the measurementsdescribed earlier (3)) and we have no explanation for the difference between the asymmetry at ~9= 45’) I#J= 4.5” and that at 19= 135”, 4 = 4.5”. Nevertheless, the fact that (c%?~)‘/~is so much smaller than (60 * ) ‘I* when C$= 4.5” shows that the distribution of the orientations of the local principal axis is much more symmetric when projected onto the bc plane than when projected onto the ab plane. The quadrupole interactions experiencedby nuclei which are situated close to strong defects may be so large that they do not contribute to the observed satellite resonances (2). It is therefore not immediately obvious that the resonancesshown in Figs. 2a and 2b are obtained from the same group of 23Nanuclei, becausethey were obtained using very different values of 4. To check this, the absorption derivative resonancesin Fig. 2 were integrated and the areas under the absorption curves determined. The values are listed in Table 2, together with the peak-to-peak modulation amplitudes used to record the resonances. The area under an absorption curve is proportional to the modulation amplitude used to record the absorption derivative and the number of nuclei that contribute to the resonance (provided other effects such as saturation are unchanged). The mean area under the absorption divided by the peak-to-peak modulation amplitude is 774 arbitrary units/G for the resonances corresponding to 4 = 90” and 778 arbitrary units/G for C$= 4.5”. This strongly suggeststhat the same nuclei contribute to the resonancesshown in Fig. 2a and Fig. 2b, so that our calculated valuesof(68*)‘/*and(66’ 3) Ii3 , for projections onto the ab and bc planes,are properties o:Fthe same group of nuclei. Interpretation of the asymmetry and linewidth in terms of the ferroelectric domain swucture. The inhomogeneous quadrupole broadening that we have observed cannot
366
HUGHES AND PANDEY TABLE 2 Areas under the Absorption Curves of the Satellite Resonance at the Various Crystal Orientations
e
4
(degrees)
(degrees)
Integrated area (arbitrary units)
Peak-to-peak modulation amplitude (G)
45 135 45 135
90 90 4.5 4.5
1512 1476 2103 2190
1.93 1.93 2.76 2.76
Note. Also shown are the peak-to-peak modulation amplitudes used to record the resonances.
be due to a large number of independent defects, since these would give rise to a symmetric distribution of the orientations of the local symmetry axis, when projected onto the ab plane. We believe that the observed asymmetric broadening is associated with the presence of ferroelectric domains (8, 9) normally found in the material. The domain walls in NaN02 are parallel to the bc plane with adjacent domains having antiparallel polarizations (8, 9). The polar axis is parallel to the principal or b axis shown in Fig. 1. It is believed that the polarization is largely due to the dipole moments of the NO2 groups (IO). For purposes of discussion, we assume that the local polarization is parallel to the symmetry axis of the NO2 groups and that this, in turn, is parallel to the orientation of the local principal axis as determined by the quadrupole interaction of the z3Na nuclei, at least in regions well away from a domain wall. It is reasonable to assume that the direction of the polarization changes gradually as the domain wall is approached, rather than abruptly just at the wall. We believe that the small changes in the orientation of the principal axis, which cause the observed inhomogeneous broadening of the satellite resonance, are precursors of the much larger changes in orientation which occur within the domain walls themselves. Figures 3a and 3b show, schematically, two possible configurations of the local electric polarization when projected onto the ab plane. Also shown, schematically, are the regions over which the quadrupole perturbations are sufficiently small for the 23Nanuclei to contribute to the observed satellite resonance. The arrangement in Fig. 3a, repeated throughout the crystal, cannot account for our data, since it implies a symmetric distribution when projected onto the ab plane. However, the arrangement shown in Fig. 3b is consistent with the observed asymmetry, as long as it is repeated through most of the crystal. We note that a single “flaw” of the type shown in Fig. 3a, situated in the middle of the crystal, would lead to a symmetric distribution of orientations of the local polarization, since each half of the crystal would make a contribution of equal magnitude but opposite sign to (~58~)‘13. We also note that the arrangement shown in Fig. 3b would seem to be energetically more favored, since the a component of the polarization is everywhere parallel, rather than antiparallel in adjacent domains as in Fig. 3a. The domain structure cannot simply be the two-dimensional arrangement shown in Fig. 3b, since it implies that (~58~)‘I* would be zero for the bc projection. However,
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WALLS IN SODIUM NITRITE
367
a
/
(I axis
l
FIG. 3. Schematic representation of two possible configurations of the local electric polarization in the nb plane of a multidomain crystal of NaNOz . Also shown, schematically, are the domain walls and, by asterisks, the regions in which the Z3Nanuclei contribute to the observed satellite resonances.
the resonancesin Fig. 2b show that there is a large, but fairly symmetric, distribution of the orientations of the local polarization when projected onto the bc plane. This suggeststhat the local polarization begins to spiral in the vicinity of a domain wall. If this is regarded as a precursor, one must conclude that the polarization also spirals within each domain wall. The structure of the domain walls in NaNOz is therefore similar to that of Bloch walls between magnetic domains ( 11). Our data provide no information about the sense of the spiral in adjacent domains. However, it seems likely that the polarization spirals in the same sensethroughout the crystal. We have seen that the value of ( 60*) ’/* is somewhat larger for the bc projection than for the ab projection. However, when it is remembered that the ab distribution is essentially single sided, so that the center of gravity of the resonancesin Fig. 2a does not coincide with the center of the unperturbed satellite, it is clear that the intrinsic widths of the ab and bc distributions are of very similar magnitude. It therefore seems likely that the “tip angles” of the polarization within the domain walls are almost equal when projected onto the ab and bc planes. This result is at variance with that of Kinase et al. ( 12)) who found theoretically that the interaction energy between the electric dipoles in adjacent atomic layers is much stronger in the a direction than in the c direction. It may be that long-range electric interactions, which were ignored by Kinase et al., are important in determining the domain wall structure in NaNO* .
368
HUGHES
+
Domain Wall
AND
PANDEY
Domain Wall
Domain Wall
a axis
FIG. 4. Schematic three-dimensional representation of the Cartesian components of the spiraling local electric polarization in a multidomain crystal of NaNOz
Figure 4 shows, schematically, the three Cartesian components of the polarization at various locations, corresponding to the proposed domain wall structure. ACKNOWLEDGMENT
The authors are grateful to the Natural Sciences and Engineering Research Council of Canada for financial support during this work. REFERENCES 1. H. BETSUYAKU, J. Phys. Sue. Jpn. 27, 1485 (1969). 2. M. H. COHEN AND F. REIF, in “Solid State Physics” (F. Seitz and D. Turnbull, Eds.), Vol. 5, p. 3 11, Academic Press, New York, 1957. 3. D. G. HUGHES AND L. PANDEY, J. Magn. Reson. 75,272 (1987). 4. G. E. ZIEGLER, Whys. Rev. 38, 1040 ( 1931). 5. A. WEISS,Z. Naturforch. A 15, 536 (1960). 6. D. G. HUGHES AND P. A. SPENCER,J. Phys. C 15,7417 (1982). 7. D. G. HUGHES, J. Chem. Phys. 74,3234 ( 1981). 8. S. NOMURA, Y. ASAO, AND S. SAWADA, J. Phys. Sot. Jpn. 16,917 ( 1961). 9. S. SUZUKI AND M. TAKAGI, J. Phys. Sot. Jpn. 21,554 (1966). 10. K. HAMANO, J. Phys. Sot. Jpn. 35, 157 (1973). Il. A. H. MORRISH, in “The Physical Principles of Magnetism,” p. 367, Wiley, New York, 1965. 12. W. KINASE, W. MAKINO, AND K. TAKAHASHI, Ferroelectrics 64, 173 ( 1985).