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Phonon properties of vanadium-substituted lanthanum niobate derived from heat-capacity measurements
ow&3697/86 s3.00 + 0.00 Pergamon Journals Ltd.
J. Phys. Chem. SoIids Vol. 47, No. 5. pp. 501-505. 1986 Printed in Great Britain.
PHONON PROPERTIES OF VANADIUM-SUBSTITUTED LANTHANUM NIOBATE DERIVED FROM HEAT-CAPACITY MEASUREMENTS M. V. NEVITTand G. S. KNAPP Materials Science and Technology Division, Argonne National Laboratory, Argonne, IL 60439, U.S.A. (Received 30 October 1985; accepted 12 December 1985)
Abstract-We
have measured the 3-4OOK heat capacity of vanadium-substituted lanthanum niobate LaNb, _ ,V,O, (0 cx Q 0.35) to determine how the relevant averaged properties, or moments, of the phonon spectrum (w”) relate to certain other lattice properties of these compounds, most noteworthy of which is a large decrease in the paraelastic-ferroelastic transformation temperature T, with x. The pertinent moments, represented by their corresponding Debye temperatures, are &,(-3) corresponding to the lowest frequency modes; 0,(O) associated with the geometric mean frequency of the entire spectrum; and O,(2) identified with the high-frequency modes that are sampled at the upper end of the temperature range. We find that 0,( - 3) falls rapidly with x and this effect can be correlated with the comparably sharp drop in T,. There is little effect of composition on e,(O) and e,(2) because the phonons that govern soft-mode behavior represent only a small fraction of the mode population. In the vicinity of T, the temperature dependences of the free energies of the tetragonal and monoclinic phases arc so similar that the discontinuity in C, is immeasurably small. Keywords: specific heat, heat capacity, lanthanum niobate, lattice dynamics, soft modes, phase transformations, phonon spectrum, Debye temperature.
IN’I’RODUCMON LaNbO, undergoes a transformation to a ferroelastic phase with monoclinic symmetry at T, x 770 K [l-3]. We have reported earlier [4] that T, is sharply lowered (Fig. la) by the partial, random substitution of the V5+ ion for the Nb’+ ion in the central position of the NbO, tetrahedron; at the limit of V-ion substitution, corresponding approximately to x = 0.35 in the generalized formula LaNb, _,rV,04, T, has dropped to the vicinity of 4 K [S]. The ferroelastic distortion involves coupled spontaneous strains which deform the square base of the tetragonal unit cell existing above T, to a parallelogram with unequal sides as the base of the monoclinic unit cell. The spontaneous strains in the monoclinic phase obey mean-field behavior in pure LaNbO, and V-substituted compounds [4,6], but decrease in magnitude as x increases because of the drop in T,.. The transformation in pure LaNbO, is driven by a soft transverse acoustic mode propagating in the basal plane of the tetragonal paraelastic phase that exists above T, [7,8]. There is a coupled low-lying optic mode observed in Raman scattering, but it does not drive the transformation: its frequency [v(k = 0) z 94 cm-‘] is essentially independent of temperature in the paraelastic phase; then it hardens monotonically for T < T, [6,7,9]. It appears that in the V-substituted samples the transformation remains acoustic-mode-driven, inasmuch as the frequency vs temperature profile for the coupled Raman-active optic mode is qualitatively the same as in pure LaNbO,, i.e. the mode has a constant value as q is
approached from above [v(k = 0) sz 90 cm-‘] [6, IO]. The paraelastic value is, of course, retained to progressively lower temperatures as T, decreases with increasing x. We expect that at x > 0.30 this optic mode remains close to its paraelastic value down to Ta4K. Taken together, these observation6 prompted us to determine how the relevant averaged properties of the
(a)
\
I
0
I
15
\
\ \
I (b)
).. I
x
Fig. I. (a) Composition dependence of T, in LaNb, _ , V, 0, compounds
(adapted from ref. [4]). (b) Composition pendence of 0,(-3).
de-
M. V. NEWT and G. S. KNAPP
502
phonon spectrum are related to the transformation behavior and other lattice characteristics of these compounds. Accordingly, we have measured the heat capacity of LaNbO, in its pure form and with various levels of V substitution between 3 and 400 K. The measurements on pure LaNbO, have been reported and analyzed [ll]. We report here the heat capacity results for the V-substituted compounds; the analytical approach we follow is that employed in the earlier paper, to which the reader is referred for a more detailed treatment.
pendence of the lattice entropy of a slightly anharmonic solid [13-l 51 in which the w, are allowed to be temperature dependent. In the high temperature limit this expression can be cast in terms of the Debye temperatures representing the even moments of the spectrum and the geometric mean frequency,
- & ANALYTICAL
k4m16 r6
T-4 + &
APPROACH
(1) where w, are the normal-mode frequencies. (w”) can be so defined for all n > -3, # 0. For convenience in making intercomparisons, we represent the moments by their associated Debye temperatures e,(n), each corresponding to a Debye spectrum that has the same n th moment as the actual spectrum. Thus <&,)Y
n > -3,
#O. (2)
As n approaches -3, (0”) is identified with f?,( - 3) which can be shown to be equal to the Debye temperature derived from the elastic constants. The Debye temperature for n +O is associated with the geometric mean frequency wn and is given by Q,(O) = e113hw,/k,.
1 .
(5)
From the heat capacity data we develop a generalized phonon-spectrum profile in terms of the appropriate n th moments of the phonon spectrum
e,(n) =;r+
kw)i4
(3)
The parameters 8,(2n) for n > 1 are good signatures of the high-frequency modes, since they emphasize the weight of the high-frequency portion of the spectrum. Thus, a simple characterization in terms of the low-frequency spectrum, the spectral geometric mean and the high-frequency region, given by 0,( - 3), e,(O) and f&(2), respectively, is independent of a detailed model, provides significant information about the lattice-dynamical behavior of a material and facilitates a comparison of related materials. We fit the C,, data for 4 < T < IO K to the usual low temperature expansion for the lattice specific heat [ 121 and thereby derive f?,( - 3): 234R CL = ]e,( _ 3)]3 T3 + bTS. To determine O,(O) and O,(2) we calculate the entropy per gram-atom from the heat capacity in the high temperature regime (above -0,/3) and we fit the experimentally derived entropy values to analytical expression representing the temperature de-
We call attention to several conditions and simplifications which are inherent in our utilization of eqn (5): (a) A temperature dependence is introduced into e,(n) by anharmonicity. Lacking compressibility data we base our analysis on C,,, so any observed anharmonicity will include a dilational contribution c,-ct, . (b) We assign the temperature dependence of S,, which is caused by anharmonicity, to the leading 0,(O) term and treat the other 0,(n) terms as constants in recognition of the fact that the lattice entropy shows substantially less sensitivity to the temperature dependence of the higher moments because their contributions are smaller. (c) We assume O,(2) = e,(4) = t?,(6). Although e,(n) can be expected to increase for n > 2, the change should occur slowly and smoothly, and above me,/3 the contributions from the e,,(4) and O,(6) terms are small fractions of the total lattice entropy. We have shown [I l] that reasonably chosen arbitrarily larger values of O,(4) and O,(6) affect the calculated value of O,(O) in an inconsequential way. Finally, our analysis of the temperature dependence of e,(O) permits an assessment of anharmonicity in the vibrational spectrum: I
A = -3Nk,--
d@,(O)
e,(o)
dT
AT is the anharmonic contribution ity in the high temperature limit.
EXPERIMENTAL
(6)
to the heat capac-
METHOD
Heat capacity measurements were made on polycrystalline LaNb, _ .V,O, samples having nominal compositions corresponding to x = 0 [l I], 0.20, 0.25, 0.30 and 0.35. The x = 0, 0.20 and 0.25 samples, prepared by the hot isostatic pressing of previously sintered crushed and resintered powders, had densities within 3% of their theoretical values. The samples corresponding to x = 0.30 and 0.35 were prepared in the same manner except the hot isostatic pressing step was omitted; their densities were ap-
503
Phonon properties of vanadium-substimted lanthanum niobate proximately 80% of the theoretical values. Previously described [4] preparation procedures were employed, beginning with 99.9 + % pure normal sesqui- and pentoxides. The specific heat at constant pressure of samples weighing approximately 2 g was measured over the temperature range 3400 K by a heat pulse method [ 161.The calorimeter, which incorporates a feedback system to regulate the temperature of concentric radiation shields surrounding the sample, was calibrated with a polycrystalline copper sample. Measurements on polycrystalline samples of Al,03 and ZrO,, as well as on a sapphire single crystal demonstrated that accepted published C,, values could be duplicated to better than 1.5%. To test internal reproducibility, we made several randomly chosen duplicate runs on the same sample and on duplicate samples. We found that results were reproducible to within 2.5% for T < 6 K and within 0.6% for 6< T<400K.
T(K)
x-o.25
T(K) Fig. 2. Specific heat of several LaNb, _ , V,O, compounds in temperature ranges spanning r,. Shaded bands are single polynomial fits with widths equal to 2S.D.
RESULTS
Data taken at T < 10 K were analyzed according to eqn (4) to yield 0,( -3) values. We determined Q,(O) as a function of temperature* for various values of o,(2) by iteratively fitting the high temperature lattice entropy, calculated from the measured C,,, to eqn (5). As was shown earlier [l 1] the value of 6,(O) does not show high sensitivity to the selected value of e,(2), but the temperature dependence of e,(O) is sensitive to e,(2). We select for e,(2) the lowest value for which e,(O) is approximately linear in T, with a negative slope, as expected from eqn (6). From this slope we calculated A, the coefficient of anharmonicity, assigning an error that takes into account the random and systematic errors in the heat capacity measurements. The results of these calculations are displayed in Table 1 and in part in Fig. l(b). We include in Table 1 the values of b the coefficient of the T5 term in eqn (4). It should be noted that e,(O) and e,(2) values are given only for x = 0,0.20 and 0.25; the data taken on samples with higher levels of V-ion substitution, having substantially lower density, were rejected from this part of the analysis because the degraded thermal conductivity of the latter samples introduced problems in achieving thermal equilibration above - 300 K. Particular care, including repetitive runs and special efforts to achieve enhanced sensitivity, was taken to monitor C’,,in the vicinity of TC. We established that there is no measurable latent heat, and that any specific heat discontinuity at T, is below our level of detection, 0.1 J/g-at.-K. Figure 2 shows typical Cp vs T traces for samples with V substitutions corresponding to .r = 0.20 and 0.25 over temperature ranges that bracket their transformations. *The conditions and simplifications cited on p. 502 should be borne in mind.
DISCUSSION
Heat capacity near T, We treat this aspect of the results first because it provides a context for the balance of the discussion. The absence of a latent heat shows that the transformation in these polycrystalline samples is of an order higher than first throughout the range of V-ion substitution. The subtle nature of the ferroelastic transformation is manifested by our observation (Fig. 2) that no discernible specific heat anomaly occurs in these V-substituted compounds, as is also the case for pure LaNbO, [3]. We conclude that near T, the monoclinic and tetragonal phases have Gibbs free energies with nearly identical first and second temperature derivatives. Even in the absence of a well-defined heat capacity anomaly, we might expect to observe precursive anharmonicity in the vicinity of T, , in addition to the normal dilational contribution. This is not the case; the A coefficients for pure LaNbO, and for the V-substituted samples (Table 1) are small and attributable entirely to dilation [I 11. Low temperature behavior 8,(-3) decreases with x (Fig. lb), reaching an essentially constant value in the vicinity of the V-ion replacement limit. The progressively greater specific heat compared to pure LaNbO,, indicated by the drop in O,( -3), we attribute to two probable increases in the low-temperature phonon density of states that can be associated with the decrease in TC. First, as x increases and T, correspondingly decreases, the temperature at which the frequency of the soft acoustic mode assumes its minimum value descends towards and finally into the T < IOK sampling regime. O,( -3) should thereby be de-
M. V. NEWT and G. S. KNAPP
504
Table 1. Phonon parameters for LaNb, _ ,V,O, X
g.et “01 [c&g-’ 1
Dg(-3>iK1
bi 10-3J-(K.at)-‘-K-61
8,WlKl
n~10-3J-f~-atf-'K-Z1
0
8.35
423 l 2
5.87 (to.431 x to-5
55" * 5
690 t 1"
1.5 l 0.5
0.20
8.39
351 * 2
9.50 (tO.68) x IO-'
544 * 5
?20 * LO
0.73 * 0.5
0.25
a.39
321 * 2
0.87 (fl.O)
x 10-5
543 l 5
720 t 10
0.78 l 0.5
0.30
8.38
305 l I
-5.55 (M.74)
x 10-5
---
___
__-
0.35
8.36
305 l 1
-4.52 (M.53)
x 10-5
---
---
___
pressed as the soft-mode influence progressively biases the acoustic spectrum to lower frequencies. Second, as mentioned in the Introduction, the lowlying coupled optic mode retains its low paraelastic value to progressively lower temperatures as T, falls. This optic mode should contribute oscillations in excess of the acoustic spectrum as T, is depressed to low temperature. The t~~~t~e-de~ndent deviation of the Debye temperature from the @,( - 3) value provides further qualitative information regarding the effect of the V-ion substitution on the low-temperature phonon spectrum. We note (Table I) that b, the coefficient of the T* term in eqn (4), decreases with x, falling to zero in the vicinity of x = 0.25 and becoming negative with larger x. The associated changes in the spectrum can be represented graphically in terms of systematic departures from the Debye model, as shown in Fig. 3. or, the Debye temperature calculated from C, for temperatures up to T z 0.15 0, ( - 3), is plotted as 8,/f& ( - 3). We infer that at the highest levels of V-ion substitution 13,(- 3) reflects maximally the T-related anomalies in the phonon density of states proposed above. With an increase in tem~rature B, departs from Q,( - 3) with a monotonic rise that is typical of many complex oxides [17]. On the other hand, at lower values of x including x =O, the aforementioned contributions to the specific heat associated with the transformation are apparently felt to a lesser extent 1.4
8,fOflKl
r
Fig. 3. Effective Debye temperature 0, [normalized to Q,(-3)] for LaNb, _ ,V,O, compounds as a function of
reduced temperature TiO,,(- 3).
in @,( -3) but become more significant at higher temperature [r -+ 0.05 @,( - 3)]. So 8, initially falls below @a(- 3), indicating the ingress of these softmode effects, then shows the same monotonic rise with a further increase in temperature. Phonon dispersion toward the Brillouin zone boundaries may also contribute to the dip in Or. If this effect is involved, it appears to lose significance as x increases. High temperature
behavior
Although 0,(-3) decreases with x, e,(O), associated with the geometric mean frequency, is within error independent of x, while oD(2), identified with the high-frequency modes, shows a slight compensatory rise with X, probably because at high temperature the heat capacity is samphng internal modes of the tetrahedra. These, on average, should be harder in the VO, tetrahedron than in the NbO, tetrahedron [18]. It is clear therefore that the soft modes involved in the ferroelastic transformation comprise only a small fraction of the total spectrum; their effect can only be seen at the lowest temperature through the influence on the low-frequency spectrum. SUMMARY
Heat capacity measurements on LaNb, _ ,V,O, compounds in the temperature range in which only the lowest-frequency modes are sampled are useful in pointing out the features in the acoustic and lowestlying-optic portions of the spectrum that are sensitive to the level of V-ion substitution. We believe the drop in e,( - 3) with increasing V-ion concentration can be correlated with the concomitant drop in T,, the temperature of the soft-mode driven transformation. At higher temperature, where the soft phonon modes represent only a small fraction of the total spectrum, heat capacity is relatively insensitive to the transformation. In the vicinity of T, the temperature dependences of the free energies of the tetragonai and monoclinic phases are so similar that the discontinuity in C, is immeasurably small. Ackno~,/edaeme~~nrs-The authors wish to thank Dr A. T. Aldred for making certain information on transformation temperatures available prior to publication and to acknowledge valuable discussions with him and with Dr S.-K. Chan.
Phonon properties of vanadium-substituted Gratitude is also expressed to Dr H. B. Radousky and Mr T. E. Klippert, respectively, for developing the computer program for data acquisition and reduction and for assistance in the experimental work. The help of Mr J. W. Downey in sample preparation is also acknowledged. This research was supported by the Office of Basic Energy Sciences-Materials Sciences of the U.S. Department -d Energy, under Contract W-3l-l09-Eng-38.
REFERENCES 1. Gingerich K. A. and Bair H. E., A&. X-Ray Anal. 7, 22 (1964). 2. Stubican V. S., J. Am. Ceram. Sot. 47, 55 (1964). 3. Brixner L. H., Whitney J. F., Zumsteg F. C. and Jones G. A., Mar. Res. Bull. 12, I7 (1977). 4. Nevitt M. V. and Aldred A. T., Am. khernl Sot. Symp. Ser. 246 (Edited bv G. S. Bamev. J. D. Navratil and W. W. Schultz), pp. 315-321 (1984). 5. Aldred A. T., to be published. 6. Aldred A. T., Chan S.-K., Grimsditch M. H. and Nevitt M. V., Mater. Res. Sot. Symp. Proc. 24, 81 (1984).
lanthanum niobate
505
7. Wada M., Nakayama Y., Sawada A., Tsunekawa S. and Ishibashi Y., J. Phys. Sot. Japan 47, 1575 (1979). 8. Hara K., Sakai A., Tsunekawa S., Sawada A., Ishibashi Y. and Yagi T., J. Phys. Sot. Japan 54, 1168 (1985). 9. Ananeva G. V., Korovkin A. M., Kudryavtsev A. B., Kuochikov A. K.. Rvskin A. I. and Sobol’ A. A.. Sou. Ph.& Solid State 23; 625 (1981).
IO. Nevitt M. V., Aldred A. T., Chan S.-K. and Grimsditch M. H., to be published. Il. Nevitt M. V.. KDaDD G. S.. Grimsditch M. H.. Chan S.-K. and AIdred-A. T., ‘High Temperatures; High Pressures
17, 233 (1985).
12. Barron T. H. K., Berg W. T. and Morrison J. A., Proc. Roy. Sot. A 242, 478 (1957). 13. Barron T. H. K., in Lattice Dynamics (Edited by R. F.
Wallace), pp. 247-254. Pergamon Press, Oxford (1965). 14. Miller A. P. and Brockhouse B. N., Can. J. Phys. 49, 704 (1971).
IS. Knapp G. S., Bader S. D. and Fisk Z., Phys. Rev. B 13, 3783 (1976).
16. Trainor R. J., Knapp G. S., Brodsky M. B., Pokomy G. J. and Snyder R. B., Rev. Sci. Instrum. 46,95 (I 975). 17. Kieffer S. W., Rev. Geophys. Space Phys. 18,862 (1980). 18. Blasse G., J. Solid State Chem. 7, 169 (1973).
J. Phys. Chem. SoIids Vol. 47, No. 5. pp. 501-505. 1986 Printed in Great Britain.
PHONON PROPERTIES OF VANADIUM-SUBSTITUTED LANTHANUM NIOBATE DERIVED FROM HEAT-CAPACITY MEASUREMENTS M. V. NEVITTand G. S. KNAPP Materials Science and Technology Division, Argonne National Laboratory, Argonne, IL 60439, U.S.A. (Received 30 October 1985; accepted 12 December 1985)
Abstract-We
have measured the 3-4OOK heat capacity of vanadium-substituted lanthanum niobate LaNb, _ ,V,O, (0 cx Q 0.35) to determine how the relevant averaged properties, or moments, of the phonon spectrum (w”) relate to certain other lattice properties of these compounds, most noteworthy of which is a large decrease in the paraelastic-ferroelastic transformation temperature T, with x. The pertinent moments, represented by their corresponding Debye temperatures, are &,(-3) corresponding to the lowest frequency modes; 0,(O) associated with the geometric mean frequency of the entire spectrum; and O,(2) identified with the high-frequency modes that are sampled at the upper end of the temperature range. We find that 0,( - 3) falls rapidly with x and this effect can be correlated with the comparably sharp drop in T,. There is little effect of composition on e,(O) and e,(2) because the phonons that govern soft-mode behavior represent only a small fraction of the mode population. In the vicinity of T, the temperature dependences of the free energies of the tetragonal and monoclinic phases arc so similar that the discontinuity in C, is immeasurably small. Keywords: specific heat, heat capacity, lanthanum niobate, lattice dynamics, soft modes, phase transformations, phonon spectrum, Debye temperature.
IN’I’RODUCMON LaNbO, undergoes a transformation to a ferroelastic phase with monoclinic symmetry at T, x 770 K [l-3]. We have reported earlier [4] that T, is sharply lowered (Fig. la) by the partial, random substitution of the V5+ ion for the Nb’+ ion in the central position of the NbO, tetrahedron; at the limit of V-ion substitution, corresponding approximately to x = 0.35 in the generalized formula LaNb, _,rV,04, T, has dropped to the vicinity of 4 K [S]. The ferroelastic distortion involves coupled spontaneous strains which deform the square base of the tetragonal unit cell existing above T, to a parallelogram with unequal sides as the base of the monoclinic unit cell. The spontaneous strains in the monoclinic phase obey mean-field behavior in pure LaNbO, and V-substituted compounds [4,6], but decrease in magnitude as x increases because of the drop in T,.. The transformation in pure LaNbO, is driven by a soft transverse acoustic mode propagating in the basal plane of the tetragonal paraelastic phase that exists above T, [7,8]. There is a coupled low-lying optic mode observed in Raman scattering, but it does not drive the transformation: its frequency [v(k = 0) z 94 cm-‘] is essentially independent of temperature in the paraelastic phase; then it hardens monotonically for T < T, [6,7,9]. It appears that in the V-substituted samples the transformation remains acoustic-mode-driven, inasmuch as the frequency vs temperature profile for the coupled Raman-active optic mode is qualitatively the same as in pure LaNbO,, i.e. the mode has a constant value as q is
approached from above [v(k = 0) sz 90 cm-‘] [6, IO]. The paraelastic value is, of course, retained to progressively lower temperatures as T, decreases with increasing x. We expect that at x > 0.30 this optic mode remains close to its paraelastic value down to Ta4K. Taken together, these observation6 prompted us to determine how the relevant averaged properties of the
(a)
\
I
0
I
15
\
\ \
I (b)
).. I
x
Fig. I. (a) Composition dependence of T, in LaNb, _ , V, 0, compounds
(adapted from ref. [4]). (b) Composition pendence of 0,(-3).
de-
M. V. NEWT and G. S. KNAPP
502
phonon spectrum are related to the transformation behavior and other lattice characteristics of these compounds. Accordingly, we have measured the heat capacity of LaNbO, in its pure form and with various levels of V substitution between 3 and 400 K. The measurements on pure LaNbO, have been reported and analyzed [ll]. We report here the heat capacity results for the V-substituted compounds; the analytical approach we follow is that employed in the earlier paper, to which the reader is referred for a more detailed treatment.
pendence of the lattice entropy of a slightly anharmonic solid [13-l 51 in which the w, are allowed to be temperature dependent. In the high temperature limit this expression can be cast in terms of the Debye temperatures representing the even moments of the spectrum and the geometric mean frequency,
- & ANALYTICAL
k4m16 r6
T-4 + &
APPROACH
(1) where w, are the normal-mode frequencies. (w”) can be so defined for all n > -3, # 0. For convenience in making intercomparisons, we represent the moments by their associated Debye temperatures e,(n), each corresponding to a Debye spectrum that has the same n th moment as the actual spectrum. Thus <&,)Y
n > -3,
#O. (2)
As n approaches -3, (0”) is identified with f?,( - 3) which can be shown to be equal to the Debye temperature derived from the elastic constants. The Debye temperature for n +O is associated with the geometric mean frequency wn and is given by Q,(O) = e113hw,/k,.
1 .
(5)
From the heat capacity data we develop a generalized phonon-spectrum profile in terms of the appropriate n th moments of the phonon spectrum
e,(n) =;r+
kw)i4
(3)
The parameters 8,(2n) for n > 1 are good signatures of the high-frequency modes, since they emphasize the weight of the high-frequency portion of the spectrum. Thus, a simple characterization in terms of the low-frequency spectrum, the spectral geometric mean and the high-frequency region, given by 0,( - 3), e,(O) and f&(2), respectively, is independent of a detailed model, provides significant information about the lattice-dynamical behavior of a material and facilitates a comparison of related materials. We fit the C,, data for 4 < T < IO K to the usual low temperature expansion for the lattice specific heat [ 121 and thereby derive f?,( - 3): 234R CL = ]e,( _ 3)]3 T3 + bTS. To determine O,(O) and O,(2) we calculate the entropy per gram-atom from the heat capacity in the high temperature regime (above -0,/3) and we fit the experimentally derived entropy values to analytical expression representing the temperature de-
We call attention to several conditions and simplifications which are inherent in our utilization of eqn (5): (a) A temperature dependence is introduced into e,(n) by anharmonicity. Lacking compressibility data we base our analysis on C,,, so any observed anharmonicity will include a dilational contribution c,-ct, . (b) We assign the temperature dependence of S,, which is caused by anharmonicity, to the leading 0,(O) term and treat the other 0,(n) terms as constants in recognition of the fact that the lattice entropy shows substantially less sensitivity to the temperature dependence of the higher moments because their contributions are smaller. (c) We assume O,(2) = e,(4) = t?,(6). Although e,(n) can be expected to increase for n > 2, the change should occur slowly and smoothly, and above me,/3 the contributions from the e,,(4) and O,(6) terms are small fractions of the total lattice entropy. We have shown [I l] that reasonably chosen arbitrarily larger values of O,(4) and O,(6) affect the calculated value of O,(O) in an inconsequential way. Finally, our analysis of the temperature dependence of e,(O) permits an assessment of anharmonicity in the vibrational spectrum: I
A = -3Nk,--
d@,(O)
e,(o)
dT
AT is the anharmonic contribution ity in the high temperature limit.
EXPERIMENTAL
(6)
to the heat capac-
METHOD
Heat capacity measurements were made on polycrystalline LaNb, _ .V,O, samples having nominal compositions corresponding to x = 0 [l I], 0.20, 0.25, 0.30 and 0.35. The x = 0, 0.20 and 0.25 samples, prepared by the hot isostatic pressing of previously sintered crushed and resintered powders, had densities within 3% of their theoretical values. The samples corresponding to x = 0.30 and 0.35 were prepared in the same manner except the hot isostatic pressing step was omitted; their densities were ap-
503
Phonon properties of vanadium-substimted lanthanum niobate proximately 80% of the theoretical values. Previously described [4] preparation procedures were employed, beginning with 99.9 + % pure normal sesqui- and pentoxides. The specific heat at constant pressure of samples weighing approximately 2 g was measured over the temperature range 3400 K by a heat pulse method [ 161.The calorimeter, which incorporates a feedback system to regulate the temperature of concentric radiation shields surrounding the sample, was calibrated with a polycrystalline copper sample. Measurements on polycrystalline samples of Al,03 and ZrO,, as well as on a sapphire single crystal demonstrated that accepted published C,, values could be duplicated to better than 1.5%. To test internal reproducibility, we made several randomly chosen duplicate runs on the same sample and on duplicate samples. We found that results were reproducible to within 2.5% for T < 6 K and within 0.6% for 6< T<400K.
T(K)
x-o.25
T(K) Fig. 2. Specific heat of several LaNb, _ , V,O, compounds in temperature ranges spanning r,. Shaded bands are single polynomial fits with widths equal to 2S.D.
RESULTS
Data taken at T < 10 K were analyzed according to eqn (4) to yield 0,( -3) values. We determined Q,(O) as a function of temperature* for various values of o,(2) by iteratively fitting the high temperature lattice entropy, calculated from the measured C,,, to eqn (5). As was shown earlier [l 1] the value of 6,(O) does not show high sensitivity to the selected value of e,(2), but the temperature dependence of e,(O) is sensitive to e,(2). We select for e,(2) the lowest value for which e,(O) is approximately linear in T, with a negative slope, as expected from eqn (6). From this slope we calculated A, the coefficient of anharmonicity, assigning an error that takes into account the random and systematic errors in the heat capacity measurements. The results of these calculations are displayed in Table 1 and in part in Fig. l(b). We include in Table 1 the values of b the coefficient of the T5 term in eqn (4). It should be noted that e,(O) and e,(2) values are given only for x = 0,0.20 and 0.25; the data taken on samples with higher levels of V-ion substitution, having substantially lower density, were rejected from this part of the analysis because the degraded thermal conductivity of the latter samples introduced problems in achieving thermal equilibration above - 300 K. Particular care, including repetitive runs and special efforts to achieve enhanced sensitivity, was taken to monitor C’,,in the vicinity of TC. We established that there is no measurable latent heat, and that any specific heat discontinuity at T, is below our level of detection, 0.1 J/g-at.-K. Figure 2 shows typical Cp vs T traces for samples with V substitutions corresponding to .r = 0.20 and 0.25 over temperature ranges that bracket their transformations. *The conditions and simplifications cited on p. 502 should be borne in mind.
DISCUSSION
Heat capacity near T, We treat this aspect of the results first because it provides a context for the balance of the discussion. The absence of a latent heat shows that the transformation in these polycrystalline samples is of an order higher than first throughout the range of V-ion substitution. The subtle nature of the ferroelastic transformation is manifested by our observation (Fig. 2) that no discernible specific heat anomaly occurs in these V-substituted compounds, as is also the case for pure LaNbO, [3]. We conclude that near T, the monoclinic and tetragonal phases have Gibbs free energies with nearly identical first and second temperature derivatives. Even in the absence of a well-defined heat capacity anomaly, we might expect to observe precursive anharmonicity in the vicinity of T, , in addition to the normal dilational contribution. This is not the case; the A coefficients for pure LaNbO, and for the V-substituted samples (Table 1) are small and attributable entirely to dilation [I 11. Low temperature behavior 8,(-3) decreases with x (Fig. lb), reaching an essentially constant value in the vicinity of the V-ion replacement limit. The progressively greater specific heat compared to pure LaNbO,, indicated by the drop in O,( -3), we attribute to two probable increases in the low-temperature phonon density of states that can be associated with the decrease in TC. First, as x increases and T, correspondingly decreases, the temperature at which the frequency of the soft acoustic mode assumes its minimum value descends towards and finally into the T < IOK sampling regime. O,( -3) should thereby be de-
M. V. NEWT and G. S. KNAPP
504
Table 1. Phonon parameters for LaNb, _ ,V,O, X
g.et “01 [c&g-’ 1
Dg(-3>iK1
bi 10-3J-(K.at)-‘-K-61
8,WlKl
n~10-3J-f~-atf-'K-Z1
0
8.35
423 l 2
5.87 (to.431 x to-5
55" * 5
690 t 1"
1.5 l 0.5
0.20
8.39
351 * 2
9.50 (tO.68) x IO-'
544 * 5
?20 * LO
0.73 * 0.5
0.25
a.39
321 * 2
0.87 (fl.O)
x 10-5
543 l 5
720 t 10
0.78 l 0.5
0.30
8.38
305 l I
-5.55 (M.74)
x 10-5
---
___
__-
0.35
8.36
305 l 1
-4.52 (M.53)
x 10-5
---
---
___
pressed as the soft-mode influence progressively biases the acoustic spectrum to lower frequencies. Second, as mentioned in the Introduction, the lowlying coupled optic mode retains its low paraelastic value to progressively lower temperatures as T, falls. This optic mode should contribute oscillations in excess of the acoustic spectrum as T, is depressed to low temperature. The t~~~t~e-de~ndent deviation of the Debye temperature from the @,( - 3) value provides further qualitative information regarding the effect of the V-ion substitution on the low-temperature phonon spectrum. We note (Table I) that b, the coefficient of the T* term in eqn (4), decreases with x, falling to zero in the vicinity of x = 0.25 and becoming negative with larger x. The associated changes in the spectrum can be represented graphically in terms of systematic departures from the Debye model, as shown in Fig. 3. or, the Debye temperature calculated from C, for temperatures up to T z 0.15 0, ( - 3), is plotted as 8,/f& ( - 3). We infer that at the highest levels of V-ion substitution 13,(- 3) reflects maximally the T-related anomalies in the phonon density of states proposed above. With an increase in tem~rature B, departs from Q,( - 3) with a monotonic rise that is typical of many complex oxides [17]. On the other hand, at lower values of x including x =O, the aforementioned contributions to the specific heat associated with the transformation are apparently felt to a lesser extent 1.4
8,fOflKl
r
Fig. 3. Effective Debye temperature 0, [normalized to Q,(-3)] for LaNb, _ ,V,O, compounds as a function of
reduced temperature TiO,,(- 3).
in @,( -3) but become more significant at higher temperature [r -+ 0.05 @,( - 3)]. So 8, initially falls below @a(- 3), indicating the ingress of these softmode effects, then shows the same monotonic rise with a further increase in temperature. Phonon dispersion toward the Brillouin zone boundaries may also contribute to the dip in Or. If this effect is involved, it appears to lose significance as x increases. High temperature
behavior
Although 0,(-3) decreases with x, e,(O), associated with the geometric mean frequency, is within error independent of x, while oD(2), identified with the high-frequency modes, shows a slight compensatory rise with X, probably because at high temperature the heat capacity is samphng internal modes of the tetrahedra. These, on average, should be harder in the VO, tetrahedron than in the NbO, tetrahedron [18]. It is clear therefore that the soft modes involved in the ferroelastic transformation comprise only a small fraction of the total spectrum; their effect can only be seen at the lowest temperature through the influence on the low-frequency spectrum. SUMMARY
Heat capacity measurements on LaNb, _ ,V,O, compounds in the temperature range in which only the lowest-frequency modes are sampled are useful in pointing out the features in the acoustic and lowestlying-optic portions of the spectrum that are sensitive to the level of V-ion substitution. We believe the drop in e,( - 3) with increasing V-ion concentration can be correlated with the concomitant drop in T,, the temperature of the soft-mode driven transformation. At higher temperature, where the soft phonon modes represent only a small fraction of the total spectrum, heat capacity is relatively insensitive to the transformation. In the vicinity of T, the temperature dependences of the free energies of the tetragonai and monoclinic phases are so similar that the discontinuity in C, is immeasurably small. Ackno~,/edaeme~~nrs-The authors wish to thank Dr A. T. Aldred for making certain information on transformation temperatures available prior to publication and to acknowledge valuable discussions with him and with Dr S.-K. Chan.
Phonon properties of vanadium-substituted Gratitude is also expressed to Dr H. B. Radousky and Mr T. E. Klippert, respectively, for developing the computer program for data acquisition and reduction and for assistance in the experimental work. The help of Mr J. W. Downey in sample preparation is also acknowledged. This research was supported by the Office of Basic Energy Sciences-Materials Sciences of the U.S. Department -d Energy, under Contract W-3l-l09-Eng-38.
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lanthanum niobate
505
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