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“On the connection between the specific heat of a Schottky defect and the curvature observed in the
Arrhenius plots in ionic materials”
P. Varotsos, K. Alexopoulos
To cite this version:
P. Varotsos, K. Alexopoulos. “On the connection between the specific heat of a Schottky defect and the curvature observed in the Arrhenius plots in ionic materials”. Journal de Physique Colloques, 1980, 41 (C6), pp.C6-526-C6-529. �10.1051/jphyscol:19806138�. �jpa-00220047�
« On the connection between the specific heat of a Schottky defect and the curvature observed in the Arrhenius plots in ionic materials »
P. Varotsos and K. Alexopoulos
Department of Physics. University of Athens, Solonos Str. 104, Athens 144, Greece
Abstract. — The formation energy u* for the production of a Schottky defect obtained from the static lattice tech- nique is found to be much smaller than the formation enthalpy h that results from conductivity experiments. This is due to the fact that these two parameters refer to a production of a defect under different conditions. The para- meter h results from isobaric experiments while u* is calculated for a solid that retains its volume during the pro- duction of the defect. In NaCl the value of u* at the melting point is only about 30 % of h. Thermodynamic conside- rations show that h increases (whereas u* decreases) with increasing temperature thus leading to an upwards curvature of the conductivity plot.
It is extensively known that a reliable fitting of the conductivity (and diffusion) curve of alkali halides has not been reported up to date. Many workers have attempted to describe its curvature by assuming besides the Schottky defects also other defects (i.e.
Frenkel defects in both sublattices trivacancies, etc.) or mechanisms (i.e. interstitialy jumping, etc.).
Silver halides show a curvature of similar form [1].
The latter was initially explained by introducing at high temperatures an interstitialcy mechanism ; how- ever the pioneering experiment of Batra and Slifkin [2]
has undoubtedly shown that this effect is due to a temperature dependent formation enthalpy and not to a new mechanism.
In the previous conference it was reported [3] that the microscopic calculations by the Haades pro- gram [3, 4] on alkali halides demand an interstitialcy mechanism. Slifkin remarked that there must be a common basis for the explanation of the curvature both in silver and alkali halides. In the same confe- rence [5] the present authors suggested that this common basis is the result of anharmonic effects (i.e. the expansivity and the temperature decrease of the bulk modulus); these effects produce a simulta- neous increase of the formation enthalpy h and entropy s and therefore a temperature decrease of the formation Gibbs energy g faster than linearly thus accounting for the upwards curvature [6-23].
This point of view [5] disagreed, therefore, basically
with the other proposals [3] advanced in the Confe- rence. Subsequently [4] the inclusion of expansivity in the microscopic calculations led to a temperature dependent formation energy thus making the assump- tion about interstitialcies unecessary. However these calculations gave a formation energy that falls with temperature. This decrease was considered [4] as being able to explain the upwards curvature. In a recent paper [23] we stressed that this last conclusion cannot hold because Thermodynamic arguments show that an upwards curvature of the conductivity plot demands an enthalpy (and hence at P = 0 a formation energy) which increases with the tempera- ture. In the present paper this discrepancy will be explained. It is probably due to the fact that para- meters from the usual microscopic calculations (even when taking an expanded lattice into consideration) when correspond to the production of a defect under constant volume do not give the parameters that appear in the analysis of the conductivity experiments under constant pressure.
1. Thermodynamics. — Consider a perfect crystal a0 in which a single defect is isothermally produced under conditions of constant pressure thus leading to a crystal a (Fig. 1). The heat needed for this process is the usual formation enthalpy h (or when P = 0 the usual formation energy u). If, on the other hand, the defect is produced under constant volume the cor- Résumé. — L'énergie u* nécessaire pour la production d'un défaut Schottky a été trouvée par la technique static Lattice beaucoup plus petite que l'enthalpie de formation h qui résulte des expériences de conductivité. Cela est dû au fait que ces deux paramètres correspondent à la production d'un défaut sous des conditions différentes. Le paramètre h résulte des expériences isobariques tandis que u* est calculé pour un solide qui retient son volume pendant la production du défaut. Dans le NaCl la valeur de u* au point de fusion n'est qu'environ 30 % de h. Des considérations thermodynamiques montrent que h augmente (tandis que u* diminue) avec une température crois- sante et produit ainsi une courbure de la conductivité.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19806138
ON THE CONNECTION BETWEEN THE SPECIFIC HEAT OF A SCHOTTKY DEFECT C6-527
perfect crlstai'
perfect c r y s t a l
Fig. 1 . - Heats for the production of a defect under isothermal conditions.
responding heat is equal to u* which is simply the difference of the internal energies of the crystals a and crO*. Simple Thermodynamical arguments lead to the relation :
u* = h - T f v PB (1) or
u* = u - Tvf DB (at P = 0) ( 2 ) where
/3
is the thermal volume expansion coefficient, B is the isothermal bulk modulus and vf the formation volume per defect i.e. the difference of the volumes of the crystals a and a'.One sees that the quantities u* and h differ by an amount Trf /3B. In alkali halides (KCl, NaCl,
. .
.) for high temperatures, we have the typical values h-
2.5 eV, T = 900 K, vf = 3 R (R = the meanvolume per atom), B R = a few eV,
/3 - 2 x grad-'
By using these values one finds that Tuf PB is 50-70
%
of / I which means that h and u* can dEtrer by a corz~i-
derable amount. The usual microscopic calculation when consider a solid in which a defect is produced under constant volume gives u*. Obviously at abso- lute zero, u* coincides with h but at higher tempe- ratures the two quantities differ.
These considerations must be extended also, to the usual (thermal) formation entropy s which results from a comparison of the crystals a and a'. This is NOT the same with the (thermal) entropy s* obtained from the process aO* -+ a. Again simple Thermodyna- mical analysis shows that s* and s are connected through the relation :
A simple application in alkali halides at high tempe- ratures shows shows that vf PB is about 50-70
%
of s.If one attempts to analyse the experimental data by inserting the parameter u* obtained from microscopic calculations (with an expanded lattice) into a formula of the form
= e - ~ * / k T eslk
(4) (where x is the molar fraction of the defects) no ther- nzodynamically consistent results are obtained. The correct expressions are x = e-h'kT eslk, or
It is certainly possible to successfully fit (the ERRO- NEOUS expression (4)) to the experimental data.
This however will give the real value of x( = e - h l k T es/k)
multiplied by the factor exp(vf PBIk) which is tempe- rature dependent and greater than unity ; obviously such a fitting does not give the correct values of h and s.
For a self-consistent analysis of experiments when using the values resulting from <( isoc~:u~ic )) micros- copic calculations the term Tvf PB must be added to u* in order to obtain the value of h for each tempe- rature. The temperature variation of s can then be found from ah
Ip
= T ;1 .
Such an analysis requiresP
the calculation of vf.
At the present time this method cannot be applied because the usual microscopic calculations of L"' with the lattice statics technique neglects the vibrational contribution [17].
At this point we note that the increase of h and the decrease of u* with increasing temperature are compatible as can be seen by studying equation (1).
The first term of the right side of this Equation increases slightly with the temperature whereas the second (Tvf PB) increases much faster so that their difference (i.e. u*) decreases with temperature exactly a5 found by microscopic calculations.
The major role of anharmonic effects in the calcula- tion of defect parameters can be visualised if one considers the Thermodynamical relation [24] :
where c =
-
' - d T p dh
1
is the isobaric specific heat per vacancy, c$ the variation of the isochoric specific heat when a defect is produced in a perfect crystal under constant volume and temperature,p0
and KO are the thermal expansion coefficient and the com- pressibility of the perfect crystal aO, and the symbolspf
and Kf denote the expansion coefficient and the compressibility of the volume vf.One can easily prove [24] that c$ is appreciably
smaller than c, and therefore equation (4) can be written as :
Even if it were
pf
=p0
andK f
= KO, by using the typical values reported above one finds c,-
3 Kfor alkali halides at T
-
1 000 K. This means that h increases by N 0.05 eV when the temperature increases by 200 K. But Thermodynamics do not imply thatpf
=Po
or Kf =KO.
Older experimental expansivity data [25, 261, when properly analysed [27] or very recent accurate experiments [31, 321 have shown that$/Po
is of the order of 10. As the ratio Kf/Ko is usuallyP '
appreciably smaller than 2
-
the termPo
becomes much larger than unity. This means that h (and hence s) increases considerably with the tempe- rature leading to an upward curvature of the conduc- tivity (or diffusion) plots.
2. The cBR-model. - We have recently proposed a model which permits the calculation of various defect parameters. This model, up to date, gave the following results :
1) It gives the formation entropies of various
3) It gives the correct formation volume for the following cases : Schottky defects in alkali halides [9, 171, cation Frenkel defects in silver halides [Ill, monovacancy in f.c.c. and b.c.c. metals [17, 131 etc. ;
4) It calculates the correct migration volume in : cation migration in alkali halides [15], anion migra- tion in fluorides [35], interstitial motion in silver halides [19], bound interstitial motion in alkaline earth fluorides [35] :
5) It gives the correct activation volumes in extreme cases like B4-AgI and Cerium [22] ;
6) This model predicts an (formation and migra- tion) enthalpy and a (formation and migration) entropy that both increase with the temperature [7, 10, 16, 181. On this basis the curvature observed in a number of diffusion and conductivity plots have been quantitatively explained. Among these cases one should mention the case of AgBr in which when one applies the cBR-model 1161
-
without using any adjustable parameter - one can explain the curved plots within the experimental error ;7) In the case where the expansion coefficient of the formation volume have been directly measured, the experimental values are in excellent agreement with the values predicted from cBR-model [17, 221 ; kinds of defects (monovacancies in metals and Rare 8) The curvature observed in isothermal Gas Solids I6,101, Schottky defects in alkali halides [6I,
(Or COndUCti~tY) PlotS ,,S. pressure can be quantita- cation Frenkel defects in silver halides (71, anion tively by the values of ~r predicted by the Frenkel defects in alkaline earth fluorides [8]) ; in
cBR-model.
all these variety of defects a close agreement with
the experimental values have been found ; The fact that in so different cases the cBR-model 2) It gives the correct migration entropies for the gives satisfactory results cannot, of course, be for- following cases : cation and anion vacancy migration tuitous because it has no any adjustable parameter in alkali halides [6, 81, interstitial (and Na+ motion) to play with. Therefore it seems that it is a good basis in silver halides [19], bound cation vacancy motion for explaining the various anomalies without having in alkaline earth fluorides [21] doped with trivalent to introduce the assumption of temperature (and
rare earth cations ; pressure) independent parameters.
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w]
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ON THE CONNECTION BETWEEN THE SPECIFIC HEAT OF A SCHOTTKY DEFECT C6-520
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