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Specific heats of crystals with the fluorite structure
W. Schröter, J. Nölting
To cite this version:
W. Schröter, J. Nölting. Specific heats of crystals with the fluorite structure. Journal de Physique Colloques, 1980, 41 (C6), pp.C6-20-C6-23. �10.1051/jphyscol:1980605�. �jpa-00220000�
Specific heats of crystals with the fluorite structure
W . Schrijter and J . Nolting
Institut fiir Physikalische Chemie der UniversitLt Gottingen und Sonderforschungsbereich 126 Gottingen-Clausthal, Tammannstrasse 6, 3400 Gottingen, W-Germany
RCsumC. - Les mesures de c, pour PbF,, SrCl, et BaF, i temperatures ClevCes indiquent des transformations continuelles prononckes de type Schottky. On calcule, B partir des courbes de transition, les Cnergies et entropies de formation pour des dtfauts isolks. A partir de l'tnergie et de I'entropie de transition, on peut en tirer une limite infkrieure et supkrieure pour des concentrations de dkfauts complexes. Ces concentrations proche des temptratures de fusion n'exc6dent guere 10 %.
Parmi les cristaux Ctudits, BaCl, constitue une exception parce qu'au lieu d'une transformation continuelle elle montre une transformation aigue k 1 195 K.
Abstract. - Pronounced diffuse phase transitions of Schottky type are found by c, measurement for fluorites PbF,, SrCl, and BaF, at higher temperatures. Formation energies and entropies are calculated for isolated defects from the transition curves. From the energy and entropy of transition an upper and a lower limit can be derived for higher defect concentrations. It is seen that these concentrations scarcely exceed 10 % immediately below the melting point.
BaCI, is distinguished among the other fluorites showing an absolutely sharp transition instead of a diffuse one at 1 195 K.
1 . Introduction. - 1 .1 GENERAL. - Crystals with the fluorite structure show interesting aspects being of scientific as well as of technical importance.
This is connected with the unusual phenomenon of extensive disorder of the anion sublattice appearing gradually with increasing temperature. The observed extremely high ionic conductivities which lead to the somehow curious but impressive name super ionic conductors, are comparable with those of molten salts. A lot of papers concerned with this remarkable disordered state have been published within the last ten years.
The interpretations of the results. however, are frequently contradictory often having a rather specula- tive character. From recent theoretical work [ l , 21 the opinion now began to crystallize that the anion sublattice is only partially disordered by a few percent.
This limited disorder above the transition tempe- rature is said to be due to the suppression of interstitial generation coming from defect repulsions above a critical interstitial concentration.
Formation energies of isolated Frenkel-defects are not known very well a t the moment. For the defect interaction term coming into play at higher concen- trations even a theoretical formulation does not exist.
1 . 2 CALORIMETRIC INVESTIGATIONS. - With calo- rimetric measurements. which will be described in this paper. under certain favourable conditions the excess energy for establishing lattice disorder can be
determined. From this the important characteristic quantities of disorder can be calculated.
Generally the following dependence of the defect quantities is valid
A H e ( T ) = A H f ( T ) . X ( T ) .
excess defect concentration (1)
energy formation of defects energy
At higher defect concentrations AH, is an averaged quantity which depends on X and which decreases with increasing X. Except of this qualitative statement n o quantitative calculation is known about defect interaction as already mentioned.
In spite of this the following limiting values can be derived from c, measurements :
1) AH: formation energy for isolated Frenkel defects ;
2) Xmin 2 AH,,AHF as lower limit for the degree of disorder. since AH: 2 AH, holds ;
3) X,,, from entropy considerations. a s described in section 2.1.3.
These cases will be dealt with in detail in the follow- ing sections.
2. Results. - 2.1 STRONTIUM CHLORIDE. - The measurements were performed using a precision high temperature adiabatic calorimeter. The probe is continuously fed with electric power of exactly known quantity. The temperature is measured with
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980605
SPECIFIC HEATS OF CRYSTALS WITH THE FLUORITE STRUCTURE C6-2 1
high accuracy at constant time intervals. usually t 300 s. From the measured temperature-time
increments c, values at about every 5 K were calculat- ed. In addition the enthalpy of the probe can be
calculated without lack of precision by summing
-
-LC--'up the increments to a desired temperature. CY
SrCl,
15
t
Fig. 1. - Specific heat of strontium chloride.
9 11
1041 T 13 t Fig. 2. - Special diagram for evaluating defect formation enthalpy and entropy for the case of isolated defects.
l2
Figure 1 shows c, values of SrCI, as an example.
The temperature dependence of c, is clearly represent- ed by the dense sequence of measuring points. They follow a slighly increasing straight line up to 800 K attributed to lattice vibrations including small anhar- monic contributions. A pronounced diffuse transition with a maximum at 1 001 K appears above 800 K.
The transition strongly represents an anomaly of the SCHOTTKY type. The melting point at 1 146 K is found to be absolutely sharp. Rising c, values usually observed just below the melting point are missing. Finally the interesting behaviour in the melting region should be pointed out. where a smooth continuation of c, is found beyond the melting point.
2 . 1
.
1 Isolated defects. - At first defect parameters can be evaluated at the beginning of the transition curve where the defect concentration is low and the defects may be considered as independent from each other. The corresponding defect formation energy AH: is easily calculated (3) and may be derived from the expressionIn ( A c , T 2 ) = A - AHfOIRT (2) Ac, is the difference of measured and of linearly from lower temperatures extrapolated specific heats.
A is independent of temperature.
Figure 2 shows fair agreement to equation (2) till about the middle of the diagram, corresponding roughly to 900 K. The scatter of points at low tempe- ratures can easily be explained by the low Ac, values in this region (see Fig. 1 !).
Formation energies and entropies for isolated defects. easily derived from the slope and the intercept of the straight line in figure 2, give
-
. /---IdfK /---- ll!46K-
AH: = 2.67 eV and ASP = 21 R . (3)
These relatively high values correspond to really isolated defects. From equation (2) it can be derived
9
- T I K
2 . 1 . 2 Lower limit for X(T) at higher defect concen- trations. - From figure 2 it is seen that the defects that at 910 K only 0.3
%
of the anions in the fluoritegradually change to a stage of increasing interaction.
For this stage of energetically dependent defects it must be supposed that the defect formation energy AH, decreases with higher values of X. Using formula (1) with AH: gives values for X which must be too small in the case of higher defect concentrations.
Therefore these calculated values can be considered as a lower estimated limit for X.
The excess energy AH, = H - H(1in) associated with the creating of disorder of one mole of crystal is presented in figure 3 against temperature. The scale at the right hand side gives values of X as the described lower limit which is assigned to- the respec- tive temperature by the enthalpy curve. It is seen from this scale that the maximum of the transition corresponds to at most 2
%
and the melting point to at most 4%
of disordered anions.400 MM 800 1000 1200 lattice are disordered.
Fig. 3. - Excess enthalpy for defect formation with lower limit values for defect concentration (right).)
2 . 1 . 3 Upper limit for X ( T ) at higher defect concen- trations. - An upper limit of the defect concentration X can be obtained from entropy considerations. The
Table I - tnthcil/)rrs ~ndentropres o f trcinsrtron ( u ) and melting (m) jbr high fen?peruture,fiuorites.
AH,, As,, A H m A s m
Compound (kJ mol-') ( J m o l - ' K - '
1
(kJ mol- ') (J mol -'
K' ')-
PbF2 SrCl, BaF2 BaCI,
apparent entropy of transition is composed of a configurative part and the excess entropy in the neighbourhood of the defect. The configuration entropy is derived straightforward from statistical thermodynamic standard formulae. The lattice part is not easy to calculate but certainly is positive.
To get an upper limit for X this lattice part is set to zero. All measured entropy is then identified with configuration entropy. Accordingly this configuration entropy is too high and the corresponding value of the defect concentration X as well. Figure 4 shows the measured excess entropy of transition together with a scale of X values at the right hand side representing upper limit values as described above. From this scale Xis read to be 5
%
at the c, maximum and 12%
at the melting point.
2.2 LEAD FLUORIDE, BARIUM FLUORIDE AND BARIUM
CHLORIDE. - PbF, and BaF, also show pronounced diffuse transitions at 711 and 1 275 K. respectively.
In table I enthalpies and entropies of transition and of melting are compiled. BaCI, is distinguished among the other fluorites showing no diffuse tran- sition but an absolutely sharp one at 1 295 K. Table I1
21-
17.-
-
-I 0f 13-
Y 7
-
\ 9.- - .- 5
V)
1
0
Table 11. - Upper und lower limits of the defect concentration of high temperatureJuorites at transition (u) and melting (m) points.
~ , , i , , = x(H) xmax = x(S)
x = AH~IAH: AS' = AS,,,^.
- -
TLI T m TI! T m
SrCI, 2.0 4.3 4.4 12.6
PbF2 2.3 5.3 4.8 (10) (*)
BaF, 4.1 6.9 6.7 -
/ Sr CL2
- :
:
: 5 s
-- --- ---
gives values for the limits of defect concentrations calculated as described in sections 2.1.2 and 2.1 .3.
LOO 600 800
[I,
*'. 2%
1000 1200T I K
Fig. 4. -Excess entropy for defect formation with upper limit values for defect concentration (right).
3. Concluding remarks. - c, measurements from this investigation show pronounced diffuse transitions of SCHOTTKY type. These transitions are connected with massive disordering of the anion sublattice.
From c, values of the transition region characteristic quantities concerning the anion disorder can be derived.
The defect concentrations in the fluorites investigat- ed here are not substantially different from each other. It can be stated as an important result of these experiments that the concentration of defects in all investigated fluorites can be enclosed to the limits of 2-7
%
at the maximum of the transition and 4-13%
at the melting points. These figures strongly support recent theoretical work [I. 21.
Acknowledgments. - We first have to thank Pro- fessor A. B. Lidiard. AERE Harwell. for stimulating discussions which induced us to begin these investi- gations.
We further thank Dr. A. V. Chadwick and Dr. R. E. Gordon. University of Kent at Canterbury.
for preparation and transmission of BaF, single crystals used for the c, measurement.
Financial support by the Qeutsche Forschungs- gemeinschaft is gratefully acknowledged.
DISCUSSION
Question. - P. A. FLEURY. reported by W. Hayes from neutron scattering studies of the same system ?
Could you please comment on the discrepancy
between the defect concentration in PbF, inferred - J . NOLTI~G.
from your specific heat data and the much larger values See reply to the comment of Dr. Hayes.
SPECIFIC HEATS OF CRYSTALS WITH THE FLUORITE STRUCTURE C6-23
Question. - W . HAYES.
Your estimates of defect concentrations in SrC1, in the reference state agree with neutron data. How- ever, your estimates for PbF, are considerably smaller than the concentrations obtslined for PbF, by neutron scattering. Perhaps there is a problem here associated with interpretation of experimental data ?
Reply. - J . NOLTING.
I agree with Dr. Hayes that the apparent discre- pancies may be due to different interpretations of the experiments and that the existing models still have to be
refined i.n estimating the upper limit of defect concen- trations. We have restricted ourselves to the appli- cation of the simple Frenkel model, with interstitials at the octahedral sites.
Comment. - F . B E N I ~ .
There is a noteworthy agreement between your results for the enthalpy and entropy of defect forma- tion in SrCl, (2.67 eV and 21 K) and those obtained independently from self-diffusion and conductivity date analysis 2.98 eV and 22 K (BCniBre et al., J. Phys.
Chem. SOC., 1979).
References
[I] DIXON, M. and GILLAN, M. J., J . Phys. C : Solid State Phys. 11 (1978) 165.
[2] CATLOW, C. R. A., COMINS, J. D., GERMANO, F. A., HARLEY, R. T.
and HAYES, W., J. Phys. C. : Solid State Phys. 11 (1978) 3197.
[3] NOELTING, J., Angew. Chem. Znt. Ed. 9 (1970) 489.
